Marstrand's Approximate Independence of Sets and Strong Differentiation of the Integral

A constructive proof is given for the existence of a function belonging to the product Hardy space $H^1(\mathsf{R} \times \mathsf{R})$ and the Orlicz space $L(\log L)^{\epsilon}(\mathsf{R}^{2})$ for all $0<\epsilon <1$, for all whose integral is not strongly differentiable almost everywhere on a set of positive measure. It consists of a modification of a non-negative function created by J. M. Marstrand. In addition, we generalize the claim concerning "approximately independent sets" that appears in his work in relation to hyperbolic-crosses. Our generalization, which holds for any sets with boundary of sufficiently low complexity in any Euclidean space, has a version of the second Borel-Cantelli Lemma as a corollary.

Duality theory of weighted Hardy spaces with arbitrary number of parameters

In this paper, we use the discrete Littlewood–Paley–Stein analysis to get the duality result of the weighted product Hardy space for arbitrary number of parameters under a rather weak condition on the product weight