Sobolev regularity of polar fractional maximal functions

TL;DR

This paper investigates the Sobolev regularity of the uncentered fractional Hardy-Littlewood maximal operator on the sphere, establishing bounds, continuity, and implications for related conjectures in harmonic analysis.

Contribution

It provides new bounds and continuity results for fractional maximal functions acting on polar Sobolev functions, and links local boundedness conjectures to regularity properties.

Findings

Proved Sobolev norm bounds for the fractional maximal operator on the sphere.

Established continuity of the gradient map for polar Sobolev functions.

Connected local boundedness conjectures to regularity and continuity results.

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 $\mathbb{R}^d$.