We study map**s with branching satisfying certain conditions of distortion for the modulus of families of paths. Under the conditions that the domain of definition of map**s has a weakly flat boundary, the mapped domain is regular, and the majorant responsible for the distortion of the modulus of families of paths is integrable, it is proved that the families of all specified map**s with one normalization condition are equicontinuous in the closure of a given domain.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Caratheodory, “Über die Begrenzung einfach zusammenhangender Gebiete,” Math. Ann., 73, 323–370 (1913).
B. P. Kufaev, “Metrization of the space of all prime ends of the domains from the family B,” Mat. Zametki, 6, No. 5, 607–618 (1969).
G. D. Suvorov, Families of Plane Topological Map**s [in Russian], Izd. Sibir. Otdel. Akad. Nauk SSSR, Novosibirsk (1965).
G. D. Suvorov, Metric Theory of Prime Ends and Boundary Properties of Plane Map**s with Bounded Dirichlet Integrals [in Russian], Naukova Dumka, Kiev (1981).
G. D. Suvorov, Generalized Principle of Length and Area in the Theory of Map**s [in Russian], Naukova Dumka, Kiev (1985).
V. I. Kruglikov, “Prime ends of space domains with varying boundaries,” Dokl. Akad. Nauk SSSR, 297, No. 5, 1047–1050 (1987).
V. M. Miklyukov, “The relative Lavrent’ev distance and prime ends on nonparametric surfaces,” Ukr. Mat. Visn., 1, No. 3, 349–372 (2004).
E. Afanas’eva, V. Ryazanov, R. Salimov, and E. Sevost’yanov, “On boundary extension of Sobolev classes with critical exponent by prime ends,” Lobachevskii J. Math., 41, No. 11, 2091–2102 (2020).
V. Gutlyanskii, V. Ryazanov, and E. Yakubov, “The Beltrami equations and prime ends,” Ukr. Mat. Visn., 12, No. 1, 27–66 (2015).
D. Kovtonyuk, I. Petkov, and V. Ryazanov, “On the boundary behavior of map**s with finite distortion in the plane,” Lobachevskii J. Math., 38, No. 2, 290–306 (2017).
D. A. Kovtonyuk and V. I. Ryazanov, “On the theory of prime ends for space map**s,” Ukr. Mat. Zh., 67, No. 4, 467–479 (2015); English translation: Ukr. Math. J., 67, No. 4, 528–541 (2015).
D. A. Kovtonyuk and V. I. Ryazanov, “Prime ends and Orlicz–Sobolev classes,” St. Petersburg Math. J., 27, No. 5, 765–788 (2016).
V. Ryazanov and S. Volkov, “Prime ends in the Sobolev map** theory on Riemann surfaces,” Mat. Stud., 48, No. 1, 24–36 (2017).
V. Ryazanov and S. Volkov, “Prime ends in the map** theory on the Riemann surfaces,” J. Math. Sci., 227, No. 1, 81–97 (2017).
V. Ryazanov and S. Volkov, “Map**s with finite length distortion and prime ends on Riemann surfaces,” J. Math. Sci., 248, No. 2, 190–202 (2020).
O. Martio, S. Rickman, and J. Väisälä, “Definitions for quasiregular map**s,” Ann. Acad. Sci. Fenn. Ser. A1, 448, 1–40 (1969).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Ser. A1, 30, No. 1, 49–69 (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Map** Theory, Springer Science + Business Media, New York (2009).
R. Näkki, “Prime ends and quasiconformal map**s,” J. Anal. Math., 35, 13–40 (1979).
Yu. G. Reshetnyak, Space Map**s with Bounded Distortion, American Mathematical Society, Providence, RI (1989).
S. Rickman, Quasiregular Map**s, Springer, Berlin (1993).
J. Väisälä, Lectures on n-Dimensional Quasiconformal Map**s, Springer, Berlin (1971).
M. Vuorinen, “Exceptional sets and boundary behavior of quasiregular map**s in n-space,” Ann. Acad. Sci. Fenn. Math. Diss., 11, 1–44 (1976).
E. A. Sevost’yanov and S. A. Skvortsov, “On the convergence of map**s in metric spaces with direct and inverse modulus conditions,” Ukr. Mat. Zh., 70, No. 7, 952–967 (2018); English translation: Ukr. Math. J., 70, No. 7, 1097–1114 (2018).
E. A. Sevost’yanov and S. A. Skvortsov, “On the local behavior of one class of inverse map**s,” Ukr. Mat. Visn., 15, No. 3, 399–417 (2018).
E. A. Sevost’yanov and S. A. Skvortsov, “On map**s whose inverses satisfy the Poletsky inequality,” Ann. Acad. Sci. Fenn. Math., 45, 259–277 (2020).
E. O. Sevost’yanov, S. O. Skvortsov, and O. P. Dovhopyatyi, “On the nonhomeomorphic map**s with the Poletsky inverse inequality,” Ukr. Mat. Visn., 17, No. 3, 414–436 (2020).
D. I. Il’yutko and E. A. Sevost’yanov, “On the boundary behavior of discrete map**s on Riemann manifolds. II,” Mat. Sb., 211, No. 4, 63–111 (2020).
G. M. Goluzin, Geometric Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).
O. Martio, S. Rickman, and J. Väisälä, “Topological and metric properties of quasiregular map**s,” Ann. Acad. Sci. Fenn. Ser. A1, 488, 1–31 (1971).
E. A. Sevost’yanov, “Boundary extensions of map**s satisfying the Poletsky inverse inequality in terms of prime ends,” Ukr. Mat. Zh., 73, No. 7, 951–963 (2021); English translation: Ukr. Math. J., 73, No. 7, 1107–1121 (2021).
J. Herron and P. Koskela, “Quasiextremal distance domains and conformal map**s onto circle domains,” Complex Var. Theory Appl., 15, 167–179 (1990).
K. Kuratowski, Topology, Vol. 2, Academic Press, New York (1968).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 6, pp. 817–825, June, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i6.6887.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ilkevych, N.S., Sevost’yanov, E.A. On Equicontinuity of the Families of Map**s with One Normalization Condition in Terms of Prime Ends. Ukr Math J 74, 936–945 (2022). https://doi.org/10.1007/s11253-022-02107-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02107-0