# Clark measures for rational inner functions

Kelly Bickel† Department of Mathematics, Bucknell University, Lewisburg, PA
17837, USA<EMAIL_ADDRESS>,  Joseph A. Cima  Department of
Mathematics, University of North Carolina, Chapel Hill, NC 27599 USA.
<EMAIL_ADDRESS>and  Alan A. Sola  Department of Mathematics, Stockholm
University, 106 91 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

We analyze the fine structure of Clark measures and Clark isometries
associated with two-variable rational inner functions on the bidisk. In the
degree $(n,1)$ case, we give a complete description of supports and weights
for both generic and exceptional Clark measures, characterize when the
associated embedding operators are unitary, and give a formula for those
embedding operators. We also highlight connections between our results and
both the structure of Agler decompositions and study of extreme points for the
set of positive pluriharmonic measures on 2-torus.

###### Key words and phrases:

Clark measure, rational inner function, unitary embedding

###### 2020 Mathematics Subject Classification:

Primary 28A25, 28A35; Secondary 32A08, 47A55

$\dagger$ Research supported in part by National Science Foundation DMS grant
#2000088.

##  1\. Introduction

A bounded analytic function $\phi\colon\mathbb{D}^{d}\to\mathbb{C}$ is said to
be inner if $|\phi(\zeta)|=1$ for almost every $\zeta\in\mathbb{T}^{d}$, where
$\mathbb{D}$ is the unit disk and $\mathbb{T}$ is the unit circle. In the one-
variable case, each inner function $\psi$ defines a class of positive Borel
measures $\\{\sigma_{\alpha}\\}_{\alpha\in\mathbb{T}}$ on $\mathbb{T}$ that
satisfy

$\frac{1-|\psi(z)|^{2}}{|\alpha-\psi(z)|^{2}}=\int_{\mathbb{T}}\frac{1-|z|^{2}}{|\zeta-z|^{2}}d\sigma_{\alpha}(\zeta),\quad\text{
for }z\in\mathbb{D}.$

These measures have a number of important applications and properties; among
other results, they are the spectral representing measures for rank $1$
unitary perturbations of certain compressed shift operators and via
Alexandrov’s theorem, they disintegrate Lebsegue measure, see [11, 16] for
comprehensive introductions to this classical theory. Generalizations of these
measures to the polydisk $\mathbb{D}^{d}$ were recently studied by E. Doubtsov
in [12]; other multivariate generalizations of Clark theory can be found in
[3, 18]. In this paper, we obtain precise information about both the two-
variable Clark measures on the bidisk defined in [12] and associated
isometries, in the setting of two-variable rational inner functions.

###  1.1. Notation and Setup

To define Clark measures on the bidisk, we need some notation. Denote the
Poisson kernel on $\mathbb{D}^{2}$ by

$P_{z}(\zeta)=P(z,\zeta):=\frac{(1-|z_{1}|^{2})(1-|z_{2}|^{2})}{|\zeta_{1}-z_{1}|^{2}|\zeta_{2}-z_{2}|^{2}},\quad\text{
for }z\in\mathbb{D}^{2},\zeta\in\mathbb{T}^{2},$

and the Cauchy kernel for the bidisk by

$C_{w}(z)=C(z,w)=\frac{1}{(1-z_{1}\overline{w}_{1})(1-z_{2}\overline{w}_{2})},\quad\text{
for }z\in\mathbb{D}^{2},w\in\overline{\mathbb{D}}^{2}.$

Recall that $C$ acts as the reproducing kernel for the Hardy space
$H^{2}(\mathbb{D}^{2})$, which consists of all analytic functions
$f\colon\mathbb{D}^{2}\to\mathbb{C}$ satisfying the norm boundedness condition

$\|f\|^{2}_{H^{2}}:=\sup_{0<r<1}\int_{\mathbb{T}^{2}}|f(r\zeta)|^{2}dm_{2}(\zeta)<\infty.$

Here, $m_{2}$ denotes normalized Lebesgue measure on $\mathbb{T}^{2}$ and
later, we will use $m$ to denote normalized Lebesgue measure on $\mathbb{T}$.
For $f\in H^{2}(\mathbb{D}^{2})$ and $\zeta\in\mathbb{T}^{2}$, we let
$f^{*}(\zeta)$ denote the non-tangential value of $f$ at $\zeta$. Recall that
$f^{*}(\zeta)$ exists for a.e. $\zeta\in\mathbb{T}^{2}$, see Chapter XVII,
Theorem 4.8 in [35]. Also, throughout this paper, we will slightly abuse
notation by using $z,\zeta$ to refer to points from both $\mathbb{C}$ and
$\mathbb{C}^{2}$, but the meaning should be clear from the context.

Let $\phi$ be a non-constant inner function on $\mathbb{D}^{2}$ and let
$\alpha\in\mathbb{T}$. Since $z\mapsto\Re[(\alpha+\phi(z))/(\alpha-\phi(z))]$
is a positive pluriharmonic function on the bidisk, there exists a unique
positive Borel measure $\sigma_{\alpha}$ on $\mathbb{T}^{2}$ called a _Clark
measure_ such that

$\Re\left(\frac{\alpha+\phi(z)}{\alpha-\phi(z)}\right)=\frac{1-|\phi(z)|^{2}}{|\alpha-\phi(z)|^{2}}=\int_{\mathbb{T}^{2}}P_{z}(\zeta)d\sigma_{\alpha}(\zeta),$

for all $z\in\mathbb{D}^{2}.$ Observe that

$\int_{\mathbb{T}^{2}}d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}^{2}}P(0,\zeta)d\sigma_{\alpha}(\zeta)=\Re\left(\frac{\alpha+\phi(0)}{\alpha-\phi(0)}\right)=\frac{1-|\phi(0)|^{2}}{|\alpha-\phi(0)|^{2}}<\infty,$

so $\sigma_{\alpha}$ is a finite measure. Since $\sigma_{\alpha}$ is a finite
Borel measure on $\mathbb{T}^{2}$, it is actually a Radon measure and basic
measure theory (see for example [14, Proposition 7.9]) implies that
$C(\mathbb{T}^{2})$ is dense in $L^{2}(\sigma_{\alpha}).$ Furthermore, as
linear combinations of the Poisson kernels $\\{P_{z}\\}_{z\in\mathbb{D}^{2}}$
are dense in $C(\mathbb{T}^{2})$, they are also dense in
$L^{2}(\sigma_{\alpha})$. Finally, as asserted in [12], the support of each
$\sigma_{\alpha}$ should be contained in the closure of the set
$\\{\zeta\in\mathbb{T}^{2}:\phi^{*}(\zeta)=\alpha\\}.$ This support condition
can also be verified directly in the case when $\phi$ is a rational inner
function.

There are close connections between the Clark measures $\sigma_{\alpha}$ and
the model space associated with the function $\phi$ defined by

$K_{\phi}:=H^{2}(\mathbb{D}^{2})\ominus\phi H^{2}(\mathbb{D}^{2}).$

Then the reproducing kernel for $K_{\phi}$ is given by

$k(z,w)=k_{w}(z):=(1-\overline{\phi(w)}\phi(z))C_{w}(z),\text{ for
}z,w\in\mathbb{D}^{2}.$

In [12], Doubtsov defined an embedding map $J_{\alpha}\colon
K_{\phi}\rightarrow L^{2}(\sigma_{\alpha})$ by first specifying it on
reproducing kernels as

$J_{\alpha}[k_{w}](\zeta):=(1-\alpha\overline{\phi(w)})C_{w}(\zeta),\quad\text{for
}w\in\mathbb{D}^{2},\zeta\in\mathbb{T}^{2},$

then showing this definition preserves inner products on linear combinations
of reproducing kernels, and finally extending it to all of $K_{\phi}$ using
density.

In one variable, it is a classical fact (see [11, Chapter 9] or [16, Chapter
11]) that the analogous embedding is in fact a unitary for each inner function
and each $\alpha\in\mathbb{T}$. In the higher-dimensional setting, Doubtsov
[12, Theorem 3.2] shows that $J_{\alpha}$ is a unitary operator if and only if
the bidisk algebra $A(\mathbb{D}^{2})$ is dense in $L^{2}(\sigma_{\alpha}).$
He also gives examples showing that the two-variable embeddings $J_{\alpha}$
can fail to be unitary. In this paper, we investigate this phenomenon as part
of our detailed study of two-variable Clark measures associated with rational
inner functions, i.e. with functions that are both rational and inner on
$\mathbb{D}^{2}$. It is worth noting that we restrict to this two-variable
situtation (rather than a more general $d$-variable setting) because many of
our key tools, which include Agler decompositions, certain model space
properties, and well-understood unimodular level set behaviors, do not extend
to even the three-variable setting, see [10, 20].

To describe the structure of two-variable rational inner functions, or RIFs,
we require some notation. For a polynomial $p\in\mathbb{C}[z_{1},z_{2}]$, let
$\deg_{i}p$ denote the degree of $p$ in $z_{i}$ and set $\deg
p=(\deg_{1}p,\deg_{2}p).$ Then if a pair of nonnegative integers $m$ satifies
$m=(m_{1},m_{2})\geq\deg p$, we can define the $m$-reflection of $p$ as

$\tilde{p}(z)=z_{1}^{m_{1}}z_{2}^{m_{2}}\overline{p\left(1/\bar{z}_{1},1/\bar{z}_{2}\right)}.$

Rudin and Stout [31, 33] showed that each RIF $\phi$ is of the form

$\phi(z)=\gamma\frac{\tilde{p}(z)}{p(z)},$

where $\gamma\in\mathbb{T}$, $p$ has no zeros on $\mathbb{D}^{2}$, $\tilde{p}$
is some $m$-reflection of $p$, and $\tilde{p},p$ share no common factors. We
define $\deg\phi=\deg\tilde{p}\geq\deg p.$ To simplify our notation, we will
assume $\gamma=1$ for the duration of the paper. It is worth noting that, in
the one-variable setting, each RIF is a finite Blaschke product and extends
analytically to some disk containing $\overline{\mathbb{D}}$ in its interior.

Unlike this one-variable situation, two-variable rational inner functions can
possess boundary singularities. For example,

(1) $\phi(z)=\frac{2z_{1}z_{2}-z_{1}-z_{2}}{2-z_{1}-z_{2}}$

has a singularity at $(1,1)\in\mathbb{T}^{2}$. More generally, if
$\phi=\tilde{p}/p$ and $p(\tau)=0$ for some $\tau\in\mathbb{T}^{2}$, then
$\tau$ is a singularity of $\phi$ in the sense that $\phi$ cannot be extended
continuously to a neighborhood of $\tau$; see [29, Corollary 1.7]. Moreover,
if $p(\tau)=0$, then $\tilde{p}(\tau)=0$ and so Bézout’s theorem gives a bound
on the number of zeros $p$ (or equivalently, the number of singularities
$\phi$) can have on $\mathbb{T}^{2}$. Specifically, if
$\deg\tilde{p}=(m_{1},m_{2})$ and $\deg p=(n_{1},n_{2})$, Bézout’s theorem
implies that $p$ and $\tilde{p}$ have exactly $n_{1}m_{2}+n_{2}m_{1}$ common
zeros in $\mathbb{C}_{\infty}\times\mathbb{C}_{\infty}$ counted according to
intersection multiplicity, where $\mathbb{C}_{\infty}$ denotes the Riemann
sphere; see p. 1287 in [21]. If all such common zeros of $\tilde{p}$ and $p$
occur on $\mathbb{T}^{2}$, we say $p$ is $\mathbb{T}^{2}$_-saturated_. As the
intersection multiplicity of such common zeros on $\mathbb{T}^{2}$ must be
even, $\phi$ can have at most $m_{1}m_{2}$ distinct singularities on
$\mathbb{T}^{2}$, see [21].

Still, these RIF singularities are somewhat mild. Indeed, if $\phi$ is a RIF,
then [21, Corollary 14.6] states that for each $\zeta\in\mathbb{T}^{2}$,
including any points $\zeta$ where $p(\zeta)=0$, the non-tangential value
$\phi^{*}(\zeta)$ exists and is unimodular. For more information about the
zero set of $p$, denoted $\mathcal{Z}_{p}$, see [1, 21].

###  1.2. Overview of Results

The body of this paper begins with Section 2, which provides some information
about the Clark measures $\sigma_{\alpha}$ associated to a general RIF $\phi$.
Specifically, Theorem 2.1 gives a simple proof that $\sigma_{\alpha}$ cannot
possess any point-masses (a fact noted earlier in [28]), and the section also
gives further information about the closed set

(2)
$\mathcal{C}_{\alpha}:=\\{\zeta\in\mathbb{T}^{2}\colon\tilde{p}(\zeta)=\alpha
p(\zeta)\\},$

which contains the support of $\sigma_{\alpha}$.

From Section 3 onward, we study RIFs $\phi=\frac{\tilde{p}}{p}$ with
$\deg\phi=(n,1).$ In a sense, these are the simplest two-variable RIFs, but
the constructions from [8, 9] and the examples in our Section 5 show that they
can still be quite complicated. First, note that for these RIFs,

(3) $p(z)=p_{1}(z_{1})+z_{2}p_{2}(z_{1})$

is a polynomial of degree at most $(n,1)$ that does not vanish on
$\mathbb{D}^{2}$,

$\tilde{p}(z):=z_{2}\tilde{p}_{1}(z_{1})+\tilde{p}_{2}(z_{1}),\text{ where
each }\tilde{p}_{i}(z_{1})=z_{1}^{n}\overline{p_{i}(1/\bar{z_{1}})},$

the polynomials $p$, $\tilde{p}$ share no common factors, and $p$ has at most
$n$ distinct zeros on $\mathbb{T}^{2}$. In Subsection 3.1, we recall some
important properties about the model spaces and formulas associated to such
RIFs. For example, such RIFs possess a specific _Agler decomposition_ or _sums
of squares formula_ of the form

(4)
$p(z)\overline{p(w)}-\tilde{p}(z)\overline{\tilde{p}(w)}=(1-z_{1}\bar{w}_{1})\sum_{j=1}^{n}R_{j}(z)\overline{R_{j}(w)}+(1-z_{2}\bar{w}_{2})Q(z)\overline{Q(w)}$

where $R_{1},\dots,R_{n},Q\in\mathbb{C}[z_{1},z_{2}]$, $\deg
R_{j}\leq(n-1,1),$ and $\deg Q\leq(n,0).$

In Subsection 3.2, we study some preliminary objects, which are key in
analyzing both the Clark measures $\sigma_{\alpha}$ and the isometric
operators $J_{\alpha}$ associated to $\phi$. Those objects are detailed in the
following definition:

######  Definition 1.1.

Fix $\phi=\tilde{p}/p$ with $\deg\phi=(n,1)$ and $\alpha\in\mathbb{T}.$ Define
the following:

  * •

The points $(\tau_{1},\lambda_{1}),\dots,(\tau_{m},\lambda_{m})$ are the zeros
of $p$ on $\mathbb{T}^{2}$. Here, $0\leq m\leq n$.

  * •

$B_{\alpha}$ is the rational function

$B_{\alpha}(z):=\frac{\tilde{p}_{1}(z)-\alpha p_{2}(z)}{\alpha
p_{1}(z)-\tilde{p}_{2}(z)},$

where any common factors of the numerator and denominator have been cancelled.

  * •

$E_{\alpha}$ and $L_{k}$ are the sets in $\mathbb{T}^{2}$ defined by
$E_{\alpha}:=\\{(\zeta,\overline{B_{\alpha}(\zeta)}):\zeta\in\mathbb{T}\\}$
and $L_{k}=\\{\tau_{k}\\}\times\mathbb{T}$ for $k=1,\dots,m$.

