On the Multilinear Singular Integrals and Commutators in the Weighted Amalgam Spaces

Liu, Feng; Wu, Huoxiong; Zhang, Daiqing

doi:https://doi.org/10.1155/2014/686017

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Volume 2014 | Article ID 686017 | https://doi.org/10.1155/2014/686017

On the Multilinear Singular Integrals and Commutators in the Weighted Amalgam Spaces

Feng Liu,¹Huoxiong Wu,¹and Daiqing Zhang¹

Academic Editor: Yoshihiro Sawano

Received28 Sept 2013

Revised05 Jan 2014

Accepted05 Jan 2014

Published04 May 2014

Abstract

This paper is concerned with the norm estimates for the multilinear singular integral operators and their commutators formed by BMO functions on the weighted amalgam spaces . Some criterions of boundedness for such operators in are given. As applications, the norm inequalities for the multilinear Calderón-Zygmund operators and multilinear singular integrals with nonsmooth kernels as well as the corresponding commutators on are obtained.

1. Introduction

Let () be the -dimensional Euclidean space equipped with the Euclidean norm and the Lebesgue measure . For , ; the amalgam spaces of and are denoted by the set of all measurable functions , which are locally in and satisfy where for and . We remark that the amalgam spaces were introduced by Fofana in [1] in connection with the study of the continuity of the fractional maximal operator of Hardy-Littlewood and of the Fourier transformation in . In [1], Fofana also considered the subspace of , which consists of measurable functions such that for , and a suitable modification version for or .

By the definitions, it is clear (also see [1]) that , , where , with and , is the classical Morrey space that consists of measurable functions such that

In this paper, we focus on the weighted version of . Precisely, letting be a weight on and , we define the weighted amalgam spaces as the space of all measurable functions satisfying and a suitable modification version for or , where is the weighted Lebesgue space.

It is easy to check that when and , the space is nothing but the weighted Morrey space , which is the set of all measurable functions such that (see [2]) As is well known, the boundedness of the classical operators in the harmonic analysis on the weighted Morrey spaces has extensively been studied (see [2–6] and references therein). In particular, Wang and Yi [6] recently showed that the -linear commutators and the iterated commutators of the -linear Calderón-Zygmund operators are bounded on weighted Morrey spaces.

Based on the above, we feel that it is natural and interesting to study the boundedness of the classical operators in harmonic analysis on the amalgam spaces and the weighted versions. Indeed, a lot of attention has recently been given to this topic (e.g., see [7–10]). Here, we will continue the investigation along this line. The main purpose of this paper is to study the boundedness of the multilinear operators on the weighted amalgam spaces .

Let be a locally integral function defined off the diagonal in and let be an -linear operator associated with the kernel in the following way: where , in with .

For , we define the -linear commutator of denoted by as follows: where each term is the commutator of and in the th entry of ; that is and , where is a smooth function with compact support on . The iterated commutator is defined by

If is associated with a distribution kernel, which coincides with the above function , then we have, at a formal level,

Also, we recall the definitions of the classical Muckenhoupt classes weights and the multilinear conditions for multiple weights.

Definition 1. A weighted on , that is, a positive locally integrable function on , belongs to for if there exists a constant such that The infimum of these constants is called the constant of and denoted by . A weight belongs to the class if there exists a constant such that and the infimum of these constants is called the constant of and is denoted by .

Definition 2. Let with and , , and . Let and . Set We say that satisfies the condition if where for .

Obviously, for , is the classical Muckenhoupt classes condition. It is not difficult to check that for (see [11]),

which implies that something more general happens for the classes. Also, the authors in [11] showed that the conditions are the largest classes of weights because all -linear Calderón-Zygmund operators are bounded on the weighted Lebesgue spaces.

To state our main results, we still need to recall and introduce some notations. For fixed and , we set . For any , let and be the characteristic function of the set . Given any positive integer and , we denote by the family of all finite subset of of different elements. For any , we also denote the complementary sequence of by given by . We remark that if and only if . Letting for a fixed and , we set and if and if . Now we can formulate our main results as follows.

Theorem 3. Let with and be an -linear operator. Let satisfy , , , and . Assume that for with and . If maps to , then the inequality holds provided that for any ball in , any and , there exist constants and such that for a.e. ,

Theorem 4. Let with , , satisfy , , , and . Assume that for with , , and . If then the inequality holds provided that for any ball in , any , , , and , there exist constants and such that for a.e. ,

Theorem 5. Let with , , satisfy , , , and . Suppose that for with , , and . If then the inequality holds provided that for any ball in , any and , there exist constants and such that for a.e. , where for any , and .

Theorem 6. Let be an -linear operator with kernel satisfying Let satisfy , , , and . Assume that , , and . Then these inequalities (17), (20)-(21), and (24) hold.

Remark 7. We remark that for , Theorems 3–6 are also true, just with the restrictive condition: . Moreover, for , that is, , we can remove the restrictive condition in Theorems 3–6. See also [12, Theorem 3.5] for the unweighted case.

The rest of this paper is organized as follows. In Section 2, we will give the proofs of our main results. Some applications will be given in Section 3. Throughout this paper, the letter , sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables. In what follows, we use the convention and .

2. The Proofs of Main Results

Let us begin with a lemma, which will be used in the proofs of our main results.

