Tauberian constants associated to centered translation invariant density bases
Volume 243 / 2018
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, 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 $0 \lt \alpha \lt 1$.
