Nikodym sets and maximal functions associated with spheres

TL;DR

This paper investigates spherical Nikodym sets and maximal functions, establishing sharp $L^p$ bounds and demonstrating that such sets must have full Hausdorff dimension, linking maximal function estimates to wave equation smoothing.

Contribution

It provides the first sharp $L^p$ estimates for spherical Nikodym maximal functions and connects these estimates to local smoothing for the wave equation on fractal measures.

Findings

Sharp $L^p$ bounds for spherical Nikodym maximal functions.

Nikodym sets for spheres have full Hausdorff dimension.

Maximal function estimates follow from wave equation local smoothing estimates.

Abstract

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp $L^p$-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that $L^p$-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures.