论文标题
希尔伯特空间中倍增措施的重新纠正性
Rectifiability of pointwise doubling measures in Hilbert Space
论文作者
论文摘要
在几何度量理论中，人们有兴趣研究量度与可重新设定集的相互作用。在这里，我们在欧几里得空间中扩展了Badger和Schul的定理，以表征希尔伯特空间中可重新合转的次数加倍措施。给定一个量度$μ$，我们构建了一个多解决家庭$ \ mathscr {c}^μ$的Windows，然后我们使用加权琼斯的功能$ \ hat {j} _2（μ，x）$来记录如何很好地记录线条如何近似于每个窗口的分布。我们表明，当$μ$可整流时，质量在每个尺度的一条线周围充分集中，并且匡威也可以保持。此外，我们提出了一种使用适当选择的$δ$ -NET构建可矫正曲线的算法。在整个过程中，我们讨论了如何克服这样一个事实，即在无限的希尔伯特空间中，即使在有限的集合中，也可能存在无限的$δ$分隔点。最后，我们证明了Lipschitz图所携带的双重倍增度量的表征。
In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure $μ$, we construct a multiresolution family $\mathscr{C}^μ$ of windows, and then we use a weighted Jones' function $\hat{J}_2(μ, x)$ to record how well lines approximate the distribution of mass in each window. We show that when $μ$ is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the converse also holds. Additionally, we present an algorithm for the construction of a rectifiable curve using appropriately chosen $δ$-nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many $δ$-separated points, even in a bounded set. Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs.
