论文标题
弱的多孔套装和粉碎事件
Weakly porous sets and Muckenhoupt $A_p$ distance functions
论文作者
论文摘要
我们检查了欧几里得空间中弱多孔集的类别。作为我们的第一个主要结果，我们表明距离重量$ w（x）= \ operatorname {dist}（x，e）^{ - α} $属于muckenhoupt类$ a_1 $，对于某些$α> 0 $，且仅当$ e \ e \ e \ e \ subset \ subset \ subset \ subbb {r}^n $ fealply pelly pelly pelly pelly pe pelly pe pelly pe pelly。我们还根据所谓的$ e $指数提供了此特征的精确定量版本。当$ e $是弱的多孔时，我们以$ 1 <p <\ infty $的价格获得了$ w \ $ w \的类似定量表征。在论文的结尾，我们给出了一个$ e \ subset \ mathbb {r} $的示例，该$ w \ in a_p \ setminus a_1 $ in $ 0 <α<1 $ and $ 1 <p <p <p <\ iffty $中的$ w \ setminus a_1 $。
We examine the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight $w(x)=\operatorname{dist}(x,E)^{-α}$ belongs to the Muckenhoupt class $A_1$, for some $α>0$, if and only if $E\subset\mathbb{R}^n$ is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of $E$. When $E$ is weakly porous, we obtain a similar quantitative characterization of $w\in A_p$, for $1<p<\infty$, as well. At the end of the paper, we give an example of a set $E\subset\mathbb{R}$ which is not weakly porous but for which $w\in A_p\setminus A_1$ for every $0<α<1$ and $1<p<\infty$.
