Weighted integrals of holomorphic functions on the polydisc

Abstract

We show that a holomorphic function on the unit polydisc Un in Cn belongs to the weighted Bergman space Aαp(Un), when p ∈ (0, 1], if and only if all weighted derivations of order |k| (with positive orders of derivations) belong to the related weighted Lebesgue space Lαp(Un). This result extends Theorem 1.8 by Benke and Chang in [Nagoya Math. J. 159 (2000), 25-43].