Commutators for multilinear singular integrals on weighted Morrey spaces
© Wang and Jiang; licensee Springer. 2014
Received: 6 September 2013
Accepted: 18 February 2014
Published: 4 March 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 .
there exist (), it extends to a bounded multilinear operator from to , where ;
- (S2)there exists a function K, defined off the diagonal in , satisfying(1)
for some and all , whenever .
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.
The following endpoint estimate will also be proved.
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)
We denote .
Definition 2.2 (Multiple weights)
where the condition in the case is understood as .
Definition 2.3 (Weighted Morrey spaces)
Definition 2.4 (Maximal function)
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 ()
for any cube.
Lemma 2.2 ()
where , .
Lemma 2.3 ()
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 )
Lemma 2.5 (Proposition 3.13 )
Lemma 2.6 (Theorem 3.2 )
for all bounded with compact support.
Lemma 2.7 (Theorem 4.1 )
By the above two inequalities, Pérez and Trujillo-González obtained the following results.
Lemma 2.8 (Theorem 1.1 )
Lemma 2.9 (Theorem 1.2 )
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.
Combining all estimates, we complete the proof of Theorem 1.1. □
We now turn to the proof of Theorem 1.2.
Thus we complete the proof of Theorem 1.2. □
4 A problem
In , Fujita and Tomita obtained the following theorem.
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.
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.
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