Duke Mathematical Journal

Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

K. Astala, A. Clop, J. Mateu, J. Orobitg, and I. Uriarte-Tuero

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The classical Painlevé theorem tells us that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general K-quasiregular mappings in planar domains, the corresponding critical dimension is 2/(K+1). We show that when K>1, unexpectedly one has improved removability. More precisely, we prove that sets E of σ-finite Hausdorff (2/(K+1))-measure are removable for bounded K-quasiregular mappings. On the other hand, dim(E)=2/(K+1) is not enough to guarantee this property.

We also study absolute continuity properties of pullbacks of Hausdorff measures under K-quasiconformal mappings: in particular, at the relevant dimensions 1 and 2/(K+1). For general Hausdorff measures Ht, 0<t<2, we reduce the absolute continuity properties to an open question on conformal mappings (see Conjecture 2.3)

Article information

Duke Math. J., Volume 141, Number 3 (2008), 539-571.

First available in Project Euclid: 15 February 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 35J15: Second-order elliptic equations 35J70: Degenerate elliptic equations


Astala, K.; Clop, A.; Mateu, J.; Orobitg, J.; Uriarte-Tuero, I. Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings. Duke Math. J. 141 (2008), no. 3, 539--571. doi:10.1215/00127094-2007-005. https://projecteuclid.org/euclid.dmj/1203087637

Export citation


  • D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996.
  • L. V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1--11.
  • —, Lectures on Quasiconformal Mappings, Wadsworth Brooks/Cole Math. Ser., Wadsworth and Brooks/Cole Adv. Books Software, Monterey, Calif., 1987.
  • K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37--60.
  • K. Astala, T. Iwaniec, P. Koskela, and G. Martin, Mappings of $BMO$-bounded distortion, Math. Ann. 317 (2000), 703--726.
  • K. Astala, T. Iwaniec, and G. Martin, Elliptic partial differential equations and quasiconformal mappings in plane, manuscript.
  • K. Astala and V. Nesi, Composites and quasiconformal mappings: New optimal bounds in two dimensions, Calc. Var. Partial Differential Equations 18 (2003), 335--355.
  • K. Astala, S. Rohde, and O. Schramm, Dimension of quasicircles, in preparation.
  • J. Becker and C. Pommerenke, On the Hausdorff dimension of quasicircles, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 329--333.
  • C. Bishop, Distortion of disks by conformal maps, preprint, 2007.
  • L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Math. Stud. 13, Van Nostrand, Princeton, 1967.
  • G. David, Unrectifiable $1$-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), 369--479.
  • J. Duoandikoetxea, Fourier Analysis, revision of the 1995 Spanish original, Grad. Stud. Math. 29, Amer. Math. Soc., Providence, 2000.
  • J. Garnett, Analytic Capacity and Measure, Lecture Notes in Math. 297, Springer, Berlin, 1972.
  • T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29--81.
  • R. Kaufman, Hausdorff measure, BMO, and analytic functions, Pacific J. Math. 102 (1982), 369--371.
  • J. KráL, ``Analytic capacity'' in Elliptische Differentialgleichungen (Rostock, East Germany, 1977), Wilhelm-Pieck-Univ., Rostock, East Germany, 1978, 133--142.
  • O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973.
  • N. G. Makarov, Conformal mapping and Hausdorff measures, Ark. Mat. 25 (1987), 41--89.
  • P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge, 1995.
  • —, On the analytic capacity and curvature of some Cantor sets with non $\sigma$-finite length, Publ. Mat. 40 (1996), 195--204.
  • A. Mori, On an absolute constant in the theory of quasi-conformal mappings, J. Math. Soc. Japan 8 (1956), 156--166.
  • A. G. O'Farrell, Hausdorff content and rational approximation in fractional Lipschitz norms, Trans. Amer. Math. Soc. 228 (1977), 187--206.
  • C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss. 299, Springer, Berlin, 1992.
  • I. Prause, A remark on quasiconformal dimension distortion on the line, Ann. Acad. Sci. Fenn. Math. 32 (2007), 341--352.
  • H. M. Reimann, Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv. 49 (1974), 260--276.
  • H. M. Reimann and T. Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Math. 487, Springer, Berlin, 1975.
  • M. Sion and D. Sjerve, Approximation properties of measures generated by continuous set functions, Mathematika 9 (1962), 145--156.
  • X. Tolsa, Painlevé's problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), 105--149.
  • —, Bilipschitz maps, analytic capacity, and the Cauchy integral, Ann. of Math. (2) 162 (2005), 1243--1304.
  • I. Uriarte-Tuero, Sharp examples for planar quasiconformal distortion of Hausdorff measures and removability, preprint,\arxiv0707.1184v3[math.CV]
  • J. Verdera, $BMO$ rational approximation and one-dimensional Hausdorff content, Trans. Amer. Math. Soc. 297, no. 1 (1986), 283--304.