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On the Behavior of One Class of Homeomorphisms at Infinity

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Ukrainian Mathematical Journal Aims and scope

We study the behavior of ring Q-homeomorphisms with respect to the p-modulus with p > n at infinity.

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References

  1. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).

    Book  MATH  Google Scholar 

  2. V. I. Ryazanov and E. A. Sevost’yanov, “Equicontinuous classes of ring Q-homeomorphisms,” Sib. Math. J., 48, No. 6, 1093–1105 (2007).

  3. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Q-homeomorphisms,” in: Complex Analysis and Dynamical Systems, Contemporary Mathematics, 364 (2004), pp. 193–203.

    Article  MATH  Google Scholar 

  4. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Math., 30, No. 1, 49–69 (2005).

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. R. Salimov, “ACL and differentiability of a generalization of quasiconformal maps,” Izv. Math., 72, No. 5, 977–984 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Golberg, “Differential properties of (α,Q)-homeomorphisms,” in: Further Progress in Analysis, Proc. 6th ISAAC Congr. (2009), pp. 218–228.

  8. A. Golberg, “Integrally quasiconformal mappings in space,” Trans. Inst. Math. NAS Ukraine, 7, No. 2, 53–64 (2010).

    MathSciNet  MATH  Google Scholar 

  9. A. Golberg and R. Salimov, “Logarithmic Hölder continuity of ring homeomorphisms with controlled p-module,” Complex Var. Elliptic Equat., 59, No. 1, 91–98 (2014).

    Article  MATH  Google Scholar 

  10. A. Golberg, R. Salimov, and E. Sevost’yanov, “Distortion estimates under mappings with controlled p-module,” Ann. Univ. Bucharest, Math. Ser., 63, No. 5, 95–114 (2014).

  11. R. Salimov, “On finitely Lipschitz space mappings,” Sib. Elecron. Math. Rep., 8, 284–295 (2011).

    MathSciNet  MATH  Google Scholar 

  12. R. Salimov, “Estimation of the measure of the image of the ball,” Sib. Math. J., 53, No. 4, 920–930 (2012).

    Article  MathSciNet  Google Scholar 

  13. R. Salimov, “To a theory of ring Q-homeomorphisms with respect to a p-modulus,” Ukr. Math. Bull., 10, No. 3, 379–396 (2013).

    MathSciNet  Google Scholar 

  14. R. R. Salimov, “One property of ring Q-homeomorphisms with respect to a p-module,” Ukr. Mat. Zh., 65, No. 5, 728–733 (2013); English translation: Ukr. Math. J., 65, No. 5, 806–813 (2013).

  15. R. Salimov and B. Klishchuk, “The extremal problem for the area of an image of a disc,” Rep. NAS Ukr., 10, 22–27 (2016).

    MATH  Google Scholar 

  16. B. Klishchuk and R. Salimov, “Lower bounds for the area of the image of a circle,” Ufa Math. J., 9, No. 2, 55–61 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Salimov and B. Klishchuk, “Extremal problem for the area of the image of a disk,” Zap. Nauch. Sem. POMI, 456, 160–171 (2017).

    Google Scholar 

  18. R. Salimov and B. Klishchuk, “An extremal problem for the volume functional,” Mat. Stud., 50, No. 1, 36–43 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  19. B. A. Klishchuk and R. R. Salimov, “Lower bounds for the volume of the image of a ball,” Ukr. Mat. Zh., 71, No. 6, 774–785 (2019); English translation: Ukr. Math. J., 71, No. 6, 883-895 (2019).

  20. M. Cristea, “Local homeomorphisms satisfying generalized modular inequalities,” Complex Var. Elliptic Equat., 59, No. 2, 232–246 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Cristea, “Some properties of open discrete generalized ring mappings,” Complex Var. Elliptic Equat.,” 61, No. 5, 623–643 (2016).

  22. M. Cristea, “Eliminability results for mappings satisfying generalized modular inequalities,” Complex Var. Elliptic Equat., 64, No. 4, 676–684 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  23. R. R. Salimov, E. A. Sevost’yanov, and A. A. Markish, “On the lower estimate of the distortion of distance for one class of mappings,” Ukr. Mat. Zh., 70, No. 11, 1553–1562 (2018); English translation: Ukr. Math. J., 70, No. 11, 1791–1802 (2019).

  24. A. Golberg, R. Salimov, and E. Sevost’yanov, “Singularities of discrete open mappings with controlled p-module,” J. Anal. Math., 127, 303–328 (2015).

  25. A. Golberg, R. Salimov, and E. Sevost’yanov, “Poletskii type inequality for mappings from the Orlicz–Sobolev classes,” Complex Anal. Oper. Theory, 10, 881–901 (2016).

  26. A. Golberg, R. Salimov, and E. Sevost’yanov, “Estimates for Jacobian and dilatation coefficients of open discrete mappings with controlled p-module,” Complex Anal. Oper. Theory, 11, No 7, 1521–1542 (2017).

  27. R. R. Salimov and E. S. Smolovaya, “On the order of growth of ring Q-homeomorphisms at infinity,” Ukr. Math. Zh., 62, No. 6, 829–836 (2010); English translation: Ukr. Math. J., 62, No. 6, 961–969 (2010).

  28. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. V. A. Shlyk, “The equality between p-capacity and p-modulus,” Sib. Math. J., 34, No. 6, 216–221 (1993).

    Article  MathSciNet  MATH  Google Scholar 

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

  32. F. W. Gehring, “Lipschitz mappings and the p-capacity of ring in n-space,” Adv. Theory Riemann Surfaces (Proc. Conf. Stonybrook, New York, 1969), Ann. Math. Stud., 66, 175–193 (1971).

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Correspondence to V. A. Klishchuk.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 10, pp. 1416–1426, October, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i10.7158.

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Salimov, R.R., Klishchuk, V.A. On the Behavior of One Class of Homeomorphisms at Infinity. Ukr Math J 74, 1617–1628 (2023). https://doi.org/10.1007/s11253-023-02158-x

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  • DOI: https://doi.org/10.1007/s11253-023-02158-x

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