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

摘要

Let H(X) be the class of all holomorphic functions on the set X subset of C-n and u is an element of H(X). We calculate 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 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((alpha) over right arrow)(p)(D-n), (alpha) over right arrow > -1, p > 0, and the Hardy space H-p(D-n), p > 0, to a weighted-type space on the unit polydisk.