Regularity of fractional maximal functions through Fourier multipliers

TL;DR

This paper establishes endpoint bounds for derivatives of fractional maximal functions with smooth kernels or lacunary radii, and demonstrates that spherical fractional maximal functions map L^p into Sobolev spaces in higher dimensions.

Contribution

It provides new endpoint bounds for derivatives of fractional maximal functions and explores their mapping properties into Sobolev spaces in higher dimensions.

Findings

Endpoint bounds for derivatives of fractional maximal functions established.

Spherical fractional maximal functions map L^p to Sobolev spaces in dimensions n ≥ 5.

Abstract

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