Abstract
We study the behavior of a certain class of map**s of a domain in Euclidean space. We prove that this class is equicontinuous both at the interior and boundary points, of the domain provided that it consists of map**s that satisfy a common normalization condition and whose quasiconformality characteristic has only tempered growth in a neighborhood of each point in the closure of the domain.
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References
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Map** Theory (Springer, New York, 2009).
V. Gutlyanskii, V. Ryazanov, and E. Yakubov, “The Beltrami equations and prime ends,” J. Math. Sci. (N. Y.) 210 (1), 22–51 (2015).
E. A. Sevost’yanov and S. A. Skvortsov, “On the equicontinuity of families of map**s in the case of variable domains.,” Ukr. Math. J. 71 (7), 1071–1086 (2019).
J. Väisälä, Lectures on \(n\)-Dimensional Quasiconformal Map**s, in Lecture Notes in Math. (Springer- Verlag, Berlin, 1971), Vol. 229.
R. Näkki and B. Palka, “Uniform equicontinuity of quasiconformal map**s,” Proc. Amer. Math. Soc. 37 (2), 427–433 (1973).
D. P. Il’yutko and E. A. Sevost’yanov, “On prime ends on Riemannian manifolds,” J. Math. Sci., New York 241 (1), 47–63 (2019).
D. A. Kovtonyuk and V. I. Ryazanov, “Prime ends and Orlicz-Sobolev classes,” St. Petersburg Math. J. 27 (5), 765–788 (2016).
E. A. Sevost’yanov, “Boundary behavior and equicontinuity for families of map**s in terms of prime ends,” St. Petersburg Math. J. 30 (6), 973–1005 (2019).
R. Näkki, “Prime ends and quasiconformal map**s,” J. Anal. Math. 35, 13–40 (1979).
E. A. Sevost’yanov, Study of Space Map**s by a Geometric Method (Naukova Dumka, Kyiv, 2014) [in Russian].
E. A. Sevost’yanov, “Equicontinuity of homeomorphisms with unbounded characteristic,” Sib. Adv. Math. 23 (2), 106–122 (2013).
O. Lehto and K. Virtanen, Quasiconformal Map**s in the Plane (Springer, New York, 1973).
R. Näkki, “Extension of Loewner’s capacity theorem,” Trans. Amer. Math. Soc. 180, 229–236 (1973).
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Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 597-607 https://doi.org/10.4213/mzm12682.
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Sevost’yanov, E.A., Skvortsov, S.A. Equicontinuity of Families of Map**s with One Normalization Condition. Math Notes 109, 614–622 (2021). https://doi.org/10.1134/S0001434621030317
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DOI: https://doi.org/10.1134/S0001434621030317