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Research ArticleOn the Theory of Multilinear Singular Operators with RoughKernels on the Weighted Morrey Spaces
Sha He and Xiangxing Tao
Department of Mathematics Zhejiang University of Science and Technology Hangzhou 310023 China
Correspondence should be addressed to Xiangxing Tao xxtaozusteducn
Received 27 February 2016 Accepted 10 July 2016
Academic Editor Shijun Zheng
Copyright copy 2016 S He and X Tao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We study some multilinear operators with rough kernels For the multilinear fractional integral operators 119879119860Ω120572
and the multilinearfractional maximal integral operators119872119860
Ω120572 we obtain their boundedness on weighted Morrey spaces with two weights 119871119901120581(119906 V)
when119863120574119860 isin Λ120573(|120574| = 119898minus1) or119863120574119860 isin BMO (|120574| = 119898minus1) For the multilinear singular integral operators 119879119860
Ωand the multilinear
maximal singular integral operators 119872119860
Ω we show they are bounded on weighted Morrey spaces with two weights 119871119901120581(119906 V) if
119863120574119860 isin Λ
120573(|120574| = 119898 minus 1) and bounded on weighted Morrey spaces with one weight 119871119901120581(119908) if 119863120574119860 isin BMO (|120574| = 119898 minus 1) for
119898 = 1 2
1 Introduction and Main Results
Let us consider the following multilinear fractional integraloperator
119879119860
Ω120572119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1119877119898(119860 119909 119910) 119891 (119910) 119889119910
0 lt 120572 lt 119899
(1)
and the multilinear fractional maximal operator
119872119860
Ω120572119891 (119909) = sup
119903gt0
1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910) 119877119898 (119860 119909 119910) 119891 (119910)1003816100381610038161003816 119889119910
0 lt 120572 lt 119899
(2)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899119860 is a function defined onR119899 and119877
119898(119860 119909 119910)denotes the
119898th order Taylor series remainder of 119860 at 119909 expanded about119910 that is
119877119898(119860 119909 119910) = 119860 (119909) minus sum
|120574|lt119898
1
120574119863120574119860 (119910) (119909 minus 119910)
120574 (3)
120574 = (1205741 120574
119899) each 120574
119894 119894 = 1 119899 is a nonnegative integer
|120574| = sum119899
119894=1120574119894 120574 = 120574
1 sdot sdot sdot 120574
119899 119909120574 = 1199091205741
1sdot sdot sdot 119909
120574119899
119899 and 119863120574 = 120597|120574|
12059712057411199091sdot sdot sdot 120597
120574119899119909119899
We notice that if 120572 = 0 the above two operators 119879119860Ω120572
119872119860
Ω120572are the multilinear singular integral operator 119879119860
Ωand
themultilinearmaximal singular integral operator119872119860
Ωwhose
definitions are given as follows respectively
119879119860
Ω119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899+119898minus1119877119898(119860 119909 119910) 119891 (119910) 119889119910 (4)
119872119860
Ω119891 (119909) = sup
119903gt0
1
119903119899+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910) 119877119898 (119860 119909 119910) 119891 (119910)1003816100381610038161003816 119889119910
(5)
For119898 = 1119879119860Ω120572
is obviously the commutator [119860 119879Ω120572] of119879
Ω120572
and 119860 [119860 119879Ω120572]119891(119909) = 119860(119909)119879
Ω120572119891(119909) minus 119879
Ω120572(119860119891)(119909) where
119879Ω120572
is the fractional integral operator given by
119879Ω120572119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572119891 (119910) 119889119910 0 lt 120572 lt 119899 (6)
There are numerous works on the study of multilinearoperators with rough kernels If 119863120574119860 isin BMO (|120574| = 119898 minus 1)
Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 4149314 13 pageshttpdxdoiorg10115520164149314
2 Journal of Function Spaces
the 119871119901 boundedness of119879119860Ωwas obtained bymeans of a good-120582
inequality by Cohen and Gosselin [1] In 1994 Hofmann [2]proved that 119879119860
Ωis a bounded operator on 119871119901(119908) when nabla119860 isin
119861119872119874 and 119908 isin 119860119901 Recently Lu et al [3] proved 119879119860
Ωand119872119860
Ω
are bounded from 119871119901 to 119871119902 (1119901 minus 1119902 = 120573119899) when 119863120574119860 isin
Λ120573(|120574| = 119898 minus 1) while for multilinear fractional integral
operators Ding and Lu [4] showed the (119871119901(119908119901) 119871119902(119908119902))boundedness of 1198791198601 119860119896
Ω120572and119872119860
1119860119896
Ω120572(their definitions will
be given later) if 119863120574119860119895isin BMO (|120574| = 119898 minus 1) 119895 = 1 119896
After that Lu and Zhang [5] proved 119879119860Ω120572
is a boundedoperator from 119871119901 to 119871119902 (1119901 minus 1119902 = (120572 + 120573)119899) when119863120574119860 isinΛ120573(|120574| = 119898 minus 1)On the other hand the classical Morrey spaces were first
introduced by Morrey [6] to study the local behavior of solu-tions to second-order elliptic partial differential equationsFrom then on a lot of works concerning Morrey spacesand some related spaces have been done see [7ndash9] and thereferences therein for details In 2009 Komori and Shirai[10] first studied the weightedMorrey spaces and investigatedsome classical singular integrals in harmonic analysis onthem such as the Hardy-Littlewood maximal operator theCalderon-Zygmund operator the fractional integral opera-tor and the fractional maximal operator Recently Wang [11]discussed the boundedness of the classical singular operatorswith rough kernels on the weighted Morrey spaces
We note that many works concerning 119879119860Ω120572
119872119860
Ω120572 119879119860Ω
and119872119860
Ωhave been done on 119871119901 spaces or weighted 119871119901 spaces
when 119863120574119860 belongs to some function spaces for |120574| = 119898 minus1 However there is not any study about these operatorson weighted Morrey spaces Therefore it is natural to askwhether they are bounded on weighted Morrey spaces Theaim of this paper is to investigate the boundedness of 119879119860
Ω120572
119872119860
Ω120572 119879119860Ω and 119872119860
Ωon weighted Morrey spaces if 119863120574119860 isin
Λ120573(|120574| = 119898 minus 1) or 119863120574119860 isin BMO (|120574| = 119898 minus 1)
When 119863120574119860 isin Λ120573(|120574| = 119898 minus 1) we show 119879
119860
Ω120572and
119879119860
Ωare controlled pointwisely by the fractional singular
integral operators 119879Ω120572+120573
and 119879Ω120573
(their definition will begiven later) respectively Thus the problem of studying theboundedness of119879119860
Ω120572and119879119860
ΩonweightedMorrey spaces with
two weights could be reduced to that of 119879Ω120572+120573
and 119879Ω120573
When 119863120574119860 isin BMO (|120574| = 119898 minus 1) the boundedness of119879119860
Ω120572on weighted Morrey spaces with two weights is proved
by standard method However we could only obtain theboundedness of 119879119860
Ωon weighted Morrey spaces with one
weight for 119898 = 1 and 119898 = 2 since we need the 119871119901(119908)boundedness of 119879119860
Ωin our proof but to the best of our
knowledge there is not such bounds hold for119879119860Ωwhen119898 ge 3
For119872119860
Ω120572and119872119860
Ω we show they are controlled pointwisely by
119879119860
Ω120572and 119879119860
Ω respectively Thus it is easy to obtain the same
results for119872119860
Ω120572and119872119860
Ωas those of 119879119860
Ω120572and 119879119860
Ω
Before stating our main results we introduce somedefinitions and notations at first
A weight is a locally integrable function on R119899 whichtakes values in (0infin) almost everywhere For a weight 119908
and a measurable set 119864 we define 119908(119864) = int119864119908(119909)119889119909 the
Lebesgue measure of 119864 by |119864| and the characteristic functionof 119864 by 120594
119864 The weighted Lebesgue spaces with respect to the
measure 119908(119909)119889119909 are denoted by 119871119901(119908) with 0 lt 119901 lt infinWe say a weight 119908 satisfies the doubling condition if thereexists a constant 119863 gt 0 such that for any ball 119861 we have119908(2119861) le 119863119908(119861) When 119908 satisfies this condition we denote119908 isin Δ
2for short
Throughout this paper 119861(1199090 119903) denotes a ball centered at
1199090with radius 119903 Let 119876 be a cube with sides parallel to the
axes For119870 gt 0119870119876 denotes the cube with the same center as119876 and side-length being119870 times longer When 120572 = 0 we willdenote 119879
Ω120572 119879119860Ω120572
119872119860
Ω120572by 119879
Ω 119879119860Ω119872119860
Ω respectively And for
any number 119886 1198861015840 stands for the conjugate of 119886 The letter 119862denotes a positive constant that may vary at each occurrencebut is independent of the essential variable
Next we give the definition of weighted Morrey spaceintroduced in [10]
Definition 1 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight Then the weighted Morrey space is defined by
119871119901120581(119908) fl 119891 isin 119871119901loc (119908)
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) lt infin (7)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) = sup
119861
(1
119908 (119861)120581int119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909) 119889119909)
1119901
(8)
and the supremum is taken over all balls 119861 in R119899
When we investigate the boundedness of the multilinearfractional integral operator we need to consider the weightedMorrey space with two weights It is defined as follows
Definition 2 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119906 V be twoweights The two weights weighted Morrey space is definedby
119871119901120581(119906 V) fl 119891 1003817100381710038171003817119891
1003817100381710038171003817119871119901120581(119906V) lt infin (9)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V) = sup
119861
(1
V (119861)120581int119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119906 (119909) 119889119909)
1119901
(10)
and the supremum is taken over all balls 119861 in R119899 If 119906 = Vthen we denote 119871119901120581(119906) for short
As is pointed out in [10] we could also define theweightedMorrey spaces with cubes instead of balls So we shall usethese two definitions of weighted Morrey spaces appropriateto calculation
Then we give the definitions of Lipschitz space and 119861119872119874space
Definition 3 The Lipschitz space of order 120573 0 lt 120573 lt 1 isdefined by
Λ120573(R119899) = 119891
1003816100381610038161003816119891 (119909) minus 119891 (119910)1003816100381610038161003816 le 119862
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
120573 (11)
and the smallest constant 119862 gt 0 is the Lipschitz norm sdot Λ120573
Journal of Function Spaces 3
Definition 4 A locally integrable function 119887 is said to be inBMO(R119899) if
119887lowast = 119887BMO = sup119861
1
|119861|int119861
1003816100381610038161003816119887 (119909) minus 1198871198611003816100381610038161003816 119889119909 lt infin (12)
where
119887119861=1
|119861|int119861
119887 (119910) 119889119910 (13)
and the supremum is taken over all balls 119861 in R119899
At last we give the definition of two weight classes
Definition 5 A weight function 119908 is in the Muckenhouptclass 119860
119901with 1 lt 119901 lt infin if there exists 119862 gt 1 such that
for any ball 119861
(1
|119861|int119861
119908 (119909) 119889119909)(1
|119861|int119861
119908 (119909)minus1(119901minus1)
119889119909)
119901minus1
le 119862 (14)
We define 119860infin= ⋃
1lt119901ltinfin119860119901
When 119901 = 1 we define 119908 isin 1198601if there exists 119862 gt 1 such
that for almost every 119909
119872119908(119909) le 119862119908 (119909) (15)
Definition 6 A weight function 119908 belongs to 119860(119901 119902) for 1 lt119901 lt 119902 lt infin if there exists 119862 gt 1 such that such that for anyball 119861
(1
|119861|int119861
119908 (119909)119902119889119909)
1119902
sdot (1
|119861|int119861
119908 (119909)minus119901(119901minus1)
119889119909)
(119901minus1)119901
le 119862
(16)
When 119901 = 1 then we define 119908 isin 119860(1 119902) with 1 lt 119902 lt infin ifthere exists 119862 gt 1 such that
(1
|119861|int119861
119908 (119909)119902119889119909)
1119902
(ess sup119909isin119861
1
119908 (119909)) le 119862 (17)
Remark 7 (see [10]) If 119908 isin 119860(119901 119902) with 1 lt 119901 lt 119902 then
(a) 119908119902 119908minus1199011015840
119908minus1199021015840
isin Δ2
(b) 119908minus1199011015840
isin 1198601199051015840 with 119905 = 1 + 1199021199011015840
Now we state the main results of this paper
Theorem 8 If 0 lt 120572 + 120573 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1)
is homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899(120572 + 120573)
1119902 = 1119901 minus (120572 + 120573)119899 0 lt 120581 lt 119901119902 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840)
119863120574119860 isin Λ
120573(|120574| = 119898 minus 1) then
10038171003817100381710038171003817119879119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (18)
10038171003817100381710038171003817119872119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (19)
Theorem 9 If 0 lt 120573 lt 1 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is
homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899120573 1119902 =
1119901 minus120573119899 0 lt 120581 lt 1199011199021199081199041015840
isin 119860(1199011199041015840 119902119904
1015840)119863120574119860 isin Λ
120573(|120574| =
119898 minus 1) then
10038171003817100381710038171003817119879119860
Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (20)
10038171003817100381710038171003817119872119860
Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (21)
Theorem 10 If 0 lt 120572 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homo-
geneous of degree zero 1 lt 1199041015840 lt 119901 lt 119899120572 1119902 = 1119901 minus 1205721198990 lt 120581 lt 119901119902 119908119904
1015840
isin 119860(1199011199041015840 119902119904
1015840) 119863120574119860 isin BMO (|120574| = 119898 minus 1)
then
10038171003817100381710038171003817119879119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (22)
10038171003817100381710038171003817119872119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (23)
When 119898 = 1 and 119898 = 2 we denote 119879119860Ω119872119860
Ωby [119860 119879
Ω]
[119860119872Ω] and
119860
Ω 119860
Ω respectively in order to distinguish
from119879119860Ωand119872119860
Ωthat are defined for any119898 isin Nlowast To bemore
precise
[119860 119879Ω] 119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899(119860 (119909) minus 119860 (119910))
sdot 119891 (119910) 119889119910
[119860119872Ω] 119891 (119909) = sup
119903gt0
1
119903119899int|119909minus119910|lt119903
Ω(119909 minus 119910)
sdot (119860 (119909) minus 119860 (119910)) 119891 (119910) 119889119910
119860
Ω119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899+1(119860 (119909) minus 119860 (119910) minus nabla119860 (119910)
sdot (119909 minus 119910)) 119891 (119910) 119889119910
119860
Ω119891 (119909) = sup
119903gt0
1
119903119899+1int|119909minus119910|lt119903
Ω(119909 minus 119910)
sdot (119860 (119909) minus 119860 (119910) minus nabla119860 (119910) (119909 minus 119910)) 119891 (119910) 119889119910
(24)
Then for the above operators we have the following resultson weighted Morrey spaces with one weight
Theorem 11 IfΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degreezero and satisfies the vanishing condition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) =
0 1 lt 1199041015840 lt 119901 lt infin 0 lt 120581 lt 1 119908 isin 1198601199011199041015840 119860 isin BMO then
1003817100381710038171003817[119860 119879Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (25)
1003817100381710038171003817[119860119872Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (26)
4 Journal of Function Spaces
Theorem 12 If Ω isin 119871infin(119878119899minus1) is homogeneous of degree zeroand satisfies the moment condition int
119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt
infin 0 lt 120581 lt 1 119908 isin 119860119901 nabla119860 isin BMO then
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)
Remark 13 Here we point out that for 119879119860Ωand 119872119860
Ω when
119863120574119860 isin BMO (|120574| = 119898 minus 1) the analogues of Theorems 11
and 12 are open for119898 ge 3
Remark 14 Define
1198791198601119860119896
Ω120572119891 (119909) = int
R119899
119896
prod
119894=1
119877119898119894
(119860119894 119909 119910)
sdotΩ (119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119873119891 (119910) 119889119910
1198721198601119860119896
Ω120572119891 (119909) = sup
119903gt0
1
119903119899minus120572+119873int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
sdot
119896
prod
119894=1
10038161003816100381610038161003816119877119898119894
(119860119894 119909 119910)
10038161003816100381610038161003816
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(29)
where 119877119898119894
(119860119894 119909 119910) = 119860
119894(119909) minus sum
|120574|lt119898119894
(1120574)119863120574119860119894(119910)(119909 minus
119910)120574 119894 = 1 119896 119873 = sum
119896
119894=1(119898119894minus 1) When 0 lt 120572 lt 119899
