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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Local and global envelopes of holomorphy of domains in $\textbf {C}^ 2$
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by Eric Bedford PDF
Trans. Amer. Math. Soc. 292 (1985), 665-674 Request permission

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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • 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