识别Carnot群中的1-可求长测度
Identifying 1-rectifiable measures in Carnot groups
AI总结：
本研究扩展了欧几里得空间中的定理，在任意Carnot群中开发测试方法以识别测度中由可求长曲线承载的部分及其奇异部分，并关联解析旅行推销员定理，同时在完备度量空间中实现了Garnett-Killip-Schul构造。
摘要：
我们继续开展几何测度论中的一个研究计划，该计划旨在识别空间中测度与空间中典型集合族之间的相互作用方式。特别地，我们扩展了第一作者与R. Schul在欧几里得空间中的一个定理，对于任意Carnot群中的任意局部有限Borel测度，我们开发了测试方法，以识别测度中由可求长曲线承载的部分以及测度中与可求长曲线奇异的部分。我们的主要结果与解析旅行推销员定理的扩展紧密相连，该定理通过称为Jones的β数的局部几何最小二乘数据，刻画了$\mathbb{R}^2$（P. Jones, 1990）、$\mathbb{R}^n$（K. Okikolu, 1992）或任意Carnot群（第二作者）中可求长曲线的子集。在次要结果中，我们在任意完备、倍测度、局部拟凸度量空间中实现了Garnett-Killip-Schul构造，该构造在$\mathbb{R}^n$中生成一个加倍测度，该测度支撑在一条可求长曲线上。
英文摘要：
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in $\mathbb{R}^2$ (P. Jones, 1990), in $\mathbb{R}^n$ (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' $β$-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in $\mathbb{R}^n$ that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space.
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