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Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 560976 5 pageshttpdxdoiorg1011552013560976
Research ArticleOn a Class of Bilinear Pseudodifferential Operators
Aacuterpaacuted Beacutenyi1 and Tadahiro Oh2
1 Department of Mathematics Western Washington University 516 High Street Bellingham WA 98225 USA2Department of Mathematics Princeton University Fine Hall Washington Road Princeton NJ 08544-1000 USA
Correspondence should be addressed to Arpad Benyi arpadbenyiwwuedu
Received 26 September 2012 Accepted 3 December 2012
Academic Editor Baoxiang Wang
Copyright copy 2013 A Benyi and T OhThis 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 provide a direct proof for the boundedness of pseudodifferential operators with symbols in the bilinear Hormander class BS01120575
0 le 120575 lt 1 The proof uses a reduction to bilinear elementary symbols and Littlewood-Paley theory
1 Introduction Main Results and Examples
Coifman andMeyerrsquos ideas onmultilinear operators and theirapplications in partial differential equations (PDEs) havehad a great impact in the future developments and growthwitnessed in the topic of multilinear singular integrals Oneof their classical results [1 Proposition 2 p 154] is about the119871119901times119871119902rarr 119871119903 boundedness of a class of translation invariant
bilinear operators (bilinear multiplier operators) given by
minus|120573|minus|120574| for all120585 120578 isin R119899 and all multi-indices 120573 120574 then 119879
120590has a bounded
extension from 119871119901 times 119871119902 into 119871119903 for all 1 lt 119901 119902 lt infin such that1119901 + 1119902 = 1119903
In fact Coifman and Meyerrsquos approach yields TheoremA only for 119903 gt 1 The optimal extension of their result tothe range 119903 gt 12 (as implied in the theorem above) canbe obtained using interpolation arguments and an end-pointestimate 1198711 times 1198711 into 11987112infin in the works of Grafakos andTorres [2] and Kenig and Stein [3]
Bilinear pseudodifferential operators are natural non-translation invariant generalizations of the translation invari-ant ones they allow symbols to depend on the space variable
119909 as well Let us then consider bilinear operators a prioridefined from S times S into S1015840 of the form
Perhaps unsurprisingly we impose then similar conditionson the derivatives of the symbol 120590 with the expectationthat they would yield indeed bounded operators 119879
120590on
appropriate spaces of functions The estimates that we havein mind define the so-called bilinear Hormander classes ofsymbols denoted by BS119898
for all 119909 120585 120578 isin R119899 and all multi-indices 120572 120573 120574 Note that weneed smoothness in 119909 as in the linear Hormander classes Asusual the notation 119886 ≲ 119887 means that there exists a positiveconstant119870 (independent of 119886 119887) such that 119886 le 119870119887
With this terminology we can restate Theorem A asfollows
If the 119909-independent symbol 120590(120585 120578) belongs to the classBS010 then119879
120590is bounded from 119871119901times119871119902 into 119871119903 for all 1 lt 119901 119902 lt
infin such that 1119901 + 1119902 = 1119903The condition of translation invariance (equivalently the
119909-independence of the symbol) is superfluous Moreover theprevious boundedness result can be shown to hold for thelarger class of symbols BS0
1120575supe BS010 where 0 le 120575 lt 1This is a
known fact that is tightly connected to the bilinear Calderon-Zygmund theory developed by Grafakos and Torres in [2]
2 Journal of Function Spaces and Applications
and the existence of a transposition symbolic calculus provedby Benyi et al [4] Let us briefly give an outline of how thisfollows First we note that the bilinear kernels associated tobilinear operators with symbols in BS0
1120575 0 le 120575 lt 1 are
bilinear Calderon-Zygmund operators in the sense of [2]Second we recall that [2 Corollary 1] which is an applicationof the bilinear 119879(1) theorem therein states the following
Theorem B If 119879 and its transposes 119879lowast1 and 119879
lowast2 havesymbols in BS0
11 then they can be extended as bounded
operators from 119871119901times 119871119902 into 119871119903 for 1 lt 119901 119902 lt infin and
1119901 + 1119902 = 1119903
Third by [4 Theorem 21] we have the following
Theorem C Assume that 0 le 120575 le 120588 le 1 120575 lt 1 and 120590 isinBS119898120588120575 Then for 119895 = 1 2119879lowast119895
120590= 119879120590lowast119895 where 120590lowast119895 isin BS119898
120588120575
Finally since BS01120575
sub BS011 we can directly combine
Theorems B and C to recover the following optimal extensionof the Coifman-Meyer result note that now the symbol isallowed to depend on 119909 while 119903 is still allowed to be in theoptimal interval (12infin)
