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

Bober, Jonathan; Carneiro, Emanuel; Hughes, Kevin; Pierce, Lillian

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

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 PDF

Proc. Amer. Math. Soc. 140 (2012), 1669-1680 Request permission

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.

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