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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Jonathan Bober, Emanuel Carneiro, Kevin Hughes and Lillian B. Pierce
Journal: Proc. Amer. Math. Soc. 140 (2012), 1669-1680
MSC (2010): Primary 42B25, 46E35
Published electronically: September 1, 2011
MathSciNet review: 2869151
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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,

$\displaystyle \operatorname{Var}(\widetilde{M} f) \leq \operatorname{Var}(f),$

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})$,

$\displaystyle \operatorname{Var}(Mf) \leq C \Vert f\Vert _{\ell^1(\mathbb{Z})}.$

This provides a positive solution to a question of Hajłasz and Onninen in the discrete one-dimensional case.


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Additional 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
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
Email: lbpierce@math.ias.edu, lillian.pierce@maths.ox.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11008-6
PII: S 0002-9939(2011)11008-6
Keywords: Maximal operators, Sobolev spaces, discrete operators, Tanaka’s theorem.
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.