Boundary behavior of functions in the Schur-Agler class of the polydisc

TL;DR

This paper extends boundary behavior analysis of functions in the Schur-Agler class on the polydisc, introducing a generalized model to study boundary singularities and directional derivatives, building on classical theorems.

Contribution

It introduces a generalized Hilbert space model for Schur-Agler functions, enabling detailed boundary behavior analysis at singular points on the polydisc.

Findings

Existence of a generalized model with enhanced boundary continuity.

Proven directional differentiability of Schur-Agler functions at boundary singularities.

Extended classical Julia-Wolff-Carathéodory results to the polydisc setting.

Abstract

We describe a generalization of the notion of a Hilbert space model of a function $\varphi$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $\varphi$ at a mild singularity $\tau$ on the $d$-torus. We prove the existence of a generalized model with an enhanced continuity property at such a singularity $\tau$. We use this result to prove the directional differentiability of a function $\varphi$ in the Schur-Agler class at a singular point on the $d$-torus for which the Carath\'eodory condition holds and to calculate the corresponding directional derivative. The results of this paper extend to the polydisc $\mathbb{D}^d$ results of Agler, McCarthy, Tully-Doyle and Young which generalized to the bidisc the classical Julia-Wolff-Carath\'{e}odory theorem about analytic self-maps of $\mathbb{D}$.