Sobolev regularity of polar fractional maximal functions

Abstract

We study the Sobolev regularity on the sphere Sd of the uncentered fractional Hardy–Littlewood maximal operator M˜β at the endpoint p=1, when acting on polar data. We first prove that if q=dd−β, 0<β<d and f is a polar W1,1(Sd) function, we have ‖∇M˜βf‖q≲d,β‖∇f‖1. We then prove that the map f↦|∇M˜βf| is continuous from W1,1(Sd) to Lq(Sd) when restricted to polar data. Our methods allow us to give a new proof of the continuity of the map f↦|∇M˜βf| from Wrad1,1(Rd) to Lq(Rd). Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy–Littlewood maximal operator Mβ implies the continuity of the map f↦|∇Mβf| from W1,1 to Lq, in the context of polar functions on Sd and radial functions on Rd.