Weighted Endpoint Estimates for Singular Integral Operators Associated with Zygmund Dilations

Abstract

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 $H^p_{\mathfrak{z},w}$ and prove the boundedness for these operators on $H^p_{\mathfrak{z},w}$ for certain $p \leq 1$, 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.