Theorem 1 If 120590 is a symbol in BS01120575 0 le 120575 lt 1 then 119879
120590has a
bounded extension from 119871119901 times 119871119902 into 119871119903 for all 1 lt 119901 119902 lt infinsuch that 1119901 + 1119902 = 1119903
Once we have the boundedness of the class BS01120575
onproducts of Lebesgue spaces a ldquoreduction methodrdquo allows usto deduce also the boundedness of the class BS119898
1120575on appro-
priate products of Sobolev spaces Moreover our estimatesin this case come in the form of Leibniz-type rules formore on these kinds of properties see the work of Bernicotet al [5] In the particular case when the bilinear operatoris just a differential operator the Leibniz-type rules arereferred to as Kato-Poncersquos commutator estimates and areknown to play a significant role in the study of the Eulerand Navier-Stokes equations see [6] see also Kenig et al[7] for further applications of commutators to nonlinearSchrodinger equations Let 119869119898 = (119868 minus Δ)
1198982 denote thelinear Fourier multiplier operator with symbol ⟨120585⟩119898 where⟨120585⟩ = (1 + |120585|
2)12 By definition we say that 119891 belongs to the
Sobolev space 119871119901119898if 119869119898119891 isin 119871119901 We have the following
Theorem 2 Let 120590 be a symbol in BS1198981120575 0 le 120575 lt 1 119898 ge 0
and let 119879120590be its associated operator Then there exist symbols
1205901and 120590
2in BS01120575
such that for all 119891 119892 isin S
119879120590(119891 119892) = 119879
1205901
(119869119898119891 119892) + 119879
1205902
(119891 119869119898119892) (4)
In particular then one has that 119879120590has a bounded extension
Theproof ofTheorem2 follows a similar path as the one inthe work of Benyi et al [8Theorem 27] For the convenience
of the reader we sketch here the main steps in the argumentLet 120601 be a 119862infin-function on R such that 0 le 120601 le 1 supp120601 sub [minus2 2] and 120601(119903) + 120601(1119903) = 1 on [0infin) Then (4) holdsif we let
One of the main reasons for the study of the Hormanderclasses of bilinear pseudodifferential operators is the factthat the conditions imposed on the symbols arise naturallyin PDEs In particular the bilinear Hormader classes BS119898
120588120575
model the product of two functions and their derivatives
Example 3 Consider first a bilinear partial differential oper-ator with variable coefficients
119863119896ℓ(119891 119892) = sum
|120573|le119896
sum
|120574|leℓ
119888120573120574(119909)
120597120573119891
120597119909120573
120597120574119892
120597119909120574 (8)
Note that119863119896ℓ= 119879120590119896ℓ
where the bilinear symbol is given by
120590119896ℓ(119909 120585 120578) = (2120587)
minus2119899sum
120573120574
119888120573120574(119909) (119894120585)
120573(119894120578)120574 (9)
Assuming that the coefficients 119888120573120574
have bounded derivativesit is easy to show that 120590
119896ℓisin BS119896+ℓ10
Example 4 The symbol in the previous example is almostequivalent to a multiplier of the form
119896120578ℓ120590minus1(120585 120578) belong to BS119896+ℓ
10
Example 6 One of the recurrent techniques in PDE estimatesis to truncate a given multiplier at the right scale Considernow
120590 (120585 120578) = 120590119898(120585 120578) sum
119886119887isinN
119888119886119887(119909) 120593 (2
minus119886120585) 120594 (2
minus119887120578) (11)
Journal of Function Spaces and Applications 3
where120593 and120594 are smooth ldquocutoffrdquo functions supported in theannulus 12 le |120585| le 2 and the coefficients satisfy derivativeestimates of the form
Elementary calculations show that 120590 isin BS1198981120575
Remark 7 Theorems 1 and 2 lead to the natural questionabout the boundedness properties of other Hormanderclasses of bilinear pseudodifferential operators An interest-ing situation arises when we consider the bilinear Calderon-Vaillancourt class BS0
00 A result of Benyi andTorres [9] shows
that in this case the 119871119901 times 119871119902 rarr 119871119903 boundedness fails
One can impose some additional conditions (besides being inBS000) on a symbol to guarantee that the corresponding bilin-
ear pseudodifferential operator is 119871119901 times 119871119902 rarr 119871119903 bounded
see for example [9] and the recent work of Bernicot andShrivastava [10] However there is a nice substitute for theLebesgue space estimates If we consider instead modulationspaces119872119901119902 (see the excellent book byGrochenig [11] for theirdefinition and basic properties) we can show for examplethat if 120590 isin BS0
00then 119879
120590 1198712times 1198712rarr 119872
1infin (which contains1198711) This and other more general boundedness results on
