Differentiation of the integral, Hardy spaces and Calderón–Zygmund operators in the product setting

Abstract

This article concerns strong differentiation and operators on product Hardy spaces. We first show that an example, created by Papoulis, of a function on R2 whose integral is strongly differentiable almost everywhere, but the integral of its absolute value fails to be strongly differentiable on a set of positive measure, belongs to the product Hardy space H1(R×R). The methods that we develop enable us to present a relaxed version of Chang–Fefferman p-atoms with a lower number of required vanishing moments and no smoothness needed on the elementary particles. In analogy with the proof of this result, we show a generalization of a theorem of R. Fefferman which concludes Hp→Lp, 0<p≤1, boundedness of multiparameter operators from their behavior on rectangle atoms. In addition, we extend a result of Pipher concerning boundedness of multiparameter Calderón–Zygmund operators from Hp to Lp.