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 $\varphi : \mathbb {D}^n \to \mathbb {D}$, $n\ge 2$, 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 $L^2(\sigma _\alpha )$, $\alpha \in \mathbb {T}$.