The weighted integral representations of functions in the polydisc and in the space \(\mathbb{C}^ n\)
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 \(\prod_{j=1}^n |z_j|^{\gamma_j} \exp(-\sigma_j|z_j|^{\rho_j})\).
scientific article; zbMATH DE number 1043213