BV continuity for the uncentered Hardy--Littlewood maximal operator

TL;DR

This paper proves that the uncentered Hardy--Littlewood maximal operator is continuous on the space of functions of bounded variation in one dimension, answering a previously open question.

Contribution

It establishes the continuity of the uncentered Hardy--Littlewood maximal operator on BV space, a novel result in harmonic analysis.

Findings

Proves continuity of the operator on BV()

Answers an open question by Carneiro, Madrid, and Pierce

Advances understanding of maximal operators in function spaces

Abstract

We prove the continuity of the map $f \mapsto \widetilde{M}f$ from $BV(\mathbb{R})$ to itself, where $\widetilde{M}$ is the uncentered Hardy--Littlewood maximal operator. This answers a question of Carneiro, Madrid and Pierce.