$C^{1,α}$参数化和可求正的充分条件
Sufficient conditions for $C^{1,α}$ parametrization and rectifiability
摘要：
我们说一个测度是$C^{1,α}$ $d$ -可求正的，如果存在一个可数的$C^{1,α}$ $d$ -曲面的并，其补测度为零。本文给出了$\mathbb{R}^n$中的Radon测度是$C^{1,α}$ $d$ -可求积的充分条件，其中有$α\in (0,1]$ .条件涉及Bishop-Jones型平方函数，并且所有陈述都是定量的，因为$C^{1,α}$常数取决于这样的函数。一路上，我们也给充分条件的$C^{1,α}$参数化Reifenberg平坦的设置在相同的平方函数。证明的关键工具来自大卫和托罗的Reifenberg参数化的集合与洞的霍尔德和利普希茨类别。
英文摘要：
We say a measure is $C^{1,α}$ $d$-rectifiable if there is a countable union of $C^{1,α}$ $d$-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in $\mathbb{R}^n$ to be $C^{1,α}$ $d$-rectifiable, with $α\in (0,1]$. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the $C^{1,α}$ constants depend on such a function. Along the way we also give sufficient conditions for $C^{1,α}$ parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and Toro's Reifenberg parametrizations of sets with holes in the Hölder and Lipschitz categories.
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