Publications [#302454] of Lillian B. Pierce

Papers Published

1. 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. 1669-1680, American Mathematical Society (AMS), ISSN 0002-9939 [doi]
   (last updated on 2025/04/03)

   Abstract:
   In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered 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 one-dimensional case.