Статья
2013
Absolute continuity of quasiconformal mapping of Carnot-Carathéodory spaces
M. V. Tryamkin
Русская математика
https://doi.org/10.3103/S1066369X13050071
Abstract / Full Text
We show that a quasiconformal mapping of Carnot-Carathéodory spaces is absolutely continuous not only on integral curves of horizontal vector fields but also on integral curves of vector fields whose degree differs from one.
Author information
- Novosibirsk State University, 2 Pirogova str., Novosibirsk, 630090, RussiaM. V. Tryamkin
References
- D. E. Menchoff, “Sur Une Généralization d’un Théorème de M. H. Bohr,” Matem. sb. 2(44), 339–356 (1937).
- Yu. G. Reshetnyak, Spatial Mappings with Bounded Distortion (Nauka, Novosibirsk, 1982) [in Russian].
- J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings (Springer, Berlin etc., 1971).
- B. Fuglede, “Extremal Length and Functional Completion,” Acta Math. 98, 171–219 (1957).
- P. Pansu, “Métriques de Carnot-Carathéodory etQuasiisométries des Espaces Symmétriques de Rang Un,” Ann. Math. 129(1), 1–60 (1989).
- G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces (Tokyo Univ. Press, Princeton, 1973).
- A. Koranyi and H.M. Reimann, “Foundations for the Theory of Quasiconformal Mappings on the Heisenberg Group,” Adv. Math. 111(1), 1–87 (1995).
- S. K. Vodop’anov and A. V. Greshnov, “Analytic Properties of Quasiconformal Mappings on Carnot Groups,” Sib. Matem. Zhurn. 36(6), 1317–1327 (1995).
- G. A. Margulis and G. D. Mostow, “The Differential of Quasiconformal Mapping of a Carnot-Carathéodory Space,” Geom. Funct. Anal. 5(2), 402–433 (1995).
- S. G. Basalaev and S. K. Vodopyanov, “Approximate Differentiability of Mappings of Carnot-Carathéodory Spaces,” Eurasian Math. J. 3(4) (2012).
- J. Mitchell, “On Carnot-Carathéodory Metrics,” J. Differ. Geom. 21(1), 35–45 (1985).
- J. Heinonen and P. Koskela, “Quasiconformal Maps in Metric Spaces with Controlled Geometry,” Acta Math. 181(1), 1–61 (1998).
- J. Heinonen and P. Koskela, “Definitions of Quasiconformality,” Invent. Math. 120(1), 61–70 (1995).
- S. G. Basalaev, “The Poincaré Inequality for C 1-Smooth Vector Fields,” Tr. Matem. Tsentra im. N. I. Lobachevskogo (Kazan, Kazansk. Gos. Univ., 2012), Vol. 45, pp. 20–23.