Small families of complex lines for testing holomorphic extendibility

TL;DR

This paper investigates the conditions under which smooth boundary functions in C^∞ extend holomorphically into the unit ball in C^2, revealing that such functions cannot exist with certain properties and characterizing pairs of points for extendibility.

Contribution

It proves the non-existence of certain smooth boundary functions with specified holomorphic extension properties and characterizes pairs of points related to holomorphic extendibility.

Findings

No such functions exist in C^∞ with the specified extension properties.

Complete description of pairs of points for holomorphic extendibility.

Smooth functions in C^∞ cannot have the same extension properties as in finite smoothness.

Abstract

Let B be the open unit ball in C^2 and let a, b be two points in B. It is known that for every positive integer k there is a function f in C^k(bB) which extends holomorphically into B along any complex line passing through either a or b yet f does not extend holomorphically through B. In the paper we show that there is no such function in C^\infty (bB). Moreover, we obtain a fairly complete description of pairs of points a, b in C^2 such that if a function f in C^\infty(bB) extends holomorphically into B along each complex line passing through either a or b that meets B, then f extends holomorphically through B.