Some remarks on boundary values of vector-valued harmonic and analytic functions
type: Item
instance of: scholarly article
title: Some remarks on boundary values of vector-valued harmonic and analytic functions
zbMATH Open document ID: 0696.46031
DOI: 10.1007/BF01200044
author: Wolfgang Hensgen
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: Let X be a complex Banach space and D the open unit disc in \({\mathbb{C}}\), \[ h^ 1(X):=\{u: D\to X\quad harmonic:\quad \sup_{r<1}\int^{2\pi}_{0}\| u(re^{i\delta})\|_ X(d\delta /2\pi)<\infty \}, \] \[ N(X):=\{f: D\to X\quad analytic:\quad \sup_{r<1}\int^{2\pi}_{0}\ell n^+\| f(re^{i\delta})\|_ X(d\delta /2\pi)<\infty \}. \] The paper consists of amendments to \textit{A. V. Bukhvalov}'s and \textit{A. A. Danilevich}'s ``Boundary properties of analytic and harmonic functions with values in Banach space'' [Mat. Zametki 31, 203-214 (1982; Zbl 0496.30029)]. Specifically it is proved that if X has RNP, then every \(u\in h^ 1(X)\) possesses angular limits a.e. on \(\partial D\). If \(u\in h^ 1(X)\) has a.e. weak radial limits, then u has a.e. strong angular limits. If X is separable and Y has RNP, then very \(u\in h^ 1(L(X,Y))\) admits a.e. angular limits in the strong operator topology. In the analytic case, the same results hold, where \(h^ 1\) can be replaced by N and RNP by ARNP.
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 4140540
zbMATH Keywords: vector-valued Hardy space
zbMATH Keywords: Radon-Nikodym property
zbMATH Keywords: analytic Radon-Nikodym property
zbMATH Keywords: strong angular limits
zbMATH Keywords: strong operator topology
zbMATH Keywords: RNP
zbMATH Keywords: ARNP
MaRDI profile type: Publication
cites work: Boundary values of functions in vector-valued Hardy spaces and geometry on Banach spaces
cites work: Hardy Classes of Banach-Space-Valued Distributions
cites work: Q4114342
cites work: Boundary properties of analytic and harmonic functions with values in Banach space
cites work: Q5517160
cites work: Q5618011
cites work: Q3673778
cites work: Q3922907
cites work: Q5645920
cites work: Q3244654
cites work: Q3874458
cites work: Characterization of Poisson integrals of vector-valued functions and measures on the unit circle
cites work: Shorter Notes: Totally Real Klein Bottles in C 2
cites work: Q5727756
cites work: Q5732890
full work available at URL: https://doi.org/10.1007/bf01200044
OpenAlex ID: W2049512570
description: scientific article; zbMATH DE number 4140540