Clark measures on the torus

Abstract

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

9 pages