Skip to main content
SpringerLink
Log in

The lower Q-homeomorphisms relative to a p-modulus

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The paper is devoted to the development of the theory of lower Q-homeomorphisms relative to a p-modulus in ℝn, n ≥ 2. 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 is proved, various theorems on estimates of distortion of the Euclidean distance are given, an estimate of the ball image measure is established, and, as a consequence, an analog of the Ikoma–Schwartz lemma is proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  Google Scholar 

  2. D. A. Kovtonyuk, R. R. Salimov, and E. A. Sevost’yanov, To the Theory of Mappings of the Sobolev and Orlicz−Sobolev Classes [in Russian], Naukova Dumka, Kiev, 2013.

    MATH  Google Scholar 

  3. 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 

  4. V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer, Berlin, 2012.

    Book  MATH  Google Scholar 

  5. V. N. Dubinin, Capacities of Capacitors and Symmetrization in the Geometric Theory of Functions of Complex Variable [in Russian], Dal’nauka, Vladivostok, 2009.

    Google Scholar 

  6. A. Vasil’ev, Moduli of Families of Curves for Conformal and Quasiconformal Mappings, Springer, Berlin, 2002.

    Book  MATH  Google Scholar 

  7. 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 

  8. 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 

  9. S. K. Vodop’yanov, “Description of composition operators of Sobolev spaces,” Doklady Mathematics, 71, No. 1, 5−9 (2005).

  10. S. K. Vodop’yanov, “Composition operators on Sobolev spaces,” in: Complex Analysis and Dynamical Systems II, edited by M. Agranovsky, K. Karp, and D. Shoikhet, AMS, Providence, RI, 2005, pp. 401−415.

  11. S. K. Vodop’yanov and A. D. Ukhlov, “Operators of superposition in Lebesgue spaces and differentiability of quasiadditive functions of a set,” Vladikav. Mat. Zh, 4, No. 1, 11−33 (2002).

    MathSciNet  MATH  Google Scholar 

  12. 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 

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

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  15. V. S. Kud’yavin, “Estimates of a distance distortion for mappings quasiconformal on the average,” Dinam. Spl. Sr., 52, 168−171 (1981).

    MathSciNet  MATH  Google Scholar 

  16. V. S. Kud’yavin, “Local and boundary properties of mappings quasiconformal on the average,” in: Collection of Scientific Works [in Russian], Inst. of Mathem. of the SD of the AS of the USSR, Novosibirsk, 1981, pp. 168−171.

  17. V. S. Kud’yavin, “The behavior of a class of mappings quasiconformal on the average at a singular isolated point,” Dokl. Akad. Nauk SSSR, 277, No. 5, 1056−1058 (1984).

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  19. L. V. Kovalev, “Monotonicity of generalized reduced modulus,” Zapiski Nauch. Sem. POMI, 276, 219−236 (2001).

    Google Scholar 

  20. G. V. Kuz’mina, “The moduli of the families of curves and quadratic differentials,” Trudy MIAN SSSR, 139, 3−241 (1980).

    MathSciNet  Google Scholar 

  21. V. M. Miklyukov, Conformal Mapping of an Irregular Surface and Its Applications [in Russian], Volgograd State Univ., Volgograd, 2005.

    Google Scholar 

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

    MATH  Google Scholar 

  23. V. I. Ryazanov and E. A. Sevost’yanov, “Equicontinuous classes of ring Q-homeomorphisms,” Sibir. Mat. Zh., 48, No. 6, 1361−1376 (2007).

    MathSciNet  MATH  Google Scholar 

  24. A. Yu. Solynin, “Moduli and extreme metric problems,” Alg. Analiz, 11, No. 1, 3−86 (1999).

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  28. V. A. Shlyk, “On the equality of a p-capacity and a p-modulus,” Sibir. Mat. Zh., 34, No. 6, 216−221 (1993).

    MathSciNet  Google Scholar 

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

  30. C. Andreian Cazacu, “On the length-area dilatation,” Complex Var. Theory Appl., 50, No. 7−11, 765−776 (2005).

  31. C. Andreian Cazacu, “Module inequalities for quasiregular mappings,” Ann. Acad. Sci. Fenn. Math., 2, 17−28 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  32. C. Andreian Cazacu, “Modules and quasiconformality,” in: Symposia Mathematica 18, Academic Press, London, 1976, pp. 519−534.

  33. C. Andreian Cazacu, “A generalization of the quasiconformality,” in: Topics in Analysis, Springer, Berlin, 1974, pp. 4−17.

  34. P. Caraman, n-Dimensional Quasiconformal Mappings, Haessner, Newfoundland, NJ, 1974.

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

    MathSciNet  MATH  Google Scholar 

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

  37. M. Cristea, “Local homeomorphisms having local ACLn inverses,” Complex Var. Ellip. Equ., 53, No. 1, 77−99 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  38. M. Cristea, “Mappings of finite distortion: Zoric’s theorem, and equicontinuity results,” Rev. Roum. Math. Pures Appl., 52, No. 5, 539−554 (2007).

    MathSciNet  MATH  Google Scholar 

  39. M. Cristea, “Mappings of finite distortion: boundary extension,” Rev. Roum. Math. Pures Appl., 51, Nos. 5−6, 607−631 (2006).

