Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

We establish continuity mapping properties of the noncentered fractional maximal operator $M_{\beta }$ in the endpoint input space $W^{1,1}({\mathbb R}^d)$ for $d \geq 2$ in the cases for which its boundedness is 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 the centered counterpart $M_\beta ^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_\beta ^c$ implies its continuity.