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- W2072870111 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." @default.
- W2072870111 created "2016-06-24" @default.
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- W2072870111 date "2012-12-18" @default.
- W2072870111 modified "2023-09-27" @default.
- W2072870111 title "Continuity of halo functions associated to homothecy invariant density bases" @default.
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