Asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains in $\mathbf{C}^2 $
Asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains in $\mathbf{C}^2 $
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$mathbf{C}^2 $ 中严格伪凸完全 Reinhardt 域的伯格曼核的渐近展开
DOI：
10.3792/pjaa.66.39
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
Noriyuki Nakazawa
中科院分区：
文献类型：
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作者：
Noriyuki Nakazawa
本文讨论了严格拟凸区域上Bergman核的渐近展开式，它是由February [2]得到的。限制自己的类完全Reinhardt域在C中，我们考虑的符号的伪微分算子，它代表的奇异性的Bergman核。我们给出该符号的积分表示（见第1节中的定理2）。通过使用该积分表示，我们确定了费曼渐近展开式的六个系数（参见第1节中的定理1和Γ）。给定C中有界严格伪凸域Ω，其C边界为δΩ，我们考虑z e Ω的Bergman核K（z），它限制在Ω x Ω的对角线上。设λ e（Ω）是Ω的负符号定义函数，在Ω中λ<0的意义上，|θΩ上的梯度λ 0。让我们回忆一下Hδrmander [5]的一个经典结果，
This paper is concerned with the asymptotic expansion, due to Fefferman [2], of the Bergman kernel for a strictly pseudoconvex domain. Restricting ourselves to the class of complete Reinhardt domains in C, we consider the symbol of a pseudodifferential operator which represents the singularity of the Bergman kernel. We give an integral representation of that symbol (see Theorem 2 in Section 1). By using that integral representation, we identify six coefficients of Fefferman's asymptotic expansion (see Theorems 1 and Γ in Section 1). Given a bounded strictly pseudoconvex domain Ω in C with C boundary δΩ, we consider the Bergman kernel K(z) for z e Ω, which is restricted to the diagonal of Ω x Ω. Let λ e C^ίΩ) be a negatively signed defining function of Ω in the sense that λ<0 in Ω and |grada λ\>0 on θΩ. Let us recall a classical result of Hδrmander [5] asserting that