Atomic Decomposition and Boundedness Criterion of Operators on Multi-parameter Hardy Spaces of Homogeneous Type

The main purpose of this paper is to derive a new (p, q)-atomic decomposition on the multi-parameter Hardy space H(X1 ×X2) for 0 < p0 < p ≤ 1 for some p0 and all 1 < q < ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L(X1 × X2) (for 1 < q < ∞) and Hardy space H(X1 × X2) (for 0 < p ≤ 1). As an application, we prove that an operator T , which is bounded on L(X1 × X2) for some 1 < q < ∞, is bounded from H(X1 ×X2) to L(X1 ×X2) if and only if T is bounded uniformly on all (p, q)-product atoms in L(X1 ×X2). The similar boundedness criterion from H(X1 ×X2) to H(X1 ×X2) is also obtained.