Tauberian constants associated to centered translation invariant density bases

Abstract

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, given $x\in {\mathbb{R}}^n$, let $\mathcal{B}={\cup }_{x\in {\mathbb{R}}^n}\mathcal{B}(x)$ be 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 that $\mathcal{B}(x+y)=\{R+y:R\in \mathcal{B}(x)\}.$ Associated to these collections is a maximal operator $M_{\mathcal{B}}$ given by $$M_{\mathcal{B}}f(x):=\underset{R\in \mathcal{B}(x)}{\sup }\frac{1}{|R|}{\int }_R|f|.$$ The Tauberian constants $C_{\mathcal{B}}(\alpha )$ associated to $M_{\mathcal{B}}$ are given by $$C_{\mathcal{B}}(\alpha ):=\underset{\genfrac{}{}{0ex}{}{E\subset {\mathbb{R}}^n}{0<|E|<\infty }}{\sup }\frac{1}{|E|}|\{x\in {\mathbb{R}}^n:M_{\mathcal{B}}{\chi }_E(x)>\alpha \}|.$$ Given $0<r<\infty$, we set ${\mathcal{B}}_r(x):=\{R\in \mathcal{B}(x):\mathrm{diam}R<r\}$, and let ${\mathcal{B}}_r:={\cup }_{x\in {\mathbb{R}}^n}{\mathcal{B}}_r(x).$ We prove that $\mathcal{B}$ is a density basis if and only if, given $0<\alpha <\infty$, there exists $r=r(\alpha )>0$ such that $C_{{\mathcal{B}}_r}(\alpha )<\infty$. Subsequently, we construct a centered translation invariant density basis $\mathcal{B}={\cup }_{x\in {\mathbb{R}}^n}\mathcal{B}(x)$ such that there does not exist any $0<r$ satisfying $C_{{\mathcal{B}}_r}(\alpha )<\infty$ for all $0<\alpha <1$.

arXiv comment7 pages, submitted for publication