On the membership of two-variable rational inner functions in spaces of Dirichlet type

We study membership of rational inner functions on the bidisk D2 in a scale of Dirichlet spaces considered by Bera, Chavan, and Ghara, and in higher-order variants of these spaces. We give a characterization for membership in terms of the geometric concept of contact order of a rational inner function at its singular points, and we record some consequences and variants of our main result.