Tauberian constants associated to centered translation invariant density bases (1705.10094v1)

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 $$M_{\mathcal{B}}f(x) :=\sup_{R \in \mathcal{B}(x)} \frac{1}{|R|} \int_R |f|.$$ The Tauberian constants $C_{\mathcal{B}}(\alpha)$ associated to $M_{\mathcal{B}}$ are given by $$C_{\mathcal{B}}(\alpha) :=\sup_{E \subset \mathbb{R}^n \atop 0 < |E| < \infty} \frac{1}{|E|}|{x \in \mathbb{R}^n :\, M_{\mathcal{B}}\chi_E(x) > \alpha}|.$$ 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 $0 < \alpha < 1$.