Weakly porous sets and Muckenhoupt $A_p$ distance functions
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914875425423360 |
|---|---|
| author | Anderson, Theresa C. Lehrbäck, Juha Mudarra, Carlos Vähäkangas, Antti V. |
| author_facet | Anderson, Theresa C. Lehrbäck, Juha Mudarra, Carlos Vähäkangas, Antti V. |
| contents | 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 $1<p<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06284 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weakly porous sets and Muckenhoupt $A_p$ distance functions Anderson, Theresa C. Lehrbäck, Juha Mudarra, Carlos Vähäkangas, Antti V. Classical Analysis and ODEs Metric Geometry 42B99 (Primary) 28A75, 28A80, 42B25 (Secondary) 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 $1<p<\infty$. |
| title | Weakly porous sets and Muckenhoupt $A_p$ distance functions |
| topic | Classical Analysis and ODEs Metric Geometry 42B99 (Primary) 28A75, 28A80, 42B25 (Secondary) |
| url | https://arxiv.org/abs/2209.06284 |