Journal of Function Spaces

Volume 2014 (2014), Article ID 162518, 12 pages

http://dx.doi.org/10.1155/2014/162518

Research Article

Commutators of Multilinear Calderón-Zygmund Operator and BMO Functions in Herz-Morrey Spaces with Variable Exponents

Canqin Tang,¹ Qing Wu,¹ and Jingshi Xu²

¹Department of Mathematics, Dalian Maritime University, Dalian 116026, China

²Department of Mathematics, Hainan Normal University, Haikou 571158, China

Received 25 December 2013; Accepted 25 April 2014; Published 25 May 2014

Academic Editor: Yongsheng S. Han

Copyright © 2014 Canqin Tang 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.

Abstract.

We obtain the boundedness of a commutator generated by multilinear Calderón-Zygmund operator and BMO functions in Herz-Morrey spaces with variable exponents.

1. Introduction

In recent decades, variable exponent function spaces have attracted much attention. Since Kováčik and Rákosník [1] clarified fundamental properties of the variable Lebesgue and Sobolev spaces, there are many spaces studied, such as Besov and Trieble-Lizorkin spaces with variable exponents, Hardy spaces with variable exponents, Bessel potential spaces with a variable exponent, and Herz-Morrey spaces with variable exponents; see [2–13]. Recently, multilinear operators and their commutators in variable exponent function spaces are also intensively studied by a significant number of authors, such as multilinear commutators of multilinear singular integral with Lipschitz functions and BMO functions, respectively, in [14–19], multilinear commutators of BMO functions and multilinear singular integral operators with nonsmooth kernels in [20, 21], a vector-estimate of higher order commutators on Herz-Morrey spaces with variable exponent in [22], maximal multilinear commutators and maximal iterated commutators generated by multilinear operators and Lipschitz functions in [23], and weighted estimates for vector-valued commutators of multilinear operators in [24].

Motivated by the above results, in this paper, we will consider the boundedness of a commutator generated by a multilinear Calderón-Zygmund operator and BMO functions on the variable Herz-Morrey spaces.

To state the main result of this paper, we need to recall more notations.

Let be a multilinear singular integral operator which is initially defined on the -fold product of the Schwartz space . Its values are taken in the space of tempered distributions such that, for , ,

where (the space of compactly supported bounded functions). Here, the kernel is a function in away from the diagonal and satisfies the standard estimates

provided that , and, for each ,

provided that , where and are positive constants.

Such kernels are called the linear Calderón-Zygmund kernels and the collection of such functions is denoted by in [25].

Let be as in (1) with an kernel. If, for some , , is bounded from to with , then we say is an linear Calderón-Zygmund operator. Grafakos and Torres in [25] showed that if is an linear Calderón-Zygmund operator, then is bounded from to for any , such that . Then, Grafakos and Torres in [26] obtained weighted norm inequalities for multilinear Calderón-Zygmund operators.

If (the set of all complex-valued locally integrable functions on ), set

where the supremum is taken over all balls in , is the mean of on , and what follows is the Lebesgue measure of measurable set in . A function is called bounded mean oscillation if and is the set of all locally integrable functions on such that .

Let and for . We will consider the commutator , which is defined for suitable functions by

where , denotes a subset of ,   is the complement of in , denotes the number of elements of , , , when , , otherwise, .

Definition 1. Let be a measurable function.

(i)The Lebesgue space with variable exponent is defined by

(ii)The space is defined by

where, and in what follows, denotes the characteristic function of a measurable set .

is a Banach function space when equipped with the norm

Letting , we denote

The set consists of all satisfying and ; consists of all satisfying and . can be similarly defined as mentioned above for . means the conjugate exponent of that means .

Let . Then the standard Hardy-Littlewood maximal function of is defined by

where is a ball and is the volume of . Let be the set of such that is bounded on . For more on , see [27, 28].

Definition 2. We say that a function is locally log-Hölder continuous if there exists a constant such that

If

then we say that is log-Hölder continuous at the origin. If there exists such that

then we say that is log-Hölder continuous at the infinity. If is both locally log-Hölder continuous and log-Hölder continuous at the infinity, then we say is global log-Hölder continuous.

We denote by and the class of all exponents which is log-Hölder continuous at the origin and at the infinity, respectively.

