B V spaces and rectifiability for Carnot-Carathéodory metrics: an introduction

Franchi, Bruno

  • Nonlinear Analysis, Function Spaces and Applications, Publisher: Czech Academy of Sciences, Mathematical Institute(Praha), page 73-132

Abstract

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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.

How to cite

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Franchi, Bruno. "$BV$ spaces and rectifiability for Carnot-Carathéodory metrics: an introduction." Nonlinear Analysis, Function Spaces and Applications. Praha: Czech Academy of Sciences, Mathematical Institute, 2003. 73-132. <http://eudml.org/doc/221113>.

@inProceedings{Franchi2003,
abstract = {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.},
author = {Franchi, Bruno},
booktitle = {Nonlinear Analysis, Function Spaces and Applications},
keywords = {Carnot-Carathéodory metrics; Carnot groups; Poincaré inequality; hypersurfaces; rectifiability; Sobolev spaces; BV spaces},
location = {Praha},
pages = {73-132},
publisher = {Czech Academy of Sciences, Mathematical Institute},
title = {$BV$ spaces and rectifiability for Carnot-Carathéodory metrics: an introduction},
url = {http://eudml.org/doc/221113},
year = {2003},
}

TY - CLSWK
AU - Franchi, Bruno
TI - $BV$ spaces and rectifiability for Carnot-Carathéodory metrics: an introduction
T2 - Nonlinear Analysis, Function Spaces and Applications
PY - 2003
CY - Praha
PB - Czech Academy of Sciences, Mathematical Institute
SP - 73
EP - 132
AB - 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.
KW - Carnot-Carathéodory metrics; Carnot groups; Poincaré inequality; hypersurfaces; rectifiability; Sobolev spaces; BV spaces
UR - http://eudml.org/doc/221113
ER -

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