Testing families of analytic discs in the unit ball

TL;DR

This paper identifies specific families of analytic discs in the unit ball that serve as testing families for continuous functions on the sphere, advancing understanding in complex analysis and geometric function theory.

Contribution

The authors improve previous results by characterizing new families of analytic discs that test continuity on the sphere, solving an open case in the literature.

Findings

Families of lines through three non-collinear points can test continuity on the sphere.

The set of lines concurrent to specific points forms a testing family.

The result extends and refines previous work by the authors and Globevnik.

Abstract

Let $a,b,c\in \mathbb{C}^2$ be three non collinear points such that their mutual joining complex lines do not intersect the unit ball $\mathbb{B}^2$ and such that the line through $a$ and $b$ is tangent to $\mathbb{B}^2$. Then the set of lines concurrent to $a,b$ and $c$ is a testing family for continuous functions on $\mathbb{S}^3$. This improves a result by the authors and solves a case left open in the literature as described by Globevnik.