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Local behavior of Q-homeomorphisms in Loewner spaces

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Ukrainian Mathematical Journal Aims and scope

We study the problem of the elimination of isolated singularities for so-called Q-homeomorphisms in Loewner spaces. We formulate several conditions for a function Q(x) under which every Q-homeomorphism admits a continuous extension to an isolated singular point. We also consider the problem of the homeomorphicity of the extension obtained. The results are applied to Riemannian manifolds and Carnot groups.

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References

  1. M. Brakalova and J. Jenkins, “On the local behavior of certain homeomorphisms,” Kodai Math. J., 17, No. 2, 201–213 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Brakalova and J. Jenkins, “On the local behavior of certain homeomorphisms,” Zap. Nauch. Sem. POMI, 337, 11–20 (1997).

    Google Scholar 

  3. V. Gutlyanskii and V. Ryazanov, “On the theory of the local behavior of quasiconformal mappings,” Izv. Akad. Nauk Rossii, Ser. Mat., 59, No. 3, 31–58 (1995).

    MathSciNet  Google Scholar 

  4. V. Gutlyanskii and V. Ryazanov, “On boundary correspondence under quasiconformal mappings,” Ann. Acad. Sci. Fenn. Al. Math., 21, No. 1, 167–178 (1996).

    MathSciNet  Google Scholar 

  5. V. Gutlyanskii, O. Martio, V. Ryazanov, and M. Vuorinen, “Infinitesimal geometry of quasiregular mappings,” Ann. Acad. Sci. Fenn. Al. Math., 25, No. 1, 101–130 (2000).

    MathSciNet  Google Scholar 

  6. V. Gutlyanskii and M. Vuorinen, “On maps almost conformal at the boundary,” Complex Variables Theory Appl., 34, 445–464 (1967).

    MathSciNet  Google Scholar 

  7. A. Ignat’ev and V. Ryazanov, “Finite mean oscillation in the mapping theory,” Ukr. Math. Bull., 2, No. 3, 403–424 (2005).

    MathSciNet  Google Scholar 

  8. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. Anal. Math., 93, 215–236 (2004).

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  10. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Al. Math., 30, 49–69 (2005).

    MATH  MathSciNet  Google Scholar 

  11. V. M. Miklyukov, “On removable singularities of quasiconformal mappings in a space,” Dokl. Akad. Nauk SSSR, 188, No. 3, 525–527 (1969).

    MATH  MathSciNet  Google Scholar 

  12. V. I. Ryazanov and E. A. Sevost’yanov, “On normal families of Q-homeomorphisms,” Tr. Inst. Prikl. Mat. Mekh., 9, 161–176 (2004).

    MATH  MathSciNet  Google Scholar 

  13. V. Ryazanov, U. Srebro, and E. Yakubov, “On the theory of BMO-quasiregular mappings,” Dokl. Ros. Akad. Nauk, 369, No. 1, 13–15 (1999).

    MathSciNet  Google Scholar 

  14. V. Ryazanov, U. Srebro, and E. Yakubov, “Plane mappings with dilatation dominated by functions of bounded mean oscillation,” Sib. Adv. Math., 11, No. 2, 94–130 (2001).

    MATH  MathSciNet  Google Scholar 

  15. V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solution of Beltrami equations,” J. Anal. Math., 96, 117–150 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  16. V. Ryazanov, U. Srebro, and E. Yakubov, “Finite mean oscillation and Beltrami equation,” Isr. J. Math., 153, 247–266 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  17. V. Ryazanov, U. Srebro, and E. Yakubov, “On the theory of the Beltrami equation,” Ukr. Mat. Zh., 58, No. 11, 1571–1583 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  18. S. K. Vodop’yanov and V. M. Gol’dshtein, “Criteria for the removability of sets for the spaces \(L^1_p\) of quasiconformal and quasiisometric mappings,” Sib. Mat. Zh., 18, No. 1, 630–633 (1977).

    Google Scholar 

  19. M. Vuorinen, “Exceptional sets and boundary behavior of quasiregular mappings in n-spaces,” Ann. Acad. Sci. Fenn. Al. Math. Dissertations, No. 11 (1976).

  20. M. Vuorinen, “Lower bounds for the moduli of path families with applications to nontangential limits of quasiconformal mappings,” Ann. Acad. Sci. Fenn. Al. Math., 2, 279–291 (1979).

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  23. J. Heinonen and P. Koskela, “Quasiconformal maps in metric spaces with controlled geometry,” Acta Math., 181, No. 1, 1–61 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  24. O. Martio, “Modern tools in the theory of quasiconformal maps,” Texts Math. B, 27, 1–43 (2000).

    MathSciNet  Google Scholar 

  25. R. R. Salimov, “On the boundary behavior of imbeddings of metric spaces into a Euclidean space,” Ukr. Mat. Zh., 59, No. 8, 1068–1074 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  26. K. Kuratowski, Topology [Russian translation], Vol. 2, Mir, Moscow (1969).

    Google Scholar 

  27. J. Heinonen, T. Kilpelainen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, New York (1993).

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  29. N. Bourbaki, Topologie Générale [Russian translation], Nauka, Moscow (1968).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. H. Federer, Geometric Measure Theory, Springer, Berlin (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

    R. R. Salimov

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  1. R. R. Salimov
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1378–1388, October, 2008.

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Salimov, R.R. Local behavior of Q-homeomorphisms in Loewner spaces. Ukr Math J 60, 1605–1617 (2008). https://doi.org/10.1007/s11253-009-0159-3

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  • DOI: https://doi.org/10.1007/s11253-009-0159-3

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