论文标题
一类积分分类测量和应用的可重新讨论性
Rectifiability of a class of integralgeometric measures and applications
论文作者
论文摘要
我们解决了费德勒（Federer）与指数p> 1的积分几何措施的重构性有关的长期开放问题，从而解决了一个自提出以来一直存在的问题。尽管主要定理与以前的版本没有变化，但已对博览会和应用进行了基本修订，以强调结果对Vitushkin对分析能力的猜想和复杂平面中可移动性的后果。作为一种应用，我们建立了与Vitushkin的猜想相关的两个新型结果：在多尺度环境中，我们为在以前未解决的小规模的偏爱长度行为中的有限积分几何措施的集合提供了肯定的答案；在单尺度的框架中，我们将Besicovitch-Federer投影定理扩展到经典的Sigma-Finite设置之外，即对于平面设置，在有限的许多点中与典型的线相交。较早版本中包含的一般ra措施的重新讨论标准已被删除，并将出现在单独的工作中。
We resolve a long-standing open problem posed by Federer concerning the rectifiability of the integral geometric measure with exponent p >1, thereby settling a question that has persisted since its formulation. While the main theorem is unchanged from previous versions, the exposition and applications have been substantially revised to highlight the result's consequences for Vitushkin's conjecture on analytic capacity and removability in the complex plane. As an application, we establish two novel results related to Vitushkin's conjecture: in a multi-scale setting, we provide an affirmative answer for sets with finite integral geometric measure within regimes of Favard length behavior at small scales not previously addressed; and in a single-scale framework, we extend the Besicovitch-Federer projection theorem beyond the classical sigma-finite setting, namely for planar sets intersecting a typical line in finitely many points. The rectifiability criterion for general Radon measures via slicing, included in earlier versions, has been removed and will appear in separate work.
