Nonlinear Differ. Equ. Appl. 18 (2011), 287–308 c 2011 Springer Basel AG 1021-9722/11/030287-22 published online January 13, 2011 DOI 10.1007/s00030-010-0096-8 Nonlinear Differential Equations and Applications NoDEA Multilinear commutators of fractional integrals over Morrey spaces with non-doubling measures Xiangxing Tao and Taotao Zheng Abstract. In this paper, the authors study the boundedness of multilin- ear fractional integrals on the product Morrey space with non-doubling measure, and investigate the Morrey boundedness properties of the multi- linear commutators generated by multilinear fractional integral operators with a tuple of RBMO functions. Mathematics Subject Classification (2010). 42B20, 42B25, 42B30. Keywords. Multi-commutator, Multilinear fractional integral, RBMO function, Morrey space, Non-doubling measure. 1. Introduction and main results The classical result of commutator is due to Coifman et al. [1]: if b ∈ BMO and T is a Calder´on–Zygmund operator, then the commutator [b, T ] is bounded on L p spaces for 1 <p< ∞. In 2001, Tolsa [2] developed the theory of Calder´ on- Zygmund operators and their commutators with RBMO functions in the set- ting of non-doubling measures. Later on, Chen and Sawyer [3] modified the definition of RBMO to investigate the commutators of the potential operators and RBMO functions. Recently, Sawano andTanaka [4, 5] defined the Morrey spaces for non-doubling measures, and obtained the boundedness of the com- mutators generated by fractional integrals with RBMO functions in Morrey spaces. Here we point out that the commutator estimates for non-doubling measures play an important role in solving the long-standing open questions, like the Painlev` e problem. In recent years, the theory on multilinear integral operators and multilin- ear commutators have attracted much attentions as a rapid developing field in This work was supported in part by the NNSF of China under Grant #10771110 and #10471069, and sponsored by the NSF of Ningbo City under Grant #2009A610090.
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Multilinear commutators of fractional integrals over ...harmonic analysis. Hu et al. [6] studied the Lp-boundedness and certain weak type endpoint estimates for multilinear commutators
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Multilinear commutators of fractionalintegrals over Morrey spaces withnon-doubling measures
Xiangxing Tao and Taotao Zheng
Abstract. In this paper, the authors study the boundedness of multilin-ear fractional integrals on the product Morrey space with non-doublingmeasure, and investigate the Morrey boundedness properties of the multi-linear commutators generated by multilinear fractional integral operatorswith a tuple of RBMO functions.
The classical result of commutator is due to Coifman et al. [1]: if b ∈ BMO andT is a Calderon–Zygmund operator, then the commutator [b, T ] is bounded onLp spaces for 1 < p < ∞. In 2001, Tolsa [2] developed the theory of Calderon-Zygmund operators and their commutators with RBMO functions in the set-ting of non-doubling measures. Later on, Chen and Sawyer [3] modified thedefinition of RBMO to investigate the commutators of the potential operatorsand RBMO functions. Recently, Sawano and Tanaka [4,5] defined the Morreyspaces for non-doubling measures, and obtained the boundedness of the com-mutators generated by fractional integrals with RBMO functions in Morreyspaces. Here we point out that the commutator estimates for non-doublingmeasures play an important role in solving the long-standing open questions,like the Painleve problem.
In recent years, the theory on multilinear integral operators and multilin-ear commutators have attracted much attentions as a rapid developing field in
This work was supported in part by the NNSF of China under Grant #10771110 and#10471069, and sponsored by the NSF of Ningbo City under Grant #2009A610090.
288 X. Tao and T. Zheng NoDEA
harmonic analysis. Hu et al. [6] studied the Lp-boundedness and certain weaktype endpoint estimates for multilinear commutators generated by multilinearCalderon-Zygmund singular integral with RBMO functions. In this paper,we will extend the works of Sawano and Tanaka [4,5,7] and Tao et al. [8] tothe context for multilinear commutators generated by multilinear fractionalintegrals and RBMO functions over the Morrey spaces with non-doublingmeasures.
Throughout this paper μ will be a positive Radon measures on Rd sat-
isfing the growth condition: there exist a constant C > 0 and n ∈ (0, d] suchthat
μ(Q) ≤ Cl(Q)n (1.1)
for any cube Q ∈ Rd with sides parallel to the coordinate axes. Q(x, l(Q)) will
be the cube centered at x with side length l(Q). For r > 0, rQ will denote thecube with the same center as Q and with l(rQ) = rl(Q). The set of all cubesQ ⊂ R
d satisfying μ(Q) > 0 is denoted by Q(μ).Before stating our main results, we fix some notations and define some
terminologies. Given βd ≥ 2d+1 large enough but depending only on the dimen-sion d, we say that a cube Q ⊂ R
d is doubling if μ(2Q) ≤ βdμ(Q). For any fixedcube Q ⊂ R
d, let N ≥ 0 be the smallest integer such that 2NQ is doubling.We denote this cube by ˜Q.
Let 0 ≤ α < n, for two cubes Q ⊂ R in Rd, we always set
K(α)Q,R = 1 +
NQ,R∑
k=1
[
μ(2kQ)l(2kQ)n
]1− αmn
,
where NQ,R is the first positive integer k such that l(2kQ) ≥ l(R). If α = 0,then K
(α)Q,R = KQ,R, The latter was introduced by Tolsa in [2].
Denote by mQf the mean value of f on Q, namely, mQf = 1μ(Q)
∫
Qf(x)dμ.
Let η > 1 be a fixed constant, we say that f ∈ L1loc(μ) is in RBMO(μ) if there
exist a constant A such that
1μ(ηQ)
∫
Q
|f(y) − m˜Qf |dμ(y) ≤ A (1.2)
for any cube Q, and
|mQf − mRf | ≤ AKQ,R (1.3)
for any two doubling cubes Q ⊂ R. The minimal constant A is the RBMO(μ)norm of f , and it will be denoted by ‖f‖∗.
