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

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 $\mathbb{D}$.