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H p →→ H p boundedness implies H p →→ L p boundedness

H p →→ H p boundedness implies H p →→ L p boundedness In this paper, we explore a general method to derive H p →→ L p boundedness from H p →→ H p boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals (((Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals))). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this paper, we will illustrate further that this method will enable us to prove the H p →→ L p boundedness on product spaces of homogeneous type in the sense of Coifman and Weiss (((Lecture Notes in Math. 242: 1971))) where maximal function characterization of Hardy spaces is not available. Moreover, we also provide a particularly easy argument in those settings such as one parameter or multiparameter Hardy spaces and where the maximal function characterization exists. The key idea is to prove ‖‖ƒƒ‖‖ L p ≤≤ C ‖‖ƒƒ ‖‖ H p for ƒƒ ∈∈ L q ∩∩ H p (1 < q < ∞∞, 0 < p ≤≤ 1). It is surprising that this simple result even in this classical setting has been absent in the literature. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

H p →→ H p boundedness implies H p →→ L p boundedness

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References (48)

Publisher
de Gruyter
Copyright
©© de Gruyter 2011
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORM.2011.026
Publisher site
See Article on Publisher Site

Abstract

In this paper, we explore a general method to derive H p →→ L p boundedness from H p →→ H p boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals (((Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals))). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this paper, we will illustrate further that this method will enable us to prove the H p →→ L p boundedness on product spaces of homogeneous type in the sense of Coifman and Weiss (((Lecture Notes in Math. 242: 1971))) where maximal function characterization of Hardy spaces is not available. Moreover, we also provide a particularly easy argument in those settings such as one parameter or multiparameter Hardy spaces and where the maximal function characterization exists. The key idea is to prove ‖‖ƒƒ‖‖ L p ≤≤ C ‖‖ƒƒ ‖‖ H p for ƒƒ ∈∈ L q ∩∩ H p (1 < q < ∞∞, 0 < p ≤≤ 1). It is surprising that this simple result even in this classical setting has been absent in the literature.

Journal

Forum Mathematicum – de Gruyter

Published: Jul 1, 2011

Keywords: Multiparameter Hardy spaces; discrete Calderóón's identity; discrete Littlewood––Paley theory; min-max type inequality; Calderóón––Zygmund singular operators

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