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Transactions of the American Mathematical Society

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



Meromorphic extensions from small families of circles and holomorphic extensions from spheres

Author: Josip Globevnik
Journal: Trans. Amer. Math. Soc. 364 (2012), 5857-5880
MSC (2010): Primary 32V25
Published electronically: May 7, 2012
MathSciNet review: 2946935
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Abstract: Let $ \mathbb{B}$ be the open unit ball in $ \mathbb{C}^2$ and let $ a, b, c$ be three points in $ \mathbb{C}^2$ which do not lie in a complex line, such that the complex line through $ a, b$ meets $ \mathbb{B}$ and such that if one of the points $ a, b$ is in $ \mathbb{B}$ and the other in $ \mathbb{C}^2\setminus \overline {\mathbb{B}}$ then $ \langle a\vert b\rangle \not = 1$ and such that at least one of the numbers $ \langle a\vert c\rangle ,\ \langle b\vert c\rangle $ is different from $ 1$. We prove that if a continuous function $ f$ on $ b\mathbb{B}$ extends holomorphically into $ \mathbb{B}$ along each complex line which meets $ \{ a, b, c\}$, then $ f$ extends holomorphically through $ \mathbb{B}$. This generalizes the recent result of L. Baracco who proved such a result in the case when the points $ a, b, c$ are contained in $ \mathbb{B}$. The proof is quite different from the one of Baracco and uses the following one-variable result, which we also prove in the paper: Let $ \Delta $ be the open unit disc in $ \mathbb{C}$. Given $ \alpha \in \Delta $ let $ \mathcal {C}_\alpha $ be the family of all circles in $ \Delta $ obtained as the images of circles centered at the origin under an automorphism of $ \Delta $ that maps 0 to $ \alpha $. Given $ \alpha , \beta \in \Delta ,\ \alpha \not = \beta $, and $ n\in \mathbb{N}$, a continuous function $ f$ on $ \overline {\Delta }$ extends meromorphically from every circle $ \Gamma \in \mathcal {C}_\alpha \cup \mathcal {C}_\beta $ through the disc bounded by $ \Gamma $ with the only pole at the center of $ \Gamma $ of degree not exceeding $ n$ if and only if $ f$ is of the form $ f(z) = a_0(z)+a_1(z)\overline z +\cdots +a_n(z)\overline z^n (z\in \Delta ) $ where the functions $ a_j, 0\leq j\leq n$, are holomorphic on $ \Delta $.

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Additional Information

Josip Globevnik
Affiliation: Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Ljubljana, Slovenia

Received by editor(s): December 30, 2010
Received by editor(s) in revised form: January 24, 2011
Published electronically: May 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.