We recall the definition of the Morrey space with non-doubling measure.
Definition 1.1. Let k > 1, and 1 ≤ q ≤ p < ∞, the Morrey space Mpq (k, μ) is
defined as
Mpq (k, μ) :=
{
f ∈ Lqloc(μ); ‖f‖Mp
q (k,μ) < ∞}
Vol. 18 (2011) Multilinear commutators with non-doubling measures 289
with the norm
‖f‖Mpq (k,μ) := sup
Q∈Q(μ)
μ(kQ)1p − 1
q
(∫
Q
|f |qdμ
)1q
.
As is easily seen, the space Mpq (k, μ) is a Banach space with its norm.
The Morrey space norm reflects local regularity of f more precisely than theLebesgue space norm. It is easy to see from the Holder inequality that Lp(μ) =Mp
p (k, μ) ⊂ Mpq1
(k, μ) ⊂ Mpq2
(k, μ) whenever 1 ≤ q2 ≤ q1 ≤ p < ∞. Moreover,the definition of the spaces is independent of the constant k > 1, and thenorms for different choice of k > 1 are equivalent, see [4,5,7] for details. Wewill denote Mp
q (2, μ) by Mpq (μ).
Denoting by �f = (f1, f2, . . . , fm), we consider in this article the multilin-ear fractional integral operator Iα,m as follows,
In case σ = {1, 2, . . . ,m}, one sees that [−→bσ , Iα,m] is just the commutator
[−→b , Iα,m].
Here we remark that the multilinear commutator [�b, Iα,m] can be equiv-alently rewritten as
[
�b, Iα,m
]
(�f)(x) = [bm, [bm−1, . . . , [b1, Iα,m
m︷ ︸︸ ︷
] · · · ]](�f)(x). (1.5)
The main results in this paper can be stated as follows.
Theorem 1.2. Suppose that 0 < α < mn and 1 < qi ≤ pi < ∞, and let1p = 1
p1+ 1
p2+ · · · + 1
pm− α
n > 0, 1q = 1
q1+ 1
q2+ · · · + 1
qm− α
n > 0, then
‖Iα,m(�f)‖Mpq (μ) ≤ C
m∏
i=1
‖fi‖Mpiqi (μ)
with the constant C independent of �f .
290 X. Tao and T. Zheng NoDEA
Remark 1.3. In the case pi = qi, Theorem 1.2 implies the Lp-boundedness forthe operator Iα,m, precisely,
‖Iα,m(�f)‖Lp(μ) ≤ Cm∏
i=1
‖fi‖Lpi (μ)
with 1p = 1
p1+ 1
p2+ · · · + 1
pm− α
n > 0 for any 1 < pi < ∞.
Theorem 1.4. Let 0 < α < mn and 1 < qi ≤ pi < ∞, and let 1p = 1
p1+ 1
p2+
· · · + 1pm
− αn > 0, 1
q = 1q1
+ 1q2
+ · · · + 1qm
− αn > 0. Suppose ‖μ‖=∞, and
that bi ∈ RBMO(μ) for i = 1, 2, . . . , m. Then, for all σ ⊆ {1, 2, . . . ,m}, thecommutators [�bσ, Iα,m] are bounded from Mp1
q1(μ) × Mp2
q2(μ) × · · · × Mp1
q1(μ) to
Mpq (μ), moreover,
∥
∥
∥
[
�bσ, Iα,m
]
(�f)∥
∥
∥
Mpq (μ)
≤ C∏
j∈σ
‖bj‖∗m∏
i=1
‖fi‖Mpiqi (μ),
and particularly,∥
∥
∥
[
�b, Iα,m
]
(�f)∥
∥
∥
Mpq (μ)
≤ C
m∏
i=1
‖bi‖∗‖fi‖Mpiqi (μ),
where C > 0 is the constant independent of �f , �b and �bσ.
Remark 1.5. In the case m = 1, Theorem 1.2 and 1.4 follow that the fractionalintegral operator Iα and its commutator [b, Iα] are all bounded from Mp1
q1(μ)
to Mpq (μ), which are just Theorem 3.3 of [4] and Theorem 4.6 of [5], so our
theorems contain their conclusions. In [8] the authors obtained the same resultas Theorem 1.2 if μ is the Lebesgue measure.
To show Theorem 1.2 in Sect. 2, we will use a maximal function Mkf(x)and give a pointwise estimate for the multilinear fractional integral Iα,m(�f)(x)in the setting of non-doubling measures. In order to derive Theorem 1.4, we willexploit the decomposition technique for the product of functions and overcomethe difficult brought from the multilinear operators and non-doubling mea-sures. We will establish a pointwise estimate for the sharp maximal functionof the commutator [�b, Iα,m](�f) by using the non-centered maximal operatorM
(α)τ,η , see Lemma 3.4 below. We will prove Theorem 1.4 in Sect. 3.
2. The proof of Theorem 1.2
Before proving the theorem, we recall the following maximal operator,
Mkf(x) = supx∈Q⊂Q(μ)
1μ(kQ)
∫
Q
|f(y)|dμ(y),
and some lemmas which will be used in this paper.
Vol. 18 (2011) Multilinear commutators with non-doubling measures 291
Lemma 2.1. [4] If 1 < q ≤ p < ∞, then there exist a constant C independentof f such that
‖Mkf‖Mpq (μ) ≤ C‖f‖Mp
q (μ).
Lemma 2.2. For 0 < α < mn, 1 < qi ≤ pi < ∞, 1h = 1
p1+ 1
p2+ · · · + 1
pm,
1p = 1
h − αn > 0, we have
|Iα,m�f(x)| ≤ C
(
m∏
i=1
‖fi‖Mpiqi (μ)
)1− hp(
m∏
i=1
Mkfi(x)
)hp
with the absolute constant C independent of �f .
