欧几里得空间任意子集的锥和抛物面点
Cone and paraboloid points of arbitrary subsets of Euclidean space
摘要：
本文给出了欧几里得空间中任意子集的锥点。给定$E \subset \mathbb{R}^n$，$x \in E$是$E$的锥点当且仅当\begin{align*} \int_{0}^1 β_{E}^{d,2}(B(x,r))^2 \frac{dr}{r} < \infty, \end{align*}直到一个零$d$-测度集。系数$β_E^{d,2}$是琼斯系数的变化。这是一个高维对应定理的主教和琼斯从1994年。我们还证明了类似的结果$α$抛物面点，这是$C^{1,α}$可求直对应锥点：$x \in E$是$α$抛物面点当且仅当\begin{align*} \int_0^1 \frac{\overlineβ_{E}^{d,2}(B(x,r))^2}{r^{2α}} \, \frac{dr}{r} < \infty \end{align*}的零$d$-措施。这里，$\overlineβ^{d,2}_E$是由Azzam和Schul引入的琼斯系数的另一种变体。
英文摘要：
In this paper we characterise cone points of arbitrary subsets of Euclidean space. Given $E \subset \mathbb{R}^n$, $x \in E$ is a cone point of $E$ if and only if \begin{align*} \int_{0}^1 β_{E}^{d,2}(B(x,r))^2 \frac{dr}{r} < \infty, \end{align*} up to a set of zero $d$-measure. The coefficients $β_E^{d,2}$ are a variation of the Jones coefficients. This is a high dimensional counterpart of a theorem of Bishop and Jones from 1994. We also prove similar results for $α$-paraboloid points, which are the $C^{1,α}$ rectifiability counterparts to cone points: $x \in E$ is an $α$-paraboloid point if and only if \begin{align*} \int_0^1 \frac{\overlineβ_{E}^{d,2}(B(x,r))^2}{r^{2α}} \, \frac{dr}{r} < \infty \end{align*} up to a set of zero $d$-measure. Here, $\overlineβ^{d,2}_E$ is another variant of the Jones coefficients, introduced by Azzam and Schul.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
