TY - JOUR

T1 - Boundary properties of green functions in the plane

AU - Baranov, Anton

AU - Hedenmalm, Håkan

PY - 2008/10/1

Y1 - 2008/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=54149102794&partnerID=8YFLogxK

U2 - 10.1215/00127094-2008-044

DO - 10.1215/00127094-2008-044

M3 - Article

AN - SCOPUS:54149102794

VL - 145

SP - 1

EP - 24

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 1

ER -