Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Clark measures on the complex sphere   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 $C_\varphi: H^2(B_1)\to
H^2(B_d)$.

28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an