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

This article is cited in 24 scientific papers (total in 24 papers)

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.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm7529

Full text: PDF file (666 kB)
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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
Received: 23.01.2009 and 19.01.2010

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
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    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  mathnet  crossref  mathscinet  isi  elib
    2. Sevost'yanov E.A., “On some properties of generalized quasiisometries with unbounded characteristic”, Ukr. Math. J., 63:3 (2011), 443–460  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi  scopus
    4. R. R. Salimov, “Estimation of the measure of the image of the ball”, Siberian Math. J., 53:4 (2012), 739–747  mathnet  crossref  mathscinet  isi
    5. Cristea M., “On generalized quasiregular mappings”, Complex Var. Elliptic Equ., 58:12 (2013), 1745–1764  crossref  mathscinet  zmath  isi  scopus
    6. R. R. Salimov, “On the Lipschitz Property of a Class of Mappings”, Math. Notes, 94:4 (2013), 559–566  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. V. I. Ryazanov, R. R. Salimov, E. A. Sevostyanov, “On convergence analysis of space homeomorphisms”, Sib. Adv. Math., 23:4 (2013), 263–293  crossref  zmath  scopus
    8. R. R. Salimov, “O koltsevykh $Q$-otobrazheniyakh otnositelno nekonformnogo modulya”, Dalnevost. matem. zhurn., 14:2 (2014), 257–269  mathnet
    9. R. R. Salimov, “Lower estimates of $p$-modulus and mappings of Sobolev's class”, St. Petersburg Math. J., 26:6 (2015), 965–984  mathnet  crossref  mathscinet  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  scopus
    11. M. Cristea, “Local homeomorphisms satisfying generalized modular inequalities”, Complex Var. Elliptic Equ., 59:10 (2014), 1363–1387  crossref  mathscinet  zmath  isi  scopus
    12. M. Cristea, “Boundary behaviour of the mappings satisfying generalized inverse modular inequalities”, Complex Var. Elliptic Equ., 60:4 (2015), 437–469  crossref  mathscinet  zmath  isi  scopus
    13. V. Tengvall, “Absolute continuity of mappings with finite geometric distortion”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 40:1 (2015), 3–15  crossref  mathscinet  isi  scopus
    14. R. R. Salimov, “O konechnoi lipshitsevosti klassov Orlicha–Soboleva”, Vladikavk. matem. zhurn., 17:1 (2015), 64–77  mathnet
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    17. Cristea M., “The limit mapping of generalized ring homeomorphisms”, Complex Var. Elliptic Equ., 61:5 (2016), 608–622  crossref  mathscinet  zmath  isi  scopus
    18. Cristea M., “Some properties of open, discrete, generalized ring mappings”, Complex Var. Elliptic Equ., 61:5 (2016), 623–643  crossref  mathscinet  zmath  isi  scopus
    19. Sevost'yanov E., “On local behavior of mappings with unbounded characteristic”, Lobachevskii J. Math., 38:2, SI (2017), 371–378  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    21. Afanas'eva E., “Ring Q-Homeomorphisms on Finsler Manifolds”, Complex Anal. Oper. Theory, 11:7, SI (2017), 1557–1567  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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