Abstract

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 $B$.

pdf

Share