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

In this paper we prove a discrete version of Tanaka’s theorem for the Hardy-Littlewood maximal operator in dimension n=1n=1, both in the non-centered and centered cases. For the non-centered maximal operator M~\widetilde {M} we prove that, given a function f:Z→Rf: \mathbb {Z} \to \mathbb {R} of boun...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2012-05, Vol.140 (5), p.1669-1680
Main Authors: Bober, Jonathan, Carneiro, Emanuel, Hughes, Kevin, Pierce, Lillian B.
Format: Article
Language:English
Subjects:
Harmonic analysis
Integers
Local maximum
Local minimum
Mathematical constants
Mathematical functions
Mathematical inequalities
Mathematical theorems
Radon
Research article
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Summary:In this paper we prove a discrete version of Tanaka’s theorem for the Hardy-Littlewood maximal operator in dimension n=1n=1, both in the non-centered and centered cases. For the non-centered maximal operator M~\widetilde {M} we prove that, given a function f:Z→Rf: \mathbb {Z} \to \mathbb {R} of bounded variation, \[ Var⁡(M~f)≤Var⁡(f),\operatorname {Var}(\widetilde {M} f) \leq \operatorname {Var}(f), \] where Var⁡(f)\operatorname {Var}(f) represents the total variation of ff. For the centered maximal operator MM we prove that, given a function f:Z→Rf: \mathbb {Z} \to \mathbb {R} such that f∈ℓ1(Z)f \in \ell ^1(\mathbb {Z}), \[ Var⁡(Mf)≤C‖f‖ℓ1(Z).\operatorname {Var}(Mf) \leq C \|f\|_{\ell ^1(\mathbb {Z})}. \] This provides a positive solution to a question of Hajłasz and Onninen in the discrete one-dimensional case.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-2011-11008-6