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On the Boundary Behavior of One Class of Mappings in Metric Spaces

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Ukrainian Mathematical Journal Aims and scope

We study the problem of extension to the boundary of continually ring Q-homeomorphisms relative to a p-module between continual domains in metric spaces with measures and formulate the conditions for the function Q and the boundaries of domains under which every continually ring Q-homeomorphism admits a continuous or homeomorphic extension to the boundary. The accumulated results yield, in particular, important applications to fractals in ℝn , n ≥ 2.

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References

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

    MATH  MathSciNet  Google Scholar 

  2. P. Koskela and J. Onninen, “Mappings of finite distortion: Capacity and modulus inequalities,” J. Reine Angew. Math., 599, 1–26 (2006).

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. J. Onninen, Mappings of Finite Distortion: Continuity, Dissertation, Jyvaskyla (2002).

  5. K. Rajala, Mappings of Finite Distortion: Removable Singularities, Dissertation, Jyvaskyla (2003).

  6. C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, On Conformal Dilatation in Space, Preprint No. 256, University of Helsinki, Helsinki (2000).

  7. C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Int. J. Math. Sci., 22, 1397–1420 (2003).

    Article  MathSciNet  Google Scholar 

  8. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn., Ser. A1, 30, No. 1, 49–69 (2005).

    MATH  MathSciNet  Google Scholar 

  9. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Q-homeomorphisms,” Contemp. Math., 364, 193–203 (2004).

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  11. A. Golberg, “Differential properties of (α,Q)-homeomorphisms,” in: Proceedings of the 6th International ISAAC Congress “Further Progress in Analysis,” World Scientific, Hackensack (2009), pp. 218–228.

    Chapter  Google Scholar 

  12. A. Golberg, “Integrally quasiconformal mappings in space,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Vol. 7, No. 2 (2010), pp. 53–64.

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

    Article  MATH  MathSciNet  Google Scholar 

  14. V. Ryazanov, U. Srebro, and E. Yakubov, “The Beltrami equation and ring homeomorphisms,” Ukr. Mat. Vestn., 4, No. 1, 97–115 (2007).

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. V. Ryazanov, U. Srebro, and E. Yakubov, “On strong solutions of the Beltrami equations,” Complex Variables Elliptic Equat., 55, No. 1-3, 219–236 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  17. V. Ryazanov, U. Srebro, and E. Yakubov, “To strong ring solutions of the Beltrami equations,” Uzbek. Math. J., No. 1, 127–137 (2009).

  18. V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: a Geometric Approach, Springer, New York (2012).

    Book  Google Scholar 

  19. E. S. Afanas’eva, V. I. Ryazanov, and R. R. Salimov, “On mappings in the Orlicz–Sobolev classes on Riemannian manifolds,” Ukr. Mat. Vestn., 8, No. 3, 319–342 (2011).

    Google Scholar 

  20. R. R. Salimov, “Estimation of measure for the image of a ball,” Sib. Mat. Zh., 53, No. 4, 920–930 (2012).

    Article  MathSciNet  Google Scholar 

  21. E. S. Smolovaya, “Boundary behavior of ring Q-homeomorphisms in metric spaces,” Ukr. Mat. Zh., 62, No. 5, 682–689 (2010); English translation: Ukr. Math. J., 62, No. 5, 785–793 (2010).

    Article  MathSciNet  Google Scholar 

  22. V. Ryazanov and P. Salimov, “Weakly plane spaces and boundaries in the mapping theory,” Ukr. Mat. Vestn., 4, No. 2, 199–234 (2007).

    MathSciNet  Google Scholar 

  23. N. Bourbaki, General Topology. Main Structures [Russian translation], Nauka, Moscow (1969).

    Google Scholar 

  24. K. Kuratowski, Topology, Vol. 2, Academic Press, New York (1969).

    Google Scholar 

  25. G. T. Whyburn, Analytic Topology, American Mathematical Society, Providence (1942).

    MATH  Google Scholar 

  26. W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton (1948).

    MATH  Google Scholar 

  27. B. Fuglede, “Extremal length and functional completion,” Acta Math., 98, 171–219 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  28. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York (2001).

    Book  MATH  Google Scholar 

  29. A. A. Ignat’ev and V. I. Ryazanov, “Finite mean oscillation in the theory of mappings,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).

    MATH  MathSciNet  Google Scholar 

  30. H. Federer, Geometric Measure Theory, Springer, New York (1969).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine

    E. S. Afanas’eva

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  1. E. S. Afanas’eva
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 1, pp. 17–29, January, 2014.

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Afanas’eva, E.S. On the Boundary Behavior of One Class of Mappings in Metric Spaces. Ukr Math J 66, 16–29 (2014). https://doi.org/10.1007/s11253-014-0908-9

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