Journal of Function Spaces
Volume 2017 (2017), Article ID 3764142, 9 pages
https://doi.org/10.1155/2017/3764142

Research Article

Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

Wei Wang¹ and Jingshi Xu²

¹School of Sciences, Central South University of Forestry and Technology, Changsha 410004, China
²Department of Mathematics, Hainan Normal University, Haikou 571158, China

Correspondence should be addressed to Jingshi Xu; jingshixu@126.com

Received 10 May 2017; Accepted 18 June 2017; Published 20 July 2017

Academic Editor: Dashan Fan

Copyright © 2017 Wei Wang and Jingshi Xu. 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 give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.

1. Introduction

Let the Calderón-Zygmund singular integral operator be defined by

where is a measurable function on and satisfies the following conditions:

(i) is a homogeneous function of degree zero on ; that is,

(ii) has mean zero on ; that is,

Here is the unit sphere in and is the area measure on it.

For a function (the set of all locally integrable functions on ), let be the corresponding multiplication operator defined by for a measurable function . Then the commutator between and is denoted by

for suitable functions Denote the bounded mean oscillation function space by

here and in the sequel

It is well known that commutators play a very important role in harmonic analysis and PDEs. Indeed, Coifman et al. [1] characterized the -boundedness of , where are the Reisz transforms and Using this characterization, the authors of [1] obtained a decomposition theorem of the real Hardy spaces Uchiyama [2] and Janson [3] showed that the Riesz transform may be replaced by the Calderón-Zygmund singular integral operator as in (1). Coifman et al. generalized the boundedness results of to Hardy spaces and gave important applications to some nonlinear PDEs in [4]. The characterization of -compactness of was obtained by Uchiyama [2]. We remark that the interest in the compactness of in complex analysis is from the connection between the commutators and the Hankel-type operators; see [5]. In recent years, Chen et al. have considered the compactness of commutators in [6–8]. Specially, the results in [2] were generalized to Morrey spaces in [8]. The Morrey space was introduced by Morrey in 1938 and it is connected to certain problems in elliptic PDE [9]. After that the Morrey spaces were found to have many important applications to the Navier-Stokes equations (see [10]), the Schrödinger equations (see [11]), and potential theory (see [12–14]).

During last three decades, the theory of variable function spaces has developed quickly; see [15–40]. We claim that the list is not exhaust. The boundedness in variable function spaces of many classical operators from harmonic analysis has been obtained; see [18, 19, 41–44]. Motivated by these works, we will consider analogous results in [8] to variable exponent situation. The structure of this paper is as follows. In Section 2, we give sufficient conditions for a set to be a precompact set in a variable Morrey space. In Section 3, we obtain the boundedness of singular integrals and their commutator in variable Morrey spaces. In Section 4, we discuss compactness of commutators in variable Morrey spaces. The remainder of this section is some notions.

Let be a measurable subset in with , where as usual is the Lebesgue measure of . Let be a measurable function on with range in The variable exponent modulus is defined for measurable functions on by

denotes the set of measurable functions on such that for some The set becomes a Banach function space when equipped with the norm

These spaces are the so-called variable Lebesgue spaces. Denote by the set of measurable functions on with range in such that

For and for , the variable Morrey space is defined as the set of integrable functions on with the finite norm

where denotes a ball centered at with radius and is the volume of the unit ball in

2. Precompact Sets in Variable Morrey Spaces

In this section, we give a compactness criterion in variable Morrey spaces. We remark here that a compactness criterion for variable exponent Lebesgue spaces was given in [45].

Theorem 1. Let and for Suppose is a subset in satisfying the following conditions:

(i)Norm boundedness uniformly is

(ii)Translation continuity uniformly is

(iii)Uniformly convergence at infinity is

where

Then is a precompact set in

To prove Theorem 1, we need the following two lemmas, which are well known; for example, see [46].

Lemma 2. A set is precompact in a Banach space if and only if it is totally bounded which means for every positive number there is a finite subset of points of such that

where denotes a ball centered at with radius . The set is called an -net of

Lemma 3 (the Ascoli-Arzela theorem). Let be a bounded domain in . A subset of is precompact in if the following two conditions hold:

(i)There exists a constant such that holds for every and (ii)For every , there exists such that for , , and

Now there is a position to prove Theorem 1.

Proof of Theorem 1. Let , we denote the mean of on by

For and

Then by Hölder’s inequality and Fubini’s Theorem, for any and ,

Thus,

Therefore, to prove that is a precompact set, we need only to prove that is precompact for small By Lemma 2, it suffices to show that has finite -net for any To do so, firstly, by Lemma 3, we show that is precompact in for each , where For any , by Hölder’s inequality,

since , and Therefore, are uniformly bounded functions by Condition (i). Then for ,

Thus, by Condition (ii) and Lemma 3, is precompact in
Finally, we verify that has finite -net for each small positive For , there exist and such that and for each

By Lemma 3, there exist such that is a finite -net in in the norm of Below we verify that is a finite -net in in the norm of
To finish the proof, we only need to show that, for , there is a such that for ,

Now, we choose such that

To show (22), we consider into three cases.
Case  1.
If ,

If ,

Case  2. Thus, we have

Case  3. , and Hence,

Here, can be estimated that by Cases and , respectively.
Therefore, has a finite -net in This completes the proof.

3. Boundedness of Singular Integrals and Their Commutators

To consider the boundedness of singular integrals, a fundamental operator is the Hardy-Littlewood maximal operator. Given a function , the maximal function is defined by

where the supremum is taken over all cubes containing . It is well known that is bounded on , However, for any , need not be bounded in Let be the set of such that is bounded on For the set , we refer the reader to [19, 41] for details. If , we will use the following results.

