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Journal of Function Spaces
Volume 2014, Article ID 430365, 9 pages
http://dx.doi.org/10.1155/2014/430365
Research Article

Marcinkiewicz Integral Operators and Commutators on Herz Spaces with Variable Exponents

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

Received 26 July 2014; Accepted 21 September 2014; Published 15 October 2014

Academic Editor: Dashan Fan

Copyright © 2014 Liwei Wang. 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

Our aim in this paper is to give the boundedness of the Marcinkiewicz integral on Herz spaces and , where the two main indices are variable. Meanwhile, we consider the boundedness of the higher order commutator generated by and a function in BMO on these spaces.

1. Introduction

Let be the unit sphere in    equipped with the normalized Lebesgue measure . Suppose that is homogeneous of degree zero on and has mean zero on , that is, Then the Marcinkiewicz integral in higher dimension is defined by where

Denote by the set of all positive integer numbers. Let and ; the higher order commutator is defined by where

Stein [1] defined the operator and proved that if   , then is of type    and of weak type . Benedek et al. [2] showed that is of type    with . Ding et al. [3] improved the previous results to the case of , where denotes the Hardy space on . Obviously, , which was defined by Torchinsky and Wang in [4]; moreover, they proved that if   , then is bounded on   . Ding et al. [5] weakened the smoothness of the kernel to a rough kernel and showed that if   , then is of type   . Ding et al. [6] established the weighted weak log type estimates for when   . Recently, Zhang [7] improved the previous result and proved that enjoys the same weighted weak log type estimates when the kernel satisfies a kind of Dini’s conditions. For further details on recent developments on this field, we refer the readers to [8, 9] and references therein.

Function spaces with variable exponents were intensively studied during the past 20 years, due to their applications to PDE with nonstandard growth conditions and so on; we mention [10, 11], for instance. Since the fundamental paper [12] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent have attracted a great attention and many interesting results have been obtained; see [1315]. Izuki [16, 17] defined the Herz spaces and with variable exponent but fixed and . Wang et al. [18, 19] obtained the boundedness of and on and . Almeida and Drihem [20] established the boundedness of a wide class of sublinear operators, which includes maximal, potential, and Calderón-Zygmund operators, on Herz spaces and , where the two main exponents and are both variable. In this paper we will give boundedness results for and on Herz spaces and .

For brevity, denotes the Lebesgue measure for a measurable set . denotes the integral average of on , that is, . stands for the conjugate exponent . . denotes a positive constant, which may have different values even in the same line. means that , and means that .

2. Preliminaries and Main Results

Let with , and let be a measurable function. Let us first recall some definitions and notations.

Definition 1. The Lebesgue space with variable exponent is defined by

This is a Banach space with the Luxemburg norm

Let ; the Hardy-Littlewood maximal operator is defined by

Denote

Let , , and be the characteristic function of the set for . For , one denotes if , and . By   , we denote the discrete Lebesgue space equipped by the usual quasinorm.

Definition 2. Let , , and with .(1)The homogeneous Herz space is defined by where (2)The inhomogeneous Herz space is defined by where with the usual modification when .

Remark 3. It is obvious that if , then and . If both and are constants, then and are classical Herz spaces; see [21, 22].

Definition 4. A function is called log-Hölder continuous at the origin, if there exists a constant such that for all . If, for some and , there holds for all , then is called log-Hölder continuous at infinity.
Let one denote for sequences of measurable functions (with the usual modification when ).

Proposition 5 (see [20]). Let , , and . If is log-Hölder continuous both at the origin and at infinity, then

Before stating the main results of this paper, we introduce some key lemmas that will be used later.

Lemma 6 (generalized Hölder’s inequality [12]). Let ; if and , then where .

We remark that the following Lemmas 79 were shown in Izuki [17, 23], and Lemma 10 was considered by Wang et al. in [18].

Lemma 7. Let ; then one has, for all balls in ,

Lemma 8. Let ; then one has, for all balls in and all measurable subsets , where and are constants with , .

Lemma 9. Let , , and   ; then one has

Lemma 10. Let   , , and ; then one has

Our results in this paper can be stated as follows.

Theorem 11. Let   , , and . And let be log-Hölder continuous both at the origin and at infinity, such that , where , are the constants appearing in Lemma 8; then the operator is bounded on and .

Theorem 12. Let   , , , and . And let be log-Hölder continuous both at the origin and at infinity, such that , where , are the constants appearing in Lemma 8; then the higher order commutator is bounded on and .

Remark 13. If is constant, then the statements corresponding to Theorems 11 and 12 can be found in [19, 24]. We consider only in Section 3. The arguments are similar in the case .

3. Proofs of the Theorems

In this section, we prove the boundedness of and on (the same arguments can be used in ); some of our decomposition techniques are similar to those used by Dong and Xu in [25].

Proof of Theorem 11. In view of Proposition 5, we have
Let ; write
Minkowski’s inequality implies that
Similarly we obtain
Thus we get where , , and .
For , Lemma 10 yields
Now we turn to estimate . Observe that if , , and , then and
Since , by Minkowski’s inequality and Lemma 6, we have
Lemmas 7 and 8 lead to
Thus we get
If , since , Hölder’s inequality implies that
If , by the well-known inequality we obtain
Similarly we have
If , since , then we get
If , since , we obtain
Thus, we arrive at
For , observe that if , , and , then and
From Minkowski’s inequality and Lemma 6, it follows that
By Lemmas 7 and 8, we have
Thus we get
Using the same arguments as that for and , we get
Hence the proof of Theorem 11 is completed.

Proof of Theorem 12. We apply Proposition 5 again and get
Let , and write
By Minkowski’s inequality, we have
By the same way, we obtain
Thus, we have where , , and .
For , by Lemma 10, we have
For , observe that if , , and , then
An application of Lemmas 7, 8, and 10 gives
For convenience below we put ; if , then we use Hölder’s inequality and obtain
If , then we get
Similarly, we put ; if , by Hölder’s inequality, we obtain