Meromorphic extensions from small families of circles and holomorphic extensions from spheres

Abstract:

Let $\mathbb {B}$ be the open unit ball in $\mathbb {C}^2$ and 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 $\langle a|b\rangle \not = 1$ and such that at least one of the numbers $\langle a|c\rangle ,\ \langle b|c\rangle$ is different from $1$. We prove that if a continuous function $f$ on $b\mathbb {B}$ extends holomorphically into $\mathbb {B}$ along each complex line which meets $\{ a, b, c\}$, then $f$ extends holomorphically through $\mathbb {B}$. This generalizes the recent result of L. Baracco who proved such a result in the case when the points $a, b, c$ are contained in $\mathbb {B}$. The proof is quite different from the one of Baracco and uses the following one-variable result, which we also prove in the paper: Let $\Delta$ be the open unit disc in $\mathbb {C}$. Given $\alpha \in \Delta$ let $\mathcal {C}_\alpha$ be the family of all circles in $\Delta$ obtained as the images of circles centered at the origin under an automorphism of $\Delta$ that maps $0$ to $\alpha$. Given $\alpha , \beta \in \Delta ,\ \alpha \not = \beta$, and $n\in \mathbb {N}$, a continuous function $f$ on $\overline {\Delta }$ extends meromorphically from every circle $\Gamma \in \mathcal {C}_\alpha \cup \mathcal {C}_\beta$ through the disc bounded by $\Gamma$ with the only pole at the center of $\Gamma$ of degree not exceeding $n$ if and only if $f$ is of the form $f(z) = a_0(z)+a_1(z)\overline z +\cdots +a_n(z)\overline z^n (z\in \Delta )$ where the functions $a_j, 0\leq j\leq n$, are holomorphic on $\Delta$.