Finely holomorphic functions and finely harmonic morphisms
type: Item
instance of: scholarly article
title: Finely holomorphic functions and finely harmonic morphisms
zbMATH Open document ID: 0647.31005
DOI: 10.1007/BF01830668
published in: Aequationes Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137236
review text: The following result is well known and easy to prove: If f is a complex valued function on a domain U in the complex plane C with the property that both f and \(f^ 2\) are harmonic in U, then either f or \(\bar f\) is holomorphic in U. An outstanding problem in the theory of finely harmonic functions, posed by the author [18th Scand. Congr. Math., Proc., Aarhus 1980, Prog. Math. 11, 22-38 (1981; Zbl 0462.30004)] is whether the fine version of this holds: If f is a complex valued function on a fine domain W in C with the property that both f and \(f^ 2\) are finely harmonic in W, does it follow that either f or \(\bar f\) is finely holomorphic in W? In this article the author shows that the answer to this question is yes under the extra hypothesis that f is a fine \(C^ 1\)-function. It is still open to what extent this hypothesis can be weakened or dropped entirely.
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 30G30
zbMATH DE Number: 4056014
zbMATH Keywords: finely harmonic morphism
zbMATH Keywords: finely holomorphic
zbMATH Keywords: fine \(C^ 1\)-function
reviewed by: B. Øksendal
MaRDI profile type: Publication
cites work: Analytic capacity and differentiability properties of finely harmonic functions
cites work: Les potentiels d'energie finie
cites work: Les espaces du type de Beppo Levi
cites work: Finely harmonic functions
cites work: Fonctions harmoniques et fonctions finement harmoniques
cites work: Finely harmonic mappings and finely holomorphic mappings
cites work: Sur les fonctions finement holomorphes
cites work: Q3943215
cites work: Value distribution of harmonic and finely harmonic morphisms and applications in complex analysis
cites work: Finely holomorphic functions
cites work: A characterization of the finely harmonic morphism in \(R^ n\)
cites work: Finely harmonic morphisms, Brownian path preserving functions and conformal martingales
author: Bent Fuglede
description: scientific article; zbMATH DE number 4056014