The weighted reproducing kernels of the Reinhardt domain

TL;DR

This paper develops a general theory for weighted Bergman spaces on Reinhardt domains, providing explicit formulas for the Bergman kernels in specific cases, advancing understanding of complex analysis in these domains.

Contribution

It introduces a general representation formula for Bergman kernels on Reinhardt domains with weights, including explicit calculations for particular domain classes.

Findings

Derived a general representation formula for weighted Bergman kernels.

Calculated explicit Bergman kernels for specific Reinhardt domains.

Enhanced understanding of weighted spaces in complex analysis.

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 $\mathbb{C}^n,$ $D_{n,m}:= \{(z, w)\in \mathbb{C}^n \times \mathbb{C}^m : \|w\|^2 <{e}^{-\mu_1\|z\|^{\mu_2}}\}$ and $V_{\eta}:=\{(z, z', w) \in \mathbb{C}^{n} \times \mathbb{C}^{m} \times \mathbb{C} : \sum_{j=1}^{n} e^{\eta_{j}|w|^{2}}|z_{j}|^{2}+\|z'\|^{2}<1\}$.