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 $\phi :B_d\to B_1$, we study the corresponding family ${\sigma }_{\alpha }[\phi ]$, $\alpha \in \partial B_1$, of Clark measures on the unit sphere $\partial B_d$. If $\phi$ 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 $U_{\alpha }^{\ast }f$ such that $f{\sigma }_{\alpha }$ is a pluriharmonic measure. Also, for an arbitrary holomorphic $\phi :B_d\to B_1$, we use the family ${\sigma }_{\alpha }[\phi ]$ to compute the essential norm of the composition operator $C_{\phi }:H^2(B_1)\to H^2(B_d)$.

22 pages