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 $$M_{\mathcal{B}}f(x):=\underset{x\in R\in \mathcal{B}}{\sup }\frac{1}{\mathrm{\mid }R\mathrm{\mid }}{\int }_R\mathrm{\mid }f\mathrm{\mid }\mathrm{.}$$ The sharp Tauberian constant of $M_{\mathcal{B}}$ associated to $\alpha$, denoted by $C_{\mathcal{B}}(\alpha )$, is defined as $$C_{\mathcal{B}}(\alpha ):=\underset{E:0<\mathrm{\mid }E\mathrm{\mid }<\mathrm{\infty }}{\sup }\frac{1}{\mathrm{\mid }E\mathrm{\mid }}\mid \{x\in {\mathbb{R}}^n:M_{\mathcal{B}}{\chi }_E(x)>\alpha \}\mid \mathrm{.}$$ 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 $$\underset{\alpha \to 1^{-}}{\lim }{C}_{\mathcal{B}}(\alpha )=1$$ holds. Similar results for iterated maximal functions are obtained, and open problems in the field of Solyanik estimates are also discussed.