Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of \(\mathbb C^n\) (Q542045)

A remarkable theorem due to Forelli claims that one can check holomorphicity by slices; i.e., if a function \(f\) in a domain \(\Omega \subset\mathbb C^n\) is holomorphic on the intersections \(\Omega\cap L\) of \(\Omega\) with every complex line \(L\) passing through a fixed point \(a \in \Omega\) and \(f\in C^\infty(\Omega)\), then \(f\) is holomorphic near \(a\). Forelli's theorem fails if the expression ``holomorphic function'' is replaced by ``boundary value of a holomorphic function''. In this article, the author proves a boundary analog of Forelli's theorem: Let \(f\in C_\omega{(\partial B^n)}\), where \(B^n\) is the unit ball in \(\mathbb C^n\). If \(a,b \in \overline{B}^n\), \(a\neq b\), such that, for every complex line \(L\) passing through one of the points \(a\) or \(b\), the restricted function \(f\big|_{L\cap \partial B^n}\) has a holomorphic extension to the intersection \(L\cap B^n\), then \(f\) is the boundary value of a holomorphic function in \( B^n\).