## 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 Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.

### Duke Scholars

##### Altmetric Attention Stats

##### Dimensions Citation Stats

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- 4904 Pure mathematics
- 0101 Pure Mathematics

### Citation

*Proceedings of the American Mathematical Society*,

*140*(5), 1669–1680. https://doi.org/10.1090/S0002-9939-2011-11008-6

*Proceedings of the American Mathematical Society*140, no. 5 (May 1, 2012): 1669–80. https://doi.org/10.1090/S0002-9939-2011-11008-6.

*Proceedings of the American Mathematical Society*, vol. 140, no. 5, May 2012, pp. 1669–80.

*Manual*, doi:10.1090/S0002-9939-2011-11008-6.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- 4904 Pure mathematics
- 0101 Pure Mathematics