Regularity of fractional maximal functions through Fourier multipliers

Abstract

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n\ge 2$. We also show that the spherical fractional maximal function maps $L^p$ into a first order Sobolev space in dimensions $n\ge 5$.

final version, to appear in Journal of Functional Analysis