Testing families of complex lines for the unit ball (Q1682120)

Let \(\mathbb B^2\) be the unit ball, and \(l\) a complex line in \(\mathbb C^2\), and let \(f:\partial\mathbb B^2\to C\) be a continuous function. Assume that \(\emptyset\neq l\cap \mathbb B^2=:\mathbb D\). Then \(f\) is said to extend holomorphically to \(l\) (or to \(\mathbb D\)), if there exists a function \(\widetilde f:\mathbb D\to\mathbb C\) holomorphic in \(\mathbb D\) and continuous up to the boundary of \(\mathbb D\) such that \(\widetilde f|\partial\mathbb D=f|\partial\mathbb D\). The main concern of the present paper is how many lines with holomorphic extensions are sufficient to guarantee that \(f\) extends holomorphically to the ball \(\mathbb B^2\). This problem has a long history, we refer to the references given in the paper. The main result of the paper is the following. Theorem 1.1. Let \( P_1\), \(P_2\) be two points in \(\mathbb C^2 \setminus \overline{\mathbb B^2}\) such that the complex line joining them is tangent to the sphere \(\partial \mathbb B^2\). If a real analytic function \(f\) on \(\partial\mathbb B^2\) extends holomorphically to every line through \( P_1\) and every line through \(P_2\), then \(f\) extends holomorphically to \(\mathbb B^2\). The authors also derive the following. Corollary 1.2. Let \(P_1\) and \(P_2\) be two points in \(\mathbb C^2\) such that the complex line joining them has non-empty intersection with the closed ball \(\overline{B^2}\). If a real analytic function \(f\) on \(\partial \mathbb B^2\) extends holomorphically to every line through \(P_1\) and to every line through \(P_2\), then \(f\) extends holomorphically to \(\mathbb B^2\).