APDE seminar@BCAM: Endpoint estimates for the fractal circular maximal function and related local smoothing

Abstract

The spherical maximal function is a relevant object in harmonic analysis, connected to the solutions of the wave equation and related smoothing properties, and variants of it have been widely studied in the literature. In recent times, there has been an increasing interest in understanding sharp forms of L^p-L^q estimates for the spherical maximal function when the supremum is taken over dilation sets of fractal dimensions of different nature. In this talk we will prove missing endpoint estimates for the fractal spherical maximal function which were open when d=2, and study closely related  L^p-L^q local smoothing estimates for the wave operator over fractal dilation sets. Our approach relies on bilinear restriction estimates for the cone due to T. Wolff and T. Tao.

This is joint work with Sanghyuk Lee, Feng Zhang, and Shuijiang Zhao.