Mathematische Zeitschrift

, Volume 257, Issue 3, pp 525–545

Doubling measures, monotonicity, and quasiconformality



We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.

Mathematics Subject Classification (2000)

30C65 28A75 42A55 47H05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aimar H., Forzani L. and Naibo V. (2002). Rectangular differentiation of integrals of Besov functions. Math. Res. Lett. 9(2–3): 173–189 MATHGoogle Scholar
  2. 2.
    Alberti G. and Ambrosio L. (1999). A geometrical approach to monotone functions in \(\mathbb{R}^n\). Math. Z. 230(2): 259–316 MATHCrossRefGoogle Scholar
  3. 3.
    Beurling A. and Ahlfors L.V. (1956). The boundary correspondence under quasiconformal mappings. Acta Math. 96: 125–142 MATHCrossRefGoogle Scholar
  4. 4.
    Bishop, C.J.: An A 1 weight not comparable with any quasiconformal Jacobian. In: Canary, R., Gilman, J., Heinonen, J., Masur, H. (eds.) Proceedings of the 2005 Ahlfors-Bers Colloquium, Contemp. Math. Amer. Math. Soc., Providence (in press)Google Scholar
  5. 5.
    Bonk M., Heinonen J. and Rohde S. (2001). Doubling conformal densities. J. Reine Angew. Math. 541: 117–141 MATHGoogle Scholar
  6. 6.
    Bonk, M., Heinonen, J., Saksman, E.: The quasiconformal Jacobian problem. In: Abikoff, W., Haas, A. (eds.) In the tradition of Ahlfors and Bers, III, Contemp. Math., 355, 77–96, Amer. Math. Soc., Providence (2004)Google Scholar
  7. 7.
    Bonk, M., Heinonen, J., Saksman, E.: Logarithmic potentials, quasiconformal flows, and , preprint (2006)Google Scholar
  8. 8.
    Buckley S.M., Hanson B. and MacManus P. (2001). Doubling for general sets. Math. Scand. 88(2): 229–245 MATHGoogle Scholar
  9. 9.
    David, G., Semmes, S.: Strong A weights, Sobolev inequalities and quasiconformal mappings. In: Sadosky C. (ed.), Analysis and partial differential equations, 101–111, Lect. Notes Pure Appl. Math. 122, 101–111 (1990)Google Scholar
  10. 10.
    Gehring F.W. (1973). The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130: 265–277 MATHCrossRefGoogle Scholar
  11. 11.
    Gutiérrez C.E. (2001). The Monge-Ampère equation. Birkhäuser, Boston MATHGoogle Scholar
  12. 12.
    Heinonen J. (2001). Lectures on analysis on metric spaces. Springer, Heidelberg MATHGoogle Scholar
  13. 13.
    Jerison D.S. and Kenig C.E. (1982). Hardy spaces, A and singular integrals on chord-arc domains. Math. Scand. 50(2): 221–247 MATHGoogle Scholar
  14. 14.
    Kovalev, L.V.: Quasiconformal geometry of monotone mappings. J. London Math. Soc. (in press)Google Scholar
  15. 15.
    Kovalev L.V. and Maldonado D. (2005). Mappings with convex potentials and the quasiconformal Jacobian problem. Illinois J. Math. 49(4): 1039–1060 MATHGoogle Scholar
  16. 16.
    Laakso T.J. (2002). Plane with A -weighted metric not bi-Lipschitz embeddable to \(\mathbb{R}^n\). Bull. London Math. Soc. 34(6): 667–676 MATHCrossRefGoogle Scholar
  17. 17.
    Rohde S. (2001). Quasicircles modulo bilipschitz maps. Rev. Mat. Iberoamericana 17(3): 643–659 MATHGoogle Scholar
  18. 18.
    Semmes S. (1996). On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A -weights. Rev. Mat. Iberoamericana 12(2): 337–410 MATHGoogle Scholar
  19. 19.
    Sobolevskii P.E. (1957). On equations with operators forming an acute angle. Dokl. Akad. Nauk SSSR (N.S.) 116: 754–757 Google Scholar
  20. 20.
    Staples S.G. (1992). Doubling measures and quasiconformal maps. Comment. Math. Helv. 67(1): 119–128 MATHCrossRefGoogle Scholar
  21. 21.
    Stein E.M. (1970). Singular integrals and differentiability properties of functions. Princeton University Press, Princeton MATHGoogle Scholar
  22. 22.
    Stein E.M. (1993). Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals. Princeton University Press, Princeton MATHGoogle Scholar
  23. 23.
    Tukia P. (1981). Extension of quasisymmetric and Lipschitz embeddings of the real line into the plane. Ann. Acad. Sci. Fenn. Ser. A I Math. 6(1): 89–94 Google Scholar
  24. 24.
    Tukia P. and Väisälä J. (1980). Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5(1): 97–114 Google Scholar
  25. 25.
    Tyson J.T. and Wu J.-M. (2005). Characterizations of snowflake metric spaces. Ann. Acad. Sci. Fenn. Math. 30(2): 313–336 MATHGoogle Scholar
  26. 26.
    Väisälä, J.: Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., vol 229. Springer, Berlin (1971)Google Scholar
  27. 27.
    Zeidler E. (1990). Nonlinear functional analysis and its applications II/B. Nonlinear monotone operators. Springer, Heidelberg MATHGoogle Scholar
  28. 28.
    Zygmund A. (2002). Trigonometric Series. Cambridge Univ. Press, Cambridge MATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Leonid V. Kovalev
    • 1
  • Diego Maldonado
    • 2
    • 3
  • Jang-Mei Wu
    • 4
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of MathematicsKansas State UniversityManhattanUSA
  4. 4.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations