Journal of Function Spaces

Volume 2017 (2017), Article ID 3764142, 9 pages

https://doi.org/10.1155/2017/3764142

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

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

Correspondence should be addressed to Jingshi Xu

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 bywhere 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 byhere 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 normThese 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 normwhere 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 iswhere **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 byFor 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 thatTo 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*

*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 satisfyingIf , , 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 haveThus,Hence, we obtain It remains to estimate on By the size estimate of , we haveThus, 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 satisfyingwhere 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 obtainHence, for , using Lemmas 4 and 5 we haveFor , by Hölder’s inequality and Lemma 6, we have Therefore, as the argument as for , we haveFinally, for by Hölder’s inequality, we have Thus, as the argument as before, we obtain thatFrom (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) andwhereIf , 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*

*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