Continuity for the one-dimensional centered Hardy-Littlewood maximal operator at the derivative level

TL;DR

This paper proves the continuity of the derivative of the centered Hardy-Littlewood maximal operator acting on Sobolev space functions, resolving an open question in harmonic analysis.

Contribution

It establishes the continuity of the map from Sobolev space to Lebesgue space for the maximal operator's derivative, answering a previously open problem.

Findings

Proves continuity of the derivative map for the maximal operator

Solves an open question in harmonic analysis

Advances understanding of maximal operators in Sobolev spaces

Abstract

We prove the continuity of the map $f \mapsto (Mf)'$ from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$, where $M$ is the centered Hardy-Littlewood maximal operator. This solves a question posed by Carneiro, Madrid and Pierce.