A note on local Hölder 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 \[ \mathsf{C}_{\mathsf{HL},w}(α) :=\sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \mathsf M χ_E(x) > α\}) \] and \[ \mathsf{C}_{\mathsf{S},w}(α) := \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \mathsf M _{\mathsf S}χ_E(x) > α\}). \] 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 Hölder class $C^{(c_n[w{]}_{A_{\infty }}{)}^{-1}}(0,1)$ and ${\mathsf{C}}_{\mathsf{S},w}$ lies in the local Hölder class $C^{(c_n[w{]}_{A_{\infty }^{\ast }}{)}^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.

8 pages, submitted for publication

References in corpus (4)

• Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
• Weighted Solyanik Estimates for the Hardy-Littlewood maximal operator and embedding of $A_{\infty }$ into $A_p$
• Weighted Solyanik estimates for the strong maximal function
• Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators