$BV$ spaces and rectifiability for Carnot-Carathéodory metrics: an introduction

summary:This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory 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-Carathéodory 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-Carathéodory metric, to the properties of related Sobolev spaces and to Poincaré 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. Gutiérrez, P. Hajłasz, P. Koskela, G. Lu and C. Pérez, deep relationships between Poincaré inequality and the geometry of Carnot-Carathéodory spaces are studied