Abstract

This work concerns strong differentiation and operators on product Hardy spaces. We show, by counterexample, that strong differentiability of the integral fails even for functions in the intersection of H_{rect}^{1}(\mathbb{R}×\mathbb{R}) with L(log⁡L)^{ε}(\mathbb{R}^{2}) for all 0<ε<1. Our example is a modification of a function that appears in a work of J. M. Marstrand, where he makes a claim concerning” approximately independent sets”. We generalize his claim and, as a corollary, we obtain a version of the second Borel-Cantelli Lemma. In addition, we prove that a function f created by Papoulis to show that the strong differentiability of ∫f does not imply the same behavior for ∫|f|, belongs to the product Hardy space H_{rect}^{p}(\mathbb{R}×\mathbb{R}). The method that we develop to approach this example allows us to relax the sufficient conditions of the Chang-Fefferman atomic decomposition. In analogy with the proof of this result, we demonstrate that a theorem of R. Fefferman, which concludes H^{p}→L^{p}, 0<p≤1, boundedness of two-parameter operators from their behavior on rectangle atoms, can be generalized to settings with more parameters. This generalization enables us to extend a theorem of Pipher concerning boundedness of multiparameter Calderón-Zygmund operators from H^{p} to L^{p}. Furthermore, we present variants of Journé's Lemma, two of which hold for the product of \mathbb{R} with a metric measure space satisfying certain conditions.