Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces

TL;DR

This paper proves a new inequality relating the variation of the centered Hardy--Littlewood maximal operator applied to certain simple functions, strengthening previous results and conjecturing broader applicability to all bounded variation functions.

Contribution

It establishes a stronger variation inequality for piecewise constant functions with gaps, extends previous results, and introduces a transference principle between continuous and discrete settings.

Findings

Proved a variation inequality for specific simple functions.

Extended the inequality to a discrete setting.

Conjectured the inequality holds for all functions of bounded variation.

Abstract

Let $M$ denote the centered Hardy--Littlewood operator on $\mathbb{R}$. We prove that \[ {\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big| \] for piecewise constant functions $f$ with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt \cite{BW} proved for indicator functions of bounded variation vanishing at $\pm\infty$. We conjecture that the inequality holds for all functions of bounded variation, representing a stronger version of the existing conjecture ${\rm Var} (Mf)\le {\rm Var} (f)$. We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.