Complex analysis of L holomorphic functions on pseudoconvex domains of finite type

从奇点理论的观点出发，研究了多元复变函数论中的多种对象。特别是，我们对平方可积的全纯函数的边界行为很感兴趣。具体地说，Bergman核和Szegoe核是非常重要的积分核，它们具有关于全纯函数边界行为的许多重要信息。众所周知，严格伪凸域的情形得到了许多关于Bergman核和Szegoe核的强有力的结果。例如，C.Fefferman的渐近展开完全揭示了它们的边界行为。我们对弱伪凸域的情形很感兴趣。一般情况很难分析，所以我们将自己限制在D‘Angelo意义下的有限类型情况下的对象。从有限类型的定义来看，代数几何和奇点理论的论证是有价值的。我们将牛顿多面体的概念引入到Bergman核的分析中，证明了它的奇异性可以用牛顿多面体的拓扑信息来表示。此外，我们还分析了任意有限类型整环的峰值函数的构造。

We studied many kinds of objects in the function theory of several complex variables from the viewpoints of the theory of singularities. In particular, we are interested in the boundary behavior of the holomorphic functions which are square integrabel. Concretely the Bergman kernel and Szegoe kernel are very important integral kernel and they have many important information of the boundary behavior of holomotphic functions. As is very well known, the case of strictly pseudoconvex domains has many strong results about the Bergman kernel and Szegoe kernel. For example, the asymptotic expansion due to C.Fefferman reveals completely their boundary behaviors. We are interested in the weakly pseudoconvex domains case. The general case is very difficult to analyze and so we restrict ourselves to the objects in the case of finite type in the sense of D'Angelo. From the definition of finite type, the argument from algebraic geometry and singularity theory are valuable. We introduced the concepts of"Newton polyhedra"into the analysis of the Bergman kernel and showed that its singularity can be expressed in terms of the topological information of the Newton polyhedra. Moreover, we analyzed the construction of peak functions of any finite type domains.