Rational inner functions and their Dirichlet type norms

2020/11/27 by Linus Bergqvist, Bergqvist, Linus · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2011.13651

openalex publication_date 2020/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \mathbbDⁿ belong to D_{1/n, … ,1/n}(\mathbbDⁿ). Furthermore, we show that if 1/p ∈ D_α,...,α, then the RIF p/p ∈ D_{α+2/n,...,α+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/p in certain Dirichlet-type spaces.

Cited by

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

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