Proof. For x, yi ∈ Rd and x = yi, i = 1, 2, . . . ,m, we can see from the Fubini
Using the Holder inequality and the growth condition (1.1) for the non-doubling measure μ, and letting cube Q = Q(x, 2l), then we have
∫ λ
0
(
m∏
i=1
1ln
∫
|(x−yi)|<l
|fi(yi)|dμ(yi)
)
lα−1dl
≤ C
∫ λ
0
(
m∏
i=1
1μ(kQ)
∫
Q
|fi(yi)|dμ(yi)
)
lα−1dl
≤ C
∫ λ
0
(
m∏
i=1
Mkfi(x)
)
lα−1dl ≤ Cλαm∏
i=1
Mkfi(x).
On the other hand, with the similar ways as above we have
1ln
∫
|(x−yi)|<l
|fi(yi)|dμ(yi) ≤ C
ln
∫
Q
|fi(yi)|dμ(yi)
≤ C
ln
(∫
Q
|fi(yi)|qidμ(yi))
1qi
μ(Q)1− 1qi
≤ Cl− n
pi ‖fi‖Mpiqi (μ)
292 X. Tao and T. Zheng NoDEA
which means that∫ ∞
λ
(
m∏
i=1
1ln
∫
|(x−yi)|<l
|fi(yi)|dμ(yi)
)
lα−1dl
≤ C
∫ ∞
λ
(
m∏
i=1
l− n
pi ‖fi‖Mpiqi (μ)
)
lα−1dl ≤ Cλ− np
m∏
i=1
‖fi‖Mpiqi (μ).
Thus, by (2.1), we obtain that
|Iα,m�f(x)| ≤ C
[
λαm∏
i=1
Mkfi(x) + λ− np
m∏
i=1
‖fi‖Mpiqi (μ)
]
.
This yields the lemma by choosing the suitable positive real number λ. �
Now we give the proof of theorem 1.2. Note 0 < α < mn, 1 < qi ≤ pi <∞, 1
h = 1p1
+ 1p2
+ · · · + 1pm
, and let 1r = 1
q1+ 1
q2+ · · · + 1
qm. By the lemma 2.2,
one can get that
‖Iα,m�f‖Mp
q (μ) ≤ C
(
m∏
i=1
‖fi‖Mpiqi (μ)
)1− hp
× supQ∈Q(μ)
|μ(kQ)| 1p − 1
q
⎧
⎨
⎩
∫
Q
(
m∏
i=1
Mkfi(y)
)qhp
dμ(y)
⎫
⎬
⎭
1q
.
Since prqh = n−rα
n−hα > 1, hence by the Holder inequality and Lemma 2.1 it follows
|μ(kQ)| 1p − 1
q
⎧
⎨
⎩
∫
Q
(
m∏
i=1
Mkfi(y)
)qhp
dμ(y)
⎫
⎬
⎭
1q
≤ C|μ(kQ)| 1p − h
pr
{
∫
Q
(
m∏
i=1
Mkfi(y)
)r
dμ(y)
}hpr
≤ C
⎧
⎨
⎩
|μ(kQ)| 1h − 1
r
{
∫
Q
(
m∏
i=1
Mkfi(y)
)r
dμ(y)
}1r
⎫
⎬
⎭
hp
≤ C
{
m∏
i=1
|μ(kQ)| 1pi
− 1qi
(∫
Q
(Mkfi(y))qi dμ(y))
1qi
}hp
≤ C
(
m∏
i=1
‖Mkfi‖Mpiqi (μ)
)hp
≤ C
(
m∏
i=1
‖fi‖Mpiqi (μ)
)hp
.
Thus the theorem is obtained.
Vol. 18 (2011) Multilinear commutators with non-doubling measures 293
3. The proof of Theorem 1.4
In this section we will use the sharp maximal estimates. Let f be a functionin L1
loc(μ), the sharp maximal function of f is defined by
M �,(α)f(x) = supQ�x
1μ(
32Q
)
∫
Q
|f(y) − m˜Qf |dμ(y)
+ supR⊃Q�xQ,R doubling
|mQf − mRf |K
(α)Q,R
. (3.1)
The non-centered doubling maximal operator is defined by
Nf(x) = supQ�x
Q doubling
1μ(Q)
∫
Q
|f(y)|dμ(y). (3.2)
By the Lebesgue differential theorem, it is easy to see that |f(x)| ≤ Nf(x) forany f ∈ L1
loc(μ) and μ-a.e. x ∈ Rd. Define the non-centered maximal operator,
M (α)τ,η f(x) = sup
Q�x
(
1μ(ηQ)1− τα
n
∫
Q
|f(y)|τdμ(y))
1τ
(3.3)
for η > 1 and τ > 1, where the supremum is taking over all the cubes Qcontaining the point x. Usually we denote M �,(0)(f) simply by M �(f) andM
(0)τ,η (f) by Mτ,η(f).
From the work in [2], we know that Mτ,η is bounded on Lq(μ) for q >τ > 1 and η > 1. Moreover, we have the following lemma.
Lemma 3.1. Let q > τ > 1, η > 1, and 1 < q ≤ p < ∞, then the operator Mτ,η
is bounded on Mpq (μ) and
‖Mτ,η(f)‖Mpq (μ) ≤ C ‖f‖Mp
q (μ)
with the constant C independent of f .
