论文标题
关于Carnot组的可纠正措施：密度的存在
On rectifiable measures in Carnot groups: existence of density
论文作者
论文摘要
In this paper we start a detailed study of a new notion of 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. 首先，我们将$ \ mathscr {p} _h $ - 重构性与以前在Carnot组的文献中所知的其他可重新讨论性概念进行比较，我们证明它严格比他们弱。其次，我们证明了$ \ mathscr {p} _h $ - retectififiable度量的几个结构属性。就是说，我们证明了$ \ Mathscr P_H $ -Rectiabile的支持几乎在满足类似圆锥形的属性的集合所涵盖的任何地方，并且在特殊情况下，在特殊情况下，我们表明，它们具有互补的切线，我们表明它们在本质上可lipsinally lipslipsible Lipschitz和不同的图表中得到了支持。这样的覆盖属性用于证明本文的主要结果：我们表明，每当切线至少接纳一个互补的亚组，每当切线接收到最高和有限的$ h $密度，几乎到处都有$ \ mathscr {p} _h $ - 可纠正的度量。
In this paper we start a detailed study of a new notion of 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. First, we compare $\mathscr{P}_h$-rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of $\mathscr{P}_h$-rectifiable measures. Namely, we prove that the support of a $\mathscr P_h$-rectifiabile measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of $\mathscr P_h$-rectifiabile measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a $\mathscr{P}_h$-rectifiable measure has almost everywhere positive and finite $h$-density whenever the tangents admit at least one complementary subgroup.