  * •

$W_{\alpha}$ is the function on $\mathbb{T}$ defined by

$W_{\alpha}(\zeta):=\frac{|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}}{|\tilde{p}_{1}(\zeta)-\alpha
p_{2}(\zeta)|^{2}}.$

Lastly, we say $\alpha\in\mathbb{T}$ is an _exceptional value_ for $\phi$ if
there is a $k$ such that $\phi^{*}(\tau_{k},\lambda_{k})=\alpha$ and
$\alpha\in\mathbb{T}$ is a generic value for $\phi$ otherwise.

Subsection 3.3 contains our first main result, the following complete
characterization of the Clark measures $\sigma_{\alpha}$ associated to a given
degree $(n,1)$ RIF $\phi$:

######  Theorem 1.2.

For $\alpha\in\mathbb{T}$, the Clark measure $\sigma_{\alpha}$ satisfies

$\int_{\mathbb{T}^{2}}f(\zeta)\
d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}}f(\zeta,\overline{B_{\alpha}(\zeta)})\
d\nu_{\alpha}(\zeta)+\sum_{k=1}^{m}c_{k}^{\alpha}\int_{\mathbb{T}}f(\tau_{k},\zeta)\
dm(\zeta)$

for all $f\in L^{1}(\sigma_{\alpha})$, where $d\nu_{\alpha}=W_{\alpha}dm$, the
functions $B_{\alpha},W_{\alpha}$ are from Definition 1.1, and the constants
$c_{k}^{\alpha}$ are nonzero (and positive) if and only if
$\phi^{*}(\tau_{k},\lambda_{k})=\alpha$.

When they are non-zero, the constants $c_{k}^{\alpha}$ can be obtained from
the formula in (17). We should note that although Clark measures can often be
computed in the one-variable case, very is known in the two-variable setting.
Indeed, this theorem can be viewed as a significant generalization of Example
4.3 in [12], which described $\sigma_{\alpha}$ for the specific degree $(1,1)$
RIF given in (1), and of Example 3 in [25], which includes the case where
$\phi(z)=z_{2}b(z_{1})$, for $b$ a finite Blaschke product with
$b(0)\in\mathbb{R}$. Because of its length, the proof of Theorem 1.2 is broken
into two pieces, Propositions 3.8 and 3.9.

In Section 3.4, we prove our other main result, a formula for the isometry
$J_{\alpha}:K_{\phi}\rightarrow L^{2}(\sigma_{\alpha})$ and an exact
characterization of when it is unitary:

######  Theorem 1.3.

Fix $\alpha\in\mathbb{T}$.

  * i.

For each each $f\in K_{\phi}$, $(J_{\alpha}f)(\zeta)=f^{*}(\zeta)\text{ for
}\sigma_{\alpha}\text{-a.e.}\ \zeta\in\mathbb{T}^{2}.$

  * ii.

$J_{\alpha}:K_{\phi}\rightarrow L^{2}(\sigma_{\alpha})$ is unitary if and only
if $\alpha$ is a generic value for $\phi$.

This result is in contrast to the one-variable case, where $J_{\alpha}$ is
always unitary. Here, by computing the non-tangential values of $\phi$ at its
finite number of singularities, this theorem allows us to easily identify
whether a given $J_{\alpha}$ is unitary. Part (i) is true in the one-variable
setting and follows from a famous (and more general) result of Poltoratski
about normalized Cauchy transforms, see [30] and [11, Theorem 10.3.1]. Thus,
our result can be viewed as a partial two-variable analogue of Poltoratski’s
result. Again, due to length, we break the proof into two pieces, Propositions
3.11 and 3.12.

Section 4 connects our $(n,1)$ results to two related areas of study. First,
in Theorem 4.1, we connect Theorem 1.2 to the theory of Agler decompositions
and use our results about $\sigma_{\alpha}$ and $J_{\alpha}$ to establish
formulas for some of the polynomials $R_{j}$ in (4). Then, we observe that
this study of Clark measures can be put into a more general context.
Specifically, let $P(\mathbb{T}^{2})$ denote the set of Borel probability
measures on $\mathbb{T}^{2}$ equipped with the topology of weak-$\star$
convergence and define

$\mathcal{P}_{2}=\\{f\in\text{Hol}(\mathbb{D}^{2}):\Re f(z)>0\text{ and
}f(0,0)=1\\},$

which is compact in the topology of uniform convergence on compact subsets of
$\mathbb{D}^{2}$. Let $M:\mathcal{P}_{2}\rightarrow P(\mathbb{T}^{2})$ denote
the map that takes each $f\in\mathcal{P}_{2}$ to the unique Borel probability
measure $\mu_{f}$ with

$f(z)=\int_{\mathbb{T}^{2}}P_{z}(\zeta)d\mu_{f}(\zeta)\quad\text{ for
}z\in\mathbb{D}^{2}.$

Then both $\mathcal{P}_{2}$ and its image $M(\mathcal{P}_{2})$ are compact
convex sets and by the Krein-Milman theorem, equal the closed, convex hull of
their extreme points. It is also easy to show that $f$ is an extreme point of
$\mathcal{P}_{2}$ if and only if $\mu_{f}$ is an extreme point of
$M(\mathcal{P}_{2})$. In [32], Rudin posed the question

“What are the extreme points of $\mathcal{P}_{2}$ (or equivalently, of
$M(\mathcal{P}_{2})$)?”

While this question is still open, a number of interesting examples and
related results (often in the $n$-variable situation) have been proved by
Forelli [15], Knese [22], and McDonald [25, 26, 27, 28]. As the Clark measures
$\sigma_{\alpha}$ are trivially in $M(\mathcal{P}_{2})$ when $\phi(0)=0$, it
makes sense to consider our investigations in the context of Rudin’s question
and these subsequent results. In particular, the following is a quick
corollary of Theorem 1.2 and a theorem from [22]:

######  Corollary 1.4.

Let $\phi=\frac{\tilde{p}}{p}$ be a degree $(n,1)$ RIF with
$\tilde{p}(0,0)=0$. If $\alpha\in\mathbb{T}$, then:

  * i.

If $\alpha$ is an exceptional value for $\phi$, then $\sigma_{\alpha}$ is not
an extreme point of $M(\mathcal{P}_{2})$.

  * ii.

If $p$ is saturated, $\deg p=\deg\tilde{p}$, and $\alpha$ is generic for
$\phi$, then $\sigma_{\alpha}$ is an extreme point of $M(\mathcal{P}_{2})$.

Part (i) of this corollary shows that in our setting, the more complicated
Clark measures cannot be extreme points of $M(\mathcal{P}_{2})$. Meanwhile
part (ii) coupled with our earlier characterizations provide explicit formulas
for some extreme points of $M(\mathcal{P}_{2})$. We should mention that these
appear somewhat related to the measures studied in [25, Example 3].

In the last section, we use our results to compute the Clark measures and
study the $J_{\alpha}$ isometries associated to several degree $(n,1)$ RIFs.
First, in Example 5.1, we use our results to recover Example 4.3 in [12]. Then
in Example 5.2, we apply our results to a more complicated degree $(2,1)$ RIF
with a single singularity and in Example 5.4, we study a degree $(3,1)$ RIF
with two singularities. Finally, in Example 5.6, we investigate a degree
$(3,3)$ RIF $\phi$ with a singularity at $(1,1)$. In particular, we show that
although $\phi^{*}(1,1)=-1$ and hence $\alpha=-1$ is an exceptional value for
$\phi$, the operator $J_{-1}$ is unitary. This demonstrates that, in its
current form, Theorem 1.3(ii) does not extend to general RIFs.

##  2\. Clark measures for RIFs

We begin with some remarks about Clark measures associated with general RIFs
on the bidisk, before focusing on the degree $(n,1)$ case. First, the
following result appears to be known, see for instance [28, p.732] and [23],
but here, we give a simple proof in the RIF case by adapting some of the
arguments from the one-variable proof of [11, Theorem 9.2.1].

######  Theorem 2.1.

If $\phi$ is a nonconstant RIF on $\mathbb{D}^{2}$ and $\alpha\in\mathbb{T}$,
then $\sigma_{\alpha}$ does not possess any point masses.

###### Proof.

Without loss of generality, we will show that $\sigma_{\alpha}$ does not
possess a point mass at $(1,1)$. First, applying [12, Proposition 2.6] with
$w=(0,0)$ yields

(5)
$\int_{\mathbb{T}^{2}}\frac{1}{(1-z_{1}\bar{\zeta}_{1})(1-z_{2}\bar{\zeta}_{2})}d\sigma_{\alpha}(\zeta)=\frac{1-\phi(z)\overline{\phi(0,0)}}{(1-\bar{\alpha}\phi(z))(1-\alpha\overline{\phi(0,0)})}.$

Observe that $\theta(z):=\phi(z,z)$ is a nonconstant finite Blaschke product.
Then for $0<r<1$, set $z=r$ and multiply both sides of (5) by $(1-r)^{2}$ to
get

(6)
$\int_{\mathbb{T}^{2}}\frac{(1-r)^{2}}{(1-r\bar{\zeta}_{1})(1-r\bar{\zeta}_{2})}d\sigma_{\alpha}(\zeta)=\frac{(1-r)^{2}(1-\theta(r)\overline{\theta(0)})}{(1-\bar{\alpha}\theta(r))(1-\alpha\overline{\theta(0)})}.$

Observe that

$\lim_{r\nearrow
1}\frac{(1-r)^{2}}{(1-r\bar{\zeta}_{1})(1-r\bar{\zeta}_{2})}=\left\\{\begin{array}[]{cc}1&\text{
if }\zeta=(1,1),\\\ 0&\text{otherwise.}\end{array}\right.$

Since $\sigma_{\alpha}$ is a finite measure, the dominated convergence theorem
implies

$\lim_{r\nearrow
1}\int_{\mathbb{T}^{2}}\frac{(1-r)^{2}}{(1-r\bar{\zeta}_{1})(1-r\bar{\zeta}_{2})}d\sigma_{\alpha}(\zeta)=\sigma_{\alpha}\\{(1,1)\\}.$

As $\theta$ is a nonconstant finite Blaschke product, $\theta(1)$ exists and
equals some $\lambda\in\mathbb{T}$ and its derivative $\theta^{\prime}(1)$
exists and is nonzero, see [17, Lemma 7.5]. If $\lambda\neq\alpha$, then

$\lim_{r\nearrow
1}\frac{(1-r)^{2}(1-\theta(r)\overline{\theta(0)})}{(1-\bar{\alpha}\theta(r))(1-\alpha\overline{\theta(0)})}=\lim_{r\nearrow
1}(1-r)^{2}\frac{1-\lambda\overline{\theta(0)}}{(1-\bar{\alpha}\lambda)(1-\alpha\overline{\theta(0)})}=0.$

If $\lambda=\alpha$, then

$\displaystyle\lim_{r\nearrow
1}\frac{(1-r)^{2}(1-\theta(r)\overline{\theta(0)})}{(1-\bar{\alpha}\theta(r))(1-\alpha\overline{\theta(0)})}$
$\displaystyle=\frac{1-\alpha\overline{\theta(0)}}{\bar{\alpha}-\overline{\theta(0)}}\cdot\lim_{r\nearrow
1}\frac{1-r}{\theta(1)-\theta(r)}\cdot\lim_{r\nearrow 1}(1-r)$
$\displaystyle=\alpha\cdot\frac{1}{\theta^{\prime}(1)}\cdot 0=0.$

Equating the two sides in (6) implies that $\sigma_{\alpha}\\{(1,1)\\}=0.$ ∎

We can say a little bit more about Clark measures associated with RIFs. The
papers [8, 9] include several results concerning boundary behavior of two-
variable RIFs, and in particular, the structure of their unimodular level
sets. Recall from (2) that for $\alpha\in\mathbb{T}$ and $\phi=\tilde{p}/p$,

$\mathcal{C}_{\alpha}=\\{\zeta\in\mathbb{T}^{2}\colon\tilde{p}(\zeta)=\alpha
p(\zeta)\\}.$

Then each $\mathcal{C}_{\alpha}$ satisifes
$\mathcal{C}_{\alpha}=\\{\zeta\in\mathbb{T}^{2}\colon\phi^{*}(\zeta)=\alpha\\}\cup(\mathcal{Z}_{p}\cap\mathbb{T}^{2})$,
and as was shown in [9, Theorem 2.8], the components of $\mathcal{C}_{\alpha}$
can be locally parametrized using one-variable analytic functions. Intuitively
speaking, this implies that, for each $\alpha$, the Clark measure
$\sigma_{\alpha}$ of any two-variable RIF has support contained in a one-
dimensional subset of $\mathbb{T}^{2}$. (We should mention that, technically
speaking, the result in [9] was proved for $\phi=\frac{\tilde{p}}{p}$ with
$\deg p=\deg\tilde{p}$, but that assumption does not appear to materially
affect the conclusions.) As an aside, we also note that general pluriharmonic
measures on $\mathbb{T}^{2}$ can have substantially larger, and even two-
dimensional support, viz. [28].

It should be noted that knowing that $\sigma_{\alpha}$ is a Clark measure
associated with some RIF and is supported on some set of the form
$\\{\zeta\in\mathbb{T}^{2}\colon\tilde{p}(\zeta)=\alpha p(\zeta)\\}$ does not
suffice to determine that measure (or its associated RIF) uniquely. Indeed,
one can exhibit (see Example 5.2) two different RIFs
$\phi_{1}=\frac{\tilde{p}_{1}}{p_{1}}$ and
$\phi_{2}=\frac{\tilde{p}_{2}}{p_{2}}$ whose Clark measures are not multiples
of each other but are both supported on the same set
$\\{\zeta\in\mathbb{T}^{2}\colon\tilde{p}_{1}(\zeta)=\alpha
p_{1}(\zeta)\\}=\\{\zeta\in\mathbb{T}^{2}\colon\tilde{p}_{2}(\zeta)=\alpha
p_{2}(\zeta)\\}$ for some $\alpha\in\mathbb{T}$. Thus, in order to study Clark
measures and isometries for two-variable RIFs, we will need to perform a
structural analysis of the measures $\sigma_{\alpha}$ that goes beyond
determining their supports.

##  3\. Clark measures for Degree $(n,1)$ RIFs

Throughout the rest of this paper, we let $\phi=\tilde{p}/p$ denote a fixed
degree $(n,1)$ rational inner function for some $n\geq 1$. Recall that we can
decompose $p$ as in (3). Then the polynomials $p$, $\tilde{p}$ share no common
factors. Moreover, $p$ has no zeros on
$\mathbb{D}^{2}\cup(\mathbb{D}\times\mathbb{T})\cup(\mathbb{T}\times\mathbb{D})$
and at most $n$ distinct zeros on $\mathbb{T}^{2}.$ See for example, Lemma
10.1, the proof of Corollary 13.5, and Appendix $C$ in [21].

######  Remark 3.1.

For a given degree $(n,1)$ RIF function $\phi$ and $\alpha\in\mathbb{T}$,
recall the objects from Definition 1.1 and define the following additional
objects:

  * •

$\nu_{\alpha}$ is the measure on $\mathbb{T}$ defined by
$d\nu_{\alpha}:=W_{\alpha}dm$.

  * •

$Q$ and $R_{1},\dots,R_{n}$ are the polynomials given in Theorem 3.2.

  * •

$b_{1},\dots,b_{n}$ are the rational functions in the disk algebra
$A(\mathbb{D})$ from Proposition 3.6.

