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

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 Hajłasz and Onninen \cite{HO} in the discrete one-dimensional case.

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