Math @ Duke

Publications [#302454] of Lillian B. Pierce
Papers Published
 Bober, J; Carneiro, E; Hughes, K; Pierce, LB, On a discrete version of Tanaka's theorem for maximal functions,
Proceedings of the American Mathematical Society, vol. 140 no. 5
(May, 2012),
pp. 16691680, ISSN 00029939 [doi]
(last updated on 2018/12/15)
Abstract: In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the HardyLittlewood maximal operator in dimension $n=1$, both in the noncentered and centered cases. For the discrete noncentered maximal operator $\widetilde{M} $ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ of bounded variation, $$\textrm{Var}(\widetilde{M} f) \leq \textrm{Var}(f),$$ where $\textrm{Var}(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ such that $f \in \ell^1(\mathbb{Z})$, $$\textrm{Var}(Mf) \leq C \f\_{\ell^1(\mathbb{Z})}.$$ This provides a positive solution to a question of Haj{\l}asz and Onninen \cite{HO} in the discrete onedimensional case.


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