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

Jonathan W Bober; Emanuel Carnerio; Kevin  Hughes; Lillian Pierce

doi:10.1090/S0002-9939-2011-11008-6

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

Jonathan W Bober, Emanuel Carnerio, Kevin Hughes, Lillian Pierce

Research output: Contribution to journal › Article (Academic Journal) › peer-review

48 Citations (Scopus)

Abstract

In this paper we prove a discrete version of Tanaka's theorem for the Hardy-Littlewood maximal operator in dimension , both in the non-centered and centered cases. For the non-centered maximal operator we prove that, given a function of bounded variation,

where represents the total variation of . For the centered maximal operator we prove that, given a function such that ,

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

Original language	English
Pages (from-to)	1669-1680
Number of pages	12
Journal	Proceedings of the American Mathematical Society
Volume	140
Issue number	5
DOIs	https://doi.org/10.1090/S0002-9939-2011-11008-6
Publication status	Published - 2012

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title = "On a discrete version of Tanaka's theorem for maximal functions",

abstract = "In this paper we prove a discrete version of Tanaka's theorem for the Hardy-Littlewood maximal operator in dimension , both in the non-centered and centered cases. For the non-centered maximal operator we prove that, given a function of bounded variation, where represents the total variation of . For the centered maximal operator we prove that, given a function such that , This provides a positive solution to a question of Haj{\l}asz and Onninen in the discrete one-dimensional case. ",

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T1 - On a discrete version of Tanaka's theorem for maximal functions

AU - Bober, Jonathan W

AU - Carnerio, Emanuel

AU - Hughes, Kevin

AU - Pierce, Lillian

PY - 2012

Y1 - 2012

N2 - In this paper we prove a discrete version of Tanaka's theorem for the Hardy-Littlewood maximal operator in dimension , both in the non-centered and centered cases. For the non-centered maximal operator we prove that, given a function of bounded variation, where represents the total variation of . For the centered maximal operator we prove that, given a function such that , This provides a positive solution to a question of Hajłasz and Onninen in the discrete one-dimensional case.

AB - In this paper we prove a discrete version of Tanaka's theorem for the Hardy-Littlewood maximal operator in dimension , both in the non-centered and centered cases. For the non-centered maximal operator we prove that, given a function of bounded variation, where represents the total variation of . For the centered maximal operator we prove that, given a function such that , This provides a positive solution to a question of Hajłasz and Onninen in the discrete one-dimensional case.

U2 - 10.1090/S0002-9939-2011-11008-6

DO - 10.1090/S0002-9939-2011-11008-6

M3 - Article (Academic Journal)

VL - 140

SP - 1669

EP - 1680

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 5

ER -

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

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