Sobolev regularity of polar fractional maximal functions

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 $$\mathrm{\parallel }\mathrm{\nabla }{\tilde{\mathcal{M}}}_{\beta }f{\mathrm{\parallel }}_q{\lesssim }_{d,\beta }\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_1\mathrm{.}$$ We then prove that the map $$f\mapsto \mid \mathrm{\nabla }{\tilde{\mathcal{M}}}_{\beta }f\mid$$ 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 $f\mapsto \mathrm{\mid }\mathrm{\nabla }{\tilde{M}}_{\beta }f\mathrm{\mid }$ 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 $f\mapsto \mathrm{\mid }\mathrm{\nabla }{M}_{\beta }f\mathrm{\mid }$ from $W^{1,1}$ to $L^q$, in the context of polar functions on ${\mathbb{S}}^d$ and radial functions on ${\mathbb{R}}^d$.