A Canonical Characterization of Normal Functions
Peter V Dovbush, Steven G Krantz
TL;DR
The paper provides a canonical characterization of normal holomorphic functions on the unit ball by reducing the problem to their behavior on complex lines through the origin and analytic discs. It develops a framework based on invariant families, the Bergman and Kobayashi metrics, and the spherical derivative to establish precise normality criteria. The central result shows that normality of a function or a family on the ball is equivalent to normality on all complex lines through the origin, with a parallel Hartogs-type theorem proving convergence from line-restrictions. These insights bridge several complex variables, invariant metrics, and classical convergence results, offering a robust radial/line-dimension reduction for normal families. The work has potential implications for function theory in several complex variables and geometric function theory, reinforcing how slice-wise behavior determines global normality.
Abstract
We characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. We then generalize this result to a characterization of normal functions according to behavior on analytic discs. A simple proof of an old theorem of Hartog's that a formal power series at 0 in $\Cn$ is convergent if its restriction to each complex line through the origin is convergent are given.