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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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  • [A1] M. L. Agranovsky: Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of $ C^n$, J. d'Analyse Math. 113 (2011) 293-304.
  • [A2] M. L. Agranovsky: Characterization of polyanalytic functions by meromorphic extensions into chains of circles, J. d'Analyse Math, 113 (2011) 305-329.
  • [A3] M. L. Agranovsky: Boundary Forelli theorem for the sphere in $ \mathbb{C}^n$ and $ n+1$ bundles of complex lines, http:/
  • [AG] M. L. Agranovsky and J. Globevnik: Analyticity on circles for rational and real-analytic functions of two real variables, J. d'Analyse Math. 91 (2003) 31-65. MR 2037401 (2004j:30003)
  • [AV] M. L. Agranovskiĭ and R. E. Val'skiĭ: Maximality of invariant algebras of functions, Sibirsk. Mat. Zh., 12 (1971) No. 1, 3-12. MR 0285911 (44:3128)
  • [B1] L. Baracco: Holomorphic extension from the sphere to the ball,
  • [B2] L. Baracco: Separate holomorphic extension along lines and holomorphic extension of a continuous function from the sphere to the ball: solution of a conjecture by M. Agranovsky
  • [G1] J. Globevnik: Holomorphic extensions from open families of circles, Trans. Amer. Math. Soc. 355 (2003) 1921-1931. MR 1953532 (2003j:30007)
  • [G2] J. Globevnik: Analyticity of functions analytic on circles, Jour. Math. Anal. Appl. 360 (2009) 363-368. MR 2561234 (2010k:30018)
  • [G3] J. Globevnik: Small families of complex lines for testing holomorphic extendibility, To appear in Amer. J Math.
  • [H] K. Hoffman, Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs (N.J.), 1962. MR 0133008 (24:A2844)
  • [KM] A. M. Kytmanov and S. G. Myslivets: On families of complex lines sufficient for holomorphic extension, Mathematical Notes 83 (2008) 500-505 (translated from Mat. Zametki 83 (2008) 545-551. MR 2431620 (2009d:32007)
  • [L] H. Lewy: On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. (2) 64 (1956) 514-522. MR 0081952 (18:473b)
  • [R] H. Rossi: A generalization of a theorem of Hans Lewy, Proc. Amer. Math. Soc. 19 (1968) 436-440. MR 0222327 (36:5379)
  • [Ru] W. Rudin: Function theory in the unit ball of $ C^n$, Springer, Berlin-Heidelberg-New York, 1980. MR 601594 (82i:32002)
  • [S] E. L. Stout: The boundary values of holomorphic functions of several complex variables, Duke Math. J. 44 (1977) 105-108. MR 0437800 (55:10722)
  • [T1] A. Tumanov: A Morera type theorem in the strip, Math. Res. Lett. 11 (2004) 23-29. MR 2046196 (2004k:30004)
  • [T2] A. Tumanov: Testing analyticity on circles, Amer. J. Math 129 (2007) 785-790. MR 2325103 (2008d:30005)
  • [Ts] M. Tsuji: Potential theory in modern function theory, Maruzen, Tokyo, 1959. MR 0114894 (22:5712)

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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.

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