Traces of analytic uniform algebras on subvarieties and test collections

Abstract

Given a complex domain $\Omega$ and analytic functions ${\phi }_1,\dots ,{\phi }_n:\Omega \to \mathbb{D}$, we give geometric conditions for $H^{\infty }(\Omega )$ to be generated by functions of the form $g\circ {\phi }_k$, $g\in H^{\infty }(\mathbb{D})$. 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.

27 pages, 3 figures

References in corpus (1)

• Multipliers of embedded discs