Sharp total variation results for maximal functions

TL;DR

This paper establishes sharp total variation inequalities for maximal functions, extending previous work by considering variable truncations and interpolations, and provides counterexamples delineating the scope of these results.

Contribution

It introduces new sharp inequalities for maximal functions' total variation, including variable truncations and interpolations, advancing understanding of variation bounds in harmonic analysis.

Findings

Sharp constants for total variation inequalities are obtained.

Counterexamples show limitations of the methods used.

Progress towards the conjecture of the best constant being one in the centered case.

Abstract

In this article, we prove some total variation inequalities for maximal functions. Our results deal with two possible generalizations of the results contained in Aldaz and P\'erez L\'azaro's work, one of whose considers a variable truncation of the maximal function, and the other one interpolates the centered and the uncentered maximal functions. In both contexts, we find sharp constants for the desired inequalities, which can be viewed as progress towards the conjecture that the best constant for the variation inequality in the centered context is one. We also provide counterexamples showing that our methods do not apply outside the stated parameter ranges.