Weighted Integrals of Holomorphic Functions on the Polydisc

Abstract

We show that a holomorphic function on the unit polydisc $U^n$ in ${\mathbf{C}}^n$ belongs to the weighted Bergman space ${\mathcal{A}}_{\alpha }^p(U^n)$, when $p\in (0,1],$ if and only if all weighted derivations of order $|k|$ (with positive orders of derivations) belong to the related weighted Lebesgue space ${\mathcal{L}}_{\alpha }^p(U^n).$ This result extends Theorem 1.8 by Benke and Chang in their recent paper which appeared in Nagoya Math. J. 159 (2000), 25–43.