Boundedness of Singular Integral Operators with Variable Kernels on Weighted Weak Hardy Spaces

Abstract

Let be the singular integral operator with variable kernel . In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of on these spaces, under some Dini type conditions imposed on the variable kernel .

1. Introduction

Let be the unit sphere in () equipped with the normalized Lebesgue measure . A function defined on is said to belong to , , if it satisfies the following conditions:(A)for all and , ;(B)for any , ;(C) ,

where for any . Set . In this paper, we consider the singular integral operator with variable kernel which is defined by

In [1, 2], Calderón and Zygmund investigated the boundedness of singular integral operators with variable kernels. They found that these operators are closely related to the problem about second order elliptic partial differential equations with variable coefficients. We will denote the conjugate exponent of by . In [3], Calderón and Zygmund proved the following theorem.

Theorem A (see [3]). Let , satisfy(i) if , or(ii) if .
Suppose that . Then there exists a constant independent of such that In particular, is bounded on for all .

In 1971, Muckenhoupt and Wheeden [4] studied the weighted norm inequalities for with power weights. In 2008, Lee et al. [5] also considered the weighted boundedness of with more general weights and showed that if the kernel satisfies the -Hörmander condition with respect to and variables, respectively, then is bounded on . More precisely, they proved the following.

Theorem B (see [5]). Let . Suppose that such that the following two inequalities hold for all . If and , then is bounded on .

It should be pointed out that the above -Hörmander conditions on the variable kernels were also considered by Rubio de Francia et al. in [6].

In [7, 8], Ding et al. introduced some definitions about the variable kernel when they studied the boundedness of Marcinkiewicz integral. Replacing the condition (C) mentioned above, they strengthened it to the condition .

For , a function is said to satisfy the -Dini condition if the conditions (A), (B), and hold and where is the integral modulus of continuity of order of defined by

In order to obtain the boundedness of , Lee et al. [5] generalized the -Dini condition by replacing (4) to the following stronger condition (see also [9]):

If satisfies (6) for some and , we say that it satisfies the -Dini condition. For the special case , it reduces to the -Dini condition. For , if satisfies the -Dini condition, then it also satisfies the -Dini condition. We thus denote by the class of all functions which satisfy the -Dini condition for all .

Theorem C (see [5]). Let and . Suppose such that (3) hold for a certain large number . If , then there exists a constant independent of such that

It is easy to check that Then for , we define

The main purpose of this paper is to study the corresponding estimates of on the weighted weak Hardy spaces (see Section 2 for its definition). We now present our main result as follows.

Theorem 1. Let , , and . Suppose such that (3) hold. Then there exists a constant independent of such that

2. Notations and Preliminaries

The definition of class was first used by Muckenhoupt [10], Hunt et al. [11], and Coifman and Fefferman [12] in the study of weighted boundedness of Hardy-Littlewood maximal functions and singular integrals. Let be a nonnegative, locally integrable function defined on ; all cubes are assumed to have their sides parallel to the coordinate axes. We say that , , if where is a positive constant which is independent of the choice of . For the case , , if The smallest value of such that the above inequalities hold is called the characteristic constant of and denoted by . For the case , , if it satisfies the condition for some .

A weight function is said to belong to the reverse Hölder class if there exist two constants and such that the following reverse Hölder inequality holds:

It is well known that if with , then for all and for some . We thus write to denote the critical index of . Moreover, if with , then there exists such that . It follows directly from Hölder’s inequality that implies for all .

Given a cube and , stands for the cube with the same center as whose side length is times that of . denotes the cube centered at with side length . For a weight function and a measurable set , we denote the Lebesgue measure of by and set the weighted measure .

We give the following results that will be used in the sequel.

Lemma 2 (see [13]). Let with . Then, for any cube , there exists an absolute constant such that In general, for any , one has where does not depend on or .

Given a weight function on , for , we denote by the weighted space of all functions satisfying When , will be taken to mean and We also denote by the weighted weak space which is formed by all measurable functions satisfying

Let us now turn to the weighted weak Hardy spaces. The (unweighted) weak spaces have first appeared in the work of Fefferman et al. [14], which are the intermediate spaces between two Hardy spaces through the real method of interpolation. The atomic decomposition characterization of weak space on was given by Fefferman and Soria in [15]. Later, Liu [16] established the weak spaces on homogeneous groups for the whole range . The corresponding results related to can be found in [17]. For the boundedness properties of some operators on weak Hardy spaces, we refer the readers to [18–24]. In 2000, Quek and Yang [25] introduced the weighted weak Hardy spaces and established their atomic decompositions. Moreover, by using the atomic decomposition theory of , Quek and Yang [25] also obtained the boundedness of Calderón-Zygmund type operators on these weighted spaces.

We write to denote the Schwartz space of all rapidly decreasing infinitely differentiable functions and to denote the space of all tempered distributions, that is, the topological dual of . Let , and . Define where , and For any given , the grand maximal function of is defined by Then we can define the weighted weak Hardy space by . Moreover, we set .

Theorem 3 (see [25]). Let and . For every , there exists a sequence of bounded measurable functions such that(i) in the sense of distributions;(ii)each can be further decomposed into , where satisfies that(a)each is supported in a cube with and . Here denotes the characteristic function of the set and ;(b), where is independent of and ;(c) for every multi-index with .
Conversely, if has a decomposition satisfying and , then . Moreover, one has .

Throughout this paper always denotes a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence.

3. Proof of Theorem 1

Following the same arguments as in the proof of Lemma  5 in [26], we can also establish the following lemma on the variable kernel (see [5, 8]).

Lemma 4. Let . Suppose that satisfies the -Dini condition in Section 1. If there exists a constant such that , then for any , one has where the constant is independent of and .

We are now in a position to give the proof of Theorem 1.

Proof of Theorem 1. For any given , we may choose such that . For every , then by Theorem 3, we can write where , , and satisfies (a)–(c) in Theorem 3. Then we have First we claim that the following inequality holds: In fact, since and according to Theorem 3, then it follows directly from Minkowski’s inequality that For each , by using the bounded overlapping property of the cubes and the fact that , we thus obtain Since and , then we have . In this case, we know that there exists a number such that . More specifically, by using the sharp reverse Hölder’s inequality for weights obtained recently in [27], we find that for , Observe that , then we are able to find a positive number large enough such that and . By the choice of , we can easily check that , which implies . Hence, by using Theorem B, we know that is bounded on . This fact together with Chebyshev’s inequality and (25) yields We now turn our attention to the estimate of . Setting where and is a fixed positive number such that . Thus, we can further decompose as Let us first deal with the term . Since , then by Lemma 2, we can deduce that On the other hand, it follows immediately from Chebyshev’s inequality that Now denote , , and An application of Hölder’s inequality gives us that Let for simplicity. Then for any and with , we can easily see that . Hence, by the cancellation condition of , we get When and , then a trivial computation shows that We also observe that and , then . Using Hölder’s inequality, the estimate (37), and Lemma 4, we can see that for any , the integral of the above expression is dominated by Recall that . From the above estimate (38), it follows that In addition, for with , then we can take a sufficiently small number such that . Thus, by using Lemma 2 again, we finally obtain Therefore Combining the above inequality (41) with (29) and (32), and then taking the supremum over all , we conclude the proof of Theorem 1.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank Professor J. Duoandikoetxea for pointing out the reference [6].