they are a class of multilinear fractional integral operatorsand multilinear fractional maximal operators When 120572 = 0they are a class of multilinear singular integral operators andmultilinear maximal singular integral operators Repeatingthe proofs of the theorems above we will find that for1198791198601119860119896
Ω120572and 119872119860
1119860119896
Ω120572 the conclusions of Theorems 8 and
9 above with the bounds 119862prod119896119894=1(sum|120574|=119898
119894minus1119863120574119860119894Λ120573
) andTheorem 10 with the bounds 119862prod119896
119894=1(sum|120574|=119898
119894minus1119863120574119860119894lowast) also
hold respectively
The organization of this paper is as follows In Section 2we give some requisite lemmas and well-known results thatare important in proving the theorems The proof of thetheorems will be shown in Section 3
2 Lemmas and Well-Known Results
Lemma 15 (see [1]) Let 119860(119909) be a function on R119899 with 119898thorder derivatives in 119871119897loc(R
119899) for some 119897 gt 119899 Then
1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898sum
|120574|=119898
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
1003816100381610038161003816119863120574119860 (119911)
1003816100381610038161003816
119897119889119911)
1119897
(30)
where 119868119910119909is the cube centered at 119909with sides parallel to the axes
whose diameter is 5radic119899|119909 minus 119910|
Lemma 16 (see [12]) For 0 lt 120573 lt 1 1 le 119902 lt infin we have
10038171003817100381710038171198911003817100381710038171003817Λ120573
asymp sup119876
1
|119876|1+120573119899
int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909
asymp sup119876
1
|119876|120573119899(1
|119876|int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816
119902119889119909)
1119902
(31)
For 119902 = infin the formula should be interpreted appropriately
Lemma 17 (see [13]) Let 1198761sub 119876
2 119892 isin Λ
120573(0 lt 120573 lt 1) Then
100381610038161003816100381610038161198921198761
minus 1198921198762
10038161003816100381610038161003816le 119862
100381610038161003816100381611987621003816100381610038161003816
120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573
(32)
Theorem 18 (see [14]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 and Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous ofdegree zero Then 119879
Ω120572is a bounded operator from 119871
119901(119908119901) to
119871119902(119908119902) if the index set 120572 119901 119902 119904 satisfies one of the following
conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840
such that 119908(119909)1199031015840
isin 119860(119901 119902)
Lemma 19 (see [10]) If 119908 isin Δ2 then there exists a constant
1198631gt 1 such that
119908 (2119861) ge 1198631119908 (119861) (33)
We call1198631the reverse doubling constant
Theorem 20 (see [4]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 Ω isin 119871
119904(119878119899minus1) (119904 gt 1) is homogeneous of
degree zero Moreover for 1 le 119894 le 119896 |120574| = 119898119894minus 1119898
119894ge 2 and
119863120574119860119894isin BMO(R119899) if the index set 120572 119901 119902 119904 satisfies one of
the following conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840such that 119908(119909)119903
1015840
isin 119860(119901 119902)
Then there is a 119862 gt 0 independent of 119891 and 119860119894 such that
(intR119899
100381610038161003816100381610038161198791198601119860119896
Ω120572119891 (119909)119908 (119909)
10038161003816100381610038161003816
119902
119889119909)
1119902
le 119862
119896
prod
119894=1
( sum
|120574|=119898119894minus1
1003817100381710038171003817119863120574119860119894
1003817100381710038171003817lowast)
sdot (intR119899
1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816
119901119889119909)
1119901
(34)
Lemma 21 (see [15]) (a) (John-Nirenberg Lemma) Let 1 le119901 lt infin Then 119887 isin BMO if and only if
1
|119876|int119876
1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816
119901119889119909 le 119862 119887
119901
lowast (35)
Journal of Function Spaces 5
(b) Assume 119887 isin BMO then for cubes 1198761sub 119876
2
100381610038161003816100381610038161198871198761
minus 1198871198762
10038161003816100381610038161003816le 119862 log(
100381610038161003816100381611987621003816100381610038161003816
100381610038161003816100381611987611003816100381610038161003816
) 119887lowast (36)
(c) If 119887 isin BMO then10038161003816100381610038161198872119895+1119861 minus 119887119861
1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)
Theorem 22 (see [16]) Suppose that Ω isin 119871119904(119878119899minus1) (119904 gt
1) is homogeneous of degree zero and satisfies the vanishingcondition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879
Ω]
is bounded on 119871119901(119908) if the index set 119901 119902 119904 satisfies one of thefollowing conditions
(a) 1199041015840 le 119901 lt infin 119901 = 1 and 119908 isin 1198601199011199041015840
(b) 1 le 119901 le 119904 119901 = infin and 1199081minus1199011015840
isin 11986011990110158401199041015840
(c) 1 le 119901 lt infin and 1199081199041015840
isin 119860119901
Theorem 23 (see [2]) If Ω isin 119871infin(119878119899minus1) is homogeneous of
degree zero and satisfies themoment condition int119878119899minus1120579Ω(120579)119889120579 =
0 119908 isin 119860119901 1 lt 119901 lt infin nabla119860 isin BMO then we have100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901(119908)
le 119862 Ωinfin nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901(119908) (38)
Lemma 24 (see [15]) The following are true(1) If 119908 isin 119860
119901for some 1 le 119901 lt infin then 119908 isin Δ
2 More
precisely for all 120582 gt 1 we have
119908 (120582119876) le 119862120582119899119901119908 (119876) (39)
(2) If 119908 isin 119860119901for some 1 le 119901 lt infin then there exist 119862 gt 0
and 120575 gt 0 such that for any cube 119876 and a measurableset 119878 sub 119876
119908 (119878)
119908 (119876)le 119862(
|119878|
|119876|)
120575
(40)
Lemma25 (see [17]) Let119908 isin 119860infinThen the normofBMO(119908)
is equivalent to the norm of BMO(R119899) where
BMO (119908) = 119887 119887lowast119908
= sup119876
1
119908 (119876)int119876
1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909
119898119876119908119887 =
1
119908 (119876)int119876
119887 (119909)119908 (119909) 119889119909
(41)
3 Proofs of the Main Results
Before proving Theorem 8 we give a pointwise estimate of119879119860
Ω120572119891(119909) at first Set
119879Ω120572+120573
119891 (119909) = intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
0 lt 120572 + 120573 lt 119899
(42)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 Then we have the following estimate
Theorem26 If 120572 ge 0 0 lt 120572+120573 lt 119899119863120574119860 isin Λ120573(|120574| = 119898minus1)
then there exists a constant 119862 independent of 119891 such that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le 119862( sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
)119879Ω120572+120573
119891 (119909) (43)
Proof For fixed 119909 isin R119899 119903 gt 0 let 119876 be a cube with center at119909 and diameter 119903 Denote 119876
119896= 2
119896119876 and set
119860119876119896
(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876119896
(119863120574119860)119910
120574 (44)
where 119898119876119896
119891 is the average of 119891 on 119876119896 Then we have when
|120574| = 119898 minus 1
119863120574119860119876119896
(119910) = 119863120574119860 (119910) minus 119898
119876119896
(119863120574119860) (45)
and it is proved in [1] that
119877119898(119860 119909 119910) = 119877
119898(119860119876119896
119909 119910) (46)
Hence
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816
le
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
flinfin
sum
119896=minusinfin
119879119896
(47)
By Lemma 15 we get
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le10038161003816100381610038161003816119877119898minus1
(119860119876119896
119909 119910)10038161003816100381610038161003816
+ 119862 sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
+ 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
(48)
6 Journal of Function Spaces
Note that if |119909 minus 119910| lt 2119896119903 then 119868119910119909sub 5119899119876
119896 By Lemmas 16
and 17 we have when |120574| = 119898 minus 1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
= (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876
119896
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
le (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119868119910
119909
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
+10038161003816100381610038161003816119898119868119910
119909
(119863120574119860) minus 119898
5119899119876119896
(119863120574119860)10038161003816100381610038161003816
+100381610038161003816100381610038161198985119899119876119896
(119863120574119860) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(49)
It is obvious that when |120574| = 119898 minus 1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(50)
Thus
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1(2119896119903)120573
sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
(51)
Therefore
119879119896le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
(2119896119903)120573
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(52)
It follows that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le
infin
sum
119896=minusinfin
(119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
119879Ω120572+120573
119891 (119909)
(53)
Thus we finish the proof of Theorem 26
The following theorem is a key theorem in proving (18) ofTheorem 8
Theorem 27 Under the same assumptions of Theorem 8119879Ω120572+120573
is bounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
Proof Fix a ball 119861(1199090 119903119861) we decompose 119891 = 119891
1+ 119891
2with
1198911= 119891120594
2119861 Then we have
(1
119908119902 (119861)120581119902119901
int119861
10038161003816100381610038161003816119879Ω120572+120573
119891 (119909)10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
le1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198911 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
+1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
fl 1198691+ 1198692
(54)
We estimate 1198691at first By Remark 7(a) we know that119908119902 isin Δ
2
Then byTheorem 18(a) and the fact that 119908119902 isin Δ2we get
1198691le
1
119908119902 (119861)120581119901
10038171003817100381710038171003817119879Ω120572+120573
1198911
10038171003817100381710038171003817119871119902(119908119902)
le119862
119908119902 (119861)120581119901
100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)
=119862
119908119902 (119861)120581119901(int2119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909)
119901119889119909)
1119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
119908119902(2119861)
120581119901
119908119902 (119861)120581119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(55)
Journal of Function Spaces 7
Now we consider the term 1198692 By Holderrsquos inequality we have
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816=
infin
sum
119895=1
int2119895+11198612119895119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
le 119862
infin
sum
119895=1
(int2119895+1119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
119904119889119910)
1119904
sdot (int2119895+11198612119895119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
(119899minus120572minus120573)1199041015840119889119910)
11199041015840
fl 119862infin
sum
119895=1
(11986811198951198682119895)
(56)
We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for
119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have
1198681119895= (int
2119895+2119861
|Ω (119911)|119904119889119911)
1119904
= (int
2119895+2
119903119861
0
int119878119899minus1
10038161003816100381610038161003816Ω (119911
1015840)10038161003816100381610038161003816
119904
1198891199111015840119903119899minus1119889119903)
1119904
= 119862 Ω119871119904(119878119899minus1)
100381610038161003816100381610038162119895+211986110038161003816100381610038161003816
1119904
(57)
where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus
1198682119895le
119862
10038161003816100381610038162119895+1119861
1003816100381610038161003816
1minus(120572+120573)119899(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
(58)
By Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
le 119862(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot
100381610038161003816100381610038162119895+111986110038161003816100381610038161003816
(119901119902minus1199041015840
119902+1199041015840
119901)1199011199021199041015840
119908119902 (2119895+1119861)1119902
(59)
Thus
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816le 119862
infin
sum
119895=1
(11986811198951198682119895)
le 119862
infin
sum
119895=1
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
1
119908119902 (2119895+1119861)1119902minus120581119901
(60)
So we get
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119908119902(119861)
1119902minus120581119901
119908119902 (2119895+1119861)1119902minus120581119901
(61)
We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)
Therefore the proof of Theorem 27 is completed
Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879
Ω120572is bounded
on weighted Morrey spaces
Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-
rems 26 and 27Then let us turn to prove (19)Set
119879119860
Ω120572119891 (119909)
= intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
0 le 120572 lt 119899
(63)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879
119860
Ω120572 On the
other hand for any 119903 gt 0 we have
119879119860
Ω120572119891 (119909)
ge int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
ge1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)
10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
(64)
Taking the supremum for 119903 gt 0 on the inequality above weget
119879119860
Ω120572119891 (119909) ge 119872
119860
Ω120572119891 (119909) (65)
Thus we can immediately obtain (19) from (65) and (18)
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 2
2 Journal of Function Spaces
the 119871119901 boundedness of119879119860Ωwas obtained bymeans of a good-120582
inequality by Cohen and Gosselin [1] In 1994 Hofmann [2]proved that 119879119860
Ωis a bounded operator on 119871119901(119908) when nabla119860 isin
119861119872119874 and 119908 isin 119860119901 Recently Lu et al [3] proved 119879119860
Ωand119872119860
Ω
are bounded from 119871119901 to 119871119902 (1119901 minus 1119902 = 120573119899) when 119863120574119860 isin
Λ120573(|120574| = 119898 minus 1) while for multilinear fractional integral
operators Ding and Lu [4] showed the (119871119901(119908119901) 119871119902(119908119902))boundedness of 1198791198601 119860119896
Ω120572and119872119860
1119860119896
Ω120572(their definitions will
be given later) if 119863120574119860119895isin BMO (|120574| = 119898 minus 1) 119895 = 1 119896
After that Lu and Zhang [5] proved 119879119860Ω120572
is a boundedoperator from 119871119901 to 119871119902 (1119901 minus 1119902 = (120572 + 120573)119899) when119863120574119860 isinΛ120573(|120574| = 119898 minus 1)On the other hand the classical Morrey spaces were first
introduced by Morrey [6] to study the local behavior of solu-tions to second-order elliptic partial differential equationsFrom then on a lot of works concerning Morrey spacesand some related spaces have been done see [7ndash9] and thereferences therein for details In 2009 Komori and Shirai[10] first studied the weightedMorrey spaces and investigatedsome classical singular integrals in harmonic analysis onthem such as the Hardy-Littlewood maximal operator theCalderon-Zygmund operator the fractional integral opera-tor and the fractional maximal operator Recently Wang [11]discussed the boundedness of the classical singular operatorswith rough kernels on the weighted Morrey spaces
We note that many works concerning 119879119860Ω120572
119872119860
Ω120572 119879119860Ω
and119872119860
Ωhave been done on 119871119901 spaces or weighted 119871119901 spaces
when 119863120574119860 belongs to some function spaces for |120574| = 119898 minus1 However there is not any study about these operatorson weighted Morrey spaces Therefore it is natural to askwhether they are bounded on weighted Morrey spaces Theaim of this paper is to investigate the boundedness of 119879119860
Ω120572
119872119860
Ω120572 119879119860Ω and 119872119860
Ωon weighted Morrey spaces if 119863120574119860 isin
Λ120573(|120574| = 119898 minus 1) or 119863120574119860 isin BMO (|120574| = 119898 minus 1)
When 119863120574119860 isin Λ120573(|120574| = 119898 minus 1) we show 119879
119860
Ω120572and
119879119860
Ωare controlled pointwisely by the fractional singular
integral operators 119879Ω120572+120573
and 119879Ω120573
(their definition will begiven later) respectively Thus the problem of studying theboundedness of119879119860
Ω120572and119879119860
ΩonweightedMorrey spaces with
two weights could be reduced to that of 119879Ω120572+120573
and 119879Ω120573
When 119863120574119860 isin BMO (|120574| = 119898 minus 1) the boundedness of119879119860
Ω120572on weighted Morrey spaces with two weights is proved
by standard method However we could only obtain theboundedness of 119879119860
Ωon weighted Morrey spaces with one
weight for 119898 = 1 and 119898 = 2 since we need the 119871119901(119908)boundedness of 119879119860
Ωin our proof but to the best of our
knowledge there is not such bounds hold for119879119860Ωwhen119898 ge 3
For119872119860
Ω120572and119872119860
Ω we show they are controlled pointwisely by
119879119860
