Continuity of the gradient of the fractional maximal operator on $W^{1,1}({\mathbb{R}}^d)$

Abstract

We establish that the map $f\mapsto |\nabla {\mathcal{M}}_{\alpha }f|$ is continuous from $W^{1,1}({\mathbb{R}}^d)$ to $L^q({\mathbb{R}}^d)$, where $\alpha \in (0,d)$, $q=\frac{d}{d-\alpha }$ and ${\mathcal{M}}_{\alpha }$ denotes either the centered or non-centered fractional Hardy--Littlewood maximal operator. In particular, we cover the cases $d>1$ and $\alpha \in (0,1)$ in full generality, for which results were only known for radial functions.

12 pages, 1 figure