Harmonic functions in non-locally convex spaces
type: Item
instance of: scholarly article
title: Harmonic functions in non-locally convex spaces
zbMATH Open document ID: 0677.31003
DOI: 10.1007/BF01193624
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let X be a topological vector space which is p normable for some \(0<p\leq 1\) i.e. X is a complex complete metrisable topological vector space whose topology is given by a quasi-norm such that \(\| x+y\|^ p\leq \| x\|^ p+\| y\|^ p.\) One defines a harmonic function h on an open set \(\Omega\) of \({\mathbb{C}}\) as the sum \(h=f_ 1(z)+f_ 2(\bar z)\) where \(f_ 1\) and \(f_ 2\) are analytic functions from \(\Omega\) to X. It is well known that in the case of a Banach space X, the maximum modulus principle for X valued analytic functions holds. In particular let f be a harmonic function in the open unit disk D of \({\mathbb{C}}\) which can be continuously extended to the closed unit disk and such that f vanishes on the boundary of D then f is identically zero in D. This property is no longer true for non locally convex topological spaces and an explicit example is given by the Poisson kernel with values in \(L^ p\), \(0<p<1.\) Hence for a p normable space one can ask at what rate one can have f(z)\(\to 0\) when \(| z| \to 1\) for an non trivial harmonic function h. The main result of this paper is the following: Let \(\beta >1/p-1\). If one has \(\| h(z)\| =O(1-| z|)^{\beta}\) then \(h\equiv 0\) on D. Assume that \(<p<1\) and let us set \(\beta =1/p-1\). If one has \(\| h(z)\| =o(1-| z|)^{\beta}\) then \(h\equiv 0\) on D.
Mathematics Subject Classification ID: 31A25
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 4109102
zbMATH Keywords: boundary values
zbMATH Keywords: harmonic function
zbMATH Keywords: maximum
zbMATH Keywords: principle
author: N. J. Kalton
reviewed by: Jean Lacroix
MaRDI profile type: Publication
cites work: Lokalkonvexe Unterräume in topologischen Vektorräumen und das \(\varepsilon\)-Produkt
cites work: Q5600026
cites work: Q4191150
cites work: Analytic functions in non-locally convex spaces and applications
cites work: Plurisubharmonic functions on quasi-Banach spaces
cites work: Q4048020
cites work: Topological vector spaces and algebras
cites work: Q3957476
full work available at URL: https://doi.org/10.1007/bf01193624
OpenAlex ID: W2111610330
description: scientific article; zbMATH DE number 4109102