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 Mat. Sb., 2010, Volume 201, Number 6, Pages 131–158 (Mi msb7529)

The theory of shell-based $Q$-mappings in geometric function theory

R. R. Salimov, E. A. Sevost'yanov

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: Open, discrete $Q$-mappings in ${\mathbb R}^n$, $n\ge2$, $Q\in L^1_{\mathrm{loc}}$, are proved to be absolutely continuous on lines, to belong to the Sobolev class $W_{\mathrm{loc}}^{1,1}$, to be differentiable almost everywhere and to have the $N^{-1}$-property (converse to the Luzin $N$-property). It is shown that a family of open, discrete shell-based $Q$-mappings leaving out a subset of positive capacity is normal, provided that either $Q$ has finite mean oscillation at each point or $Q$ has only logarithmic singularities of order at most $n-1$. Under the same assumptions on $Q$ it is proved that an isolated singularity $x_0\in D$ of an open discrete shell-based $Q$-map $f\colon D\setminus\{x_0\}\to\overline{\mathbb R} ^n$ is removable; moreover, the extended map is open and discrete. On the basis of these results analogues of the well-known Liouville, Sokhotskii-Weierstrass and Picard theorems are obtained.
Bibliography: 34 titles.

Keywords: quasiconformal mappings and their generalizations, moduli of families of curves, capacity, removing singularities of maps, theorems of Liouville, Sokhotskii and Picard type.
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DOI: https://doi.org/10.4213/sm7529

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English version:
Sbornik: Mathematics, 2010, 201:6, 909–934

Bibliographic databases:

UDC: 517.548.2+517.548.9+517.547.26
MSC: 30C65

Citation: R. R. Salimov, E. A. Sevost'yanov, “The theory of shell-based $Q$-mappings in geometric function theory”, Mat. Sb., 201:6 (2010), 131–158; Sb. Math., 201:6 (2010), 909–934

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. E. A. Sevost'yanov, “On the branch points of mappings with the unbounded coefficient of quasiconformality”, Siberian Math. J., 51:5 (2010), 899–912
2. Sevost'yanov E.A., “On some properties of generalized quasiisometries with unbounded characteristic”, Ukr. Math. J., 63:3 (2011), 443–460
3. Salimov R.R., Sevost'yanov E.A., “Estimation of dilatations for mappings more general than quasiregular mappings”, Ukr. Math. J., 62:11 (2011), 1775–1782
4. R. R. Salimov, “Estimation of the measure of the image of the ball”, Siberian Math. J., 53:4 (2012), 739–747
5. Cristea M., “On generalized quasiregular mappings”, Complex Var. Elliptic Equ., 58:12 (2013), 1745–1764
6. R. R. Salimov, “On the Lipschitz Property of a Class of Mappings”, Math. Notes, 94:4 (2013), 559–566
7. V. I. Ryazanov, R. R. Salimov, E. A. Sevostyanov, “On convergence analysis of space homeomorphisms”, Sib. Adv. Math., 23:4 (2013), 263–293
8. R. R. Salimov, “O koltsevykh $Q$-otobrazheniyakh otnositelno nekonformnogo modulya”, Dalnevost. matem. zhurn., 14:2 (2014), 257–269
9. R. R. Salimov, “Lower estimates of $p$-modulus and mappings of Sobolev's class”, St. Petersburg Math. J., 26:6 (2015), 965–984
10. T. V. Lomako, “Theorem on closure and the criterion of compactness for the classes of solutions of the Beltrami equations”, Ukr. Math. J., 65:12 (2014), 1834–1844
11. M. Cristea, “Local homeomorphisms satisfying generalized modular inequalities”, Complex Var. Elliptic Equ., 59:10 (2014), 1363–1387
12. M. Cristea, “Boundary behaviour of the mappings satisfying generalized inverse modular inequalities”, Complex Var. Elliptic Equ., 60:4 (2015), 437–469
13. V. Tengvall, “Absolute continuity of mappings with finite geometric distortion”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 40:1 (2015), 3–15
14. R. R. Salimov, “O konechnoi lipshitsevosti klassov Orlicha–Soboleva”, Vladikavk. matem. zhurn., 17:1 (2015), 64–77
15. E. A. Sevost'yanov, “Analog of the Montel Theorem for Mappings of the Sobolev Class with Finite Distortion”, Ukrainian Math. J., 67:6 (2015), 938–947
16. D. P. Il'yutko, E. A. Sevost'yanov, “Open discrete mappings with unbounded coefficient of quasi-conformality on Riemannian manifolds”, Sb. Math., 207:4 (2016), 537–580
17. Cristea M., “The limit mapping of generalized ring homeomorphisms”, Complex Var. Elliptic Equ., 61:5 (2016), 608–622
18. Cristea M., “Some properties of open, discrete, generalized ring mappings”, Complex Var. Elliptic Equ., 61:5 (2016), 623–643
19. Sevost'yanov E., “On local behavior of mappings with unbounded characteristic”, Lobachevskii J. Math., 38:2, SI (2017), 371–378
20. Golberg A. Salimov R. Sevost'yanov E., “Estimates For Jacobian and Dilatation Coefficients of Open Discrete Mappings With Controlled P-Module”, Complex Anal. Oper. Theory, 11:7, SI (2017), 1521–1542
21. Afanas'eva E., “Ring Q-Homeomorphisms on Finsler Manifolds”, Complex Anal. Oper. Theory, 11:7, SI (2017), 1557–1567
22. Salimov R.R., Sevost'yanov E.A., “On the Absolute Continuity of Mappings Distorting the Moduli of Cylinders”, Ukr. Math. J., 69:6 (2017), 1001–1006
23. D. P. Il'yutko, E. A. Sevost'yanov, “Boundary behaviour of open discrete mappings on Riemannian manifolds”, Sb. Math., 209:5 (2018), 605–651
24. Golberg A., Salimov R., “Regularity of Mappings With Integrally Restricted Moduli”, Complex Analysis and Dynamical Systems: New Trends and Open Problems, Trends in Mathematics, eds. Agranovsky M., Golberg A., Jacobzon F., Shoikhet D., Zalcman L., Birkhauser Verlag Ag, 2018, 129–140
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