Proof. The lemma could be proved along the similar lines as that ofTheorem 2.3 in [4], i.e. Lemma 2.1 of this paper. Fix a cube Q0 ∈ Q(μ)and put L = l(Q0)
2 , and let f1 = χ η+7η−1 Q0
f and f2 = f − f1, then we have forall y ∈ Q0 that
Mτ,ηf(y) ≤ Mτ,ηf1(y) + Mτ,ηf2(y). (3.4)
Noting for y ∈ Q0,
Mτ,ηf2(y) ≤ supy∈Q∈Q(μ)l(Q)≥ 8L
η−1
{
1μ(ηQ)
∫
Q
|f |τdμ
}1τ
≤ supQ0⊂Q∈Q(μ)
⎧
⎨
⎩
1
μ(
2η1+η Q
)
∫
Q
|f |τdμ
⎫
⎬
⎭
1τ
(3.5)
294 X. Tao and T. Zheng NoDEA
because of the fact that y ∈ Q0
⋂
Q and l(Q) ≥ 8L(η−1) yields Q0 ⊂ 1+η
2 Q.Thus, from the inequalities (3.4) and (3.5), the Lq(μ)-boundedness for Mτ,η,and the Holder inequality we get that
μ
(
2η(η + 7)η2 − 1
Q0
)1p − 1
q(∫
Q0
(Mτ,ηf)qdμ
)1q
≤ μ
(
2η(η + 7)η2 − 1
Q0
)1p − 1
q(∫
Q0
(Mτ,ηf1)qdμ
)1q
+μ (Q0)1p − 1
q
(∫
Q0
(Mτ,ηf2)qdμ
)1q
≤ μ
(
2η(η + 7)η2 − 1
Q0
)1p − 1
q
(
∫
η+7η−1 Q0
|f |qdμ
)1q
+ supQ0⊂Q∈Q(μ)
μ(Q0)1p
μ(
2η1+η Q
)1τ
(∫
Q
|f |τdμ
)1τ
≤ C supQ0⊂Q∈Q(μ)
μ
(
2η
1 + ηQ
)1p − 1
q(∫
Q
|f |qdμ
)1q
which implies the lemma. �
Lemma 3.2. Let 1 < τ < nα , 1 < τ < q ≤ p < ∞, 1 < t ≤ s < ∞, 1
s = 1p − α
n ,qp = t
s , then
‖M (α)τ,η f‖Ms
t (μ) ≤ C‖f‖Mpq (μ)
with the constant C independent of f .
Proof. Noting the condition qp = t
s , it is enough to show the following pointwiseestimates
M (α)τ,η (f)(x) ≤ C
(‖f‖Mpq (μ)
)1− ps (Mτ,ηf(x))
ps . (3.6)
To do this, we fixed the point x such that M(α)τ,η (f)(x) > 0. We choose βx =
(‖f‖M
pq (μ)
Mτ,ηf(x) )p and then write
M (α)τ,η f(x) ≤ sup
x∈Q∈Q(μ)μ(ηQ)≤βx
(
1μ(ηQ)1− τα
n
∫
Q
|f(y)|τdμ(y))
1τ
+ supx∈Q∈Q(μ)μ(ηQ)>βx
(
1μ(ηQ)1− τα
n
∫
Q
|f(y)|τdμ(y))
1τ
≤ (βx)αn sup
x∈Q∈Q(μ)μ(ηQ)≤βx
(
1μ(ηQ)
∫
Q
|f(y)|τdμ(y))
1τ
Vol. 18 (2011) Multilinear commutators with non-doubling measures 295
+ supx∈Q∈Q(μ)μ(ηQ)>βx
μ(ηQ)−( 1p − α
n )‖f‖Mpq (μ)
by using the Holder inequality. Hence we have
M (α)τ,η f(x) ≤ (βx)
αn Mτ,ηf(x) + (βx)− 1
s ‖f‖Mpq (μ),
which yields the desired inequality (3.6). The lemma is proved. �
Lemma 3.3. [5] Suppose that 1 < q ≤ p < ∞, and there exists an increasingsequence of concentric doubling cubes, I0 ⊂ I1 ⊂ · · · ⊂ Ik ⊂ · · ·, such that
limk→∞
mIk(f) = 0, and
∞⋃
k=0
Ik = Rd.
Then there exist a constant C > 0 independent on f such that
‖Nf‖Mpq (μ) ≤ C‖M �,(α)f‖Mp
q (μ).
In order to investigate the multilinear commutators [�b, Iα,m] and [�bσ,Iα,m],we introduce the following decomposition technique. For any
hence we can obtain from the equality (3.7) that[−→
b , Iα,m
]
(�f)(x) = (−1)m+1m∏
j=1
(bj(x) − λj)Iα,m(�f)(x)
+ (−1)mIα,m
(
(�b − �λ)�f)
(x)
+m−1∑
i=1
∑
σ∈Cmi
Cm,i
∏
σj∈σ
(bσj(x) − λσj
)
×[−→bσ′ , Iα,m
]
(�f)(x), (3.8)
where Cm,i are constants depending only on m and i.
296 X. Tao and T. Zheng NoDEA
Remember these notations and the equality (3.8) in mind, we turn toprove the following lemma, which is crucial to us.
Lemma 3.4. Let 0 < α < mn, τ > 1, si > 1, bi ∈ RBMO(μ) and fi ∈ Lqi(μ),i = 1, 2, . . . ,m, then there exist a constant C > 0 independent of bi and fi
such that
M �,(α)([
�b, Iα,m
]
(�f))
(x) ≤ C
[
‖b1‖∗ . . . ‖bm‖∗Mτ, 32(Iα,m(�f))(x)
+m−1∑
i=1
∑
σ∈Cmi
∏
j∈σ
‖bj‖∗Mτ, 32
×([−→
bσ′ , Iα,m
]
(�f))
(x)
+m∏
i=1
‖bi‖∗M( α
m )si,
98
fi(x)
]
. (3.9)
Proof. For simplicity, we denote by R(�b, �f)(x) the quantities on the right handside of the inequality (3.9). Recall the definition of the sharp maximal operatorM �,(α), and use the standard technique, see [3] for example, we only need toprove that, for any x ∈ R
d and a cube Q � x, there is a quantity hQ such that
1μ(
32Q
)
∫
Q
∣
∣
∣[�b, Iα,m](�f)(z) − hQ
∣
∣
∣ dμ(z) ≤ CR(�b, �f)(x) (3.10)
and
|hQ − hR| ≤ C(
KQ,RK(α)Q,R
)m
R(�b, �f)(x) (3.11)
with the absolute constant C independent of �b, �f,Q and R, where R is anydoubling cube with Q ⊂ R. In fact, we take
hQ = (−1)mmQ
(
Iα,m
(
(m˜Q(b1) − b1)f1χRd\ 4
3 Q, . . . , (m˜Q(bm)
−bm)fmχRd\ 4
3 Q
))
,
and clearly
hR = (−1)mmR
(
Iα,m
(
(mR(b1) − b1)f1χRd\ 43 R, . . . , (mR(bm)
−bm)fmχRd\ 4
3 R
))
.