Ω120572and 119879119860
Ω respectively Thus it is easy to obtain the same
results for119872119860
Ω120572and119872119860
Ωas those of 119879119860
Ω120572and 119879119860
Ω
Before stating our main results we introduce somedefinitions and notations at first
A weight is a locally integrable function on R119899 whichtakes values in (0infin) almost everywhere For a weight 119908
and a measurable set 119864 we define 119908(119864) = int119864119908(119909)119889119909 the
Lebesgue measure of 119864 by |119864| and the characteristic functionof 119864 by 120594
119864 The weighted Lebesgue spaces with respect to the
measure 119908(119909)119889119909 are denoted by 119871119901(119908) with 0 lt 119901 lt infinWe say a weight 119908 satisfies the doubling condition if thereexists a constant 119863 gt 0 such that for any ball 119861 we have119908(2119861) le 119863119908(119861) When 119908 satisfies this condition we denote119908 isin Δ
2for short
Throughout this paper 119861(1199090 119903) denotes a ball centered at
1199090with radius 119903 Let 119876 be a cube with sides parallel to the
axes For119870 gt 0119870119876 denotes the cube with the same center as119876 and side-length being119870 times longer When 120572 = 0 we willdenote 119879
Ω120572 119879119860Ω120572
119872119860
Ω120572by 119879
Ω 119879119860Ω119872119860
Ω respectively And for
any number 119886 1198861015840 stands for the conjugate of 119886 The letter 119862denotes a positive constant that may vary at each occurrencebut is independent of the essential variable
Next we give the definition of weighted Morrey spaceintroduced in [10]
Definition 1 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119908 be aweight Then the weighted Morrey space is defined by
119871119901120581(119908) fl 119891 isin 119871119901loc (119908)
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) lt infin (7)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) = sup
119861
(1
119908 (119861)120581int119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909) 119889119909)
1119901
(8)
and the supremum is taken over all balls 119861 in R119899
When we investigate the boundedness of the multilinearfractional integral operator we need to consider the weightedMorrey space with two weights It is defined as follows
Definition 2 Let 1 le 119901 lt infin let 0 lt 120581 lt 1 and let 119906 V be twoweights The two weights weighted Morrey space is definedby
119871119901120581(119906 V) fl 119891 1003817100381710038171003817119891
1003817100381710038171003817119871119901120581(119906V) lt infin (9)
where
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V) = sup
119861
(1
V (119861)120581int119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119906 (119909) 119889119909)
1119901
(10)
and the supremum is taken over all balls 119861 in R119899 If 119906 = Vthen we denote 119871119901120581(119906) for short
As is pointed out in [10] we could also define theweightedMorrey spaces with cubes instead of balls So we shall usethese two definitions of weighted Morrey spaces appropriateto calculation
Then we give the definitions of Lipschitz space and 119861119872119874space
Definition 3 The Lipschitz space of order 120573 0 lt 120573 lt 1 isdefined by
Λ120573(R119899) = 119891
1003816100381610038161003816119891 (119909) minus 119891 (119910)1003816100381610038161003816 le 119862
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
120573 (11)
and the smallest constant 119862 gt 0 is the Lipschitz norm sdot Λ120573
Journal of Function Spaces 3
Definition 4 A locally integrable function 119887 is said to be inBMO(R119899) if
119887lowast = 119887BMO = sup119861
1
|119861|int119861
1003816100381610038161003816119887 (119909) minus 1198871198611003816100381610038161003816 119889119909 lt infin (12)
where
119887119861=1
|119861|int119861
119887 (119910) 119889119910 (13)
and the supremum is taken over all balls 119861 in R119899
At last we give the definition of two weight classes
Definition 5 A weight function 119908 is in the Muckenhouptclass 119860
119901with 1 lt 119901 lt infin if there exists 119862 gt 1 such that
for any ball 119861
(1
|119861|int119861
119908 (119909) 119889119909)(1
|119861|int119861
119908 (119909)minus1(119901minus1)
119889119909)
119901minus1
le 119862 (14)
We define 119860infin= ⋃
1lt119901ltinfin119860119901
When 119901 = 1 we define 119908 isin 1198601if there exists 119862 gt 1 such
that for almost every 119909
119872119908(119909) le 119862119908 (119909) (15)
Definition 6 A weight function 119908 belongs to 119860(119901 119902) for 1 lt119901 lt 119902 lt infin if there exists 119862 gt 1 such that such that for anyball 119861
(1
|119861|int119861
119908 (119909)119902119889119909)
1119902
sdot (1
|119861|int119861
119908 (119909)minus119901(119901minus1)
119889119909)
(119901minus1)119901
le 119862
(16)
When 119901 = 1 then we define 119908 isin 119860(1 119902) with 1 lt 119902 lt infin ifthere exists 119862 gt 1 such that
(1
|119861|int119861
119908 (119909)119902119889119909)
1119902
(ess sup119909isin119861
1
119908 (119909)) le 119862 (17)
Remark 7 (see [10]) If 119908 isin 119860(119901 119902) with 1 lt 119901 lt 119902 then
(a) 119908119902 119908minus1199011015840
119908minus1199021015840
isin Δ2
(b) 119908minus1199011015840
isin 1198601199051015840 with 119905 = 1 + 1199021199011015840
Now we state the main results of this paper
Theorem 8 If 0 lt 120572 + 120573 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1)
is homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899(120572 + 120573)
1119902 = 1119901 minus (120572 + 120573)119899 0 lt 120581 lt 119901119902 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840)
119863120574119860 isin Λ
120573(|120574| = 119898 minus 1) then
10038171003817100381710038171003817119879119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (18)
10038171003817100381710038171003817119872119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (19)
Theorem 9 If 0 lt 120573 lt 1 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is
homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899120573 1119902 =
1119901 minus120573119899 0 lt 120581 lt 1199011199021199081199041015840
isin 119860(1199011199041015840 119902119904
1015840)119863120574119860 isin Λ
120573(|120574| =
119898 minus 1) then
10038171003817100381710038171003817119879119860
Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (20)
10038171003817100381710038171003817119872119860
Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (21)
Theorem 10 If 0 lt 120572 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homo-
geneous of degree zero 1 lt 1199041015840 lt 119901 lt 119899120572 1119902 = 1119901 minus 1205721198990 lt 120581 lt 119901119902 119908119904
1015840
isin 119860(1199011199041015840 119902119904
1015840) 119863120574119860 isin BMO (|120574| = 119898 minus 1)
then
10038171003817100381710038171003817119879119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (22)
10038171003817100381710038171003817119872119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (23)
When 119898 = 1 and 119898 = 2 we denote 119879119860Ω119872119860
Ωby [119860 119879
Ω]
[119860119872Ω] and
119860
Ω 119860
Ω respectively in order to distinguish
from119879119860Ωand119872119860
Ωthat are defined for any119898 isin Nlowast To bemore
precise
[119860 119879Ω] 119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899(119860 (119909) minus 119860 (119910))
sdot 119891 (119910) 119889119910
[119860119872Ω] 119891 (119909) = sup
119903gt0
1
119903119899int|119909minus119910|lt119903
Ω(119909 minus 119910)
sdot (119860 (119909) minus 119860 (119910)) 119891 (119910) 119889119910
119860
Ω119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899+1(119860 (119909) minus 119860 (119910) minus nabla119860 (119910)
sdot (119909 minus 119910)) 119891 (119910) 119889119910
119860
Ω119891 (119909) = sup
119903gt0
1
119903119899+1int|119909minus119910|lt119903
Ω(119909 minus 119910)
sdot (119860 (119909) minus 119860 (119910) minus nabla119860 (119910) (119909 minus 119910)) 119891 (119910) 119889119910
(24)
Then for the above operators we have the following resultson weighted Morrey spaces with one weight
Theorem 11 IfΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degreezero and satisfies the vanishing condition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) =
0 1 lt 1199041015840 lt 119901 lt infin 0 lt 120581 lt 1 119908 isin 1198601199011199041015840 119860 isin BMO then
1003817100381710038171003817[119860 119879Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (25)
1003817100381710038171003817[119860119872Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (26)
4 Journal of Function Spaces
Theorem 12 If Ω isin 119871infin(119878119899minus1) is homogeneous of degree zeroand satisfies the moment condition int
119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt
infin 0 lt 120581 lt 1 119908 isin 119860119901 nabla119860 isin BMO then
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)
Remark 13 Here we point out that for 119879119860Ωand 119872119860
Ω when
119863120574119860 isin BMO (|120574| = 119898 minus 1) the analogues of Theorems 11
and 12 are open for119898 ge 3
Remark 14 Define
1198791198601119860119896
Ω120572119891 (119909) = int
R119899
119896
prod
119894=1
119877119898119894
(119860119894 119909 119910)
sdotΩ (119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119873119891 (119910) 119889119910
1198721198601119860119896
Ω120572119891 (119909) = sup
119903gt0
1
119903119899minus120572+119873int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
sdot
119896
prod
119894=1
10038161003816100381610038161003816119877119898119894
(119860119894 119909 119910)
10038161003816100381610038161003816
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(29)
where 119877119898119894
(119860119894 119909 119910) = 119860
119894(119909) minus sum
|120574|lt119898119894
(1120574)119863120574119860119894(119910)(119909 minus
119910)120574 119894 = 1 119896 119873 = sum
119896
119894=1(119898119894minus 1) When 0 lt 120572 lt 119899
they are a class of multilinear fractional integral operatorsand multilinear fractional maximal operators When 120572 = 0they are a class of multilinear singular integral operators andmultilinear maximal singular integral operators Repeatingthe proofs of the theorems above we will find that for1198791198601119860119896
Ω120572and 119872119860
1119860119896
Ω120572 the conclusions of Theorems 8 and
9 above with the bounds 119862prod119896119894=1(sum|120574|=119898
119894minus1119863120574119860119894Λ120573
) andTheorem 10 with the bounds 119862prod119896
119894=1(sum|120574|=119898
119894minus1119863120574119860119894lowast) also
hold respectively
The organization of this paper is as follows In Section 2we give some requisite lemmas and well-known results thatare important in proving the theorems The proof of thetheorems will be shown in Section 3
2 Lemmas and Well-Known Results
Lemma 15 (see [1]) Let 119860(119909) be a function on R119899 with 119898thorder derivatives in 119871119897loc(R
119899) for some 119897 gt 119899 Then
1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898sum
|120574|=119898
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
1003816100381610038161003816119863120574119860 (119911)
1003816100381610038161003816
119897119889119911)
1119897
(30)
where 119868119910119909is the cube centered at 119909with sides parallel to the axes
whose diameter is 5radic119899|119909 minus 119910|
Lemma 16 (see [12]) For 0 lt 120573 lt 1 1 le 119902 lt infin we have
10038171003817100381710038171198911003817100381710038171003817Λ120573
asymp sup119876
1
|119876|1+120573119899
int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909
asymp sup119876
1
|119876|120573119899(1
|119876|int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816
119902119889119909)
1119902
(31)
For 119902 = infin the formula should be interpreted appropriately
Lemma 17 (see [13]) Let 1198761sub 119876
2 119892 isin Λ
120573(0 lt 120573 lt 1) Then
100381610038161003816100381610038161198921198761
minus 1198921198762
10038161003816100381610038161003816le 119862
100381610038161003816100381611987621003816100381610038161003816
120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573
(32)
Theorem 18 (see [14]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 and Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous ofdegree zero Then 119879
Ω120572is a bounded operator from 119871
119901(119908119901) to
119871119902(119908119902) if the index set 120572 119901 119902 119904 satisfies one of the following
conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840
such that 119908(119909)1199031015840
isin 119860(119901 119902)
Lemma 19 (see [10]) If 119908 isin Δ2 then there exists a constant
1198631gt 1 such that
119908 (2119861) ge 1198631119908 (119861) (33)
We call1198631the reverse doubling constant
Theorem 20 (see [4]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 Ω isin 119871
119904(119878119899minus1) (119904 gt 1) is homogeneous of
degree zero Moreover for 1 le 119894 le 119896 |120574| = 119898119894minus 1119898
119894ge 2 and
119863120574119860119894isin BMO(R119899) if the index set 120572 119901 119902 119904 satisfies one of
the following conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840such that 119908(119909)119903
1015840
isin 119860(119901 119902)
Then there is a 119862 gt 0 independent of 119891 and 119860119894 such that
(intR119899
100381610038161003816100381610038161198791198601119860119896
Ω120572119891 (119909)119908 (119909)
10038161003816100381610038161003816
119902
119889119909)
1119902
le 119862
119896
prod
119894=1
( sum
|120574|=119898119894minus1
1003817100381710038171003817119863120574119860119894
1003817100381710038171003817lowast)
sdot (intR119899
1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816
119901119889119909)
1119901
(34)
Lemma 21 (see [15]) (a) (John-Nirenberg Lemma) Let 1 le119901 lt infin Then 119887 isin BMO if and only if
1
|119876|int119876
1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816
119901119889119909 le 119862 119887
119901
lowast (35)
Journal of Function Spaces 5
(b) Assume 119887 isin BMO then for cubes 1198761sub 119876
2
100381610038161003816100381610038161198871198761
minus 1198871198762
10038161003816100381610038161003816le 119862 log(
100381610038161003816100381611987621003816100381610038161003816
100381610038161003816100381611987611003816100381610038161003816
) 119887lowast (36)
(c) If 119887 isin BMO then10038161003816100381610038161198872119895+1119861 minus 119887119861
1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)
Theorem 22 (see [16]) Suppose that Ω isin 119871119904(119878119899minus1) (119904 gt
1) is homogeneous of degree zero and satisfies the vanishingcondition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879
Ω]
is bounded on 119871119901(119908) if the index set 119901 119902 119904 satisfies one of thefollowing conditions
(a) 1199041015840 le 119901 lt infin 119901 = 1 and 119908 isin 1198601199011199041015840
(b) 1 le 119901 le 119904 119901 = infin and 1199081minus1199011015840
isin 11986011990110158401199041015840
(c) 1 le 119901 lt infin and 1199081199041015840
isin 119860119901
Theorem 23 (see [2]) If Ω isin 119871infin(119878119899minus1) is homogeneous of
degree zero and satisfies themoment condition int119878119899minus1120579Ω(120579)119889120579 =
0 119908 isin 119860119901 1 lt 119901 lt infin nabla119860 isin BMO then we have100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901(119908)
le 119862 Ωinfin nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901(119908) (38)
Lemma 24 (see [15]) The following are true(1) If 119908 isin 119860
119901for some 1 le 119901 lt infin then 119908 isin Δ
2 More
precisely for all 120582 gt 1 we have
119908 (120582119876) le 119862120582119899119901119908 (119876) (39)
(2) If 119908 isin 119860119901for some 1 le 119901 lt infin then there exist 119862 gt 0
and 120575 gt 0 such that for any cube 119876 and a measurableset 119878 sub 119876
119908 (119878)
119908 (119876)le 119862(
|119878|
|119876|)
120575
(40)
Lemma25 (see [17]) Let119908 isin 119860infinThen the normofBMO(119908)
is equivalent to the norm of BMO(R119899) where
BMO (119908) = 119887 119887lowast119908
= sup119876
1
119908 (119876)int119876
1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909
119898119876119908119887 =
1
119908 (119876)int119876
119887 (119909)119908 (119909) 119889119909
(41)
3 Proofs of the Main Results
Before proving Theorem 8 we give a pointwise estimate of119879119860
Ω120572119891(119909) at first Set
119879Ω120572+120573
119891 (119909) = intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
0 lt 120572 + 120573 lt 119899
(42)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 Then we have the following estimate
Theorem26 If 120572 ge 0 0 lt 120572+120573 lt 119899119863120574119860 isin Λ120573(|120574| = 119898minus1)
then there exists a constant 119862 independent of 119891 such that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le 119862( sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
)119879Ω120572+120573
119891 (119909) (43)
Proof For fixed 119909 isin R119899 119903 gt 0 let 119876 be a cube with center at119909 and diameter 119903 Denote 119876
119896= 2
119896119876 and set
119860119876119896
(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876119896
(119863120574119860)119910
120574 (44)
where 119898119876119896
119891 is the average of 119891 on 119876119896 Then we have when
|120574| = 119898 minus 1
119863120574119860119876119896
(119910) = 119863120574119860 (119910) minus 119898
119876119896
