Siberian Mathematical Journal

, Volume 59, Issue 6, pp 983–1005 | Cite as

Differentiability of Mappings of the Sobolev Space Wn−11 with Conditions on the Distortion Function

  • S. K. VodopyanovEmail author


We define two scales of the mappings that depend on two real parameters p and q, with n−1 ≤ qp < ∞, as well as a weight function θ. The case q = p = n and θ ≡ 1 yields the well-known mappings with bounded distortion. The mappings of a two-index scale are applied to solve a series of problems of global analysis and applications. The main result of the article is the a.e. differentiability of mappings of two-index scales.


quasiconformal analysis Sobolev space capacity estimate differentiability Liouville theorem 


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  1. 1.
    Vodopyanov S. K., “Basics of the quasiconformal analysis of a two-index scale of spatial mappings,” Sib. Math. J., vol. 59, No. 5, 805–834 (2018).CrossRefGoogle Scholar
  2. 2.
    Reshetnyak Yu. G., Space Mappings with Bounded Distortion, Amer. Math. Soc., Providence (1989).CrossRefzbMATHGoogle Scholar
  3. 3.
    Stepanoff W., “Sur les conditions de l’existence de la differentiate totale,” Mat. Sb., vol. 32, 511–526 (1925).Google Scholar
  4. 4.
    Saks S., Theory of the Integral, Hafner Publishing Company, New York (1937).zbMATHGoogle Scholar
  5. 5.
    Whitney H., “On totally differentiable and smooth functions,” Pacific J. Math., vol. 1, 143–159 (1951).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Federer H., Geometric Measure Theory, Springer-Verlag, New York (1969).zbMATHGoogle Scholar
  7. 7.
    Vodopyanov S. K., “Quasiconformal analysis of two-indexed scale of spatial mappings and its applications,” in: Conference on Complex Analysis and Its Applications: Int. Conf. Materials Dedicated to the 90th Anniversary of I. O. Mityuk, Krasnodar, June, 02–09, Prosveshcheniye-Yug, Krasnodar, 2018, 25–27.Google Scholar
  8. 8.
    Rickman S., Quasiregular Mappings, Springer-Verlag, Berlin etc. (1993).CrossRefzbMATHGoogle Scholar
  9. 9.
    Heinonen Ju., Kilpelainen T., and Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford etc. (1993).zbMATHGoogle Scholar
  10. 10.
    Baykin A. N. and Vodopyanov S. K., “Capacity estimates, Liouville’s Theorem, and singularity removal for mappings with bounded (P, Q)-distortion,” Sib. Math. J., vol. 56, No. 2, 237–261 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Vodopyanov S. K., “Spaces of differential forms and maps with controlled distortion,” Izv. Math., vol. 74, No. 4, 663–689 (2010).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Vodopyanov S. K. and Ukhlov A. D., “Sobolev spaces and (P,Q)-quasiconformal mappings of Carnot groups,” Sib. Math. J., vol. 39, No. 4, 665–682 (1998).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Vodopyanov S. K., “Composition operators on Sobolev spaces,” in: Complex Analysis and Dynamical Systems. II, Amer. Math. Soc., Providence, 2005, 327–342 (Contemp. Math.; vol. 382).Google Scholar
  14. 14.
    Ukhlov A. and Vodopyanov S. K., “Mappings with bounded (P,Q)-distortion on Carnot groups,” Bull. Sci. Mat., vol. 134, No. 6, 605–634 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Csörnyei M., Hencl S., and Mal´y Y., “Homeomorphisms in the Sobolev space W1,n−1,” J. Reine Angew. Math., vol. 644, 221–235 (2010).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Tengvall V., “Differentiability in the Sobolev space W1,n−1,” Calculus of Variations and Partial Differential Equations, vol. 51, No. 1–2, 381–399 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Vodopyanov S. K., “Regularity of mappings inverse to Sobolev mappings,” Sb. Math., vol. 203, No. 10, 1383–1410 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Haj–lasz P., “Change of variable formula under the minimal assumptions,” Colloq. Math., vol. 64, No. 1, 93–101 (1993).MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ukhlov A. and Vodopyanov S. K., “Mappings associated with weighted Sobolev spaces,” in: Complex Analysis and Dynamical Systems, II, Amer. Math. Soc., Providence, 2008, 369–382 (Contemp. Math.; vol. 455).Google Scholar
  20. 20.
    Poletskii E. A., “The modulus method for nonhomeomorphic quasiconformal mappings,” Math. USSR-Sb., vol. 12, No. 2, 260–270 (1970).CrossRefGoogle Scholar
  21. 21.
    Vodopyanov S. K., “On the regularity of the Poletskii function under weak analytic assumptions on the given mapping,” Dokl. Math., vol. 89, No. 2, 157–161 (2014).MathSciNetCrossRefGoogle Scholar
  22. 22.
    Martio O., Rickman S., and Väisälä J., “Definitions for quasiregular mappings,” Ann. Acad. Sc. Fenn., vol. 448, 5–40 (1969).MathSciNetzbMATHGoogle Scholar
  23. 23.
    Mazya V. G., Sobolev Spaces, Springer-Verlag, Berlin (2011).CrossRefGoogle Scholar
  24. 24.
    Mazya V. G. and Havin V. P., “Non-linear potential theory,” Russian Math. Surveys, vol. 27, No. 6, 71–148 (1972).CrossRefGoogle Scholar
  25. 25.
    Radó N. and Reichelderfer P. V., Continuous Transformation in Analysis, Springer–Verlag, Berlin (1955). vol.CrossRefzbMATHGoogle Scholar
  26. 26.
    Vodopyanov S. K. and Ukhlov A. D., “Set functions and its applications in the theory of Lebesgue and Sobolev spaces. I,” Siberian Adv. Math., vol. 14, No. 4, 78–125 (2004).MathSciNetGoogle Scholar
  27. 27.
    Kruglikov V. I., “Capacity of condensers and spatial mappings quasiconformal in the mean,” Math. USSR-Sb., vol. 58, No. 1, 185–205 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Koskela P. and Mal´y J., “Mappings of finite distortion: The zero set of the Jacobian,” J. Eur. Math. Soc., vol. 5, No. 2, 95–105 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vodopyanov S. K. and Ukhlov A. D., “Superposition operators in Sobolev spaces,” Russian Math. (Iz. VUZ), vol. 46, No. 10, 9–31 (2002).MathSciNetzbMATHGoogle Scholar
  30. 30.
    Salimov R. and Sevostyanov E., “ACL and differentiability of open discrete ring (p;Q)-mappings,” Mat. Stud., vol. 35, No. 1, 28–36 (2011).MathSciNetzbMATHGoogle Scholar
  31. 31.
    Reshetnyak Yu. G., “Sobolev-type classes of functions with values in a metric space,” Sib. Math. J., vol. 38, No. 3, 567–582 (1997).CrossRefGoogle Scholar
  32. 32.
    Troyanov M. and Vodopyanov S. K., “Liouville type theorems for mappings with bounded (co)-distortion,” Ann. Inst. Fourier (Grenoble), vol. 52, No. 6, 1753–1784 (2001).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

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