Boundary behavior of analytic functions of two variables via generalized models

TL;DR

This paper introduces a generalized model for Schur class functions of two variables, enabling analysis of boundary behavior and singularities on the 2-torus, with applications to derivatives and representation theorems.

Contribution

It develops a new generalized Hilbert space model for bidisc Schur functions, facilitating the study of boundary singularities and related function-theoretic problems.

Findings

Existence of a generalized model for boundary singularities.

Characterization of directional derivatives at boundary points.

Representation theorem for two-variable Pick class functions.

Abstract

We describe a generalization of the notion of a Hilbert space model of a function in the Schur class of the bidisc. This generalization is well adapted to the investigation of boundary behavior at a mild singularity of the function on the 2-torus. We prove the existence of a generalized model with certain properties corresponding to such a singularity and use this result to solve two function-theoretic problems. The first of these is to characterise the directional derivatives of a function in the Schur class at a singular point on the torus for which the Carath\'eodory condition holds. The second is to obtain a representation theorem for functions in the two-variable Pick class analogous to the refined Nevanlinna representation of functions in the one-variable Pick class.