Abstract
We extend a well-known theorem of Jones and Makarov [8] on the singularity of boundary distortion of planar conformal mappings. Using a different technique, we recover the previous result and generalize the result to quasiconformal mappings of the unit ball \( \mathbb{B}^n \) ⊂_∝n, n ≥ 2. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere ∂\( \mathbb{B}^n \) and show that this estimate is essentially sharp.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Astala and F. W. Gehring, Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood, Michigan Math. J. 32 (1985), 99–107.
K. Astala and P. Koskela, Quasiconformal mappings and global integrability of the derivative, J. Anal. Math. 57 (1991), 203–220.
H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg, 1969.
M. Bonk, P. Koskela and S. Rohde, Conformal metrics on the unit ball in euclidean space, Proc. London Math. Soc. (3) 77 (1998), 635–664.
F. W. Gehring and B. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50–74.
S. Hencl and P. Koskela, Quasihyperbolic boundary conditions and capacity: Uniform continuity of quasiconformal mappings, J. Anal. Math. 96 (2005), 19–35.
J. Heinonen and S. Rohde, The Gehring-Hayman inequality and quasi-hyperbolic geodesics, Math. Proc. Cambridge Philos. Soc. 114 (1993) 393–405.
P. W. Jones and N. Makarov, Density properties of harmonic measure, Ann. of Math. (2) 142 (1995), 427–455.
P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), 593–609.
T. Nieminen, Generalized mean porosity and dimension, Ann. Acad. Sci. Fenn. Math. 31 (2006), 143–172.
T. Nieminen and T. Tossavainen, Boundary behavior of conformal deformations, Conform.Geom. Dyn. 11 (2007), 56–64
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, American Mathematical Society, Providence, RI, 1989.
C. A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge, 1970.
E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.
J. Väisälä, Quasiconformal maps and positive boundary measure, Analysis 9 (1991), 205–216.
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
W. P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nieminen, T., Uriarte-Tuero, I. Quasiconformal mappings and singularity of boundary distortion. J Anal Math 107, 377–389 (2009). https://doi.org/10.1007/s11854-009-0014-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-009-0014-3