Small families of complex lines for testing holomorphic extendibility

2009/11/26 by Josip Globevnik, Globevnik, Josip
Mathematics · #32A40 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:32A40

paper · pdf · doi:10.48550/arxiv.0911.5088

17 pages, an error in the last step of the proof of the main theorem has been corrected

openalex publication_date 2009/11/26 · arxiv created 2009/12/03 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B be the open unit ball in C² and let a, b be two points in B. It is known that for every positive integer k there is a function f in 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. In the paper we show that there is no such function in C^∞ (bB). Moreover, we obtain a fairly complete description of pairs of points a, b in C² such that if a function f in C^∞(bB) extends holomorphically into B along each complex line passing through either a or b that meets B, then f extends holomorphically through B.

Related

• Meromorphic extensions from small families of circles and holomorphic extensions from spheres
• Orthogonal testing families and holomorphic extension from the sphere to the ball
• Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball
• Holomorphic extendability in \mathbf Cⁿ as a rare phenomenon
• Holomorphic mappings between domains in \mbb C²