Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 784983, 9 pages

http://dx.doi.org/10.1155/2013/784983

Research Article

Boundedness of Sublinear Operators with Rough Kernels on Weighted Morrey Spaces

Shaoguang Shi and Zunwei Fu

Department of Mathematics, Linyi University, Linyi 276005, China

Received 18 January 2013; Accepted 11 March 2013

Academic Editor: Dashan Fan

Copyright © 2013 Shaoguang Shi and Zunwei Fu. 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.

The aim of this paper is to get the boundedness of a class of sublinear operators with rough kernels on weighted Morrey spaces under generic size conditions, which are satisfied by most of the operators in classical harmonic analysis. Applications to the corresponding commutators formed by certain operators and BMO functions are also obtained.

1. Introduction and Main Results

Given a function over the unit sphere of equipped with the normalized Lebesgue measure and , a Calderón-Zygmund singular integral operator with rough kernel was given by

and a related maximal operator

where is homogeneous of degree zero and satisfies

When is a smooth kernel and a standard Calderón-Zygmund singular integral operator which has been fully studied by many papers, a classical survey work; see, for example, [1].

For simplicity of notation, is always homogeneous of degree zero and satisfies (3) and (4) throughout this paper if there are no special instructions. Here and in what follows, for , , and , denotes the ball centered at with radius and . When satisfies some size conditions, the kernel of the operator has no regularity, and so the operator is called rough singular integral operator. In recent years, a variety of operators related to the singular integrals for Calderón-Zygmund, but lacking the smoothness required in the classical theory, have been studied. Duoandikoetxea [2] studied the norm inequalities for in homogeneous case on weighted spaces. For more corresponding works, we refer the reader to [3–8] and the references therein.

In [9], Hu et al. considered some more general sublinear operators with rough kernels which satisfy

for with compact support. Condition (5) was first introduced by Soria and Weiss [10]. Inequality (5) is satisfied by many operators with rough kernels in classical harmonic analysis, such as (see [11]) and the oscillatory singular integral operator

where the phase is a polynomial. The boundedness of on weighted spaces was fully studied by Ojanen in his doctoral dissertation [12].

Let and let for . Throughout this paper, we will denote by the characteristic function of the set . Inspired by the works of [6, 13], in this paper, we consider some sublinear operators under some size conditions (the following (7) and (8)) which are more general than (5):

when and with and

when and with , respectively. It is worth pointing out that satisfies conditions (7) and (8). Also, condition (5) implies the size conditions (7) and (8) since when and while and imply .

The topic of this paper is intended as an attempt to study the boundedness of sublinear operators with rough kernels which satisfy (7) and (8) on weighted Morrey spaces. We first recall some definitions and notations for weighted spaces. The Muckenhoupt classes and [14] contain the functions which satisfy

respectively, where . For , the and weights are defined by

respectively. Here ess and the following ess are the abbreviations of essential supremum and essential infimum, respectively. Clearly, if and only if there is a constant such that

In [15], Komori and Shirai introduced a weighted Morrey space, which is a natural generalization of weighted Lebesgue space, and investigated the boundedness of classical operators in harmonic analysis. Let ,   and let be a weight function. Then the weighted Morrey space is defined by

where . For , if , then while implies .

Now, we formulate our major results of this paper as follows.

Theorem 1. Let , , and and let a sublinear operator satisfy (7) and (8). If is bounded on with , then is bounded on .

When , we have the following theorem.

Theorem 2. Let and and let satisfy (7) and (8). Then if is bounded from to with , there exists a constant such that for all and all balls ,

In the fractional case, we need to consider a weighted Morrey space with two weights which is also introduced by Komori and Shirai in [15]. Let ,  . For two weights and ,

If , we write .

We can get similar results for fractional integrals following the line of Theorems 1 and 2.

Theorem 3. Let , , and . Suppose that a sublinear operator satisfies the size conditions

when and with and

when and with . Then one has the following.

(a)If maps into with , then is bounded from to , where , and .(b)If is bounded from to with and , then there exists a constant such that for all and all balls ,

where .

We emphasize that (15) and (16) are weaker conditions than the following condition:

for any integral function with compact support. Condition (18) is satisfied by most fractional integral operators with rough kernels, such as the fractional integral operators of Muckenhoupt and Wheeden [16]:

For some mapping properties of on various kinds of function spaces, see [17–19] and the references therein.

We end this section with the outline of this paper. Section 2 contains the proofs of Theorems 1 and 3; this part is partly motivated by the methods in [20] dealing with the case of the Lebesgue measure. In Section 3, we extend the corresponding results to commutators of certain sublinear operators.

2. Boundedness of Sublinear Operators

Proofs of Theorems 1 and 3 depend heavily on some properties of weights, which can be found in any papers or any books dealing with weighted boundedness for operators in harmonic analysis, such as [1]. For the convenience of the reader we collect some relevant properties of weights without proofs, thus making our exposition self-contained.

Lemma 4. Let and . Then the following statements are true.

