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

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