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On the noncompactness of classes of mappings with restrictions on dilation in measure

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Abstract

We establish that the classes of mappings with restrictions in measure cannot be compact except for certain degenerate cases. In particular, this implies that any David class ΓI is noncompact.

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References

  1. L. V. Ahlfors, “On quasiconformal mappings,” J. Anal. Math., 3, 1–58 (1953/1954).

    Article  MathSciNet  Google Scholar 

  2. P. A. Biluta, “Some extremal problems for mappings quasiconformal in the mean,” Sib. Mat. Zh., 6, No. 4, 717–726 (1965).

    MATH  MathSciNet  Google Scholar 

  3. V. I. Kruglikov, “On the existence and uniqueness of mappings quasiconformal in the mean,” in: Metric Problems of the Theory of Functions and Mappings [in Russian], Naukova Dumka, Kiev (1973).

    Google Scholar 

  4. V. I. Kruglikov and V. M. Miklyukov, “On some classes of plane topologic mappings,” in: Metric Problems of the Theory of Functions and Mappings [in Russian], Naukova Dumka, Kiev (1973).

    Google Scholar 

  5. V. I. Kruglikov and S. M. Borchuk, “Convergent sequences of mappings with distortion bounded in the mean,” Dokl. Akad. Nauk Ukr. SSR, No. 10, 6–9 (1990).

    Google Scholar 

  6. S. L. Krushkal’ and R. Kühnau, Quasiconformal Mappings. New Methods and Applications [Russian translation], Nauka, Novosibirsk (1984).

    Google Scholar 

  7. V. C. Kud’yavin, “Local structure of plane mappings quasiconformal in the mean,” Dokl. Akad. Nauk Ukr., No. 3, 10–12 (1991).

  8. V. M. Maslyukov and G. D. Suvorov, “On the existence and uniqueness of quasiconformal mappings with unbounded characteristics,” in: Investigations in the Theory of Functions of a Complex Variable and Its Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1972), pp. 45–53.

    Google Scholar 

  9. M. Perovich, “On a global homeomorphism of mappings quasiconformal in the mean,” Dokl. Akad. Nauk SSSR, 230, No. 4, 781–784 (1976).

    MathSciNet  Google Scholar 

  10. I. N. Pesin, “Mappings quasiconformal in the mean,” Dokl. Akad. Nauk SSSR, 187, No. 4, 740–742 (1969).

    MathSciNet  Google Scholar 

  11. V. I. Ryazanov, “On quasiconformal mappings with restrictions in measure,” Ukr. Mat. Zh., 45, No. 7, 1009–1019 (1993).

    Article  MathSciNet  Google Scholar 

  12. V. I. Ryazanov, “On mappings quasiconformal in the mean,” Sib. Mat. Zh., 37, No. 2, 378–388 (1996).

    MathSciNet  Google Scholar 

  13. G. David, “Solution de l’equation de Beltrami avec ¦ μ¦ ∞ = 1,” Ann. Acad. Sci Fenn., Math., 13, No. 1, 25–75 (1988).

    MATH  Google Scholar 

  14. P. Tukia, “Compactness properties of μ-homeomorphisms,” Ann. Acad. Sci. Fenn., Math., 16, No 1, 47–69 (1991).

    MathSciNet  Google Scholar 

  15. O. Lehto and K. J. Virtanen, Quasiconformal Mappings in the Plane, Springer, Berlin (1973).

    MATH  Google Scholar 

  16. N. Dunford and J. T. Schwartz, Linear Operators. General Theory [Russian translation], Inostrannaya Literatura, Moscow (1962).

    Google Scholar 

  17. V. I. Ryazanov, “Some questions of convergence and compactness for quasiconformal mappings,” Amer. Math. Soc. Trans., 131, No. 2, 7–19 (1986).

    MathSciNet  Google Scholar 

  18. V. I. Ryazanov, “On the compactification of classes with integral restrictions on the Lavrent’ev characteristics,” Sib. Mat Zh., 33, No. 1, 87–104 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  19. V. Y. Gutlianskii and V. I. Ryazanov, “On boundary correspondence under quasiconformal mappings,” Ann. Acad. Sci. Fenn., Ser A1, Math., 21, 167–178 (1996).

    Google Scholar 

  20. V. L. Potemkin and V. I. Ryazanov, “On homeomorphisms of the Sobolev class with restrictions on dilation in measure,” Dokl. Akad. Nauk Ukr., No. 2, 8–11 (1996).

    Google Scholar 

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Authors and Affiliations

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

    V. L. Potemkin & V. I. Ryazanov

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  1. V. L. Potemkin
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  2. V. I. Ryazanov
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1522–1531, November, 1998.

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Potemkin, V.L., Ryazanov, V.I. On the noncompactness of classes of mappings with restrictions on dilation in measure. Ukr Math J 50, 1738–1749 (1998). https://doi.org/10.1007/BF02524480

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  • DOI: https://doi.org/10.1007/BF02524480

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