Abstract
An upper bound for the measure of the image of a ball under mappings of a certain class generalizing the class of branched spatial quasi-isometries is determined. As a corollary, an analog of Schwarz’ classical lemma for these mappings is proved under an additional constraint of integral character. The obtained results have applications to the classes of Sobolev and Orlicz–Sobolev spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Vôisôlô, Lectures on n-Dimensional QuasiconformalMappings, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1971), Vol.229.
R. R. Salimov and E. A. Sevost’yanov, “The Poletskii and Vôisôlôinequalities for the mappings with (p, q)-distortion,” Complex Var. Elliptic Equ. 59 (2), 217–231 (2014).
D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “Toward the theory of Orlicz–Sobolev classes,” Algebra Anal. 25 (6), 50–102 (2013) [St. Petersbg. Math. J. 25 (6), 929–963 (2014)].
F. Gehring, “Lipschitz mappings and p-capacity of rings in n-space,” in Annals ofMathematical Studies, Vol. 66: Advances in the Theory of Riemann Surfaces (Proceedings of the 1969 Stony Brook Conference) (Princeton Univ. Press, Princeton, NJ, 1971), pp. 175–193.
K. Ikoma, “On the distortion and correspondence under quasiconformal mappings in space,” Nagoya Math. J. 25, 175–203 (1965).
O. Martio, S. Rickman, and J. Vôisôlô, “Definitions for quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A I 448, 1–40 (1969).
V. M. Gol’dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives, and Quasiconformal Mappings (Nauka, Moscow, 1983) [in Russian].
V. I. Kruglikov, “Capacity of condensers and spatial mappings quasiconformal in the mean,” Mat. Sb. 130 (172) (2 (6)), 185–206 (1986) [Math. USSR-Sb. 58 (1), 185–205 (1987)].
V. G. Maz’ya, Sobolev Spaces (Izd. Leningrad. Univ., Leningrad, 1985) [in Russian].
S. Rickman, Quasiregular Mappings, in Ergeb. Math. Grenzgeb. (3) (Springer-Verlag, Berlin, 1993), Vol.26.
R. R. Salimov and E. A. Sevost’yanov, “Analogs of the Ikoma–Schwartz lemma and Liouville theorem for mappings with unbounded characteristic,” Ukrain.Mat. Zh. 63 (10), 1368–1380 (2011) [UkrainianMath. J. 63 (10), 1551–1565 (2012)].
O. Martio, S. Rickman, and J. Vôisôlô, “Distortion and singularities of quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A I 465, 1–13 (1970).
R. R. Salimov, “Estimation of the measure of the image of the ball,” Sibirsk. Mat. Zh. 53 (4), 920–930 (2012) [SiberianMath. J. 53 (4), 739–747 (2012)].
M. A. Lavrent’ev, VariationalMethods for Boundary-Value Problems for Systems of Elliptic Equations (Izd. Akad. Nauk SSSR, Moscow, 1962; Noordhoff, Groningen, 1963).
S. Saks, Theory of the Integral (Courier Corporation, New York–Chicago, 1947; Inostrannaya Literatura, Moscow, 1949).
K. Kuratowski, Topology (Academic, New York–London; PWN, Warsaw, 1968;Mir, Moscow, 1969), Vol.2.
R. Miniowitz, “Normal families of quasimeromorphic mappings,” Proc. Amer. Math. Soc. 84 (1), 35–43 (1982).
A. Golberg, R. Salimov, and E. Sevost’yanov, “Normal families of discrete open mappings with controlled p-module,” in Contemp. Math., Vol. 667: Complex Analysis and Dynamical Systems VI (Israel Mathematical Conference Proceedings) (Amer.Math. Soc., Providence, RI, 2016), Part 2, pp. 83–103.
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory (Springer, New York, 2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © R. R. Salimov, E. A. Sevost’yanov, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 4, pp. 594–610.
Rights and permissions
About this article
Cite this article
Salimov, R.R., Sevost’yanov, E.A. On local properties of spatial generalized quasi-isometries. Math Notes 101, 704–717 (2017). https://doi.org/10.1134/S0001434617030294
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434617030294