The pointwise multipliers of Bloch type space \(\beta^{p}\) and Dirichlet type space \(D_{q}\) on the unit ball of \(\mathbb{C}^{n}\).
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) \(M(D_n, \beta^1) = I_1 = \{\varphi\in H^\infty: \sup_{z\in B} (1-| z| ^2) (\log \frac{1}{1-| z| ^2} )^{1/2} | R\varphi(z)| <\infty\}\).
scientific article; zbMATH DE number 1992552