Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w ∈ A∞. In this paper, we introduce weighted-(p, q) atomic Hardy spaces Hwp,q(ℤn ×ℤm) for 0 < p ⩽ 1, q >qw and show that the weighted Hardy space Hwp (ℤn × ℤm) defined via Littlewood-Paley square functions coincides with Hwp,q (ℤn × ℤm) for 0 < p ⩽ 1, q > qw. As applications, we get a general principle on the Hwp (ℤn × ℤm) to Lwp(ℤn ×ℤm) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.