Rational Inner Functions and their Dirichlet Type Norms

Abstract

AbstractWe study membership of rational inner functions in Dirichlet-type spaces in polydiscs. In particular, we prove a theorem relating such inclusions to $$H^p$$ H p integrability of partial derivatives of an RIF, and as a corollary we prove that all rational inner functions on $${\mathbb {D}}^n$$ D n belong to $${\mathcal {D}}_{1/n, \ldots ,1/n}({\mathbb {D}}^n)$$ D 1 / n , … , 1 / n ( D n ) . Furthermore, we show that if $$1/p \in {\mathcal {D}}_{\alpha ,\ldots ,\alpha }$$ 1 / p ∈ D α , … , α , then the RIF $$\tilde{p}/p \in {\mathcal {D}}_{\alpha +2/n,\ldots ,\alpha +2/n}$$ 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.