Vol. 18 (2011) Multilinear commutators with non-doubling measures 297
Recall the equality (3.8), for any z ∈ Q, we have that
∣
∣
∣
[−→b , Iα,m
]
(�f)(z) − hQ
∣
∣
∣ ≤∣
∣
∣
∣
∣
∣
m∏
j=1
(bj(x) − m˜Q(bj))Iα,m(�f)(z)
∣
∣
∣
∣
∣
∣
+
∣
∣
∣
∣
∣
∣
m−1∑
i=1
∑
σ∈Cmi
Cm,i
∏
σj∈σ
(bσj(z) − m
˜Q(bσj))
×[−→bσ′ , Iα,m
]
(�f)(z)∣
∣
∣
+∣
∣
∣(−1)mIα,m
(
(�b − m˜Q(�b))�f
)
(z) − hQ
∣
∣
∣
=: I(z) + II(z) + III(z). (3.12)
In order to show the inequality (3.10), we will calculate the integralsfor the three functions above, respectively. Firstly, for τ > 1, by the Holderinequality one sees that
Similarly, for τ > 1, by the Holder inequality, we also deduce that
1μ(
32Q
)
∫
Q
|II(z)|dμ(z)
≤ C
m−1∑
i=1
∑
σ∈Cmi
∏
σj∈σ
‖bσj‖∗Mτ, 3
2
([−→bσ′ , Iα,m
]
(�f))
(x). (3.14)
To estimate the integral related to the function III(z), we split each fi
as fi = f0i + f∞
i , where f0i = fχ 4
3 Q and f∞i = fi − f0
i , this yields
m∏
j=1
fj(yj) =∑
α1,α2,...,αm∈{0,∞}fα11 (y1) . . . fαm
m (ym)
=m∏
j=1
f0j (yj) +
∑
∗fα11 (y1) . . . fαm
m (ym) +m∏
j=1
f∞j (yj), (3.15)
where each term in∑
∗ satisfies that αj1 = αj2 = · · · = αjλ= 0, for some
1 ≤ λ < m and some {j1, j2, . . . , jλ} ⊂ {1, 2, . . . ,m}. So we can decompose
298 X. Tao and T. Zheng NoDEA
the function III(z) further into three parts as follows
III(z) ≤∣
∣
∣Iα,m
(
(b1 − m˜Q(b1))f0
1 , . . . , (bm − m˜Q(bm))f0
m
)
(z)∣
∣
∣
+∑
∗
∣
∣
∣Iα,m
(
(b1 − m˜Q(b1))fα1
1 , . . . , (bm − m˜Q(bm))fαm
m
)
(z)∣
∣
∣
+∣
∣
∣(−1)mIα,m
(
(b1 − m˜Q(b1))f∞
1 , . . . , (bm
−m˜Q(bm))f∞
m
)
(z) − hQ
∣
∣
∣
=: III1(z) + III2(z) + III3(z). (3.16)
For si > 1, we can take 1 < ui < si such that 1v = 1
u1+ 1
u2+ · · · +
1um
− αn > 0 and v > 1. Let 1
ui= 1
si+ 1
tifor each i = 1, 2, . . . ,m, then
1 < ti < ∞. Using Theorem 1.2, we know that the operator Iα,m is boundedfrom Lu1(μ) × Lu2(μ) × · · · × Lum(μ) to Lv(μ). Hence, by this boundednessand the Holder inequality, we have
1μ(
32Q
)
∫
Q
|III1(z)|dμ(z)
≤ μ(Q)1− 1v
μ(
32Q
) ‖Iα,m
(
(b1 − m˜Q(b1))f0
1 , . . . , (bm − m˜Q(bm))f0
m
)
(z)‖Lv(μ)
≤ Cμ
(
32Q
)− 1v
m∏
i=1
‖(bi − m˜Q(bi))f0
i ‖Lui (μ)
≤ Cμ
(
32Q
)− 1v
m∏
i=1
(
∫
( 43 Q)
|fi|sidμ(yi)
)1si
×(
∫
( 43 Q)
|bi − m˜Q(bi)|tidμ(yi)
)1ti
≤ C
m∏
i=1
⎛
⎝
1
μ(
32Q
)1− αsimn
∫
( 43 Q)
|fi|sidμ
⎞
⎠
1si
×(
1μ(
32Q
)
∫
( 43 Q)
|bi − m˜Q(bi)|tidμ
)1ti
≤ C
m∏
i=1
‖bi‖∗M( α
m )si,
98
fi(x). (3.17)
In order to estimate the integral of terms III2(z) and III3(z) over Q, wewill give their point-wise estimates. In fact, for z ∈ Q, since 1 ≤ λ ≤ m − 1 weobserve that
Vol. 18 (2011) Multilinear commutators with non-doubling measures 299
III2(z) =∑
∗
∣
∣
∣Iα,m
(
(b1 − m˜Q(b1))fα1
1 , . . . , (bm − m˜Q(bm))fαm
m
)
(z)∣
∣
∣
≤ C∑
∗
∏
j∈{j1,...,jλ}
∫
43 Q
|bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)
×∫
(Rd\ 43 Q)m−λ
∏
j /∈{j1,...,jλ} |bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)
(
∑
j /∈{j1,...,jλ} |z − yj |)mn−α
≤ C∑
∗μ
(
32Q
)λ− αλmn ∏
j∈{j1,...,jλ}
×⎛
⎝
1
μ(
32Q
)1− αsjmn
∫
43 Q
|fj(yj)|sj dμ(yj)
⎞
⎠
1sj
×(
1μ(
32Q
)
∫
43 Q
|bj(yj) − m˜Q(bj)|sj
′dμ(yj)
)1
sj′
×∞∑
k=1
∫
(2k 43 Q)m−λ
∏
j /∈{j1,...,jλ} |bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)
l(
2k 43Q
)mn−α
≤ C∑
∗
⎛
⎝
∏
j∈{j1,...,jλ}‖bj‖∗M
( αm )
sj , 98
fj(x)
⎞
⎠
∞∑
k=1
2−(k−1)(n− αm )λ
×∫
(2k 43 Q)m−λ
∏
j /∈{j1,...,jλ} |bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)
l(
2k 43Q
)(n− αm )(m−λ)
≤ C∑
∗
⎛
⎝
∏
j∈{j1,...,jλ}‖bj‖∗M
( αm )
sj , 98
fj(x)
⎞
⎠
∞∑
k=1
2−(k−1)(n− αm )λ
×∏
j /∈{j1,...,jλ}
1
l(
2k 43Q
)(n− αm )
∫
(2k 43 Q)
|bj(yj) − m˜Q(bj)||fj(yj)|dμ(yj)
≤ C∑
∗
⎛
⎝
∏
j∈{j1,...,jλ}‖bj‖∗M
( αm )
sj , 98
fj(x)
⎞
⎠
∞∑
k=1
2−(k−1)(n− αm )λ
×∏
j /∈{j1,...,jλ}
⎡
⎣
(
1l(
2k 43Q
)n
∫
(2k 43 Q)
|bj(yj) − m˜Q(bj)|s′
j dμ(yj)
)1
s′j
×⎛
⎝
1
l(
2k 43Q
)(n− αsjm )
∫
(2k 43 Q)
|fj(yj)|sj dμ(yj)
⎞
⎠
1sj
⎤
⎥
⎦
300 X. Tao and T. Zheng NoDEA
≤ C∑
∗
∏
j∈{j1,...,jλ}‖bj‖∗M
( αm )
sj , 98
fj(x)∞∑
k=1
2−k(n− αm )λ
×∏
j /∈{j1,...,jλ}‖bj‖∗(k + 1)M( α
m )sj , 9
8fj(x)
≤ C
m∏
j=1
‖bj‖∗M( α
m )sj , 9
8fj(x),
where we have used the fact (see [2]) that, there is an absolute constant Csuch that, for any b ∈ RBMO, integer k ≥ 0 and cubes Q,
|m˜2k 4
3 Q(b) − m
˜Q(b)| ≤ C‖b‖∗K˜Q,˜2k 4
3 Q≤ C‖b‖∗KQ,2k 4
3 Q ≤ Ck‖b‖∗.(3.18)
On the other hand, for III3(z), we note for any z, y ∈ Q that∣
∣
∣Iα,m
(
(b1 − m˜Q(b1))f∞
1 , . . . , (bm − m˜Q(bm))f∞
m
)
(z)
− Iα,m
(
(b1 − m˜Q(b1))f∞
1 , . . . , (bm − m˜Q(bm))f∞
m
)
(y)∣
∣
∣
≤∫
(Rd\ 43 Q)m
∣
∣
∣
∣
1|(z − y1, . . . , z − ym)|mn−α
− 1|(y − y1, . . . , y − ym)|mn−α
∣
∣
∣
∣
×∣
∣
∣
∣
∣
m∏
i=1
(bi(yi) − m˜Q(bi))f∞
i
∣
∣
∣
∣
∣
dμ(y1)dμ(y2) · · · dμ(ym)
≤ C
∫
(Rd\ 43 Q)m
|z − y|∏mi=1 |(bi(yi) − m
˜Q(bi))f∞i |
|(y − y1, . . . , y − ym)|mn−α+1
×dμ(y1)dμ(y2) · · · dμ(ym)
≤ Cm∏
i=1
∞∑
k=1
∫
2k 43 Q\2k−1 4
3 Q
l(Q)1m
l(
2k 32Q
)n− αm + 1
m
|(bi(yi) − m˜Q(bi))fi|dμ(yi)
≤ C
m∏
i=1
∞∑
k=1
2−k/m
(
1l(2k 3
2Q)n
∫
2k 43 Q
|(bi(yi) − m˜Q(bi))|si
′dμ(yi)
)1
si′
×(
1l(2k 3
2Q)n−(αsi/m)
∫
2k 43 Q
|fi(yi)|sidμ(yi)
)1si
≤ C
m∏
i=1
∞∑
k=1
2−k/m(k + 1)‖bi‖∗M( α
m )si,
98
fi(x) ≤ C
m∏
i=1
‖bi‖∗M( α
m )si,
98
fi(x),
where we have used the fact (3.18) again.Taking the mean over y ∈ Q, we can obtain that
1μ(
32Q
)
∫
Q
(|III2(z)| + |III3(z)|) dμ(z) ≤ C
m∏
i=1
‖bi‖∗M( α
m )si,
98
fi(x). (3.19)
Vol. 18 (2011) Multilinear commutators with non-doubling measures 301
Combing the inequalities (3.12), (3.13), (3.14), (3.16), (3.17) and (3.19),we see from the estimates of I, II, III1, III2 and III3 that the desired inequal-ity (3.10) holds.
