Endpoint Sobolev continuity of the fractional maximal function in higher dimensions

Abstract

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

18 pages. Revised version incorporating referee suggestions. To appear in Int. Math. Res. Not. IMRN