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

  • Raquel Cabral

Abstract

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.
Published
2016-08-19
How to Cite
Cabral, R. (2016). Marstrand’s Approximate Independence of Sets and Strong Differentiation of the Integral. MATHEMATICA SCANDINAVICA, 119(1), 92-112. https://doi.org/10.7146/math.scand.a-24186
Section
Articles