  * •

For $f\in K_{\phi}$, $h,g_{1},\dots,g_{n}$ are the $H^{2}(\mathbb{D})$
functions given in Theorem 3.2.

Finally, recall that $\alpha\in\mathbb{T}$ is an _exceptional value_ for
$\phi$ if there is a $k$ such that $\phi^{*}(\tau_{k},\lambda_{k})=\alpha$ and
$\alpha\in\mathbb{T}$ is a _generic value_ for $\phi$ otherwise.

###  3.1. Model Space Preliminaries

Clark measures are closely related to the model space $K_{\phi}$ and so, we
pause to record some known facts about $K_{\phi}$ in the degree $(n,1)$ case.

######  Theorem 3.2.

There are polynomials $Q,R_{1},\dots,R_{n}\in\mathbb{C}[z_{1},z_{2}]$ such
that $\deg R_{j}\leq(n-1,1)$, $\deg Q\leq(n,0)$ and for
$z,w\in\mathbb{C}^{2}$,

(7)
$p(z)\overline{p(w)}-\tilde{p}(z)\overline{\tilde{p}(w)}=(1-z_{1}\bar{w}_{1})\sum_{j=1}^{n}R_{j}(z)\overline{R_{j}(w)}+(1-z_{2}\bar{w}_{2})Q(z)\overline{Q(w)}.$

Furthermore, each $R_{j}$ and $Q$ vanish at each $(\tau_{k},\lambda_{k})$ and
a function $f\in K_{\phi}$ if and only if there exist $g_{1},\dots,g_{n},h\in
H^{2}(\mathbb{D})$ such that

(8)
$f(z)=\tfrac{Q}{p}(z)h(z_{1})+\sum_{j=1}^{n}\tfrac{R_{j}}{p}(z)g_{j}(z_{2})\quad\text{
for }z\in\mathbb{D}^{2}.$

Finally, if $f\in K_{\phi}$ is written as in (8), then

$\|f\|_{K_{\phi}}^{2}=\|f\|^{2}_{H^{2}(\mathbb{D}^{2})}=\|h\|^{2}_{H^{2}(\mathbb{D})}+\sum_{j=1}^{n}\|g_{j}\|^{2}_{H^{2}(\mathbb{D})}.$

###### Proof.

As this result is not new, we just give some intuition and references for the
different components of the theorem. First, note that on
$H^{2}(\mathbb{D}^{2}),$ there are two shift operators, $M_{z_{1}}$ and
$M_{z_{2}}$, defined by $(M_{z_{i}}f)(z)=z_{i}f(z)$ for $i=1,2.$ Let
$S^{\mathrm{max}}_{1}$ be the maximal $M_{z_{1}}$-invariant subspace of
$K_{\phi}$, where $M_{z_{1}}$ is multiplication by $z_{1}$. Then, while not
obvious, it is true that $S^{\mathrm{min}}_{2}:=K_{\phi}\ominus
S^{\mathrm{max}}_{1}$ is invariant under $M_{z_{2}}$, see [4, 5]. Let
$K_{1},K_{2}$ denote the reproducing kernels of the two Hilbert spaces

$S^{\mathrm{max}}_{1}\ominus
M_{z_{1}}S^{\mathrm{max}}_{1}:=\mathcal{H}(K_{1})\ \ \text{ and }\ \
S^{\mathrm{min}}_{2}\ominus
M_{z_{2}}S^{\mathrm{min}}_{2}:=\mathcal{H}(K_{2}),$

respectively. This yields the Agler decomposition

$1-\phi(z)\overline{\phi(w)}=(1-z_{1}\bar{w}_{1})K_{2}(z,w)+(1-z_{2}\bar{w}_{2})K_{1}(z,w).$

Since $\phi$ is a degree $(n,1)$ RIF, one can show that
$\dim\mathcal{H}(K_{1})=1$ and $\dim\mathcal{H}(K_{2})=n$. Let $Q/p$ be an
orthonormal basis for $\mathcal{H}(K_{1})$ and $R_{1}/p,\dots,R_{n}/p$ be an
orthonormal basis for $\mathcal{H}(K_{2})$. One can show that the $Q,R_{j}$
are polynomials with $\deg Q\leq(n,0)$ and $\deg R_{j}\leq(n-1,1)$ and each
$Q,R_{j}$ vanishes at each $(\tau_{k},\lambda_{k})$. Furthermore,

$K_{1}(z,w)=\frac{Q(z)\overline{Q(w)}}{p(z)\overline{p(w)}}\text{ and
}K_{2}(z,w)=\sum_{j=1}^{n}\frac{R_{j}(z)\overline{R_{j}(w)}}{p(z)\overline{p(w)}},$

and substituting the formulas into the Agler decomposition and multiplying
through by the denominator gives (7). For the details, see for example [4, 6,
19] and the references within.

The characterization of functions in $K_{\phi}$ from (8) follows from the fact
that the reproducing kernel $k(z,w)$ of $K_{\phi}$ satisfies

$k(z,w)=\frac{1}{1-z_{2}\bar{w}_{2}}\sum_{j=1}^{n}\frac{R_{j}(z)\overline{R_{j}(w)}}{p(z)\overline{p(w)}}+\frac{1}{1-z_{1}\bar{w}_{1}}\frac{Q(z)\overline{Q(w)}}{p(z)\overline{p(w)}}$

and from standard properties of reproducing kernels. A proof of the formula
for the norm of functions in $K_{\phi}$ can be found, for example, in Remark
2.3 in [7]. ∎

###  3.2. Support Sets and Consequences

In this subsection, we obtain some information about the objects from
Definition 1.1 and Remark 3.1. First, recall that $\mathcal{C}_{\alpha}$ from
(2) contains the support of $\sigma_{\alpha}.$ This theorem gives additional
information about $\mathcal{C}_{\alpha}$ and $B_{\alpha}$.

######  Theorem 3.3.

Let $\alpha\in\mathbb{T}$.

  * i.

Then $B_{\alpha}$ is a finite Blaschke product.

  * ii.

Let $\alpha$ be a generic value of $\phi$. Then $\deg B_{\alpha}=n$ and
$\mathcal{C}_{\alpha}$ equals $E_{\alpha}$.

  * iii.

Let $\alpha$ be an exceptional value of $\phi$ and (after reordering if
necessary), assume $\phi^{*}(\tau_{k},\lambda_{k})=\alpha$ for
$k=1,\dots,\ell$. Then $\deg B_{\alpha}=n-\ell$ and $\mathcal{C}_{\alpha}$ is
$E_{\alpha}\cup(\cup_{k=1}^{\ell}L_{k})$.

###### Proof.

Fix $\alpha\in\mathbb{T}$ and observe that $\mathcal{C}_{\alpha}$ is the set
of $\zeta\in\mathbb{T}^{2}$ satisfying

$\displaystyle 0=\tilde{p}(\zeta)-\alpha p(\zeta)$
$\displaystyle=\zeta_{2}(\tilde{p}_{1}-\alpha
p_{2})(\zeta_{1})+(\tilde{p}_{2}-\alpha p_{1})(\zeta_{1})$
$\displaystyle=(\tilde{p}_{1}-\alpha
p_{2})(\zeta_{1})\left(\zeta_{2}-1/B_{\alpha}(\zeta_{1})\right).$

Too see that $B_{\alpha}$ is a finite Blaschke product, observe that if

$r=\alpha p_{1}-\tilde{p}_{2},\text{ then
}\tilde{r}=\bar{\alpha}\tilde{p_{1}}-p_{2}.$

Since $|r|=|\tilde{r}|$ on $\mathbb{T}$, this implies $|B_{\alpha}(\zeta)|=1$
on $\mathbb{T}$. As $\phi$ is nonconstant,
$|\phi(z_{1},0)|=|\frac{\tilde{p}_{2}}{p_{1}}(z_{1})|<1$ on $\mathbb{D}$ and
so, $\alpha p_{1}-\tilde{p}_{2}$ is nonvanishing on $\mathbb{D}$. This implies
$B_{\alpha}$ is a finite Blaschke product and the common zeros of its original
numerator and denominator are exactly the zeros of $\tilde{p}_{1}-\alpha
p_{2}$ in $\mathbb{T}$. Denote those zeros by
$\gamma_{1},\dots,\gamma_{\ell}$. Then

(9) $\tilde{p}(z)-\alpha
p(z)=q(z_{1})\left(z_{2}-1/B_{\alpha}(z_{1})\right)\prod_{k=1}^{\ell}(z_{1}-\gamma_{k}),$

where $q\in\mathbb{C}[z]$ is the numerator of $B_{\alpha}$ once common terms
have been cancelled. Equation (9) shows that $\mathcal{C}_{\alpha}$ is the set
of lines $\\{\gamma_{k}\\}\times\mathbb{T}$ for $k=1,\dots,\ell$ and
$E_{\alpha}.$ By Lemma 3.5(i), we can further assume that
$\gamma_{1},\dots,\gamma_{\ell}$ are distinct.

To establish $\deg B_{\alpha}$, note that $\deg(\tilde{p}_{1}-\alpha
p_{2})=n$. This occurs because since $|\phi|<1$ on $\mathbb{D}^{2},$ we have
$|p_{1}(0)|>|\tilde{p}_{2}(0)|$ and thus, the coefficient of the degree $n$
term in $\tilde{p}_{1}-\alpha p_{2}$ is nonzero. Thus, $\deg
B_{\alpha}=n-\ell$, its original degree minus the number of cancelled terms or
equivalently, the number of lines of the form $\\{\gamma\\}\times\mathbb{T}$
in $\mathcal{C}_{\alpha}$.

For (ii), let $\alpha$ be generic. By Lemma 3.4 below, $\mathcal{C}_{\alpha}$
cannot contain any lines of the form $\\{\gamma\\}\times\mathbb{T}$ and so,
our arguments give $\mathcal{C}_{\alpha}=E_{\alpha}$ and $\deg B_{\alpha}=n$.

For (iii), let $\alpha$ be exceptional and (after reordering if necessary),
assume $\phi^{*}(\tau_{k},\lambda_{k})=\alpha$ for $k=1,\dots,\ell$. Then
Lemma 3.4 implies that $L_{1},\dots,L_{\ell}$ are exactly the lines of the
form $\\{\gamma\\}\times\mathbb{T}$ in $\mathcal{C}_{\alpha}$. Then the above
arguments imply $\mathcal{C}_{\alpha}=E_{\alpha}\cup(\cup_{k=1}^{\ell}L_{k})$
and $\deg B_{\alpha}=n-\ell.$ ∎

In the above proof, we used the following two lemmas.

######  Lemma 3.4.

For $\gamma\in\mathbb{T}$, the set $\mathcal{C}_{\alpha}$ contains
$\\{\gamma\\}\times\mathbb{T}$ if and only if $\gamma=\tau_{k}$ for some $k$
and $\phi^{*}(\tau_{k},\lambda_{k})=\alpha.$

###### Proof.

Observe that $\\{\gamma\\}\times\mathbb{T}\subseteq\mathcal{C}_{\alpha}$ if
and only if $\phi^{*}(\gamma,\zeta_{2})\equiv\alpha$ for all
$\zeta_{2}\in\mathbb{T}$, except maybe at one $\zeta_{2}$ where
$p(\gamma,\zeta_{2})=0.$

Thus, for the forward direction, we can assume
$\phi^{*}(\gamma,\zeta_{2})\equiv\alpha$ (except maybe at one $\zeta_{2})$. As
our assumptions imply $\deg\tilde{p}(\gamma,\zeta_{2})=1$, the polynomials
$p(\gamma,\cdot),\tilde{p}(\gamma,\cdot)$ must share a common factor with a
zero on $\mathbb{T}$, say $(z_{2}-\beta).$ This implies that
$p(\gamma,\beta)=\tilde{p}(\gamma,\beta)=0$. Thus,
$(\gamma,\beta)=(\tau_{k},\lambda_{k})$ for some $k.$

Set $\tau:=(\tau_{k},\lambda_{k})$ and write

$p(z)=\sum_{j=M}^{n+1}P_{j}(\tau-z)\ \text{ and
}\tilde{p}(z)=\sum_{j=M}^{n+1}Q_{j}(\tau-z),$

where $P_{j}$ and $Q_{j}$ are homogeneous polynomials in $z_{1},z_{2}$ of
degree $j$ and $M\geq 1$. Define $\lambda=\phi^{*}(\tau)$. By [21,
Propositions 14.3, 14.5], $P_{M}=\lambda Q_{M}$ and since $\tilde{p},p$ share
no common factors, we can conclude that $M=1$ and $P_{1}$ contains a term
$cz_{2}$ with $c\neq 0$. Then

$\alpha\equiv\phi(\tau_{k},\zeta_{2})=\frac{\lambda
c(\lambda_{k}-\zeta_{2})}{c(\lambda_{k}-\zeta_{2})}=\lambda,$

so $\alpha=\phi^{*}(\tau).$ Similarly, if $\alpha=\phi^{*}(\tau)$ for some
$\tau=(\tau_{k},\lambda_{k})$, the above equality and arguments imply that we
also have $\alpha\equiv\phi^{*}(\tau_{k},\zeta_{2})$ for all
$\zeta_{2}\in\mathbb{T}\setminus\\{\lambda_{k}\\}$ and so
$\\{\tau_{k}\\}\times\mathbb{T}$ is in $\mathcal{C}_{\alpha}$. ∎

The following lemma describes finer behavior of $B_{\alpha}$ and $\phi$
related to the singularities $(\tau_{k},\lambda_{k})$ for $k=1,\dots,m$.

######  Lemma 3.5.

For $\alpha\in\mathbb{T}$,

  * i.

$\tilde{p}-\alpha p$ and $\tilde{p}_{1}-\alpha p_{2}$ do not possess repeated
linear factors $(z_{1}-\gamma)^{2}$ with $\gamma\in\mathbb{T}.$

  * ii.

$\overline{B_{\alpha}(\tau_{k})}=\lambda_{k}$ for $k=1,\dots,m$.

  * iii.

If $\mathcal{C}_{\alpha}$ contains $L_{k}$, then for all
$z_{2}\in\overline{\mathbb{D}}$ with $z_{2}\neq\lambda_{k}$,
$\frac{\partial\phi}{\partial z_{1}}(\tau_{k},z_{2})$ equals a fixed nonzero
constant $C$.

###### Proof.

By the factorization in (9), it is easy to see that for $k\in\mathbb{N}$,
$(z_{1}-\gamma)^{k}$ is a factor of $\tilde{p}-\alpha p$ if and only if it is
a factor of $\tilde{p}_{1}-\alpha p_{2}.$ Then, by way of contradiction,
assume $\tilde{p}-\alpha p$ is divisible by $(z_{1}-\gamma)^{2}$. Then,
$\gamma=\tau_{k}$ for some $k$ and for
$\zeta_{2}\in\mathbb{T}\setminus\\{\lambda_{k}\\}$, we have
$\phi(\gamma,\zeta_{2})\equiv\alpha$ and

$\tfrac{\partial\phi}{\partial
z_{1}}(\gamma,\zeta_{2})=\frac{p\frac{\partial\tilde{p}}{\partial
z_{1}}-\tilde{p}\frac{\partial p}{\partial
z_{1}}}{p^{2}}(\gamma,\zeta_{2})=\frac{\frac{\partial\tilde{p}}{\partial
z_{1}}-\alpha\frac{\partial p}{\partial z_{1}}}{p}(\gamma,\zeta_{2})=0.$

Fix any $\lambda\in\mathbb{T}$ with
$\lambda\not\in\\{\lambda_{1},\dots,\lambda_{m}\\}$. Then
$\phi_{\lambda}(z_{1}):=\phi(z_{1},\lambda)$ is a nonconstant finite Blaschke
product and so, $\phi^{\prime}_{\lambda}(\gamma)\neq 0$. See, for example,
Lemma 7.5 in [17]. Since

$0\neq\phi^{\prime}_{\lambda}(\gamma)=\tfrac{\partial\phi}{\partial
z_{1}}(\gamma,\lambda)=0$

by the above argument, we obtain the requisite contradiction.

