Traces of analytic uniform algebras on subvarieties and test collections

Abstract

Given a complex domain $Ω$ and analytic functions $φ_1,\ldots,φ_n : Ω\to \mathbb{D}$, we give geometric conditions for $H^\infty(Ω)$ to be generated by functions of the form $g \circ φ_k$, $g \in H^\infty(\mathbb{D})$. We apply these results to the extension of bounded functions on an analytic one-dimensional complex subvariety of the polydisk $\mathbb{D}^n$ to functions in the Schur-Agler algebra of $\mathbb{D}^n$, with an estimate on the norm of the extension. Our proofs use some extension of the techniques of separation of singularities by Havin, Nersessian and Ortega-Cerdá.