Josip Globevnik
Small families of complex lines for testing holomorphic extendibility from spheres
Let $\mathbb B$ be the open unit ball in $\mathbb C^2$. Let $f$ be a continuous function on $b\mathbb B$. If $L$ is a complex line that meets $\mathbb B$ then we say that the function $f$ extends holomorphically into $\mathbb B$ along $L$ if $f|_{L\cap b\mathbb B}$ extends holomorphically through $L\cap\mathbb B$. We consider the question about along how many complex lines should $f$ extend holomorphically into $\mathbb B$ in order that $f$ extends holomorphically through $\mathbb B$. Denote by $\mathcal L(a)$ the set of
all complex lines passing through $a$.
Theorem 1. Let $a,b$ be two points in $\mathbb C^2$ such that the complex line through $a$ and $b$ meets $\mathbb B$ and such that $\langle a|b\rangle\neq1$ if one of the points is contained in $\mathbb B$ and the other in $\mathbb C^2\setminus\overline{\mathbb B}$. If a function $f\in\mathcal C^{\infty}(b\mathbb B)$ extends holomorphically into $\mathbb B$ along each $L\in\mathcal L(a)\cup\mathcal L(b)$ then $f$ extends holomorphically through $\mathbb B$.
When $a,b\in\overline{\mathbb B}$ and when $f$ is real analytic, this theorem was proved
by M. Agranovsky. Such a theorem fails to hold for functions in $\mathcal C^k(b\mathbb B)$.
Theorem 2. Let $a,b,c$ be three points in $\mathbb C^2$ which do not lie in a complex line, such that the complex line through $a,b$ meets $\mathbb B$ and such that if one of the points $a,b$ is in $\mathbb B$ and the other in $\mathbb C^2\setminus\overline{\mathbb B}$ then $\langle a|b\rangle\neq1$ and such that at least one of the numbers $\langle a|c\rangle,\ \langle b|c\rangle$ is different from $1$. If a continuous function $f$ on $b\mathbb B$ extends holomorphically into $\mathbb B$ along each $L\in\mathcal L(a)\cup\mathcal L(b)\cup\mathcal L(c)$ then $f$ extends holomorphically through $\mathbb B$.
In the special case when $a,b,c$ are contained in $\mathbb B$ this theorem was proved by L. Baracco.
In the talk we indicate how to prove these theorems and on the way we present two new one variable results.