Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru
DOI: 10.46698/o8833-7719-4418-f
Stoilow Factorization of the Heisenberg Group
Dorokhin, D. K.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 3.
Abstract: In this article we study the properties of quasiconformal mappings on the Heisenberg group \(\Bbb H^1\) and consider the definition of quasiconformal mappings in terms of the Beltrami equation. In particular, we obtain an explicit expression for the Beltrami coefficient for the composition of two quasiconformal mappings and we prove an analogue of the Stoilow factorization theorem on the plane. Namely, if the Beltrami coefficients of two quasiconformal mappings are equal almost everywhere, then there exists a conformal mapping such that by acting on one of the given quasiconformal mappings from the left, we obtain another given mapping. As an application of these results on the Heisenberg group \(\Bbb H^1\) we compute the Beltrami coefficients of some quasiconformal mappings and we prove a theorem on the images of quasi-Brownian motions. In specific examples we demonstrate the invariance of the Beltrami coefficient under the action of the composition of a conformal function on the corresponding left mapping. Using the Stoilow factorization on the Heisenberg group, we show that if two quasi-Brownian motions have the corresponding Beltrami coefficients equal almost everywhere, then their trajectories are equivalent only if the conformal map in the Stoilov factorization is a map obtained from a composition of translations, rotations and dilations.
For citation: Dorokhin, D. K. Stoilow Factorization of the Heisenberg Group, Vladikavkaz Math. J., 2025, vol. 27, no. 3, pp. 50-59. DOI 10.46698/o8833-7719-4418-f
1. Astala, K., Iwaniec, T. and Martin, G. Elliptic Partial Differential Equations and Quasiconformal
Mappings in the Plane, Princeton, Princeton University Press, 2009, 696 p.
DOI: 10.1515/ 9781400830114.
2. Stoilow, S. Lecons Sur les Principes Topologiques de la Theorie des Fonctions
Analytiques Professees a la Sorbonne et a l'Universite de Cernauti,
Paris, Gauthier-Villars, 1938.
3. Reshetnyak, Y. G. Space Mappings with Bounded Distortion,
Siberian Mathematical Journal, 1967, vol. 8, pp. 466-487.
DOI: 10.1007/BF02196429.
4. Mostow, G. D. Strong Rigidity of Locally Symmetric Spaces,
Princeton University Press, 1973, 204 p.
5. Mostow, G. D. Quasi-Conformal Mappings in $n$-Spaces and the Rigidity of Hyperbolic
Space Forms, Publications Mathematiques de l'Institut des Hautes Scientifiques,
1968, vol. 34, pp. 53-104. DOI: 10.1007/BF02684590.
6. Gromov, M. Groups of Polynomial Growth and Expanding Maps,
Publications Mathematiques de l'Institut des Hautes Scientifiques,
1981, vol. 53, pp. 53-78. DOI: 10.1007/BF02698687.
7. Gromov, M. Structures Metriques Pour les Varietes Riemanniennes,
United States, Boston, Birkhauser, 1999, 585 p.
8. Mitchell, J. On Carnot-Caratheodory Metrics, Journal of Differential Geometry, 1981,
vol. 21, no. 1, pp. 35-45. DOI: 10.4310/jdg/1214439462.
9. Gromov, M. Carnot-Caratheodory Spaces Seen from Within,
Sub-Riemannian Geometry, Progress in Mathematics, vol. 144,
Basel, Birkhauser, 1981, pp. 79-323.DOI: 10.1007/978-3-0348-9210-0_2.
10. Pansu, P. Metriques de Carnot-Caratheodory et Quasiisometries des Espaces
Symetriques de Rang un, Annals of Mathematics,
1989, vol. 129, pp. 1-60. DOI: 10.2307/1971484.
11. Koranyi, A. and Reimann, H. M. Foundations for the Theory of Quasiconformal Mappings
on the Heisenberg Group, Advances in Mathematics, 1995, vol. 111, pp. 1-87. DOI: 10.1006/aima.1995.1017.
