Geometry of Carnot-Carathéodory spaces, quasiconformal analysis, and geometric measure theory. (Russian) Zbl 1048.43011

P. Pansu [Ann. Math. (2) 129, No. 1, 1–60 (1989; Zbl 0678.53042)] extended Rademacher’s theorem [H. Rademacher, Math. Ann. 79, 340–359 (1919); 81, 52–63 (1920; JFM 47.0243.01)] on the differentiability almost everywhere of a Lipschitz mapping to Carnot groups. In the same paper Pansu formulated a generalization of Stepanov’s theorem [W. Stepanoff, Math. Ann. 90, 318–320 (1923; JFM 49.0183.01)] to Carnot groups. Unfortunately, Pansu left the latter result without a detailed proof while it turns out that the arguments used in the Euclidean case do not work in the case of Carnot groups. A generalization of Stepanov’s theorem to Carnot groups was established in [S. K. Vodopyanov and A. D. Ukhlov, Sib. Math. J. 37, No. 1, 62–78 (1996; Zbl 0870.43005)].
In the paper under review it is demonstrated how the concept of differentiability can be extended to Lipschitz maps of Carnot-Carathéodory spaces. Some applications to geometric measure theory and the theory of quasiconformal maps are given.


43A85 Harmonic analysis on homogeneous spaces
46G05 Derivatives of functions in infinite-dimensional spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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