Solyanik estimates and local H\"older continuity of halo functions of geometric maximal operators

Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in ${\mathbb{R}}^n$, and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}}f(x):=\underset{x\in R\in \mathcal{B}}{\sup }\frac{1}{\mathrm{\mid }R\mathrm{\mid }}{\int }_R\mathrm{\mid }f\mathrm{\mid }$$ and the halo function ${\varphi }_{\mathcal{B}}(\alpha )$ on $(1,\mathrm{\infty })$ by $${\varphi }_{\mathcal{B}}(\alpha ):=\underset{E\subset {\mathbb{R}}^n:0<\mathrm{\mid }E\mathrm{\mid }<\mathrm{\infty }}{\sup }\frac{1}{\mathrm{\mid }E\mathrm{\mid }}\mathrm{\mid }\{x\in {\mathbb{R}}^n:M_{\mathcal{B}}{\chi }_E(x)>1\mathrm{/}\alpha \}\mathrm{\mid }\mathrm{.}$$ It is shown that if ${\varphi }_{\mathcal{B}}(\alpha )$ satisfies the Solyanik estimate $\phi_{\mathcal B}(\alpha) - 1 \leq C (1 - \frac{1}{\alpha})^p$for$\alpha\in(1,\infty)$sufficientlycloseto1then$\phi_{$\mathcal{B}$}$liesintheH\ddot{o}lderclass$ C^p(1,\infty)$. As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on ${\mathbb{R}}^n$ lie in the H"older class $C^{1\mathrm{/}n}(1,\mathrm{\infty })$.