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Mappings with finite length distortion

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Abstract

General properties of mappings of finite metric distortion and of finite length distortion are studied. Uniqueness, equicontinuity, boundary behavior and removability of singularities are obtained under minimal additional assumptions.

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References

  • [MV] O. Martio and J. Väisälä,Elliptic equations and maps of bounded metric distortion, Math. Ann.282 (1988), 423–443.

    Article  MATH  MathSciNet  Google Scholar 

  • [MV1] J. J. Manfredi and E. Villamor,Mappings with integrable dilatation in higher dimensions, Bull. Amer. Math. Soc.32 (1995), 235–240.

    MATH  MathSciNet  Google Scholar 

  • [MV2] J. J. Manfredi and E. Villamor,An extension of Reshetnyak's theorem, Indiana Univ. Math. J.47 (1998), 1131–1145.

    MATH  MathSciNet  Google Scholar 

  • [Pol] E. A. Poletskii,The modulus method for non-homeomorphic quasiconformal mappings, Mat. Sb.83 (125) (1970), 261–272.

    MathSciNet  Google Scholar 

  • [Pon1] S. P. Ponomarev,On the N-property of homeomorphisms of the class W p1 , Sibirsk. Mat. Zh.28, No. 2 (1987), 140–148.

    MATH  MathSciNet  Google Scholar 

  • [Pon2] S. P. Ponomarev,The N −1 -property of mappings, and Lusin's (N) condition, Mat. Zametki58 (1995), 411–418; transl. in Math. Notes58 (1995), 960–965.

    MathSciNet  Google Scholar 

  • [RR] T. Rado and P. V. Reichelderfer,Continuous Transformations in Analysis, Springer-Verlag, Berlin, 1955.

    MATH  Google Scholar 

  • [Re1] Yu. G. Reshetnyak,Space Mappings with Bounded Distortion, Transl. Math. Monographs 73, Amer. Math. Soc., Providence, RI, 1989.

    MATH  Google Scholar 

  • [Re2] Yu. G. Reshetnyak,Liouville's conformal mapping theorem under minimal regularity hypotheses, Sibirsk. Math. Zh.8 (1967), 835–840.

    MATH  Google Scholar 

  • [Ri] S. Rickman,Quasiregular Mappings, Springer, Berlin, 1993.

    MATH  Google Scholar 

  • [Sa] S. Saks,Theory of the Integral, Dover, New York, 1964.

    Google Scholar 

  • [Se] J. Serrin,On the differentiability of functions of several variables, Arch. Rational Mech. Anal.7 (1961), 359–372.

    Article  MATH  MathSciNet  Google Scholar 

  • [SY] U. Srebro and E. Yakubov,Branched folded maps and alternating Beltrami equations, J. Analyse Math.70 (1996), 65–90.

    MATH  MathSciNet  Google Scholar 

  • [TY] C. J. Titus and G. S. Young,The extension of interiority, with some applications, Trans. Amer. Math. Soc.103 (1962), 329–340.

    Article  MATH  MathSciNet  Google Scholar 

  • [Va1] J. Väisälä,Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math.229, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

  • [Va2] J. Väisälä,On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser A I298 (1961), 1–36.

    Google Scholar 

  • [Vu] M. Vuorinen,Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math.1319, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  • [Whi] H. Whitney,On totally differentiable and smooth functions, Pacific J. Math.1 (1951), 143–159.

    MATH  MathSciNet  Google Scholar 

  • [Ki] M. D. Kirszbraun,Uber die zusammenziehende und Lipschitzsche Transformationen, Fund. Math.22 (1934), 77–108.

    MATH  Google Scholar 

  • [KO] P. Koskela and J. Onninen,Mappings of finite distortion: capacity and modulus inequalities, Dept. Math. Stat., University of Jyväskylä, Preprint 257 (2002), 1–32.

  • [Ma] V. Mayer,Quasiregular analogues of critically finite rational functions with parabolic orbifold, J. Analyse Math.75 (1998), 105–119.

    Article  MATH  Google Scholar 

  • [MM] J. Maly and O. Martio,Lusin's condition (N) and mappings of the class W 1,n, J. Reine Angew. Math.458 (1995), 19–36.

    MATH  MathSciNet  Google Scholar 

  • [MRV] O. Martio, S. Rickman and J. Väisälä,Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math.448 (1969), 1–40.

    Google Scholar 

  • [MRSY1] O. Martio, V. Ryazanov, U. Srebro and E. Yakubov,Q-homeomorphisms, Contemp. Math.364 (2004), 93–103.

    MathSciNet  Google Scholar 

  • [MRSY2] O. Martio, V. Ryazanov, U. Srebro and E. Yakubov,On Q-homeomorphisms, submitted.

  • [McSh] E. J. McShane,Extension of range of functions, Bull. Amer. Math. Soc.40 (1934), 837–842.

    MATH  MathSciNet  Google Scholar 

  • [MV] O. Martio and J. Väisälä,Elliptic equations and maps of bounded metric distortion, Math. Ann.282 (1988), 423–443.

    Article  MATH  MathSciNet  Google Scholar 

  • [MV1] J. J. Manfredi and E. Villamor,Mappings with integrable dilatation in higher dimensions, Bull. Amer. Math. Soc.32 (1995), 235–240.

    MATH  MathSciNet  Google Scholar 

  • [MV2] J. J. Manfredi and E. Villamor,An extension of Reshetnyak's theorem, Indiana Univ. Math. J.47 (1998), 1131–1145.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Helsinki, Yliopistonk. 5, P.O. Box 4, FIN-00014, Finland

    O. Martio

  2. Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 83114, Donetsk, Ukraine

    V. Ryazanov

  3. Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    U. Srebro

  4. Holon Academic Institute of Technology, 52 Golomb St., P.O. Box 305, 58102, Holon, Israel

    E. Yakubov

Authors
  1. O. Martio
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  2. V. Ryazanov
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  3. U. Srebro
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  4. E. Yakubov
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Correspondence to O. Martio.

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Martio, O., Ryazanov, V., Srebro, U. et al. Mappings with finite length distortion. J. Anal. Math. 93, 215–236 (2004). https://doi.org/10.1007/BF02789308

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  • DOI: https://doi.org/10.1007/BF02789308

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