论文标题
关于Carnot组的可纠正措施：Marstand-Mattila可重新讨论标准
On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion
论文作者
论文摘要
在本文中，我们继续研究Carnot组的$ \ Mathscr {p} $ - 可重差性的概念。 We say that a Radon measure is $\mathscr{P}_h$-rectifiable, for $h\in\mathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples.在本文中，我们证明了$ \ mathscr {p} $的任意Carnot组中的Marstrand-Mattila可重新讨论标准，可使用切线的平面进行正常互补子组的切线。也就是说，在这种共同的情况下，即使先验地点处的切线平面在不同的尺度上可能并不相同，后验也几乎到处都有独特的切线。 由于Carnot组的每个水平亚组都有正常的补体，因此我们的标准适用于切线是一维水平亚组的特殊情况。因此，作为我们的马斯特兰（Marstrand）的直接结果 - 马蒂拉（Mattila）的重新讨论标准，以及chousionis--magnani-tyson的结果，我们在第一个海森伯格集团$ \ mathbb h^1 $中获得了一维的Preiss定理。更确切地说，我们表明，$ \ mathbb h^1 $上的ra量度$ ϕ $与koranyi相对于koranyi的距离具有积极和有限的一密度，对于一维的hausdorff量$ \ mathcal {h}^1 $，绝对是连续的Lipschitz从$ a \ subseteq \ mathbb r $到$ \ mathbb h^1 $。
In this paper we continue the study of the notion of $\mathscr{P}$-rectifiability in Carnot groups. We say that a Radon measure is $\mathscr{P}_h$-rectifiable, for $h\in\mathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. In this paper we prove a Marstrand--Mattila rectifiability criterion in arbitrary Carnot groups for $\mathscr{P}$-rectifiable measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if a priori the tangent planes at a point might not be the same at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup of a Carnot group has a normal complement, our criterion applies in the particular case in which the tangents are one-dimensional horizontal subgroups. Hence, as an immediate consequence of our Marstrand--Mattila rectifiability criterion and a result of Chousionis--Magnani--Tyson, we obtain the one-dimensional Preiss's theorem in the first Heisenberg group $\mathbb H^1$. More precisely, we show that a Radon measure $ϕ$ on $\mathbb H^1$ with positive and finite one-density with respect to the Koranyi distance is absolutely continuous with respect to the one-dimensional Hausdorff measure $\mathcal{H}^1$, and it is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps from $A\subseteq \mathbb R$ to $\mathbb H^1$.
