PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Tauberian constants associated to centered translation invariant density bases

Volume 243 / 2018

Paul Hagelstein, Ioannis Parissis Fundamenta Mathematicae 243 (2018), 169-177 MSC: Primary 42B25. DOI: 10.4064/fm409-2-2018 Published online: 20 July 2018


This paper provides a necessary and sufficient condition on Tauberian constants associated to a centered translation invariant differentiation basis so that the basis is a density basis. More precisely, let $\mathcal{B} = \bigcup_{x \in \mathbb{R}^n} \mathcal{B}(x)$ where $\mathcal{B}(x)$ is a collection of bounded open sets in $\mathbb{R}^n$ containing $x$. Suppose moreover that these collections are translation invariant in the sense that, for any two points $x$ and $y$ in $\mathbb{R}^n$ we have $\mathcal{B}(x + y) = \{R + y :\, R \in \mathcal{B}(x)\}.$ Associated to these collections is a maximal operator $M_{\mathcal{B}}$ given by $$M_{\mathcal{B}}f(x) := \sup_{R \in \mathcal{B}(x)} \frac{1}{|R|} \int_R |f|.$$ The Tauberian constants $C_{\mathcal{B}}(\alpha)$ associated to $M_{\mathcal{B}}$ are given by $$C_{\mathcal{B}}(\alpha) := \sup_{\substack{E \subset \mathbb{R}^n \\ 0 \lt |E| \lt \infty}} \frac{1}{|E|}|\{x \in \mathbb{R}^n : M_{\mathcal{B}}\chi_E(x) \gt \alpha\}|.$$ Given $0 \lt r \lt \infty$, we set $\mathcal{B}_r(x) := \{R \in \mathcal{B}(x) : \operatorname{diam}(R) \lt r\}$, and let $\mathcal{B}_r := \bigcup_{x \in \mathbb{R}^n} \mathcal{B}_r (x).$ We prove that $\mathcal{B}$ is a density basis if and only if, given $0 \lt \alpha \lt \infty$, there exists $ r = r(\alpha) \gt 0$ such that $C_{\mathcal{B}_r}(\alpha) \lt \infty$. Subsequently, we construct a centered translation invariant density basis $\mathcal{B} = \bigcup_{x \in \mathbb{R}^n} \mathcal{B}(x)$ such that there is no $ r \gt 0$ satisfying $C_{\mathcal{B}_{r}}(\alpha) \lt \infty$ for all $0 \lt \alpha \lt 1$.


  • Paul HagelsteinDepartment of Mathematics
    Baylor University
    Waco, TX 76798, U.S.A.
  • Ioannis ParissisDepartamento de Matemáticas
    Universidad del Pais Vasco
    Aptdo. 644
    48080 Bilbao, Spain
    Ikerbasque, Basque Foundation for Science
    Bilbao, Spain

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image