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

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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.