Journal of Function Spaces
Volume 2016 (2016), Article ID 1570109, 5 pages
http://dx.doi.org/10.1155/2016/1570109

Research Article

Logarithmic Bounds for Oscillatory Singular Integrals on Hardy Spaces

Hussain Al-Qassem,¹ Leslie Cheng,² and Yibiao Pan³

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 28 December 2015; Accepted 7 February 2016

Academic Editor: Maria Alessandra Ragusa

Copyright © 2016 Hussain 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.

Abstract.

We establish a logarithmic bound for oscillatory singular integrals with quadratic phases on the Hardy space . The logarithmic rate of growth is the best possible.

1. Introduction

For , let be a Calderón-Zygmund kernel on and let be a polynomial of variables with real coefficients. Consider the following oscillatory singular integral operator:

It is well known that is bounded from to when and also from to . Additionally, and bounds are dependent on the degree of the phase polynomial only, not its coefficients (see [1, 2]).

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

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

Theorem 1 (see [5]). Let , , and be a polynomial of degree in with real coefficients. Let be a Calderón-Zygmund kernel and let be given as in (1). Then, there exists a positive constant such that

for all . The constant may depend on , and but is independent of the coefficients of .

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

Does

hold for all ?

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 and be a quadratic polynomial in with real coefficients. Let be a Calderón-Zygmund kernel and let be given as in (1). Then, there exists a positive constant such that

for all . The constant may depend on and but is independent of the coefficients of .

We point out that 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 and , let and denote the Euclidean volume of .

Let be a function in the Schwartz space such that . For each and , we let

where .

Definition 3. For a nonnegative, locally integrable function on , the Hardy space is given by

with .

Definition 4. A measurable function on is called atom if there exist and such that

Lemma 5 (see [9, 10]). For each , there exist atoms and coefficients such that

Definition 6. A function is called a Calderón-Zygmund kernel if the following are true:

(i)There exists such that

 holds for all .(ii)For all ,

Lemma 7. Let for and . Define operator by

Then, there exists independent of such that

holds for all and .

Proof. We start by treating the more difficult case . The other case, , will be briefly considered later.
Write

with for . Then, there exist such that

Thus, we have

For , let

Then, there are polynomials , on , , on , and on such that

Let for and if . Then,

where

Since and , we have

The treatment of the case only involves the Fourier transform step of the preceding argument. Details are omitted.

Lemma 8. Let and be a quadratic polynomial in with real coefficients. Let be a Calderón-Zygmund kernel satisfying (11)-(12) and let be given as in (1). Then, there exists a positive constant such that

for every atom which satisfies (7)–(9) with and . The constant may depend on and but is independent of , , and .

Proof. By the uniform boundedness of on and (7)-(8),

By (11), we have

Let . It follows from (11) and (7)-(8) and Lemma 7 that

If , then (23) follows from (24) and (26).
Thus, we may assume that . To finish the proof, it suffices to show that

We will establish (27) by discussing two cases.
Case 1 (). In this case, we have

Case 2 (). In this case, we let

It follows from Theorem of [3] that

For and , we have

By (30)-(31) and

we have

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 , let denote the th Riesz transform; that is,

Then, there exist such that

for , and

for all .

We will now give the proof of Theorem 2.

Proof. For , let be a sequence of complex numbers and let be a sequence of atoms such that

For each , let and such that and . Then,

where and . Observe that, for each , satisfies (11)-(12) with the same constant and satisfies (7)–(9) with , . Since

by Lemma 8,

which implies that

It follows from Lemma 5 that

By the translation invariance of and (42) and (35), we have

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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