Bergman kernel function and canonical domains

本研究的主要目的是研究各种类型的正则偶极子的再生核、性质及其在各种极值问题中的应用。首先，我们利用SIROV边界上的正交系构造了有界完全圆域上的Szego核，并利用Szego核给出了SIROV边界上的L_2-极小问题的解。作为应用，我们得到了Y.Kubota极值定理的合理推广，它是有界对称域上Riemann映射定理的高维推广.其次，利用规范域的性质和我们研究中得到的任意两个有界齐次域之间的广义Schwarz-Pick型引理，我们给出了任意两个经典Cartan域之间的广义Schwarz常数(引理中的最佳可能常数).我们可以证明，对于Cartan域，Yau给出的Schwarz引理(对于Kahler流形)的每个Ahlfors型系数都与我们的Schwarz常数重合，因此是最佳可能的。顺便说一句，我们略微改进了K.H.Look给出的乘积Cartan域的Schwarz常数。

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.