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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Local and global envelopes of holomorphy of domains in $\textbf {C}^ 2$


Author: Eric Bedford
Journal: Trans. Amer. Math. Soc. 292 (1985), 665-674
MSC: Primary 32D10; Secondary 32D15
DOI: https://doi.org/10.1090/S0002-9947-1985-0808745-2
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 \[ \Omega (\Phi ) = \{ (z,w) \in {{\mathbf {C}}^2}:\operatorname {Re} w + {r^k}\Phi (\theta ) < 0\} ,\] where $r = |z|$, $\theta = \arg (z)$. 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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Keywords: Envelope of holomorphy, local extendibility, subharmonic minorant, CR extendibility
Article copyright: © Copyright 1985 American Mathematical Society