  40. 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 

  41. A. Golberg, “Directional dilatations in space,” Complex Var. Ellip. Equ., 55, Nos. 1−3, 13−29 (2010).

  42. A. Golberg, “Homeomorphisms with finite mean dilatations,” Contemp. Math., 382, 177−186 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  43. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  46. 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 

  47. J. Hesse, “A p-extremal length and p-capacity equality,” Arc. Mat., 13, 131−144 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  48. K. Ikoma, “On the distortion and correspondence under quasiconformal mappings in space,” Nagoya Math. J., 25, 175−203 (1965).

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  50. V. Maz’ya, “Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces,” Contemp. Math., 338, 307−340 (2003).

    Article  MathSciNet  MATH  Google Scholar 

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

  52. J. Vӓisӓlӓ, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin, 1971.

    Google Scholar 

  53. J. Vӓisӓlӓ, “Two new characterizations for quasiconformality,” Ann. Acad. Sci. Fenn. Ser. A1 Math., 362, 1−12 (1965).

  54. M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Springer, Berlin, 1988.

    MATH  Google Scholar 

  55. W. P. Ziemer, “Extremal length and p-capacity,” Michigan Math. J., 16, No. 1, 43−51 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  56. V. G. Maz’ya, Sobolev Spaces, Leningrad State Univ., Leningrad, 1985.

    MATH  Google Scholar 

  57. E. S. Afanas’eva, V. I. Ryazanov, and R. R. Salimov, “On mappings in Orlicz-−Sobolev classes on Rie-mannian manifolds,” Ukr. Mat. Visn., 8, No. 3, 319−342 (2011).

    MathSciNet  Google Scholar 

  58. T. Iwaniec, P. Koskela, and J. Onninen, “Mappings of finite distortion: Compactness,” Ann. Acad. Sci. Fenn. Ser. A1. Math., 27, No. 2, 391−417 (2002).

  59. D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “On mappings in the Orlicz-Sobolev classes,” Ann. Univ. Bucharest, Ser. Math., 3(LXI), No. 1, 67−78 (2012).

  60. D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “The boundary behavior of the Orlicz−Sobolev classes,” Matem. Zam., 95, No. 4, 564−576 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  61. 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 

  62. R. R. Salimov, “Lower bounds of a p-modulus and mappings of the Sobolev class,” Alg. Analiz, 26, No. 6, 143−171 (2014).

    MathSciNet  Google Scholar 

  63. T. Lomako, R. Salimov, and E. Sevost’yanov, “On equicontinuity of solutions to the Beltrami equations,” Ann. Univ. Bucharest, Math. Ser., LIX, No. 2, 263−274 (2010).

  64. V. Ryazanov, R. Salimov, U. Srebro, and E. Yakubov, “On boundary value problems for the Beltrami equations,” Contemp. Math., 591, 211−242 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  65. D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov, and R. R. Salimov, “Boundary behavior and the Dirichlet problem for the Beltrami equations,” Alg. Analiz, 25, No. 4, 101−124 (2013).

    MathSciNet  MATH  Google Scholar 

  66. R. R. Salimov, “The absolute continuity on lines and the differentiability of a generalization of quasicon-formal mappings,” Izv. Ross. Akad. Nauk. Ser. Mat., 72, No. 5, 141−148 (2008).

    Article  MathSciNet  Google Scholar 

  67. R. R. Salimov, “On an estimate of the measure of the image of a ball,” Sibir. Mat. Zh., 53, No. 4, 920−930 (2012).

    MathSciNet  Google Scholar 

  68. R. R. Salimov, “On finitely Lipschitz space mappings,” Sibir. Elektr. Mat. Izv., 8, (2011).

  69. R. R. Salimov, “On the Lipschitz property of a class of mappings,” Matem. Zam., 94, No. 4, 591−599 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  70. R. R. Salimov, “On ring Q-mappings relative to a nonconformal modulus,” Dal’nevost. Mat. Zh., 14, No. 2, 257−269 (2014).

  71. R. R. Salimov and E. A. Sevost’yanov, “The theory of ring Q-mappings in the geometric theory of functions,” Matem. Sb., 201, No. 6, 131−158 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  72. E. A. Sevost’yanov, “On the spatial mappings with integral restrictions imposed on a characteristic,” Alg. Analiz, 24, No. 1, 131−156 (2012).

    MathSciNet  Google Scholar 

  73. S. Saks, Theory of the Integral, Dover, New York, 1964.

    MATH  Google Scholar 

  74. T. Ransford, Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  75. M. A. Lavrent’ev, Variational Methods for Boundary Value Problems for Systems of Elliptic Equations, Dover, New York, 2015.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ruslan R. Salimov

Authors
  1. Ruslan R. Salimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruslan R. Salimov.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 12, No. 4, pp. 484–510, September–December, 2015.

Translated from Russian by V. V. Kukhtin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salimov, R.R. The lower Q-homeomorphisms relative to a p-modulus. J Math Sci 218, 47–68 (2016). https://doi.org/10.1007/s10958-016-3010-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-016-3010-z

Keywords

Search

Navigation