A note on local Hölder continuity of weighted Tauberian functions (1503.02898v1)

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 $n$.