# 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.

## Key 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 $n \geq 5$.