Journal of Geometric Analysis

, Volume 20, Issue 3, pp 670–689

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

Article

DOI: 10.1007/s12220-010-9123-6

Cite this article as:
Han, Y., Lu, G. & Zhao, K. J Geom Anal (2010) 20: 670. doi:10.1007/s12220-010-9123-6

Abstract

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

Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsAuburn UniversityAuburnUSA
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA
  3. 3.College of MathematicsQingdao UniversityQingdaoChina

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