## On a discrete version of Tanaka’s theorem for maximal functions

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- by Jonathan Bober, Emanuel Carneiro, Kevin Hughes and Lillian B. Pierce
- Proc. Amer. Math. Soc.
**140**(2012), 1669-1680 - DOI: https://doi.org/10.1090/S0002-9939-2011-11008-6
- Published electronically: September 1, 2011
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Corrigendum: Proc. Amer. Math. Soc.

**143**(2015), 5471-5473.

## Abstract:

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

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## Bibliographic Information

**Jonathan Bober**- Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
- Address at time of publication: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195-4350
- Email: bober@math.ias.edu, jwbober@uw.edu
**Emanuel Carneiro**- Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
- Address at time of publication: Instituto de Matematica Pura e Aplicada–IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil
- Email: ecarneiro@math.ias.edu, carneiro@impa.br
**Kevin Hughes**- Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544
- MR Author ID: 962878
- ORCID: 0000-0002-8621-8259
- Email: kjhughes@math.princeton.edu
**Lillian B. Pierce**- Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
- Address at time of publication: Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, United Kingdom
- MR Author ID: 757898
- Email: lbpierce@math.ias.edu, lillian.pierce@maths.ox.ac.uk
- Received by editor(s): May 6, 2010
- Received by editor(s) in revised form: January 14, 2011
- Published electronically: September 1, 2011
- Communicated by: Thomas Schlumprecht
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**140**(2012), 1669-1680 - MSC (2010): Primary 42B25, 46E35
- DOI: https://doi.org/10.1090/S0002-9939-2011-11008-6
- MathSciNet review: 2869151