Weighted Estimates for Oscillatory Singular Integrals

Abstract

We establish uniform bounds for oscillatory singular integrals as well as oscillatory singular integral operators. We allow the singular kernel to be given by a function in the Hardy space , while such results were known previously only for kernels in log , a proper subspace of . One of our results established a bound for certain weights. At the same time, it provides a solution to an open problem in Lu (2005).

1. Introduction

In this paper we establish uniform bounds for oscillatory singular integrals. We consider two types of oscillatory singular integrals, which will be described later.

Let and denote the unit sphere in equipped with the induced Lebesgue measure . For an integrable function satisfying

we define where for . For , let

Type I. An oscillatory integral of type I is given by where is given by (2) and is a polynomial on . For a given and the main concern is to establish a bound for

Previous results in this regard include Stein [1] for and Papadimitrakis and Parissis [2] for which improved Stein's result.

Type II. A type II oscillatory singular integral is actually an integral operator of the form where is given by (2) and is a real-valued polynomial on . Ricci and Stein [3] showed that, if , is bounded on . Subsequently Lu and Zhang [4] and Jiang and Lu [5] established the same bounds for under the weaker conditions and , respectively.

We will now state our main results, beginning with oscillatory singular integrals of Type II.

A set in is called a rectangle if there is an orthonormal basis of (which may depend on ) such that

In other words, what we call a rectangle in is simply any rotation of an arbitrary -cell . Let denote the collection of all rectangles in .

Definition 1. Let , and let be a nonnegative, locally integrable function on . We say that is in the weight class if

It is easy to see that is a subcollection of the well-known weight class of Muckenhoupt [6, 7]. Examples of weights in include all weights of the form , where is a polynomial in and .

Theorem 2. Let be a real-valued polynomial on . Suppose that , and satisfies (1). Then the operator is bounded on for , with a bound on its norm which may depend on the degree of but is otherwise independent of the coefficients of .

The space is the Hardy space on the unit sphere. Since is a proper subspace of , Theorem 2 represents an improvement over results mentioned earlier. By taking , it answers an open question in [8, page 52] in the affirmative.

Our second result has the same flavor as the first, but it concerns Type I oscillatory singular integrals instead.

Theorem 3. Suppose that and satisfies (1). Then where is a constant independent of and .

Our result in this regard is built on the work of Papadimitrakis and Parissis who gave the following bound in [2]: They also showed the logarithmic growth of the bound in to be best possible. Our bound, while dependent on the dimension , provides an improvement over the factor .

2. Proofs of Theorems 2 and 3

We will begin by recalling the atomic decomposition for .

Definition 4. A measurable function on is called a regular atom if it satisfies the following:(i),(ii) for some and , where ,(iii).
An exceptional atom is just an function on satisfying .

The following result is from [9, 10].

Lemma 5. For every there exist and atoms (both regular and exceptional) such that and .

Proof of Theorem 2. Let . It suffices to show that, for , there exists a such that for all . Since the sum in (11) converges in the sense of distribution, by Lemma 5, we only need to prove
when is a regular atom.
Below we will assume that satisfies Conditions (i)–(iii) in Definition 4. Obviously we may also assume that . We also extend to be a homogeneous function of degree 0 by setting for . Let be an orthogonal matrix such that . We define the linear transformation on by where denotes the identity matrix and . By letting , , , and we get
If for some , then by (i) we have which implies that Thus, Therefore we have
By its definition and a well-known argument, is homogeneous of degree 0 and satisfies (1). Also observe that and is an weight with an bound independent of . Thus, by Theorem  5 in [11] and Theorem  5 of [5], there is a such that This proves Theorem 2.

Proof of Theorem 3. Let , and let . For a which satisfies (1), we write where are regular atoms and . By the proof of Theorem 2, for each , there exist a and a function on which satisfies (1) and such that where . By (23) and Theorem 1 in [2], we have which proves Theorem 3.

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Copyright

Copyright © 2013 H. Al-Qassem et al. 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.