(119863120574119860) (45)
and it is proved in [1] that
119877119898(119860 119909 119910) = 119877
119898(119860119876119896
119909 119910) (46)
Hence
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816
le
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
flinfin
sum
119896=minusinfin
119879119896
(47)
By Lemma 15 we get
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le10038161003816100381610038161003816119877119898minus1
(119860119876119896
119909 119910)10038161003816100381610038161003816
+ 119862 sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
+ 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
(48)
6 Journal of Function Spaces
Note that if |119909 minus 119910| lt 2119896119903 then 119868119910119909sub 5119899119876
119896 By Lemmas 16
and 17 we have when |120574| = 119898 minus 1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
= (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876
119896
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
le (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119868119910
119909
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
+10038161003816100381610038161003816119898119868119910
119909
(119863120574119860) minus 119898
5119899119876119896
(119863120574119860)10038161003816100381610038161003816
+100381610038161003816100381610038161198985119899119876119896
(119863120574119860) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(49)
It is obvious that when |120574| = 119898 minus 1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(50)
Thus
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1(2119896119903)120573
sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
(51)
Therefore
119879119896le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
(2119896119903)120573
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(52)
It follows that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le
infin
sum
119896=minusinfin
(119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
119879Ω120572+120573
119891 (119909)
(53)
Thus we finish the proof of Theorem 26
The following theorem is a key theorem in proving (18) ofTheorem 8
Theorem 27 Under the same assumptions of Theorem 8119879Ω120572+120573
is bounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
Proof Fix a ball 119861(1199090 119903119861) we decompose 119891 = 119891
1+ 119891
2with
1198911= 119891120594
2119861 Then we have
(1
119908119902 (119861)120581119902119901
int119861
10038161003816100381610038161003816119879Ω120572+120573
119891 (119909)10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
le1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198911 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
+1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
fl 1198691+ 1198692
(54)
We estimate 1198691at first By Remark 7(a) we know that119908119902 isin Δ
2
Then byTheorem 18(a) and the fact that 119908119902 isin Δ2we get
1198691le
1
119908119902 (119861)120581119901
10038171003817100381710038171003817119879Ω120572+120573
1198911
10038171003817100381710038171003817119871119902(119908119902)
le119862
119908119902 (119861)120581119901
100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)
=119862
119908119902 (119861)120581119901(int2119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909)
119901119889119909)
1119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
119908119902(2119861)
120581119901
119908119902 (119861)120581119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(55)
Journal of Function Spaces 7
Now we consider the term 1198692 By Holderrsquos inequality we have
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816=
infin
sum
119895=1
int2119895+11198612119895119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
le 119862
infin
sum
119895=1
(int2119895+1119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
119904119889119910)
1119904
sdot (int2119895+11198612119895119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
(119899minus120572minus120573)1199041015840119889119910)
11199041015840
fl 119862infin
sum
119895=1
(11986811198951198682119895)
(56)
We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for
119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have
1198681119895= (int
2119895+2119861
|Ω (119911)|119904119889119911)
1119904
= (int
2119895+2
119903119861
0
int119878119899minus1
10038161003816100381610038161003816Ω (119911
1015840)10038161003816100381610038161003816
119904
1198891199111015840119903119899minus1119889119903)
1119904
= 119862 Ω119871119904(119878119899minus1)
100381610038161003816100381610038162119895+211986110038161003816100381610038161003816
1119904
(57)
where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus
1198682119895le
119862
10038161003816100381610038162119895+1119861
1003816100381610038161003816
1minus(120572+120573)119899(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
(58)
By Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
le 119862(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot
100381610038161003816100381610038162119895+111986110038161003816100381610038161003816
(119901119902minus1199041015840
119902+1199041015840
119901)1199011199021199041015840
119908119902 (2119895+1119861)1119902
(59)
Thus
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816le 119862
infin
sum
119895=1
(11986811198951198682119895)
le 119862
infin
sum
119895=1
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
1
119908119902 (2119895+1119861)1119902minus120581119901
(60)
So we get
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119908119902(119861)
1119902minus120581119901
119908119902 (2119895+1119861)1119902minus120581119901
(61)
We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)
Therefore the proof of Theorem 27 is completed
Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879
Ω120572is bounded
on weighted Morrey spaces
Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-
rems 26 and 27Then let us turn to prove (19)Set
119879119860
Ω120572119891 (119909)
= intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
0 le 120572 lt 119899
(63)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879
119860
Ω120572 On the
other hand for any 119903 gt 0 we have
119879119860
Ω120572119891 (119909)
ge int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
ge1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)
10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
(64)
Taking the supremum for 119903 gt 0 on the inequality above weget
119879119860
Ω120572119891 (119909) ge 119872
119860
Ω120572119891 (119909) (65)
Thus we can immediately obtain (19) from (65) and (18)
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 3
Journal of Function Spaces 3
Definition 4 A locally integrable function 119887 is said to be inBMO(R119899) if
119887lowast = 119887BMO = sup119861
1
|119861|int119861
1003816100381610038161003816119887 (119909) minus 1198871198611003816100381610038161003816 119889119909 lt infin (12)
where
119887119861=1
|119861|int119861
119887 (119910) 119889119910 (13)
and the supremum is taken over all balls 119861 in R119899
At last we give the definition of two weight classes
Definition 5 A weight function 119908 is in the Muckenhouptclass 119860
119901with 1 lt 119901 lt infin if there exists 119862 gt 1 such that
for any ball 119861
(1
|119861|int119861
119908 (119909) 119889119909)(1
|119861|int119861
119908 (119909)minus1(119901minus1)
119889119909)
119901minus1
le 119862 (14)
We define 119860infin= ⋃
1lt119901ltinfin119860119901
When 119901 = 1 we define 119908 isin 1198601if there exists 119862 gt 1 such
that for almost every 119909
119872119908(119909) le 119862119908 (119909) (15)
Definition 6 A weight function 119908 belongs to 119860(119901 119902) for 1 lt119901 lt 119902 lt infin if there exists 119862 gt 1 such that such that for anyball 119861
(1
|119861|int119861
119908 (119909)119902119889119909)
1119902
sdot (1
|119861|int119861
119908 (119909)minus119901(119901minus1)
119889119909)
(119901minus1)119901
le 119862
(16)
When 119901 = 1 then we define 119908 isin 119860(1 119902) with 1 lt 119902 lt infin ifthere exists 119862 gt 1 such that
(1
|119861|int119861
119908 (119909)119902119889119909)
1119902
(ess sup119909isin119861
1
119908 (119909)) le 119862 (17)
Remark 7 (see [10]) If 119908 isin 119860(119901 119902) with 1 lt 119901 lt 119902 then
(a) 119908119902 119908minus1199011015840
119908minus1199021015840
isin Δ2
(b) 119908minus1199011015840
isin 1198601199051015840 with 119905 = 1 + 1199021199011015840
Now we state the main results of this paper
Theorem 8 If 0 lt 120572 + 120573 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1)
is homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899(120572 + 120573)
1119902 = 1119901 minus (120572 + 120573)119899 0 lt 120581 lt 119901119902 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840)
119863120574119860 isin Λ
120573(|120574| = 119898 minus 1) then
10038171003817100381710038171003817119879119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (18)
10038171003817100381710038171003817119872119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (19)
Theorem 9 If 0 lt 120573 lt 1 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is
homogeneous of degree zero 1 lt 1199041015840lt 119901 lt 119899120573 1119902 =
1119901 minus120573119899 0 lt 120581 lt 1199011199021199081199041015840
isin 119860(1199011199041015840 119902119904
1015840)119863120574119860 isin Λ
120573(|120574| =
119898 minus 1) then
10038171003817100381710038171003817119879119860
Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (20)
10038171003817100381710038171003817119872119860
Ω11989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (21)
Theorem 10 If 0 lt 120572 lt 119899 Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homo-
geneous of degree zero 1 lt 1199041015840 lt 119901 lt 119899120572 1119902 = 1119901 minus 1205721198990 lt 120581 lt 119901119902 119908119904
1015840
isin 119860(1199011199041015840 119902119904
1015840) 119863120574119860 isin BMO (|120574| = 119898 minus 1)
then
10038171003817100381710038171003817119879119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (22)
10038171003817100381710038171003817119872119860
Ω12057211989110038171003817100381710038171003817119871119902120581119902119901(119908119902)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (23)
When 119898 = 1 and 119898 = 2 we denote 119879119860Ω119872119860
Ωby [119860 119879
Ω]
[119860119872Ω] and
119860
Ω 119860
Ω respectively in order to distinguish
from119879119860Ωand119872119860
Ωthat are defined for any119898 isin Nlowast To bemore
precise
[119860 119879Ω] 119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899(119860 (119909) minus 119860 (119910))
sdot 119891 (119910) 119889119910
[119860119872Ω] 119891 (119909) = sup
119903gt0
1
119903119899int|119909minus119910|lt119903
Ω(119909 minus 119910)
sdot (119860 (119909) minus 119860 (119910)) 119891 (119910) 119889119910
119860
Ω119891 (119909) = int
R119899
Ω(119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899+1(119860 (119909) minus 119860 (119910) minus nabla119860 (119910)
sdot (119909 minus 119910)) 119891 (119910) 119889119910
119860
Ω119891 (119909) = sup
119903gt0
1
119903119899+1int|119909minus119910|lt119903
Ω(119909 minus 119910)
sdot (119860 (119909) minus 119860 (119910) minus nabla119860 (119910) (119909 minus 119910)) 119891 (119910) 119889119910
(24)
Then for the above operators we have the following resultson weighted Morrey spaces with one weight
Theorem 11 IfΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degreezero and satisfies the vanishing condition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) =
0 1 lt 1199041015840 lt 119901 lt infin 0 lt 120581 lt 1 119908 isin 1198601199011199041015840 119860 isin BMO then
1003817100381710038171003817[119860 119879Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (25)
1003817100381710038171003817[119860119872Ω] 1198911003817100381710038171003817119871119901120581(119908) le 119862 119860lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (26)
4 Journal of Function Spaces
Theorem 12 If Ω isin 119871infin(119878119899minus1) is homogeneous of degree zeroand satisfies the moment condition int
119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt
infin 0 lt 120581 lt 1 119908 isin 119860119901 nabla119860 isin BMO then
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)
Remark 13 Here we point out that for 119879119860Ωand 119872119860
Ω when
119863120574119860 isin BMO (|120574| = 119898 minus 1) the analogues of Theorems 11
and 12 are open for119898 ge 3
Remark 14 Define
1198791198601119860119896
Ω120572119891 (119909) = int
R119899
119896
prod
119894=1
119877119898119894
(119860119894 119909 119910)
sdotΩ (119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119873119891 (119910) 119889119910
1198721198601119860119896
Ω120572119891 (119909) = sup
119903gt0
1
119903119899minus120572+119873int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
sdot
119896
prod
119894=1
10038161003816100381610038161003816119877119898119894
(119860119894 119909 119910)
10038161003816100381610038161003816
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(29)
where 119877119898119894
(119860119894 119909 119910) = 119860
119894(119909) minus sum
|120574|lt119898119894
(1120574)119863120574119860119894(119910)(119909 minus
119910)120574 119894 = 1 119896 119873 = sum
119896
119894=1(119898119894minus 1) When 0 lt 120572 lt 119899
they are a class of multilinear fractional integral operatorsand multilinear fractional maximal operators When 120572 = 0they are a class of multilinear singular integral operators andmultilinear maximal singular integral operators Repeatingthe proofs of the theorems above we will find that for1198791198601119860119896
Ω120572and 119872119860
1119860119896
Ω120572 the conclusions of Theorems 8 and
9 above with the bounds 119862prod119896119894=1(sum|120574|=119898
119894minus1119863120574119860119894Λ120573
) andTheorem 10 with the bounds 119862prod119896
119894=1(sum|120574|=119898
119894minus1119863120574119860119894lowast) also
hold respectively
The organization of this paper is as follows In Section 2we give some requisite lemmas and well-known results thatare important in proving the theorems The proof of thetheorems will be shown in Section 3
2 Lemmas and Well-Known Results
Lemma 15 (see [1]) Let 119860(119909) be a function on R119899 with 119898thorder derivatives in 119871119897loc(R
119899) for some 119897 gt 119899 Then
1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898sum
|120574|=119898
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
1003816100381610038161003816119863120574119860 (119911)
1003816100381610038161003816
119897119889119911)
1119897
(30)
where 119868119910119909is the cube centered at 119909with sides parallel to the axes
whose diameter is 5radic119899|119909 minus 119910|
Lemma 16 (see [12]) For 0 lt 120573 lt 1 1 le 119902 lt infin we have
10038171003817100381710038171198911003817100381710038171003817Λ120573
asymp sup119876
1
|119876|1+120573119899
int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909
asymp sup119876
1
|119876|120573119899(1
|119876|int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816
119902119889119909)
1119902
(31)
For 119902 = infin the formula should be interpreted appropriately
Lemma 17 (see [13]) Let 1198761sub 119876
2 119892 isin Λ
120573(0 lt 120573 lt 1) Then
100381610038161003816100381610038161198921198761
minus 1198921198762
10038161003816100381610038161003816le 119862
100381610038161003816100381611987621003816100381610038161003816
120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573
(32)
Theorem 18 (see [14]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 and Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous ofdegree zero Then 119879
Ω120572is a bounded operator from 119871
119901(119908119901) to
119871119902(119908119902) if the index set 120572 119901 119902 119904 satisfies one of the following
conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840
such that 119908(119909)1199031015840
isin 119860(119901 119902)
Lemma 19 (see [10]) If 119908 isin Δ2 then there exists a constant
1198631gt 1 such that
119908 (2119861) ge 1198631119908 (119861) (33)
We call1198631the reverse doubling constant
Theorem 20 (see [4]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 Ω isin 119871
119904(119878119899minus1) (119904 gt 1) is homogeneous of
degree zero Moreover for 1 le 119894 le 119896 |120574| = 119898119894minus 1119898
119894ge 2 and
119863120574119860119894isin BMO(R119899) if the index set 120572 119901 119902 119904 satisfies one of
the following conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840such that 119908(119909)119903
1015840
isin 119860(119901 119902)
Then there is a 119862 gt 0 independent of 119891 and 119860119894 such that
(intR119899
100381610038161003816100381610038161198791198601119860119896
Ω120572119891 (119909)119908 (119909)
10038161003816100381610038161003816
119902
119889119909)
1119902
le 119862
119896
prod
119894=1
( sum
|120574|=119898119894minus1
1003817100381710038171003817119863120574119860119894
1003817100381710038171003817lowast)
sdot (intR119899
1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816
119901119889119909)
1119901
(34)
Lemma 21 (see [15]) (a) (John-Nirenberg Lemma) Let 1 le119901 lt infin Then 119887 isin BMO if and only if
1
|119876|int119876
1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816
119901119889119909 le 119862 119887
119901
lowast (35)
Journal of Function Spaces 5
(b) Assume 119887 isin BMO then for cubes 1198761sub 119876
2
100381610038161003816100381610038161198871198761
