Weighted integrals of holomorphic functions on the polydisc II

Abstract

Let ℒαp(Un) denote the class of all measurable functions defined on the unit polydisc Un = {z ε Cn |zi| < 1, i = 1, ..., n} such that ∥f∥ℒαp(Un) = ∫U n |f(z)|p π|zj|2) αjdm(zj) < ∞, where αj > - 1, j = 1,..., n, and dm(zj) is the normalized area measure on the unit disk U, H(Un) the class of all holomorphic functions on Un, and let Aαp(Un) = ℒαp(Un) ∩ H(Un) (the weighted Bergman space). In this paper we prove that for p ε (0, ∞), f ε Aαp(Un) if and only if the functions π(1 - |zj|2) ∂|S|f/π jεS ∂zj (χS (1) z1, χS(2)z2,..., χS(n) zn) belong to the space ℒαp(Un) for every S ⊆ {1, 2,..., n}, where χS(·) is the characteristic function of S, |S| is the cardinal number of S, and πjεS ∂z ji = ∂zji...∂zj|S|, where j k ε S, k = 1,..., |S|. This result extends Theorem 22 of Kehe Zhu in Trans. Amer. Math. Soc. 309 (1988) (1), 253-268, when p ε (0, 1). Also in the case p ε [1, ∞), we present a new proof.