Research Article | Open Access
Multilinear Singular Integral Operators on Generalized Weighted Morrey Spaces
The purpose of this paper is to discuss the boundedness properties of multilinear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces.
1. Introduction and Results
Multilinear Calderón-Zygmund theory is a natural generalization of the linear case. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in  and was later systematically studied by Grafakos and Torres in [2–4]. In 2009, the weighted estimates of multilinear Calderón-Zygmund singular integral operator and its commutator were established in  by Lerner et al. In 2013, the results of  were extended to the weighted Morrey space (see [6, 7]). In this paper, we will discuss the boundedness properties of multilinear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces.
Let be the -dimensional Euclidean space, and let be the -fold product space . We denote by the space of all Schwartz functions on and denote by its dual space of the set of all tempered distributions on .
We say that a locally integrable function defined away from the diagonal in is a kernel in the class - if it satisfies the size estimate for some and all with for some , . Moreover, assume that for some and all it satisfies the smoothness estimates whenever
Let be a multilinear operator initially defined on the -fold product of Schwartz spaces, and, by taking values into the space of tempered distributions, Following , we say that is an -linear Calderón-Zygmund operator if, for some and with , it extends to a bounded multilinear operator from into and if there exists a kernel function in the class -, defined away from the diagonal in , satisfying for all and are functions with compact support.
Theorem 1 (see ). Let and let be an -linear Calderón-Zygmund operator. Suppose that , satisfies the condition, and .(1)If , , then (2)If , , and at least one of the , then
The purpose of this paper is to discuss the boundedness properties of -linear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces .
The classical Morrey spaces were originally introduced by Morrey in  to study the local behavior of solutions to the second-order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the readers to [9–11]. Moreover, various Morrey spaces are defined in the process of study. Mizuhara  introduced the generalized Morrey space ; Komori and Shirai  defined the weighted Morrey spaces ; Guliyev  gave a concept of generalized weighted Morrey space which could be viewed as extension of both and . The boundedness of some operators on these Morrey spaces can be seen in [12–16].
Our first result can be formulated as follows.
Theorem 2. Let , let be an -linear Calderón-Zygmund operator, and let , . Suppose that , satisfies the condition with , and satisfies the condition (1)If , , then (2)If , , and at least one of the , then where and .
Given a locally integrable vector function , the commutator of and the -linear Calderón-Zygmund operator , denoted here by , was introduced in  and is defined via where And the iterated commutator was introduced in  and is defined by
To clarify the notation, if is associated in the usual way with a Calderón-Zygmund kernel , then, at a formal level,
For is an -linear Calderón-Zygmund operator, , and with and , Lerner et al.  proved that bounded from to and Pérez et al.  extended the result to . Our second result is to extend their results to the generalized weighted Morrey space.
Theorem 3. Let and let be an -linear Calderón-Zygmund operator. Suppose that , with , satisfies the condition with , and satisfies the condition If,, then there exist constants independent of such that where and .
Corollary 5 (see ). Let and let be an -linear Calderón-Zygmund operator. Suppose that , , satisfies the condition with , and .(1)If , , then (2)If , , and at least one of the , then
Corollary 6. Let and let be an -linear Calderón-Zygmund operator. Suppose that , , satisfies the condition with , and . If , , then there exist constants independent of such that
2. Definitions and Preliminaries
A weight is a nonnegative, locally integrable function on . Let denote the ball with the center and radius . For any ball and , denotes the ball concentric with whose radius is times as long. For a given weight function and a measurable set , we also denote the Lebesgue measure of by and set the weighted measure .
Let , let be a positive measurable function on , and let be a nonnegative measurable function on . Following , we denote by the generalized weighted Morrey space and the space of all functions with finite norm where Furthermore, by we denote the weak generalized weighted Morrey space of all functions for which where (1)If and with , then is the classical Morrey space.(2)If , then is the weighted Morrey space.(3)If , then is the two weighted Morrey space.(4)If , then is the generalized Morrey space.(5)If , then .
A weight is said to belong to , for , if there exists a constant such that, for every ball , where is the dual of such that . The class is defined by replacing the above inequality with
A weight is said to belong to if there are positive numbers and so that for all balls and all measurable . It is well known that
The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal function in .
Now, let us recall the definition of multiple weights. For exponents , we write . Let , and let with , . Given , set . We say that satisfies the condition if it satisfies When , is understood as .
Lemma 8 (see ). Let , and let . Then, if and only if where and the condition in the case is understood as .
Lemma 9 (see ). Let , and let with . Assume that and . Then, for any ball , there exists a constant such that
Let us recall the definition and some properties of . Following , a locally integrable function is said to be in if where .
Lemma 10 (see ). Suppose that and . Then, for any , one has
Lemma 11 (see ). Let , , and . Then, where is independent of , , , and .
Lemma 12. Suppose that and . Then, for any and , one has
3. Proof of Theorem 2
We first prove the following conclusions.
Theorem 13. Let , let be an -linear Calderón-Zygmund operator, and let satisfy the condition with . If and with , then, for any , If , , and with , then, for any ,
Proof. We represent as , where , , and denotes the characteristic function of . Then,
where each term of contains at least one . Since is an -linear operator, then
Then, by Theorem 1, if , , we get
If , , then
Applying Hölder’s inequality, for , , we have
Thus, for ,
From (35) and Lemma 9, we get
Then, for , ,
This gives that and are majored by
For the other term, let us first consider the case when . By the size condition (1), for any , we obtain Applying Hölder’s inequality, it can be found that is less than Hence, Substituting (50) into above, we obtain
Using Hölder’s inequality, From (56) and (57), we know that and are not greater than (52) for , .
Now, we consider the case where exactly of the are for some . We only give the arguments for one of the cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Applying the size condition again, we deduce that, for any , Similar to the estimates for , we get Then, and are all less than Combining the above estimates, we complete the proof of Theorem 13.
Now, we can give the proof of Theorem 2. From the definition of generalized Morrey space, the norm of on equals By (37), we get Combining (42) and (62), Since , then, by (31) and the fact that are all nondecreasing function of , we get Then, By condition (8), we get Combining (61), (63), and (66), the proof of the first part of Theorem 2 is completed.
4. Proof of Theorem 3
Theorem 14. Let , let be an -linear Calderón-Zygmund operator, and let satisfy the condition with . If and with , then, for any ,
Proof. We will give the proof for because the proof for is very similar but easier. Moreover, for simplicity of the expansion, we only present the case .
We represent as , where , , and denotes the characteristic function of . Then, Since