On equicontinuity of families of mappings with one normalization condition by the prime ends
Abstract
UDC 517.5
We study mappings with branching that satisfy certain conditions of distortion for the modulus of paths families. Under the conditions that the domain of definition of mappings has a weakly flat boundary, the mapped domain is regular, and the majorant responsible for the distortion of modulus of the families of paths is integrable; it is proved that the families of all specified mappings with one normalization condition are equicontinuous in the closure of the given domain.
References
C. Caratheodory, Über die Begrenzung einfach zusammenhängender Gebiete. (German), Math. Ann., 73, 323 – 370 (1913), https://doi.org/10.1007/BF01456699 DOI: https://doi.org/10.1007/BF01456699
B. P. Kufarev, Metrizaciya prostranstva vsekh prostyh koncov oblastej semejstva frakB , Mat. zametki, 6, № 5, 607 – 618 (1969).
G. D. Suvorov, Semejstva ploskih topologicheskih otobrazhenij, Izd-vo SO AN SSSR, Novosibirsk (1965).
G. D. Suvorov, Metricheskaya teoriya prostyh koncov i granichnye svojstva ploskih otobrazhenij s ogranichennymi integralami Dirihle/, Nauk. dumka, Kiev (1981).
G. D. Suvorov, Obobshchennyj princip dliny i ploshchadi v teorii otobrazhenij, Nauk. dumka, Kiev (1985).
V. I. Kruglikov, Prostye koncy prostranstvennyh oblastej s peremennymi granicami, Dokl. AN SSSR, 297, № 5, 1047 – 1050 (1987).
V. M. Miklyukov, Otnositel'noe rasstoyanie M. A. Lavrent'eva i prostye koncy na neparametricheskih poverhnostyah, Ukr. mat. visn., 1, № 3, 349 – 372 (2004).
E. Afanas’eva, V. Ryazanov, R. Salimov, E. Sevost’yanov, On boundary extension of Sobolev classes with critical exponent by prime ends, Lobachevskii J. Math., 41, № 11, 2091 – 2102 (2020), https://doi.org/10.1134/s1995080220110025 DOI: https://doi.org/10.1134/S1995080220110025
V. Gutlyanskii, V. Ryazanov, E. Yakubov, The Beltrami equations and prime ends, Укр. мат. вiсн., 12, № 1, 27 – 66 (2015), https://doi.org/10.1007/s10958-015-2546-7 DOI: https://doi.org/10.1007/s10958-015-2546-7
D. Kovtonyuk, I. Petkov, V. Ryazanov, On the boundary behavior of mappings with nite distortion in the plane, Lobachevskii J. Math., 38, № 2, 290 – 306 (2017), https://doi.org/10.1134/S1995080217020123 DOI: https://doi.org/10.1134/S1995080217020123
D. A. Kovtonyuk, V. I. Ryazanov, K teorii prostyh koncov dlya prostranstvennyh oblastej, Ukr. mat. zhurn., 67, № 4, 467 – 479 (2015). DOI: https://doi.org/10.1007/s11253-015-1098-9
D. A. Kovtonyuk, V. I. Ryazanov, Prime ends and Orlicz – Sobolev classes, St. Petersburg Math. J., 27, № 5, 765 – 788 (2016), https://doi.org/10.1090/spmj/1416 DOI: https://doi.org/10.1090/spmj/1416
V. Ryazanov, S. Volkov, Prime ends in the Sobolev mapping theory on Riemann surfaces, Mat. Stud., 48, № 1, 24 – 36 (2017), https://doi.org/10.15330/ms.48.1.24-36 DOI: https://doi.org/10.15330/ms.48.1.24-36
V. Ryazanov, S. Volkov, Prime ends in the mapping theory on the Riemann surfaces, J. Math. Sci., 227, № 1, 81 – 97 (2017). DOI: https://doi.org/10.1007/s10958-017-3575-1
V. Ryazanov, S. Volkov, Mappings with nite length distortion and prime ends on Riemann surfaces, J. Math. Sci., 248, № 2, 190 – 202 (2020), https://doi.org/10.1007/s10958-020-04869-z DOI: https://doi.org/10.1007/s10958-020-04869-z
O. Martio, S. Rickman, J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 448, 1 – 40 (1969). DOI: https://doi.org/10.5186/aasfm.1969.448
O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, On $Q$-homeomorphisms, Ann. Acad. Sci. Fenn. Ser. A1, 30, № 1, 49 – 69 (2005).
