The pointwise multipliers of Bloch type space \(\beta^{p}\) and Dirichlet type space \(D_{q}\) on the unit ball of \(\mathbb{C}^{n}\).
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| Název: | The pointwise multipliers of Bloch type space \(\beta^{p}\) and Dirichlet type space \(D_{q}\) on the unit ball of \(\mathbb{C}^{n}\). |
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| Autoři: | Zhang, Xuejun |
| Informace o vydavateli: | Elsevier, San Diego, CA |
| Témata: | Dirichlet type space, Bloch type space, pointwise multiplier, Bloch functions, normal functions of several complex variables, unit ball |
| Popis: | 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 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 \(n+2-2q1\), \(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) |
| Druh dokumentu: | Article |
| Popis souboru: | application/xml |
| DOI: | 10.1016/s0022-247x(03)00404-9 |
| Přístupová URL adresa: | https://zbmath.org/1992552 |
| Přístupové číslo: | edsair.c2b0b933574d..251fb0b81484c852f721e90bc97032fa |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | 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 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 \(n+2-2q1\), \(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) |
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| DOI: | 10.1016/s0022-247x(03)00404-9 |