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-Functions in Unbounded Balanced Domains.

Peter Pflug1, Włodzimierz Zwonek2

  • 11Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, Postfach 2503, 26111 Oldenburg, Germany.

Journal of Geometric Analysis
|March 7, 2019
PubMed
Summary
This summary is machine-generated.

We solved problems concerning square integrable holomorphic functions on balanced domains. This work provides criteria for positive Bergman kernels and metrics in pseudoconvex balanced domains.

Keywords:
Balanced domainBergman spaceLelong number

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Area of Science:

  • Complex Analysis
  • Functional Analysis
  • Several Complex Variables

Background:

  • Investigating holomorphic functions on unbounded domains is crucial in complex analysis.
  • Square integrable functions and their properties are fundamental in functional analysis.
  • Balanced domains and their relation to holomorphy present significant theoretical challenges.

Purpose of the Study:

  • To address the existence of square integrable holomorphic functions on unbounded balanced domains.
  • To solve Wiegerinck's problem for two-dimensional balanced domains.
  • To characterize -domains of holomorphy within balanced domains and provide algebraic criteria for square integrability.

Main Methods:

  • Utilizing techniques from functional analysis to study function spaces.
  • Applying methods from several complex variables to analyze properties of domains.
  • Developing algebraic criteria for specific classes of polynomials within pseudoconvex balanced domains.

Main Results:

  • The problem of Wiegerinck is solved for balanced domains in dimension two.
  • A description of -domains of holomorphy is provided for balanced domains.
  • A purely algebraic criterion is established for homogeneous polynomials to be square integrable in pseudoconvex balanced domains.

Conclusions:

  • The findings facilitate the determination of pseudoconvex balanced domains with positive Bergman kernels.
  • This research aids in identifying pseudoconvex balanced domains that admit the Bergman metric.
  • The study contributes to a deeper understanding of holomorphic function spaces on complex domains.