minus 1198871198762
10038161003816100381610038161003816le 119862 log(
100381610038161003816100381611987621003816100381610038161003816
100381610038161003816100381611987611003816100381610038161003816
) 119887lowast (36)
(c) If 119887 isin BMO then10038161003816100381610038161198872119895+1119861 minus 119887119861
1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)
Theorem 22 (see [16]) Suppose that Ω isin 119871119904(119878119899minus1) (119904 gt
1) is homogeneous of degree zero and satisfies the vanishingcondition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879
Ω]
is bounded on 119871119901(119908) if the index set 119901 119902 119904 satisfies one of thefollowing conditions
(a) 1199041015840 le 119901 lt infin 119901 = 1 and 119908 isin 1198601199011199041015840
(b) 1 le 119901 le 119904 119901 = infin and 1199081minus1199011015840
isin 11986011990110158401199041015840
(c) 1 le 119901 lt infin and 1199081199041015840
isin 119860119901
Theorem 23 (see [2]) If Ω isin 119871infin(119878119899minus1) is homogeneous of
degree zero and satisfies themoment condition int119878119899minus1120579Ω(120579)119889120579 =
0 119908 isin 119860119901 1 lt 119901 lt infin nabla119860 isin BMO then we have100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901(119908)
le 119862 Ωinfin nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901(119908) (38)
Lemma 24 (see [15]) The following are true(1) If 119908 isin 119860
119901for some 1 le 119901 lt infin then 119908 isin Δ
2 More
precisely for all 120582 gt 1 we have
119908 (120582119876) le 119862120582119899119901119908 (119876) (39)
(2) If 119908 isin 119860119901for some 1 le 119901 lt infin then there exist 119862 gt 0
and 120575 gt 0 such that for any cube 119876 and a measurableset 119878 sub 119876
119908 (119878)
119908 (119876)le 119862(
|119878|
|119876|)
120575
(40)
Lemma25 (see [17]) Let119908 isin 119860infinThen the normofBMO(119908)
is equivalent to the norm of BMO(R119899) where
BMO (119908) = 119887 119887lowast119908
= sup119876
1
119908 (119876)int119876
1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909
119898119876119908119887 =
1
119908 (119876)int119876
119887 (119909)119908 (119909) 119889119909
(41)
3 Proofs of the Main Results
Before proving Theorem 8 we give a pointwise estimate of119879119860
Ω120572119891(119909) at first Set
119879Ω120572+120573
119891 (119909) = intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
0 lt 120572 + 120573 lt 119899
(42)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 Then we have the following estimate
Theorem26 If 120572 ge 0 0 lt 120572+120573 lt 119899119863120574119860 isin Λ120573(|120574| = 119898minus1)
then there exists a constant 119862 independent of 119891 such that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le 119862( sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
)119879Ω120572+120573
119891 (119909) (43)
Proof For fixed 119909 isin R119899 119903 gt 0 let 119876 be a cube with center at119909 and diameter 119903 Denote 119876
119896= 2
119896119876 and set
119860119876119896
(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876119896
(119863120574119860)119910
120574 (44)
where 119898119876119896
119891 is the average of 119891 on 119876119896 Then we have when
|120574| = 119898 minus 1
119863120574119860119876119896
(119910) = 119863120574119860 (119910) minus 119898
119876119896
(119863120574119860) (45)
and it is proved in [1] that
119877119898(119860 119909 119910) = 119877
119898(119860119876119896
119909 119910) (46)
Hence
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816
le
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
flinfin
sum
119896=minusinfin
119879119896
(47)
By Lemma 15 we get
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le10038161003816100381610038161003816119877119898minus1
(119860119876119896
119909 119910)10038161003816100381610038161003816
+ 119862 sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
+ 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
(48)
6 Journal of Function Spaces
Note that if |119909 minus 119910| lt 2119896119903 then 119868119910119909sub 5119899119876
119896 By Lemmas 16
and 17 we have when |120574| = 119898 minus 1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
= (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876
119896
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
le (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119868119910
119909
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
+10038161003816100381610038161003816119898119868119910
119909
(119863120574119860) minus 119898
5119899119876119896
(119863120574119860)10038161003816100381610038161003816
+100381610038161003816100381610038161198985119899119876119896
(119863120574119860) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(49)
It is obvious that when |120574| = 119898 minus 1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(50)
Thus
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1(2119896119903)120573
sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
(51)
Therefore
119879119896le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
(2119896119903)120573
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(52)
It follows that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le
infin
sum
119896=minusinfin
(119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
119879Ω120572+120573
119891 (119909)
(53)
Thus we finish the proof of Theorem 26
The following theorem is a key theorem in proving (18) ofTheorem 8
Theorem 27 Under the same assumptions of Theorem 8119879Ω120572+120573
is bounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
Proof Fix a ball 119861(1199090 119903119861) we decompose 119891 = 119891
1+ 119891
2with
1198911= 119891120594
2119861 Then we have
(1
119908119902 (119861)120581119902119901
int119861
10038161003816100381610038161003816119879Ω120572+120573
119891 (119909)10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
le1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198911 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
+1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
fl 1198691+ 1198692
(54)
We estimate 1198691at first By Remark 7(a) we know that119908119902 isin Δ
2
Then byTheorem 18(a) and the fact that 119908119902 isin Δ2we get
1198691le
1
119908119902 (119861)120581119901
10038171003817100381710038171003817119879Ω120572+120573
1198911
10038171003817100381710038171003817119871119902(119908119902)
le119862
119908119902 (119861)120581119901
100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)
=119862
119908119902 (119861)120581119901(int2119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909)
119901119889119909)
1119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
119908119902(2119861)
120581119901
119908119902 (119861)120581119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(55)
Journal of Function Spaces 7
Now we consider the term 1198692 By Holderrsquos inequality we have
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816=
infin
sum
119895=1
int2119895+11198612119895119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
le 119862
infin
sum
119895=1
(int2119895+1119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
119904119889119910)
1119904
sdot (int2119895+11198612119895119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
(119899minus120572minus120573)1199041015840119889119910)
11199041015840
fl 119862infin
sum
119895=1
(11986811198951198682119895)
(56)
We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for
119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have
1198681119895= (int
2119895+2119861
|Ω (119911)|119904119889119911)
1119904
= (int
2119895+2
119903119861
0
int119878119899minus1
10038161003816100381610038161003816Ω (119911
1015840)10038161003816100381610038161003816
119904
1198891199111015840119903119899minus1119889119903)
1119904
= 119862 Ω119871119904(119878119899minus1)
100381610038161003816100381610038162119895+211986110038161003816100381610038161003816
1119904
(57)
where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus
1198682119895le
119862
10038161003816100381610038162119895+1119861
1003816100381610038161003816
1minus(120572+120573)119899(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
(58)
By Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
le 119862(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot
100381610038161003816100381610038162119895+111986110038161003816100381610038161003816
(119901119902minus1199041015840
119902+1199041015840
119901)1199011199021199041015840
119908119902 (2119895+1119861)1119902
(59)
Thus
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816le 119862
infin
sum
119895=1
(11986811198951198682119895)
le 119862
infin
sum
119895=1
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
1
119908119902 (2119895+1119861)1119902minus120581119901
(60)
So we get
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119908119902(119861)
1119902minus120581119901
119908119902 (2119895+1119861)1119902minus120581119901
(61)
We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)
Therefore the proof of Theorem 27 is completed
Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879
Ω120572is bounded
on weighted Morrey spaces
Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-
rems 26 and 27Then let us turn to prove (19)Set
119879119860
Ω120572119891 (119909)
= intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
0 le 120572 lt 119899
(63)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879
119860
Ω120572 On the
other hand for any 119903 gt 0 we have
119879119860
Ω120572119891 (119909)
ge int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
ge1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)
10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
(64)
Taking the supremum for 119903 gt 0 on the inequality above weget
119879119860
Ω120572119891 (119909) ge 119872
119860
Ω120572119891 (119909) (65)
Thus we can immediately obtain (19) from (65) and (18)
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 4
4 Journal of Function Spaces
Theorem 12 If Ω isin 119871infin(119878119899minus1) is homogeneous of degree zeroand satisfies the moment condition int
119878119899minus1120579Ω(120579)119889120579 = 0 1 lt 119901 lt
infin 0 lt 120581 lt 1 119908 isin 119860119901 nabla119860 isin BMO then
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (27)
100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901120581(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908) (28)
Remark 13 Here we point out that for 119879119860Ωand 119872119860
Ω when
119863120574119860 isin BMO (|120574| = 119898 minus 1) the analogues of Theorems 11
and 12 are open for119898 ge 3
Remark 14 Define
1198791198601119860119896
Ω120572119891 (119909) = int
R119899
119896
prod
119894=1
119877119898119894
(119860119894 119909 119910)
sdotΩ (119909 minus 119910)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119873119891 (119910) 119889119910
1198721198601119860119896
Ω120572119891 (119909) = sup
119903gt0
1
119903119899minus120572+119873int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
sdot
119896
prod
119894=1
10038161003816100381610038161003816119877119898119894
(119860119894 119909 119910)
10038161003816100381610038161003816
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(29)
where 119877119898119894
(119860119894 119909 119910) = 119860
119894(119909) minus sum
|120574|lt119898119894
(1120574)119863120574119860119894(119910)(119909 minus
119910)120574 119894 = 1 119896 119873 = sum
119896
119894=1(119898119894minus 1) When 0 lt 120572 lt 119899
they are a class of multilinear fractional integral operatorsand multilinear fractional maximal operators When 120572 = 0they are a class of multilinear singular integral operators andmultilinear maximal singular integral operators Repeatingthe proofs of the theorems above we will find that for1198791198601119860119896
Ω120572and 119872119860
1119860119896
Ω120572 the conclusions of Theorems 8 and
9 above with the bounds 119862prod119896119894=1(sum|120574|=119898
119894minus1119863120574119860119894Λ120573
) andTheorem 10 with the bounds 119862prod119896
119894=1(sum|120574|=119898
119894minus1119863120574119860119894lowast) also
hold respectively
The organization of this paper is as follows In Section 2we give some requisite lemmas and well-known results thatare important in proving the theorems The proof of thetheorems will be shown in Section 3
2 Lemmas and Well-Known Results
Lemma 15 (see [1]) Let 119860(119909) be a function on R119899 with 119898thorder derivatives in 119871119897loc(R
119899) for some 119897 gt 119899 Then
1003816100381610038161003816119877119898 (119860 119909 119910)1003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898sum
|120574|=119898
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
1003816100381610038161003816119863120574119860 (119911)
1003816100381610038161003816
119897119889119911)
1119897
(30)
where 119868119910119909is the cube centered at 119909with sides parallel to the axes
whose diameter is 5radic119899|119909 minus 119910|
Lemma 16 (see [12]) For 0 lt 120573 lt 1 1 le 119902 lt infin we have
10038171003817100381710038171198911003817100381710038171003817Λ120573
asymp sup119876
1
|119876|1+120573119899
int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816 119889119909
asymp sup119876
1
|119876|120573119899(1
|119876|int119876
1003816100381610038161003816119891 (119909) minus 119887119876 (119891)1003816100381610038161003816
119902119889119909)
1119902
(31)
For 119902 = infin the formula should be interpreted appropriately
Lemma 17 (see [13]) Let 1198761sub 119876
2 119892 isin Λ
120573(0 lt 120573 lt 1) Then
100381610038161003816100381610038161198921198761
minus 1198921198762
10038161003816100381610038161003816le 119862
100381610038161003816100381611987621003816100381610038161003816
120573119899 10038171003817100381710038171198921003817100381710038171003817Λ120573
(32)
Theorem 18 (see [14]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 and Ω isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous ofdegree zero Then 119879
Ω120572is a bounded operator from 119871
119901(119908119901) to
119871119902(119908119902) if the index set 120572 119901 119902 119904 satisfies one of the following
conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840
such that 119908(119909)1199031015840
isin 119860(119901 119902)
Lemma 19 (see [10]) If 119908 isin Δ2 then there exists a constant
1198631gt 1 such that
119908 (2119861) ge 1198631119908 (119861) (33)
We call1198631the reverse doubling constant
Theorem 20 (see [4]) Suppose that 0 lt 120572 lt 119899 1 lt 119901 lt 1198991205721119902 = 1119901 minus 120572119899 Ω isin 119871
119904(119878119899minus1) (119904 gt 1) is homogeneous of
degree zero Moreover for 1 le 119894 le 119896 |120574| = 119898119894minus 1119898
119894ge 2 and
119863120574119860119894isin BMO(R119899) if the index set 120572 119901 119902 119904 satisfies one of
the following conditions
(a) 1199041015840 lt 119901 and 119908(119909)1199041015840
isin 119860(1199011199041015840 119902119904
1015840)
(b) 119904 gt 119902 and 119908(119909)minus1199041015840
isin 119860(11990210158401199041015840 11990110158401199041015840)
(c) 120572119899+1119904 lt 1119901 lt 11199041015840 and there is 119903 1 lt 119903 lt 119904(119899120572)1015840such that 119908(119909)119903
1015840
isin 119860(119901 119902)
Then there is a 119862 gt 0 independent of 119891 and 119860119894 such that
(intR119899
100381610038161003816100381610038161198791198601119860119896
Ω120572119891 (119909)119908 (119909)
10038161003816100381610038161003816
119902
119889119909)
1119902
le 119862
119896
prod
119894=1
( sum
|120574|=119898119894minus1
1003817100381710038171003817119863120574119860119894
1003817100381710038171003817lowast)
sdot (intR119899
1003816100381610038161003816119891 (119909)119908 (119909)1003816100381610038161003816
119901119889119909)
1119901
(34)
Lemma 21 (see [15]) (a) (John-Nirenberg Lemma) Let 1 le119901 lt infin Then 119887 isin BMO if and only if
1
|119876|int119876
1003816100381610038161003816119887 minus 1198871198761003816100381610038161003816
119901119889119909 le 119862 119887
119901
lowast (35)
Journal of Function Spaces 5
(b) Assume 119887 isin BMO then for cubes 1198761sub 119876
2
100381610038161003816100381610038161198871198761
minus 1198871198762
10038161003816100381610038161003816le 119862 log(
100381610038161003816100381611987621003816100381610038161003816
100381610038161003816100381611987611003816100381610038161003816
) 119887lowast (36)
(c) If 119887 isin BMO then10038161003816100381610038161198872119895+1119861 minus 119887119861
1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)
Theorem 22 (see [16]) Suppose that Ω isin 119871119904(119878119899minus1) (119904 gt
1) is homogeneous of degree zero and satisfies the vanishingcondition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879
Ω]
is bounded on 119871119901(119908) if the index set 119901 119902 119904 satisfies one of thefollowing conditions
(a) 1199041015840 le 119901 lt infin 119901 = 1 and 119908 isin 1198601199011199041015840
(b) 1 le 119901 le 119904 119901 = infin and 1199081minus1199011015840
isin 11986011990110158401199041015840
(c) 1 le 119901 lt infin and 1199081199041015840
isin 119860119901
Theorem 23 (see [2]) If Ω isin 119871infin(119878119899minus1) is homogeneous of
degree zero and satisfies themoment condition int119878119899minus1120579Ω(120579)119889120579 =
0 119908 isin 119860119901 1 lt 119901 lt infin nabla119860 isin BMO then we have100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901(119908)
