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 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.