The pointwise multipliers of Bloch type space \(\beta^{p}\) and Dirichlet type space \(D_{q}\) on the unit ball of \(\mathbb{C}^{n}\).
type: Item
instance of: scholarly article
title: The pointwise multipliers of Bloch type space \(\beta^{p}\) and Dirichlet type space \(D_{q}\) on the unit ball of \(\mathbb{C}^{n}\).
zbMATH Open document ID: 1040.32002
DOI: 10.1016/S0022-247X(03)00404-9
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-10-14T00:00:00Z
review text: Let \(X\), \(Y\) be two spaces of holomorphic functions on the unit ball \(B\) of \(\mathbb{C}^n\). We call \(\varphi\) a pointwise multipliers from \(X\) to \(Y\) if \(\varphi f\in Y \) for all \(f\in X\). The collection of all pointwise multipliers from \(X\) to \(Y\) is denoted by \(M(X, Y)\). Let \(f\in H(B)\), the class of all functions holomorphic on \(B\). For \(p\geq 0\), \(f\) is said to be in the Bloch type spaces \(\beta^p\) provided that \( \sup_{z\in B} (1-| z| ^2)^p | Rf(z)| < \infty,\) where \(Rf(z)= \sum_{j=1}^n z_j \frac{\partial f(z)}{\partial z_j}. \) For \(p\in (-\infty, \infty)\), \(f\) is said to be in the Dirichlet type spaces \(D_p\) if and only if \[ \sum_{| \alpha| \geq 0} (n+| \alpha| )^p | b_\alpha| ^2 \omega_\alpha <\infty, \] where \(f(z)= \sum_{| \alpha| \geq 0} b_\alpha z^\alpha\), \(\omega_\alpha = \int_{\partial B} | \xi^\alpha| ^2 \,d \sigma(\xi)\), \(\alpha=(\alpha_1,\dots, \alpha_n)\). The author proves the following Theorem. (1) \(\varphi \in M(D_p, \beta^q)\) if and only if \(\varphi\equiv 0\) for \( 0\leq q < \frac{n+2-p}{2}\) or \( \begin{cases} p=n+2;\\ q=0; \end{cases} \) (2) \(M(D_p, \beta^q) = \beta^q\) for \( \begin{cases} p>n;\\ q\geq \max\{0, (n+2-p)/2\}, \end{cases} \) except \( \begin{cases} p=n+2;\\ q=0; \end{cases} \) (3) \(M(D_p, \beta^q) = \beta^{q-(n-p)/2}\) for \(n+2-2q<p<n\); (4) \(M(D_p, \beta^q) = H^\infty\) for \( \begin{cases} p<n;\\ p+2q=n+2; \end{cases} \) (5) \(q>1\), \(M(D_n, \beta^q) = I_q = \{\varphi\in H(B): \sup_{z\in B} (1-| z| ^2)^q (\text{log}\frac{1}{1-| z| ^2} )^{1/2} | R\varphi(z)| <\infty\}\); (6) \(M(D_n, \beta^1) = I_1 = \{\varphi\in H^\infty: \sup_{z\in B} (1-| z| ^2) (\log \frac{1}{1-| z| ^2} )^{1/2} | R\varphi(z)| <\infty\}\).
Mathematics Subject Classification ID: 32A18
zbMATH DE Number: 1992552
zbMATH Keywords: unit ball
zbMATH Keywords: Bloch type space
zbMATH Keywords: Dirichlet type space
zbMATH Keywords: pointwise multiplier
author: Xuejun Zhang
reviewed by: Zengjian Lou
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1992552