Traces of analytic uniform algebras on subvarieties and test collections

TL;DR

This paper establishes geometric conditions under which the algebra of bounded analytic functions on a domain can be generated by compositions with certain analytic maps, and applies these to extend functions on subvarieties within the polydisk.

Contribution

It provides new geometric criteria for generating $H^ty(\u03a9)$ via compositions and extends bounded functions from subvarieties to the Schur-Agler algebra with norm estimates.

Findings

Geometric conditions for algebra generation by compositions.

Extension of functions from subvarieties to the Schur-Agler algebra.

Norm estimates for the extensions.

Abstract

Given a complex domain $\Omega$ and analytic functions $\varphi_1,\ldots,\varphi_n : \Omega \to \mathbb{D}$, we give geometric conditions for $H^\infty(\Omega)$ to be generated by functions of the form $g \circ \varphi_k$, $g \in H^\infty(\mathbb{D})$. 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-Cerd\'a.