Schur-Agler class and Carath\'eodory extremal functions

We study the role of Carath\'eodory 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 Carath\'eodory extremals in the unit ball of the multiplier algebra of the Drury-Arveson space and give operator-theoretic Herglotz representations for any Carath\'eodory hyperbolic domain.