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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Local and global envelopes of holomorphy of domains in $ {\bf C}\sp 2$

Author: Eric Bedford
Journal: Trans. Amer. Math. Soc. 292 (1985), 665-674
MSC: Primary 32D10; Secondary 32D15
MathSciNet review: 808745
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Abstract: A criterion is given for a smoothly bounded domain $ D \subset {{\mathbf{C}}^2}$ to be locally extendible to a neighborhood of a point $ {z_0} \in \partial D$. (This result may also be formulated in terms of extension of CR functions on $ \partial D$.) This is related to the envelope of holomorphy of the semitubular domain

$\displaystyle \Omega (\Phi ) = \{ (z,w) \in {{\mathbf{C}}^2}:\operatorname{Re} w + {r^k}\Phi (\theta ) < 0\} ,$

where $ r = \vert z\vert$, $ \theta = \arg (z)$. Necessary and sufficient conditions are given for the envelope of holomorphy of $ \Omega (\Phi )$ to be $ {{\mathbf{C}}^2}$. These conditions are equivalent to the existence of a subharmonic minorant for $ {r^k}\Phi (\theta )$.

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Additional Information

PII: S 0002-9947(1985)0808745-2
Keywords: Envelope of holomorphy, local extendibility, subharmonic minorant, CR extendibility
Article copyright: © Copyright 1985 American Mathematical Society