维数大于1的集的分析家旅行商定理
An Analyst's Traveling Salesman Theorem for sets of dimension larger than one
摘要：
在他1990年的Inventions论文中，P. Jones通过$β$数的多尺度和来描述平面上可求长曲线的子集。这些$β$ -数是几何量，测量给定集合在每个尺度和位置处偏离最佳拟合线的程度。琼斯的结果是一种定量的方式说，一条曲线是可求长的，当且仅当它几乎在每一点都有切线。此外，计算曲线的平方和将曲线的长度返回到乘法常数。K. Okikiolu延长他的结果从子集的飞机子集的欧几里得空间。G. David和S.塞姆斯在阿尔福斯正则性的假设下，使用琼斯的$β$数的一个变体，将讨论扩展到包括大于1的（整数）维集合。本文对欧几里得空间中的任意（整数）维集合给出了P. Jones定理的一个版本。我们用$β$型数的类似和来估计一个集合的$d$维Hausdorff测度.没有假设Ahlfors正则性，而是只有Hausdorff内容的下界。我们通过使用Choquet积分来重新定义Jones的$β$ -数，从而适应David和Semmes的版本。证明中的一个关键工具是G.大卫和T. Toro的参数化Reifenberg平坦集（与洞）。
英文摘要：
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $β$-numbers. These $β$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones' $β$ numbers. In this paper we give a version of P. Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean space. We estimate the $d$-dimensional Hausdorff measure of a set in terms of an analogous sum of $β$-type numbers. There is no assumption of Ahlfors regularity, but rather, only of a lower bound on the Hausdorff content. We adapt David and Semmes' version of Jones' $β$-numbers by redefining them using a Choquet integral. A key tool in the proof is G. David and T. Toro's parametrization of Reifenberg flat sets (with holes).
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