Circular average relative to fractal measures

TL;DR

This paper establishes new $L^p$-$L^q$ estimates for circular and spherical averages over fractal measures, unifying various maximal estimates and linking them to wave operator smoothing effects.

Contribution

It introduces novel $L^p$-$L^q$ estimates for averages over fractal measures, connecting these to wave operator smoothing, and extends results to spherical averages.

Findings

New $L^p$-$L^q$ estimates for fractal measure averages

Unified framework for maximal estimates of circular averages

Connections between averaging estimates and wave operator smoothing

Abstract

We prove new $L^p$- $L^q$ estimates for averages over dilates of the circle with respect to $\alpha$-dimensional fractal measure, which unify different types of maximal estimates for the circular average. Our results are consequences of $L^p$- $L^q$ smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.