Clark Measures Associated with Rational Inner Functions on Bounded Symmetric Domains

TL;DR

This paper characterizes Clark measures associated with rational inner functions on bounded symmetric domains, providing explicit formulas and conditions for when the associated Hardy spaces coincide with L^2 spaces.

Contribution

It derives explicit formulas for Clark measures on bounded symmetric domains and characterizes when the Hardy space equals L^2 for these measures, especially on polydiscs.

Findings

Explicit Clark measure formula involving the gradient of the inner function

Characterization of when H^2(μ_α) equals L^2(μ_α) on polydiscs

Necessary and sufficient conditions for general domains

Abstract

Given a bounded symmetric domain $D$ in $\mathbb C^n$, we consider the Clark measures $\mu_\alpha$, $\alpha\in \mathbb T$, associated with a rational inner function $\varphi$ from $D$ into the unit disc in $\mathbb C$. We show that $\mu_\alpha=c|\nabla \varphi|^{-1}\chi_{\mathrm b D \cap \varphi^{-1}(\alpha)}\cdot \mathcal H^{m-1}$, where $m$ is the dimension of the Shilov boundary $\mathrm b D$ of $D$ and $c$ is a suitable constant. Denoting with $H^2(\mu_\alpha)$ the closure of the space of holomorphic polynomials in $L^2(\mu_\alpha)$, we characterize the $\alpha$ for which $H^2(\mu_\alpha)=L^2(\mu_\alpha)$ when $D$ is a polydisc; we also provide some necessary and some sufficient conditions for general domains.