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

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