Sharp inequalities for the variation of the discrete maximal function

TL;DR

This paper derives new optimal bounds for the derivatives of discrete maximal functions, addressing an open question and enhancing understanding of their variation in both centered and uncentered cases.

Contribution

It provides the first optimal bounds for the derivative of discrete maximal functions, solving a previously open problem.

Findings

Established new optimal bounds for the derivative of discrete maximal functions

Solved an open problem posed by Bober, Carneiro, Hughes, and Pierce

Enhanced understanding of variation bounds in discrete harmonic analysis

Abstract

In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, both in the centered and uncentered versions. In particular, we solve a question originally posed by Bober, Carneiro, Hughes and Pierce.