Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball

TL;DR

This paper proves that a continuous function on the sphere, which extends holomorphically along lines through three interior points, must be the boundary trace of a holomorphic function in the ball.

Contribution

It confirms a conjecture by Agranovsky, establishing conditions under which separate holomorphic extension implies a global holomorphic extension.

Findings

Positive answer to Agranovsky's conjecture.

Holomorphic extension along lines through three points implies extension to the ball.

Characterization of boundary functions as traces of holomorphic functions.

Abstract

We give positive answer to a conjecture by Agranovsky. A continuous function on the sphere which has separate holomorphic extension along the set of complex lines passing through three non aligned interior points, is the trace of a holomorphic function in the ball.