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

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.

56 pages, 2 figures

References in corpus (4)

• Facial behaviour of analytic functions on the bidisk
• Properties of Beurling-Type Submodules via Agler Decompositions
• Integrability and regularity of rational functions
• An inductive Julia-Caratheodory theorem for Pick functions in two variables

Cited by in corpus (2)

• Function theory on quotient domains related to the polydisc
• Stable polynomials and admissible numerators in product domains