# Sobolev regularity of polar fractional maximal functions

Research paper by **Cristian González-Riquelme**

Indexed on: **24 Feb '21**Published on: **12 Oct '19**Published in: **arXiv - Mathematics - Classical Analysis and ODEs**

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#### Abstract

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 $\mathbb{R}^d$.