Advertisement

Ukrainian Mathematical Journal

, Volume 70, Issue 11, pp 1791–1802 | Cite as

On the Lower Estimate of the Distortion of Distance for One Class of Mappings

  • R. R. Salimov
  • E. A. Sevost’yanov
  • A. A. Markish
Article
  • 9 Downloads

We study the behavior of one class of mappings with finite distortion in the vicinity of the origin. Under certain conditions imposed on the characteristic of quasiconformality, we establish a lower estimate for the distortion of distance under mappings of the indicated kind.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. R. Salimov and E. A. Sevost’yanov, “Analogs of the Ikoma–Schwartz lemma and Liouville theorem for mappings with unbounded characteristic,” Ukr. Mat., Zh., 63, No. 10, 1368–1380 (2011); English translation: Ukr. Math. J., 63, No. 10, 1551–1565 (2012).Google Scholar
  2. 2.
    K. Ikoma, “On the distortion and correspondence under quasiconformal mappings in space,” Nagoya Math. J., 25, 175–203 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    R. R. Salimov, “On the ring Q-mappings with respect to a nonconformal module,” Dal’nevost. Mat. Zh., 14, No. 2, 257–269 (2014).MathSciNetzbMATHGoogle Scholar
  4. 4.
    R. R. Salimov and E. A. Sevost’yanov, “Some properties of the generalized space quasiisometries,” Mat. Zametki, 101, No. 4, 594–610 (2017).CrossRefGoogle Scholar
  5. 5.
    V. Mazya, “Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces,” Contemp. Math., 338, 307–340 (2003).MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Gehring, “Lipschitz mappings and p-capacity of rings in n-space,” Ann. Math. Stud., 66, 175–193 (1971).MathSciNetGoogle Scholar
  7. 7.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer Science + Business Media, LLC, New York (2009).zbMATHGoogle Scholar
  8. 8.
    V. Ya. Gutlyanskii, V. I. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer, New York (2012).CrossRefzbMATHGoogle Scholar
  9. 9.
    V. Gutlyanskii, V. Ryazanov, and E. Yakubov, “The Beltrami equations and prime ends,” Ukr. Mat. Visn., 12, No. 1, 27–66 (2015).MathSciNetzbMATHGoogle Scholar
  10. 10.
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).CrossRefzbMATHGoogle Scholar
  11. 11.
    V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad Univ., Leningrad (1985).zbMATHGoogle Scholar
  12. 12.
    R. R. Salimov, “On the estimation of the measure of an image of the ball,” Sib. Mat. Zh. 53, No. 4, 920–930 (2012).MathSciNetCrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    D. P. Il’yutko and E. A. Sevost’yanov, “On the open discrete mappings with unbounded characteristic on Riemannian manifolds,” Mat. Sb., 207, No. 4, 65–112 (2016).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • R. R. Salimov
    • 1
  • E. A. Sevost’yanov
    • 2
  • A. A. Markish
    • 2
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.I. Franko Zhitomir State UniversityZhitomirUkraine

Personalised recommendations