SMALL FAMILIES OF COMPLEX LINES FOR TESTING HOLOMORPHIC EXTENDIBILITY

Let B be the open unit ball in ℂ 2 . This paper deals with the analog of Hartogs' separate analyticity theorem for CR functions on the sphere bB. We prove such a theorem for functions in C ∞ (bB): If a, b ∈ B, a ≠ b and if f ∈ C ∞ (bB) extends holomorphically into B along any complex line passi...

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Bibliographic Details
Published in:American journal of mathematics 2012-12, Vol.134 (6), p.1473-1490
Main Author: Globevnik, Josip
Format: Article
Language:English
Subjects:
Analyticity
Automorphisms
Circles
Continuous functions
Geometric lines
Geometry
Linear equations
Mathematical functions
Mathematical problems
Mathematical theorems
Parallel lines
Theorems
Unit ball
Variables
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Summary:Let B be the open unit ball in ℂ 2 . This paper deals with the analog of Hartogs' separate analyticity theorem for CR functions on the sphere bB. We prove such a theorem for functions in C ∞ (bB): If a, b ∈ B, a ≠ b and if f ∈ C ∞ (bB) extends holomorphically into B along any complex line passing through either a or b, then f extends holomorphically through B. On the other hand, for each k ∈ ℕ there is a function f ∈ C k (bB) which extends holomorphically into B along any complex line passing through either a or b yet f does not extend holomorphically through B. More generally, in the paper we obtain a fairly complete description of pairs of points a, b ∈ ℂ 2 , a ≠ b, such that if f ∈ C ∞ (bB) extends holomorphically into B along every complex line passing through either a or b that meets B, then f extends holomorphically through B.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2012.0045