Clark measures on the torus

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}.