Products of radial derivative and multiplication operators from \(F(p,q,s)\) to weighted-type spaces on the unit ball
type: Item
instance of: scholarly article
title: Products of radial derivative and multiplication operators from \(F(p,q,s)\) to weighted-type spaces on the unit ball
zbMATH Open document ID: 1284.47027
author: Jie Zhou
author: Yongmin Liu
published in: Taiwanese Journal of Mathematics
publication date: 2013-04-25T00:00:00Z
full work available at URL: http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2127
review text: Given \(0<p,s<\infty\), \(-n-1<q<\infty\), and a weight \(\mu\) on \([0,1)\), that is, a positive and continuous function, the authors deal with the weighted-type spaces \[ H^{\infty}_{\mu}(\mathbb{B}):=\{f\in H(\mathbb{B}) \text{ such that } \|f\|:=\sup_{z\in \mathbb{B}}\mu(|z|)|f(z)|<\infty\}, \] \[ H^{\infty}_{\mu,0}(\mathbb{B}):=\{f\in H(\mathbb{B}) \text{ such that } \lim_{|z|\rightarrow 1}\mu(|z|)|f(z)|=0\}, \] and the general function space \[ \begin{multlined} F(p,q,s)(\mathbb{B}):=\{f\in H(\mathbb{B}) \text{ such that } \|f\|^p:\\ =|f(0)|^p+\sup_{a\in \mathbb{B}}\int_{\mathbb{B}}|\mathcal{R}f(z)|^p(1-|z|^2)^qg^s(z,a)\, dV(z)<\infty\},\end{multlined} \] where \(H(\mathbb{B})\) denotes the class of all holomorphic functions on the open unit ball in \(\mathbb{C}^n,\) \(g(z,a)=\log|\varphi_a(z)|^{-1}\) is the Green function for \(\mathbb{B}\) with logarithmic singularity at \(a,\) \(dV\) is the normalized Lebesgue measure on \(\mathbb{C}^n,\) and \(\mathcal{R}\) is the radial derivative of \(f\) at \(z\) defined by \(\mathcal{R}f(z)=\sum_{j=1}^nz_j\frac{\delta f}{\delta z_j}(z)\) for every \(f\in H(\mathbb{B})\). For special parameters \(p, q, s\), we can get many function spaces, such as the BMOA space, the Bergman space, the Hardy space, or the Bloch space. For normal weights, that is, such that there is \(\delta\in [0,1)\) and \(a\) and \(b\), \(0<a<b\), such that \[ \frac{\mu(r)}{(1-r)^a} \text{ is decreasing on } [\delta, 1) \text{ and } \lim_{r\rightarrow 1}\frac{\mu(r)}{(1-r)^a}=0, \] \[ \frac{\mu(r)}{(1-r)^b} \text{ is increasing on } [\delta, 1) \text{ and } \lim_{r\rightarrow 1}\frac{\mu(r)}{(1-r)^b}=\infty, \] the authors obtain a complete characterization of the boundedness and compactness of the products of the radial derivative and the multiplication operator \(\mathcal{R}M_u,\) \(u\) holomorphic, from \(F(p,q,s)\) to the weighted-type spaces of the unit ball in terms of conditions on \(u\) and \(\mu\). Observe that, for every holomorphic function…
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32A18
zbMATH DE Number: 6157596
zbMATH Keywords: unit ball
zbMATH Keywords: general space
zbMATH Keywords: weighted-type space
zbMATH Keywords: radial derivative operator
zbMATH Keywords: multiplication operator
reviewed by: María J. Beltrán
MaRDI profile type: Publication
OpenAlex ID: W1967224183
DOI: 10.11650/TJM.17.2013.2127
description: scientific article; zbMATH DE number 6157596