Holomorphic extension from the unit sphere in ${\mathbb{C}}^n$ into complex lines passing through a finite set

Abstract

Let $B^n$ be the $n$-dimensional unit complex ball and let $a$ and $b$ be two distinct points in its closure. Let $f$ be a real-analytic function on the complex unit sphere $\partial B^n.$ Suppose that for any complex line $L,$ meeting the two points set $\{a,b\},$ the function $f$ admits one-dimensional holomorphic extension in the cross-section $L\cap B^n.$ Then $f$ is the boundary value of a function holomorphic in $B^n$. Two points can not be replaced by a single point. The proof essentially uses recent result of the author about characterization of polyanalytic functions in the complex plane.

A stronger version of the main result is presented. Some minor corrections are made

Cited by in corpus (1)

• Holomorphic extension from the sphere to the ball