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Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball. (English) Zbl 1186.32004

Let \(H(X)\) be the class of all holomorphic functions on the set \(X\subset\mathbb{C}^{n}\) and \(u\in H(X).\)
The author calculates operator norms of the multiplication operators \(M_{u}(f)=uf,\) on the weighted Bergman space \(A_{\alpha}^{p}(X),\) as well as on the Hardy space \(H^{p}(X),\) where \(X\) is the unit polydisk \(\mathbb{D}^{n}\) or the unit ball \(\mathbb{B}\) in \(\mathbb{C}^{n}.\) The author also calculates the norm of the weighted composition operator from the weighted Bergman space \(A_{\overrightarrow{\alpha}}^{p}(\mathbb{D}^{n}),\) \(\overrightarrow{\alpha}>-1,\) \(p>0,\) and the Hardy space \(H^{p}(\mathbb{D}^{n}),\) \(p>0,\) to a weighted-type space on the unit polydisk.

MSC:

32A36 Bergman spaces of functions in several complex variables
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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