Clark measures on the torus

Abstract

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