Regularity of fractional maximal functions through Fourier multipliers

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$\mathrm{.}Wealsoshowthatthesphericalfractionalmaximalfunctionmaps$L^{p}$intoafirstorderSobolevspaceindimensions$n \geq 5$.