Sunrise strategy for the continuity of maximal operators

TL;DR

This paper introduces a novel 'sunrise' strategy inspired by harmonic analysis to prove the $W^{1,1}$-continuity of various maximal operators at the gradient level, extending recent results and applying to multiple settings.

Contribution

The paper develops a new decomposition technique for maximal operators, enabling the proof of their $W^{1,1}$-continuity, including for radial and polar functions across different dimensions.

Findings

Proves the $W^{1,1}$-continuity of the uncentered Hardy-Littlewood maximal operator on radial functions in $ ext{dim} \, d \\geq 2$.

Establishes the continuity of the gradient map for maximal operators in various geometric contexts.

Extends the 'sunrise' strategy to non-tangential and spherical maximal operators.

Abstract

In this paper we address the $W^{1,1}$-continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into "lateral" maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy-Littlewood maximal operator $\widetilde{M}$ acting on $W^{1,1}_{\rm rad}(\mathbb{R}^d)$, the subspace of $W^{1,1}(\mathbb{R}^d)$ consisting of radial functions. In dimension $d\geq 2$ it was recently established by H. Luiro that the map $f \mapsto \nabla \widetilde{M} f$ is bounded from $W^{1,1}_{\rm rad}(\mathbb{R}^d)$ to $L^1(\mathbb{R}^d)$, and we show that such map is also continuous. Further applications of the sunrise strategy in connection with the $W^{1,1}$-continuity problem include non-tangential maximal operators on $\mathbb{R}^d$ acting on radial functions when $d\geq 2$ and general functions when $d=1$, and the uncentered Hardy-Littlewood maximal operator on the sphere $\mathbb{S}^d$ acting on polar functions when $d\geq 2$ and general functions when $d=1$.