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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 543748, 3 pages
Weighted Estimates for Oscillatory Singular Integrals
1Department of Mathematics and Physics, Qatar University, Doha, Qatar
2Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA
3Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
Received 25 January 2013; Accepted 15 March 2013
Academic Editor: Alberto Fiorenza
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.
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).
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
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  showed that, if , is bounded on . Subsequently Lu and Zhang  and Jiang and Lu  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
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 : 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 .
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 .
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  and Theorem 5 of , 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 , we have which proves Theorem 3.
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