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The Scientific World Journal

Volume 2014 (2014), Article ID 790671, 10 pages

http://dx.doi.org/10.1155/2014/790671
Research Article

Littlewood-Paley Operators on Morrey Spaces with Variable Exponent

College of Mathematics and Statistics Science, Northwest Normal University, Lanzhou 730070, China

Received 23 June 2014; Accepted 21 July 2014; Published 7 August 2014

Academic Editor: Jόzef Banas

Copyright © 2014 Shuangping Tao and Lijuan 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

By applying the vector-valued inequalities for the Littlewood-Paley operators and their commutators on Lebesgue spaces with variable exponent, the boundedness of the Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley -functions and -functions, and their commutators generated by BMO functions, is obtained on the Morrey spaces with variable exponent.

1. Introduction and Main Results

Let and satisfy the following:(i); (ii); (iii),where are all positive constants. Denote with and . Given a function , the Lusin area integral of is defined by where denote the usual cone of aperture one As , we denote as .

Now let us turn to the introduction of the other two Littlewood-Paley operators. It is well known that the Littlewood-Paley operators include also the Littlewood-Paley -functions and the Littlewood-Paley -functions besides the Lusin area integrals. The Littlewood-Paley -functions, which can be viewed as a “zero-aperture” version of , and -functions, which can be viewed as an “infinite-aperture” version of , are, respectively, defined by If we take to be the Poisson kernel, then the functions defined above are the classical Littlewood-Paley operators.

Letting ,  , the corresponding -order commutators of Littlewood-Paley operators above generated by a function are defined by where .

The Littlewood-Paley operators are a class of important integral operators. Due to the fact that they play very important roles in harmonic analysis, PDE, and the other fields (see [13]), people pay much more attention to this class of operators. In 1995, Lu and Yang investigated the behavior of Littlewood-Paley operators in the space CBMOp   in [4]. In 2005, Zhang and Liu proved the commutator is bounded on in [5]. In 2009, Xue and Ding gave the weighted estimate for Littlewood-Paley operators and their commutators (see [6]). There are some other results about Littlewood-Paley operators in [79] and so forth.

On the other hand, Lebesgue spaces with variable exponent become one class of important research subject in analysis filed due to the fundamental paper [10] by Kováčik and Rákosník. In the past twenty years, the theory of these spaces has made progress rapidly, and the study of which has many applications in fluid dynamics, elasticity, calculus of variations, and differential equations with nonstandard growth conditions (see [1115]). In [16], Cruz-Uribe et al. stated that the extrapolation theorem leads the boundedness of some classical operators including the commutator on . Karlovich and Lerner also independently obtained the boundedness of the singular integrals commutator on Lebesgue spaces with variable exponent in [17]. In 2009 and 2010, Izuki considered the boundedness of vector-valued sublinear operators and fractional integrals on Herz-Morrey spaces with variable exponent in [18, 19], respectively. In 2013, Ho in [20] introduced a class of Morrey spaces with variable exponent and studied the boundedness of the fractional integral operators on .

Inspired by the results mentioned previously, in this paper we will consider the vector-valued inequalities of the Littlewood-Paley operators and their -order commutators on Morrey spaces with variable exponent. Before stating our main results, we need to recall some relevant definitions and notations.

Let be a Lebesgue measurable set in with measure .

Definition 1 (see [10]). Let be a measurable function.

The Lebesgue space with variable exponent is defined by

The space is defined by

The Lebesgue space is a Banach space with the norm defined by

Remark 2. (1) Note that if the function is a constant function, then equals . This implies that the Lebesgue spaces with variable exponent generalize the usual Lebesgue spaces. And they have many properties in common with the usual Lebesgue spaces.

(2) Denote ,  . Then consists of all satisfying and .

(3) The Hardy-Littlewood maximal operator is defined by Denote to be the set of all functions satisfying the condition that is bounded on .

(4) Let . Denote and is the conjugate exponent of (see [20]).

Definition 3 (see [20]). Let , . If there exists a constant such that, for any and , Lebesgue measurable function satisfying then one says is a Morrey weight function for . One denotes the class of Morrey weight functions by .

Definition 4 (see [20]). Let , . Then the Morrey spaces with variable exponent are defined by where

Remark 5. (1) If , then is the Lebesgue spaces with variable exponent .