For (ii), let $\tau:=(\tau_{k},\lambda_{k})$. First assume that
$\phi^{*}(\tau)\neq\alpha$. As $\tilde{p}(\tau)=p(\tau)=0$, it follows by
definition that $\tau\in\mathcal{C}_{\alpha}$. Thus, Theorem 3.3 implies
$\lambda_{k}=\overline{B_{\alpha}(\tau_{k})}.$ Now if $\phi^{*}(\tau)=\alpha$,
then as in the proof of Lemma 3.4, we can write:

$(p-\alpha\tilde{p})(z)=\sum_{j=2}^{n+1}(P_{j}-\alpha
Q_{j})(z-\tau)=(\tau_{k}-z_{1})G(z),$

where $Q_{j},P_{j}$ are homogeneous polynomials of degree $j$ and $G$ is a
polynomial. Here, we used the fact that $P_{1}=\alpha Q_{1}$ and
$\deg(p-\alpha\tilde{p})\leq(n,1)$. From this, it is clear that $G(\tau)=0.$
Using the proof of Theorem 3.3, we know $(z_{1}-\tau_{k})$ divides
$\tilde{p}_{1}-\alpha p_{2}$ and so,

$(\tau_{k}-z_{1})G(z)=(\tilde{p}_{1}-\alpha
p_{2})(z_{1})\left(z_{2}-1/B_{\alpha}(z_{1})\right)=r(z_{1})(\tau_{k}-z_{1})\left(z_{2}-1/B_{\alpha}(z_{1})\right).$

for some $r\in\mathbb{C}[z]$. By (i), $r(\tau_{k})\neq 0$. Dividing through by
$(\tau_{k}-z_{1})$ and plugging in $\tau$ implies
$\lambda_{k}=1/B_{\alpha}(\tau_{k})=\overline{B_{\alpha}(\tau_{k})}.$

For (iii), by the assumptions, there is some $r\in\mathbb{C}[z]$ with
$r(\tau_{k})\neq 0$ such that

$(\tilde{p}-\alpha p)(z)=(z_{1}-\tau_{k})r(z_{1})(z_{2}-1/B_{\alpha}(z_{1})).$

Then for $z_{2}\in\mathbb{C}\setminus\\{\lambda_{k}\\}$, we have
$\phi(\tau_{k},z_{2})\equiv\alpha$ and we can use (ii) to conclude

$\tfrac{\partial\phi}{\partial
z_{1}}(\tau_{k},z_{2})=\frac{\frac{\partial\tilde{p}}{\partial
z_{1}}-\alpha\frac{\partial p}{\partial
z_{1}}}{p}(\tau_{k},z_{2})=\frac{r(\tau_{k})(z_{2}-\lambda_{k})}{c(z_{2}-\lambda_{k})}=\frac{r(\tau_{k})}{c},$

for some $c\neq 0$. ∎

We can also use the set $\mathcal{C}_{\alpha}$ to refine our understanding of
the polynomials $R_{1},\dots,R_{n},Q$ from Theorem 3.2 as follows:

######  Proposition 3.6.

$R_{1},\dots,R_{n},Q$ satisfy the following properties:

  * i.

For $\zeta_{1}\in\mathbb{T}$,
$|Q(\zeta_{1})|^{2}=|p_{1}(\zeta_{1})|^{2}-|p_{2}(\zeta_{1})|^{2}$.

  * ii.

For $z\in\overline{\mathbb{D}}^{2}$,
$R_{j}(z)=r_{j}(z_{1})\left(1-B_{\alpha}(z_{1})z_{2}\right)+z_{2}Q(z_{1})b_{j}(z_{1})$,
for some unique $r_{j}\in\mathbb{C}[z]$ with $\deg r_{j}\leq(n-1)$ and
rational $b_{j}\in A(\mathbb{D}).$

###### Proof.

Part (i) follows from some algebra; substituting $z=w=(\zeta_{1},z_{2})$ into
(7) gives

$|p(\zeta_{1},z_{2})|^{2}-|\tilde{p}(\zeta_{1},z_{2})|^{2}=(1-|z_{2}|^{2})|Q(\zeta_{1})|^{2}.$

The left-hand-side becomes

$|p_{1}(\zeta_{1})+z_{2}p_{2}(\zeta_{1})|^{2}-|z_{2}\overline{p_{1}(\zeta_{1})}+\overline{p_{2}(\zeta_{1})}|^{2}=(1-|z_{2}|^{2})\left(|p_{1}(\zeta_{1})|^{2}-|p_{2}(\zeta_{1})|^{2}\right),$

and dividing by $(1-|z_{2}|^{2})$ gives the desired formula.

For (ii), recall that $\tilde{p}(z)=\alpha p(z)$ whenever
$z_{2}=1/B_{\alpha}(z_{1})$. Substituting $z_{2}=1/B_{\alpha}(z_{1})$ and
$w_{2}=1/B_{\alpha}(w_{1})$ into (7) gives

$0=(1-z_{1}\bar{w}_{1})\sum_{j=1}^{n}R_{j}(z_{1},1/B_{\alpha}(z_{1}))\overline{R_{j}(w_{1},1/B_{\alpha}(w_{1}))}+(1-1/B_{\alpha}(z_{1})\overline{1/B_{\alpha}(w_{1})})Q(z_{1})\overline{Q(w_{1})},$

for all $z_{1},w_{1}$ where $B_{\alpha}(z_{1})\neq 0$. We can rewrite this as

$B_{\alpha}(z_{1})\overline{B_{\alpha}(w_{1})}\sum_{j=1}^{n}R_{j}(z_{1},1/B_{\alpha}(z_{1}))\overline{R_{j}(w_{1},1/B_{\alpha}(w_{1}))}=\frac{1-B_{\alpha}(z_{1})\overline{B_{\alpha}(w_{1})}}{1-z_{1}\bar{w}_{1}}Q(z_{1})\overline{Q(w_{1})},$

for $z_{1},w_{1}\in\mathbb{D}.$ Then the right-hand-side is the reproducing
kernel of the one-variable model space
$\hat{K}_{B_{\alpha}}:=H^{2}(\mathbb{D})\ominus B_{\alpha}H^{2}(\mathbb{D})$
(which is composed of rational functions in $A(\mathbb{D})$, see [16, Chapter
5]) times the $Q$ term. To finish the proof, fix $R_{j}$ and by Theorem 3.2,
write

$R_{j}(z)=r_{j}(z_{1})+z_{2}q_{j}(z_{1}),$

for $r_{j},q_{j}\in\mathbb{C}[z]$. Then standard properties of reproducing
kernels imply that for some $b_{j}\in\hat{K}_{B_{\alpha}}$,

$B_{\alpha}(z_{1})r_{j}(z_{1})+q_{j}(z_{1})=b_{j}(z_{1})Q(z_{1})$

for $z_{1}\in\overline{\mathbb{D}}.$ Solving this for $q_{j}$ and substituting
back into the formula for $R_{j}$ yields the desired result. ∎

In what follows, we will also require information about the weight function
$W$

(10)
$W(x,\zeta)=W_{x}(\zeta)=\frac{|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}}{|\tilde{p}_{1}(\zeta)-xp_{2}(\zeta)|^{2}}\text{
for }(x,\zeta)\in\mathbb{T}^{2}.$

######  Lemma 3.7.

The function $W$ from (10) satisfies the following properties:

  * i.

$W$ is well defined and continuous on $\mathbb{T}^{2}$, except possibly at the
finite set of points $(\alpha,\tau_{k})$ where $\alpha$ is exceptional for
$\phi$ and $L_{k}\subseteq\mathcal{C}_{\alpha}.$

  * ii.

For $\alpha\in\mathbb{T}$, $W_{\alpha}$ has at most a finite number of
discontinuities on $\mathbb{T}$, all of which are removable. So, $W_{\alpha}$
equals a bounded, continuous function $m$-a.e. on $\mathbb{T}.$

###### Proof.

For (i), it follows from the proof of Theorem 3.3 that
$\tilde{p}_{1}(\zeta)-xp_{2}(\zeta)$ only vanishes at some
$(\zeta,x)\in\mathbb{T}^{2}$ if $x$ is exceptional, $\zeta=\tau_{k}$ for some
$k$, and $L_{k}\subseteq\mathcal{C}_{x}$.

For (ii), first observe that if $\alpha$ is generic, then (i) implies
$W_{\alpha}$ is continuous, and hence bounded, on $\mathbb{T}$. If $\alpha$ is
exceptional, after reordering the singularities of $\phi$, assume
$L_{1},\dots,L_{\ell}$ are exactly the lines in $\mathcal{C}_{\alpha}$. Then
by the proof of Theorem 3.3,

$(\tilde{p}_{1}-\alpha p_{2})(z)=q(z)\prod_{k=1}^{\ell}(z-\tau_{k}),$

for some $q\in\mathbb{C}[z]$ that is non-vanishing on $\mathbb{T}$. Similarly,
Theorem 3.2 and Proposition 3.6 imply that for $\zeta\in\mathbb{T}$,

$|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}=|Q(\zeta)|^{2}=\prod_{k=1}^{\ell}|\zeta-\tau_{k}|^{2}|r(\zeta)|^{2},$

for some $r\in\mathbb{C}[z]$. Thus, for $\zeta\neq\tau_{1},\dots\tau_{\ell}$,

$W_{\alpha}(\zeta)=\left|\tfrac{r}{q}(\zeta)\right|^{2}.$

This shows that the only possible singularities $W_{\alpha}$ could have on
$\mathbb{T}$ are at $\tau_{1},\dots,\tau_{\ell}$ and any such singularities
must be removable. ∎

###  3.3. Clark Measure Formulas

Recall that the Clark measures associated to $\phi$ are characterized via
Theorem 1.2. We split the proof into two propositions; the first considers
generic values for $\phi$ and the second considers exceptional values for
$\phi$.

######  Proposition 3.8.

Let $\alpha\in\mathbb{T}$ be a generic value for $\phi$. Then for all $f\in
L^{1}(\sigma_{\alpha})$,

(11) $\int_{\mathbb{T}^{2}}f(\zeta)\
d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}}f(\zeta,\overline{B_{\alpha}(\zeta)})\
d\nu_{\alpha}(\zeta),$

where $d\nu_{\alpha}=W_{\alpha}dm$ and $B_{\alpha},W_{\alpha}$ are from
Definition 1.1.

###### Proof.

For each fixed $z_{2}\in\mathbb{D}$, define the function

$\psi^{\alpha}_{z_{2}}(z_{1}):=\frac{\alpha+\phi(z_{1},z_{2})}{\alpha-\phi(z_{1},z_{2})}\text{
for }z_{1}\in\mathbb{D}.$

Recall that $\phi$ has no singularities on $\mathbb{T}\times\mathbb{D}$.
Furthermore, since $\alpha$ is generic, if $\phi(\zeta_{1},z_{2})=\alpha$ for
$\zeta_{1}\in\mathbb{T}$, then $z_{2}=1/B_{\alpha}(\zeta_{1})\in\mathbb{T}$.
Thus, $\psi^{\alpha}_{z_{2}}\in A(\mathbb{D})$ and for all
$z_{1}\in\mathbb{D}$,

$\Re\left(\psi^{\alpha}_{z_{2}}(z_{1})\right)=\int_{\mathbb{T}}\frac{1-|z_{1}|^{2}}{|z_{1}-\zeta|^{2}}\Re\left(\psi^{\alpha}_{z_{2}}(\zeta)\right)dm(\zeta).$

If $z_{1}\in\mathbb{D}$ and $\zeta\in\mathbb{T}$, then the computation in the
proof of Proposition 3.6(i) gives

$|p(\zeta,z_{2})|^{2}-|\tilde{p}(\zeta,z_{2})|^{2}=(1-|z_{2}|^{2})\left(|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}\right),$

which one can use to obtain

$\displaystyle\Re\left(\psi^{\alpha}_{z_{2}}(\zeta)\right)=\frac{1-|\phi(\zeta,z_{2})|^{2}}{|\alpha-\phi(\zeta,z_{2})|^{2}}=\frac{|p(\zeta,z_{2})|^{2}-|\tilde{p}(\zeta,z_{2})|^{2}}{|\alpha
p(\zeta,z_{2})-\tilde{p}(\zeta,z_{2})|^{2}}=\frac{1-|z_{2}|^{2}}{|z_{2}-\overline{B_{\alpha}(\zeta)}|^{2}}\frac{|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}}{|\alpha
p_{2}(\zeta)-\tilde{p}_{1}(\zeta)|^{2}}.$

This implies that

$\displaystyle\Re\left(\psi^{\alpha}_{z_{2}}(z_{1})\right)$
$\displaystyle=\int_{\mathbb{T}}\frac{1-|z_{1}|^{2}}{|z_{1}-\zeta|^{2}}\frac{1-|z_{2}|^{2}}{|z_{2}-\overline{B_{\alpha}(\zeta)}|^{2}}\frac{|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}}{|\alpha
p_{2}(\zeta)-\tilde{p}_{1}(\zeta)|^{2}}dm(\zeta)$
$\displaystyle=\int_{\mathbb{T}}P_{z}(\zeta,\overline{B_{\alpha}(\zeta)})W_{\alpha}(\zeta)dm(\zeta).$

Thus, by the definition of $\sigma_{\alpha}$, we can conclude that

$\int_{\mathbb{T}^{2}}P_{z}(\zeta)d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}}P_{z}(\zeta,\overline{B_{\alpha}(\zeta)})d\nu_{\alpha}(\zeta).$

Since finite linear combinations of Poisson functions $P_{z}$ are dense in
$C(\mathbb{T}^{2})$, this formula extends to all functions in
$C(\mathbb{T}^{2})$. Since $W_{\alpha}$ is bounded by Proposition 3.7, the
formula extends to $f\in L^{1}(\sigma_{\alpha})$ by Lemma 3.10.∎

Let us now consider the exceptional $\alpha$ values for $\phi$.

######  Proposition 3.9.

Let $\alpha\in\mathbb{T}$ be an exceptional value for $\phi$ and (after re-
ordering the zeros of $p$ on $\mathbb{T}^{2}$ if necessary and applying
Theorem 3.3), assume
$\mathcal{C}_{\alpha}=E_{\alpha}\cup(\cup_{k=1}^{\ell}L_{k})$. Then for all
$f\in L^{1}(\sigma_{\alpha})$,

(12)
$\int_{\mathbb{T}^{2}}f(\zeta)d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}}f(\zeta,\overline{B_{\alpha}(\zeta)})d\nu_{\alpha}(\zeta)+\sum_{k=1}^{\ell}c^{\alpha}_{k}\int_{\mathbb{T}}f(\tau_{k},\zeta)\
dm(\zeta),$

where $d\nu_{\alpha}=W_{\alpha}dm$, $B_{\alpha},W_{\alpha}$ are from
Definition 1.1, and $c^{\alpha}_{1},\dots,c^{\alpha}_{\ell}>0$.

