# Clark measures on the complex sphere

## TL;DR

This paper investigates Clark measures associated with holomorphic functions on the complex unit ball, introduces related unitary operators for inner functions, and uses these measures to analyze the essential norm of composition operators.

## Contribution

It introduces a new framework for Clark measures on the complex sphere and characterizes associated unitary operators, extending the understanding of composition operators in several complex variables.

## Key findings

- Explicit characterization of unitary operators related to Clark measures.
- Analysis of pluriharmonic measures via these operators.
- Calculation of the essential norm of composition operators.

## 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 $C_\varphi: H^2(B_1)\to H^2(B_d)$.