论文标题
ra措施和Lipschitz图
Radon measures and Lipschitz graphs
论文作者
论文摘要
对于所有$ 1 \ leq m \ leq n-1 $，我们调查了$ \ mathbb {r}^n $与$ m $ $ $ dimensional lipschitz图形的本地有限度量的相互作用。例如，我们表征了rad尺寸的$μ$，在存在图中，Lipschitz将其显示为$γ_1，γ_2，\点$，使得$μ（\ Mathbb {r}^n \ setMinus \ setMinus \ bigCup_1^\ bigCup_1^\inftyγ_i）= 0 $，仅使用许多量度评估。几何度量理论中的这个问题是在较小的度量类别中进行了经典研究的，例如〜对于$ m $ $二维的Hausdorff度量$ \ MATHCAL {h}^m $ to $ e \ e \ subseteq \ subseteq \ mathbb {r}^n $ ting $ 0 <\ natercal $ 0 <\ \ nathcal {h H}^M（e）然而，Csörnyei，Käenmäki，Rajala和Suomala的一个例子表明，经典方法不足以检测一般测量何时收取Lipschitz图。为了开发Lipschitz图对任意ra的量度的重构性的表征，我们研究了与圆锥形Annuli相交的二聚体立方体上的粗加倍比率的行为。通过模仿Badger和Schul通过对rad子测量的表征，通过模仿在表征ra径的方法中，通过那不勒斯通过那不勒斯来扩展了图形重差的表征。
For all $1\leq m\leq n-1$, we investigate the interaction of locally finite measures in $\mathbb{R}^n$ with the family of $m$-dimensional Lipschitz graphs. For instance, we characterize Radon measures $μ$, which are carried by Lipschitz graphs in the sense that there exist graphs $Γ_1,Γ_2,\dots$ such that $μ(\mathbb{R}^n\setminus\bigcup_1^\inftyΓ_i)=0$, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, e.g.~for the restrictions of $m$-dimensional Hausdorff measure $\mathcal{H}^m$ to $E\subseteq \mathbb{R}^n$ with $0<\mathcal{H}^m(E)<\infty$. However, an example of Csörnyei, Käenmäki, Rajala, and Suomala shows that classical methods are insufficient to detect when a general measure charges a Lipschitz graph. To develop a characterization of Lipschitz graph rectifiability for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes that intersect conical annuli. This extends a characterization of graph rectifiability for pointwise doubling measures by Naples by mimicking the approach used in the characterization of Radon measures carried by rectifiable curves by Badger and Schul.
