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The Bergman Kernel on Certain Decoupled Domains

  • Joe Kamimoto
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)

Abstract

In this paper we study the singularities of the Bergman kernel of a decoupled tube domain of finite type:\({\Omega _m} = \left\{ {z \in {C^{n + 1}};{I_{{z_{n + 1}}}} > \sum\nolimits_{j = 1}^n {{a_j}{{[I{z_j}]}^{2{m_j}}}} } \right\}\)where aj > 0, mj ∈ ℕ and m n ≠ 1. Suppose k 0 is the cardinality of the set {j; m j ≠ 1}. Note that O = (0,..., 0) is a weakly pseudoconvex point. First the singularities of the Bergman kernel at O on the diagonal is essentially expressed by using (k 0 + 1)-variables; moreover the structure of singularities can be understood completely by using at most k 0-times real blowing-ups. Next in order to investigate the singularities of the Bergman kernel off the diagonal, we give an integral representation by using countably many functions whose singularities are understood directly. We also give an analogous result in the case of the Szegö kernel.

Keywords

Integral Representation Finite Type Recursive Formula Pseudoconvex Domain Bergman Kernel 
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Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Joe Kamimoto
    • 1
  1. 1.Department of MathematicsKumamoto UniversityKumamotoJapan

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