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The coarea formula for real‐valued Lipschitz maps on stratified groups
Author(s) -
Magnani Valentino
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310334
Subject(s) - mathematics , lipschitz continuity , differentiable function , lipschitz domain , hausdorff dimension , euclidean geometry , pure mathematics , group (periodic table) , hausdorff measure , class (philosophy) , hausdorff distance , dimension (graph theory) , measure (data warehouse) , heisenberg group , packing dimension , mathematical analysis , minkowski–bouligand dimension , geometry , fractal dimension , fractal , chemistry , organic chemistry , database , artificial intelligence , computer science
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)