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

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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 .

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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.