Boundary Forelli theorem for the sphere in $\mathbb C^n$ and $n+1$ bundles of complex lines

TL;DR

This paper proves a boundary extension theorem for functions on the sphere in complex n-space, showing that certain holomorphic extension conditions imply the function is holomorphic inside the ball, confirming a prior conjecture.

Contribution

The paper establishes a sharp boundary extension criterion involving affinely independent points and complex line restrictions, confirming a conjecture in several complex variables.

Findings

Boundary value functions extend holomorphically under specified line conditions.

The conditions on points are proven to be sharp.

The result confirms a conjecture from arXiv:0910.3592.

Abstract

Let $B^n$ be the unit ball in $\mathbb C^n$ and let the points $a_1,...,a_{n+1} \in B^n $ are affinely independent. If $f \in C(\partial B^n)$ and for any complex line $L$, containing at least one of the points $a_j$, the restriction $f|_{L \cap \partial B^n}$ extends holomorphically in the disc $L \cap B^n$, then $f$ is the boundary value of a holomorphic function in $B^n$. The condition for the points $a_j$ is sharp. The result confirms a conjecture from the preprint arXiv:0910.3592 by the author.