Continuity of halo functions associated to homothecy invariant density 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 $$\underset{k\to \infty }{\lim }\frac{1}{|R_k|}{\int }_{R_k}{\chi }_A={\chi }_A(x)$$ 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} $\varphi$ of $\mathcal{B}$ is defined on $(1,\infty )$ by $$\varphi (u)=\sup \{\frac{1}{|A|}|\{x\in {\mathbb{R}}^n:M_{\mathcal{B}}{\chi }_A(x)>\frac{1}{u}\}|:0<|A|<\infty \}$$ and on $[0,1]$ by $\varphi (u)=u$. 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.