Lemma 8 (cf. [6, Lemma 3.1]). Let , and with . Assume that and , then for any ball , there exists a constant such that

Proof of Theorem 3. For fixed , we can write The boundedness of from to , (17) and Hölder’s inequality lead to Note that for any and , there exists a constant that depends only on , , such that Hence, multiplying both sides of (28) by , note that and ; by Lemma 8 and (29) we obtain which combined with the fact that for all leads to Theorem 3 is proved.

Proof of Theorem 4. For fixed , by linearity we can write Invoking (18), (20)-(21) and Hölder’s inequality, we have By a similar argument as in getting (31), we can conclude that which completes the proof of Theorem 4.

Proof of Theorem 5. For fixed , we can write Applying (22), (24) and Hölder’s inequality, we get that By similar arguments as in getting (31) again, we can deduce that This completes the proof of Theorem 5.

Proof of Theorem 6. For fixed , it is easy to check that Since , we have for any and , By Hölder’s inequality, (25) and (38), writing , we have It follows from (39)–(42) that This implies (17) in the case of that and or and .
For , and , we have from (25) and (38) that Since , thus with . By the properties of functions in and Hölder’s inequality, we have for any ball and , It follows from (39) and (45) that for any and , Let for some . We now consider two cases:
Case 1 (). We have which satisfies (20) in the case of that and or and .
Case 2 (). We have which satisfies (20) in the case of that and or .
For , it follows from (25) and (38) that This together with (39)–(41), (45) and Hölder’s inequality leads to which satisfies (21) in the case of that and or and .
For , , , and , we have where For fixed , and , we set Then by (25) and (38), we have It is easy to check that for any . This combining (39)–(41), (45) with (54) yields that This together with (51) implies (24) in the case of that and , or and , or and , and completes the proof of Theorem 6.

3. Applications

3.1. On the Multilinear Calderón-Zygmund Operators

An -linear operator associated with is said to be an -linear Calderón-Zygmund operator if, for some , it extends to a bounded multilinear operator from to , where and the kernel satisfies (25) and the regularity conditions for some and all , whenever . We denote by the collection of all kernels satisfying (25) and (57).

As is well known, the multilinear version of the Calderón-Zygmund theory originated in the works of Coifman and Meyer in the 1970s; see, for example, [13, 14], and it was oriented towards the study of the Calderón commutator. Later on the topic was retaken by several authors, including Christ and Journé [15], Kenig and Stein [16], and Grafakos and Torres [17, 18]. Moreover, commutators of multilinear singular integral operators with BMO functions have been the subject of many recent articles (see [11, 19–21] et al.). The following results, which will be used in the next theorem, follow from [11, 20].

Lemma 9 (cf. [11, 20]). Let with and be an -linear Calderón-Zygmund operator. Let with , , and . Suppose that and satisfies the condition. Then where .

This lemma together with Theorems 3–6 directly leads to the following result.

Theorem 10. Let with and be an -linear Calderón-Zygmund operator. Let satisfy , , , and . Suppose that , , and . Then where depends only on and .

Furthermore, by Remark 7 and Lemma 9, we have the following.

Theorem 11. Let with and be an -linear Calderón-Zygmund operator. Let satisfy , and . Assume that and . Then where depends only on and .

3.2. On the Multilinear Singular Integrals with Nonsmoothness Kernels

Let be a class of integral operators, which play the roles of approximate identities (see [22]). We always assumed that the operators are associated with kernels in the sense that for all and , and the kernels satisfy the following conditions: where is a positive fixed constant and is a positive, bounded, decreasing function satisfying that for some , Recall that the th transpose of the th linear operator is defined via for all in . Notice that the kernel of is related to the kernel of via the identity If an -linear operator maps a product of Banach spaces to another Banach space , then the transpose maps to . Moreover, the norm of and is equal. To maintain uniform notation, we may occasionally denote by and by .

Assumption 12. Assume that for each , there exist operators with kernels that satisfy conditions (62)-(63) with constants , and that for every , there exist kernels such that for all in with . Also assume that there exists a function with and a constant so that for every and every , we have whenever .
We say that is an -linear operator with generalized Calderón-Zygmund kernel if satisfies Assumption 12. We denote by - the set of functions satisfying (25), (66)-(67) with parameters , and . We also say is of class - if has an associated kernel in -.

Assumption 13. Assume that there exist operators with kernels that satisfy conditions (66)-(67) with constants and . Let whenever , and whenever and .
It should be pointed out that the condition (67) is weaker than the condition (57) (see [23, Proposition 2.1]). Similarly, we can verify that Assumption 13 is weaker than the condition (57). These assumptions were introduced by Duong et al. in [23, 24]. An important example for satisfying these assumptions is the th Calderón commutator. For in - with kernel satisfying Assumption 13 and the corresponding commutators and , lots of attention has been given (e.g., see [23–29] et al.). In particular, following from [26, 28], we have the following.

Lemma 14 (cf. [26, 28]). Assume that is a multilinear operator in - with kernel satisfying Assumption 13, . If there exist some and some with , such that maps to , then for , with , , and , where .

Invoking this lemma and Theorems 3–6, one has the following results.

Theorem 15. Let with and be an -linear operator in - with the kernel satisfying Assumption 13. Let satisfy , , , and . Assume that , , and . If there exist some and some with , such that maps to , then where depends only on and .

In particular, by Remark 7 again and Lemma 14, one has the following.

Theorem 16. Let with and be an -linear operator in - with the kernel satisfying Assumption 13. Let satisfy , , and . Assume that , . If there exist some and some with , such that maps to , then where depends only on and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the NNSF of China (nos. 11071200, 11371295).

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Copyright © 2014 Feng Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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