Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w is an element of A(infinity). In this paper, we introduce weighted-(p,q) atomic Hardy spaces H-w(p,q) (R-n x R-m) for 0 < p <= 1, q > q(w) and show that the weighted Hardy space H-w(p) (R-n x R-m) defined via Littlewood-Paley square functions coincides with H-w(p,q) (R-n x R-m) for 0 < p <= 1, q > q(w). As applications, we get a general principle on the H-w(p) (R-n x R-m) to L-w(p)(R-n x R-m) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.