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

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