Bergman kernel function and canonical domains

Bergman 核函数和规范域

• 批准号: 61540098
• 负责人: MATSUURA Shozo
• 金额: $ 1.6万
• 依托单位: Nagoya Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1986
• 资助国家: 日本
• 起止时间: 1986 至 1987
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-61540098/
• 关键词: Analytic mapping; Bergman kernel; Szego kernel; HL_2-minimum problem; Canonical domain; Circular domain; シュワルツ定数; 多変数解析関数; 代表領域; 典型カルタン領域; シュワルツのレンマ

The main objects of this research are to study the reproducing kernels, properties of many types of canonical dimains and their applications to various extremal problems. First we construct the Szego kernel on a bounded complete circular domain using of an orthonormal system on its Silov boundary and give the solutions of L_2-minimum problems on the Silov boundary in terms of the Szego kernel. As its application we get a reasonable generalization of Y. Kubota's extremal theorem which is a higher dimensional generalization of the Riemann mapping theorem for bounded symmetric domains.Next, using properties of canonical domains and a generalized Schwarz-Pick type lemma between arbitrary two bounded homogeneous domains obtained in our research, we give the generalized Schwarz constant (the best possible constant in the lemma) between two arbitrary classical Cartan domains. We can show that in the case of Cartan domains each Ahlfors type coefficient of the Schwarz lemma (for Kahler manifolds) given by Yau coincides with our Schwarz constant and so is best possible. Incidentally we slightly ammend thw Schwarz constants of product Cartan domains given by K. H. Look.

本论文的主要目的是研究再生核、各种典型二次模的性质及其在极值问题中的应用。首先利用有界完备圆域上Silov边界上的标准正交系构造了Szego核，并利用Szego核给出了Silov边界上L_2-极小问题的解.作为其应用，我们得到了Y. Kubota极值定理是有界对称域上Riemann映射定理的高维推广.其次，利用典型域的性质和我们所得到的任意两个有界齐次域之间的一个推广的Schwarz-Pick型引理，给出了任意两个经典Cartan域之间的广义施瓦茨常数（引理中的最佳常数）.我们可以证明，在Cartan域的情况下，每个Ahlfors型系数的施瓦茨引理（Kahler流形）由丘与我们的施瓦茨常数重合，所以是最好的可能。顺便对K给出的积Cartan域的施瓦茨常数作了一些修正。H.听着

项目成果

Nobushige Toda: "On the growth of meromorphic solutions of some algebraic differential equations" Tohoku Math. Jour.38. 599-608 (1986)

Nobushige Toda：“关于一些代数微分方程的亚纯解的增长”东北数学。

Kazuhiro Yamamoto: "Singularities of solutions to the boundary value problems for elastic and Maxwell's equation" Japanese Jour. Math.14. (1988)

Kazuhiro Yamamoto：“弹性和麦克斯韦方程边值问题解的奇异性”日本期刊。

Moses Glasner & Mitsuru Nakai: Complex Variables. 6. 81-97 (1986)

摩西·格拉斯纳

Mikihiro Hayashi & Mitsuru Nakai: Pacific Jour.Math.

林干宏

Moses Glasner & Mitsuru Nakai: Israel Jour.Math.55. 15-32 (1986)

摩西·格拉斯纳