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13.
Small families of complex lines for testing holomorphic extendibility
Globevnik, Josip, 1945-
Let $B$ be the open unit ball in ▫${\Bbb C}^2$▫. 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 ▫$C^\infty (bB)$▫: If ▫$a, b \in \overline B$▫, ▫$a \ne b$▫ and if ▫$f \in C^\infty (bB)$▫ 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 ▫$k \in \Bbb{N}$▫ 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 ▫$B$▫. More generally, in the paper we obtain a fairly complete description of pairs of points ▫$a, b \in {\Bbb C}^2$, $a \ne b$▫, such that if ▫$f \in C^\infty (bB)$▫ extends holomorphically into ▫$B$▫ along every complex line passing through either ▫$a$▫ or ▫$b$▫ that meets ▫$B$▫, then ▫$f$▫ extends holomorphically through ▫$B$▫.
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