Abstract
We study the geometry of balls in the Carnot-Carathéodory metric and find the optimal equivalence constants in the Ball-Box Theorem on the Heisenberg group algebras ℍ n α .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agrachev A., Barilari D., and Boscain U., “On the Hausdorff volume in sub-Riemannian geometry,” Calc. Var., 43, 355–388 (2012).
Postnikov M. M., Lie Groups and Lie Algebras [in Russian], Nauka, Moscow (1982).
Gromov M., “Carnot-Carathéodory spaces seen from within,” in: Sub-Riemannian Geometry (Vol. 144), Birkhäuser, Basel, 1996, pp. 79–323.
Korányi A. and Reimann H. M., “Foundations for the theory of quasiconformal mappings on the Heisenberg group,” Adv. Math., 111, No. 1, 1–87 (1995).
Greshnov A. V., “Local approximation of uniformly regular Carnot-Carathéodory quasispaces by their tangent cones,” Siberian Math. J., 48, No. 2, 229–248 (2007).
Greshnov A. V., “Differentiability of horizontal curves in Carnot-Carathéodory quasispaces,” Siberian Math. J., 49, No. 1, 53–67 (2008).
Nagel A., Stein E. M., and Wainger S., “Balls and metrics defined by vector fields. I: Basic properties,” Acta Math., 155, No. 1–2, 103–147 (1985).
Vodop’yanov 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 Math. J., 36, No. 5, 873–901 (1995).
Berestovskiĭ V. N. and Zubareva I. A., “Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups,” Siberian Math. J., 42, No. 4, 613–628 (2001).
Grushin V. V., “On a class of hypoelliptic operators,” Math. USSR-Sb., 12, No. 3, 458–476 (1970).
Grushin V. V., “On a class of elliptic pseudodifferential operators degenerate on a submanifold,” Math. USSR-Sb., 13, No. 2, 155–185 (1971).
Bellaïche A., “The tangent space in sub-Riemannian geometry,” in: Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, pp. 1–78.
Faĭzullin R. R., “Connection between the nonholonomic metric on the Heisenberg group and the Grushin metric,” Siberian Math. J., 44, No. 6, 1085–1090 (2003).
Capogna L., Danielli D., Pauls S. D., and Tyson J. T., An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Birkhäuser, Basel, Boston, and Berlin (2007).
Vershik A. M. and Gershkovich V. Ya., “Nonholonomic dynamical systems, geometry of distributions and variational problems,” in: Dynamical Systems. VII. Encycl. Math. Sci. Vol. 16, 1994, pp. 1–81.
Belykh A. V. and Greshnov A. V., “The condition of the cc-homogeneous cone and cc-balls on the Heisenberg groups,” Vestnik NGU. Ser. Mat., Mekh., Inform., 11, No. 4, 8–20 (2011).
Maz’ya V. G., Sobolev Spaces, Springer-Verlag, Berlin (2011).
Pontryagin L. S., Boltyanskiĭ V. G., Gamkrelidze R. V., and Mishchenko E. F., Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969).
Berestovskiĭ V. N., “Geodesics of nonholonomic left-invariant intrinsic metrics on the Heisenberg group and isoperimetric curves on the Minkowski plane,” Siberian Math. J., 35, No. 1, 1–8 (1994).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2014 Greshnov A.V.
The author was partially supported by a Grant of the Government of the Russian Federation for the State Maintenance of Scientific Research (Agreement 14.B25.31.0029).
To Yuriĭ Grigor’evich Reshetnyak on his 85th birthday.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 5, pp. 1040–1058, September–October, 2014.
Rights and permissions
About this article
Cite this article
Greshnov, A.V. The geometry of cc-balls and the constants in the Ball-Box theorem on Heisenberg group algebras. Sib Math J 55, 849–865 (2014). https://doi.org/10.1134/S003744661405005X
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S003744661405005X