Weighted mixed-norm estimates for circular averages and exceptional set estimates for the wave equation

TL;DR

This paper establishes new mixed-norm estimates for circular averages over fractal measures in the plane, leading to novel results on the regularity of wave equation solutions and exceptional set estimates.

Contribution

It introduces sharp and improved mixed-norm estimates for fractal measures, extending Wolff's and Bourgain's circular maximal functions, with applications to wave equation regularity.

Findings

Sharp mixed-norm estimates for -dimensional fractal measures.

Improved estimates for  <   for  < .

First results on Hf6lder regularity in time for wave solutions.

Abstract

We prove mixed-norm estimates for circular averages with respect to $\alpha$-dimensional fractal measures on $\mathbb{R}^2$, using circle tangency bounds when $\alpha \in (0,1]$ and a $\delta$-discretized slicing lemma for fractals when $\alpha \in (1,2]$. The former estimate is sharp, while the latter improves previous results for $\alpha \in (\frac{3}{2},2]$. These estimates can be viewed as X-ray-type extensions of Wolff's and Bourgain's circular maximal functions. As applications, we obtain new exceptional set estimates for the radial integrability of functions in Lebesgue spaces, as well as for the H\"older regularity in time of solutions to the linear wave equation on $\mathbb{R}^2$. The latter results are the first of their kind.