`Journal of Function Spaces and ApplicationsVolume 2013 (2013), Article ID 257537, 7 pageshttp://dx.doi.org/10.1155/2013/257537`
Research Article

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

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 [13] 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 [57]. 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, 1214], 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).

#### References

1. D. R. Adams, “A note on Riesz potentials,” Duke Mathematical Journal, vol. 42, no. 4, pp. 765–778, 1975.
2. S. Lu and D. Yang, “Hardy-Littlewood-Sobolev theorems of fractional integration on Herz-type spaces and its applications,” Canadian Journal of Mathematics, vol. 48, no. 2, pp. 363–380, 1996.
3. S. G. Shi and Z. W. Fu, “Boundedness of sublinear operators with rough kernels on weighted Morrey spaces,” Journal of Function Spaces and Applications, vol. 2013, Article ID 784983, 9 pages, 2013.
4. O. Kováčik and J. Rákosník, “On spaces ${L}^{p\left(x\right)}$ and ${W}^{k,p\left(x\right)}$,” Czechoslovak Mathematical Journal, vol. 41, no. 4, pp. 592–618, 1991.
5. L. Diening and M. Růžička, “Calderón-Zygmund operators on generalized Lebesgue spaces ${L}^{p\left(·\right)}$ and problems related to fluid dynamics,” Journal für die Reine und Angewandte Mathematik, vol. 563, pp. 197–220, 2003.
6. C. Capone, D. Cruz-Uribe, and A. Fiorenza, “The fractional maximal operator and fractional integrals on variable ${L}^{p}$ spaces,” Revista Mathemática Iberoamericana, vol. 23, no. 3, pp. 743–770, 2007.
7. M. Izuki, “Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent,” Rendiconti del Circolo Matematico di Palermo, vol. 59, no. 3, pp. 461–472, 2010.
8. L. Diening, “Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces ${L}^{p\left(·\right)}$ and ${W}^{k,p\left(·\right)}$,” Mathematische Nachrichten, vol. 268, pp. 31–43, 2004.
9. S. Z. Lu, D. C. Yang, and G. E. Hu, Herz Type Spaces and Their Applications, Science Press, Beijing, China, 2008.
10. Z.-W. Fu, Z.-G. Liu, S.-Z. Lu, and H.-B. Wang, “Characterization for commutators of $n$-dimensional fractional Hardy operators,” Science in China A, vol. 50, no. 10, pp. 1418–1426, 2007.
11. S. Lu, Q. Wu, and D. Yang, “Boundedness of commutators on Hardy type spaces,” Science in China A, vol. 45, no. 8, pp. 984–997, 2002.
12. T. Zhou, Commutators of Fractional Integrals on Spaces With Variable Exponent, Dalian Maritime University, 2012.
13. H. B. Wang, Z. W. Fu, and Z. G. Liu, “Higher order commutators of Marcinkiewicz integrals on variable Lebesgue spaces,” Acta Mathematica Scientia A, vol. 32, no. 6, pp. 1092–1101, 2012.
14. H. B. Wang, Function Spaces with Variable Exponent and Related Topics, China University of Mining and Technology, 2012.
15. A. Nekvinda, “Hardy-Littlewood maximal operator on ${L}^{p\left(x\right)}$$\left({ℝ}^{n}\right)$,” Mathematical Inequalities & Applications, vol. 7, no. 2, pp. 255–265, 2004.
16. M. Izuki, “Boundedness of commutators on Herz spaces with variable exponent,” Rendiconti del Circolo Matematico di Palermo, vol. 59, no. 2, pp. 199–213, 2010.