Solyanik estimates and local H\"older continuity of halo functions of geometric maximal operators

TL;DR

This paper establishes that certain geometric maximal operators' halo functions are locally Hölder continuous near 1, based on their Solyanik estimates, with implications for classical maximal operators like Hardy-Littlewood.

Contribution

It proves that halo functions satisfying Solyanik estimates are Hölder continuous, linking geometric maximal operator behavior to regularity properties.

Findings

Halo functions are in the Hölder class C^p near 1.

Hardy-Littlewood and strong maximal operators' halo functions are in C^{1/n}.

Establishes a connection between Solyanik estimates and function regularity.

Abstract

Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in $\mathbb{R}^n$, and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$ M_{\mathcal{B}} f(x) :=\sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f| $$ and the halo function $\phi_{\mathcal{B}}(\alpha)$ on $(1,\infty)$ by $$\phi_{\mathcal B}(\alpha) :=\sup_{E \subset \mathbb{R}^n :\, 0 < |E| < \infty}\frac{1}{|E|}|\{x\in \mathbb{R}^n : M_{\mathcal{B}} \chi_E (x) >1/\alpha\}|. $$ It is shown that if $\phi_{\mathcal{B}}(\alpha)$ satisfies the Solyanik estimate $\phi_{\mathcal B}(\alpha) - 1 \leq C (1 - \frac{1}{\alpha})^p$ for $\alpha\in(1,\infty)$ sufficiently close to 1 then $\phi_{\mathcal{B}}$ lies in the H\"older class $ C^p(1,\infty)$. As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on $\mathbb{R}^n$ lie in the H\"older class $C^{1/n}(1,\infty)$.