## Duke Mathematical Journal

### Boundary properties of Green functions in the plane

#### Abstract

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 $L^p$-spaces

#### Article information

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

Dates
First available in Project Euclid: 17 September 2008

https://projecteuclid.org/euclid.dmj/1221656860

Digital Object Identifier
doi:10.1215/00127094-2008-044

Mathematical Reviews number (MathSciNet)
MR2451287

Zentralblatt MATH identifier
1157.35327

#### Citation

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

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