Clark measures on the complex sphere

Abstract

Let $B_d$ denote the unit ball of $\mathbb{C}^d$, $d\ge 1$. Given a holomorphic function $\varphi: B_d \to B_1$, we study the corresponding family $\sigma_\alpha[\varphi]$, $\alpha\in\partial B_1$, of Clark measures on the unit sphere $\partial B_d$. If $\varphi$ is an inner function, then we introduce and investigate related unitary operators $U_\alpha$ mapping analogs of model spaces onto $L^2(\sigma_\alpha)$, $\alpha\in\partial B_1$. In particular, we explicitly characterize the set of $U_\alpha^* f$ such that $f\sigma_\alpha$ is a pluriharmonic measure. Also, for an arbitrary holomorphic $\varphi: B_d \to B_1$, we use the family $\sigma_\alpha[\varphi]$ to compute the essential norm of the composition operator $C_\varphi: H^2(B_1)\to H^2(B_d)$.