Arkiv för Matematik

  • Ark. Mat.
  • Volume 38, Number 2 (2000), 263-279.

Removability theorems for Sobolev functions and quasiconformal maps

Peter W. Jones and Stanislav K. Smirnov

Full-text: Open access


We establish several conditions, sufficient for a set to be (quasi)conformally removable, a property important in holomorphic dynamics. This is accomplished by proving removability theorems for Sobolev spaces in Rn. The resulting conditions are close to optimal.


The first author is supported by N.S.F. Grant No. DMS-9423746.


The second author is supported by N.S.F. Grants No. DMS-9304580 and DMS-9706875.

Article information

Ark. Mat., Volume 38, Number 2 (2000), 263-279.

Received: 29 January 1999
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2000 © Institut Mittag-Leffler


Jones, Peter W.; Smirnov, Stanislav K. Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat. 38 (2000), no. 2, 263--279. doi:10.1007/BF02384320.

Export citation


  • [A] Ahlfors, L. V., Lectures on Quasiconformal Mappings, Van Nostrand Mathematical Studies 10D, Van Nostrand, Princeton, N. J., 1966.
  • [AB] Ahlfors, L., and Beurling, A., Conformal invariants and function-theoretic nullsets, Acta Math. 83 (1950), 101–129.
  • [Be] Besicovitch, A. S., On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated points, Proc. London Math. Soc. 32 (1931), 1–9.
  • [Bi] Bishop, C. J., Some homeomorphisms of the sphere conformal off a curve, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 323–338.
  • [C] Carleson, L., On null-sets for continuous analytic functions, Ark. Mat. 1 (1951), 311–318.
  • [G] Gehring, F. W., The definitions and exceptional sets for quasiconformal mappings, Ann. Acad. Sci. Fenn Ser. A I Math. 281 (1960), 1–28.
  • [GS1] Graczyk, J. and Smirnov, S., Collet, Eckmann and Hölder, Invent. Math. 133 (1998), 69–96.
  • [GS2] Graczyk, J. and Smirnov, S., Non-uniform hyperbolicity in complex dynamics, Preprint, 1997–2000.
  • [Je] Jerison, D., The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503–523.
  • [Jo] Jones, P. W., On removable sets for Sobolev spaces in the plane, in Essays on Fourier Analysis in Honor of Elias M. Stein (Fefferman, C., Fefferman, R. and Wainger, S., eds.), Princeton Math. Ser., 42, pp. 250–267, Princeton Univ. Press, Princeton, N. J., 1995.
  • [JM] Jones, P. W. and Makarov, N. G., Density properties of harmonic measure, Ann. of Math. 142 (1995), 427–455.
  • [Ka] Kaufman, R., Fourier-Stieltjes coefficients and continuation of functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 27–31.
  • [KW] Kaufman, R. and Wu, J.-M., On removable sets for quasiconformal mappings, Ark. Mat. 34 (1996), 141–158.
  • [Ko] Koskela, P., Old and new on the quasihyperbolic metric, in Quasiconformal Mappings and Analysis (Duren, P., Heinonen, J., Osgood, B. and Palka, B., eds.), pp. 205–219, Springer-Verlag, New York, 1998.
  • [PR] Przytycki, F. and Rohde, S., Rigidity of holomorphic Collet-Eckmann repellers, Ark. Mat. 37 (1999), 357–371.
  • [U] Uy, N. X., Removable sets of analytic functions satisfying a Lipschitz condition, Ark. Mat. 17 (1979), 19–27.
  • [V] Väisälä, J., Lectures on n-dimensional Quasiconformal Mappings, Lecture Notes in Math. 229, Springer-Verlag, Berlin-Heidelberg, 1971.
  • [Z] Ziemer, W. P., Weakly Differentiable Functions, Grad. Texts in Math., 120, Springer-Verlag, New York, 1989.