Weighted Integrals of Holomorphic Functions on the Polydisc II

  • Stevo Stevic

    Serbian Academy of Science, Beograd, Serbia


\def\a{\alpha} \def\pt{\partial} Let \( {\cal L}^p_{\a}(U^n) \) denote the class of all measurable functions defined on the unit polydisc such that

\[ \|f\|^p_{{\cal L}_{\a}(U^n)}=\int_{U^n}|f(z)|^p\prod_{j=1}^n (1-|z_j|^2)^{\a_j}dm(z_j)<\infty, \]

where \( \a_j>-1 \), , and is the normalized area measure on the unit disk , the class of all holomorphic functions on , and let \( {\cal A}^p_{\a}(U^n)={\cal L}^p_{\a}(U^n) \cap H(U^n) \) (the weighted Bergman space). In this paper we prove that for \( f\in {\cal A}^p_{\a}(U^n) \) if and only if the functions

\[ \prod_{j\in S}(1-|z_j|^2)\frac{\pt ^{|S|} f} {\prod_{j\in S}\pt z_j}\big(\chi_S(1)z_1, \chi_S(2)z_2,..., \chi_S(n)z_n\big) \]

belong to the space \( {\cal L}^p_{\a}(U^n) \) for every where is the characteristic function of is the cardinal number of and \( \prod_{j\in S}\pt z_j=\pt z_{j_1}\cdots\pt z_{j_{|S|}}, \) where This result extends Theorem 22 of Kehe Zhu in Trans. Amer. Math. Soc. 309 (1988) (1), 253 -- 268, when Also in the case , we present a new proof.

Cite this article

Stevo Stevic, Weighted Integrals of Holomorphic Functions on the Polydisc II. Z. Anal. Anwend. 23 (2004), no. 4, pp. 775–782

DOI 10.4171/ZAA/1222