Rational inner functions and their Dirichlet type norms

Abstract

We study membership of rational inner functions in Dirichlettype 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 D n belong to D 1/n,...,1/n (D n ). Furthermore, we show that if 1/p ∈ Dα,...,α, then the RIF p/p ∈ D α+2/n,...,α+2/n . Finally we illustrate how these results can be applied through several examples, and how the Łojasiewicz inequality can sometimes be applied to determine inclusion of 1/p in certain Dirichlet-type spaces.