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

TL;DR

This paper investigates the local Hölder continuity of weighted Tauberian functions associated with Hardy-Littlewood and strong maximal operators, establishing their regularity properties using weighted Solyanik estimates.

Contribution

It proves that these weighted Tauberian functions are locally Hölder continuous with explicit exponents depending on Muckenhoupt weights and dimension, extending previous unweighted results.

Findings

Weighted Tauberian functions are in specific Hölder classes.

Hölder exponents depend on $A_$ constants of weights.

Results apply to both cube and strong maximal operators.

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 $n$.