Sobolev regularity of polar fractional maximal functions

Abstract

We study the Sobolev regularity on the sphere S d of the uncentered fractional Hardy-Littlewood maximal operator M β at the endpoint p = 1, when acting on polar data. We firstWe then prove that the mapwhen restricted to polar data. Our methods allow us to give a new proof of the continuity of the map f → |∇ M β f | from W 1,1 rad (R d ) to L q (R d ). Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy-Littlewood maximal operator M β implies the continuity of the map f → |∇M β f | from W 1,1 to L q , in the context of polar functions on S d and radial functions on R d .

Sunrise strategy for the continuity of maximal operators

“…In this section we develop the first part of our overall strategy of the proof of Theorem 1, by establishing a control of the convergence near the origin. This is inspired in an argument of [12]. 1 Recall that we allow for the possibility B 0 (x) = {x}.…”

Continuity of the gradient of the fractional maximal operator on $W^{1,1}(\mathbb{R}^d)$

“…The proof in [24] is based on the corresponding bound for the dyadic maximal operator in the non-fractional case α = 0 in [25]. Other interesting related results in the context of fractional maximal functions have recently been proven in [4,9,12,21,22].…”