le 119862 Ωinfin nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901(119908) (38)
Lemma 24 (see [15]) The following are true(1) If 119908 isin 119860
119901for some 1 le 119901 lt infin then 119908 isin Δ
2 More
precisely for all 120582 gt 1 we have
119908 (120582119876) le 119862120582119899119901119908 (119876) (39)
(2) If 119908 isin 119860119901for some 1 le 119901 lt infin then there exist 119862 gt 0
and 120575 gt 0 such that for any cube 119876 and a measurableset 119878 sub 119876
119908 (119878)
119908 (119876)le 119862(
|119878|
|119876|)
120575
(40)
Lemma25 (see [17]) Let119908 isin 119860infinThen the normofBMO(119908)
is equivalent to the norm of BMO(R119899) where
BMO (119908) = 119887 119887lowast119908
= sup119876
1
119908 (119876)int119876
1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909
119898119876119908119887 =
1
119908 (119876)int119876
119887 (119909)119908 (119909) 119889119909
(41)
3 Proofs of the Main Results
Before proving Theorem 8 we give a pointwise estimate of119879119860
Ω120572119891(119909) at first Set
119879Ω120572+120573
119891 (119909) = intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
0 lt 120572 + 120573 lt 119899
(42)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 Then we have the following estimate
Theorem26 If 120572 ge 0 0 lt 120572+120573 lt 119899119863120574119860 isin Λ120573(|120574| = 119898minus1)
then there exists a constant 119862 independent of 119891 such that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le 119862( sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
)119879Ω120572+120573
119891 (119909) (43)
Proof For fixed 119909 isin R119899 119903 gt 0 let 119876 be a cube with center at119909 and diameter 119903 Denote 119876
119896= 2
119896119876 and set
119860119876119896
(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876119896
(119863120574119860)119910
120574 (44)
where 119898119876119896
119891 is the average of 119891 on 119876119896 Then we have when
|120574| = 119898 minus 1
119863120574119860119876119896
(119910) = 119863120574119860 (119910) minus 119898
119876119896
(119863120574119860) (45)
and it is proved in [1] that
119877119898(119860 119909 119910) = 119877
119898(119860119876119896
119909 119910) (46)
Hence
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816
le
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
flinfin
sum
119896=minusinfin
119879119896
(47)
By Lemma 15 we get
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le10038161003816100381610038161003816119877119898minus1
(119860119876119896
119909 119910)10038161003816100381610038161003816
+ 119862 sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
+ 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
(48)
6 Journal of Function Spaces
Note that if |119909 minus 119910| lt 2119896119903 then 119868119910119909sub 5119899119876
119896 By Lemmas 16
and 17 we have when |120574| = 119898 minus 1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
= (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876
119896
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
le (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119868119910
119909
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
+10038161003816100381610038161003816119898119868119910
119909
(119863120574119860) minus 119898
5119899119876119896
(119863120574119860)10038161003816100381610038161003816
+100381610038161003816100381610038161198985119899119876119896
(119863120574119860) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(49)
It is obvious that when |120574| = 119898 minus 1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(50)
Thus
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1(2119896119903)120573
sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
(51)
Therefore
119879119896le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
(2119896119903)120573
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(52)
It follows that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le
infin
sum
119896=minusinfin
(119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
119879Ω120572+120573
119891 (119909)
(53)
Thus we finish the proof of Theorem 26
The following theorem is a key theorem in proving (18) ofTheorem 8
Theorem 27 Under the same assumptions of Theorem 8119879Ω120572+120573
is bounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
Proof Fix a ball 119861(1199090 119903119861) we decompose 119891 = 119891
1+ 119891
2with
1198911= 119891120594
2119861 Then we have
(1
119908119902 (119861)120581119902119901
int119861
10038161003816100381610038161003816119879Ω120572+120573
119891 (119909)10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
le1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198911 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
+1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
fl 1198691+ 1198692
(54)
We estimate 1198691at first By Remark 7(a) we know that119908119902 isin Δ
2
Then byTheorem 18(a) and the fact that 119908119902 isin Δ2we get
1198691le
1
119908119902 (119861)120581119901
10038171003817100381710038171003817119879Ω120572+120573
1198911
10038171003817100381710038171003817119871119902(119908119902)
le119862
119908119902 (119861)120581119901
100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)
=119862
119908119902 (119861)120581119901(int2119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909)
119901119889119909)
1119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
119908119902(2119861)
120581119901
119908119902 (119861)120581119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(55)
Journal of Function Spaces 7
Now we consider the term 1198692 By Holderrsquos inequality we have
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816=
infin
sum
119895=1
int2119895+11198612119895119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
le 119862
infin
sum
119895=1
(int2119895+1119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
119904119889119910)
1119904
sdot (int2119895+11198612119895119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
(119899minus120572minus120573)1199041015840119889119910)
11199041015840
fl 119862infin
sum
119895=1
(11986811198951198682119895)
(56)
We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for
119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have
1198681119895= (int
2119895+2119861
|Ω (119911)|119904119889119911)
1119904
= (int
2119895+2
119903119861
0
int119878119899minus1
10038161003816100381610038161003816Ω (119911
1015840)10038161003816100381610038161003816
119904
1198891199111015840119903119899minus1119889119903)
1119904
= 119862 Ω119871119904(119878119899minus1)
100381610038161003816100381610038162119895+211986110038161003816100381610038161003816
1119904
(57)
where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus
1198682119895le
119862
10038161003816100381610038162119895+1119861
1003816100381610038161003816
1minus(120572+120573)119899(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
(58)
By Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
le 119862(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot
100381610038161003816100381610038162119895+111986110038161003816100381610038161003816
(119901119902minus1199041015840
119902+1199041015840
119901)1199011199021199041015840
119908119902 (2119895+1119861)1119902
(59)
Thus
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816le 119862
infin
sum
119895=1
(11986811198951198682119895)
le 119862
infin
sum
119895=1
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
1
119908119902 (2119895+1119861)1119902minus120581119901
(60)
So we get
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119908119902(119861)
1119902minus120581119901
119908119902 (2119895+1119861)1119902minus120581119901
(61)
We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)
Therefore the proof of Theorem 27 is completed
Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879
Ω120572is bounded
on weighted Morrey spaces
Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-
rems 26 and 27Then let us turn to prove (19)Set
119879119860
Ω120572119891 (119909)
= intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
0 le 120572 lt 119899
(63)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879
119860
Ω120572 On the
other hand for any 119903 gt 0 we have
119879119860
Ω120572119891 (119909)
ge int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
ge1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)
10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
(64)
Taking the supremum for 119903 gt 0 on the inequality above weget
119879119860
Ω120572119891 (119909) ge 119872
119860
Ω120572119891 (119909) (65)
Thus we can immediately obtain (19) from (65) and (18)
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 5
Journal of Function Spaces 5
(b) Assume 119887 isin BMO then for cubes 1198761sub 119876
2
100381610038161003816100381610038161198871198761
minus 1198871198762
10038161003816100381610038161003816le 119862 log(
100381610038161003816100381611987621003816100381610038161003816
100381610038161003816100381611987611003816100381610038161003816
) 119887lowast (36)
(c) If 119887 isin BMO then10038161003816100381610038161198872119895+1119861 minus 119887119861
1003816100381610038161003816 le 2119899(119895 + 1) 119887lowast (37)
Theorem 22 (see [16]) Suppose that Ω isin 119871119904(119878119899minus1) (119904 gt
1) is homogeneous of degree zero and satisfies the vanishingcondition int
119878119899minus1Ω(119909
1015840)119889120590(119909
1015840) = 0 If 119887 isin BMO(R119899) then [119887 119879
Ω]
is bounded on 119871119901(119908) if the index set 119901 119902 119904 satisfies one of thefollowing conditions
(a) 1199041015840 le 119901 lt infin 119901 = 1 and 119908 isin 1198601199011199041015840
(b) 1 le 119901 le 119904 119901 = infin and 1199081minus1199011015840
isin 11986011990110158401199041015840
(c) 1 le 119901 lt infin and 1199081199041015840
isin 119860119901
Theorem 23 (see [2]) If Ω isin 119871infin(119878119899minus1) is homogeneous of
degree zero and satisfies themoment condition int119878119899minus1120579Ω(120579)119889120579 =
0 119908 isin 119860119901 1 lt 119901 lt infin nabla119860 isin BMO then we have100381710038171003817100381710038171003817119860
Ω119891100381710038171003817100381710038171003817119871119901(119908)
le 119862 Ωinfin nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901(119908) (38)
Lemma 24 (see [15]) The following are true(1) If 119908 isin 119860
119901for some 1 le 119901 lt infin then 119908 isin Δ
2 More
precisely for all 120582 gt 1 we have
119908 (120582119876) le 119862120582119899119901119908 (119876) (39)
(2) If 119908 isin 119860119901for some 1 le 119901 lt infin then there exist 119862 gt 0
and 120575 gt 0 such that for any cube 119876 and a measurableset 119878 sub 119876
119908 (119878)
119908 (119876)le 119862(
|119878|
|119876|)
120575
(40)
Lemma25 (see [17]) Let119908 isin 119860infinThen the normofBMO(119908)
is equivalent to the norm of BMO(R119899) where
BMO (119908) = 119887 119887lowast119908
= sup119876
1
119908 (119876)int119876
1003816100381610038161003816119887 (119909) minus 1198981198761199081198871003816100381610038161003816 119908 (119909) 119889119909
119898119876119908119887 =
1
119908 (119876)int119876
119887 (119909)119908 (119909) 119889119909
(41)
3 Proofs of the Main Results
Before proving Theorem 8 we give a pointwise estimate of119879119860
Ω120572119891(119909) at first Set
119879Ω120572+120573
119891 (119909) = intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
0 lt 120572 + 120573 lt 119899
(42)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 Then we have the following estimate
Theorem26 If 120572 ge 0 0 lt 120572+120573 lt 119899119863120574119860 isin Λ120573(|120574| = 119898minus1)
then there exists a constant 119862 independent of 119891 such that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le 119862( sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
)119879Ω120572+120573
119891 (119909) (43)
Proof For fixed 119909 isin R119899 119903 gt 0 let 119876 be a cube with center at119909 and diameter 119903 Denote 119876
119896= 2
119896119876 and set
119860119876119896
(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876119896
(119863120574119860)119910
120574 (44)
where 119898119876119896
119891 is the average of 119891 on 119876119896 Then we have when
|120574| = 119898 minus 1
119863120574119860119876119896
(119910) = 119863120574119860 (119910) minus 119898
119876119896
(119863120574119860) (45)
and it is proved in [1] that
119877119898(119860 119909 119910) = 119877
119898(119860119876119896
119909 119910) (46)
Hence
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816
le
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
flinfin
sum
119896=minusinfin
119879119896
(47)
By Lemma 15 we get
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le10038161003816100381610038161003816119877119898minus1
(119860119876119896
119909 119910)10038161003816100381610038161003816
+ 119862 sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119898minus1
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
+ 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816
(48)
6 Journal of Function Spaces
Note that if |119909 minus 119910| lt 2119896119903 then 119868119910119909sub 5119899119876
119896 By Lemmas 16
and 17 we have when |120574| = 119898 minus 1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
= (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876
119896
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
le (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119868119910
119909
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
+10038161003816100381610038161003816119898119868119910
119909
(119863120574119860) minus 119898
5119899119876119896
(119863120574119860)10038161003816100381610038161003816
+100381610038161003816100381610038161198985119899119876119896
(119863120574119860) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(49)
It is obvious that when |120574| = 119898 minus 1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(50)
Thus
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1(2119896119903)120573
sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
(51)
Therefore
119879119896le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
(2119896119903)120573
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(52)
It follows that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le
infin
sum
119896=minusinfin
(119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
119879Ω120572+120573
119891 (119909)
(53)
Thus we finish the proof of Theorem 26
The following theorem is a key theorem in proving (18) ofTheorem 8
Theorem 27 Under the same assumptions of Theorem 8119879Ω120572+120573
is bounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
Proof Fix a ball 119861(1199090 119903119861) we decompose 119891 = 119891
1+ 119891
2with
1198911= 119891120594
2119861 Then we have
(1
119908119902 (119861)120581119902119901
int119861
10038161003816100381610038161003816119879Ω120572+120573
119891 (119909)10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
le1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198911 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
+1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
fl 1198691+ 1198692
(54)
We estimate 1198691at first By Remark 7(a) we know that119908119902 isin Δ
2
Then byTheorem 18(a) and the fact that 119908119902 isin Δ2we get
1198691le
1
119908119902 (119861)120581119901
10038171003817100381710038171003817119879Ω120572+120573
1198911
10038171003817100381710038171003817119871119902(119908119902)
le119862
119908119902 (119861)120581119901
100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)
=119862
119908119902 (119861)120581119901(int2119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909)
119901119889119909)
1119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
119908119902(2119861)
120581119901
119908119902 (119861)120581119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(55)
Journal of Function Spaces 7
Now we consider the term 1198692 By Holderrsquos inequality we have
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816=
infin
sum
119895=1
int2119895+11198612119895119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
le 119862
infin
sum
119895=1
(int2119895+1119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
119904119889119910)
1119904
sdot (int2119895+11198612119895119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
(119899minus120572minus120573)1199041015840119889119910)
11199041015840
fl 119862infin
sum
119895=1
(11986811198951198682119895)
(56)
We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for
119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have
1198681119895= (int
2119895+2119861
|Ω (119911)|119904119889119911)
1119904
= (int
2119895+2
119903119861
0
int119878119899minus1
10038161003816100381610038161003816Ω (119911
1015840)10038161003816100381610038161003816
119904
1198891199111015840119903119899minus1119889119903)
1119904
= 119862 Ω119871119904(119878119899minus1)
100381610038161003816100381610038162119895+211986110038161003816100381610038161003816
1119904
(57)
where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus
1198682119895le
119862
10038161003816100381610038162119895+1119861
1003816100381610038161003816
1minus(120572+120573)119899(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
(58)
By Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
le 119862(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot
100381610038161003816100381610038162119895+111986110038161003816100381610038161003816
(119901119902minus1199041015840
