The one-dimensional centred maximal function diminishes the variation of indicator functions

TL;DR

This paper establishes precise bounds on the variation of one-dimensional centered Hardy--Littlewood maximal functions of indicator functions, characterizing maximizers and extending results to broader function classes in both continuous and discrete contexts.

Contribution

It provides sharp local and global variation bounds for the maximal functions of indicator functions, including characterizations of maximizers and extensions to larger function classes.

Findings

Sharp local and global variation bounds established

Maximizers characterized in both continuous and discrete settings

Results extended to a broader class of functions

Abstract

We prove sharp local and global variation bounds for the centred Hardy--Littlewood maximal functions of indicator functions in one dimension. We characterise maximisers, treat both the continuous and discrete settings and extend our results to a larger class of functions.