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Boundary properties of Green functions in the plane. (English) Zbl 1157.35327

Summary: 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 P. W. Jones and N. G. Makarov [Ann. Math. (2) 142, No. 3, 427–455 (1995; Zbl 0842.31001)]. 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

MSC:

35B65 Smoothness and regularity of solutions to PDEs
30C35 General theory of conformal mappings
30C55 General theory of univalent and multivalent functions of one complex variable
30C85 Capacity and harmonic measure in the complex plane

Citations:

Zbl 0842.31001
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References:

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