The weighted integral representations of functions in the polydisc and in the space \(\mathbb{C}^ n\)
type: Item
instance of: scholarly article
title: The weighted integral representations of functions in the polydisc and in the space \(\mathbb{C}^ n\)
zbMATH Open document ID: 0889.32004
author: Albert I. Petrosyan
published in: Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences
publication date: 1998-06-09T00:00:00Z
review text: The author generalizes to \(n\) dimensions two one-dimensional weighted Cauchy integral formulas with remainder due to \textit{M. M. Dzhrbashyan} [J. Contemp. Math. Anal., Armen. Acad. Sci. 28, No. 4, 1-27 (1993; Zbl 0832.30025)]. For functions \(f\) on the polydisc in \(\mathbb{C}^n\) for which \(|f(z)|^p\) is integrable with respect to Lebesgue measure weighted by the factor \(\prod_{j=1}^n (1-|z_j|^{\rho_j})^{\alpha_j} |z_j|^{\gamma_j}\), the author finds explicit integral operators \(P_{\rho,\alpha,\gamma}\) and \(T_{\rho,\alpha,\gamma}\) in terms of Mittag-Leffler functions such that \(f=P_{\rho,\alpha,\gamma}(f) + T_{\rho,\alpha,\gamma}(\overline\partial f)\). The author gives a similar integral representation for functions \(f\) on \(\mathbb{C}^n\) for which \(|f(z)|^p\) is integrable with respect to the weight \(\prod_{j=1}^n |z_j|^{\gamma_j} \exp(-\sigma_j|z_j|^{\rho_j})\).
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 1043213
zbMATH Keywords: Cauchy-Green formula
zbMATH Keywords: weighted integral representations
zbMATH Keywords: polydisc
zbMATH Keywords: space \(\mathbb{C}^ n\)
reviewed by: Harold P. Boas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1043213