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Sobolev regularity of quasiconformal mappings on domains

  • Martí PratsEmail author
Article

Abstract

Consider a Lipschitz domain Ω and a measurable function μ supported in \(\overline{\Omega}\)with ‖μL < 1. Then the derivatives of a quasiconformal solution of the Beltrami equation \(\overline{\partial}f=\mu\;\partial{f}\) inherit the Sobolev regularity Wn,p(Ω) of the Beltrami coefficient μ as long as Ω is regular enough. The condition obtained is that the outward unit normal vector N of the boundary of the domain is in the trace space, that is, \(N\in{B}_{p,p}^{n-1/p}(\partial\Omega)\).

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Notes

Acknowledgement

The author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 320501. Also, he was partially supported by grants 2014-SGR-75 (Generalitat de Catalunya), MTM-2010-16232 and MTM-2013-44304-P (Spanish government) and by a FI-DGR grant from the Generalitat de Catalunya, (2014FI-B2 00107).

The author would like to thank Xavier Tolsa for advice on his Ph.D. thesis, which gave rise to this work; Cruz,Mateu, Orobitg andVerdera for their advice and interest; and the editor and referee for their patient work and valuable comments.

References

  1. [AF03]
    R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.zbMATHGoogle Scholar
  2. [AIM09]
    K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, NJ, 2009.zbMATHGoogle Scholar
  3. [AIS01]
    K. Astala, T. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), 27–56.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Ast94]
    K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 7–60.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [CF12]
    G. Citti and F. Ferrari, A sharp regularity result of solutions of a transmission problem, Proc. Amer. Math. Soc. 140 (2012), 615–620.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [CFM+09]
    [CFM+09]_A. Clop, D. Faraco, J. Mateu, J. Orobitg and X. Zhong, Beltrami equations with coefficient in the Sobolev space W 1,p, Publ. Mat. 53 (2009), 197–230.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [CFR10]
    A. Clop, D. Faraco and A. Ruiz, Stability of Calderón’s inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging 4 (2010), 49–91.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [CMO13]
    V. Cruz, J. Mateu and J. Orobitg, Beltrami equation with coefficient in Sobolev and Besov spaces, Canad. J. Math. 65 (2013), 1217–1235.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [CT12]
    V. Cruz and X. Tolsa, Smoothness of the Beurling transform in Lipschitz domains, J. Funct. Anal. 262 (2012), 4423–4457.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Eva98]
    L. C. Evans, Partial Differential Equations, Oxford University Press, Oxford, 1998.zbMATHGoogle Scholar
  11. [Iwa92]
    T. Iwaniec, L p-theory of quasiregular mappings, in Quasiconformal Space Mappings, Springer, Berlin-Heidelberg, 1992, pp. 39–64.CrossRefGoogle Scholar
  12. [Jon81]
    P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71–88.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [MOV09]
    J. Mateu, J. Orobitg, and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl. 91 (2009), 402–431.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Pra15]
    M. Prats, Sobolev regularity of the Beurling transform on planar domains, Publ. Mat. 2 (2017), 291–336.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [PT15]
    M. Prats and X. Tolsa, A T(P) theorem for Sobolev spaces on domains, J. Funct. Anal. 268 (2015), 2946–2989.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [RS96]
    T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter; Berlin-New York, 1996.CrossRefzbMATHGoogle Scholar
  17. [Sch02]
    M. Schechter, Principles of Functional Analysis, American Mathematical Society, Providence, RI, 2002.zbMATHGoogle Scholar
  18. [Ste70]
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.zbMATHGoogle Scholar
  19. [Tol13]
    X. Tolsa, Regularity of C1 and Lipschitz domains in terms of the Beurling transform, J. Math. Pures Appl. 100 (2013), 137–165.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Tri78]
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.zbMATHGoogle Scholar
  21. [Tri83]
    H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.CrossRefzbMATHGoogle Scholar
  22. [Ver01]
    J. Verdera, L 2 boundedness of the Cauchy integral and Menger curvature in Harmonic Analysis and Boundary Value Problems, American Mathematical Society, Providence, RI, 2001, pp. 139–158.CrossRefzbMATHGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaCataloniaSpain

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