modulation spaces for the class BS000
were obtained by Benyiet al [12] Then this particular boundedness result with thereduction method employed in Theorem 2 allows us to alsoobtain the boundedness of the class BS119898
00from 1198712
119898times 1198712
119898into
1198721infin Interestingly we can also obtain the 119871119901 times 119871119902 rarr 119871
119903
boundedness of the class BS11989800 but we have to require in this
case the order 119898 to depend on the Lebesgue exponents seethe work of Benyi et al [13] also Miyachi and Tomita [14] forthe optimality of the order 119898 and the extension of the resultin [13] below 119903 = 1 The most general case of the classes BS119898
120588120575
is also given in [13]
In the remainder of the paper we will provide an alternateproof ofTheorem 1 that does not use sophisticated tools suchas the symbolic calculus The proof is in the original spiritof the work of Coifman and Meyer that made use of theLittlewood-Paley theory As such we will only be concernedhere with the boundedness into the target space 119871119903 with 119903 gt1 Of course obtaining the full result for 119903 gt 12 is thenpossible because of the bilinear Calderon-Zygmund theorywhich applies to our case We will borrow some of the ideasfrom Benyi and Torres [15] which in turn go back to the niceexposition (in the linear case) by Journe [16] by making useof the so-called bilinear elementary symbols
2 Proof of Theorem 1
We start with two lemmas that provide the anticipateddecomposition of our symbol into bilinear elementary sym-bols Since they are the immediate counterparts of [15Lemma 1 and Lemma 2] to our class BS0
1120575 we will skip their
proofs see also [16 pp 72ndash75] and [1 pp 55ndash57] The firstreduction is as follows
Lemma 8 Fix a symbol 120590 in the class BS01120575 0 le 120575 lt 1 and
an arbitrary large positive integer 119873 Then for any 119891 119892 isin S119879120590(119891 119892) can be written in the form
119879120590(119891 119892) = sum
119896ℓisinZ119899
119889119896ℓ119879120590119896ℓ
(119891 119892) + 119877 (119891 119892) (13)
where 119889119896ℓ is an absolutely convergent sequence of numbers
and 119877 is a bounded operator from 119871119901 times 119871119902 into 119871119903 for 1119901 +1119902 = 1119903 1 lt 119901 119902 119903 lt infin
Now if 120590119896ℓ
is any of the symbols in (14) and we knew apriori that119879
120590119896ℓ
are bounded from119871119901times119871119902 into119871119903 with operatornorms depending only on the implicit constants from (15)the fact that the sequence 119889
119896ℓ is absolutely convergent
immediately implies the 119871119901 times 119871119902 rarr 119871119903 boundedness of 119879
120590
Our first step has thus reduced the study of generic symbolsin the class BS0
1120575to symbols of the form
120590 (119909 120585 120578) =
infin
sum
119895=0
119898119895(119909) 120595 (2
minus119895120585 2minus119895120578) (16)
where ||120597120572119898119895||119871infin ≲ 2
119895120575|120572| and 120595 is supported in 13 le
max(|120585| |120578|) le 1Our second step is to further reduce the simpler looking
symbol given in (16) to a sum of bilinear elementary symbols
Lemma9 Let 120590 be as in (16) One can further reduce the studyto symbols of the form
120590 = 1205901+ 1205902+ 1205903 (17)
where the elementary symbols 120590119896 119896 = 1 2 3 are defined via
This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817
wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892
119896ℎ119896as being dyadic
blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control
uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order
to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on
119878119894however follows from the uniform estimate on the 119892
119896rsquos and
an immediate application of Littlewood-Paley theory
Acknowledgments
A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant
References
[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010
[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf
[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988
[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003
[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006
[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004
[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011
[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001
[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005
[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf
[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press
[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003
[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983
and the existence of a transposition symbolic calculus provedby Benyi et al [4] Let us briefly give an outline of how thisfollows First we note that the bilinear kernels associated tobilinear operators with symbols in BS0
