Clark measures on the torus

TL;DR

This paper investigates Clark measures associated with holomorphic functions on the polydisc and introduces related isometric operators on the torus, extending classical one-variable results to higher dimensions.

Contribution

It extends the theory of Clark measures to several complex variables and introduces isometric operators linked to inner functions on the polydisc.

Findings

Characterization of Clark measures on the torus for multivariable functions

Introduction of isometric operators related to model spaces in higher dimensions

Analysis of properties of these operators and measures

Abstract

Let $\mathbb{D}$ denote the unit disc of $\mathbb{C}$ and let $\mathbb{T}= \partial\mathbb{D}$. Given a holomorphic function $\varphi: \mathbb{D}^n \to \mathbb{D}$, $n\ge 2$, we study the corresponding family $\sigma_\alpha[\varphi]$, $\alpha\in\mathbb{T}$, of Clark measures on the torus $\mathbb{T}^n$. If $\varphi$ is an inner function, then we introduce and investigate related isometric operators $T_\alpha$ mapping analogs of model spaces into $L^2(\sigma_\alpha)$, $\alpha\in\mathbb{T}$.