Orthogonal testing families and holomorphic extension from the sphere to the ball

TL;DR

This paper proves that an analytic function on the sphere in complex two-space, which extends holomorphically in each variable and along lines through a point outside the ball, can be extended holomorphically into the entire ball.

Contribution

It establishes a new extension theorem for functions on the sphere based on separate and line-wise holomorphic extendability.

Findings

Functions extend holomorphically into the ball under specified conditions.

Extension holds when functions are separately holomorphic and along lines through an external point.

Provides a new criterion for holomorphic extension from boundary data.

Abstract

Let $\mathbb{B}^2$ denote the open unit ball in $\mathbb{C}^2$, and let $p\in \mathbb{C}^2\setminus\overline{\mathbb{B}^2}$. We prove that if $f$ is an analytic function on the sphere $\partial\mathbb{B}^2$ that extends holomorphically in each variable separately and along each complex line through $p$, then $f$ is the trace of a holomorphic function in the ball.