Regularity of the centered fractional maximal function on radial functions

TL;DR

This paper investigates the regularity of the centered fractional maximal function for radial functions, establishing boundedness and continuity of its gradient in certain Sobolev spaces, and extends results to the one-dimensional case.

Contribution

It proves the boundedness and continuity of the gradient map of the centered fractional maximal function for radial functions, including the one-dimensional case without radiality.

Findings

Boundedness of the gradient map in the endpoint case for radial functions.

Continuity of the gradient map in the Sobolev space.

Extension of results to the one-dimensional case without radiality.

Abstract

We study the regularity properties of the centered fractional maximal function $M_{\beta}$. More precisely, we prove that the map $f \mapsto |\nabla M_\beta f|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-\beta)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.