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.
Categories: math.FA