Boundedness of One-Sided Oscillatory Integral Operators on Weighted Lebesgue Spaces

Abstract

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.

1. Introduction and Main Results

Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject; three chapters are devoted to them in the celebrated Stein's book [1]. Many operators in harmonic analysis or partial differential equations are related to some versions of oscillatory integrals, such as the Fourier transform, the Bochner-Riesz means, and the Radon transform which has important applications in the CT technology. Among numerous papers dealing with oscillatory singular integral operators in some function spaces, we refer to [2–7] and the references therein. More generally, let us now consider a class of oscillatory integrals defined by Ricci and Stein [8]: where is a real-valued polynomial defined on and the function is a Calderón-Zygmund kernel. That means satisfies Throughout this paper, the letter will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.

We state a celebrated result of Ricci and Stein on oscillatory integrals as follows.

Theorem 1 (see [8]). Let , satisfy (2) and (3). Then for any real-valued polynomial , the oscillatory integral operator is of type and its norm depends on the total degree of , but not on the coefficients of in other respects.

Weighted inequalities arise naturally in harmonic analysis, but their use is best justified by the variety of applications in which they appear. It is worth pointing out that many authors are interested in the inequalities when the weight functions belong to the Muckenhoupt classes ([9]), which are denoted by classes for simplicity. This class consists of positive locally integrable functions (weight functions) for which where the supremum is taken over all intervals and .

In 1992, Lu and Zhang [10] established the weighted version of Theorem 1.

Theorem 2. Let and be as in Theorem 1. Then the oscillatory singular integral operator is of type , with . Here its operator norm is bounded by a constant depending on the total degree of , but not on the coefficients of in other respects.

We point out that Theorems 1 and 2 also hold for dimension . We choose the results for here in order to introduce the one-sided operators which were defined on . Theorems 1 and 2 are also true for more general kernels, that is, nonconvolution kernels, under the -boundedness assumption on the corresponding Calderón-Zygmund singular integral operators: However, this topic exceeds the scope of this paper. For more information about this work, see [8, 10], for example.

The study of weights for one-sided operators was motivated not only by the generalization of the theory of both-sided ones, but also by their natural appearance in harmonic analysis; for example, they are required when we treat the one-sided Hardy-Littlewood maximal operator [11]: arising in the ergodic maximal function. Sawyer first introduced the classical one-sided weight classes in [11]. The general definitions of and were introduced in [12] as where , ; also, for ,

The smallest constant for which the above inequalities are satisfied will be denoted by and , . () will be called the (resp., ) constant of . By Lebesgue's differentiation theorem, we can easily prove (resp., . In [13], the class was introduced as (see also [14]). It is easy to see that for , , , and .

Theorem 3 (see [11]). Let . Then(1) is bounded in if and only if ;(2) is bounded in if and only if .

The one-sided weight classes are of interest, not only because they control the boundedness of the one-sided Hardy-Littlewood maximal operator, but also because they are the right classes for the weighted estimates of one-sided Calderón-Zygmund singular integral operators [15], which are defined by where is the one-sided Calderón-Zygmund kernel with support in and , respectively. We say a function is a one-sided Calderón-Zygmund kernel if satisfies (2) and with support in or . An example of such a kernel is where denotes the characteristic function of a set .

Theorem 4 (see [15]). Let and be a one-sided Calderón-Zygmund kernel. Then(1) is bounded in if and only if ;(2) is bounded in if and only if .

Theorem 4 is the one-sided version of weighted norm inequality of singular integral due to Coifman and Fefferman [9].

Highly inspired by the above statements for oscillatory singular integral operators and one-sided operator theory, in [16], the authors had introduced the one-sided oscillatory singular integral operators and studied the weighted weak type norm inequalities for these operators. In this paper, we will further study the one-sided Muckenhoupt weight classes and give the one-sided version of Theorem 2. It is well known that the property of the one-sided Muckenhoupt weight classes is worse than the Muckenhoupt weight classes (see also [17]). For example, both the reverse Hölder inequality and the doubling condition are not true for the one-sided case. Therefore, some new methods are needed to deal with some new difficulties.

We first recall the definition of one-sided oscillatory integral operator as where is a real-valued polynomial defined on and the kernel is a one-sided Calderón-Zygmund kernel with support in and , respectively. Now, we formulate our results as follows.

Theorem 5. Let and be a one-sided Calderón-Zygmund kernel. Then for any real-valued polynomial ,(1)there exists constant such that    where and the operator norm depend on the total degree of and , but not on the coefficients of in other respects;(2)there exists constant such that    where and the operator norm depend on the total degree of and , but not on the coefficients of in other respects.

The rest of this paper is devoted to the argument for Theorem 5. Section 2 contains some preliminaries which are essential to our proof. In Section 3, we will give the proof of Theorem 5.

2. Preliminaries

Lemma 6 (see [11, 18]). Let and be locally integrable. Then the following statements are equivalent:(1);(2);(3)there exist and such that .