O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Sci. + Business Media, LLC, New York (2009).
R. Nakki, Prime ends and quasiconformal mappings, J. Anal. Math., 35, 13 – 40 (1979), https://doi.org/10.1007/BF02791061 DOI: https://doi.org/10.1007/BF02791061
Yu. G. Reshetnyak, Space mappings with bounded distortion, Transl. Math. Monogr., 73 (1989), https://doi.org/10.1090/mmono/073 DOI: https://doi.org/10.1090/mmono/073
S. Rickman, Quasiregular mappings, Springer-Verlag, Berlin (1993), https://doi.org/10.1007/978-3-642-78201-5 DOI: https://doi.org/10.1007/978-3-642-78201-5
J. Väisälä, Lectures on$ n$-dimensional quasiconformal mappings, Lect. Notes Math., 229/, Springer-Verlag, Berlin etc. (1971). DOI: https://doi.org/10.1007/BFb0061216
M. Vuorinen, Exceptional sets and boundary behavior of quasiregular mappings in $n$-space, Ann. Acad. Sci. Fenn. Math. Diss., 11, 1 – 44 (1976).
E. A. Sevost'yanov, S. A. Skvorcov, O skhodimosti otobrazhenij v metricheskih prostranstvah s pryamymi i obratnymi modul'nymi usloviyami, Ukr. mat. zhurn., 70, № 7, 952 – 967 (2018).
E. A. Sevost'yanov, S. A. Skvorcov, O lokal'nom povedenii odnogo klassa obratnyh otobrazhenij, Ukr. mat. vestn., 15, № 3, 399 – 417 (2018).
E. A. Sevost’yanov, S. A. Skvortsov, On mappings whose inverse satisfy the Poletsky inequality, Ann. Acad. Sci. Fenn. Math., 45, 259 – 277 (2020), https://doi.org/10.5186/aasfm.2020.4520 DOI: https://doi.org/10.5186/aasfm.2020.4520
Є. O. Sevost'yanov, S. O. Skvorcov, O. P. Dovgopyatij, Pro negomeomorfni vidobrazhennya z obernenoyu nerivnistyu Polec'kogo, Ukr. mat. visn., 17, № 3, 414 – 436 (2020).
D. P. Il'yutko, E. A. Sevost'yanov, O granichnom povedenii otkrytyh diskretnyh otobrazhenij na rimanovyh mnogoobraziyah. II, Mat. sb., 211, № 4, 63 – 111 (2020). DOI: https://doi.org/10.4213/sm9228
G. M. Goluzin, Geometricheskaya teoriya funkcij kompleksnogo peremennogo, Fizmatgiz, Moskva (1966).
O. Martio, S. Rickman, J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 488, 1 – 31 (1971). DOI: https://doi.org/10.5186/aasfm.1971.488
E. A. Sevost’yanov, Boundary extension of mappings that satisfy the Poletsky inverse inequality in terms of prime ends, Укр. мат. журн., 73, № 7, 951 – 963 (2021), https://doi.org/10.37863/umzh.v73i7.6507 DOI: https://doi.org/10.1007/s11253-021-01980-5
J. Herron, P. Koskela, Quasiextremal distance domains and conformal mappings onto circle domains, Complex Var. Theory and Appl., 15, 167 – 179 (1990), https://doi.org/10.1080/17476939008814448 DOI: https://doi.org/10.1080/17476939008814448
K. Kuratovskij, Topologiya, t. 2, Mir, Moskva (1969).
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