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

In this paper we establish a discrete Calderón’s identity which converges in both Lq(Rn+m) (1 < q <∞) and Hardy space H(R ×Rm) (0 <p ≤ 1). Based on this identity, we derive a new atomic decomposition into (p, q)-atoms (1 < q <∞) on H(R × R) 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.