Ukrainian Mathematical Journal

, Volume 65, Issue 9, pp 1394–1405 | Cite as

On the Orlicz–Sobolev Classes and Mappings with Bounded Dirichlet Integral

  • V. I. Ryazanov
  • R. R. Salimov
  • E. A. Sevost’yanov
Article
  • 38 Downloads

It is shown that homeomorphisms f in \( {{\mathbb{R}}^n} \), n ≥ 2, with finite Iwaniec distortion of the Orlicz–Sobolev classes W 1,φ loc under the Calderon condition on the function φ and, in particular, the Sobolev classes W 1,φ loc, p > n - 1, are differentiable almost everywhere and have the Luzin (N) -property on almost all hyperplanes. This enables us to prove that the corresponding inverse homeomorphisms belong to the class of mappings with bounded Dirichlet integral and establish the equicontinuity and normality of the families of inverse mappings.

Keywords

Carnot Group Sobolev Class Total Differential Coordinate Hyperplane Ordinary Partial Derivative 
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References

  1. 1.
    G. D. Suvorov, Generalized Principle of Length and Area in the Theory of Mappings [in Russian], Naukova Dumka, Kiev (1985).Google Scholar
  2. 2.
    Yu. G. Reshetnyak, Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).Google Scholar
  3. 3.
    S. K. Vodop’yanov and V. M. Gol’dshtein, Sobolev Spaces and Special Classes of Mappings [in Russian], Novosibirsk University, Novosibirsk (1981).Google Scholar
  4. 4.
    V. M. Gol’dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings [in Russian], Nauka, Novosibirsk (1983).Google Scholar
  5. 5.
    S. K. Vodop’yanov, “Mappings with bounded distortion and finite distortion on Carnot groups,” Sib. Mat. Zh., 40, No. 4, 764–804 (1999).CrossRefMathSciNetGoogle Scholar
  6. 6.
    T. Iwaniec and V. Sverák, “On mappings with integrable dilatation,” Proc. Amer. Math. Soc., 118, 181–188 (1993).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    T. Iwaniec and G. Martin, Geometrical Function Theory and Non-Linear Analysis, Clarendon Press, Oxford (2001).Google Scholar
  8. 8.
    G. Federer, Geometric Theory of Measure [in Russian], Nauka, Moscow (1987).Google Scholar
  9. 9.
    M. A. Krasnosel’skii and Ya.V. Rutitskii, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).Google Scholar
  10. 10.
    V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad University, Leningrad (1985).Google Scholar
  11. 11.
    W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton (1948).MATHGoogle Scholar
  12. 12.
    A. P. Calderon, “On the differentiability of absolutely continuous functions,” Riv. Mat. Univ. Parma, 2, 203–213 (1951).MATHMathSciNetGoogle Scholar
  13. 13.
    S. Saks, Theory of the Integral, Państwowe Wydawnictwo Naukowe, Warsaw (1937).Google Scholar
  14. 14.
    A. G. Fadell, “A note on a theorem of Gehring and Lehto,” Proc. Amer. Math. Soc., 49, 195–198 (1975).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    F.W. Gehring and O. Lehto, “On the total differentiability of functions of a complex variable,” Ann. Acad. Sci. Fenn. Ser. A1. Math., 272, 3–8 (1959).MathSciNetGoogle Scholar
  16. 16.
    D. Menchoff, “Sur les differencelles totales des fonctions univalentes,” Math. Ann., 105, 75–85 (1931).CrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Väisälä, “Two new characterizations for quasiconformality,” Ann. Acad. Sci. Fenn. Ser. A1. Math., 362, 1–12 (1965).Google Scholar
  18. 18.
    P. S. Aleksandrov, A. I. Markushevich, and A.Ya. Khinchin, Encyclopedia of Elementary Mathematics. Book Four. Geometry [in Russian], Fizmatgiz, Moscow (1963).Google Scholar
  19. 19.
    S. K. Vodop’yanov, “Mappings with finite distortion and the classes of Sobolev functions,” Dokl. Akad. Nauk, 440, No. 3, 301–305 (2008).Google Scholar
  20. 20.
    P. Koskela and J. Maly, “Mappings of finite distortion: The zero set of Jacobian,” J. Eur. Math. Soc., 5, No. 2, 95–105 (2003).CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    S. P. Ponomarev, “N -1-property of functions and the Luzin (N) condition,” Mat. Zametki, 58, Issue 3, 411–418 (1995).MathSciNetGoogle Scholar
  22. 22.
    V. A. Zorich, Mathematical Analysis [in Russian], Vol. 1, Nauka, Moscow (1981).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • V. I. Ryazanov
    • 1
  • R. R. Salimov
    • 1
  • E. A. Sevost’yanov
    • 1
  1. 1.Institute of Applied Mathematics and MechanicsUkrainian National Academy of SciencesDonetskUkraine

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