(a) There exists a constant such that

    where satisfies this condition; one says satisfies the doubling condition.(b) There exists a constant such that

    where satisfies this condition; one says satisfies the reverse doubling condition.(c) There exist two constants and such that the following reverse Hölder inequality holds for every ball :

(d) For all , one has

(e) There exist two constants and such that for any measurable set

    if satisfies (24); one says .(f) For all , one has

The following lemma about the rough kernel is essential to our proofs. One can find its proof in [21].

Lemma 5. Let with . Then the following statements are true.

(a)If   and , then .(b)If and , then .

Proof of Theorem 1. Let , , and . Our task is to show

For a fixed ball , there is no loss of generality in assuming . We decompose . Since is a sublinear operator, so we get

By the assumption on and (25), we can obtain

For the term , by (8) we have

where

We distinguish two cases according to the size of and to get the estimates for .

Case 1   (). In this case, implies that

By (31), Hölder’s inequality, and Lemma 5, we have

Case 2  (). In this case, implies that

which in combination with the Hölder inequality and Lemma 5 yields that

Substituting (32) and (34) into (29), we can assert that

where we have used (21) in the last inequality. Combining (28) and (29), we obtain the proof of Theorem 1.

Proof of Theorem 2. The task is now to show the following inequality:

In order to get this inequality, it will be necessary to decompose with as in Theorem 1. Since is a sublinear operator, we can rewrite

An application of (20) and the weighted weak type estimates for yield that

To estimate the term , we note that

By (22), (33), and the Hölder inequality, we can estimate as

Combining these inequalities for and , we have completed the proof of Theorem 2.

Proof of Theorem 3. We can use the similar arguments as in the proof of Theorem 1 and Theorem 2. For the proof of , it suffices to show that

For a fixed ball , we decompose . Since is a sublinear operator, we get

To estimate the term , using the fact that is bounded from to with , we can get

which implies that

For the term , by the similar arguments as that of Theorem 1, we obtain

We have completed the proof of .

We will omit the proof of since we can prove it by using condition and the weak type estimates of similar to the proof of Theorem 2.

3. Boundedness of Commutators

We say that is a function if the following sharp maximal function is finite:

where the supreme is taken over all balls and . This means . An early work about space can be attributed to John and Nirenberg [22]. For , there is a close relation between and weights:

Given an operator acting on a generic function and a function , the commutator is formally defined as

Since , the boundedness of is worse than (e.g., the singularity; see also [23]). Therefore, many authors want to know whether shares the similar boundedness with . There are a lot of articles that deal with the topic of commutators of different operators with functions on Lebesgue spaces. The first results for this commutator were obtained by Coifman et al. [24] in their study of certain factorization theorems for generalized Hardy spaces. In the present section, we will extend the boundedness of and to and , respectively.

Theorem 6. Let , , , and be as in Theorem 1. Suppose that the sublinear operator satisfies condition (5) for any integral function with compact support. If is bounded on with , then is bounded on .

Theorem 7. Let , , , , , and be as in Theorem 3 and let the sublinear operator satisfy condition (18) for any integral function with compact support. If maps into with , then is bounded from to .

The following lemmas about functions will help us to prove Theorems 6 and 7.

Lemma 8 (see [25, Theorem ]). Let and . Then for any ball , the following statements are true.

(a)There exist constants and such that for all

(b)Inequality (49) is called John-Nirenberg inequality:

Lemma 9 ([1, Proposition ] (see also [14, Theorem ])). Let and . Then the following statements are equivalent:

(a), (b), (c), where and .

Lemma 10. Let , , , and be as in Theorem 6 and let be a generic fixed ball. Then the inequality

holds for every , where .

Proof. We will consider two cases.

Case   (). In this case, . Using Hölder’s inequality and Lemma 5 to the left-hand side of (51), we have

Set

where . Thus

Lemma 9 implies that

We are now in a position to deal with ; by (50), we have

Combining (23) with (49), we have

In the same manner we can see that

It follows immediately that

Therefore

A further use of (21) and allow us to obtain

where is a constant that appeared in (21).

Case 2  (). In this case, . We can prove (51) by a similar analysis as in the proof of Theorem 1 (in the case ) and Case .

Having disposed of the previous preliminary step, we can now return to the proofs of Theorems 6 and 7.

Proof of Theorem 6. The task is now to find a constant such that for fixed ball , we can obtain

We decompose and consider the corresponding splitting

It follows from the boundedness of and that

Then a further use of (5) derives that

Therefore

By Lemma 10, we have

We proceed to estimate . Without loss of generality, we assume that . Taking into account (20), (22), and Lemma 9, we have

Hence

According to (64) and (69), we have completed the proof of Theorem 6.

Proof of Theorem 7. The proof of Theorem 7 is similar to that of Theorem 6, except using . We omit its proof here.

Acknowledgments

The authors thank Professor Shanzhen Lu and the referee for their valuable suggestions. This work was partially supported by NSF of China (Grants nos. 11271175, 10901076, and 11171345) and NSF of Shandong Province (Grant no. ZR2012AQ026).

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