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.