Linus Lidman Bergqvist: Rational inner functions and their Dirichlet type norms

Abstract

In this talk, I go through a recent paper in which I study membership of rational inner functions (RIF) in Dirichlet-type spaces in polydisks.

In particular, we prove a theorem relating such inclusions to \(H^p\) integrability of partial derivatives of a RIF, and as a corollary we prove that all rational inner functions on \(\mathbb{D}^n\) belong to \(\mathcal{D}_{1/n, \ldots ,1/n}(\mathbb{D}^n)\).

If time permits, we also show that if \(1/p \in \mathcal{D}_{\alpha,...,\alpha}\), then the RIF \(\tilde{p}/p \in \mathcal{D}_{\alpha+2/n,...,\alpha+2/n}\).