Calculus of Variations and Geometric Measure Theory

V. Magnani

The coarea formula for real-valued Lipschitz maps on stratified groups

created on 21 May 2003
modified by magnani on 24 Apr 2006


Published Paper

Inserted: 21 may 2003
Last Updated: 24 apr 2006

Journal: Math. Nachr.
Volume: 278
Number: 14
Pages: 1689-1705
Year: 2005


We establish a coarea formula for real-valued Lipschitz maps on stratified groups when the domain is endowed with a homogeneous distance and level sets are measured by the $Q$-1 dimensional spherical Hausdorff measure. The number $Q$ is the Hausdorff dimension of the group with respect to its Carnot-Carathéodory distance. We construct a Lipschitz map on the Heisenberg group which is not approximately differentiable on a set of positive measure, provided that the Euclidean notion of differentiability is adopted. The coarea formula for stratified groups also applies to this map, where the Euclidean one clearly fails. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz maps which arises from the geometry of the group and that this class may be strictly larger than the usual one.

Keywords: Carnot-Carathéodory distance, stratified groups, coarea formula, lipschitz maps