Weighted endpoint estimates for singular integral operators associated with zygmund dilations

The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. We develop the theory of the weighted multi-parameter Hardy space H _z,w ^p and prove the boundedness for these operators on H _z,w ^p for certain p ≤ 1, which provide endpoint estimates for those singular integral operators studied by Ricci-Stein [31] and Fefferman-Pipher [15]. We also establish the Calderón-Zygmund decomposition and interpolation theorem in this setting.