1120575 0 le 120575 lt 1 are
bilinear Calderon-Zygmund operators in the sense of [2]Second we recall that [2 Corollary 1] which is an applicationof the bilinear 119879(1) theorem therein states the following
Theorem B If 119879 and its transposes 119879lowast1 and 119879
lowast2 havesymbols in BS0
11 then they can be extended as bounded
operators from 119871119901times 119871119902 into 119871119903 for 1 lt 119901 119902 lt infin and
1119901 + 1119902 = 1119903
Third by [4 Theorem 21] we have the following
Theorem C Assume that 0 le 120575 le 120588 le 1 120575 lt 1 and 120590 isinBS119898120588120575 Then for 119895 = 1 2119879lowast119895
120590= 119879120590lowast119895 where 120590lowast119895 isin BS119898
120588120575
Finally since BS01120575
sub BS011 we can directly combine
Theorems B and C to recover the following optimal extensionof the Coifman-Meyer result note that now the symbol isallowed to depend on 119909 while 119903 is still allowed to be in theoptimal interval (12infin)
Theorem 1 If 120590 is a symbol in BS01120575 0 le 120575 lt 1 then 119879
120590has a
bounded extension from 119871119901 times 119871119902 into 119871119903 for all 1 lt 119901 119902 lt infinsuch that 1119901 + 1119902 = 1119903
Once we have the boundedness of the class BS01120575
onproducts of Lebesgue spaces a ldquoreduction methodrdquo allows usto deduce also the boundedness of the class BS119898
1120575on appro-
priate products of Sobolev spaces Moreover our estimatesin this case come in the form of Leibniz-type rules formore on these kinds of properties see the work of Bernicotet al [5] In the particular case when the bilinear operatoris just a differential operator the Leibniz-type rules arereferred to as Kato-Poncersquos commutator estimates and areknown to play a significant role in the study of the Eulerand Navier-Stokes equations see [6] see also Kenig et al[7] for further applications of commutators to nonlinearSchrodinger equations Let 119869119898 = (119868 minus Δ)
1198982 denote thelinear Fourier multiplier operator with symbol ⟨120585⟩119898 where⟨120585⟩ = (1 + |120585|
2)12 By definition we say that 119891 belongs to the
Sobolev space 119871119901119898if 119869119898119891 isin 119871119901 We have the following
Theorem 2 Let 120590 be a symbol in BS1198981120575 0 le 120575 lt 1 119898 ge 0
and let 119879120590be its associated operator Then there exist symbols
1205901and 120590
2in BS01120575
such that for all 119891 119892 isin S
119879120590(119891 119892) = 119879
1205901
(119869119898119891 119892) + 119879
1205902
(119891 119869119898119892) (4)
In particular then one has that 119879120590has a bounded extension
Theproof ofTheorem2 follows a similar path as the one inthe work of Benyi et al [8Theorem 27] For the convenience
of the reader we sketch here the main steps in the argumentLet 120601 be a 119862infin-function on R such that 0 le 120601 le 1 supp120601 sub [minus2 2] and 120601(119903) + 120601(1119903) = 1 on [0infin) Then (4) holdsif we let
One of the main reasons for the study of the Hormanderclasses of bilinear pseudodifferential operators is the factthat the conditions imposed on the symbols arise naturallyin PDEs In particular the bilinear Hormader classes BS119898
120588120575
model the product of two functions and their derivatives
Example 3 Consider first a bilinear partial differential oper-ator with variable coefficients
119863119896ℓ(119891 119892) = sum
|120573|le119896
sum
|120574|leℓ
119888120573120574(119909)
120597120573119891
120597119909120573
120597120574119892
120597119909120574 (8)
Note that119863119896ℓ= 119879120590119896ℓ
where the bilinear symbol is given by
120590119896ℓ(119909 120585 120578) = (2120587)
minus2119899sum
120573120574
119888120573120574(119909) (119894120585)
120573(119894120578)120574 (9)
Assuming that the coefficients 119888120573120574
have bounded derivativesit is easy to show that 120590
119896ℓisin BS119896+ℓ10
Example 4 The symbol in the previous example is almostequivalent to a multiplier of the form
119896120578ℓ120590minus1(120585 120578) belong to BS119896+ℓ
10
Example 6 One of the recurrent techniques in PDE estimatesis to truncate a given multiplier at the right scale Considernow
120590 (120585 120578) = 120590119898(120585 120578) sum
119886119887isinN
119888119886119887(119909) 120593 (2
minus119886120585) 120594 (2
minus119887120578) (11)
Journal of Function Spaces and Applications 3
where120593 and120594 are smooth ldquocutoffrdquo functions supported in theannulus 12 le |120585| le 2 and the coefficients satisfy derivativeestimates of the form
Elementary calculations show that 120590 isin BS1198981120575
