Espaces de Hardy et domains de Denjoy. (Hardy spaces and Denjoy domains)
type: Item
instance of: scholarly article
title: Espaces de Hardy et domains de Denjoy. (Hardy spaces and Denjoy domains)
zbMATH Open document ID: 0682.30030
DOI: 10.1007/BF02386381
author: Michel Zinsmeister
published in: Arkiv för Matematik
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Omega\) be a Denjoy domain in the plane, that is, a domain of the form \(\Omega ={\mathbb{C}}\setminus K\), where \(K\subseteq {\mathbb{R}}\) is a compact set of positive Lebesgue measure. Let \(H^ 1(\Omega)\) be the Hardy space of all functions analytic in \(\Omega\) satisfying \(\lim_{| z| \to \infty}zF(z)=0\) and \[ \| F\|_{H^ 1(\Omega)}:=\sup_{\delta >0}\int_{\Gamma_{\delta}}| F(z)| | dz| <\infty, \] where \(\Gamma_{\delta}\) is the boundary of the union of all squares centered on K with length \(\delta\). The author studies the problem for which compact sets \(K\subseteq {\mathbb{R}}\) there exists a constant \(C>0\) such that \[ \int_{L\cap \Omega}| F(z)| | dz| \leq C\| F\|_{H^ 1(\Omega)},\quad \Omega ={\mathbb{C}}\setminus K, \] for every \(F\in H^ 1(\Omega)\) and every line L in \({\mathbb{C}}\). A positive answer is given for the so-called homogeneous sets. This notion was introduced by \textit{L. Carleson} [Harmonic analysis, Conf. in Honor A. Zygmund, Chicago 1981, Vol. 2, 349-372 (1983; Zbl 0527.46048)] in relation with the corona theorem for Denjoy domains. An n- dimensional version of the results is briefly sketched in the last section.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 32A35
zbMATH DE Number: 4116898
zbMATH Keywords: homogeneous compact sets
zbMATH Keywords: Denjoy domains
MaRDI profile type: Publication
cites work: Level sets of univalent functions
cites work: Q3308586
cites work: Q5640160
cites work: Q5607484
cites work: Extensions of Hardy spaces and their use in analysis
cites work: Régularité d'accès des bouts et frontière de Martin d'un domaine euclidiens
cites work: \(H^p\) spaces of several variables
cites work: Q3252865
Recommended article: Critical points of Green's function, harmonic measure, and the corona problem
Recommended article: The Corona theorem for Denjoy domains
Recommended article: Q3374396
Recommended article: Homogeneous subsets of a Lipschitz graph and the corona theorem
Recommended article: A new approach to the corona theorem for domains bounded by a C^1+α curve
Recommended article: Q3766037
Recommended article: Q4952564
Recommended article: Hausdorff dimension of harmonic measures in the plane
Recommended article: Ahlfors Functions on Denjoy Domains
Recommended article: On finitely generated closed ideals in \(H^{\infty}(D)\)
description: scientific article; zbMATH DE number 4116898