(2) Notice that if , is a constant function, then formula (12) can be rewritten as an integral in form. To be precise, formula (12) can be rewritten in the following form (see [20]): Let . By the the conditions of Morrey weight functions mentioned in [21] and Hölder’s inequality, via simple calculation, we have From this, it follows that if , is a constant function, then condition (12) is weaker than condition (16). Thus, the class of the Morrey spaces introduced in Definition 4 is more wide than that satisfying condition in [21]. More studies of common Morrey spaces can be seen in [22, 23] and so forth.

(3) If ,  , then the space mentioned in Definition 4 is the Morrey space with variable exponent introduced in [24]. And when , it is easy to see satisfying condition (12). That is because it follows from ,  , that (see [20])

For Littlewood-Paley operators , and , in this paper, we have the following results.

Theorem 6. Suppose that function satisfies (i)–(iii) and is defined by (1). If , , , then there exists a constant independent of such that, for any function sequences with , the following inequality holds:

Theorem 7. Suppose that is defined by (3). Then under the same condition as the one in Theorem 6, there exists a constant independent of such that, for any function sequences with , the following inequality holds:

Theorem 8. Suppose that is defined by (4) and ,  . Then under the same condition as the one in Theorem 6, there exists a constant independent of such that, for any function sequences with , the following inequality holds:

For commutators ,  , and , we have the following results.

Theorem 9. Suppose that function satisfies (i)–(iii) and is defined by (5). Let , , ,  . If, for any and , function satisfies then there exists a constant independent of such that, for any function sequences with , the following inequality holds:

Theorem 10. Suppose that is defined by (6). Then under the same condition as the one in Theorem 9, there exists a constant independent of such that, for any function sequences with , the following inequality holds:

Theorem 11. Suppose that is defined by (7), , and . Then under the same condition as the one in Theorem 9, there exists a constant independent of such that, for any function sequences with , the following inequality holds:

Remark 12. (1) It is easy to see that condition (22) in Theorem 9 is stronger than condition (12) in Definition 3. Therefore, if a function satisfies condition (22), then .

(2) The function which satisfies (22) exists. In fact, if we take such that function satisfies then . That is because, for any , there exists such that . Hence, it follows from (18) that

We end this section by introducing some conventional notations which will be used later. Throughout this paper, given a function , we denote the mean value of on by . means the conjugate exponent of ; namely, holds. always means a positive constant independent of the main parameters and may change from one occurrence to another.

2. Preliminary Lemmas

In this section, we introduce some conclusions which will be used in the proofs of our main results.

Lemma 13 (see [10] (generalized Hölder’s inequality)). Let .(1)For any , where .(2)For any , when , one has where .

Lemma 14 (see [17]). If , then there exist constants , such that, for all balls and all measurable subsets ,

Remark 15. From formula (12), it follows that Thus, by Lemma 14, we have, ,

Lemma 16 (see [18]). If , then there exists constant , such that, for all balls ,

Lemma 17 (see [25]). Let ;   is a positive integer. There exist constants , such that, for any with ,(1);(2).

Lemma 18 (see [26]). Let satisfy (i)–(iii). If ,  , then for all bounded compactly support functions such that , that is, , the following vector-valued inequalities hold:(1),(2),(3), where .

Lemma 19 (see [26]). Let satisfy (i)–(iii), . If , then for all bounded compactly support functions such that , that is, , the following vector-valued inequalities hold:(1),(2),(3),where ,  .

3. Proofs of Main Results

Next, let us show the proofs of Theorems 611, respectively.

Proof of Theorem 6. Let ; for any , , denote where ,  .

Noting that, in order to prove Theorem 6, it is enough to show that the following inequality holds: Thus, For the term , notice that supp ; using Lemma 18 and (32), it is easy to see that We now turn to estimate . To do this, we need to consider first. Without loss of generality, we may assume that . Let Then, by Minkowski’s inequality, we have Observe that if ,  ,  , then Therefore, it follows from condition (ii) that On the other hand, we denote Note that if , then . Thus, by condition (iii), we get And by condition (ii), similar to the estimate of , we obtain Hence, from the estimates above, it follows that Thus, Therefore, applying the generalized Hölder’s inequality, Lemma 16, and (12), we have