Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

Abstract

The main purpose of this paper is to derive a new (p, q)-atomic decomposition on the multi-parameter Hardy space H p (X 1 × X 2 ) for 0 < p 0 < p ≤ 1 for some p 0 and all 1 < q < ∞, where X 1 × X 2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L q (X 1 × X 2 ) (for 1 < q < ∞) and Hardy space H p (X 1 × X 2 ) (for 0 < p ≤ 1). As an application, we prove that an operator T , which is bounded on L q (X 1 × X 2 ) for some 1 < q < ∞, is bounded from H p (X 1 × X 2 ) to L p (X 1 × X 2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in L p (X 1 × X 2 ). The similar boundedness criterion from H p (X 1 × X 2 ) to H p (X 1 × X 2 ) is also obtained.