###### Proof.

Recall that $\sigma_{\alpha}$ is supported on
$\mathcal{C}_{\alpha}=E_{\alpha}\cup(\cup_{k=1}^{\ell}L_{k})$. Since the
$L_{k}$ are disjoint and $E_{\alpha}\cap
L_{k}=\\{(\tau_{k},\overline{B_{\alpha}(\tau_{k})})\\}$, which has
$\sigma_{\alpha}$ measure $0$, we only need to show

(13)
$\displaystyle\int_{\mathbb{T}^{2}}f(\zeta)\chi_{E_{\alpha}}(\zeta)d\sigma_{\alpha}(\zeta)$
$\displaystyle=\int_{\mathbb{T}}f(\zeta,\overline{B_{\alpha}(\zeta)})W_{\alpha}(\zeta)dm(\zeta)$
(14)
$\displaystyle\int_{\mathbb{T}^{2}}f(\zeta)\chi_{L_{k}}(\zeta)d\sigma_{\alpha}(\zeta)$
$\displaystyle=c^{\alpha}_{k}\int_{\mathbb{T}}f(\tau_{k},\zeta)dm(\zeta)$

for $f\in L^{1}(\sigma_{\alpha})$ and $k=1,\dots,\ell$, where $\chi_{E}$
denotes the characteristic function of a set $E\subseteq\mathbb{T}^{2}.$ Then
Lemma 3.7 implies $W_{\alpha}$ is bounded on $\mathbb{T}$ (with at most $\ell$
removable singularities) and by Lemma 3.10, we need only establish (13) and
(14) for $f\in C(\mathbb{T}^{2})$.

To ease notation, throughout this proof, for any $F$ defined on
$\mathbb{T}^{2}$, we will use $F_{\alpha}$ to denote
$F_{\alpha}(\zeta):=F(\zeta,\overline{B_{\alpha}(\zeta)})$ and $F_{k}$ to
denote $F_{k}(\zeta):=F(\tau_{k},\zeta).$

Part 1. We first prove (13). To that end, fix a small $\epsilon>0$ and define

$S_{\epsilon}=\\{\zeta\in\mathbb{T}:\min_{k}|\zeta-\tau_{k}|<\epsilon\\}$

and define $S_{\epsilon/2}$ analogously. By Lemma 3.7 and the definition of
$B_{\alpha}$, we can find a small arc $A_{\alpha}\subseteq\mathbb{T}$ centered
at $\alpha$ such that both $W(x,\zeta)$ and $B(x,\zeta):=B_{x}(\zeta)$ are
uniformly continuous on $A_{\alpha}\times(\mathbb{T}\setminus
S_{\epsilon/2}).$ Choose $(\alpha_{n})\subseteq\mathbb{T}$ such that each
$\alpha_{n}$ is generic and $(\alpha_{n})\rightarrow\alpha$. Then by [12,
Corollary 2.2], $(\sigma_{\alpha_{n}})$ converges weak-$\star$ to
$\sigma_{\alpha}$.

To exploit that fact, let $\Psi_{\epsilon}$ be a continuous function on
$\mathbb{T}$ such that

$\Psi_{\epsilon}\equiv 1\text{ on }\mathbb{T}\setminus S_{\epsilon},\ \
\Psi_{\epsilon}\equiv 0\text{ on }S_{\epsilon/2},\ \ 0\leq\Psi_{\epsilon}\leq
1\text{ on }S_{\epsilon}\setminus S_{\epsilon/2}.$

Fix $f\in C(\mathbb{T}^{2})$. By our assumptions and by Proposition 3.8,

$\displaystyle\int_{\mathbb{T}^{2}}f(\zeta)\Psi_{\epsilon}(\zeta_{1})d\sigma_{\alpha}(\zeta)$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{2}}f(\zeta)\Psi_{\epsilon}(\zeta_{1})d\sigma_{\alpha_{n}}(\zeta)$
(15)
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{T}}f_{\alpha_{n}}(\zeta)\
\Psi_{\epsilon}(\zeta)\
d\nu_{\alpha_{n}}(\zeta)=\int_{\mathbb{T}}f_{\alpha}(\zeta)\
\Psi_{\epsilon}(\zeta)\ d\nu_{\alpha}(\zeta),$

where the last equality follows because

$\int_{\mathbb{T}}\left|f_{\alpha_{n}}(\zeta)W_{\alpha_{n}}(\zeta)-f_{\alpha}(\zeta)W_{\alpha}(\zeta)\right|\Psi_{\epsilon}(\zeta)dm(\zeta)\leq\sup_{\zeta\in\mathbb{T}\setminus
S_{\epsilon/2}}\left|\left(f_{\alpha_{n}}W_{\alpha_{n}}-f_{\alpha}W_{\alpha}\right)(\zeta)\right|\rightarrow
0,$

as $n\rightarrow\infty$ because $f_{x}(\zeta)W(x,\zeta)$ is uniformly
continuous on $A_{\alpha}\times(\mathbb{T}\setminus S_{\epsilon/2}).$
Furthermore, observe that since $\Psi_{\epsilon}(\zeta_{1})\equiv 0$ on each
$L_{k}$, we have

$\displaystyle\left|\int_{\mathbb{T}^{2}}f(\zeta)\chi_{E_{\alpha}}(\zeta)d\sigma_{\alpha}(\zeta)-\int_{\mathbb{T}^{2}}f(\zeta)\Psi_{\epsilon}(\zeta_{1})d\sigma_{\alpha}(\zeta)\right|$
$\displaystyle\leq\int_{\mathbb{T}^{2}}|f(\zeta)|(1-\Psi_{\epsilon}(\zeta_{1}))\chi_{E_{\alpha}}(\zeta)d\sigma_{\alpha}(\zeta)$
$\displaystyle\leq\|f\|_{L^{\infty}(\mathbb{T}^{2})}\sigma_{\alpha}((S_{\epsilon}\times\mathbb{T})\cap
E_{\alpha}).$

Here, $(S_{\epsilon}\times\mathbb{T})\cap E_{\alpha}$ is the intersection of
the curve $E_{\alpha}$ with thin strips in $\mathbb{T}^{2}$, see Figure 1.

Figure 1. This shows the set $(S_{\epsilon}\times\mathbb{T})\cap E_{\alpha}$
graphed on $[-\pi,\pi]^{2}$. $E_{\alpha}$ is the red curve,
$\tau_{1},\tau_{2},\tau_{3}$ are the black tics on the horizontal axis, and
$S_{\epsilon}\times\mathbb{T}$ is the union of thin gray strips.

Letting $\epsilon\searrow 0$ and using the dominated convergence theorem gives

$\lim_{\epsilon\searrow 0}\sigma_{\alpha}((S_{\epsilon}\times\mathbb{T})\cap
E_{\alpha})=\sigma_{\alpha}\left(\cup_{k=1}^{\ell}(\tau_{k},\overline{B_{\alpha}(\tau_{k})})\right)=0.$

This in turn implies that

$\int_{\mathbb{T}^{2}}f(\zeta)\chi_{E_{\alpha}}(\zeta)d\sigma_{\alpha}(\zeta)=\lim_{\epsilon\searrow
0}\int_{\mathbb{T}^{2}}f(\zeta)\Psi_{\epsilon}(\zeta_{1})d\sigma_{\alpha}(\zeta).$

As $f_{\alpha}$ and $W_{\alpha}$ are both bounded, we can also conclude that

$\lim_{\epsilon\searrow
0}\int_{\mathbb{T}}f_{\alpha}(\zeta)\Psi_{\epsilon}(\zeta)\
d\nu_{\alpha}(\zeta)=\int_{\mathbb{T}}f_{\alpha}(\zeta)\
d\nu_{\alpha}(\zeta).$

Combining these last two equalities with (15) yields (13).

Part 2. Now we prove (14). We will show that for $z\in\mathbb{D}^{2}$, (14)
holds for $P_{z}$. Since linear combinations of these are dense in
$C(\mathbb{T}^{2})$, the result will follow. To that end, fix $0<r<1$. Then
the definition of $\sigma_{\alpha}$ gives

(16)
$\int_{\mathbb{T}^{2}}P_{(r\tau_{k},z_{2})}(\zeta)d\sigma_{\alpha}(\zeta)=\Re\left(\frac{\alpha+\phi(r\tau_{k},z_{2})}{\alpha-\phi(r\tau_{k},z_{2})}\right).$

We will multiply both sides by $(1-r)$ and let $r\nearrow 1$. First, observe
that for $\zeta\in\mathbb{T}^{2}$,

$\lim_{r\nearrow
1}(1-r)P_{(r\tau_{k},z_{2})}(\zeta)=\left\\{\begin{array}[]{cc}0&\text{ if
}\zeta_{1}\neq\tau_{k},\\\ 2P_{z_{2}}(\zeta_{2})&\text{ if
}\zeta_{1}=\tau_{k}.\end{array}\right.$

Then by the dominated convergence theorem,

$\lim_{r\nearrow
1}\int_{\mathbb{T}^{2}}(1-r)P_{(r\tau_{k},z_{2})}(\zeta)d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}^{2}}2P_{z_{2}}(\zeta_{2})\chi_{L_{k}}(\zeta)d\sigma_{\alpha}(\zeta).$

Observe that $L_{k}\subseteq\mathcal{C}_{\alpha}$ actually implies that
$\phi(\tau_{k},z_{2})=\alpha$ for all $z_{2}\in\mathbb{D}$. Furthermore, since
$\phi$ is analytic at each $(\tau_{k},z_{2})$, we have

$\lim_{z_{1}\rightarrow\tau_{k}}\phi(z_{1},z_{2})=\alpha\text{ and
}\lim_{z_{1}\rightarrow\tau_{k}}\frac{\phi(z_{1},z_{2})-\alpha}{z_{1}-\tau_{k}}=\tfrac{\partial\phi}{\partial
z_{1}}(\tau_{k},z_{2}):=C\neq 0,$

by Lemma 3.5. Then Carathéodory’s theorem, see (VI-3) in [34], implies

$\lim_{r\nearrow
1}\frac{1-|\phi(r\tau_{k},z_{2})|}{1-r}=C\tau_{k}\bar{\alpha}=|C|$

and so

$\displaystyle\lim_{r\nearrow
1}\Re\left(\frac{(1-r)(\alpha+\phi(r\tau_{k},z_{2}))}{\alpha-\phi(r\tau_{k},z_{2})}\right)$
$\displaystyle=\lim_{r\nearrow
1}(1-r)\frac{1-|\phi(r\tau_{k},z_{2})|^{2}}{|\alpha-\phi(r\tau_{k},z_{2})|^{2}}$
$\displaystyle=\lim_{r\nearrow
1}2\left|\frac{\tau_{k}-r\tau_{k}}{\alpha-\phi(r\tau_{k},z_{2})}\right|^{2}\frac{1-|\phi(r\tau_{k},z_{2})|}{1-r}=\frac{2}{|C|}.$

Now set

(17) $c^{\alpha}_{k}=\frac{1}{|C|}=\frac{1}{|\tfrac{\partial\phi}{\partial
z_{1}}(\tau_{k},z_{2})|}>0.$

Then (16) and our subsequent computations combine to give

$\int_{\mathbb{T}^{2}}P_{z_{2}}(\zeta_{2})\chi_{L_{k}}(\zeta)d\sigma_{\alpha}(\zeta)=c^{\alpha}_{k}=c^{\alpha}_{k}\int_{\mathbb{T}}P_{z_{2}}(\zeta)dm(\zeta).$

Multiplying both sides by $P_{z_{1}}(\tau_{k})$ establishes (14) for $f=P_{z}$
and completes the proof. ∎

The following lemma, which was used in the above proof, is a consequence of
standard measure-theory facts. We include its proof here for the ease of the
reader.

######  Lemma 3.10.

Let $\sigma$ be a Borel measure on $\mathbb{T}^{2}$ and let
$\zeta_{2}=g(\zeta_{1})$ be a continuous curve in $\mathbb{T}^{2}$. If $W$ is
a continuous function defined on $\mathbb{T}$ such that

(18)
$\int_{\mathbb{T}^{2}}f(\zeta)d\sigma(\zeta)=\int_{\mathbb{T}}f(\zeta,g(\zeta))W(\zeta)dm(\zeta)\
\text{ for all }f\in C(\mathbb{T}^{2}),$

then (18) holds for all $f\in L^{1}(\sigma).$

###### Proof.

For ease of notation, set $d\nu=Wdm$. Then $\nu$ is a Borel measure on
$\mathbb{T}$. Furthermore, if $E\subseteq\mathbb{T}^{2}$ is a Borel set, then

(19) $E_{g}:=\\{\zeta_{1}\in\mathbb{T}:\text{there exists
}\zeta_{2}\in\mathbb{T}\text{ with }(\zeta_{1},\zeta_{2})\in
E\cap\\{(\zeta,g(\zeta)):\zeta\in\mathbb{T}\\}\\}$

is the projection of a Borel set in $\mathbb{T}^{2}$ onto its first
coordinate. This implies that $E_{g}$ is an analytic set and its
characteristic function is Lesbesgue measurable and hence, $\nu$-measurable,
see for example Chapter 13 in [13]. We will use $E_{g}$ frequently because
$\chi_{E}(\zeta,g(\zeta))=\chi_{E_{g}}(\zeta)$ for $\zeta\in\mathbb{T}$. The
following proof has three steps.

Step 1: Establish (18) for $f=\chi_{U}$, where $U$ is an arbitrary open set in
$\mathbb{T}^{2}$. Let $\\{K_{n}\\}_{n=1}^{\infty}$ be a sequence of nested
compact sets with $U=\bigcup_{n}K_{n}$. Then, by Urysohn’s lemma, there exists
a sequence $\\{f_{n}\\}_{n=1}^{\infty}$ of continuous functions on
$\mathbb{T}^{2}$ having $0\leq f_{n}\leq 1$ on $\mathbb{T}^{2}$ and

$f_{n}=1\quad\textrm{on}\quad K_{n}\quad\textrm{and}\quad
f_{n}=0\quad\textrm{on}\quad U^{c}.$

Then $\lim_{n\to\infty}f_{n}(\zeta)=\chi_{U}(\zeta)$ for every
$\zeta\in\mathbb{T}^{2}$, and since $\nu$ is a finite measure, we can apply
the dominated convergence theorem to obtain

$\sigma(U)=\lim_{n\to\infty}\int_{\mathbb{T}^{2}}f_{n}(\zeta)d\sigma(\zeta)\\\
=\lim_{n\to\infty}\int_{\mathbb{T}}f_{n}(\zeta,g(\zeta))d\nu(\zeta)=\int_{\mathbb{T}}\chi_{U}(\zeta,g(\zeta))d\nu(\zeta),$

as desired.

