Solyanik Estimates in Harmonic Analysis

Abstract

Let $\mathcal{B}$ denote a collection of open bounded sets in $\mathbb{R}^{n}$, and define the associated maximal operator $M_{\mathcal{B}}$ by

$$\displaystyle{M_{\mathcal{B}}f(x)\,:=\,\sup _{x\in R\in \mathcal{B}} \frac{1} {\vert R\vert }\int _{R}\vert f\vert.}$$

The sharp Tauberian constant of $M_{\mathcal{B}}$ associated with α, denoted by $C_{\mathcal{B}}(\alpha )$, is defined as

$$\displaystyle{C_{\mathcal{B}}(\alpha )\,:=\,\sup _{E:\,0<\vert E\vert <\infty } \frac{1} {\vert E\vert }\big\vert \big\{x \in \mathbb{R}^{n}:\, M_{ \mathcal{B}}\chi _{E}(x) >\alpha \big\}\big \vert.}$$

Motivated by previous work of A. A. Solyanik, we show that if $M_{\mathcal{B}}$ is the uncentered Hardy–Littlewood maximal operator associated with balls, the estimate

$$\displaystyle{\lim _{\alpha \rightarrow 1^{-}}C_{\mathcal{B}}(\alpha ) = 1}$$

holds. Similar results for iterated maximal functions are obtained, and open problems in the field of Solyanik estimates are also discussed.