A note on local H\"older continuity of weighted Tauberian functions

Abstract

Let $\mathsf M$ and $\mathsf M _{\mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $\mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $\mathbb{R}^n$. We define the associated Tauberian functions $\mathsf{C}_{\mathsf{HL},w}(\alpha)$ and $\mathsf{C}_{\mathsf{S},w}(\alpha)$ on $(0,1)$ by \[ \mathsf{C}_{\mathsf{HL},w}(\alpha) :=\sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \mathsf M \chi_E(x) > \alpha\}) \] and \[ \mathsf{C}_{\mathsf{S},w}(\alpha) := \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \mathsf M _{\mathsf S}\chi_E(x) > \alpha\}). \] Utilizing weighted Solyanik estimates for $\mathsf M$ and $\mathsf M_{\mathsf S}$, we show that the function $\mathsf{C}_{\mathsf{HL},w} $ lies in the local H\"older class $C^{(c_n[w]_{A_{\infty}})^{-1}}(0,1)$ and $\mathsf{C}_{\mathsf{S},w} $ lies in the local H\"older class $C^{(c_n[w]_{A_{\infty}^\ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.