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

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 $\tilde{M}$ we prove that, given a function $f:\mathbb{Z}\to \mathbb{R}$ of bounded variation, $$\text{Var}(\tilde{M}f)\le \text{Var}(f),$$ where $\text{Var}(f)$ 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 $f\in {\ell }^1(\mathbb{Z})$, $$\text{Var}(Mf)\le C\Vert f{\Vert }_{{\ell }^1(\mathbb{Z})}.$$ This provides a positive solution to a question of Hajłasz and Onninen \cite{HO} in the discrete one-dimensional case.

V4 - Proof of Lemma 3 updated

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