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.