12. Vodopyanov, S. K. Monotone Functions and Quasiconformal Mappings on Carnot
Groups, Siberian Mathematical Journal, 1996, vol. 37, no. 6,
pp. 1113-1136. DOI: 10.1007/BF02106736.
13. Vodopyanov, S. K and Greshnov, A. V. On Extension of Functions of Bounded Mean
Oscillation from Domains in a Space of Homogeneous Type with Intrinsic Metric,
Siberian Mathematical Journal, 1995, vol. 36, no. 5, pp. 873-901. DOI: 10.1007/BF02112531.
14. Vodopyanov, S. K and Greshnov, A. V. Analytical Properties of Quasiconformal
Mappings on Carnot Groups, Siberian Mathematical Journal,
1995, vol. 36, no. 6, pp. 1142-1151. DOI: 10.1007/BF02106836.
15. Vodopyanov, S. K and Greshnov, A. V. Continuation of Differentiable Functions and Quasiconformal Mappings on Carnot Groups, Doklady Akademii Nauk, 1996, vol. 348, no. 1, pp. 15-18.
16. Vodopyanov, S. K and Ukhlov, A. D. Approximately Differentiable Transformations and
Replacement Variables on Nilpotent Groups, Siberian Mathematical Journal, 1996, vol. 37, no. 1, pp. 70-89.
DOI: 10.1007/BF02104760.
17. Koranyi, A. and Reimann, H. M. Quasiconformal Mappings on the Heisenberg Group,
Inventiones Mathematicae, 1985, vol. 80, pp. 309-338.
18. Wu, Q. Ya and Wang, W. The Beltrami Equations for Quasiconformal Mappings on Strictly
Pseudoconvex Hyperplanes, Siberian Mathematical Journal, 2012,
vol. 53, no. 2, pp. 316-334. DOI: 10.1134/ S0037446612020140.
19. Isangulova, D. V. The Class of Mappings with Bounded Specific Oscillation,
and Integrability of Mappings with Bounded Distortion on Carnot Groups,
Siberian Mathematical Journal, 2007, vol. 48, no. 2, pp. 249-267.
DOI: 10.1007/s11202-007-0025-1.
20. Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F. Stratified Lie Groups and Potential Theory for Their Sub-Laplacian,
Berlin-Heidelberg, Springer-Verlag, 2007.
21. Vodopyanov, S. K. Closure of Classes of Mappings with Bounded Distortion on
Carnot Groups, Siberian Advances in Mathematics, 2004, vol. 14, no. 1, pp. 84-125.
22. Platis, I. D. Quasiconformal Mappings on the Heisenberg Group: An Overview,
Handbook of Teichmuller Theory, IRMA Lectures in Mathematics and Theoretical Physics,
2016, vol. 6, pp. 375-393. DOI: 10.4171/161-1/12.
23. Bernard, A., Campbell, E. A. and Davie, A. M. Brownian Motions and Generalized Analytic and Inner Functions,
Annales de l'Institut Fourier, 1979, vol. 29, no. 1, pp. 207-228.
24. Oksendal, B. Dirichlet Forms, Quasiregular Functions and Brownian Motion,
Inventiones Mathematicae, 1988, vol. 91, pp. 273-297.
25. Evseev, N. A. Brownian Path Preserving Mappings on the Heisenberg Group,
Journal of Mathematical Analysis and Applications,
2025, vol. 548, no. 2, 129388. DOI: 10.1016/j.jmaa.2025.129388.
26. Zijian, Le. Brownian Motion, Quasiconformal Mappings and the Beltrami
Equation, A Dissertation Submitted in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy, University of Washington, 2021.
27. Calin, O. Transience of Diffusions on Heisenberg and Grushin Distributions,
Journal of Geometry and Physics, 2014, vol. 77, pp. 131-142. DOI: 10.1016/j.geomphys.2013.12.010.
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
