On \(\alpha\)-Bloch spaces and multipliers of Dirichlet spaces
type: Item
instance of: scholarly article
title: On \(\alpha\)-Bloch spaces and multipliers of Dirichlet spaces
zbMATH Open document ID: 0892.30030
author: Rauno Aulaskari
author: Peter Lappan
author: Jie Xiao
author: Ruhan Zhao
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-07-19T00:00:00Z
review text: Suppose \(B^\alpha, \alpha\in (0, \infty); Q_p, p\in (0, \infty)\) and \(D_\alpha, \alpha\in (-\infty, 2)\) are the spaces of all analytic functions \(f\) in the unit disc \(D:| z| <1\), respectively with \(\sup_D (1-| | ^2)^\alpha | f'(z)| <\infty; \sup_{w\in D}\int_D | f'(z)| ^2 (log| \frac{1-\bar w z}{w-z}| )^p dx dy<\infty\) and \(\int_D | f'(z)| ^2 (1-| z| )^{1-\alpha}dx dy<\infty\). Also suppose \(M(D_\alpha)\) is the set of all multipliers of \(D_\alpha\). This paper mainly establishes the following 1. An inclusion between \(B^\alpha\) and \(Q_p\). 2. Some conditions for an analytic function with non-negative Taylor coefficients to be in \(B^\alpha\). 3. Relations between \(B^\alpha\) and \(D_\alpha\). 4. Inclusions between \(B^\beta \) and \(M(D_\alpha)\) and the answer to a question of Brown and Shields on cyclic vectors of the Dirichlet space. 5. A criterion for \(f\in Q_p\) in term of the Möbius transform of \(f\). 6. A sharp version of a result of Baernstein about univalent Bloch functions. 7. The meromorphic case of 1.
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 1031556
zbMATH Keywords: \(\alpha\)-Bloch space
zbMATH Keywords: BMOA
zbMATH Keywords: \(Q_p\) space
zbMATH Keywords: multiplier space
zbMATH Keywords: Dirichlet space
zbMATH Keywords: Bloch functions
reviewed by: Zengjian Lou
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5345
OpenAlex ID: W2045002818
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DOI: 10.1006/JMAA.1997.5345
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description: scientific article; zbMATH DE number 1031556