Weighted Integrals of Holomorphic Functions on the Polydisc II

Abstract

Let {\mathcal L}^p_{\alpha}(U^n) denote the class of all measurable functions defined on the unit polydisc U^n=\{z\in {\bf C}^n\, \big| \;|z_i|<1,\ i=1,...,n\} such that \|f\|^p_{{\mathcal L}_{\alpha}(U^n)}=\int_{U^n}|f(z)|^p\prod_{j=1}^n (1-|z_j|^2)^{\alpha_j}dm(z_j)<\infty, where \alpha_j>-1 , j=1,...,n , and dm(z_j) is the normalized area measure on the unit disk U , H(U^n) the class of all holomorphic functions on U^n , and let {\mathcal A}^p_{\alpha}(U^n)={\mathcal L}^p_{\alpha}(U^n) \cap H(U^n) (the weighted Bergman space). In this paper we prove that for p\in (0,\infty), f\in {\mathcal A}^p_{\alpha}(U^n) if and only if the functions \prod_{j\in S}(1-|z_j|^2)\frac{\partial ^{|S|} f} {\prod_{j\in S}\partial z_j}\big(\chi_S(1)z_1, \chi_S(2)z_2,..., \chi_S(n)z_n\big) belong to the space {\cal L}^p_{\alpha}(U^n) for every S\subseteq \{1,2,...,n\}, where \chi_S(\cdot) is the characteristic function of S, |S| is the cardinal number of S, and \prod_{j\in S}\partial z_j=\partial z_{j_1}\cdots\partial z_{j_{|S|}}, where j_k\in S, \, k=1,...,|S|. This result extends Theorem 22 of Kehe Zhu in Trans. Amer. Math. Soc. 309 (1988) (1), 253–268, when p\in (0,1). Also in the case p\in [1,\infty) , we present a new proof.