Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball

Abstract

Let H (X) be the class of all holomorphic functions on the set X ⊂ Cn and u ∈ H (X). We calculate operator norms of the multiplication operators Mu (f) = uf, on the weighted Bergman space Aαp (X), as well as on the Hardy space Hp (X), where X is the unit polydisk Dn or the unit ball B in Cn. We also calculate the norm of the weighted composition operator from the weighted Bergman space Aover(α, →)p (Dn), over(α, →) > - 1, p > 0, and the Hardy space Hp (Dn), p > 0, to a weighted-type space on the unit polydisk.