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

Abstract

We establish that the map $f \mapsto {\lvert \nabla \mathcal{M}_\alpha f \rvert}$ 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 $M_\alpha$ denotes either the centered or non-centered fractional Hardy–Littlewood maximal operator. In particular, we cover the cases $d \gt 1$ and $\alpha \in (0, 1)$ in full generality, for which results were only known for radial functions.