Step 2: Establish (18) for $f=\chi_{E}$, where $E$ is an arbitrary Borel set
in $\mathbb{T}^{2}$. Since $\sigma$ is a finite Borel measure on
$\mathbb{T}^{2}$, and hence is Radon, for each $n\in\mathbb{N}$ there exists a
compact set $K_{n}$ and an open set $U_{n}$ such that $K_{n}\subseteq
E\subseteq U_{n}$ and $\sigma(U_{n}\setminus K_{n})<1/n$. Urysohn’s lemma
again guarantees the existence of a sequence $\\{f_{n}\\}_{n=1}^{\infty}$ of
continuous functions with $0\leq f_{n}\leq 1$, $f_{n}=1$ on $K_{n}$, and
$f_{n}=0$ on $U_{n}^{c}$. Since

$\int_{\mathbb{T}^{2}}|(f_{n}-\chi_{E})(\zeta)|\
d\sigma(\zeta)\leq\int_{\mathbb{T}^{2}}\chi_{U_{n}\setminus
K_{n}}(\zeta)d\sigma(\zeta)=\sigma(U_{n}\setminus K_{n})<\tfrac{1}{n},$

we have $\|f-f_{n}\|_{L^{1}(\sigma)}<1/n$ and $f_{n}\to f$ in $L^{1}(\sigma)$
as $n\to\infty$. Since the $f_{n}$ are continuous, this implies

$\int_{\mathbb{T}^{2}}\chi_{E}(\zeta)d\sigma(\zeta)=\lim_{n\to\infty}\int_{\mathbb{T}^{2}}f_{n}(\zeta)d\sigma(\zeta)=\lim_{n\to\infty}\int_{\mathbb{T}}f_{n}(\zeta,g(\zeta))d\nu(\zeta).$

Because $\chi_{K_{n}}\leq f_{n}\leq\chi_{U_{n}}$ on $\mathbb{T}^{2}$, we then
obtain

(20)
$\int_{\mathbb{T}}\chi_{K_{n}}(\zeta,g(\zeta))d\nu(\zeta)\leq\int_{\mathbb{T}^{2}}f_{n}(\zeta)d\sigma(\zeta)\leq\int_{\mathbb{T}}\chi_{U_{n}}(\zeta,g(\zeta))d\nu(\zeta)$

and as $\chi_{K_{n}}\leq\chi_{E}\leq\chi_{U_{n}}$ on $\mathbb{T}^{2}$,

(21)
$\int_{\mathbb{T}}\chi_{K_{n}}(\zeta,g(\zeta))d\nu(\zeta)\leq\int_{\mathbb{T}}\chi_{E}(\zeta,g(\zeta))d\nu(\zeta)\leq\int_{\mathbb{T}}\chi_{U_{n}}(\zeta,g(\zeta))d\nu(\zeta).$

Combining (20) and (21) gives

$\displaystyle\left|\int_{\mathbb{T}^{2}}\chi_{E}(\zeta)d\sigma(\zeta)-\int_{\mathbb{T}}\chi_{E}(\zeta,g(\zeta))d\nu(\zeta)\right|$
$\displaystyle\leq\|\chi_{E}-f_{n}\|_{L^{1}(\sigma)}$
$\displaystyle+\left|\int_{\mathbb{T}^{2}}f_{n}(\zeta)d\sigma(\zeta)-\int_{\mathbb{T}}\chi_{E}(\zeta,g(\zeta))d\nu(\zeta)\right|$
$\displaystyle\leq\tfrac{1}{n}+\int_{\mathbb{T}}(\chi_{U_{n}}-\chi_{K_{n}})(\zeta,g(\zeta))d\nu(\zeta)$
$\displaystyle=\tfrac{1}{n}+\sigma(U_{n}\setminus K_{n})<\tfrac{2}{n}$

for all $n$, where we used Step $1$ applied to $U_{n}\setminus K_{n}=U_{n}\cap
K_{n}^{c}$. Letting $n\rightarrow\infty$ gives (18) for $f=\chi_{E}$.

Step 3: Establish (18) for a general $f\in L^{1}(\sigma)$. Pick a sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ of continuous functions on $\mathbb{T}^{2}$ such
that $f_{n}\to f$ in $L^{1}(\sigma)$ and pointwise $\sigma$-almost everywhere
on $\mathbb{T}^{2}$. Then there exists some Borel set $E\subset\mathbb{T}^{2}$
with $\sigma(E)=0$ such that if $f_{n}(\zeta)\nrightarrow f(\zeta)$,
$n\to\infty$, then $\zeta\in E$. Then, if $f_{n}(\zeta,g(\zeta))\nrightarrow
f(\zeta,g(\zeta))$ then $\zeta\in E_{g}$, where $E_{g}$ is defined in (19). By
Step 2, $\nu(E_{g})=\sigma(E)=0$. Hence

(22) $f_{n}(\zeta,g(\zeta))\to
f(\zeta,g(\zeta))\quad\textrm{for}\quad\nu-\mathrm{a.e}\,\,\zeta\in\mathbb{T}.$

Since the $f_{n}$ are continuous, we have

$\int_{\mathbb{T}^{2}}|f_{n}(\zeta)-f_{m}(\zeta)|d\sigma(\zeta)=\int_{\mathbb{T}}|f_{n}(\zeta,g(\zeta))-f_{m}(\zeta,g(\zeta))|d\nu(\zeta).$

This implies that $\\{f_{n}(\zeta,g(\zeta))\\}_{n=1}^{\infty}$ is a Cauchy
sequence and hence has a limit $F$ in $L^{1}(\nu)$, and by (22) we must have
$F=f$ in $L^{1}(\nu)$. Then $L^{1}$-convergence gives

$\int_{\mathbb{T}^{2}}f(\zeta)d\sigma(\zeta)=\lim_{n\to\infty}\int_{\mathbb{T}^{2}}f_{n}(\zeta)d\sigma(\zeta)=\lim_{n\to\infty}\int_{\mathbb{T}}f_{n}(\zeta,g(\zeta))d\nu(\zeta)=\int_{\mathbb{T}}f(\zeta,g(\zeta))d\nu(\zeta),$

which gives (18) for $f$. ∎

###  3.4. Properties of $J_{\alpha}$

Recall that the isometry $J_{\alpha}\colon K_{\phi}\rightarrow
L^{2}(\sigma_{\alpha})$ is obtained by first defining the operator on
reproducing kernels $k_{w}$ as

$J_{\alpha}[k_{w}](\zeta):=(1-\alpha\overline{\phi(w)})C_{w}(\zeta),\quad\text{for
}w\in\mathbb{D}^{2},\zeta\in\mathbb{T}^{2},$

and then extending it to the rest of $K_{\phi}$. Theorem 1.3 details our main
results about $J_{\alpha}$, which are proved below in two propositions.

First, unlike the one-variable case, these isometries $J_{\alpha}$ need not be
unitary. The exact situation in our setting is encoded in the following
result:

######  Proposition 3.11.

The isometric embedding $J_{\alpha}:K_{\phi}\rightarrow
L^{2}(\sigma_{\alpha})$ is unitary if and only if $\alpha$ is a generic value
for $\phi$.

###### Proof.

($\Rightarrow$) Assume that $\alpha$ is generic. By Theorem 3.2 in [12], we
need only show that $A(\mathbb{D}^{2})$ is dense in $L^{2}(\sigma_{\alpha})$.
Since $\sigma_{\alpha}$ is a finite Radon measure, $C(\mathbb{T}^{2})$ is
dense in $L^{2}(\sigma_{\alpha})$ and by the Stone-Weierstrass theorem, the
set of two-variable trigonometric polynomials is dense in $C(\mathbb{T}^{2})$
and hence, in $L^{2}(\sigma_{\alpha})$. Thus, to show $J_{\alpha}$ is unitary,
we need only show that each two-variable trigonometric polynomial agrees with
some function in $A(\mathbb{D}^{2})$ on $E_{\alpha}$, which contains the
support of $\sigma_{\alpha}.$

Let $h(\zeta)=\zeta_{1}^{m}\zeta_{2}^{n}$ be an arbitrary trigonometric
monomial. To construct a function in $A(\mathbb{D}^{2})$ that agrees with $h$
on $E_{\alpha}$, first define $t_{1},t_{2}\in A(\mathbb{D}^{2})$ by
$t_{1}(z)=z_{1}$ and $t_{2}(z)=z_{2}$. Then, recall from Theorem 3.3 that
$\deg B_{\alpha}=n$. Write $B_{\alpha}=\gamma\prod_{j=1}^{n}b_{a_{j}}$, where
$\gamma\in\mathbb{T}$ and each $b_{a_{j}}=(z-a_{j})/(1-\bar{a}_{j}z)$ is the
Blaschke factor with zero $a_{j}\in\mathbb{D}$. Let Let
$b^{-1}_{\overline{a}_{1}}$ denote the inverse function of
$b_{\overline{a}_{1}}$ and define $s_{1},s_{2}\in A(\mathbb{D}^{2})$ by

$s_{1}(z)=b_{\overline{a}_{1}}^{-1}\Big{[}\Big{(}\gamma\prod_{j=2}^{n}b_{a_{j}}(z_{1})\Big{)}z_{2}\Big{]},$

and $s_{2}(z)=B_{\alpha}(z_{1})$. Then, restricting to $E_{\alpha}$, we have

$s_{1}(\zeta,\overline{B_{\alpha}(\zeta)})=b_{\overline{a}_{1}}^{-1}\Big{[}\Big{(}\gamma\prod_{j=2}^{n}b_{a_{j}}(\zeta)\Big{)}\overline{\Big{(}\gamma\prod_{j=1}^{n}b_{a_{j}}(\zeta)\Big{)}}\Big{]}=b_{\overline{a}_{1}}^{-1}\left[b_{\overline{a}_{1}}(\bar{\zeta})\right]=\bar{\zeta}$

and $s_{2}(\zeta,\overline{B_{\alpha}(\zeta)})=B_{\alpha}(\zeta).$ As

$h(\zeta,\overline{B_{\alpha}(\zeta)})=\zeta^{m}\overline{B_{\alpha}(\zeta)}^{n},$

$h$ agrees with one of $t_{1}^{|m|}t_{2}^{|n|},$ $t_{1}^{|m|}s_{2}^{|n|},$
$s_{1}^{|m|}t_{2}^{|n|},$ $s_{1}^{|m|}s_{2}^{|n|}$ on $E_{\alpha}$. Taking
linear combinations of these shows that every two-variable trigonometric
polynomial agrees with some $F\in A(\mathbb{D}^{2})$ on $E_{\alpha}$ and
completes the proof of this forward direction.

($\Leftarrow$) Assume that $\alpha$ is exceptional and
$\phi^{*}(\tau_{k},\lambda_{k})=\alpha$. By way of contradiction, assume that
$A(\mathbb{D}^{2})$ is dense in $L^{2}(\sigma_{\alpha})$. Let
$f(\zeta)=\bar{\zeta}_{2}$. By assumption, there is a sequence
$(f_{n})\subseteq A(\mathbb{D}^{2})$ that converges to $f$ in
$L^{2}(\sigma_{\alpha})$. Then by Theorem 1.2, there is a $c^{\alpha}_{k}>0$
such that

$\int_{\mathbb{T}}|f(\tau_{k},\zeta)-f_{n}(\tau_{k},\zeta)|^{2}dm(\zeta)\leq\tfrac{1}{c^{\alpha}_{k}}\|f-f_{n}\|^{2}_{L^{2}(\sigma_{\alpha})}\rightarrow
0,$

as $n\rightarrow\infty$. Since each $f_{n}(\tau_{k},\cdot)$ is in
$H^{2}(\mathbb{D})$, so is the limit function $f(\tau_{k},\cdot)$. Since
$f(\tau_{k},\zeta)=\bar{\zeta}$, it is clearly not in $H^{2}(\mathbb{D})$ and
so, we obtain the needed contradiction. ∎

We can also identify the exact form of the isometric operator $J_{\alpha}$.

######  Proposition 3.12.

For each each $f\in K_{\phi}$, the isometry $J_{\alpha}:K_{\phi}\rightarrow
L^{2}(\sigma_{\alpha})$ satisfies

$(J_{\alpha}f)(\zeta)=f^{*}(\zeta)\text{ for }\sigma_{\alpha}\text{-a.e.}\
\zeta\in\mathbb{T}^{2}.$

###### Proof.

Fix $f\in K_{\phi}$. By Lemma 3.13, $f^{*}$ exists and equals a Borel-
measurable function $\sigma_{\alpha}$-a.e. on $\mathbb{T}^{2}$. We claim that
$f^{*}\in L^{2}(\sigma_{\alpha})$ and

(23) $\|f^{*}\|_{L^{2}(\sigma_{\alpha})}\lesssim\|f\|_{K_{\phi}},$

where the implied constant does not depend on $f$. To see this, use Theorem
3.2 to write

$f(z)=\tfrac{Q}{p}(z)h(z_{1})+\sum_{j=1}^{n}\tfrac{R_{j}}{p}(z)g_{j}(z_{2})\quad\text{
for }z\in\mathbb{D}^{2}$

and $g_{1},\dots,g_{n},h\in H^{2}(\mathbb{D})$. By the proof of Lemma 3.13,
this formula extends to $\mathbb{T}^{2}$ via non-tangential limits both
Lebesgue and $\sigma_{\alpha}$-a.e. By Proposition 3.6, there is a $b_{j}\in
A(\mathbb{D})$ such that

$\left|(\tfrac{R_{j}}{p})(\zeta,\overline{B_{\alpha}(\zeta)})\right|=\left|(\tfrac{Q}{p})(\zeta,\overline{B_{\alpha}(\zeta)})\right||b_{j}(\zeta)|,$

for all $\zeta\in\mathbb{T}$. Working through the definitions and applying
Proposition 3.6 give

$|(\tfrac{Q}{p})(\zeta,\overline{B_{\alpha}(\zeta)})|^{2}W_{\alpha}(\zeta)=\frac{|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}}{(|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2})^{2}}|(\tilde{p}_{1}-\alpha
p_{2})(\zeta)|^{2}\frac{|p_{1}(\zeta)|^{2}-|p_{2}(\zeta)|^{2}}{|(\tilde{p}_{1}-\alpha
p_{2})(\zeta)|^{2}}=1$

for all $\zeta\in\mathbb{T}\setminus\\{\tau_{1},\dots,\tau_{m}\\}$. If
$B_{\alpha}$ is non-constant, this immediately implies that

$\displaystyle\int_{\mathbb{T}}$
$\displaystyle|f^{*}(\zeta,\overline{B_{\alpha}(\zeta)})|^{2}d\nu_{\alpha}(\zeta)$
$\displaystyle\lesssim\int_{\mathbb{T}}|(\tfrac{Q}{p})(\zeta,\overline{B_{\alpha}(\zeta)})|^{2}\Big{(}|h^{*}(\zeta)|^{2}+\sum_{j=1}^{n}|b_{j}(\zeta)|^{2}|g^{*}_{j}(\overline{B_{\alpha}(\zeta)})|^{2}\Big{)}W_{\alpha}(\zeta)dm(\zeta)$
$\displaystyle=\int_{\mathbb{T}}|h^{*}(\zeta)|^{2}+\sum_{j=1}^{n}|b_{j}(\zeta)|^{2}|\bar{g}^{*}_{j}(B_{\alpha}(\zeta))|^{2}dm(\zeta)$
$\displaystyle\lesssim\Big{(}\|h\|^{2}_{H^{2}}+\sum_{j=1}^{n}\|\overline{g}_{j}\circ
B_{\alpha}\|_{H^{2}}^{2}\Big{)}\lesssim\|f\|^{2}_{K_{\phi}},$

where $\bar{g}$ is the function in $H^{2}(\mathbb{D})$ whose Taylor
coefficients are the complex conjugates of those of $g$. In this computation,
we used Theorem 3.2 and the well-known fact that composition by a non-constant
finite Blaschke product $B_{\alpha}$ induces a bounded operator on
$H^{2}(\mathbb{D})$, see Theorem 5.1.5 in [24]. Here, the implied constant
does not depend on $f$. If $B_{\alpha}$ is constant, then the one-variable
model space $\hat{K}_{B_{\alpha}}=\\{0\\}$, so each $b_{j}\equiv 0$ and

$\int_{\mathbb{T}}|f^{*}(\zeta,\overline{B_{\alpha}(\zeta)}))|^{2}W_{\alpha}(\zeta)dm(\zeta)=\|h\|^{2}_{H^{2}}\leq\|f\|^{2}_{K_{\phi}}.$

Similarly, for each $1\leq j\leq n$ and $1\leq k\leq m$, Proposition 3.6 gives
constants $M_{jk}$ and $d^{\alpha}_{jk}$ such that for $\zeta\neq\lambda_{k}$,

$\tfrac{Q}{p}(\tau_{k},\zeta)\equiv 0\text{ and
}\tfrac{R_{j}}{p}(\tau_{k},\zeta)=M_{jk}\frac{1-B_{\alpha}(\tau_{k})\zeta}{p(\tau_{k},\zeta)}=:d^{\alpha}_{jk},$

since both the numerator and denominator are linear and by Lemma 3.5, vanish
at $\lambda_{k}$. This shows

$c^{\alpha}_{k}\int_{\mathbb{T}}|f^{*}(\tau_{k},\zeta)|^{2}dm(\zeta)\lesssim\sum_{j=1}^{n}(d_{jk}^{\alpha})^{2}\int_{\mathbb{T}}|g_{j}(\zeta)|^{2}dm(\zeta)\lesssim\|f\|_{K_{\phi}}^{2}.$

By Lemma 3.13, this shows $f^{*}\in L^{2}(\sigma_{\alpha})$. Furthermore, if
we define a linear map $T_{\alpha}:K_{\phi}\rightarrow L^{2}(\sigma_{\alpha})$
by $(T_{\alpha}f)=f^{*}$, then $T_{\alpha}$ is bounded. Moreover, observe that
for
$\zeta\in\mathcal{C}_{\alpha}\setminus\\{(\tau_{1},\lambda_{1}),\dots,(\tau_{m},\lambda_{m})\\}$,
we have

$T_{\alpha}[k_{w}](\zeta)=(1-\alpha\overline{\phi(w)})C_{w}(\zeta)=J_{\alpha}[k_{w}](\zeta).$

Thus, these functions are equal in $L^{2}(\sigma_{\alpha})$. Since
$T_{\alpha}$ and $J_{\alpha}$ agree on a dense set of functions in $K_{\phi}$,
it follows that $T_{\alpha}=J_{\alpha}$, which completes the proof. ∎

The proof of Proposition 3.12 required the following lemma.

