# Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

## TL;DR

This paper investigates the boundary singularities of rational inner functions on the bidisk, characterizing their derivatives' integrability in Hardy and Dirichlet spaces through contact order and boundary behavior.

## Contribution

It provides a new characterization of derivative space membership for rational inner functions based on contact order and boundary regularity.

## Key findings

- Derivatives of rational inner functions are not in $H^{\mathfrak{p}}$ for $\mathfrak{p} \ge 3/2$.
- Higher boundary regularity reduces derivative integrability.
- Certain rational inner functions do not belong to the unweighted Dirichlet space.

## Abstract

We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative $H^{\mathfrak{p}}$ membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in $H^{\mathfrak{p}}$ for $\mathfrak{p}\ge\frac{3}{2}$ and that higher non-tangential regularity of a rational inner function paradoxically reduces the $H^{\mathfrak{p}}$ integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for $H^{\mathfrak{p}}$. Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of rational inner functions fails to belong to the unweighted Dirichlet space.