Ukrainian Mathematical Journal

, Volume 69, Issue 11, pp 1821–1834 | Cite as

On the Equicontinuity of Homeomorphisms of Orlicz and Orlicz–Sobolev Classes in the Closure of a Domain

  • E. A. Sevost’yanov
  • E. A. Petrov

We study the behavior of homeomorphisms of Orlicz–Sobolev classes in the closure of a domain. The theorems on equicontinuity of the indicated classes are obtained in terms of the prime ends of regular domains. In particular, it is shown that indicated classes are equicontinuous in domains with certain restrictions imposed on their boundaries provided that the corresponding inner dilatation of order p has a majorant of finite mean oscillation at every point. We also prove theorems on the (pointwise) equicontinuity of the analyzed classes in the case of locally connected boundaries.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. Gutlyanskii, V. Ryazanov, and E. Yakubov, “The Beltrami equations and prime ends,” Ukr. Mat. Visn., 12, No. 1, 27–66 (2015).MathSciNetzbMATHGoogle Scholar
  2. 2.
    D. A. Kovtonyuk and V. I. Ryazanov, “On the theory of prime ends for space mappings,” Ukr. Mat. Zh., 67, No. 4, 467–479 (2015); English translation : Ukr. Math. J., 67, No. 4, 528–541 (2015).Google Scholar
  3. 3.
    R. Näkki, “Prime ends and quasiconformal mappings,” J. Anal. Math., 35, 13–40 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    R. Näkki and B. Palka, “Uniform equicontinuity of quasiconformal mappings,” Proc. Amer. Math. Soc., 37, No. 2, 427–433 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E. A. Sevost’yanov, Investigation of Space Mappings by the Geometric Method [in Russian], Naukova Dumka, Kiev (2014).Google Scholar
  6. 6.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York (2009).zbMATHGoogle Scholar
  7. 7.
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).CrossRefzbMATHGoogle Scholar
  8. 8.
    J. Heinonen, Lectures on Analysis on Metric Spaces, Springer Sci. + Business Media, New York (2001).Google Scholar
  9. 9.
    F. W. Gehring, “Rings and quasiconformal mappings in space,” Trans. Amer. Math. Soc., 103, 353–393 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E. A. Sevost’yanov, R. R. Salimov, and E. A. Petrov, “On the elimination of singularities in the Orlicz–Sobolev classes,” Ukr. Mat. Vestn., 13, No. 3, 324–349 (2016).Google Scholar
  11. 11.
    D. A. Kovtonyuk, R. R. Salimov, and E. A. Sevost’yanov, On the Theory of Mappings of the Sobolev and Orlicz–Sobolev Classes [in Russian], Naukova Dumka, Kiev (2013).zbMATHGoogle Scholar
  12. 12.
    D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “On the theory of Orlicz–Sobolev classes,” Alg. Anal., 25, No. 6, 50–102 (2013).MathSciNetzbMATHGoogle Scholar
  13. 13.
    T. Adamowicz and N. Shanmugalingam, “Nonconformal Loewner type estimates for modulus of curve families,” Ann. Acad. Sci. Fenn. Math., 35, 609–626 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. Rickman, Quasiregular Mappings, Springer, Berlin (1993).CrossRefzbMATHGoogle Scholar
  15. 15.
    A. Golberg, R. Salimov, and E. Sevost’yanov, “Normal families of discrete open mappings with controlled p-module,” Contemp. Math., 667, 83–103 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    E. S. Afanas’eva, “On the boundary behavior of one class of mappings in metric spaces,” Ukr. Mat. Zh., 66, No. 1, 17–29 (2014); English translation : Ukr. Math. J., 66, No. 1, 16–29 (2014).Google Scholar
  17. 17.
    W. P. Ziemer, “Change of variables for absolutely continuous functions,” Duke Math. J., 36, 171–178 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    H. Federer, Geometric Measure Theory, Springer, New York (1969).zbMATHGoogle Scholar
  19. 19.
    J. Väisälä, “Two new characterizations for quasiconformality,” Ann. Acad. Sci. Fenn. Ser. A1. Math., 362, 1–12 (1965).MathSciNetzbMATHGoogle Scholar
  20. 20.
    R. R. Salimov and E. A. Sevost’yanov, “The Poletskii and Väisälä inequalities for the mappings with (p, q)-distortion,” Complex Var. Elliptic Equat., 59, No. 2, 217–231 (2014).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • E. A. Sevost’yanov
    • 1
  • E. A. Petrov
    • 2
  1. 1.Franko Zhitomir State UniversityZhitomirUkraine
  2. 2.Institute of Applied Mathematics and MechanicsUkrainian National Academy of SciencesSlavyanskUkraine

Personalised recommendations