Advertisement

Journal d'Analyse Mathématique

, Volume 137, Issue 1, pp 251–268 | Cite as

Quasiconformal maps with controlled Laplacian

  • David KalajEmail author
  • Eero Saksman
Article
  • 29 Downloads

Abstract

We establish that every K-quasiconformal mapping w of the unit disk \(\mathbb{D}\) onto a C2-Jordan domain Ω is Lipschitz provided that ΔwLp(\(\mathbb{D}\)) for some p > 2. We also prove that if in this situation K → 1 with ||Δw||Lp(\(\mathbb{D}\)) → 0, and Ω→\(\mathbb{D}\) in C1,α-sense with α > 1/2, then the bound for the Lipschitz constant tends to 1. In addition, we provide a quasiconformal analogue of the Smirnov theorem on absolute continuity over the boundary.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure and Appl. Math. 12, (1959) 623–727.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. Ahlfors, Lectures on Quasiconformal mappings, D. Van Nostrand, Inc. Princeton, 1966.zbMATHGoogle Scholar
  3. [3]
    A. B. Aleksandrov, J. M. Anderson, and A Nicolau, Inner functions, Bloch spaces and symmetric measures, Proc. London Math. Soc. 79 (1999), 318–352.CrossRefzbMATHGoogle Scholar
  4. [4]
    K. Astala, T. Iwaniec, and G. J. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, 2009.zbMATHGoogle Scholar
  5. [5]
    K. Astala and V. Manojlović, On Pavlović theorem in space, Potential Analysis 43 (2015), 361–370.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    K. Astala, T. Iwaniec, I. Prause, and E. Saksman, Bilipschitz and quasiconformal rotation, stretching and multifractal spectra, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 113–154.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    R. A. Fefferman, C. E. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65–124.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, BerlinHeidelbergNew YorkTokyo, 1983.CrossRefzbMATHGoogle Scholar
  9. [9]
    G. L. Goluzin, Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, RI, 1969.CrossRefzbMATHGoogle Scholar
  10. [10]
    J. P. Kahane, Trois notes sur les ensembles parfait linearés, Enseign. Math. 15 (1969), 185–192.zbMATHGoogle Scholar
  11. [11]
    D. Kalaj, Harmonic mappings and distance function, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), 669–681.MathSciNetzbMATHGoogle Scholar
  12. [12]
    D. Kalaj, On boundary correspondences under quasiconformal harmonic mappings between smooth Jordan domains, Math. Nachr. 285 (2012), 283–294.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Kalaj, A priori estimate of gradient of a solution to certain differential inequality and quasiregular mappings, J. Anal. Math. 119 (2013), 63–88.Google Scholar
  14. [14]
    D. Kalaj, M. Markovic, and M. Mateljević, Carathéodory and Smirnov type theorems for harmonic mappings of the unit disk onto surfaces, Ann. Acad. Sci. Fenn. Math. 38 (2013), 565–580.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Kalaj and M. Pavlović, On quasiconformal self-mappings of the unit disk satisfying the Poisson equation, Trans. Amer. Math. Soc. 363 (2011) 4043–4061.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin-Heidelberg, 1992.CrossRefzbMATHGoogle Scholar
  17. [17]
    D. Partyka and K. Sakan, On bi-Lipschitz type inequalities for quasiconformal harmonic mappings, Ann. Acad. Sci. Fenn.Math. 32 (2007), 579–594.MathSciNetzbMATHGoogle Scholar
  18. [18]
    M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disc, Ann. Acad. Sci. Fenn. 27 (2002) 365–372.zbMATHGoogle Scholar
  19. [19]
    G. Piranian, Two monotonic, singular, uniformly almost smooth functions, Duke Math. J. 33 (1966), 255–262.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    W. Rudin, Real and Complex Analysis, Third edition. McGraw-Hill 1986.zbMATHGoogle Scholar
  21. [21]
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.zbMATHGoogle Scholar
  22. [22]
    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.zbMATHGoogle Scholar
  23. [23]
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2. Auflage. Barth, Heidelberg, 1995.zbMATHGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.University of Montenegro Faculty of Natural Sciences and MathematicsPodgoricaMontenegro
  2. 2.Department Of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

Personalised recommendations