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 ${\mathcal{D}}_{1/n,\dots ,1/n}({\mathbb{D}}^n)$. 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.

20 pages