Article

Journal of Geometric Analysis

, Volume 20, Issue 3, pp 670-689

First online:

Discrete Calderón’s Identity, Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

  • Y. HanAffiliated withDepartment of Mathematics, Auburn University
  • , G. LuAffiliated withDepartment of Mathematics, Wayne State University Email author 
  • , K. ZhaoAffiliated withCollege of Mathematics, Qingdao University

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Abstract

In this paper we establish a discrete Calderón’s identity which converges in both L q (ℝn+m) (1<q<∞) and Hardy space H p (ℝ n ×ℝ m ) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on H p (ℝ n ×ℝ m ) for 0<p≤1. As an application, we prove that an operator T, which is bounded on L q (ℝn+m) for some 1<q<∞, is bounded from H p (ℝ n ×ℝ m ) to L p (ℝn+m) if and only if T is bounded uniformly on all (p,q)-product atoms in L p (ℝn+m). The similar result from H p (ℝ n ×ℝ m ) to H p (ℝ n ×ℝ m ) is also obtained.

Keywords

Boundedness Calderón-Zygmund operator Calderón’s identity Multiparameter Hardy spaces Atomic decomposition Boundedness criterion of operators

Mathematics Subject Classification (2000)

42B30 42B20