On Poletsky type inequality for mappings of Riemannian surfaces
Abstract
We obtain upper estimates for the distortion of the modulus of families of paths under the Sobolev class mappings, whose dilatation is locally integrable. As a consequence, we prove theorems on local and boundary behavior for these mappings.
References
Martio, O.; Rickman, S.; Väisälä, J. Distortion and singularities of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I, no. 465, 1970, 13 pp. https://doi.org/10.5186/aasfm.1971.488 DOI: https://doi.org/10.5186/aasfm.1971.488
Martio, Olli; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard. Moduli in modern mapping theory. Springer Monographs in Mathematics. Springer, New York, 2009. xii+367 pp. ISBN: 978-0-387-85586-8 https://doi.org/10.1007/978-0-387-85588-2_3 DOI: https://doi.org/10.1007/978-0-387-85588-2_3
Väisälä, Jussi. Lectures on $n$-dimensional quasiconformal mappings. Lecture Notes in Mathematics, Vol. 229. Springer-Verlag, Berlin-New York, 1971. xiv+144 pp. DOI: https://doi.org/10.1007/BFb0061216
Ryazanov, Vladimir; Volkov, Sergei. On the boundary behavior of mappings in the class $W^{1,1}_{{loc}}$ on Riemann surfaces. Complex Anal. Oper. Theory 11 (2017), no. 7, 1503–1520. https://doi.org/10.1007/s11785-016-0618-4 DOI: https://doi.org/10.1007/s11785-016-0618-4
Ryazanov, V.; Volkov, S. Prime ends in the Sobolev mapping theory on Riemann surfaces. Mat. Stud. 48 (2017), no. 1, 24–36.https://doi.org/10.15330/ms.48.1.24-36 DOI: https://doi.org/10.15330/ms.48.1.24-36
Väisälä, Jussi. Modulus and capacity inequalities for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A. I 1972, no. 509, 14 pp. https://doi.org/10.5186/aasfm.1972.509 DOI: https://doi.org/10.5186/aasfm.1972.509
Näkki, Raimo. Boundary behavior of quasiconformal mappings in $n$-space. Ann. Acad. Sci. Fenn. Ser. A I No. 484 1970 50 pp.
Poleckij, E. A. Метод модулей для негомеоморфных квазиконформных отображений. (Russian) [Metod modulej dlja negomeomorfnyh kvazikonformnyh otobrazhenij], Mat. sb., 83 (1970), no. 2, 261–272.
Krushkal', S. L.; Apanasov, B. N.; Gusevskij, N. A. Униформизация и клейновы группы. (Russian) [Uniformizacija i klejnovy gruppy]. Novosib. gos. un-t, Novosibirsk (1979).
Berdon, A. Геометрия дискретных групп. (Russian) [The geometry of discrete groups] Translated from the English and with a foreword by A. S. Solodovnikov. "Nauka", Moscow, 1986. 301 pp. https://www.twirpx.com/file/1049045/
Heinonen, Juha. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001. x+140 pp. ISBN: 0-387-95104-0 https://link.springer.com/book/10.1007%2F978-1-4613-0131-8
Malý, Jan; Martio, Olli. Lusin's condition (N) and mappings of the class $W^{1,n}$. J. Reine Angew. Math. 458 (1995), 19–36. https://doi.org/10.1515/crll.1995.458.19 DOI: https://doi.org/10.1515/crll.1995.458.19
Kuratovskij, K. Топология, т. 1. (Russian) [Topologija, t. 1]. Mir, Moskva (1966). http://biblioclub.ru/index.php?page=book_red&id=464136&razdel=213
Federer, G. Геометрическая теория меры. (Russian) [Geometricheskaja teorija mery], Nauka, Moskva (1987). https://www.twirpx.com/file/571043/
Saks, Stanislaw. Теория интеграла. (Russian) [Theory of the Integral]. Izd-vo inostr. lit., Moskva (1949). https://www.amazon.com/Theory-Integral-Dover-Books-Mathematics/dp/0486446484
Ignat'ev, A.; Rjazanov, V. Конечное среднее колебание в теории отображений. (Russian) [Konechnoe srednee kolebanie v teorii otobrazhenij]. Укр. мат. вестн. [Ukr. mat. vestn.], 2, no. 3, 395–417 (2005).
Sevost'yanov, E.; Markysh, A. On Sokhotski–Casorati–Weierstrass theorem on metric spaces. Complex Var. Elliptic Equ. 64 (2019), no. 12, 1973–1993. https://doi.org/10.1080/17476933.2018.1557155 DOI: https://doi.org/10.1080/17476933.2018.1557155
Севостьянов, Е. А. О локальном и граничном поведении отображений в метрических пространствах. (Russian) [O lokal'nom i granichnom povedenii otobrazhenij v metricheskih prostranstvah], Алгебра и анализ [Algebra i analiz], 28, no. 6, 118–146 (2016).
Copyright (c) 2020 Євген Олександрович Севостьянов
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.