Commutators of Singular Integral Operators and Lipschitz Functions on Amalgam Spaces

Abstract

In this paper, we obtain the boundedness of commutators generated by singular integral operators and Lipschitz functions on amalgam spaces and weighted amalgam spaces.

1. Introduction

Let b be a locally integrable function on and be a Calderón–Zygmund singular integral operator, then the commutator generated by b and is defined as

Coifman et al. [1] proved that is bounded on for if and only if . Then, Janson [2] obtained that is bounded from to if and only if , where , , and . Recently, Wang [3] showed that is bounded on weighted amalgam spaces when . On the contrary, in [4], the boundedness for the commutator generated by the singular integral operator and the weighted Lipschitz function on Lebesgue spaces is obtained. The aim of this paper is to establish the boundedness of the commutator generated by the Calderón–Zygmund singular integral operator and the weighted Lipschitz function on amalgam spaces.

Amalgam spaces were first introduced by Wiener [5, 6]. He considered the special cases , and , . A systematic study on general amalgam spaces goes back to the work of Holland [7]. He defined the spaces to be the sets of all locally p-th power summable functions f on such thatwith the usual convention applied when p or q is infinity.

Stewart [8] gave the definition of spaces over a locally compact Abelian group G. By the structure theorem, we may write , where a is a nonnegative integer and is a group which contains a compact open subgroup H (if is compact, we take ; if is discrete and infinite, we take ; otherwise, H is arbitrary but fixed). Let and , where , with and (the collection of t’s being a transversal of in H, i.e., is a coset decomposition of ). Then, in terms of the disjoint union , where J = { for and }, the amalgam space of and on G was defined bywhere

We refer the reader to the work of Fournier and Steward [9] for more information about these spaces.

Fofana introduced in [10] the amalgam spaces as follows:wherehere and is its Lebesgue measure. In [10], the author showed that the space is nontrivial if and only if ; thus, we will always assume throughout the present article.

We recall the following interesting facts about the amalgam spaces :(1)For , we can easily see that , where is Wiener’s amalgam space (see [7])(2)If and , then is just the classical Morrey space (3)If and , then is just the usual Lebesgue space

The first main result of this paper is as follows.

Theorem 1. Let , , , and . Assume that and . If , then is bounded from to .
We also need the definition of the weighted amalgam spaces.

Definition 1 (see [3]). Let and and μ be two weights on . Then, the weighted amalgam spaces are defined aswherewith the usual modification when . Moreover, the spaces defined above are Banach spaces.

Definition 2 (see [3]). Let and u, , and μ be three weights on . Then, the weighted amalgam spaces are defined aswherewith the usual modification when . Moreover, the spaces defined above are Banach spaces.
The second main result is the following:

Theorem 2. Let , , , , , and . Suppose that , is a weight, and , then is bounded from to .
Throughout the paper, we will assume that the Calderón–Zygmund singular integral operator is defined bywhere is a homogeneous function of degree zero, and .

1.1. Some Preliminaries and Notations

We denote by the difference operator. That is,

For a cube and a locally integrable function f, we write

For , the Lipschitz space consists of functions f such that

For and , we say if there is a constant such thatfor any x, . The minimal constant C mentioned above is called the norm of f, denoted by .

Remark 1. When , ; when

Lemma 1 (see [11]). If and , then

Lemma 2 (see [11]). If , then
From Lemma 1, we see thatThe following result was established in [2].

Theorem 3 (see [2]). Let and . Then, the following statements are equivalent:(a)(b) is bounded from to , where A weight is said to belong to the Muckenhoupt class for , , if there exists a constant C such thatfor any ball . The class is defined by replacing the above inequality withfor any ball and (see [12]).
Let and be a weight function, then the weighted Lipschitz space is defined bywhere

Lemma 3 (see [4, 12]). If , , , and , then we have

Lemma 4 (see [4, 12]). If , , , and is a cube, then there exists a constant , independent of b and , such that

Lemma 5 (see [4, 12]). If , , , and is a cube, then there exists a constant , independent of b and , such that

Theorem 4 (see [4]). Let , for and . Then, if and only if is bounded from to .

1.2. Proofs of the Main Results

Proof of Theorem 1. Let , , , , , , and . Fix and and set and . Let and We writeWe will estimate and separately. Using Theorem 3, we haveSince and , we have . Consequently,Now we treat . It is obvious that , when and . We haveDecomposing into a geometrically increasing sequence of concentric balls, we haveFrom these estimates, it follows thatUsing Hölder’s inequality and Lemma 1, we haveAs for , we getIt follows from Hölder’s inequality and Lemma 1 thatFor the last term , we use Hölder’s inequality and Lemma 2 to getSummarizing the above estimates, we conclude thatTaking the -norm and using Minkowski’s inequality, we getThus, by taking the supremum over all , the proof is completed.

Proof of Theorem 2. Let , , , , , , and . Fix and and set and . Let and . We writeNow we will estimate and separately. Using Theorem 4, we haveNoticing that implies that is a doubling measure and that and , and it follows thatSo we haveSimilar to the proof of Theorem 1, we decompose asNotice that implies for all . Using Hölder’s inequality and Lemma 4, we haveFrom the fact thatand , we getUsing the estimatewe conclude thatSince , there exists a constant such thatHence, we haveAs for , we haveIt follows from Hölder’s inequality, the fact that and Lemma 4, then we haveFor the last term, applying Lemma 5 and the property, we haveBy an argument similar to that used in estimating , we getTaking the -norm and using Minkowski’s inequality, we getFinally, by taking the supremum over all , the proof is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors thank Prof. Xinfeng Wu for his very helpful comments. This work was supported in part by the National Natural Science Foundation of China (grant no. 11671397).