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Calculus on Carnot groups. (English) Zbl 0863.22009

Kilpeläinen, T. (ed.), Fall school in analysis, Jyväskylä, Finland, October 3-7, 1994. Jyväskylä: University of Jyväskylä. Ber., Univ. Jyväskylä, Math. Inst. 68, 1-31 (1995).
A Carnot group is a simply connected nilpotent Lie group whose Lie algebra admits a grading. The geometric significance is that the grading can be used to define a left-invariant metric on the group in such a way that, with respect to this metric, the group is self-similar. Consequently there exist natural dilations on the group much like Euclidean similarities. There arises thereby a natural notion of conformality or quasiconformality exposited by the author in the last section of this survey paper. This quasiconformality-conformality was developed and exploited by Mostow in his proof of the famous Mostow rigidity theorem. The connection with Mostow’s setting is this: for almost all of the noncompact rank 1 symmetric spaces of negative sectional curvature the sphere at infinity is the one-point compactification of a Carnot group. The author also deals with other important analytic properties of Carnot groups: Sobolev inequalities, isoperimetric inequalities (consequences of the Sobolev inequalities), and Poincaré inequalities. The author includes good references to the foundations of the subject and instructive exercises.
Reviewer: J.W.Cannon (Provo)

MSC:

22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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