The weighted reproducing kernels of the Reinhardt domain

Abstract

In this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domains ${\mathbb{C}}^n,$ $D_{n,m}:=\{(z,w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m:\Vert w{\Vert }^2<e^{-{\mu }_1\Vert z{\Vert }^{{\mu }_2}}\}$ and $V_{\eta }:=\{(z,z^{\prime },w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m\times \mathbb{C}:{\sum }_{j=1}^n{e}^{{\eta }_j|w{|}^2}|z_j{|}^2+\Vert z^{\prime }{\Vert }^2<1\}$.

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