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 $\phi$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $\phi$ 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 $\phi$ 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 ${\mathbb{D}}^d$ results of Agler, McCarthy, Tully-Doyle and Young which generalized to the bidisc the classical Julia-Wolff-Carathéodory theorem about analytic self-maps of $\mathbb{D}$.

75 pages. We have added some proofs and corrected some small mistakes