Logarithmic Bounds for Oscillatory Singular Integrals on Hardy Spaces

Abstract

We establish a logarithmic bound for oscillatory singular integrals with quadratic phases on the Hardy space [superscript] H 1 [/superscript] ( [superscript] R n [/superscript] ) . The logarithmic rate of growth is the best possible.

Hussain Al-Qassem 1 and Leslie Cheng 2 and Yibiao Pan 3

Academic Editor:Maria Alessandra Ragusa

1, Department of Mathematics and Physics, Qatar University, Doha, Qatar
2, Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA
3, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received 28 December 2015; Accepted 7 February 2016

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.

1. Introduction

For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a Calderon-Zygmund kernel on [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be a polynomial of [figure omitted; refer to PDF] variables with real coefficients. Consider the following oscillatory singular integral operator: [figure omitted; refer to PDF] It is well known that [figure omitted; refer to PDF] is bounded from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] when [figure omitted; refer to PDF] and also from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] . Additionally, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] bounds are dependent on the degree of the phase polynomial [figure omitted; refer to PDF] only, not its coefficients (see [1, 2]).

However, for [figure omitted; refer to PDF] boundedness of [figure omitted; refer to PDF] , the answers are not nearly as clear-cut. First, it was shown in [3] that, in general, [figure omitted; refer to PDF] may fail to be bounded on [figure omitted; refer to PDF] and when the coefficients of the first-order terms of [figure omitted; refer to PDF] vanish, [figure omitted; refer to PDF] is bounded from [figure omitted; refer to PDF] to itself with a bound independent of the higher order coefficients of [figure omitted; refer to PDF] .

More recent work can be found in [4, 5], including the following.

Theorem 1 (see [5]).

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] be a polynomial of degree [figure omitted; refer to PDF] in [figure omitted; refer to PDF] with real coefficients. Let [figure omitted; refer to PDF] be a Calderon-Zygmund kernel and let [figure omitted; refer to PDF] be given as in (1). Then, there exists a positive constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . The constant [figure omitted; refer to PDF] may depend on [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] but is independent of the coefficients [figure omitted; refer to PDF] of [figure omitted; refer to PDF] .

In order to determine the optimal bound on [figure omitted; refer to PDF] , an example was given in [5] to show that, as [figure omitted; refer to PDF] , any bound on [figure omitted; refer to PDF] must increase at least at the rate of [figure omitted; refer to PDF] . This naturally leads to the following question.

Does [figure omitted; refer to PDF] hold for all [figure omitted; refer to PDF] ?

In this paper, we will prove that the answer to the above question is affirmative for all quadratic polynomials. Namely, we have the following.

Theorem 2.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be a quadratic polynomial in [figure omitted; refer to PDF] with real coefficients. Let [figure omitted; refer to PDF] be a Calderon-Zygmund kernel and let [figure omitted; refer to PDF] be given as in (1). Then, there exists a positive constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . The constant [figure omitted; refer to PDF] may depend on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] but is independent of the coefficients [figure omitted; refer to PDF] of [figure omitted; refer to PDF] .

We point out that [figure omitted; refer to PDF] denotes an absolute constant whose value may change from line to line.

2. Some Definitions and Lemmas

Many of the tools we use are known. For readers who wish to see the definitions and some of their properties, the following references are suggested: [6-12].

For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the Euclidean volume of [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be a function in the Schwartz space [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we let [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Definition 3.

For a nonnegative, locally integrable function [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , the Hardy space [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Definition 4.

A measurable function [figure omitted; refer to PDF] on [figure omitted; refer to PDF] is called [figure omitted; refer to PDF] atom if there exist [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Lemma 5 (see [9, 10]).

For each [figure omitted; refer to PDF] , there exist [figure omitted; refer to PDF] atoms [figure omitted; refer to PDF] and coefficients [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Definition 6.

A [figure omitted; refer to PDF] function [figure omitted; refer to PDF] is called a Calderon-Zygmund kernel if the following are true:

(i) There exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: holds for all [figure omitted; refer to PDF] .

(ii) For all [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Lemma 7.

Let [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Define operator [figure omitted; refer to PDF] by [figure omitted; refer to PDF] Then, there exists [figure omitted; refer to PDF] independent of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Proof.

We start by treating the more difficult case [figure omitted; refer to PDF] . The other case, [figure omitted; refer to PDF] , will be briefly considered later.

Write [figure omitted; refer to PDF] with [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Then, there exist [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Thus, we have [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] Then, there are polynomials [figure omitted; refer to PDF] , [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] on [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] if [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] The treatment of the case [figure omitted; refer to PDF] only involves the Fourier transform step of the preceding argument. Details are omitted.

Lemma 8.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be a quadratic polynomial in [figure omitted; refer to PDF] with real coefficients. Let [figure omitted; refer to PDF] be a Calderon-Zygmund kernel satisfying (11)-(12) and let [figure omitted; refer to PDF] be given as in (1). Then, there exists a positive constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] atom [figure omitted; refer to PDF] which satisfies (7)-(9) with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The constant [figure omitted; refer to PDF] may depend on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] but is independent of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Proof.

By the uniform boundedness of [figure omitted; refer to PDF] on [figure omitted; refer to PDF] and (7)-(8), [figure omitted; refer to PDF]

By (11), we have [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] . It follows from (11) and (7)-(8) and Lemma 7 that [figure omitted; refer to PDF]

If [figure omitted; refer to PDF] , then (23) follows from (24) and (26).

Thus, we may assume that [figure omitted; refer to PDF] . To finish the proof, it suffices to show that [figure omitted; refer to PDF]

We will establish (27) by discussing two cases.

Case 1 ( [figure omitted; refer to PDF] ) . In this case, we have [figure omitted; refer to PDF]

Case 2 ( [figure omitted; refer to PDF] ) . In this case, we let [figure omitted; refer to PDF] It follows from Theorem [figure omitted; refer to PDF] of [3] that [figure omitted; refer to PDF]

For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By (30)-(31) and [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Thus, (27) holds in both cases.

3. Proof of Main Theorem

To finish the proof, we recall the following result concerning Riesz transforms and Hardy spaces.

Lemma 9 (see [10, 13]).

For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] denote the [figure omitted; refer to PDF] th Riesz transform; that is, [figure omitted; refer to PDF] Then, there exist [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

We will now give the proof of Theorem 2.

Proof.

For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a sequence of complex numbers and let [figure omitted; refer to PDF] be a sequence of [figure omitted; refer to PDF] atoms such that [figure omitted; refer to PDF] For each [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Observe that, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] satisfies (11)-(12) with the same constant [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfies (7)-(9) with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] by Lemma 8, [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] It follows from Lemma 5 that [figure omitted; refer to PDF] By the translation invariance of [figure omitted; refer to PDF] and (42) and (35), we have [figure omitted; refer to PDF] By applying (36), (42), and (43), we obtain (4).

Conflict of Interests

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

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