######  Lemma 3.13.

If $f\in K_{\phi}$, then $f^{*}$ exists and agrees with a Borel measurable
function $\sigma_{\alpha}$-a.e. on $\mathbb{T}^{2}$ and

$\int_{\mathbb{T}^{2}}|f^{*}(\zeta)|^{2}d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}}|f^{*}(\zeta,\overline{B_{\alpha}(\zeta)})|^{2}d\nu_{\alpha}(\zeta)+\sum_{k=1}^{m}c^{\alpha}_{k}\int_{\mathbb{T}}|f^{*}(\tau_{k},\zeta)|^{2}dm(\zeta),$

where $d\nu_{\alpha}=W_{\alpha}dm$, the functions $B_{\alpha},W_{\alpha}$ are
from Definition 1.1, and the $c^{\alpha}_{k}$ are from Theorem 1.2.

###### Proof.

By Theorem 3.2, there exist $g_{1},\dots,g_{n},h\in H^{2}(\mathbb{D})$ such
that

$f(z)=\tfrac{Q}{p}(z)h(z_{1})+\sum_{j=1}^{n}\tfrac{R_{j}}{p}(z)g_{j}(z_{2}),\quad\text{
for }z\in\mathbb{D}^{2}.$

Let $A\subseteq\mathbb{T}$ be a Borel set with $m(A)=0$ such that
$h,g_{1},\dots,g_{n}$ have non-tangential limits at all
$\zeta\in\mathbb{T}\setminus A$. To finish the set-up, assume
$(z^{n})=(z_{1}^{n},z_{2}^{n})\rightarrow(\tau_{k},\zeta_{2})\in\mathbb{T}^{2}$
non-tangentially, where $\zeta_{2}\neq\lambda_{k}$. Then since
$(z_{1}-\tau_{k})$ is a factor of $Q$ and $Q/p$ is continuous near
$(\tau_{k},\zeta_{2})$,

(24)
$\lim_{n\rightarrow\infty}\left|\tfrac{Q}{p}(z^{n})h(z^{n}_{1})\right|\lesssim\lim_{n\rightarrow\infty}|z^{n}_{1}-\tau_{k}|\|h\|_{H^{2}}\tfrac{1}{\sqrt{1-|z^{n}_{1}|^{2}}}\lesssim\lim_{n\rightarrow\infty}\sqrt{1-|z^{n}_{1}|}=0,$

where we also used the reproducing property of $H^{2}$ and the non-tangential
property of $(z^{n})$. Similarly, if $B_{\alpha}$ equals some constant
$\gamma\in\mathbb{T},$ then each $\lambda_{k}=\bar{\gamma}$ and in Proposition
3.6, each $b_{j}\equiv 0$ and $(z_{2}-\overline{\gamma})$ divides $R_{j}$.
Thus, arguments analogous to those in (24) imply that if
$(z^{n})=(z_{1}^{n},z_{2}^{n})\rightarrow(\zeta_{1},\overline{\gamma})\in\mathbb{T}^{2}$
non-tangentially with
$\zeta_{1}\in\mathbb{T}\setminus\\{\tau_{1},\dots,\tau_{m}\\}$, then

$\lim_{n\rightarrow\infty}\left|\tfrac{R_{j}}{p}(z^{n})g_{j}(z^{n}_{2})\right|=0,\text{
for }j=1,\dots,n.$

This implies that $f^{*}(\zeta)$ exists for all
$\zeta\in\mathbb{T}^{2}\setminus\hat{A}$, where

$\hat{A}:=\\{(\tau_{k},\lambda_{k}):k=1,\dots,m\\}\cup((A\setminus\\{\tau_{1},\dots,\tau_{m}\\})\times\mathbb{T})\cup(\mathbb{T}\times
A\setminus\\{\bar{\gamma}\\}),$

where we only include $\bar{\gamma}$ if $B_{\alpha}$ is constant. By
definition, $\hat{A}$ is a Borel set and we claim
$\sigma_{\alpha}(\hat{A})=0$. Since $\sigma_{\alpha}$ has no point masses, it
is immediate that

$\sigma_{\alpha}(\\{(\tau_{k},\lambda_{k}):k=1,\dots,m\\})=0.$

Set $A_{1}=(A\setminus\\{\tau_{1},\dots,\tau_{m}\\})\times\mathbb{T}.$ Then as
$A_{1}\cap L_{k}=\emptyset$ for each $k$ and Lemma 3.7 shows $W_{\alpha}$ is
bounded, we can use Theorem 1.2 to compute

$\sigma_{\alpha}(A_{1})=\int_{\mathbb{T}}\chi_{A_{1}}(\zeta,\overline{B_{\alpha}(\zeta)})d\nu_{\alpha}(\zeta)\lesssim\int_{\mathbb{T}}\chi_{A}(\zeta)dm(\zeta)=0.$

If $B_{\alpha}$ is non-constant, set $A_{2}=\mathbb{T}\times A$. Again by
Theorem 1.2, there are constants $c_{k}^{\alpha}$ such that

$\displaystyle\sigma_{\alpha}(A_{2})$
$\displaystyle=\int_{\mathbb{T}}\chi_{A_{2}}(\zeta,\overline{B_{\alpha}(\zeta)})d\nu_{\alpha}(\zeta)+\sum_{k=1}^{m}c^{\alpha}_{k}\int_{\mathbb{T}}\chi_{A_{2}}(\tau_{k},\zeta)dm(\zeta)$
$\displaystyle\lesssim m(\\{\zeta\in\mathbb{T}:\overline{B_{\alpha}(\zeta)}\in
A\\})+\sum_{k=1}^{m}c^{\alpha}_{k}m(A)=0,$

where the first set has Lebesgue measure $0$ because non-constant finite
Blaschke products are smooth, have non-zero derivatives on $\mathbb{T}$, and
are locally invertible on $\mathbb{T}$. Hence, the preimage
$\bar{B}_{\alpha}^{-1}(A)$ must have measure $0$ because $A$ does. If
$B_{\alpha}=\gamma$ is constant, set
$A_{2}=\mathbb{T}\times(A\setminus\\{\bar{\gamma}\\}).$ Then

$\displaystyle\sigma_{\alpha}(A_{2})=\int_{\mathbb{T}}\chi_{A_{2}}(\zeta,\bar{\gamma})d\nu_{\alpha}(\zeta)+\sum_{k=1}^{m}c^{\alpha}_{k}\int_{\mathbb{T}}\chi_{A_{2}}(\tau_{k},\zeta)dm(\zeta)=0$

by the definition of $A_{2}$. Thus, $f^{*}$ exists $\sigma_{\alpha}$-a.e. on
$\mathbb{T}^{2}$. Finally, observe that

$F(\zeta)=\limsup_{r\nearrow 1}\Re(f(r\zeta))+i\limsup_{r\nearrow
1}\Im(f(r\zeta))$

is Borel measurable since each $f_{r}(\zeta):=f(r\zeta)$ is continuous on
$\mathbb{T}^{2}$ and $F=f^{*}$ on $\mathbb{T}^{2}\setminus\hat{A}$ and hence
$\sigma_{\alpha}$-a.e. Our prior arguments also imply
$f^{*}(\zeta,\overline{B_{\alpha}(\zeta)})=F(\zeta,\overline{B_{\alpha}(\zeta)})$
for $\nu_{\alpha}$-a.e. $\zeta\in\mathbb{T}$ and
$f^{*}(\tau_{k},\zeta)=F(\tau_{k},\zeta)$ for $m$-a.e. $\zeta\in\mathbb{T}.$
To finish the proof, for each $n\in\mathbb{N}$, define the Borel set

$D_{n}=\\{\zeta\in\mathbb{T}^{2}:|F(\zeta)|<n\\}.$

Then Theorem 1.2 combined with the monotone convergence theorem gives

$\displaystyle\int_{\mathbb{T}^{2}}$
$\displaystyle|f^{*}(\zeta)|^{2}d\sigma_{\alpha}(\zeta)=\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{2}}|F(\zeta)|^{2}\chi_{D_{n}}(\zeta)d\sigma_{\alpha}(\zeta)$
$\displaystyle=\lim_{n\rightarrow\infty}\left(\int_{\mathbb{T}}|(F\chi_{D_{n}}(\zeta,\overline{B_{\alpha}(\zeta)})|^{2}\
d\nu_{\alpha}(\zeta)+\sum_{k=1}^{m}c_{k}^{\alpha}\int_{\mathbb{T}}|(F\chi_{D_{n}})(\tau_{k},\zeta)|^{2}dm(\zeta)\right)$
$\displaystyle=\int_{\mathbb{T}}|f^{*}(\zeta,\overline{B_{\alpha}(\zeta)})|^{2}\
d\nu_{\alpha}(\zeta)+\sum_{k=1}^{m}c_{k}^{\alpha}\int_{\mathbb{T}}|f^{*}(\tau_{k},\zeta)|^{2}dm(\zeta),$

which is what we needed to show. ∎

##  4\. Applications

The results from Section 3 have implications for the structure of Agler
decompositions and connections to the study of extreme measures from [22, 25,
28] and the references therein. In this section, we again assume
$\phi=\frac{\tilde{p}}{p}$ is a degree $(n,1)$ rational inner function and
throughout, will use the notation denoted earlier in Definition 1.1 and Remark
3.1.

###  4.1. Agler Decompositions

Recall that each such $\phi$ possesses an Agler decomposition from Theorem 3.2
arising from a particular orthonormal list in $K_{\phi}$. Moreover, the
polynomial $Q$ in that decomposition can be computed directly on $\mathbb{T}$
via Proposition 3.6. In the case of exceptional $\alpha$, we can apply Theorem
1.2 to specify some of the remaining polynomials $R_{1},\dots,R_{n}$ from (7).

######  Theorem 4.1.

Let $\alpha\in\mathbb{T}$ be exceptional for $\phi$ and (after reordering if
necessary) assume $\phi^{*}(\tau_{k},\lambda_{k})=\alpha$ for
$k=1,\dots,\ell$. Using Theorem 3.3, write
$B_{\alpha}=b^{1}_{\alpha}/b^{2}_{\alpha},$ where each $\deg
b_{\alpha}^{i}=n-\ell.$ Then in (7), we can take

(25)
$R_{j}(z)=d^{\alpha}_{j}\Big{(}b_{\alpha}^{2}(z_{1})-z_{2}b_{\alpha}^{1}(z_{1})\Big{)}\prod_{\begin{subarray}{c}1\leq
k\leq\ell\\\ k\neq j\end{subarray}}(z_{1}-\tau_{k}),$

for $j=1,\dots,\ell,$ where each $d^{\alpha}_{j}>0$ is chosen so
$c^{\alpha}_{j}\|R_{j}/p(\tau_{j},\cdot)\|^{2}_{H^{2}(\mathbb{D})}=1$ and
$c^{\alpha}_{j}$ is from Proposition 3.9.

###### Proof.

By the proof of Theorem 3.2, the $R_{1},\dots,R_{n}$ from (7) are exactly
obtained by specifying that $R_{1}/p,\dots,R_{n}/p$ be an orthonormal basis
for $\mathcal{H}(K_{2})$. Thus, we need only show that for the $R_{j}$ defined
in (25), $R_{1}/p,\dots,R_{\ell}/p$ are in $\mathcal{H}(K_{2})$ and form an
orthonormal set there.

To that end, as in Theorem 3.2, let $\hat{R}_{1}/p,\dots,\hat{R}_{n}/p$ be
some orthonormal basis for $\mathcal{H}(K_{2})$. Recall that
$\hat{K}_{B_{\alpha}}:=H^{2}(\mathbb{D})\ominus B_{\alpha}H^{2}(\mathbb{D})$
denotes the one variable model space associated to $B_{\alpha}.$ Then
Proposition 3.6 implies that for each $j$, there is a unique polynomial
$\hat{r}_{j}$ with $\deg\hat{r}_{j}\leq n-1$ and function
$b_{j}\in\hat{K}_{B_{\alpha}}$ such that

$\hat{R}_{j}(z)=\hat{r}_{j}(z_{1})\Big{(}1-B_{\alpha}(z_{1})z_{2}\Big{)}+z_{2}Q(z_{1})b_{j}(z_{1}).$

Define a linear map
$T\colon\text{Span}\\{\hat{R}_{1},\dots,\hat{R}_{n}\\}\rightarrow\hat{K}_{B_{\alpha}}$
by first specifying $T(\hat{R}_{j})=b_{j}$ and then extending by linearity. As
$\dim\hat{K}_{B_{\alpha}}=n-\ell$, it follows that $\dim(\ker T)\geq\ell$. If
$R\in\ker(T)$, then for some $r$ with $\deg r<n$,

(26)
$R(z)=r(z_{1})(1-B_{\alpha}(z_{1})z_{2})=\frac{r(z_{1})}{b^{2}_{\alpha}(z_{1})}\Big{(}b_{\alpha}^{2}(z_{1})-z_{2}b_{\alpha}^{1}(z_{1})\Big{)}=q(z_{1})\Big{(}b_{\alpha}^{2}(z_{1})-z_{2}b_{\alpha}^{1}(z_{1})\Big{)},$

where $q\in\mathbb{C}[z]$ with $\deg q<\ell$. Note that the set of such $R$
has dimension $\ell$. By comparing dimensions, each $R$ given in (26) must be
in $\ker(T)$ and hence, each $R$ given in (26) satifies
$R/p\in\mathcal{H}(K_{2})$. In particular, this implies that each $R_{j}$ from
(25) satisfies $R_{j}/p\in\mathcal{H}(K_{2})$.

