Authors: Stević, Stevo 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Weighted integrals of holomorphic functions on the polydisc II
Journal: Zeitschrift fur Analysis und ihre Anwendung
Volume: 23
Issue: 4
First page: 775
Last page: 782
Issue Date: 1-Jan-2004
Rank: M23
ISSN: 0232-2064
DOI: 10.4171/ZAA/1222
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.
Keywords: Holomorphic function | Polydisc | Weighted Bergman space
Publisher: European Mathematical Society

Show full item record

SCOPUSTM   
Citations

14
checked on Nov 4, 2024

Page view(s)

13
checked on Nov 4, 2024

Google ScholarTM

Check

Altmetric

Altmetric


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.