Remark 7 Theorems 1 and 2 lead to the natural questionabout the boundedness properties of other Hormanderclasses of bilinear pseudodifferential operators An interest-ing situation arises when we consider the bilinear Calderon-Vaillancourt class BS0
00 A result of Benyi andTorres [9] shows
that in this case the 119871119901 times 119871119902 rarr 119871119903 boundedness fails
One can impose some additional conditions (besides being inBS000) on a symbol to guarantee that the corresponding bilin-
ear pseudodifferential operator is 119871119901 times 119871119902 rarr 119871119903 bounded
see for example [9] and the recent work of Bernicot andShrivastava [10] However there is a nice substitute for theLebesgue space estimates If we consider instead modulationspaces119872119901119902 (see the excellent book byGrochenig [11] for theirdefinition and basic properties) we can show for examplethat if 120590 isin BS0
00then 119879
120590 1198712times 1198712rarr 119872
1infin (which contains1198711) This and other more general boundedness results on
modulation spaces for the class BS000
were obtained by Benyiet al [12] Then this particular boundedness result with thereduction method employed in Theorem 2 allows us to alsoobtain the boundedness of the class BS119898
00from 1198712
119898times 1198712
119898into
1198721infin Interestingly we can also obtain the 119871119901 times 119871119902 rarr 119871
119903
boundedness of the class BS11989800 but we have to require in this
case the order 119898 to depend on the Lebesgue exponents seethe work of Benyi et al [13] also Miyachi and Tomita [14] forthe optimality of the order 119898 and the extension of the resultin [13] below 119903 = 1 The most general case of the classes BS119898
120588120575
is also given in [13]
In the remainder of the paper we will provide an alternateproof ofTheorem 1 that does not use sophisticated tools suchas the symbolic calculus The proof is in the original spiritof the work of Coifman and Meyer that made use of theLittlewood-Paley theory As such we will only be concernedhere with the boundedness into the target space 119871119903 with 119903 gt1 Of course obtaining the full result for 119903 gt 12 is thenpossible because of the bilinear Calderon-Zygmund theorywhich applies to our case We will borrow some of the ideasfrom Benyi and Torres [15] which in turn go back to the niceexposition (in the linear case) by Journe [16] by making useof the so-called bilinear elementary symbols
2 Proof of Theorem 1
We start with two lemmas that provide the anticipateddecomposition of our symbol into bilinear elementary sym-bols Since they are the immediate counterparts of [15Lemma 1 and Lemma 2] to our class BS0
1120575 we will skip their
proofs see also [16 pp 72ndash75] and [1 pp 55ndash57] The firstreduction is as follows
Lemma 8 Fix a symbol 120590 in the class BS01120575 0 le 120575 lt 1 and
an arbitrary large positive integer 119873 Then for any 119891 119892 isin S119879120590(119891 119892) can be written in the form
119879120590(119891 119892) = sum
119896ℓisinZ119899
119889119896ℓ119879120590119896ℓ
(119891 119892) + 119877 (119891 119892) (13)
where 119889119896ℓ is an absolutely convergent sequence of numbers
and 119877 is a bounded operator from 119871119901 times 119871119902 into 119871119903 for 1119901 +1119902 = 1119903 1 lt 119901 119902 119903 lt infin
Now if 120590119896ℓ
is any of the symbols in (14) and we knew apriori that119879
120590119896ℓ
are bounded from119871119901times119871119902 into119871119903 with operatornorms depending only on the implicit constants from (15)the fact that the sequence 119889
119896ℓ is absolutely convergent
immediately implies the 119871119901 times 119871119902 rarr 119871119903 boundedness of 119879
120590
Our first step has thus reduced the study of generic symbolsin the class BS0
1120575to symbols of the form
120590 (119909 120585 120578) =
infin
sum
119895=0
119898119895(119909) 120595 (2
minus119895120585 2minus119895120578) (16)
where ||120597120572119898119895||119871infin ≲ 2
119895120575|120572| and 120595 is supported in 13 le
max(|120585| |120578|) le 1Our second step is to further reduce the simpler looking
symbol given in (16) to a sum of bilinear elementary symbols
Lemma9 Let 120590 be as in (16) One can further reduce the studyto symbols of the form
120590 = 1205901+ 1205902+ 1205903 (17)
where the elementary symbols 120590119896 119896 = 1 2 3 are defined via
This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817
wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892
119896ℎ119896as being dyadic
blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control
uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order
to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on
119878119894however follows from the uniform estimate on the 119892
119896rsquos and
