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

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

Published electronically:
May 7, 2012

MathSciNet review:
2946935

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\langle a|b\rangle \not = 1$ and such that at least one of the numbers $\langle a|c\rangle ,\ \langle b|c\rangle$ 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 $\Delta$.

- 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.
- M. L. Agranovsky: Characterization of polyanalytic functions by meromorphic extensions into chains of circles, J. d’Analyse Math, 113 (2011) 305-329.
- M. L. Agranovsky: Boundary Forelli theorem for the sphere in $\mathbb {C}^n$ and $n+1$ bundles of complex lines, http:/arxiv.org/abs/1003.6125.
- 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**, DOI https://doi.org/10.1007/BF02788781 - M. L. Agranovskiĭ and R. È. Val′skiĭ,
*Maximality of invariant algebras of functions*, Sibirsk. Mat. Ž.**12**(1971), 3–12 (Russian). MR**0285911** - L. Baracco: Holomorphic extension from the sphere to the ball, http://arxiv.org/ abs/0911.2560.
- 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.
- Josip Globevnik,
*Holomorphic extensions from open families of circles*, Trans. Amer. Math. Soc.**355**(2003), no. 5, 1921–1931. MR**1953532**, DOI https://doi.org/10.1090/S0002-9947-03-03241-0 - Josip Globevnik,
*Analyticity of functions analytic on circles*, J. Math. Anal. Appl.**360**(2009), no. 2, 363–368. MR**2561234**, DOI https://doi.org/10.1016/j.jmaa.2009.06.002 - J. Globevnik: Small families of complex lines for testing holomorphic extendibility, To appear in Amer. J Math. http://arxiv.org/abs/0911.5088.
- Kenneth Hoffman,
*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008** - 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**, DOI https://doi.org/10.1134/S0001434608030231 - 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**81952**, DOI https://doi.org/10.2307/1969599 - Hugo Rossi,
*A generalization of a theorem of Hans Lewy*, Proc. Amer. Math. Soc.**19**(1968), 436–440. MR**222327**, DOI https://doi.org/10.1090/S0002-9939-1968-0222327-0 - Walter Rudin,
*Function theory in the unit ball of ${\bf C}^{n}$*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594** - Edgar Lee Stout,
*The boundary values of holomorphic functions of several complex variables*, Duke Math. J.**44**(1977), no. 1, 105–108. MR**437800** - A. Tumanov,
*A Morera type theorem in the strip*, Math. Res. Lett.**11**(2004), no. 1, 23–29. MR**2046196**, DOI https://doi.org/10.4310/MRL.2004.v11.n1.a3 - A. Tumanov,
*Testing analyticity on circles*, Amer. J. Math.**129**(2007), no. 3, 785–790. MR**2325103**, DOI https://doi.org/10.1353/ajm.2007.0015 - M. Tsuji,
*Potential theory in modern function theory*, Maruzen Co., Ltd., Tokyo, 1959. MR**0114894**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
32V25

Retrieve articles in all journals with MSC (2010): 32V25

Additional Information

**Josip Globevnik**

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

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

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.