About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 987214, 7 pages
http://dx.doi.org/10.1155/2014/987214
Research Article

Boundedness for a Class of Singular Integral Operators on Both Classical and Product Hardy Spaces

Department of Mathematics, Shantou University, Shantou 515063, China

Received 10 September 2013; Accepted 2 January 2014; Published 27 February 2014

Academic Editor: Henryk Hudzik

Copyright © 2014 Chaoqiang Tan. 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 found that the classical Calderón-Zygmund singular integral operators are bounded on both the classical Hardy spaces and the product Hardy spaces. The purpose of this paper is to extend this result to a more general class. More precisely, we introduce a class of singular integral operators including the classical Calderón-Zygmund singular integral operators and show that they are bounded on both the classical Hardy spaces and the product Hardy spaces.

1. Introduction

The classical Hardy spaces and the product Hardy spaces play important roles in Harmonic analysis, which are due to the original work of Fefferman and Stein [1] and Gundy and Stein [2], respectively. It is well known that these two Hardy spaces are essentially different. For instance, see [35]. It has been known that the classical Calderón-Zygmund singular integral operators are bounded on the classical Hardy spaces and the product singular integral operators are bounded on the product Hardy spaces. Surprisingly, in [6], we found that the classical Calderón-Zygmund singular integral operators are also bounded on the product Hardy spaces. More precisely, if is a bounded operator on with , where the kernel and satisfies for and , then is bounded on both the classical Hardy spaces and the product Hardy space .

A natural question arises: weather there exist a more general class of operators that are bounded both on the classical Hardy spaces and the product Hardy spaces. The purpose of this paper is to answer this question. Now we first recall the definitions of the classical Hardy spaces (see [1] for more details) and the product Hardy spaces (see [2, 7] for more details).

We let be the set including all that satisfy and for all . And let be the set including all that satisfy and for all .

Given an , the Littlewood-Paley-Stein square function of is defined by , where , . And the discrete square function is defined by , where are dyadic cubes in with the side length and the center and is the characteristic function. It is well known that if , then and, for different , .

The classical Hardy space is then defined by where denotes the space of distributions modulo polynomials. The norm is defined by .

Similarly, given a , the product square function of is defined by , where , . And the discrete square function is defined by , where are dyadic intervals in with the side length , and the center , respectively. Also, for , and, for different ,  .

The product Hardy space is then defined by The norm is defined by .

The following theorem is our main result.

Theorem 1. If , and is an operator bounded on with , where the kernel and satisfies for all , then is bounded on both and .

Remark 2. (I) In [8], we have shown that the operator is bounded on for all .
(II) It is easy to verify that the classical Calderón-Zygmund singular integral operators are contained in our class. Moreover, some more operators will be in our class. For example, the operator with .

Throughout this paper, we do the following conventions.(a)The notation means that for some positive constants .(b)If is a cube or interval, then we denote by its center and by its side length.(c)For and a large positive integer , we denote the set , where are dyadic cubes in with side length and .(d)For and a large positive integer , we denote the set , where and are dyadic intervals in with side length and , respectively} and .(e) means the minimum of and .

2. Proof of Theorem 1

The first crucial tool in the proof of Theorem 1 is to apply the following discrete Calderón identity (see [9] for more details).

Lemma 3. If , then consider the following.
() Suppose that with . Then for all , there exist a large positive integer (depending only on ) and a function such that , where the series converges in . Moreover, and .
() Suppose that with . Then for all , there exist a large positive integer (depending only on ) and a function such that , where the series converges in . Moreover, and .

For the proof, we refer readers to [9].

The second crucial tool in the proof of Theorem 1 is the following orthogonal estimates.

Lemma 4. Suppose that and is the kernel as in Theorem 1; then
() for with , one has , for all and , where is a constant independent of and ;
() for with , one has , for all and , where is a constant independent of , , and .