an immediate application of Littlewood-Paley theory
Acknowledgments
A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant
References
[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010
[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf
[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988
[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003
[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006
[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004
[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011
[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001
[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005
[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf
[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press
[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003
[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983
where120593 and120594 are smooth ldquocutoffrdquo functions supported in theannulus 12 le |120585| le 2 and the coefficients satisfy derivativeestimates of the form
Elementary calculations show that 120590 isin BS1198981120575
Remark 7 Theorems 1 and 2 lead to the natural questionabout the boundedness properties of other Hormanderclasses of bilinear pseudodifferential operators An interest-ing situation arises when we consider the bilinear Calderon-Vaillancourt class BS0
00 A result of Benyi andTorres [9] shows
that in this case the 119871119901 times 119871119902 rarr 119871119903 boundedness fails
One can impose some additional conditions (besides being inBS000) on a symbol to guarantee that the corresponding bilin-
ear pseudodifferential operator is 119871119901 times 119871119902 rarr 119871119903 bounded
see for example [9] and the recent work of Bernicot andShrivastava [10] However there is a nice substitute for theLebesgue space estimates If we consider instead modulationspaces119872119901119902 (see the excellent book byGrochenig [11] for theirdefinition and basic properties) we can show for examplethat if 120590 isin BS0
00then 119879
120590 1198712times 1198712rarr 119872
1infin (which contains1198711) This and other more general boundedness results on
modulation spaces for the class BS000
were obtained by Benyiet al [12] Then this particular boundedness result with thereduction method employed in Theorem 2 allows us to alsoobtain the boundedness of the class BS119898
00from 1198712
119898times 1198712
119898into
1198721infin Interestingly we can also obtain the 119871119901 times 119871119902 rarr 119871
119903
boundedness of the class BS11989800 but we have to require in this
case the order 119898 to depend on the Lebesgue exponents seethe work of Benyi et al [13] also Miyachi and Tomita [14] forthe optimality of the order 119898 and the extension of the resultin [13] below 119903 = 1 The most general case of the classes BS119898
120588120575
is also given in [13]
In the remainder of the paper we will provide an alternateproof ofTheorem 1 that does not use sophisticated tools suchas the symbolic calculus The proof is in the original spiritof the work of Coifman and Meyer that made use of theLittlewood-Paley theory As such we will only be concernedhere with the boundedness into the target space 119871119903 with 119903 gt1 Of course obtaining the full result for 119903 gt 12 is thenpossible because of the bilinear Calderon-Zygmund theorywhich applies to our case We will borrow some of the ideasfrom Benyi and Torres [15] which in turn go back to the niceexposition (in the linear case) by Journe [16] by making useof the so-called bilinear elementary symbols
2 Proof of Theorem 1
We start with two lemmas that provide the anticipateddecomposition of our symbol into bilinear elementary sym-bols Since they are the immediate counterparts of [15Lemma 1 and Lemma 2] to our class BS0
1120575 we will skip their
proofs see also [16 pp 72ndash75] and [1 pp 55ndash57] The firstreduction is as follows
Lemma 8 Fix a symbol 120590 in the class BS01120575 0 le 120575 lt 1 and
an arbitrary large positive integer 119873 Then for any 119891 119892 isin S119879120590(119891 119892) can be written in the form
119879120590(119891 119892) = sum
119896ℓisinZ119899
119889119896ℓ119879120590119896ℓ
(119891 119892) + 119877 (119891 119892) (13)
where 119889119896ℓ is an absolutely convergent sequence of numbers
and 119877 is a bounded operator from 119871119901 times 119871119902 into 119871119903 for 1119901 +1119902 = 1119903 1 lt 119901 119902 119903 lt infin
Now if 120590119896ℓ
is any of the symbols in (14) and we knew apriori that119879
120590119896ℓ
are bounded from119871119901times119871119902 into119871119903 with operatornorms depending only on the implicit constants from (15)the fact that the sequence 119889
119896ℓ is absolutely convergent
immediately implies the 119871119901 times 119871119902 rarr 119871119903 boundedness of 119879
120590
Our first step has thus reduced the study of generic symbolsin the class BS0
1120575to symbols of the form
120590 (119909 120585 120578) =
infin
sum
119895=0
119898119895(119909) 120595 (2
minus119895120585 2minus119895120578) (16)
where ||120597120572119898119895||119871infin ≲ 2
119895120575|120572| and 120595 is supported in 13 le
max(|120585| |120578|) le 1Our second step is to further reduce the simpler looking
symbol given in (16) to a sum of bilinear elementary symbols
Lemma9 Let 120590 be as in (16) One can further reduce the studyto symbols of the form
120590 = 1205901+ 1205902+ 1205903 (17)
where the elementary symbols 120590119896 119896 = 1 2 3 are defined via
This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817
wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892
119896ℎ119896as being dyadic
blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control
uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order
to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on
119878119894however follows from the uniform estimate on the 119892
119896rsquos and
an immediate application of Littlewood-Paley theory
Acknowledgments
A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant
References
[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010
[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf
[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988
[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003
[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006
[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004
[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011
[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001
[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005
[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf
[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press
[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003
[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983
This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817
wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892
119896ℎ119896as being dyadic
blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control
uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order
to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on
119878119894however follows from the uniform estimate on the 119892
119896rsquos and
an immediate application of Littlewood-Paley theory
Acknowledgments
A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant
References
[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010
[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf
[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988
[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003
[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006
[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004
[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011
[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001
[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005
[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf
[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press
[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003
[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983
This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817
wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892
119896ℎ119896as being dyadic
blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control
uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order
to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on
119878119894however follows from the uniform estimate on the 119892
119896rsquos and
an immediate application of Littlewood-Paley theory
Acknowledgments
A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant
References
[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978
[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002
[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999
[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010
[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf
[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988
[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003
[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006
[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004
[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011
[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001
[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005
[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf
[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press
[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003
[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983