On a discrete version of Tanaka's theorem for maximal functions

TL;DR

This paper establishes a discrete analogue of Tanaka's theorem for the Hardy-Littlewood maximal operator in one dimension, showing total variation bounds for both centered and non-centered cases, thus solving a discrete case of a known problem.

Contribution

It proves a discrete version of Tanaka's theorem for the maximal operator in one dimension, including bounds on total variation for non-centered and centered cases.

Findings

Total variation of non-centered maximal function is bounded by that of the original function.

Centered maximal function's variation is bounded by a constant times the L1 norm of the function.

Provides a positive answer to a discrete one-dimensional case of a previously open question.

Abstract

In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\widetilde{M} $ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ of bounded variation, $$\textrm{Var}(\widetilde{M} f) \leq \textrm{Var}(f),$$ where $\textrm{Var}(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ such that $f \in \ell^1(\mathbb{Z})$, $$\textrm{Var}(Mf) \leq C \|f\|_{\ell^1(\mathbb{Z})}.$$ This provides a positive solution to a question of Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.