Continuity of halo functions associated to homothecy invariant density bases

TL;DR

This paper proves that halo functions linked to homothecy invariant density bases are continuous on (1,∞), but provides an example where the halo function is discontinuous at 1, revealing nuanced behavior of these functions.

Contribution

It establishes the continuity of halo functions for all homothecy invariant density bases and presents a counterexample showing discontinuity at 1.

Findings

Halo functions are continuous on (1,∞) for any homothecy invariant density basis.

An example exists where the halo function is discontinuous at 1.

The study clarifies the boundary behavior of halo functions associated with these bases.

Abstract

Let $\mathcal{B}$ be a collection of open sets in $\mathbb{R}^{n}$ such that, for any $x \in \mathbb{R}^{n}$, there exists a set $U \in \mathcal{B}$ of arbitrarily small diameter {containing $x$.} $\mathcal{B}$ is said to be a \emph{density basis} provided that, given a measurable set $A \subset \mathbb{R}^{n}$, for a.e. $x \in \mathbb{R}^{n}$ we have $$\lim_{k \rightarrow \infty}\frac{1}{|R_{k}|}\int_{R_{k}}\chi_{A} = \chi_{A}(x)$$ holds for any sequence of sets $\{R_{k}\}$ in $\mathcal{B}$ containing $x$ whose diameters tend to 0. The geometric maximal operator $M_{\mathcal{B}}$ associated to $\mathcal{B}$ is defined on $L^{1}(\mathbb{R}^n)$ by $M_{\mathcal{B}}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_{R}|f|$. The \emph{halo function} $\phi$ of $\mathcal{B}$ is defined on $(1,\infty)$ by $$\phi(u) = \sup \{\frac{1}{|A|}|\{x \in \mathbb{R}^{n} : M_{\mathcal{B}}\chi_{A}(x) > \frac{1}{u}\}| : 0 < |A| < \infty\}\;$$ and on $[0,1]$ by $\phi(u) = u$. It is shown that the halo function associated to any homothecy invariant density basis is a continuous function on $(1,\infty)$. However, an example of a homothecy invariant density basis is provided such that the associated halo function is not continuous at 1.