$BV$ spaces and rectifiability for Carnot-Carathéodory metrics: an introduction
$BV$ spaces and rectifiability for Carnot-Carathéodory metrics: an introduction
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发表时间：
2003
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通讯作者：
B. Franchi
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作者：
B. Franchi
本文的意思是作为一个（短和部分）介绍研究的几何卡诺集团，更一般地说，卡诺-Caratheodory空间与一个家庭的Lipschitz连续向量场。我个人对这一领域的兴趣可以追溯到与E. Lanconelli，在那里，这个概念被利用来研究退化椭圆型偏微分方程弱解的逐点正则性。如标题所述，在这里，我们主要关注的主题几何测量理论在卡诺集团，特别是与rectifiability理论在此设置。因此，本文的核心包括第3节（致力于研究关于卡诺-Caratheodory度量的BV函数），第4节（更具体地致力于卡诺群的理论，特别是与微分丛的微分结构相关的微积分）和第5节（致力于卡诺群中的内在超曲面理论和可求正性理论）。这些部分基本上依赖于与R。Serapioni和F.塞拉卡萨诺，从1996年开始。另一方面，第2节和第6节致力于Carnot-Caratheodory度量的概念，相关Sobolev空间的性质和与Lipschitz连续向量场族相关的Poincare不等式。特别地，依靠与R. L. Wheeden，S.加洛角Gutierrez，P. Hajlasz，P. Koelia，G. Lu和C. Perez等人的工作，研究了Poincare不等式与Carnot-Caratheodory空间几何之间的深层关系。
This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Caratheodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned with topics of Geometric Measure Theory in Carnot groups and in particular with rectifiability theory in this setting. Thus, the core of the paper consists of Section 3 (dedicated to the study of BV functions with respect to Carnot-Caratheodory metrics), of Section 4 (dedicated more specifically to the theory of Carnot groups and, in particular, to the calculus associated with their differential structure as differential bundles) and of Section 5 (dedicated to the theory of intrinsic hypersurfaces and to rectifiability theory in Carnot groups). These sections rely basically on a group of results obtained in several papers in collaboration with R. Serapioni and F. Serra Cassano, starting from 1996. On the other hand, Section 2 and 6 are dedicated to the notion of Carnot-Caratheodory metric, to the properties of related Sobolev spaces and to Poincare inequality associated with a family of Lipschitz continuous vector fields. In particular, relying on a group of joint papers with R. L. Wheeden, S. Gallot, C. Gutierrez, P. Hajlasz, P. Koskela, G. Lu and C. Perez, deep relationships between Poincare inequality and the geometry of Carnot-Caratheodory spaces are studied.