Generalization of one Poletskii lemma to classes of space mappings

  • E. A. Sevost’yanov

The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings fW 1,n loc such that their outer dilatation K O (x, f) belongs to L n−1 loc and the measure of the set B f of branching points of f is equal to zero have finite length distortion. In other words, the images of almost all curves γ in the domain D under the considered mappings f : D → ℝ n , n ≥ 2, are locally rectifiable, f possesses the (N)-property with respect to length on γ, and, furthermore, the (N)-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii lemma proved for quasiregular mappings.


Space Mapping Quasiconformal Mapping Topological Index Absolute Continuity Normal Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    E. A. Poletskii, “A method of moduli for nonhomeomorphic quasiconformal mappings,” Mat. Sb., 83, No. 2, 261–272 (1970).MathSciNetGoogle Scholar
  2. 2.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. Anal. Math., 93, 215–236 (2004).CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    P. Koskela and J. Onninen, “Mappings of finite distortion: Capacity and modulus inequalities,” J. Reine Angew. Math., 599, 1–26 (2006).MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. Heinonen and P. Koskela, “Sobolev mappings with integrable dilatations,” Arch. Ration. Mech. Anal., 125, 81–97 (1993).CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    J. Maly and O. Martio, “Lusin’s condition and mappings of the class,” J. Reine Angew. Math., 458, 19–36 (1995).MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. J. Manfredi and E. Villamor, “Mappings with integrable dilatation in higher dimensions,” Bull. Amer. Math. Soc., 32, No. 2, 235–240 (1995).CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    J. J. Manfredi and E. Villamor, “An extension of Reshetnyak’s theorem,” Indiana Univ. Math. J., 47, No. 3, 1131–1145 (1998).MathSciNetzbMATHGoogle Scholar
  8. 8.
    P. Koskela and J. Maly, “Mappings of finite distortion: The zero set of Jacobian,” J. Eur. Math. Soc., 5, No. 2, 95–105 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Yu. G. Reshetnyak, “Generalized derivatives and almost-everywhere differentiability,” Mat. Sb., 75, No. 3, 323–334 (1968).MathSciNetGoogle Scholar
  10. 10.
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).zbMATHGoogle Scholar
  11. 11.
    T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, Berlin (1955).zbMATHGoogle Scholar
  12. 12.
    S. Rickman, “Quasiregular mappings,” Results Math. Relat. Areas, 26, No. 3 (1993).Google Scholar
  13. 13.
    C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Int. J. Math. Math. Sci., 22, 1397–1420 (2003).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • E. A. Sevost’yanov
    • 1
  1. 1.Institute of Applied Mathematics and MechanicsUkrainian National Academy of SciencesDonetskUkraine

Personalised recommendations