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Journal of Function Spaces
Volume 2014, Article ID 720875, 7 pages
http://dx.doi.org/10.1155/2014/720875
Research Article

Lipschitz Estimates for One-Sided Cohen’s Commutators on Weighted One-Sided Triebel-Lizorkin Spaces

1School of Sciences, China University of Mining & Technology, Beijing 100083, China
2Department of Basic Curriculum, North China Institute of Science and Technology, Hebei 065201, China
3Department of Mathematics, Linyi University, Linyi 276005, China
4School of Sciences, Shandong Normal University, Jinan 250014, China

Received 19 September 2014; Accepted 20 November 2014; Published 8 December 2014

Academic Editor: Guozhen Lu

Copyright © 2014 Xiao Jin Zhang and Xian Ming Hou. 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 introduce one-sided Cohen’s commutators of singular integral operators and fractional integral operators, respectively. Using the extrapolation of one-sided weights, we establish the boundedness of these operators from weighted Lebesgue spaces to weighted one-sided Triebel-Lizorkin spaces.

1. Introduction

The one-sided commutators considered in this paper are related to the commutators studied by Calderón in [1]. Cohen [2] defined the Cohen type commutators of Calderón-Zygmund singular integrals (for convenience, we only consider the -dimensional case) by where satisfies certain homogeneity, smoothness, and symmetry conditions. Chen and Lu [3] proved the boundedness of the commutators from Lebesgue spaces to Triebel-Lizorkin spaces for . A function , , if it satisfies Similar to an unbounded function , the functions in are not necessarily bounded either (e.g., ). Therefore, it is also nontrivial to investigate the commutators generated by operators and Lipschitz functions.

In the one-sided case, we will study the weighted boundedness of the commutators from weighted Lebesgue spaces to weighted Triebel-Lizorkin spaces. The one-sided operators were motivated as not only the generalization of the theory of both-sided ones but also the requirement in ergodic theory. Lots of results show that, for a class of smaller operators (one-sided operators) and a class of wider weights (one-sided weights), many results in harmonic analysis still hold; see [414]. However, for one-sided weights, classical reverse Hölder’s inequality does not hold.

A function is called a one-sided Calderón-Zygmund kernel (OCZK) if satisfies with support in or . An example of such a kernel is where denotes the characteristic function of a set . In [15], Aimar et al. introduced the one-sided Calderón-Zygmund singular integrals which are defined by where the kernels are OCZKs.

The study of weights for one-sided operators is motivated by their natural appearance in harmonic analysis, such as the one-sided Hardy-Littlewood maximal operator:

Recently, Sawyer [13] introduced the one-sided classes , , which are defined by when ; also, for , for some constant .

Very recently, Fu and Lu [16] introduced a class of one-sided Triebel-Lizorkin spaces and their weighted version.

Definition 1. For , , and an appropriate weight , the weighted one-sided Triebel-Lizorkin spaces and are defined by

In [16], the authors proved the boundedness for the one-sided commutators (with symbols ) of Calderón-Zygmund singular integral, , and fractional integral, , respectively. and are defined as follows:

Let be locally integrable functions on . Denote by the th order remainder of the Taylor series of at about , precisely:

Cohen’s commutators of one-sided singular integrals are defined by

Obviously, when , . Therefore, the results of this paper are the extension of [16].

Theorem 2. Assume that and . Let , . Then, one gets the following.(i)If , there exists the constant such that (ii)If , there exists the constant such that

Theorem 3. Assume that and . Let , . Then one gets the following.(i)If , there exists the constant such that (ii)If , there exists the constant such that

The other main objects in this paper are one-sided Cohen’s commutators of fractional integral operators, which are defined by Obviously, when , .

Theorem 4. Assume that and . Let , . Then one gets the following.(i)If , there exists the constant such that (ii)If , there exists the constant such that

Theorem 5. Assume that , and . Let , . Then one gets the following.(i)If , there exists the constant such that (ii)If , there exists the constant such that

Throughout this paper the letter will denote a positive constant that may vary from line to line.

2. Estimates for the One-Sided Cohen Type Commutators of Singular Integrals

This section begins with some necessary lemmas.

Lemma 6 (see [17]). If is a function with derivatives of order in , then, for the th remainder of , there is a constant such that

The primary tool in the proof of Theorem 3 is an extrapolation theorem that appeared in [18].

Lemma 7. Let be a sublinear operator defined in satisfying for all . Then, for , holds whenever .

Lemma 8 (see [9]). Suppose that ; then there exists such that, for all , .

Proof of Theorem 2. For convenience, we only prove case . By Lemma 6 and assumption (4), it is easy to prove that where . In the last inequality, we use the boundedness of that appeared in [19].

Proof of Theorem 3. Without loss of generality, we only prove case (i). Let , , and . Write , where . Then
To estimate , we have
Noting the fact that and using Lemma 6 and (4) of kernel , we consider , , and , respectively: where we use the differential mean value theorem for .
Combining the above estimate, we have
Consider the following two sublinear operators defined on : The above inequalities imply that Thus, we will discuss the boundedness of these two operators.
For , by Hölder’s inequality and Theorem 2, we get where , for .
Next, we have By Lemma 7, we obtain for all .
For , set , . Then where , for . Then By Lemma 7, we have for all .
Combining estimates (36) and (39), the proof is completed.

3. Estimates for the One-Sided Commutators of Cohen Type of Fractional Integrals

In order to prove Theorems 4 and 5, we will introduce the one-sided extrapolation lemma.

Lemma 9 (see [18]). Let and let be sublinear operator defined in satisfying for every and ; then, for every , , and , the inequality holds.

Lemma 10 (see [20]). Suppose that ; then and   for all .

Proof of Theorem 4. For convenience, we only give the proof of case (i). Using Lemma 6 and the boundedness of one-sided fractional integral operators that appeared in [19], we get where .

Proof of Theorem 5. Let , . Write , where , . Then By Lemma 6, we have
Consider the following two sublinear operators defined on :
We conclude from (43) and (45) that
If , then ; see [16]. By Lemma 8, there exists , such that . Let and . Using the boundedness of from to , together with Lemma 6, we have In the last inequality, we use the fact for all .
By Lemma 9, there exists such that thus for all .
For , let , . Then where () for all . Then By Lemma 9, the inequlity holds for .
This completes the proof of case (i). For case (ii), we omit the details since they are similar to those of the proof of (i) with instead of .

Conflict of Interests

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

Acknowledgments

The authors cordially thank the referees for their careful reading and helpful comments. This work was partially supported by FRFCU (Grant nos. 3142013027 and 3142014127), North China Institute of Science and Technology (Grant no. HKXJZD201402), and NSF of China (Grant nos. 11271175 and 11171345).

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