JournalszaaVol. 23 , No. 4DOI 10.4171/zaa/1222

Weighted Integrals of Holomorphic Functions on the Polydisc II

  • Stevo Stevic

    Serbian Academy of Science, Beograd, Serbia
Weighted Integrals of Holomorphic Functions on the Polydisc II cover

Abstract

\def\a{\alpha} \def\pt{\partial} Let L\ap(Un){\cal L}^p_{\a}(U^n) denote the class of all measurable functions defined on the unit polydisc Un={zCn  zi<1, i=1,...,n}U^n=\{z\in {\bf C}^n\, \big| \;|z_i|<1,\ i=1,...,n\} such that

fL\a(Un)p=Unf(z)pj=1n(1zj2)\ajdm(zj)<,\|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 \aj>1\a_j>-1, j=1,...,nj=1,...,n, and dm(zj)dm(z_j) is the normalized area measure on the unit disk UU, H(Un)H(U^n) the class of all holomorphic functions on UnU^n, and let A\ap(Un)=L\ap(Un)H(Un){\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 p(0,),p\in (0,\infty), fA\ap(Un)f\in {\cal A}^p_{\a}(U^n) if and only if the functions

jS(1zj2)\ptSfjS\ptzj(χS(1)z1,χS(2)z2,...,χS(n)zn)\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 L\ap(Un){\cal L}^p_{\a}(U^n) for every S{1,2,...,n},S\subseteq \{1,2,...,n\}, where χS()\chi_S(\cdot) is the characteristic function of S,S, S|S| is the cardinal number of S,S, and jS\ptzj=\ptzj1\ptzjS,\prod_{j\in S}\pt z_j=\pt z_{j_1}\cdots\pt z_{j_{|S|}}, where jkS,k=1,...,S.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(0,1).p\in (0,1). Also in the case p[1,)p\in [1,\infty), we present a new proof.