Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 257537, 7 pages

http://dx.doi.org/10.1155/2013/257537

Research Article

Higher Order Commutators of Fractional Integral Operator on the Homogeneous Herz Spaces with Variable Exponent

Liwei Wang,¹ Meng Qu,² and Lisheng Shu²

¹School of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, China

²School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China

Received 28 March 2013; Accepted 20 May 2013

Academic Editor: Dachun Yang

Copyright © 2013 Liwei Wang 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.

By decomposing functions, we establish estimates for higher order commutators generated by fractional integral with BMO functions or the Lipschitz functions on the homogeneous Herz spaces with variable exponent. These estimates extend some known results in the literatures.

1. Introduction

Let be a locally integrable function, , and ; the higher order commutators of fractional integral operator are defined by

Obviously, and . The famous Hardy-Littlewood-Sobolev theorem tells us that the fractional integral operator is a bounded operator from the usual Lebesgue spaces to when and . Also, many generalized results about and the commutator on some function spaces have been studied; see [1–3] for details.

It is well known that the main motivation for studying the spaces with variable exponent arrived in the nonlinear elasticity theory and differential equations with nonstandard growth. Since the fundamental paper [4] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent have been extensively investigated. In the recent twenty years, boundedness of some important operators, for example, the Calderón-Zygmund operators, fractional integrals, and commutators, on has been obtained; see [5–7]. Recently, Diening [8] extended the boundedness of to the Lebesgue spaces with variable exponent. Izuki [7] first introduced the Herz spaces with variable exponent , which is a generalized space of the Herz space ; see [9, 10], and in case of , he obtained the boundedness properties of the commutator . The paper [11] by Lu et al. indicates that the commutator with and with has many different properties. In 2012, Zhou [12] studied the boundedness of on the Herz spaces with variable exponent and proved that the boundedness properties of the commutator also hold in case of . The higher order commutators are recently considered by Wang et al. in the paper [13, 14]; they established the and the Lipschitz estimates for on the Lebesgue spaces with variable exponent . Motivated by [7, 12–14], in this note, we establish the boundedness of the higher order commutators on the Herz spaces with variable exponent.

For brevity, denotes the Lebesgue measure for a measurable set , and denotes the mean value of on . The exponent means the conjugate of , that is, . denotes a positive constant, which may have different values even in the same line. Let us first recall some definitions and notations.

Definition 1. For , the Lipschitz space is the space of functions satisfying

Definition 2. For , the bounded mean oscillation space is the space of functions satisfying

where the supremum is taken over all balls in .

Definition 3. Let be a measurable function.

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

(2)The space with variable exponent is defined by

The Lebesgue space is a Banach space with the Luxemburg norm

We denote

where the Hardy-Littlewood maximal operator is defined by

where .

Proposition 4 (see [15]). If satisfies

then one has .

Let , , and be the characteristic function of the set for . For , we denote if , and .

Definition 5 (see [7]). For and .

(1)The homogeneous Herz spaces are defined by

where

(2)The nonhomogeneous Herz spaces are defined by

where

In this note, we obtain the following results.

Theorem 6. Suppose that , satisfies conditions (9) in Proposition 4. If , , , , and , then the higher order commutators are bounded from to .

Theorem 7. Suppose that , satisfies conditions (9) in Proposition 4. If , , , , and , then the higher order commutators are bounded from to .

Remark A. The previous main results generalize the boundedness of the higher order commutators in [13] to the case of the Herz spaces with variable exponent. If , our conclusions coincide with the corresponding results in [7, 12]. Moreover, the same boundedness also holds for the nonhomogeneous case.

2. Proof of Theorems 6 and 7

To prove our main results, we need the following lemmas.

Lemma 8 (see [4]). Let ; if and , then

where .

Lemma 9 (see [7]). Let ; then for all balls in ,

Lemma 10 (see [7]). Let ; then for all balls in and all measurable subsets , one can take a constant , so that

Lemma 11 (see [8]). Suppose that satisfies conditions (9) in Proposition 4, and ; then

Lemma 12 (see [13]). Suppose that , .

(1)Let , . If satisfies conditions (9) in Proposition 4 and , then

(2)Let , . If satisfies conditions (9) in Proposition 4 and , then

Lemma 13 (see [16]). Let , ; one has

(1);(2)  .

Proof of Theorem 6. Let ; we can write

For , applying the inequality

we obtain

We first estimate . Noting that if , and , then , we get

By Hölder’s inequality, Lemmas 9 and 10, we have

Note that

By Lemmas 8 and 11, we obtain

Combining (24) and (26), we have the estimate

Thus,

If , noting that , by Hölder’s inequality, we have

If , by inequality (21), we have

Next, we estimate . By Lemma 12(2), we obtain

If , by Hölder’s inequality, we have

If , by inequality (21), we have

Combining the estimates for and , the proof of Theorem 6 is completed.

Proof of Theorem 7. Let ; we can write

By inequality (21), we obtain

For , using Hölder’s inequality and Lemma 8, we have

By Lemmas 9, 10, and 13, we have

Note that

By Lemmas 8 and 11, we obtain

Combining (37) and (39), we have the estimate

Thus,

In case of , noting that , by Hölder’s inequality, we have

In case of , by inequality (21), we have

For , by Lemma 12(1), we obtain

If , by Hölder’s inequality, we have

If , by inequality (21), we have

Combining the estimates for and , consequently, we have proved Theorem 7.

Acknowledgments

The authors thank the referees for their valuable comments to the original version of this note. This paper is supported by the NSF of China (no. 11201003); the Natural Science Foundation of Anhui Higher Education Institutions of China (no. KJ2011A138; no. KJ2013B034).

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