The coarea formula for real-valued Lipschitz maps on stratified groups
✍ Scribed by Valentino Magnani
- Publisher
- John Wiley and Sons
- Year
- 2005
- Tongue
- English
- Weight
- 274 KB
- Volume
- 278
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
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 function 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 function, where the Euclidean one clearly fails. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz functions which arises from the geometry of the group and that this class may be strictly larger than the usual one. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)