Meromorphic extensions from small families of circles and holomorphic extensions from spheres
Author:
Josip Globevnik
Journal:
Trans. Amer. Math. Soc. 364 (2012), 58575880
MSC (2010):
Primary 32V25
Published electronically:
May 7, 2012
MathSciNet review:
2946935
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Abstract: Let be the open unit ball in and let be three points in which do not lie in a complex line, such that the complex line through meets and such that if one of the points is in and the other in then and such that at least one of the numbers is different from . We prove that if a continuous function on extends holomorphically into along each complex line which meets , then extends holomorphically through . This generalizes the recent result of L. Baracco who proved such a result in the case when the points are contained in . The proof is quite different from the one of Baracco and uses the following onevariable result, which we also prove in the paper: Let be the open unit disc in . Given let be the family of all circles in obtained as the images of circles centered at the origin under an automorphism of that maps 0 to . Given , and , a continuous function on extends meromorphically from every circle through the disc bounded by with the only pole at the center of of degree not exceeding if and only if is of the form where the functions , are holomorphic on .
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M. L. Agranovsky: Boundary Forelli theorem for the sphere in and bundles of complex lines, http:/arxiv.org/abs/1003.6125.
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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 , J. d'Analyse Math. 113 (2011) 293304.
 [A2]
 M. L. Agranovsky: Characterization of polyanalytic functions by meromorphic extensions into chains of circles, J. d'Analyse Math, 113 (2011) 305329.
 [A3]
 M. L. Agranovsky: Boundary Forelli theorem for the sphere in and bundles of complex lines, http:/arxiv.org/abs/1003.6125.
 [AG]
 M. L. Agranovsky and J. Globevnik: Analyticity on circles for rational and realanalytic functions of two real variables, J. d'Analyse Math. 91 (2003) 3165. 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, 312. MR 0285911 (44:3128)
 [B1]
 L. Baracco: Holomorphic extension from the sphere to the ball, http://arxiv.org/
abs/0911.2560.
 [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 http://arxiv.org/abs/1003.4705.
 [G1]
 J. Globevnik: Holomorphic extensions from open families of circles, Trans. Amer. Math. Soc. 355 (2003) 19211931. MR 1953532 (2003j:30007)
 [G2]
 J. Globevnik: Analyticity of functions analytic on circles, Jour. Math. Anal. Appl. 360 (2009) 363368. MR 2561234 (2010k:30018)
 [G3]
 J. Globevnik: Small families of complex lines for testing holomorphic extendibility, To appear in Amer. J Math. http://arxiv.org/abs/0911.5088.
 [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) 500505 (translated from Mat. Zametki 83 (2008) 545551. 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) 514522. MR 0081952 (18:473b)
 [R]
 H. Rossi: A generalization of a theorem of Hans Lewy, Proc. Amer. Math. Soc. 19 (1968) 436440. MR 0222327 (36:5379)
 [Ru]
 W. Rudin: Function theory in the unit ball of , Springer, BerlinHeidelbergNew 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) 105108. MR 0437800 (55:10722)
 [T1]
 A. Tumanov: A Morera type theorem in the strip, Math. Res. Lett. 11 (2004) 2329. MR 2046196 (2004k:30004)
 [T2]
 A. Tumanov: Testing analyticity on circles, Amer. J. Math 129 (2007) 785790. 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
Email:
josip.globevnik@fmf.unilj.si
DOI:
http://dx.doi.org/10.1090/S000299472012056698
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.