119902+1199041015840
119901)1199011199021199041015840
119908119902 (2119895+1119861)1119902
(59)
Thus
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816le 119862
infin
sum
119895=1
(11986811198951198682119895)
le 119862
infin
sum
119895=1
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
1
119908119902 (2119895+1119861)1119902minus120581119901
(60)
So we get
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119908119902(119861)
1119902minus120581119901
119908119902 (2119895+1119861)1119902minus120581119901
(61)
We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)
Therefore the proof of Theorem 27 is completed
Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879
Ω120572is bounded
on weighted Morrey spaces
Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-
rems 26 and 27Then let us turn to prove (19)Set
119879119860
Ω120572119891 (119909)
= intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
0 le 120572 lt 119899
(63)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879
119860
Ω120572 On the
other hand for any 119903 gt 0 we have
119879119860
Ω120572119891 (119909)
ge int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
ge1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)
10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
(64)
Taking the supremum for 119903 gt 0 on the inequality above weget
119879119860
Ω120572119891 (119909) ge 119872
119860
Ω120572119891 (119909) (65)
Thus we can immediately obtain (19) from (65) and (18)
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 6
6 Journal of Function Spaces
Note that if |119909 minus 119910| lt 2119896119903 then 119868119910119909sub 5119899119876
119896 By Lemmas 16
and 17 we have when |120574| = 119898 minus 1
(1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860119876119896
(119911)10038161003816100381610038161003816
119897
119889119911)
1119897
= (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898119876
119896
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
le (1
1003816100381610038161003816119868119910
119909
1003816100381610038161003816
int119868119910
119909
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119868119910
119909
(119863120574119860)10038161003816100381610038161003816
119897
119889119911)
1119897
+10038161003816100381610038161003816119898119868119910
119909
(119863120574119860) minus 119898
5119899119876119896
(119863120574119860)10038161003816100381610038161003816
+100381610038161003816100381610038161198985119899119876119896
(119863120574119860) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(49)
It is obvious that when |120574| = 119898 minus 1
10038161003816100381610038161003816119863120574119860119876119896
(119910)10038161003816100381610038161003816=10038161003816100381610038161003816119863120574119860 (119910) minus 119898
119876119896
(119863120574119860)10038161003816100381610038161003816
le 11986210038161003816100381610038161198761198961003816100381610038161003816
120573119899 10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
le 119862 (2119896119903)120573 1003817100381710038171003817119863
1205741198601003817100381710038171003817Λ120573
(50)
Thus
10038161003816100381610038161003816119877119898(119860119876119896
119909 119910)10038161003816100381610038161003816
le 1198621003816100381610038161003816119909 minus 119910
1003816100381610038161003816
119898minus1(2119896119903)120573
sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
(51)
Therefore
119879119896le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
(2119896119903)120573
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
(52)
It follows that
10038161003816100381610038161003816119879119860
Ω120572119891 (119909)
10038161003816100381610038161003816le
infin
sum
119896=minusinfin
(119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910)
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
sdot
infin
sum
119896=minusinfin
int2119896minus1119903le|119909minus119910|lt2
119896119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817Λ120573
119879Ω120572+120573
119891 (119909)
(53)
Thus we finish the proof of Theorem 26
The following theorem is a key theorem in proving (18) ofTheorem 8
Theorem 27 Under the same assumptions of Theorem 8119879Ω120572+120573
is bounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
Proof Fix a ball 119861(1199090 119903119861) we decompose 119891 = 119891
1+ 119891
2with
1198911= 119891120594
2119861 Then we have
(1
119908119902 (119861)120581119902119901
int119861
10038161003816100381610038161003816119879Ω120572+120573
119891 (119909)10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
le1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198911 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
+1
119908119902 (119861)120581119901(int119861
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816
119902
119908119902(119909) 119889119909)
1119902
fl 1198691+ 1198692
(54)
We estimate 1198691at first By Remark 7(a) we know that119908119902 isin Δ
2
Then byTheorem 18(a) and the fact that 119908119902 isin Δ2we get
1198691le
1
119908119902 (119861)120581119901
10038171003817100381710038171003817119879Ω120572+120573
1198911
10038171003817100381710038171003817119871119902(119908119902)
le119862
119908119902 (119861)120581119901
100381710038171003817100381711989111003817100381710038171003817119871119901(119908119901)
=119862
119908119902 (119861)120581119901(int2119861
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901119908 (119909)
119901119889119909)
1119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
119908119902(2119861)
120581119901
119908119902 (119861)120581119901
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(55)
Journal of Function Spaces 7
Now we consider the term 1198692 By Holderrsquos inequality we have
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816=
infin
sum
119895=1
int2119895+11198612119895119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
le 119862
infin
sum
119895=1
(int2119895+1119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
119904119889119910)
1119904
sdot (int2119895+11198612119895119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
(119899minus120572minus120573)1199041015840119889119910)
11199041015840
fl 119862infin
sum
119895=1
(11986811198951198682119895)
(56)
We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for
119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have
1198681119895= (int
2119895+2119861
|Ω (119911)|119904119889119911)
1119904
= (int
2119895+2
119903119861
0
int119878119899minus1
10038161003816100381610038161003816Ω (119911
1015840)10038161003816100381610038161003816
119904
1198891199111015840119903119899minus1119889119903)
1119904
= 119862 Ω119871119904(119878119899minus1)
100381610038161003816100381610038162119895+211986110038161003816100381610038161003816
1119904
(57)
where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus
1198682119895le
119862
10038161003816100381610038162119895+1119861
1003816100381610038161003816
1minus(120572+120573)119899(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
(58)
By Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
le 119862(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot
100381610038161003816100381610038162119895+111986110038161003816100381610038161003816
(119901119902minus1199041015840
119902+1199041015840
119901)1199011199021199041015840
119908119902 (2119895+1119861)1119902
(59)
Thus
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816le 119862
infin
sum
119895=1
(11986811198951198682119895)
le 119862
infin
sum
119895=1
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
1
119908119902 (2119895+1119861)1119902minus120581119901
(60)
So we get
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119908119902(119861)
1119902minus120581119901
119908119902 (2119895+1119861)1119902minus120581119901
(61)
We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)
Therefore the proof of Theorem 27 is completed
Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879
Ω120572is bounded
on weighted Morrey spaces
Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-
rems 26 and 27Then let us turn to prove (19)Set
119879119860
Ω120572119891 (119909)
= intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
0 le 120572 lt 119899
(63)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879
119860
Ω120572 On the
other hand for any 119903 gt 0 we have
119879119860
Ω120572119891 (119909)
ge int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
ge1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)
10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
(64)
Taking the supremum for 119903 gt 0 on the inequality above weget
119879119860
Ω120572119891 (119909) ge 119872
119860
Ω120572119891 (119909) (65)
Thus we can immediately obtain (19) from (65) and (18)
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 7
Journal of Function Spaces 7
Now we consider the term 1198692 By Holderrsquos inequality we have
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816=
infin
sum
119895=1
int2119895+11198612119895119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910
le 119862
infin
sum
119895=1
(int2119895+1119861
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
119904119889119910)
1119904
sdot (int2119895+11198612119895119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
(119899minus120572minus120573)1199041015840119889119910)
11199041015840
fl 119862infin
sum
119895=1
(11986811198951198682119895)
(56)
We will estimate 1198681119895 1198682119895 respectively Let 119911 = 119909 minus 119910 then for
119909 isin 119861 119910 isin 2119895+1119861 we have 119911 isin 2119895+2119861 Noticing that Ω ishomogeneous of degree zero andΩ isin 119871119904(119878119899minus1) then we have
1198681119895= (int
2119895+2119861
|Ω (119911)|119904119889119911)
1119904
= (int
2119895+2
119903119861
0
int119878119899minus1
10038161003816100381610038161003816Ω (119911
1015840)10038161003816100381610038161003816
119904
1198891199111015840119903119899minus1119889119903)
1119904
= 119862 Ω119871119904(119878119899minus1)
100381610038161003816100381610038162119895+211986110038161003816100381610038161003816
1119904
(57)
where 1199111015840 = 119911|119911| For 119909 isin 119861 119910 isin (2119861)119888 we have |119909 minus 119910| sim|1199090minus 119910| Thus
1198682119895le
119862
10038161003816100381610038162119895+1119861
1003816100381610038161003816
1minus(120572+120573)119899(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
(58)
By Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
1199041015840
119889119910)
11199041015840
le 119862(int2119895+1119861
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot (int2119895+1119861
119908 (119910)minus1199011199041015840
(119901minus1199041015840
)119889119910)
(119901minus1199041015840
)1199011199041015840
le 11986210038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) 119908
119902(2119895+1119861)120581119901
sdot
100381610038161003816100381610038162119895+111986110038161003816100381610038161003816
(119901119902minus1199041015840
119902+1199041015840
119901)1199011199021199041015840
119908119902 (2119895+1119861)1119902
(59)
Thus
10038161003816100381610038161003816119879Ω120572+120573
1198912 (119909)
10038161003816100381610038161003816le 119862
infin
sum
119895=1
(11986811198951198682119895)
le 119862
infin
sum
119895=1
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
1
119908119902 (2119895+1119861)1119902minus120581119901
(60)
So we get
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119908119902(119861)
1119902minus120581119901
119908119902 (2119895+1119861)1119902minus120581119901
(61)
We know from Remark 7(a) and Lemma 19 that 119908119902 satisfiesinequality (33) so the above series converges since the reversedoubling constant is larger than one Hence
1198692le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902) (62)
Therefore the proof of Theorem 27 is completed
Remark 28 It is worth noting that Theorem 27 is essentiallyverifying the multilinear fractional operator 119879
Ω120572is bounded
on weighted Morrey spaces
Now we are in the position of provingTheorem 8We will obtain (18) immediately in combination ofTheo-
rems 26 and 27Then let us turn to prove (19)Set
119879119860
Ω120572119891 (119909)
= intR119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
0 le 120572 lt 119899
(63)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899 It is easy to see inequality (18) also holds for 119879
119860
Ω120572 On the
other hand for any 119903 gt 0 we have
119879119860
Ω120572119891 (119909)
ge int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120572+119898minus1
1003816100381610038161003816119877119898 (119860 119909 119910)10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
ge1
119903119899minus120572+119898minus1
sdot int|119909minus119910|lt119903
1003816100381610038161003816Ω (119909 minus 119910)10038161003816100381610038161003816100381610038161003816119877119898 (119860 119909 119910)
10038161003816100381610038161003816100381610038161003816119891 (119910)
1003816100381610038161003816 119889119910
(64)
Taking the supremum for 119903 gt 0 on the inequality above weget
119879119860
Ω120572119891 (119909) ge 119872
119860
Ω120572119891 (119909) (65)
Thus we can immediately obtain (19) from (65) and (18)
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
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Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 8
8 Journal of Function Spaces
Similarly as before we give the following theorem at firstbefore provingTheorem 9 since it plays an important role inthe proof of Theorem 9 Set
119879Ω120573119891 (119909) = int
R119899
1003816100381610038161003816Ω (119909 minus 119910)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119899minus120573
1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (66)
whereΩ isin 119871119904(119878119899minus1) (119904 gt 1) is homogeneous of degree zero inR119899
Theorem 29 Under the assumptions of Theorem 9 119879Ω120573
isbounded from 119871119901120581(119908119901 119908119902) to 119871119902120581119902119901(119908119902)
The proof of Theorem 29 can be treated as that ofTheorem 27 with only slight modifications we omit its proofhere
Now let us prove Theorem 9 It is not difficult to see that(20) can be easily obtained from Theorems 26 and 29 Thenwe can immediately arrive at (21) from (65) and (20)
From now on we are in the place of showingTheorem 10We prove (22) at first Fixing any cube119876 with center at 119909 anddiameter 119903 denote 119876 = 2119876 and set
119860119876(119910) = 119860 (119910) minus sum
|120574|=119898minus1
1
120574119898119876(119863120574119860)119910
120574 (67)
Noticing that equality (67) is the special case of equality (44)when 119896 = 1 Thus equalities (45) and (46) also hold for119860119876(119910) We decompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 = 119891
1+ 119891
2
Then we have
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198911(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
+1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
fl 119868 + 119868119868
(68)
ByTheorem 20(a) and Remark 7(a) that 119908119902 isin Δ2 we have
119868 le119862
119908119902 (119876)120581119901
sdot sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast (int
119876
1003816100381610038161003816119891 (119910)1003816100381610038161003816
119901119908 (119910)
119901119889119910)
1119901
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)(
119908119902(119876)
119908119902 (119876))
120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(69)
Next we consider the term 119879119860
Ω1205721198912(119910) contained in 119868119868 By
Lemma 15 and equality (45) (46) we have
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816
le int(119876)119888
10038161003816100381610038161003816119877119898(119860119876 119910 119911)
100381610038161003816100381610038161003816100381610038161003816119910 minus 119911
1003816100381610038161003816
119898minus1
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572
1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911
le 119862int(119876)119888
sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860119876(119905)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862int(119876)119888
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
sdot
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
1+ 119868119868
2
(70)
We estimate 1198681198681and 119868119868
2 respectively By Lemma 21(a) and (b)
Holderrsquos inequality and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
1198681198681le 119862 sum
|120574|=119898minus1
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
int119868119911
119910
10038161003816100381610038161003816119863120574119860 (119905) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
119897
119889119905)
1119897
sdot
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
[
[
(1
10038161003816100381610038161003816119868119911119910
10038161003816100381610038161003816
sdot int119868119911
119910
100381610038161003816100381610038161003816119863120574119860 (119905) minus 119898119868119911
119910
(119863120574119860)100381610038161003816100381610038161003816
119897
119889119905)
1119897
+100381610038161003816100381610038161003816119898119868119911
119910
(119863120574119860)
minus 1198985119899119876(119863120574119860)100381610038161003816100381610038161003816+100381610038161003816100381610038161198985119899119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816]
]
sdot
infin
sum
119895=1
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+11198762119895119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
