Solyanik estimates in harmonic analysis

Let $\mathcal{B}$ denote a collection of open bounded sets in $\mathbb{R}^n$, and define the associated maximal operator $M_{\mathcal{B}}$ by $$ M_{\mathcal{B}}f(x) := \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|. $$ The sharp Tauberian constant of $M_{\mathcal{B}}$ associated to $\alpha$, denoted by $C_{\mathcal{B}}(\alpha)$, is defined as $$ C_{\mathcal{B}}(\alpha) := \sup_{E :\, 0 < |E| < \infty}\frac{1}{|E|}\big|\big\{x \in \mathbb{R}^n:\, M_{\mathcal{B}}\chi_E (x) > \alpha\big\}\big|.$$ Motivated by previous work of A. A. Solyanik, we show that if $M_{\mathcal{B}}$ is the uncentered Hardy-Littlewood maximal operator associated to balls, the estimate $$ \lim_{\alpha \rightarrow 1^-}C_{\mathcal{B}}(\alpha) = 1 $$ holds. Similar results for iterated maximal functions are obtained, and open problems in the field of Solyanik estimates are also discussed.