Circular average relative to fractal measures

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.