On the generalized Hardy spaces
type: Item
instance of: scholarly article
title: On the generalized Hardy spaces
zbMATH Open document ID: 1191.30021
DOI: 10.1155/2010/803230
author: B. E. Eshmatov
published in: Abstract and Applied Analysis
publication date: 2010-04-08T00:00:00Z
full work available at URL: https://eudml.org/doc/228420
review text: Let \(H(U)\) be the space of all analytic functions on the unit disk \(U\) of the complex plane. Given \(1\leq p<\infty\) and a one-to-one linear operator \(F:H(U)\to H(U)\), the author introduces the space \(H_{F, p}(U)\) consisting of all functions \(f\in H(U)\) such that \[ \|f\|^p_{H_{F,p}}:=\sup_{0<r<1}\int_0^{2\pi}|F(f)(re^{i\theta})|^p\, {{d\theta}\over{2\pi}}<\infty. \] Clearly, if \(F\) is the identity operator, then \(H_{F, p}(U)\) is the classical Hardy space. The author obtains some sufficient conditions that imply some basic properties of such a space, namely, the property of being a Banach space and the continuity of point evaluations. The author also introduces the notion of Carleson measures for \(H_{F,p}(U)\) and uses them to obtain results concerning boundedness and compactness of composition operators on \(H_{F,p}(U)\).
reviewed by: Boo Rim Choe
Mathematics Subject Classification ID: 30H10
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 5690931
zbMATH Keywords: generalized Hardy space
zbMATH Keywords: composition operator
MaRDI profile type: Publication
OpenAlex ID: W1986731122
Wikidata QID: Q58649930
cites work: Q5618011
cites work: Q4888816
cites work: Subordinate \(H^ P\) functions
cites work: Q5588637
cites work: Q4402890
cites work: Composition operators and classical function theory
cites work: Composition operators acting on holomorphic Sobolev spaces
cites work: Composition operators on small spaces
cites work: Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball
cites work: Q2714230
cites work: Weighted composition operators on some weighted spaces in the unit ball
cites work: Weighted composition operators from Bergman-type spaces into Bloch spaces
cites work: Weighted Composition Operators from α-Bloch Space to H∞on the Polydisc
cites work: Weighted composition operators from \(H^{\infty }\) to the Bloch space on the polydisc
cites work: Weighted composition operators from Zygmund spaces into Bloch spaces
cites work: Angular Derivatives and Compact Composition Operators on the Hardy and Bergman Spaces
cites work: On generalized weighted bergman spaces
cites work: Composition operators between \(H^\infty\) and \(a\)-Bloch spaces on the polydisc
cites work: Weighted composition operators between mixed norm spaces and \(H_{\alpha }^{\infty }\) spaces in the unit ball
cites work: Essential norms of weighted composition operators from the \(\alpha \)-Bloch space to a weighted-type space on the unit ball
cites work: Q3070312
cites work: Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane
cites work: COMPOSITION OPERATORS ON THE PRIVALOV SPACES OF THE UNIT BALL OF ℂn
cites work: Q3601313
cites work: Compact weighted composition operators and multiplication operators between Hardy spaces
cites work: Q5447425
cites work: Q5297735
cites work: Q3011707
cites work: Compact composition operators on \(H^ p(B_ N)\)
cites work: Q4340161
cites work: Q4067293
cites work: Q4769223
description: scientific article; zbMATH DE number 5690931