Clark Measures Associated with Rational Inner Functions on Bounded Symmetric Domains
Mattia Calzi
TL;DR
This work extends Clark measure theory to rational inner functions on bounded symmetric domains by deriving an explicit density formula $μ_α=\frac{2π}{c|∇φ|}\chi_{φ_1^{-1}(α)}\,\mathcal{H}^{m-1}$ for the Clark measures on the boundary ${\mathrm{b}D}$ and establishing a disintegration framework $μ_α=\int μ_{α,ξ}\,d\widehat{β}(ξ)$. It then characterizes when the Hardy–type space $H^2(μ_α)$ fills the whole $L^2(μ_α)$ in the polydisc case via a fiber-projection criterion, and extends the analysis to general domains using Jordan triple systems to describe boundary geometry. The paper provides sufficient and necessary conditions for the general equality, including a notable phenomenon where the equality can fail even for irreducible tube domains, and includes explicit structural results for rational inner functions on polydiscs. Overall, the work ties Clark theory to several complex variables and operator theory on bounded symmetric domains, offering concrete criteria and illustrating the nuanced boundary behavior in higher dimensions.
Abstract
Given a bounded symmetric domain $D$ in $\mathbb C^n$, we consider the Clark measures $μ_α$, $α\in \mathbb T$, associated with a rational inner function $\varphi$ from $D$ into the unit disc in $\mathbb C$. We show that $μ_α=c|\nabla \varphi|^{-1}χ_{\mathrm b D \cap \varphi^{-1}(α)}\cdot \mathcal H^{m-1}$, where $m$ is the dimension of the Shilov boundary $\mathrm b D$ of $D$ and $c$ is a suitable constant. Denoting with $H^2(μ_α)$ the closure of the space of holomorphic polynomials in $L^2(μ_α)$, we characterize the $α$ for which $H^2(μ_α)=L^2(μ_α)$ when $D$ is a polydisc; we also provide some necessary and some sufficient conditions for general domains.