Proof. (a) Since is single-parameter dilation invariant, that is, for each , satisfies the same hypotheses, with the same bounds as . We just need to show that , for all . Without loss of generality, we may assume that . To get the required estimate, we will discuss it in the following three cases: (I) ; (II) or ; (III) .
For case (I), , by the moment condition of , we have
For case (II), or , we have
For case (III), , we let with and when and when . We have
(b) Without loss of generality, we may assume that . The required estimate will be discussed in the following four cases: (I) ; (II) ; (III) ; (IV) .
For case (I), , by the moment condition of , we have
For case (II), , similarly, we have
The cases (II) and (III) are symmetric, so case (III) follows.
For case (IV), , we let with and when and when . Then
This completes the proof of Lemma 4.

As a consequence of Lemma 4, we have the following.

Lemma 5. (a) Under hypothesis (a) of Lemma 4, one has for all and .
(b) Under hypothesis (b) of Lemma 4, one has for all , , and and .

The proof of Lemma 5 is based on the following two observations: convolution operation is commutative; that is, (or );    (or ) satisfies the same estimate as (or ) in Lemma 4 with the bound (or ). The details are left to the readers.

The last crucial tool in the proof of Theorem 1 is the following strongly maximal function estimates.

Lemma 6. Suppose that , , , , , and , and . Then consider the following.
() If , , and , then one has where is the strongly maximal operator.
() If , , and , then one has

For the proof, we refer readers to [10].

Proof of Theorem 1. Firstly we show that is bounded on the classical Hardy space . Since is dense in , we just need to show that, for all , we have ; that is, for a fixed ,
Note that
For , , and , applying (a) of Lemma 3, we have
By (a) of Lemma 5, we have
Applying (a) of Lemma 6 with and , we have
Therefore,
Applying Fefferman-Stein’s vector-valued strong maximal inequality (see [11] for more details) on , we have
The proof of ’s boundedness on is almost the same as above; that is, we just need to replace (a) of the required lemmas to (b). Here we omit the details.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Chaoqiang Tan is supported by SRF for the Doctoral Program of Higher Education (Grant no. 20104402120002).

References

  1. C. Fefferman and E. M. Stein, “Hp spaces of several variables,” Acta Mathematica, vol. 129, no. 3-4, pp. 137–193, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. F. Gundy and E. M. Stein, “Hp theory for the poly-disc,” Proceedings of the National Academy of Sciences of the United States of America, vol. 76, no. 3, pp. 1026–1029, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Carleson, “A counterexample for measures bounded on Hp for the bi-disc,” Mittag Leffler Report 7, 1974.
  4. S. Y. Chang and R. Fefferman, “A continuous version of duality of H1 with BMO on the bidisc,” Annals of Mathematics, vol. 112, no. 1, pp. 179–201, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  5. S. Y. Chang and R. Fefferman, “Some recent developments in Fourier analysis and Hp-theory on product domains,” Bulletin of the American Mathematical Society, vol. 12, no. 1, pp. 1–43, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  6. C. Q. Tan, “Boundedness of classical Calderón-Zygmund convolution operators on product Hardy space,” Mathematical Research Letters, vol. 20, no. 3, pp. 591–599, 2013. View at Publisher · View at Google Scholar
  7. R. Fefferman, “Calderón-Zygmund theory for product domains: Hp spaces,” Proceedings of the National Academy of Sciences of the United States of America, vol. 83, no. 4, pp. 840–843, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  8. C. Q. Tan, “Singular integral operators between the classical type and the product type,” Acta Scientiarum Naturalium Universitatis Sunyatseni, vol. 51, no. 5, pp. 58–62, 2012. View at Zentralblatt MATH · View at MathSciNet
  9. Y. S. Han, G. Z. Lu, and K. Zhao, “Discrete Calderón’s identity, atomic decomposition and boundedness criterion of operators on multiparameter hardy spaces,” Journal of Geometric Analysis, vol. 20, no. 3, pp. 670–689, 2010.
  10. Y.-S. Han, “Plancherel-Pólya type inequality on spaces of homogeneous type and its applications,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3315–3327, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. E. M. Stein, “Note on singular integrals,” Proceedings of the American Mathematical Society, vol. 8, pp. 250–254, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet