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An Analog of the Väisälä Inequality for Surfaces

  • E. Sevost’yanovEmail author
Article
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Abstract

The spatial mappings of the domain of the Euclidean space into itself are studied. Here we obtain the conditions under which these mappings satisfy the upper and lower estimates of the distortion of the modulus of families of surfaces. The above statements generalize the well-known Poletskii and Väisälä inequalities for quasiregular mappings.

Keywords

Mappings Moduli inequalities Surfaces 

Mathematics Subject Classification

Primary 30C65 Secondary 30C62, 31A15 

Notes

References

  1. 1.
    Martio, O., Rickman, S., Väisälä, J.: Distortion and singularities of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A1(465), 1–13 (1970)zbMATHGoogle Scholar
  2. 2.
    Rickman, S.: Quasiregular Mappings. Results in Mathematic and Related Areas (3), vol. 26. Springer, Berlin (1993)CrossRefGoogle Scholar
  3. 3.
    Miklyukov, V.M.: Boundary properties of \(n\)-dimensional quasiconformal mappings. Dokl. Akad. Naukd SSSR 93(3), 525–527 (1970). English transl., Soviet Math. Dokl. 1970. V. 11. P. 969–971MathSciNetGoogle Scholar
  4. 4.
    Väisälä, J.: Modulus and capacity inequalities for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A 1 Math. 509, 1–14 (1972)zbMATHGoogle Scholar
  5. 5.
    Poletskii, E.A.: The modulus method for non-homeomorphic quasiconformal mappings. Math. Sb. 83(2), 261–272 (1970). (in Russian)zbMATHGoogle Scholar
  6. 6.
    Koskela, P., Onninen, J.: Mappings of finite distortion: capacity and modulus inequalities. J. Reine Angew. Math. 599, 1–26 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rajala, K.: The local homeomorphism property of spatial quasiregular mappings with distortion close to one. GAFA, Geom. funct. anal. 15, 1100–1127 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Martio, O., Ryazanov, V., Srebro, U., Yakubov, E.: Moduli in Modern Mapping Theory. Springer, New York (2009)zbMATHGoogle Scholar
  9. 9.
    Fuglede, B.: Extremal length and functional completion. Acta Math. 98, 171–219 (1957)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Golberg, A.: Homeomorphisms with integrally restricted moduli. Contemp. Math. 553, 83–98 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Holopainen, I., Pankka, P.: Mappings of finite distortion: global homeomorphism theorem. Ann. Acad. Sci. Fenn. Math. 29, 59–80 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Salimov, R.R., Sevost’yanov, E.A.: The Poletskii and Väisälä inequalities for the mappings with \((p, q)\)-distortion. Complex Var. Elliptic Equ. 59(2), 217–231 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  14. 14.
    Saks, S.: Theory of the Integral. Dover Publ. Inc., New York (1964)zbMATHGoogle Scholar
  15. 15.
    Sevost’yanov, E.: On estimates of moduli of families of surfaces, p. 9. www.arxiv.org, arXiv:1602.02395

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Zhytomyr Ivan Franko State UniversityZhytomyrUkraine

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