Research Article | Open Access

# Logarithmic Bounds for Oscillatory Singular Integrals on Hardy Spaces

**Academic Editor:**Maria Alessandra Ragusa

#### 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 thatfor 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.

Doeshold 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 thatfor 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 letwhere .

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

*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 byThen, there exists independent of such thatholds for all and .*

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

Writewith for . Then, there exist such thatThus, we haveFor , letThen, there are polynomials , on , , on , and on such thatLet for and if . Then, whereSince 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 thatfor 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 haveLet . It follows from (11) and (7)-(8) and Lemma 7 thatIf , then (23) follows from (24) and (26).

Thus, we may assume that . To finish the proof, it suffices to show thatWe will establish (27) by discussing two cases.*Case** 1 (**)*. In this case, we have*Case** 2 (**)*. In this case, we letIt follows from Theorem of [3] thatFor and , we haveBy (30)-(31) andwe haveThus, (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 thatfor , andfor 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 thatFor 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 , . Sinceby Lemma 8,which implies thatIt follows from Lemma 5 thatBy the translation invariance of and (42) and (35), we haveBy 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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#### Copyright

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.