Spaces of analytic functions on the disc where the growth of \(M_ p(F,r)\) depends on a weight
type: Item
instance of: scholarly article
title: Spaces of analytic functions on the disc where the growth of \(M_ p(F,r)\) depends on a weight
zbMATH Open document ID: 0712.30043
DOI: 10.1016/0022-247X(90)90372-M
author: Oscar Blasco
author: Geraldo de Soares Souza
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: The authors consider some function spaces of analytic functions on the disc D, depending on a weight \(\rho\) (t), nonnegative and nondecreasing on [0,1). They treat the following spaces (1\(\leq p\leq \infty)\) \[ HL^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad M_ p(F',r)=O(\frac{\rho (1-r)}{1-r})\}, \] \[ Z^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad M_ p(F'',r)=O(\frac{\rho (1-r)}{(1-r)^ 2})\}, \] \[ B^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad \int^{1}_{0}\frac{\rho (1-r)}{1-r}M_ p(F,r)dr<\infty \}, \] \[ J^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad \int^{1}_{0}\rho (1- r)M_ p(F',r)dr<\infty \}, \] where \(M_ p(F,r)=(1/2\pi \int^{\pi}_{-\pi}| F(re^{i\theta}|^ pd\theta)^{1/p}\). They find conditions on \(\rho\) to get results analogous to those by Hardy and Littlewood for \(\rho (t)=t^{\alpha}(0<\alpha <1)\) and by Zygmund for \(\rho (t)=u\). Under certain assumptions they show that the boundary value functions of the above function spaces must satisfy some conditions on the first or second difference, and that the predual of \(HL^ q_{\rho}\) is \(J^ q_{\rho}\) and that of \(Z^ q_{\rho}\) is \(B^ p_{\rho}\).
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 4171315
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4171315