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.

25pages

References in corpus (3)

• Spherical maximal functions and fractal dimensions of dilation sets
• Fractal Strichartz estimate for the wave equation
• Dimension of divergence set of the wave equation

Cited by in corpus (1)

• $L^p-L^q$ local smoothing estimates for the wave equation via $k$-broad Fourier restriction