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(ℝn×ℝm) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on H p(ℝn×ℝm) 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(ℝn×ℝm) 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(ℝn×ℝm) to H p(ℝn×ℝm) is also obtained.
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Communicated by Richard Rochberg.
G. Lu is partly supported by US NSF grants DMS0500853 and DMS0901761 and NSFC of China grant No. 10710207. K. Zhao is partly supported by NNSF-China No. 10671115.
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Han, Y., Lu, G. & Zhao, K. Discrete Calderón’s Identity, Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces. J Geom Anal 20, 670–689 (2010). https://doi.org/10.1007/s12220-010-9123-6
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DOI: https://doi.org/10.1007/s12220-010-9123-6
Keywords
- Boundedness
- Calderón-Zygmund operator
- Calderón’s identity
- Multiparameter Hardy spaces
- Atomic decomposition
- Boundedness criterion of operators