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Commutators for multilinear singular integrals on weighted Morrey spaces
Journal of Inequalities and Applications volume 2014, Article number: 109 (2014)
In this paper we study the iterated commutators for multilinear singular integrals on weighted Morrey spaces. A strong type estimate and a weak endpoint estimate for the commutators are obtained. In the last section we present a problem for the multilinear Fourier multiplier with limited smooth condition.
MSC: 42B20, 42B25.
As an important direction of harmonic analysis, the theory of multilinear Calderón-Zygmund singular integral operators has attracted more and more attention, which originated from the work of Coifman and Meyer , and it systematically was studied by Grafakos and Torres [2, 3]. The literature of the standard theory of multilinear Calderón-Zygmund singular integrals is by now quite vast, for example see [2, 4–6]. In 2009, the authors  introduced the new multiple weights and new maximal functions and obtained some weighted estimates for multilinear Calderón-Zygmund singular integrals. They also resolved some problems opened up in  and .
Let and be the Schwartz spaces of all rapidly decreasing functions and tempered distributions, respectively. Having fixed , let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions,
Following , the m-multilinear Calderón-Zygmund operator T satisfies the following conditions:
there exist (), it extends to a bounded multilinear operator from to , where ;
there exists a function K, defined off the diagonal in , satisfying(1)
for all and , where
for some and all , whenever .
We also use some notation following . Given a locally integrable vector function , the commutator of b and the m-linear Calderón-Zygmund operator T, denoted here by , was introduced by Pérez and Torres in  and is defined via
The iterated commutator is defined by
To clarify the notations, if T is associated in the usual way with a Calderón-Zygmund kernel K, then at a formal level
It was shown in  that if , then an m-linear Calderón-Zygmund operator T maps from to , when for all ; and from to , when for all , and . The weighted strong and weak boundedness of T is also true for weights in the class which will be introduced in next section (see Corollary 3.9 ). It was proved in  that is bounded from to for all indices satisfying with and , . The result was extended in  to all . In fact, the authors obtained the weighted -version bounds as follows, for all :
As may be expected from the situation in the linear case, is not bounded from to . However, a sharp weak-type estimate in a very general sense was obtained in , that is, for all ,
The weighted Morrey spaces was introduced by Komori and Shirai . Moreover, they showed that some classical integral operators and corresponding commutators are bounded in weighted Morrey spaces. Some other authors have been interested in this space for sublinear operators, see [12–14]. In , Ye proved two results similar to Pérez and Trujillo-González  for the multilinear commutators of the normal Calderón-Zygmund operators on weighted Morrey spaces. Wang and Yi  considered the multilinear Calderón-Zygmund operators on weighted Morrey spaces and obtained some results similar to weighted Lebesgue spaces.
We will prove the following strong type bound for on weighted Morrey spaces.
Theorem 1.1 Let T be an m-linear Calderón-Zygmund operator; with
and , ; and . Then, for any , there exists a constant C such that
The following endpoint estimate will also be proved.
Theorem 1.2 Let T be an m-linear Calderón-Zygmund operator; , , and . Then, for any and cube Q, there exists a constant C such that
where, and .
Remark 1.1 Here we remark that the above estimate is also valid for .
2 Some definitions and results
In this section, we introduce some definitions and results used later.
Definition 2.1 ( weights)
A weight ω is a nonnegative, locally integrable function on . Let , a weight function ω is said to belong to the class , if there is a constant C such that for any cube Q,
and to the class , if there is a constant C such that for any cube Q,
We denote .
Definition 2.2 (Multiple weights)
For m exponents , we often write p for the number given by and denote by the vector . A multiple weight is said to satisfy the condition if for
when , is understood as . As remarked in , is strictly contained in , moreover, in general does not imply for any j, but instead
where the condition in the case is understood as .
Definition 2.3 (Weighted Morrey spaces)
Let , , and ω be a weight function on . The weighted Morrey space is defined by
The weighted weak Morrey space is defined by
Definition 2.4 (Maximal function)
For and a cube Q in we will consider the average of a function f given by the Luxemburg norm
and the corresponding maximal is naturally defined by
and the multilinear maximal operator is given by
The following pointwise equivalence is very useful:
We say that a weight ω satisfies the doubling condition, simply denoted , if there is a constant such that holds for any cube Q. If with , we know that for all ; then .
Lemma 2.1 ()
Suppose , then there exists a constant such that
for any cube.
Lemma 2.2 ()
If , then for any cube Q, we have
where , .
Lemma 2.3 ()
Suppose , then . Here
From the fact and Lemma 2.3, we deduce that . The following lemma is the multilinear version of the Fefferman-Stein type inequality.
Lemma 2.4 (Theorem 3.12 )
Assume that is a weight in for all , and set . Then
Lemma 2.5 (Proposition 3.13 )
Let . If , , then
Lemma 2.6 (Theorem 3.2 )
Let and let ω be a weight in . Suppose that . Then there exist (independent of b) and such that
for all bounded with compact support.
Lemma 2.7 (Theorem 4.1 )
Let . Then there exists a constant C such that
By the above two inequalities, Pérez and Trujillo-González obtained the following results.
Lemma 2.8 (Theorem 1.1 )
Let T be an m-linear Calderón-Zygmund operator; with
and , ; and . Then there exists a constant C such that
Lemma 2.9 (Theorem 1.2 )
Let T be an m-linear Calderón-Zygmund operator; , and . Then, for any and cube Q, there exists a constant C such that
3 Proofs of theorems
We only present the case for simplicity, but, as the reader will immediately notice, a complicated notation and a similar procedure can be followed to obtain the general case. Our arguments will be standard.
Proof of Theorem 1.1 For any cube Q, we split into , where and , . Then we only need to verify the following inequalities:
From Lemma 2.8 and Lemma 2.2, we get
Since II and III are symmetric we only estimate II. Taking , the operator can be divided into four parts:
Using the size condition (2) of K, Definition 2.2, and Lemma 2.2, we deduce that for any ,
Taking the above estimate together with Hölder’s inequality and Lemma 2.3, we have
where the last inequality is obtained by the property of : there is a constant such that
For , from the size condition (2) of K, the condition, Lemma 2.2, and Lemma 2.3, it follows that
The third inequality can be deduced by the fact that
Hölder’s inequality and Lemma 2.3 tell us
Similarly, we get
The term is estimated in a similar way and we deduce
Finally, we still split into four terms:
Because each term of is completely analogous to , with a small difference, we only estimate :
Combining all estimates, we complete the proof of Theorem 1.1. □
We now turn to the proof of Theorem 1.2.
Proof of Theorem 1.2 By homogeneity, we may assume that and we only need to prove that
To prove the above inequality, we can write
for any cube Q. Employing Lemma 2.9 and Lemma 2.2, we have
From Lemma 2.6 and Lemma 2.4, we deduce that
where the last inequality holds by the (3.10) and (3.11) in . Then from Lemma 2.2 and the fact that , we have
A similar statement follows:
Thus we complete the proof of Theorem 1.2. □
4 A problem
Fix . Let , for some positive integer L, satisfying the following condition:
for all and , where and . The multilinear Fourier multiplier operator is defined by
for all , where . If is an integrable function, then this can also be written as
In , Fujita and Tomita obtained the following theorem.
Theorem 4.1 Let , and for . Assume and for . If satisfies
then is bounded from to , where
where Ψ is the Schwarz function and satisfies
A natural problem is whether the Lebesgue spaces and can be replaced by and . It should be pointed out that the method in this paper may not be suitable to address this problem.
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The authors would like to thank the referee for some very valuable suggestions. This research was supported by NSF of China (no. 11161044, no. 11261055) and by XJUBSCX-2012004.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Wang, S., Jiang, Y. Commutators for multilinear singular integrals on weighted Morrey spaces. J Inequal Appl 2014, 109 (2014). https://doi.org/10.1186/1029-242X-2014-109
- multilinear singular integrals
- multiple weights