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Erschienen in:
Introductory Ideas Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

6. Additional Analytic Topics

verfasst von : Steven G. Krantz

Erschienen in: Geometric Analysis of the Bergman Kernel and Metric

Verlag: Springer New York

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Abstract

The concept of “domain of holomorphy” is central to the function theory of several complex variables. The celebrated solution of the Levi problem tells us that a connected open set (a domain) is a domain of holomorphy if and only if it is pseudoconvex. For us, in the present book, pseudoconvexity is Levi pseudoconvexity; this is defined in terms of the positive semi-definiteness of the Levi form.

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Vorheriges Kapitel Further Geometric Explorations
Nächstes Kapitel Curvature of the Bergman Metric
Fußnoten
1
Here the \(\overline{\partial }\)-Neumann operator N is the natural right inverse to the \(\overline{\partial }\)-Laplacian □  \(={ \overline{\partial }}^{{\ast}}\overline{\partial } + \overline{\partial }{\overline{\partial }}^{{\ast}}\).
 
2
At the time that Fefferman wrote his paper, it really was necessary to assume that the entire domain was strictly pseudoconvex in order to get certain global estimates for the \(\overline{\partial }\)-Neumann problem. However, more recent results of Catlin [CAT1, CAT2] and others show that one need only assume that the boundary is strictly pseudoconvex near the boundary point in question. The more global hypotheses can be something considerably weaker—like finite type.
 
3
Thus \(\mathcal{G}\) is the solution operator for the elliptic boundary value problem \(\square (\mathcal{G}f) = f\) on Ω and \(\mathcal{G}f = 0\) on \(\partial \it\Omega \), while \(\mathcal{R}\) is the solution operator for the elliptic boundary value problem \(\square (\mathcal{R}v) = 0\) on Ω and \(\mathcal{R}v = v\) on \(\partial \it\Omega \).
 
4
The characteristic variety of a pseudodifferential operator is the conic subset of the cotangent bundle on which its principal symbol vanishes.
 
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Metadaten
Titel
Additional Analytic Topics
verfasst von
Steven G. Krantz
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Geometric Analysis of the Bergman Kernel and Metric

Print ISBN: 978-1-4614-7923-9

Electronic ISBN: 978-1-4614-7924-6

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-7924-6_6