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We establish continuity mapping properties of the non-centered fractionalmaximal operator $M_{\beta}$ in the endpoint input space$W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundednessis known. More precisely, we prove that for $q=d/(d-\beta)$ the map $f \mapsto|\nabla M_\beta f|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to$L^{q}(\mathbb{R}^d)$ for $ 0 < \beta < 1$ if $f$ is radial and for $1 \leq\beta < d$ for general $f$. The results for $1\leq \beta < d$ extend to thecentered counterpart $M_\beta^c$. Moreover, if $d=1$, we show that theconjectured boundedness of that map for $M_\beta^c$ implies its continuity.

Endpoint Sobolev continuity of the fractional maximal function in higher dimensions