(119899minus120572)1199041015840119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 9
Journal of Function Spaces 9
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901
sdot 119908 (119911)119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(71)
For 119910 isin 119876 119911 isin (119876)119888 we have |119910 minus 119911| sim |119909 minus 119911| so we obtain
1198681198682le 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
+ 119862
infin
sum
119895=1
int2119895+11198762119895119876
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876(119863120574119860) minus 119898
119876(119863120574119860)10038161003816100381610038161003816
sdot1003816100381610038161003816Ω (119910 minus 119911)
1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911 fl 119868119868
21+ 119868119868
22
(72)
By Holderrsquos inequality we get
11986811986821le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)10038161003816100381610038161003816100381610038161003816Ω (119910 minus 119911)
10038161003816100381610038161003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904(119878119899minus1)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus
120572
119899
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
(73)
We estimate the part containing the function119863120574119860 as follows
(int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119911) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)119908 (119911)
minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862(int2119895+1119876
sum
|120574|=119898minus1
10038161003816100381610038161003816119863120574119860 (119911) minus 119898
2119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860)10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
+ sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
fl 119868119868119868 + 119868119881
(74)
For the term 119868119868119868 since 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we then have
119908minus1199011199041015840
(119901minus1199041015840
)isin 119860
1199051015840 sub 119860
infinby Remark 7(b) Thus by Lemma 25
that the norm of BMO(119908minus1199011199041015840
(119901minus1199041015840
)) is equivalent to the norm
of BMO(R119899) and 1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we have
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Journal of
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Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Page 10
10 Journal of Function Spaces
119868119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(75)
For the term 119868119881 by Lemma 21(a) there exist 1198621 1198622gt 0 such
that for any cube 119876 and 119904 gt 0100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816 gt 119904
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 1198621
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816119890minus1198622119904(sum|120574|=119898minus1
119863120574
119860lowast)
(76)
sincesum|120574|=119898minus1
(119863120574119860) isin BMOThen by Lemma 24(2) we have
119908(
119905 isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
) le 119862119908(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)
(77)
for some 120575 gt 0 Hence it implies
sum
|120574|=119898minus1
100381610038161003816100381610038161198982119895+1119876119908minus1199011199041015840(119901minus1199041015840)(119863120574119860) minus 119898
2119895+1119876(119863120574119860)10038161003816100381610038161003816
le1
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int2119895+1119876
sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905)
minus 1198982119895+1119876(119863120574119860)1003816100381610038161003816 119908minus1199011199041015840
(119901minus1199041015840
)(119905) 119889119905
=119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(
119905
isin 2119895+1119876 sum
|120574|=119898minus1
1003816100381610038161003816119863120574119860 (119905) minus 1198982119895+1119876 (119863
120574119860)1003816100381610038161003816
gt 119904
)119889119904 le119862
119908minus1199011199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int
infin
0
119908minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876) 119890
minus1198622119904120575(sum
|120574|=119898minus1119863120574
119860lowast)119889119904
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
(78)
As a result
119868119881 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast 119908
minus1199011199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
= 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
100381610038161003816100381610038162119895+111987610038161003816100381610038161003816
(119901minus1199041015840
)1199011199041015840
+1119902
119908119902 (2119895+1119876)1119902
(79)
Thus
11986811986821le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
1
119908119902 (2119895+1119876)1119902minus120581119901
(80)
For the term 11986811986822 by Lemma 21(c) Holderrsquos inequality and
1199081199041015840
isin 119860(1199011199041015840 119902119904
1015840) we get
11986811986822le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
sdot
infin
sum
119895=1
119895 int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899minus120572119889119911
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast Ω119871
119904(119878119899minus1)
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
infin
sum
119895=1
119895
sdot
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
1minus120572119899(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911)
119901119889119911)
1119901
sdot (int2119895+1119876
119908 (119911)minus1199011199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(81)
Hence
10038161003816100381610038161003816119879119860
Ω1205721198912(119910)10038161003816100381610038161003816le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895
119908119902 (2119895+1119876)1119902minus120581119901
(82)
Therefore
119868119868 le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
sdot
infin
sum
119895=1
119895119908119902(119876)
1119902minus120581119901
119908119902 (2119895+1119876)1119902minus120581119901
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 11
Journal of Function Spaces 11
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
infin
sum
119895=1
119895
(119863119895+1
1)1119902minus120581119901
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(83)
where1198631gt 1 is the reverse doubling constant Consequently
1
119908119902 (119876)120581119901(int119876
10038161003816100381610038161003816119879119860
Ω120572119891 (119910)
10038161003816100381610038161003816
119902
119908 (119910)119902119889119910)
1119902
le 119862 sum
|120574|=119898minus1
10038171003817100381710038171198631205741198601003817100381710038171003817lowast
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908119901 119908119902)
(84)
Taking supremum over all cubes in R119899 on both sides ofthe above inequality we complete the proof of (22) ofTheorem 10
It is not difficult to see that inequality (23) is easy to getfrom (22) and (65)
Proof of Theorem 11 We consider (25) firstly Let 119876 be thesame as in the proof of (22) and denote 119876 = 2119876 wedecompose 119891 as 119891 = 119891120594
119876+ 119891120594
(119876)119888 fl 119891
1+ 119891
2 Then we have
1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 119891 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198911 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(85)
ByTheorem 22(a) and Lemma 24(1) that 119908 isin Δ2 we get
119868 le1
119908 (119876)120581119901
1003817100381710038171003817[119860 119879Ω] 11989111003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
= 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(86)
For |[119860 119879Ω]1198912(119910)| by Holderrsquos inequality we obtain
1003816100381610038161003816[119860 119879Ω] 1198912 (119910)1003816100381610038161003816 le
infin
sum
119895=1
int2119895+11198762119895119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
1003816100381610038161003816119910 minus 1199111003816100381610038161003816
119899
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816 119889119911
le 119862
infin
sum
119895=1
1
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816Ω (119910 minus 119911)1003816100381610038161003816
119904119889119911)
1119904
sdot (int2119895+1119876
1003816100381610038161003816119860 (119910) minus 119860 (119911)1003816100381610038161003816
1199041015840
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
sdot (int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
11199041015840
+ 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199041015840
sdot1003816100381610038161003816119891 (119911)
1003816100381610038161003816
1199041015840
119889119911)
11199041015840
fl 1198681198681(119910) + 119868119868
2
(87)
Next we estimate 1198681198681(119910) and 119868119868
2 respectively By Holderrsquos
inequality and 119908 isin 1198601199011199041015840 we have
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
= 119862Ω119871119904
119908 (119876)120581119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
1199041015840
119889119911)
1199011199041015840
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)
120581
119901
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
sdot (int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le 119862
10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908 (119876)120581119901
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
(88)
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 12
12 Journal of Function Spaces
We estimate the part containing1198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) as follows
(int119876
10038161003816100381610038161003816119860 (119910) minus 119898
2119895+1119876119908minus1199041015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le (int119876
1003816100381610038161003816119860 (119910) minus 119898119876119908 (119860)1003816100381610038161003816
119901119908 (119910) 119889119910)
1119901
+10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816119908 (119876)
1119901
fl 119868119868119868 + 119868119881
(89)
For the term 119868119868119868 notice that 119908 isin 1198601199011199041015840 sub 119860
infin we thus get by
Lemma 25 that
119868119868119868 le 119860lowast 119908 (119876)1119901 (90)
Next we estimate 119868119881 By Lemmas 21(c) and 25 we have
10038161003816100381610038161003816119898119876119908 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816
le1003816100381610038161003816119898119876119908 (119860) minus 119898119876 (119860)
1003816100381610038161003816 +1003816100381610038161003816119898119876 (119860) minus 1198982119895+1119876 (119860)
1003816100381610038161003816
+100381610038161003816100381610038161198982119895+1119876 (119860) minus 1198982119895+1119876119908minus119904
1015840(119901minus1199041015840)(119860)10038161003816100381610038161003816le
1
119908 (119876)
sdot int119876
1003816100381610038161003816119860 (119905) minus 119898119876 (119860)1003816100381610038161003816 119908 (119905) 119889119905 + 2
119899(119895 + 1) 119860lowast
+1
119908minus1199041015840(119901minus119904
1015840) (2119895+1119876)
sdot int2119895+1119876
1003816100381610038161003816119860 (119905) minus 1198982119895+1119876 (119860)1003816100381610038161003816 119908minus1199041015840
(119901minus1199041015840
)(119905) 119889119905
le 119862 (119895 + 1) 119860lowast
(91)
Hence
119868119881 le 119862 (119895 + 1) 119860lowast 119908 (119876)1119901 (92)
As a result
1
119908 (119876)120581119901(int119876
1198681198681(119910)
119901119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
(119895 + 1)119908 (119876)
(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(93)
For 1198681198682 by Holderrsquos inequality and 119908 isin 119860
1199011199041015840 we get
1198681198682le 119862 Ω119871119904
sdot
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
(int2119895+1119876
1003816100381610038161003816119891 (119911)1003816100381610038161003816
119901119908 (119911) 119889119911)
1119901
sdot (int2119895+1119876
100381610038161003816100381610038161198982119895+1119876119908minus1199041015840(119901minus1199041015840)(119860) minus 119860 (119911)
10038161003816100381610038161003816
1199011199041015840
(119901minus1199041015840
)
sdot 119908 (119911)minus1199041015840
(119901minus1199041015840
)119889119911)
(119901minus1199041015840
)1199011199041015840
le 119862 Ω119871119904 119860lowast
sdot10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
100381610038161003816100381610038162119895+211987610038161003816100381610038161003816
1119904
100381610038161003816100381621198951198761003816100381610038161003816
119908 (2119895+1119876)120581119901
sdot 119908minus1199041015840
(119901minus1199041015840
)(2119895+1119876)(119901minus1199041015840
)1199011199041015840
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
sdot
infin
sum
119895=1
1
119908 (2119895+1119876)(1minus120581)119901
(94)
Therefore
1
119908 (119876)120581119901(int119876
119868119868119901
2119908 (119910) 119889119910)
1119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
infin
sum
119895=1
119908 (119876)(1minus120581)119901
119908 (2119895+1119876)(1minus120581)119901
le 119862 119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(95)
So far we have completed the proof of (25)
Inequality (26) can be immediately obtained from (65)and (25)
Proof ofTheorem 12 As before we prove (27) at first Assume119876 to be the same as in the proof of (22) denote 119876 = 2119876 andset
119860119876(119910) = 119860 (119910) minus 119898
119876(nabla119860) 119910 (96)
We also decompose 119891 according to 119876 119891 = 119891120594119876+ 119891120594
(119876)119888 fl
1198911+ 119891
2 Then we get
1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω119891 (119910)
100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
le1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198911(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
+1
119908 (119876)120581119901(int119876
100381610038161003816100381610038161003816119860
Ω1198912(119910)100381610038161003816100381610038161003816
119901
119908 (119910) 119889119910)
1119901
fl 119868 + 119868119868
(97)
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 13
Journal of Function Spaces 13
For 119868 Theorem 23 and Lemma 24(1) imply
119868 le1
119908 (119876)120581119901
100381710038171003817100381710038171003817119860
Ω1198911
100381710038171003817100381710038171003817119871119901(119908)
le119862
119908 (119876)120581119901Ωinfin nabla119860lowast
100381710038171003817100381711989111003817100381710038171003817119871119901(119908)
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
119908(119876)120581119901
119908 (119876)120581119901
le 119862 nabla119860lowast10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)
(98)
We will omit the proof for 119868119868 since it is similar to and eveneasier than the part of 119868119868 in the proof of (22) except that weuse the conditions 119908 isin 119860
119901 Ω isin 119871
infin(119878119899minus1) 119898 = 2 and 119891 isin
119871119901120581(119908) For inequality (28) it can be easily proved by (27)
and (65) Thus we complete the proof of Theorem 12
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The research was partially supported by the National NatureScience Foundation of China under Grant no 11171306 andno 11571306 and sponsored by the Scientific Project ofZhejiang Provincial Science Technology Department underGrant no 2011C33012 and the Scientific Research Fund ofZhejiang Provincial Education Department under Grant noZ201017584
References
[1] J Cohen and J Gosselin ldquoA BMO estimate for multilinearsingular integralsrdquo Illinois Journal of Mathematics vol 30 no3 pp 445ndash464 1986
[2] S Hofmann ldquoOn certain nonstandard Calderon-Zygmundoperatorsrdquo StudiaMathematica vol 109 no 2 pp 105ndash131 1994
[3] S Z Lu H XWu and P Zhang ldquoMultilinear singular integralswith rough kernelrdquo Acta Mathematica Sinica (English Series)vol 19 no 1 pp 51ndash62 2003
[4] Y Ding and S Z Lu ldquoWeighted boundedness for a class ofroughmultilinear operatorsrdquoActaMathematica SinicamdashEnglishSeries vol 17 no 3 pp 517ndash526 2001
[5] S Z Lu and P Zhang ldquoLipschitz estimates for generalizedcommutators of fractional integrals with rough kernelrdquoMathe-matische Nachrichten vol 252 no 1 pp 70ndash85 2003
[6] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938
[7] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987
[8] D Fan S Lu and D Yang ldquoRegularity in Morrey spaces ofstrong solutions to nondivergence elliptic equations with VMOcoefficientsrdquo Georgian Mathematical Journal vol 5 no 5 pp425ndash440 1998
[9] J Peetre ldquoOn the theory of 119871120588120582
spacesrdquo Journal of FunctionalAnalysis vol 4 no 1 pp 71ndash87 1969
[10] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009
[11] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 4 pp 589ndash600 2012 (Chinese)
[12] M Paluszynski ldquoCharacterization of the Besov spaces viathe commutator operator of Coifman Rochberg and WeissrdquoIndiana University Mathematics Journal vol 44 no 1 pp 1ndash171995
[13] RADeVore andRC Sharpley ldquoMaximal functionsmeasuringsmoothnessrdquo Memoirs of the American Mathematical Societyvol 47 no 293 1984
[14] Y Ding and S Z Lu ldquoWeighted norm inequalities for fractionalintegral operators with rough kernelrdquo Canadian Journal ofMathematics vol 50 no 1 pp 29ndash39 1998
[15] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009
[16] S Z Lu Y Ding and D Y Yan Singular Integrals aand RelatedTopics World Scientific 2007
[17] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Page 14
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of