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 $\widetilde{\mathcal{M}}_{\beta}$ 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 $$\|\nabla \widetilde{\mathcal{M}}_{\beta}f\|_q\lesssim_{d,\beta}\|\nabla f\|_1.$$ We then prove that the map $$f\mapsto \big | \nabla \widetilde{\mathcal{M}}_{\beta}f \big |$$ 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 $f\mapsto |\nabla \widetilde{M}_{\beta}f|$ from $W^{1,1}_{\text{rad}}(\mathbb{R}^d)$ 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 $\mathbb{R}^d$.