Holomorphic extension from the sphere to the ball

TL;DR

The paper proves that real analytic functions on the sphere boundary with certain holomorphic extension properties can be extended holomorphically to the entire ball, using an elementary proof based on power expansions.

Contribution

Provides an elementary proof for holomorphic extension from the sphere to the ball, simplifying previous CR geometry arguments.

Findings

Holomorphic extension from boundary to interior established.

Elementary proof based on power series expansion.

Extension criterion based on separate holomorphic extension along complex lines.

Abstract

Real analytic functions on the boundary of the sphere which have separate holomorphic extension along the complex lines through a boundary point have holomorphic extension to the ball. This was proved in a previous preprint by an argument of CR geometry. We give here an elementary proof based on the expansion in holomorphic and antiholomorphic powers.