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 α-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. 2020 Mathematics Subject Classification. 42B25. Key words and phrases. circular average, general measures.