Next we turn to estimate the inequality (3.11). For any cubes Q ⊂ Rwith x ∈ Q, where R is doubling. We denote NQ,R + 2 by N , then 2NQ ⊃ 2Qand 2NQ ⊃ 2R. We recall the equality (3.15) and let f0
where we have used the fact that the cubes R and 2NQ are comparable, whichimplies KR,2N Q ≤ C and KR,2k+N Q ≤ C(1 + k). Using the inequality (3.25)above and the identity (3.23), we obtain that, for τ > 1,
A2 ≤ 1μ(R)
∫
R
(|B1(z)| + |B2(z)|) dμ(z)
≤ Cm∑
i=1
∑
σ∈Cmi
⎛
⎝
∏
j∈σ
KQ,R‖bj‖∗
⎞
⎠
×(
1μ(R)
∫
R
∣
∣
∣
[
�bσ′ , Iα,m
]
(�fχRd\2N Q)(z)∣
∣
∣
τ
dμ(z))
1τ
≤ C
m∑
i=1
∑
σ∈Cmi
⎛
⎝
∏
j∈σ
KQ,R‖bj‖∗
⎞
⎠Mτ, 32
([
�bσ′ , Iα,m
]
(�f))
(x)
+C
m∏
j=1
KQ,R‖bj‖∗M( α
m )sj , 9
8fj(x).
The estimates of A3 and A4 is very similar to the one used in the estimateof A2. In fact, repeating the similar procedures used in (3.24) and (3.25) forτ = 1, and noting that KQ,2k+N Q ≤ KQ,R + KR,2k+N Q ≤ C(1 + k) + KQ,R
since 2N−3Q ⊂ 2R ⊂ 2NQ by the definition of N , we can deduce that∑
α1,α2,...,αm∈{0,∞}at least one αi =∞
and one αj =0
∣
∣
∣mQ
[
Iα,m
(
(m˜Q(b1)−b1)fα1
1 , . . . , (m˜Q(bm)−bm)fαm
m
)]∣
∣
∣
+∑
α1,α2,...,αm∈{R,∞}at least one αi =∞
|mR [Iα,m ((mR(b1) − b1)fα11 , . . . ,
× (mR(bm) − bm)fαmm )]|
≤ C
m∏
j=1
KQ,R‖bj‖∗M( α
m )sj , 9
8fj(x). (3.26)
It is left to estimate the term in A3 of the case α1 = α2 = · · · = αm = 0.A small modification is needed to estimate this term. For z ∈ Q and x ∈ Q,one sees
Iα,m
(
(m˜Q(b1) − b1)f0
1 , . . . , (m˜Q(bj) − bj)f0
j , . . . , (m˜Q(bm) − bm)f0
m
)
Vol. 18 (2011) Multilinear commutators with non-doubling measures 305
≤ CN∑
k=1
m∏
j=1
1
l(
2k 43Q
)n− αm
∫
2k 43 Q
|(m˜Q(bj) − bj(yj))fj(yj)|dμ(yj)
≤ CN∑
k=1
m∏
j=1
1
l(
2k 43Q
)n− αm
(
∫
2k 43 Q
|(m˜Q(bj) − bj)|s′
j dμ(yj)
)1
s′j
×(
∫
2k 43 Q
|fj |sj dμ(yj)
)1
sj
≤ C
N∑
k=1
m∏
j=1
μ(
2k 32Q
)1− αmn
l(
2k 43Q
)n− αm
(
1 + KQ, ˜Q + KQ,2k 43 Q
)
‖bj‖∗M( α
m )sj , 9
8fj(x)
≤ C(KQ,RKαQ,R)m
m∏
j=1
‖bj‖∗M( α
m )sj , 9
8fj(x).
This and the inequality (3.26) follows
A3 + A4 ≤ C(KQ,RK(α)Q,R)m
m∏
j=1
‖bj‖∗M( α
m )sj , 9
8fj(x).
Moreover, combing the estimates of A1, A2, A3 and A4, we obtain the desiredinequality (3.11).
Finally, let us show how to acquire the inequality (3.9) from the twoinequalities (3.10) and (3.11). Fix the point x and let Q be any cube such thatx ∈ Q, notice KQ, ˜Q ≤ C and K
(α)
Q, ˜Q≤ C, hence we see from the inequalities
(3.10) and (3.11) that
1μ(
32Q
)
∫
Q
∣
∣
∣
[
�b, Iα,m
]
(�f)(z) − m˜Q
([
�b, Iα,m
]
(�f))∣
∣
∣ dμ(z)
≤ 1μ(
32Q
)
∫
Q
∣
∣
∣
[
�b, Iα,m
]
(�f)(z) − hQ
∣
∣
∣ dμ(z) +1
μ(
32Q
)
∫
Q
∣
∣
∣hQ − h˜Q
∣
∣
∣ dμ(z)
+1
μ(
32Q
)
∫
Q
∣
∣
∣m˜Q
[
�b, Iα,m
]
(�f) − h˜Q
∣
∣
∣ dμ(z)
≤ CR(�b, �f)(x). (3.27)
On the other hand, for all doubling cubes Q ⊂ R with x ∈ Q such thatKQ,R ≤ K
(α)Q,R ≤ P0, where P0 is the constant in Lemma 6 in [3], using (3.11),
we have
|hQ − hR| ≤ CK(α)Q,RP 2m−1
0 R(�b, �f)(x), (3.28)
306 X. Tao and T. Zheng NoDEA
and moreover the inequality (3.28) holds for any doubling cubes Q,R withQ ⊂ R. Therefore,
∣
∣
∣mQ
([
�b, Iα,m
]
(�f))
− mR
([
�b, Iα,m
]
(�f))∣
∣
∣
≤∣
∣
∣mQ
([
�b, Iα,m
]
(�f))
− hQ
∣
∣
∣ +∣
∣
∣hR − mR
([
�b, Iα,m
]
(�f))∣
∣
∣
+ |hQ − hR|≤ CK
(α)Q,RR(�b, �f)(x). (3.29)
According to the estimates (3.27), (3.29) and the definition of the sharp max-imal function, we deduce the inequality (3.9) and so finish the proof of thelemma. �
Corollary 3.5. Using the same notations in Lemma 3.4, and letting σ ⊂{1, 2, . . . ,m}, we have
M �,(α)([
�bσ, Iα,m
]
(�f))
(x) ≤ C
⎡
⎣
∏
j∈σ
‖bj‖∗Mτ, 32(Iα,m(�f))(x)
+∑
σ1∪σ2=σσ1 =∅,σ2 =∅
∏
j∈σ1
‖bj‖∗Mτ, 32
([−→bσ2 , Iα,m
]
(�f))
(x)
+∏
j∈σ
‖bj‖∗m∏
j=1
M( α
m )sj , 9
8fj(x)
⎤
⎦ (3.30)
and particular,
M �,(α)([bi, Iα,m](�f))(x) ≤ C‖bi‖∗
⎡
⎣Mτ, 32(Iα,m(�f))(x) +
m∏
j=1
M( α
m )sj , 9
8fj(x)
⎤
⎦ .
