Regularity of the centered fractional maximal function on radial functions

Abstract

We study the regularity properties of the centered fractional maximal function $M_{\hat{I}}^2$. More precisely, we prove that the map $f\mapsto |\nabla M_{\hat{I}}^{2}f|$ is bounded and continuous from $W^{1,1}({\mathbb{R}}^d)$ to $L^q({\mathbb{R}}^d)$ in the endpoint case $q=d/(d-{\hat{I}}^2)$ 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.

22 pages, 1 figure