Addendum to the paper "a note on weighted Bergman spaces and the Cesàro operator"

Abstract

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let script L signαp,q(Dn), where p,q > 0, α = (α1. .., αnwith αj > -1, j = 1,...,n, be the class of all measurable functions f defined on Dn such that ∫(0,1)nMpq(f, r) Πj=1n (1-rj)αj drj < ∞ where Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on Dn by Aαp,q(Dn) = Lαp,q(D n)∩H(Dn). We provide a characterization for a function f being in Aαp,q(Dn). Using the characterization we prove the following result: Let p > 1, then the Cesàro operator is bounded on the space Aαp,p(Dn).