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

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 ${\mathsf{C}}_{\mathsf{H}\mathsf{L},w}(\alpha )$ and ${\mathsf{C}}_{\mathsf{S},w}(\alpha )$ on $(0,1)$ by $${\mathsf{C}}_{\mathsf{H}\mathsf{L},w}(\alpha ):=\underset{\begin{array}{c}E\subset {\mathbb{R}}^n\\ 0<w(E)<\mathrm{\infty }\end{array}}{\sup }\frac{1}{w(E)}w(\{x\in {\mathbb{R}}^n:\mathsf{M}{\chi }_E(x)>\alpha \})$$ and $${\mathsf{C}}_{\mathsf{S},w}(\alpha ):=\underset{\begin{array}{c}E\subset {\mathbb{R}}^n\\ 0<w(E)<\mathrm{\infty }\end{array}}{\sup }\frac{1}{w(E)}w(\{x\in {\mathbb{R}}^n:{\mathsf{M}}_{\mathsf{S}}{\chi }_E(x)>\alpha \})\mathrm{.}$$ Utilizing weighted Solyanik estimates for $\mathsf{M}$ and ${\mathsf{M}}_{\mathsf{S}}$, we show that the function ${\mathsf{C}}_{\mathsf{H}\mathsf{L},w}$ lies in the local H"older class $C^{(c_n[w{]}_{A_{\mathrm{\infty }}}{)}^{-1}}(0,1)$ and ${\mathsf{C}}_{\mathsf{S},w}$ lies in the local H"older class $C^{(c_n[w{]}_{A_{\mathrm{\infty }}^{\ast }}{)}^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.