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Metric Properties of Orlicz–Sobolev Classes

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Abstract

The homeomorphisms of the Orlicz–Sobolev class W 1,φloc under a condition of the Calderón type on φ in ℝ n, n ≥ 3 are considered. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property, as well as various theorems on estimates of a distortion of the Euclidean distance are proved. In particular, the results hold for the homeomorphisms of the Sobolev classes W 1,ploc with p > n − 1.

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References

  1. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009.

    MATH  Google Scholar 

  2. B. Bojarski, V. Gutlyanskii, O. Martio, and V. Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, EMS, Zürich, 2013.

    Book  MATH  Google Scholar 

  3. V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer, New York, 2012.

    Book  MATH  Google Scholar 

  4. V. V. Aseev, “The moduli of the families of locally quasisymmetric surfaces,” Sibir. Mat. Zh., 30, No. 3, 9–15 (1989).

    MathSciNet  MATH  Google Scholar 

  5. S. K. Vodop’yanov and A. D. Ukhlov, “Operators of superposition in Sobolev spaces,” Izv. Vyssh. Ucheb. Zav. Mat., 10, 11–33 (2002).

    MATH  Google Scholar 

  6. 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 

  7. V. I. Kruglikov, “Capacities of capacitors and spatial mappings quasiconformal on the average,” Matem. Sb., 130, No. 2, 185–206 (1986).

    MathSciNet  Google Scholar 

  8. S. L. Krushkal and R. Kühnau, Quasiconformal Mappings: New Methods and Applications [in Russian], Nauka, Novosibirsk, 1984.

  9. D. Kovtonyuk and V. Ryazanov, “To the theory of lower Q-homeomorphisms,” Ukr. Mat. Visn., 5, No. 2, 157–181 (2008).

    MathSciNet  Google Scholar 

  10. Yu. G. Reshetnyak, Spatial Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk, 1982.

    MATH  Google Scholar 

  11. R. R. Salimov, “The lower Q-homeomorphisms relative to a p-modulus,” Ukr. Mat. Visn., 12, No. 4, 484–510 (2015).

    Google Scholar 

  12. A. V. Sychev, Moduli and Spatial Quasiconformal Mappings [in Russian], Nauka, Novosibirsk, 1983.

    MATH  Google Scholar 

  13. B. V. Shabat, “The method of moduli in the space,” Dokl. Akad. Nauk SSSR, 130, 1210–1213 (1960).

    Google Scholar 

  14. B. V. Shabat, “To the theory of quasiconformal mappings in the space,” Dokl. Akad. Nauk SSSR, 132, 1045–1048 (1960).

    MathSciNet  Google Scholar 

  15. C. Andreian Cazacu, “Foundations of quasiconformal mappings,” in: Handbook of Complex Analysis: Geometric Function Theory, Vol. 2, Elsevier, Amsterdam, 2005, pp. 687–753.

  16. M. Cristea, “Dilatations of homeomorphisms satisfying some modular inequalities,” Rev. Roum. Math. Pures Appl., 56, No. 4, 275–282 (2011).

    MathSciNet  MATH  Google Scholar 

  17. M. Cristea, “Open discrete mapping having local ACLn inverses,” Complex Var. Ellip. Equ., 55, No. 1–3, 61–90 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Golberg, “Homeomorphisms with integrally restricted moduli. Complex analysis and dynamical systems IV. Part 1,” Contemp. Math., 553, 83–98 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Golberg and V. Gutlyanskii, “On Lipschitz continuity of quasiconformal mappings in space,” J. Anal. Math., 109, 233–251 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  20. F. W. Gehring, “Rings and quasiconformal mappings in space,” Trans. Amer. Math. Soc., 103,, 353–393 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  21. F. W. Gehring, Quasiconformal Mappings in Complex Analysis and Its Applications, V. 2, Intern. Atomic Energy Agency, Vienna, 1976.

  22. F. W. Gehring, “Lipschitz mappings and the p-capacity of ring in n-space,” Ann. of Math. Stud., 66, 175–193 (1971).

    MathSciNet  Google Scholar 

  23. O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York, 1973.

    Book  MATH  Google Scholar 

  24. O. Martio, S. Rickman, and J. Väisälä, “Definitions for quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A1. Math., 448, 1–40 (1969).

  25. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin, 1971.

    Book  MATH  Google Scholar 

  26. J. Väisälä, “Two new characterizations for quasiconformality,” Ann. Acad. Sci. Fenn. Ser. A1 Math., 362, 1–12 (1965).

  27. T. Iwaniec and V. Sverák, “On mappings with integrable dilatation,” Proc. Amer. Math. Soc., 118, 181–188 (1993).

  28. T. Iwaniec and G. Martin, Geometrical Function Theory and Non-Linear Analysis, Clarendon Press, Oxford, 2001.

    Google Scholar 

  29. M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.

  30. V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad State Univ., Leningrad, 1985.

    MATH  Google Scholar 

  31. D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “To the theory of Orlicz–Sobolev classes,” Alg. Analiz, 25, No. 6, 50–102 (2013).

    MathSciNet  MATH  Google Scholar 

  32. D. Menchoff, “Sur les differencelles totales des fonctions univalentes,” Math. Ann., 105, 75–85 (1931).

    Article  MathSciNet  MATH  Google Scholar 

  33. 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).

  34. H. Federer, Geometric Measure Theory, Springer, Berlin, 1996.

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of the NAS of Ukraine, Kiev, Ukraine

    Ruslan R. Salimov

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  1. Ruslan R. Salimov
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Correspondence to Ruslan R. Salimov.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 13, No. 1, pp. 129–141, January–March, 2016.

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Salimov, R.R. Metric Properties of Orlicz–Sobolev Classes. J Math Sci 220, 633–642 (2017). https://doi.org/10.1007/s10958-016-3206-2

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