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

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.