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 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 one-variable 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 .

**[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) 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 and bundles of complex lines,`http:/arxiv.org/abs/1003.6125`.**[AG]**Mark L. Agranovsky and Josip Globevnik,*Analyticity on circles for rational and real-analytic functions of two real variables*, J. Anal. Math.**91**(2003), 31–65. MR**2037401**, 10.1007/BF02788781**[AV]**M. L. Agranovskiĭ and R. È. Val′skiĭ,*Maximality of invariant algebras of functions*, Sibirsk. Mat. Ž.**12**(1971), 3–12 (Russian). MR**0285911****[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]**Josip Globevnik,*Holomorphic extensions from open families of circles*, Trans. Amer. Math. Soc.**355**(2003), no. 5, 1921–1931 (electronic). MR**1953532**, 10.1090/S0002-9947-03-03241-0**[G2]**Josip Globevnik,*Analyticity of functions analytic on circles*, J. Math. Anal. Appl.**360**(2009), no. 2, 363–368. MR**2561234**, 10.1016/j.jmaa.2009.06.002**[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]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[KM]**A. M. Kytmanov and S. G. Myslivets,*On families of complex lines that are sufficient for holomorphic extension*, Mat. Zametki**83**(2008), no. 4, 545–551 (Russian, with Russian summary); English transl., Math. Notes**83**(2008), no. 3-4, 500–505. MR**2431620**, 10.1134/S0001434608030231**[L]**Hans 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****[R]**Hugo Rossi,*A generalization of a theorem of Hans Lewy*, Proc. Amer. Math. Soc.**19**(1968), 436–440. MR**0222327**, 10.1090/S0002-9939-1968-0222327-0**[Ru]**Walter Rudin,*Function theory in the unit ball of 𝐶ⁿ*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594****[S]**Edgar Lee Stout,*The boundary values of holomorphic functions of several complex variables*, Duke Math. J.**44**(1977), no. 1, 105–108. MR**0437800****[T1]**A. Tumanov,*A Morera type theorem in the strip*, Math. Res. Lett.**11**(2004), no. 1, 23–29. MR**2046196**, 10.4310/MRL.2004.v11.n1.a3**[T2]**A. Tumanov,*Testing analyticity on circles*, Amer. J. Math.**129**(2007), no. 3, 785–790. MR**2325103**, 10.1353/ajm.2007.0015**[Ts]**M. Tsuji,*Potential theory in modern function theory*, Maruzen Co., Ltd., Tokyo, 1959. MR**0114894**

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Additional Information

**Josip Globevnik**

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

Email:
josip.globevnik@fmf.uni-lj.si

DOI:
https://doi.org/10.1090/S0002-9947-2012-05669-8

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.