Rational inner functions and their Dirichlet type norms

TL;DR

This paper investigates the inclusion of rational inner functions in Dirichlet-type spaces on polydisks, establishing new relationships with $H^p$ integrability and providing examples and applications of these theoretical results.

Contribution

It proves a theorem linking Dirichlet space membership to $H^p$ integrability of derivatives and shows all rational inner functions on $ ext{D}^n$ belong to a specific Dirichlet space, with further implications.

Findings

All rational inner functions on $ ext{D}^n$ are in $ ext{D}_{1/n,\

The inclusion of $ ilde{p}/p$ in Dirichlet spaces depends on the space containing $1/p$.

Examples demonstrate applications of the theoretical results and the use of the Lojasiewicz inequality.

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.