Hp → Hp boundedness implies Hp → Lp boundedness

Yongsheng Han, Ji Li, Guozhen Lu*, Peiyong Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

Abstract

In this paper, we explore a general method to derive Hp → Lp boundedness from Hp → Hp boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([19]). 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 Hp → Lp boundedness on product spaces of homogeneous type in the sense of Coifman and Weiss ([5]) 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 Hp(ℝn) and H p(ℝn × ℝm) where the maximal function characterization exists. The key idea is to prove ∥f∥L p ≤ C ∥f∥Hp for f ∈ Lq ∩ Hp (1 < q < ∞, 0 < p ≤ 1). It is surprising that this simple result even in this classical setting has been absent in the literature.

Original languageEnglish
Pages (from-to)729-756
Number of pages28
JournalForum Mathematicum
Volume23
Issue number4
DOIs
Publication statusPublished - Jul 2011
Externally publishedYes

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