Rational inner functions and their Dirichlet type norms

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}$^n)$\mathrm{.}Furthermore,weshowthatif$1/p \in $\mathcal{D}${\alpha,…,\alpha}$,thentheRIF$\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 $1\mathrm{/}p$ in certain Dirichlet-type spaces.