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Bi-Lipschitz Property of HQC Mappings

  • Vesna Todorčević
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Abstract

The inverse of a K-quasiconformal homeomorphism is also K-quasiconformal. By the Schwarz lemma for K-quasiconformal mappings we know that both mappings are Hölder continuous in the Euclidean metric with exponent K1∕(1−n), and the Gehring–Osgood result yields the same conclusion in the quasihyperbolic metric. The class of harmonic K-quasiconformal interpolates between the classes of conformal maps and general quasiconformal maps. In this chapter we study the modulus of continuity of harmonic quasiconformal mappings relative to the quasihyperbolic metric and prove that both the mapping and its inverse are Lipschitz-continuous.

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References

  1. 15.
    M. Arsenović, V. Božin, V. Manojlović, Moduli of continuity of harmonic quasiregular mappings in \(\mathbf {\mathbb B}^n\). Potential Anal. 34(3), 283–291 (2011)Google Scholar
  2. 19.
    K. Astala, F.W. Gehring, Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood. Mich. Math. J. 32(1), 99–107 (1985)MathSciNetCrossRefGoogle Scholar
  3. 21.
    K. Astala, V. Manojlović, On Pavlović’s theorem in space. Potential Anal. 43(3), 361–370 (2015)MathSciNetCrossRefGoogle Scholar
  4. 23.
    S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory. Graduate Texts in Mathematics, vol. 137 (Springer, New York, 1992)CrossRefGoogle Scholar
  5. 42.
    P. Duren, Harmonic Mappings in the Plane (Cambridge University Press, Cambridge, 2004)CrossRefGoogle Scholar
  6. 47.
    J.B. Garnett, D.E. Marshall, Harmonic Measure (Cambridge University Press, Cambridge, 2005)CrossRefGoogle Scholar
  7. 50.
    F.W. Gehring, The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)MathSciNetCrossRefGoogle Scholar
  8. 53.
    F.W. Gehring, B.G. Osgood, Uniform domains and the quasihyperbolic metric. J. Anal. Math. 36, 50–74 (1979)MathSciNetCrossRefGoogle Scholar
  9. 55.
    D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1998)zbMATHGoogle Scholar
  10. 56.
    S. Gleason, T. Wolff, Lewy’s harmonic gradient maps in higher dimensions. Commun. Partial Differ. Equ. 16(12), 1925–1968 (1991)MathSciNetCrossRefGoogle Scholar
  11. 64.
    J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs (Oxford University Press, New York, 1993)zbMATHGoogle Scholar
  12. 65.
    S. Hencl, P. Koskela, Lectures on Mappings of Finite Distortion. Lecture Notes in Mathematics, vol. 2096 (Springer, Berlin, 2013)zbMATHGoogle Scholar
  13. 70.
    T. Iwaniec, J. Onninen, Radó–Kneser–Choquet theorem. Bull. Lond. Math. Soc. 46(6), 1283–1291 (2014)MathSciNetCrossRefGoogle Scholar
  14. 73.
    D. Kalaj, Quasiconformal harmonic functions between convex domains. Publ. Inst. Math. 75(89), 139–146 (2004)MathSciNetCrossRefGoogle Scholar
  15. 76.
    D. Kalaj, A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings. J. Anal. Math. 119, 63–88 (2013)MathSciNetCrossRefGoogle Scholar
  16. 79.
    D. Kalaj, M. Pavlović, Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane. Ann. Acad. Sci. Fenn. Math. 30(1), 159–165 (2005)MathSciNetzbMATHGoogle Scholar
  17. 93.
    E.C. Lawrence, Partial Differential Equations. Graduated Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)Google Scholar
  18. 95.
    H. Lewy, On the non-vanishing of the Jacobian of a homeomorphism by harmonic gradients. Ann. Math. 88, 518–529 (1968)MathSciNetCrossRefGoogle Scholar
  19. 99.
    V. Manojlović, Bi-Lipschicity of quasiconformal harmonic mappings in the plane. Filomat 23(1), 85–89 (2009)MathSciNetCrossRefGoogle Scholar
  20. 103.
    V. Manojlović, On biLipschicity of quasiconformal mappings. Novi Sad J. Math. 45(1), 105–109 (2015)MathSciNetCrossRefGoogle Scholar
  21. 114.
    O. Martio, J. Väisälä, Quasihyperbolic geodesics in convex domains. II. Pure Appl. Math. Q. 7(2), 395–409 (2011). Special Issue: In Honor of Frederick W. Gehring, Part IIMathSciNetCrossRefGoogle Scholar
  22. 116.
    M. Mateljević, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry. Rev. Roumaine Math. Pures Appl. 51(5–6), 711–722 (2006)MathSciNetzbMATHGoogle Scholar
  23. 121.
    D. Partyka, K. Sakan, On bi-Lipschitz type inequalities for quasiconformal harmonic mappings. Ann. Acad. Sci. Fenn. Math. 32(2), 579–594 (2007)MathSciNetzbMATHGoogle Scholar
  24. 123.
    M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk. Ann. Acad. Sci. Fenn. Math. 27(2), 365–372 (2002)MathSciNetzbMATHGoogle Scholar
  25. 124.
    M. Pavlović, Function Classes on the Unit Disc – An Introduction. Studies in Mathematics, vol. 52 (De Gruyter, Berlin 2014)Google Scholar
  26. 150.
    E. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)zbMATHGoogle Scholar
  27. 155.
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics, vol. 229 (Springer, Berlin, 1971)CrossRefGoogle Scholar
  28. 158.
    M. Vuorinen, Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319 (Springer, Berlin, 1988)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vesna Todorčević
    • 1
    • 2
  1. 1.Faculty of Organizational SciencesUniversity of BelgradeBelgradeSerbia
  2. 2.Mathematical InstituteSerbian Academy of Sciences and ArtsBelgradeSerbia

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