On the membership of two-variable Rational Inner Functions in spaces of Dirichlet-type

TL;DR

This paper characterizes when two-variable rational inner functions belong to certain Dirichlet spaces on the bidisk, using geometric contact order at singular points.

Contribution

It provides a new characterization of membership in Dirichlet spaces for rational inner functions based on contact order at singularities.

Findings

Membership characterized by contact order at singular points

Results apply to higher-order Dirichlet space variants

Consequences and variants of the main characterization are discussed

Abstract

We study membership of rational inner functions on the bidisk $\mathbb{D}^2$ 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 further record some consequences and variants of our main result.