Next, we recall the definitions of Herz spaces and Herz-Morrey spaces with variable exponents. We use the following notations. For each , we define

Definition 3 (see [29, Definition  2]). Let , , and with . The homogeneous Herz space is defined as the set of all such that

with the usual modifications when .

Definition 4 (see [30, Definition  2]). Let , , and with . The Herz-Morrey space with variable exponents is defined by

where

If is a constant, then was defined in [30]. If , then . If both and are constant and , then is the classical Herz space in [31].

Lemma 5 (see [30, Lemma  1 and (10)]). If , then there exist constants , and such that, for all balls in and all measurable subsets ,

There is a position to state our result.

Theorem 6. Let and , satisfy , , for some , . Let , and for with

where are the constants appearing in (18). Suppose that , , and . Then

with the constant independent of .

Remark 7. Let , and then the commutator is bounded from the product of variable exponents Herz spaces to variable exponents Herz space when .

Finally, we point out that denotes a positive constant which may be different at different occurrences.

2. Proof of the Main Result

To give our proof, we need some lemmas.

Lemma 8 (see [32, Proposition  2]). If , , , and , then

Lemma 9 (see [30, Lemma  2]). If , then there exists a constant such that, for all balls in ,

Lemma 10 10 (see [29, Lemma  3]). Let be a positive integer. Then one has that, for all and all , with ,

In fact, if , then, from inequalities (23), for all balls and , with , we have

Lemma 11 (see [1, Theorem  2.1] ). If , then, for all and all , one has

where .

Lemma 12 (see [14, Theorem 2.3]). Let such that . Then there exists a constant independent of the functions and such that

holds for every and .

Lemma 13 13 (see [14, Corollary  2.2]). Let be a 2-linear Calderón-Zygmund operator and let and be functions. Let such that there exists with . If such that , then there exists a constant independent of functions for such that

Proof of Theorem 6. Although our method is suitable for any multilinear operator, for simplicity, we only consider 2-linear operators. Let and be BMO functions. Since the set of all bounded compactly supported functions is dense in Herz-Morrey spaces with variable exponents, we let and be bounded compactly supported functions; then, for , we write

We write

Then, for each , if , from Lemma 11, we obtain

and by using inequalities (18), (22), and (24), it follows that

Similarly, if , we have

As for the case , we get

By Lemma 8, we know

where

Since the estimate of is essentially similar to that of , so it suffices to prove that is bounded in Herz-Morrey spaces with variable exponents. It is easy to see that

where

Using the symmetry of and , we only need to estimate , , , , , because the estimates of , , and are analogous to those of , , and , respectively. In what follows, we divide it into 6 steps.

Step 1. To estimate the term of , we note that for . Thus, for , ,

Then, for , we get

Therefore,

By Lemma 12 and inequality (31), we obtain

Since , it follows that

where

Here, we denote for short. It follows from condition (19) that . To continue calculations for (43), we consider the two cases and .

If , since, for , ,

then

If , by the Hölder inequality, we obtain

Thus, for any ,

Step 2. To estimate , for , , and , , we have

Since , it follows from inequality (31) that

For , combining the above term and inequalities (32) and (33), we have

Here, we used and in the first inequality.

Obviously,

Therefore, we only need to estimate :

Step 3. To estimate , for , , , and ,, then we have

Thus, for , we get

From inequalities (31) and (32), we obtain

Consequently, it follows that

Note that

so we only compute . From inequality (19) and , we obtain

Therefore,

Step 4. It turns to estimate the term . Applying the Hölder inequality and Lemma 13, we have

Step 5. Now it goes to the estimate of .

It is clear that, for and and ,

Therefore,

Here, the estimate of is similar to that of and .

Step 6. Finally, we will finish the estimation of the last term . Note that and for , , : we get

Applying the Hölder inequality to the last integral, we obtain

Thus,

Here, the estimate of , , is similar to that of .

Combing all estimates for , together, we get

This finishes the proof of Theorem 6.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referee for his suggestion. The first author was supported by the National Natural Science Foundation of China (Grant no. 11021043) and the Fundamental Research Funds for the Central Universities. The third author was supported by the National Natural Science Foundation of China (Grants nos. 11071064 and 11361020) and the Natural Science Foundation of Hainan Province (no. 113004).

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