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

Abstract

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