Schur-Agler class and Carathéodory extremal functions

We study the role of Carathéodory extremal functions in the Schur-Agler class generated by a collection of test functions. We show that under certain conditions, $\mathbb{D}$ and ${\mathbb{D}}^2$ are the only domains where finitely many test functions can generate the Schur class. As applications, we give a description of the Carathéodory extremals in the unit ball of the multiplier algebra of the Drury-Arveson space and give operator-theoretic Herglotz representations for any Carathéodory hyperbolic domain.