The generalized cesàro operator on the unit polydisk

Abstract

Let Dn = {(z1,..., zn) ∈ C n : |zj| < 1, j = 1, ..., n} be the unit polydisk in Cn. The aim of this paper is to prove the boundedness of the generalized Cesàro operators Cγ→ on H p(Dn) (Hardy) and Aμ→p,q(Dn) (the generalized Bergman) spaces, for 0 < p, q < ∞ and γ→ = (γ1,...,γn) with Re (γj) > 1, j = 1,...,n. Here μ→ = (μ1,...,μn) and each μj is a positive Borel measure on the interval [0,1). Also we present a class of invariant spaces under the action of this operator.