According to the definition of , we can easily obtain the following lemma.

Lemma 7. Let and . Then , where for all .

Proof. For , if , then For , , , , and , we have The proof is complete.

We say a weight satisfies the one-sided reverse Hölder condition [18] if there exists such that for any and , where is the classical Hardy-Littlewood maximal operator. The smallest such constant will be called the constant of and will be denoted by . Corresponding to the classical reverse Hölder inequality, (17) is named the weak reverse Hölder inequality. For , we say a weight satisfies the one-sided reverse Hölder condition if there exists such that for almost all where is the one-sided minimal operator defined as The smallest such constant will be called the constant of and will be denoted by . It is clear that . In [18], the authors give several characterizations of where the constants are not necessary the same.

Lemma 8. Let , , and be locally integrable. Then the following statements are equivalent:(1);(2) with ;(3) with ;(4) with ;(5) with , .

Lemma 9 (see [18]). A weight for if and only if there exist and a constant such that for with , the following inequality holds:

Combining the results in [12, 15, 18, 19], we can deduce Lemma 10. In what follows, we will include its proof with slight modifications for the sake of completeness.

Lemma 10. Let . Then there exists such that .

Proof. By Lemma 6, we have with and . For fixed interval , we next claim that for all with . In fact, we consider the truncation of at height defined by which also satisfies condition (with a constant ). Therefore, if and , then we have Indeed, it is straightforward if since We now assume and fix and an open set such that with . Let , which is connected. There are two cases; that is, and . In the first case, it is easy to check that is not contained in . By the definition of , , we have , while the second case is handled as the case since . Thus . Adding up with , we get Therefore, we obtain (20). For fixed , multiply both sides of (20) by and integrate from to infinity; we can obtain Now if , then , which implies
The inequality implies . Therefore, if , then we have Hence by the monotone convergence theorem. Since , we next claim that . In fact, for any interval , we have by Hölder's inequality and the condition. For almost every , we have Thus, which implies our claim. Hence, where and . By Lemma 8, we obtain . Hence, for all by Lemma 6.
Let us fix and choose such that (e.g., we choose , ). Following from the five points , we have four intervals, namely, By Lemma 8, we have Thus, by Lemma 9. Choosing , then we complete the proof of the lemma.

To prove Theorem 5, we still need a celebrated interpolation theorem of operators with change of measures.

Lemma 11 (see [20]). Suppose that are positive weight functions and , . Assume sublinear operator satisfies Then, holds for any and , where , , and .

Lemmas 10 and 11 are the main tools in proving Theorem 5.

3. Proof of Theorem 5

In this section, we will prove Theorem 5 by induction, which is partly motivated by [8, 10]. We begin with the proof of (1). For any nonzero real polynomial in and , there are such that with and We will write and . Below we will carry out the argument by using a double induction on and .

If and is arbitrary, then and can be written as where . Therefore, the conclusion of Theorem 5 follows from Theorem 4.

Let and assume that the conclusion of Theorem 5 holds for all with and arbitrary.

We will now prove that the conclusion of Theorem 5 holds for all with and arbitrary.

If and , then with . By taking the factor out of the integral sign, we see that this case follows from the above inductive hypothesis.

Suppose and the desired bound holds when and .

Now, let be a polynomial with and , as given in (34).

Case 1 (). Write Take any , and write where the polynomial satisfies the induction assumption and the coefficients of depend on .
We consider first the estimates for . It is easy to check that Now we split into three parts as
Observe that if , then Thus, it follows from the induction assumption that where is independent of and the coefficients of .
Notice that if , then . Thus, So we have where is independent of and the coefficients of .
Again observe that if and , then . Thus,
Combining (43), (45), and (46), we get where is independent of and the coefficients of .
Evidently, if and , then Therefore, when , we have It follows from Theorem 3 that where is independent of and the coefficients of .
From (47) and (50), it follows that the inequality holds uniformly in , which implies where is independent of the coefficients of and .
We proceed with the proof of Theorem 5 with the estimates for . Because of the size condition (2), we observe that for where is independent of . By Lemma 10, we know that there exists such that . Thus we have where is independent of . We now only need to recall Lemma 3.7 in [16] to see that where depends only on the total degree of and . It follows from (54), (55), and Lemma 11 that where , is independent of , and depends only on the total degree of .
From (52) and (56), it is clear that when , where depends only on the total degree of .

Case 2 (). In this case, we write and Therefore, where and It is easy to check that satisfies (2) and (10). We have thus established that with similar statements as in Case 1. By Lemma 7, we have that is, where depends on the total degree of but not on the coefficients of .

(2) We omit the details, since they are very similar to that of the proof of (1) with instead of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by NSF of China (Grant nos. 11271175, 10931001, and 11301249), NSF of Shandong Province (Grant no. ZR2012AQ026), the AMEP and DYSP of Linyi University, and the Key Laboratory of Mathematics and Complex System (Beijing Normal University), Ministry of Education, China.