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Research Article | Open Access

Volume 2016 |Article ID 1570109 | 5 pages | https://doi.org/10.1155/2016/1570109

# Logarithmic Bounds for Oscillatory Singular Integrals on Hardy Spaces

Accepted07 Feb 2016
Published24 Feb 2016

#### 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  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 ). 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  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: .

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 thatholds 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 haveCase 2 (). In this case, we letIt follows from Theorem of  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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