度量空间中可求长曲线的一个尖锐必要条件
A sharp necessary condition for rectifiable curves in metric spaces
摘要：
在他1990年的Inventiones论文中，P. Jones使用现在被称为Jones $β$的多尺度和来表征平面中的可求长曲线的子集-数字，在给定尺度和位置测量平坦度的数字。这项工作被Okikiolu推广到R^n，被第二作者推广到希尔伯特空间，并且在各种度量设置中有许多变体。值得注意的是，在2005年，Hahlomaa给出了度量空间的子集包含在可求长曲线中的充分条件。我们证明了加倍曲线的Hahlomaa定理的最尖锐的可能的逆，然后推导出度量和巴拿赫空间的子集，以及海森堡群的一些推论。
英文摘要：
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $β$-numbers, numbers measuring flatness in a given scale and location. This work was generalized to R^n by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa's theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group.}
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