(3.31)
Proof. The proof of (3.30) and (3.31) is very similar to (3.9), so we omit thedetails. �
Now we are ready to give the proof of theorem 1.4. Using Lemma 3.2,Lemma 3.3, Corollary 3.5 and Theorem 1.2, we get that
∥
∥
∥[bi, Iα,m](�f)∥
∥
∥
Mpq (μ)
≤ C∥
∥
∥N([bi, Iα,m](�f))∥
∥
∥
Mpq (μ)
≤ C∥
∥
∥M �,(α)([bi, Iα,m](�f))∥
∥
∥
Mpq (μ)
≤ C‖bi‖∗
∥
∥
∥
∥
∥
Mτ, 32(Iα,m(�f))(x) +
m∏
i=1
M( α
m )si,
98
fi
∥
∥
∥
∥
∥
Mpq (μ)
≤ C‖bi‖∗∥
∥
∥Mτ, 32(Iα,m(�f))
∥
∥
∥
Mpq (μ)
+ C‖bi‖∗
Vol. 18 (2011) Multilinear commutators with non-doubling measures 307
×∥
∥
∥
∥
∥
m∏
i=1
M( α
m )si,
98
fi
∥
∥
∥
∥
∥
Mpq (μ)
≤ C‖bi‖∗∥
∥
∥Iα,m(�f)∥
∥
∥
Mpq (μ)
+ C‖bi‖∗m∏
i=1
‖fi‖Mpiqi (μ)
≤ C‖bi‖∗m∏
j=1
‖fj‖Mpjqj (μ)
.
Similarly, applying the inequality (3.30) in Corollary 3.5, for σ ⊆ {1, 2, . . . ,m},we have
∥
∥
∥[�bσ, Iα,m](�f)∥
∥
∥
Mpq (μ)
≤ C‖N([�bσ, Iα,m](�f))‖Mpq (μ)
≤ C‖M �,(α)[�bσ, Iα,m](�f)‖Mpq (μ)
≤ C
∥
∥
∥
∥
∥
∥
∏
j∈σ
‖bj‖∗Mτ, 32(Iα,m(�f))
+∑
σ1∪σ2=σσ1 =∅,σ2 =∅
∏
j∈σ1
‖bj‖∗Mτ, 32
([−→bσ2 , Iα,m
]
(�f))
+∏
j∈σ
‖bj‖∗m∏
j=1
M( α
m )sj , 9
8fj
∥
∥
∥
∥
∥
∥
Mpq (μ)
≤ C∏
j∈σ
‖bj‖∗∥
∥
∥Mτ, 32(Iα,m(�f))
∥
∥
∥
Mpq (μ)
+C∑
σ1∪σ2=σσ1 =∅,σ2 =∅
∏
j∈σ1
‖bj‖∗∥
∥
∥Mτ, 32
([−→bσ2 , Iα,m
]
(�f))∥
∥
∥
Mpq (μ)
+C∏
j∈σ
‖bj‖∗
∥
∥
∥
∥
∥
∥
m∏
j=1
M( α
m )sj , 9
8fj
∥
∥
∥
∥
∥
∥
Mpq (μ)
≤ C∏
j∈σ
‖bj‖∗m∏
j=1
‖fj‖Mpjqj (μ)
+C∑
σ1∪σ2=σσ1 =∅,σ2 =∅
∏
j∈σ1
‖bj‖∗∥
∥
∥Mτ, 32
([−→bσ2 , Iα,m
]
(�f))∥
∥
∥
Mpq (μ)
≤ C∏
j∈σ
‖bj‖∗m∏
j=1
‖fj‖Mpjqj (μ)
+C∑
σ1∪σ2=σσ1 =∅,σ2 =∅
∏
j∈σ1
‖bj‖∗∥
∥
∥
[−→bσ2 , Iα,m
]
(�f)∥
∥
∥
Mpq (μ)
308 X. Tao and T. Zheng NoDEA
where σ1 and σ2 are two nonempty subsets of σ and σ1 ∩ σ2 = ∅. Hence, wecan make use of induction on σ ⊆ {1, 2, . . . ,m} to get that
∥
∥
∥
[
�bσ, Iα,m
]
(�f)∥
∥
∥
Mpq (μ)
≤ C∏
j∈σ
‖bj‖∗m∏
i=1
‖fi‖Mpiqi (μ).
This completes the proof of Theorem 1.4.
Acknowledgments
The authors thank the anonymous referee for reading the paper carefully andgiving several useful suggestions.
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X. Tao, T. ZhengDepartment of Mathematics, Zhejiang University of Science and Technology,Hangzhou 310023, People‘s Republic of Chinae-mail: [email protected]