Rational inner functions and their Dirichlet type norms

Abstract

We study membership of rational inner functions 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 $\mathcal{D}_{1/n, \ldots ,1/n}(\mathbb{D}^n)$. Furthermore, we show that if $1/p \in \mathcal{D}_{\alpha,...,\alpha}$, then the RIF $\tilde{p}/p \in \mathcal{D}_{\alpha+2/n,...,\alpha+2/n}$. Finally we illustrate how these results can be applied through several examples, and how the Lojasiewicz inequality can sometimes be applied to determine inclusion of $1/p$ in certain Dirichlet-type spaces.