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Small families of complex lines for testing holomorphic extendibility
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 6, December 2012
- pp. 1473-1490
- 10.1353/ajm.2012.0045
- Article
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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
$a, b\in\overline B$, $a\not=b$ 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 $a, b\in{\Bbb C}^2$, $a\not= b$, 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 $B$.