Duke Mathematical Journal

Boundary properties of Green functions in the plane

Anton Baranov and Håkan Hedenmalm

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


We study the boundary properties of the Green function of bounded simply connected domains in the plane. Essentially, this amounts to studying the conformal mapping taking the unit disk onto the domain in question. Our technique is inspired by a 1995 article of Jones and Makarov [11]. The main tools are an integral identity as well as a uniform Sobolev embedding theorem. The latter is in a sense dual to the exponential integrability of Marcinkiewicz-Zygmund integrals. We also develop a Grunsky identity, which contains the information of the classical Grunsky inequality. This Grunsky identity is the case where p=2 of a more general Grunsky identity for Lp-spaces

Article information

Duke Math. J., Volume 145, Number 1 (2008), 1-24.

First available in Project Euclid: 17 September 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B65: Smoothness and regularity of solutions 30C35: General theory of conformal mappings
Secondary: 30C55: General theory of univalent and multivalent functions 30C85: Capacity and harmonic measure in the complex plane [See also 31A15]


Baranov, Anton; Hedenmalm, Håkan. Boundary properties of Green functions in the plane. Duke Math. J. 145 (2008), no. 1, 1--24. doi:10.1215/00127094-2008-044. https://projecteuclid.org/euclid.dmj/1221656860

Export citation


  • D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996.
  • L. Carleson and P. W. Jones, On coefficient problems for univalent functions and conformal dimension, Duke Math. J. 66 (1992), 169--206.
  • O. DragičEvić and A. Volberg, Bellman function, Littlewood-Paley estimates and asymptotics for the Ahlfors-Beurling operator in $L\sp p(\mathbb C)$, Indiana Univ. Math. J. 54 (2005), 971--995.
  • P. L. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer, New York, 1983.
  • L. Grafakos, Classical and Modern Fourier Analysis, Pearson, Upper Saddle River, N.J., 2004.
  • T. H. Gronwall, Some remarks on conformal representation, Ann. of Math. (2) 16 (1914/15), 72--76.
  • H. Hedenmalm and S. Shimorin, Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J. 127 (2005), 341--393.
  • —, On the universal integral means spectrum of conformal mappings near the origin, Proc. Amer. Math. Soc. 135 (2007), 2249--2255.
  • H. Hedenmalm and A. Sola, Spectral notions for conformal maps: A survey, Comput. Methods Funct. Theory 8 (2008), 447--474.
  • T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), 1--16.
  • P. W. Jones and N. G. Makarov, Density properties of harmonic measure, Ann. of Math. (2) 142 (1995), 427--455.
  • N. G. Makarov, Fine structure of harmonic measure, St. Petersburg Math. J. 10 (1999), 217--268.
  • C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss. 299, Springer, Berlin, 1992.
  • A. Zygmund, On certain lemmas of Marcinkiewicz and Carleson, J. Approximation Theory 2 (1969), 249--257.