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

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 Aα→p(Dn),α→>-1,p>0, and the Hardy space Hp(Dn),p>0, to a weighted-type space on the unit polydisk.