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 ⊂ C n and u ∈ H ( X ) . We calculate operator norms of the multiplication operators M u ( f ) = uf , on the weighted Bergman space A α p ( X ) , as well as on the Hardy space H p ( X ) , where X is the unit polydisk D n or the unit ball B in C n . We also calculate the norm of the weighted composition operator from the weighted Bergman space A α → p ( D n ) , α → > - 1 , p > 0 , and the Hardy space H p ( D n ) , p > 0 , to a weighted-type space on the unit polydisk.