Sobolev regularity of polar fractional maximal functions

Abstract

We study the Sobolev regularity on the sphere ${\mathbb{S}}^d$ of the uncentered fractional Hardy-Littlewood maximal operator ${\tilde{\mathcal{M}}}_{\beta }$ at the endpoint $p=1$, when acting on polar data. We first prove that if $q=\frac{d}{d-\beta }$, $0<\beta <d$ and $f$ is a polar $W^{1,1}({\mathbb{S}}^d)$ function, we have $$\Vert \nabla {\tilde{\mathcal{M}}}_{\beta }f{\Vert }_q{\lesssim }_{d,\beta }\Vert \nabla f{\Vert }_1.$$ We then prove that the map $$f\mapsto |\nabla {\tilde{\mathcal{M}}}_{\beta }f|$$ is continuous from $W^{1,1}({\mathbb{S}}^d)$ to $L^q({\mathbb{S}}^d)$ when restricted to polar data. Our methods allow us to give a new proof of the continuity of the map $f\mapsto |\nabla {\tilde{M}}_{\beta }f|$ from $W_{\text{rad}}^{1,1}({\mathbb{R}}^d)$ to $L^q({\mathbb{R}}^d)$. Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy-Littlewood maximal operator $M_{\beta }$ implies the continuity of the map $f\mapsto |\nabla M_{\beta }f|$ from $W^{1,1}$ to $L^q$, in the context of polar functions on ${\mathbb{S}}^d$ and radial functions on ${\mathbb{R}}^d$.

arXiv comment20 pages. To appear in Nonlinear Analysis