Rational Inner Functions and their Dirichlet Type Norms

Abstract We study membership of rational inner functions in Dirichlet-type spaces polydiscs. In particular, we prove a theorem relating such inclusions to $$H^p$$ H p integrability partial derivatives an RIF, and as corollary that all on $${\mathbb {D}}^n$$ D n belong $${\mathcal {D}}_{1/n, \ldots ,1/n}({\mathbb {D}}^n)$$ 1 / , … ( ) . Furthermore, show if $$1/p \in {\mathcal {D}}_{\alpha ,\ldots ,\alpha }$$ ∈ α , then the RIF $$\tilde{p}/p +2/n,\ldots +2/n}$$ ~ + 2 Finally illustrate how these results can be applied through several examples, Łojasiewicz inequality sometimes determine inclusion 1/ p certain spaces.