THE GENERALIZED CES`ARO OPERATOR ON THE UNIT POLYDISK

Abstract

Let $D_n=\{(z_1,\dots, z_n)\in {\bf C}^n:\, |z_j|\lt 1,\,j=1,\dots,n\}$ be the unit polydisk in ${\bf C}^n$. The aim of this paper is to prove the boundedness of the generalized Ces\` aro operators ${\cal C}^{\vec\gamma}$ on $H^p(D_n)$ (Hardy) and ${\cal A}^{p,q}_{\vec \mu}(D_n)$ (the generalized Bergman) spaces, for $01$, $j=1,\dots,n$. Here $\vec \mu=(\mu_1,\dots,\mu_n)$ and each $\mu_j$ is a positive Borel measure on the interval $[0,1)$. Also we present a class of invariant spaces under the action of this operator.