Lemma 4 (see [22]). Let Then there exist depending only on and such that for balls in and all measurable subsets

Lemma 5 (see [22]). Let Then there exists a positive constant such that for balls in ,

where and what follows is the conjugate exponent of , which means

Lemma 6 (see [23, 24]). Suppose , and then there exists a positive constant such that for each

Theorem 7. Let for Suppose is a linear or sublinear operator satisfying

If , , where is as in Lemma 4 and the operator is bounded on , then is also bounded on That means

where the constant is independent of

Proof. Let , pick any , and write , where ,
Firstly, we estimate on By the boundedness of on , we have

Thus,

Hence, we obtain

It remains to estimate on By the size estimate of , we have

Thus, by Lemmas 4 and 5 and Hölder’s inequality, we have

Thus,

This finishes the proof of Theorem 7.

Next we turn to the boundedness of commutators in variable Morrey spaces. Many authors have studied it; see [44], but they considered that it restricts the underlying space with finite measure.

Theorem 8. Suppose is a linear operator satisfying

where is a bounded measurable function. Let , and and If the commutator is bounded on , then is also bounded on

Proof. For any and , let and write as in the proof of Theorem 7. By the -boundedness of , we obtain

For and , we write

where and in what follows, for , is defined by

By the well-known fact that, for any and , (see [47, Proposition  7.1.5(i)]), we obtain

Hence, for , using Lemmas 4 and 5 we have

For , by Hölder’s inequality and Lemma 6, we have

Therefore, as the argument as for , we have

Finally, for by Hölder’s inequality, we have

Thus, as the argument as before, we obtain that

From (45), (47), and (49), we get

Hence,

This finishes the proof of Theorem 8.

Corollary 9. Let Suppose that is a bounded homogeneous function of degree and satisfies conditions (2) and

where

If , then the Calderón-Zygmund singular integral operator defined by (1) and its commutator with are both bounded on

Proof. Corollary 9 is the result of the following lemmas. Indeed, the boundedness of on is the direct result of Theorem 7, Lemmas 12 and 13. For the commutator, if , , then by Corollary  9.2.6 in [47] there exists such that and Then we choose in Lemma 11; for by Lemma 10 we obtain

for bounded functions and compactly supported functions Finally, using the similar argument for Theorem  7.5.6 in [47], we obtain that the last inequality holds for any Thus, by Lemma 13, we obtain that is bounded on Consequently, by Theorem 8, is bounded on

We remark here that the boundedness in variable Lebesgue spaces of commutator has been proved in [44] by another method when is an infinitely differentiable function on

Lemma 10 (see Lemma  2.1 in [48]). Let and (Muckenhoupt weight); then there exists a positive constant such that

for functions such that the left-hand side is finite.

Here we say , if for every cube

For properties of , we refer the reader to [47].

Lemma 11 (see Lemma  2.4.1 in [49]). Let and Then for any , there exists a constant , independent of and , such that

Lemma 12 (see Theorem  2.1.6 in [49]). Suppose that is a bounded homogeneous function of degree and satisfies conditions (1) and (52). If and , then there exists a constant , independent of , such that

Lemma 13 (see Corollary  1.11 in [18]). Given that denotes a family of ordered pairs of nonnegative, measurable functions on , assume that

holds for some , for every and for all Let Then

4. Compactness of Commutators

Now we obtain sufficient conditions for the commutator to be a compact operator on

Theorem 14. Let for and such that is a constant function. Suppose that is a bounded homogeneous function of degree of 0 and satisfies (2) and for some

where denotes the integral modulus of continuity of order of defined by

and is a rotation in and If (the closure of the set of compactly supported infinite differential functions in ), then the commutator is a compact operator on

Lemma 15. Let Suppose that is a bounded homogeneous function of degree of 0 and satisfies (2) and (52). For , let

Then for , , where is independent of and

Proof. Lemma 15 is a direct consequence of Theorem 7. In fact, by Theorem  2.1.8 in [49], given that , if , then

holds uniformly in Using Lemma 13, we can get that is bounded on uniformly in Now all conditions in Theorem 7 are fulfilled.

Lemma 16 (see Lemma  2.2 in [8]). Suppose that , satisfies (2), and , where . Then there exist such that for an and with

Proof of Theorem 14. We will use the method in [8]. Let be the unit ball in . By density, we only need to prove that when , the set is a precompact in By Theorem 1, it is sufficient to show that (11)–(13) hold uniformly in
Notice that Applying Corollary 9, we have

This shows that (11) holds.
Next we show that (13) holds. To do so, we suppose that taken so large that For any , we take such that Below we show that for every and ,  

In fact, for any and every , by Hölder’s inequality we have

Then for every and , by the Minkowski inequality and the choice of , we get

Thus, we get (67), which shows that (13) holds for in uniformly.
Finally, we show that the translation continuity condition (12) holds for the commutator in uniformly. We need to prove that, for any , if is sufficiently small depending only on , then for every

Now for , we write

Since , we have Since  and applying Lemma 15, we get

As for , for every and , using Lemma 16 and the Minkowski inequality, we get

Thus, we have

Regarding , we have by Thus,

By the Minkowski inequality, for every and , we have

Thus,

Finally, by , we have

Using the same argument for , it is easy to check that

From (72), (74), (77), and (79), and taking to be sufficiently small, we can get

Therefore, we show that the translation continuity (12) holds for the commuator in uniformly and this completes the proof of Theorem 14.

We remark that in Theorem 14 the condition is necessary by Theorem  1.2 in [8].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author was supported by the TianYuan Special Funds of the National Natural Science Foundation of China (Grant no. 11426221) and the High Level Introduction of Talent Research Start-Up Fund by Central South University of Forestry and Technology (Grant no. 1040212). The second author is supported by the National Natural Science Foundation of China (Grant no. 11361020).

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