To show $R_{1}/p,\dots,R_{\ell}/p$ are orthonormal in $K_{\phi}$, we use
Proposition 3.9 and Theorem 3.12. First, observe that those two results
combine to imply that $R_{j}/p(\tau_{j},\cdot)\in
H^{2}(\mathbb{D})\setminus\\{0\\}$, so $d^{\alpha}_{j}$ is well defined. Then,
one can use the fact that each $R_{j}$ vanishes on $E_{\alpha}$ and each
$L_{k}$ with $1\leq k\leq\ell$ and $k\neq j$ to conclude:

$\displaystyle\left\langle\tfrac{R_{i}}{p},\tfrac{R_{j}}{p}\right\rangle_{K_{\phi}}$
$\displaystyle=\left\langle
J_{\alpha}\left(\tfrac{R_{i}}{p}\right),J_{\alpha}\left(\tfrac{R_{j}}{p}\right)\right\rangle_{L^{2}(\sigma_{\alpha})}$
$\displaystyle=\int_{\mathbb{T}}\tfrac{R_{i}}{p}(\zeta,\overline{B_{\alpha}(\zeta)})\overline{\tfrac{R_{j}}{p}(\zeta,\overline{B_{\alpha}(\zeta)})}d\nu_{\alpha}(\zeta)+\sum_{k=1}^{\ell}c^{\alpha}_{k}\int_{\mathbb{T}}\tfrac{R_{i}}{p}(\tau_{k},\zeta)\overline{\tfrac{R_{j}}{p}(\tau_{k},\zeta)}\
dm(\zeta)$ $\displaystyle=0+\sum_{k=i\text{ or
}k=j}c^{\alpha}_{k}\int_{\mathbb{T}}\tfrac{R_{i}}{p}(\tau_{k},\zeta)\overline{\tfrac{R_{j}}{p}(\tau_{k},\zeta)}\
dm(\zeta)$ $\displaystyle=\left\\{\begin{array}[]{cc}1&\text{ if }i=j\\\
0&\text{ if }i\neq j.\end{array}\right.$

Thus, $\\{R_{1}/p,\dots,R_{\ell}/p\\}$ is an orthonormal set in $K_{\phi}$ and
hence in $\mathcal{H}(K_{2})$, which completes the proof. ∎

###  4.2. Extreme Points

Recall that $\mathcal{P}_{2}=\\{f\in\text{Hol}(\mathbb{D}^{2}):\Re
f(z)>0\text{ and }f(0,0)=1\\}$ and $M:\mathcal{P}_{2}\rightarrow
P(\mathbb{T}^{2})$ is the map that takes $f\in\mathcal{P}_{2}$ to the unique
Borel probability measure $\mu_{f}$ on $\mathbb{T}^{2}$ with

$f(z)=\int_{\mathbb{T}^{2}}P_{z}(\zeta)d\mu_{f}(\zeta)\quad\text{ for
}z\in\mathbb{D}^{2}.$

for some $f\in\mathcal{P}_{2}$ and $f$ is an extreme point of
$\mathcal{P}_{2}$ if and only if $\mu_{f}$ is an extreme point of
$M(\mathcal{P}_{2})$. As mentioned in the introduction, Forelli, McDonald, and
Knese have proved a number of interesting results related to such extreme
points. For example, Knese proved the following result in [22, Theorem 1.5]:

######  Theorem 4.2.

Let $q$ be a polynomial with no zeros on $\mathbb{D}^{2}$ and let $\tilde{q}$
be the reflection of $q$ with $\deg\tilde{q}=\deg q$. Assume that $q$ is
$\mathbb{T}^{2}$-saturated, $\tilde{q},q$ share no common factors,
$\tilde{q}(0,0)=0$, and $q-\tilde{q}$ is irreducible. Then
$f:=\frac{q+\tilde{q}}{q-\tilde{q}}$ is an extreme point of $\mathcal{P}_{2}$.

As mentioned in the introduction, our results in the $(n,1)$ setting coupled
with Theorem 4.2 yield Corollary 1.4, which we restate here for convenience.

######  Corollary.

1.4. Assume $\tilde{p}(0,0)=0$ and let $\alpha\in\mathbb{T}$. Then

  * i.

If $\alpha$ is an exceptional value for $\phi$, then $\sigma_{\alpha}$ is not
an extreme point of $M(\mathcal{P}_{2})$.

  * ii.

If $\deg p=\deg\tilde{p}$, $p$ is $\mathbb{T}^{2}$-saturated, and $\alpha$ is
generic for $\phi$, then $\sigma_{\alpha}$ is an extreme point of
$M(\mathcal{P}_{2})$.

###### Proof.

For (i), without loss of generality, assume
$\phi^{*}(\tau_{k},\lambda_{k})=\alpha$ for $k=1,\dots,\ell$. By Proposition
3.9, we can write

$\sigma_{\alpha}(\zeta)=\mu_{\alpha}(\zeta)+c^{\alpha}_{1}\left(\delta_{\tau_{1}}(\zeta_{1})\otimes
m(\zeta_{2})\right),$

for a positive Borel measure $\mu_{\alpha}$ on $\mathbb{T}^{2}$ and
$c^{\alpha}_{1}>0$. As $\phi(0,0)=0$, we have

$1=\sigma_{\alpha}(\mathbb{T}^{2})=\mu_{\alpha}(\mathbb{T}^{2})+c_{1}^{\alpha},$

and as $\mu_{\alpha}(\mathbb{T}^{2})>0$, we have $c_{1}^{\alpha}<1$. Then
$\hat{\mu}_{\alpha}:=\frac{1}{1-c^{\alpha}_{1}}\mu_{\alpha}$ is a probability
measure and

(27)
$\sigma_{\alpha}(\zeta)=(1-c^{\alpha}_{1})\hat{\mu}_{\alpha}(\zeta)+c^{\alpha}_{1}\left(\delta_{\tau_{1}}(\zeta_{1})\otimes
m(\zeta_{2})\right),$

so $\sigma_{\alpha}$ is a convex combination of two probability measures on
$\mathbb{T}^{2}$. Clearly, the second one is in $M(\mathcal{P}_{2})$, as

$\Re\left(\frac{\tau_{1}+z_{1}}{\tau_{1}-z_{1}}\right)=\frac{1-|z_{1}|^{2}}{|z_{1}-\tau_{1}|^{2}}=\int_{\mathbb{T}^{2}}P_{z}(\zeta)\
d\left(\delta_{\tau_{1}}(\zeta_{1})\otimes m(\zeta_{2})\right).$

For the first, observe that for each $z\in\mathbb{D}^{2}$,

$\tfrac{1}{1-c^{\alpha}_{1}}\Re\left(\frac{\alpha+\phi(z)}{\alpha-\phi(z)}-c^{\alpha}_{1}\frac{\tau_{1}+z_{1}}{\tau_{1}-z_{1}}\right)=\int_{\mathbb{T}^{2}}P_{z}(\zeta)d\hat{\mu}_{\alpha}(\zeta)>0.$

This implies that $\hat{\mu}_{\alpha}\in M(\mathcal{P}_{2})$ and by (27),
$\sigma_{\alpha}$ is not an extreme point in $M(\mathcal{P}_{2})$.

For (ii), choose $\lambda\in\mathbb{T}$ with $\lambda^{2}=\alpha$, define
$q=\lambda p$, and set

$f:=\frac{\alpha+\phi}{\alpha-\phi}=\frac{\alpha p+\tilde{p}}{\alpha
p-\tilde{p}}=\frac{q+\tilde{q}}{q-\tilde{q}}.$

Note that $\tilde{q}-q=\bar{\lambda}(\tilde{p}-\alpha p)$ must be irreducible
by the characterization of $C_{\alpha}$ from Theorem 3.3. By Theorem 4.2, $f$
is an extreme point of $\mathcal{P}_{2}$ and so $\sigma_{\alpha}$ from Theorem
1.2 is extreme in $M(\mathcal{P}_{2}).$ ∎

##  5\. Examples

We illustrate our results by examining some specific RIFs and their associated
Clark measures in detail. For the first example, we can confirm our general
findings at exceptional values $\alpha$ via direct computation.

######  Example 5.1.

Let

$\phi(z)=\frac{\tilde{p}(z)}{p(z)}=\frac{2z_{1}z_{2}-z_{1}-z_{2}}{2-z_{1}-z_{2}},$

essentially the example considered in [12]. We have the sums of squares
decomposition

$|p(z)|^{2}-|\tilde{p}(z)|^{2}=(1-|z_{2}|^{2})2|1-z_{1}|^{2}+(1-|z_{1}|^{2})2|1-z_{2}|^{2}$

and for each $\alpha\in\mathbb{T}$, the associated $B_{\alpha}$ is

$B_{\alpha}(z_{1})=\frac{2z_{1}-1+\alpha}{2\alpha-\alpha z_{1}+z_{1}}.$

Note that if $\alpha=-1$, then $B_{-1}\equiv 1$. If $\alpha\neq-1$, then
$2\alpha-\alpha z_{1}+z_{1}$ does not vanish on $\mathbb{T}$. Thus if
$\alpha\neq-1$, then by Proposition 3.8, for all $f\in
L^{2}(\sigma_{\alpha})$, we have

$\int_{\mathbb{T}^{2}}f(\zeta)d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}}f(\zeta,\overline{B_{\alpha}(\zeta)})\frac{2|1-\zeta|^{2}}{|2\zeta-1+\alpha|^{2}}\
dm(\zeta),$

and by Theorem 3.11, the isometric embedding $J_{\alpha}:K_{\phi}\rightarrow
L^{2}(\sigma_{\alpha})$ is unitary. Finally, if $\alpha=-1$, then
$|2z_{1}-1+\alpha|^{2}=4|z_{1}-1|^{2}$. By the given sums of squares
decomposition,

$\displaystyle\frac{1-|\phi(z)|^{2}}{|\alpha-\phi(z)|^{2}}$
$\displaystyle=\frac{|p(z)|^{2}-|\tilde{p}(z)|^{2}}{|\alpha
p(z)-\tilde{p}(z)|^{2}}$
$\displaystyle=\frac{|p(z)|^{2}-|\tilde{p}(z)|^{2}}{|z_{2}-1|^{2}|-p_{2}(z_{1})-\tilde{p}_{1}(z_{1})|^{2}}$
$\displaystyle=\frac{(1-|z_{2}|^{2})2|1-z_{1}|^{2}+(1-|z_{1}|^{2})2|1-z_{2}|^{2}}{|z_{2}-1|^{2}\cdot
4|z_{1}-1|^{2}}$
$\displaystyle=\frac{1}{2}\left(\frac{1-|z_{2}|^{2}}{|z_{2}-1|^{2}}+\frac{1-|z_{1}|^{2}}{|z_{1}-1|^{2}}\right),$

which shows $\sigma_{\alpha}=\frac{1}{2}(\delta_{1}(\zeta_{1})\otimes
m(\zeta_{2})+m(\zeta_{1})\otimes\delta_{1}(\zeta_{2})).$ This was observed in
[12], and confirms the contents of Theorem 1.2. Note in particular that
$\frac{\partial\phi}{\partial
z_{1}}(z_{1},z_{2})=-2\frac{(z_{2}-1)^{2}}{(2-z_{1}-z_{2})^{2}}$, so that
$\frac{\partial\phi}{\partial z_{1}}(1,z_{2})=-2$ independent of $z_{2}$.

See Figure 2(a) for a visual representation of the sets
$\mathcal{C}_{\alpha}$. $\hfill\blacklozenge$

Now let us consider a RIF that was not studied in [12], and again illustrate
how the exceptional measure $\sigma_{\alpha}$ can be identified using both our
results and concrete Agler decompositions.

######  Example 5.2.

Let $\phi=\frac{\tilde{p}}{p}$, where

$p(z)=4-z_{2}-3z_{1}-z_{1}z_{2}+z_{1}^{2}\text{ and
}\tilde{p}(z)=4z_{1}^{2}z_{2}-z_{1}^{2}-3z_{1}z_{2}-z_{1}+z_{2}.$

This example was introduced by Agler-McCarthy-Young in [2]. In [21, Section
15], Knese provides the following sums of squares decomposition:

$|p(z)|^{2}-|\tilde{p}(z)|^{2}=4(1-|z_{2}|^{2})|1-z_{1}|^{4}+4(1-|z_{1}|^{2})\left(|1-z_{1}|^{2}|1-z_{2}|^{2}+2|1-z_{1}z_{2}|^{2}\right).$

The only singularity of $\phi$ occurs at $(1,1)$. For each
$\alpha\in\mathbb{T},$ setting $\phi(z)=\alpha$ and solving for $z_{2}$ yields
$z_{2}=1/B_{\alpha}(z_{1})$, where

$B_{\alpha}(z_{1})=\frac{4z_{1}^{2}-3z_{1}+1+\alpha+\alpha
z_{1}}{4\alpha-3z_{1}\alpha+z_{1}^{2}\alpha+z_{1}^{2}+z_{1}}.$

As $\phi$ has only one singularity, by previous discussions, the denominator
of $B_{\alpha}$ can vanish at a point on $\mathbb{T}$ for at most one
$\alpha$. This occurs at $\alpha=-1$, where $B_{-1}$ reduces to $B_{-1}(z)=z$
and $\phi=-1$ has the additional solution $z_{1}=1$. Thus, we can apply
Proposition 3.8 to $\alpha\neq-1$ to obtain: for all $f\in
L^{1}(\sigma_{\alpha})$,

$\int_{\mathbb{T}^{2}}f(\zeta)d\sigma_{\alpha}(\zeta)=\int_{\mathbb{T}}f(\zeta,\overline{B_{\alpha}(\zeta)})\
\frac{4|\zeta-1|^{4}}{|4\zeta^{2}-3\zeta+1+\alpha+\alpha\zeta|^{2}}\
dm(\zeta).$

By Theorem 3.11, the isometric embedding $J_{\alpha}:K_{\phi}\rightarrow
L^{2}(\sigma_{\alpha})$ is unitary for every $\alpha\neq-1$.

(a) Level curves for $\phi=(2z_{1}z_{2}-z_{1}-z_{2})/(2-z_{1}-z_{2})$
corresponding to $\alpha=1$ (black), $\alpha=e^{i\pi/4}$ (gray),
$\alpha=e^{i\pi/2}$ (orange), and $\alpha=e^{3i\pi/4}$ (pink). Level set
corresponding to exceptional value $\alpha=-1$ marked in red. (b) Level curves
for
$\phi=(4z_{1}^{2}z_{2}-z_{1}^{2}-3z_{1}z_{2}-z_{1}+z_{2})/(4-z_{2}-3z_{1}-z_{1}z_{2}+z_{1}^{2})$
corresponding to $\alpha=1$ (black), $\alpha=e^{i\pi/4}$ (gray),
$\alpha=e^{i\pi/2}$ (orange), and $\alpha=e^{3i\pi/4}$ (pink). Level set
corresponding to exceptional value $\alpha=-1$ marked in red.

Figure 2. Supports of $\sigma_{\alpha}$ for two different RIFs.

Let us now examine the exceptional value $\alpha=-1=\phi^{*}(1,1)$. A
computation shows

$p(z)+\tilde{p}(z)=4(1-z_{1})(1-z_{1}z_{2}),$

and hence, for $\alpha=-1$, we have

$\frac{1-|\phi(z)|^{2}}{|\alpha-\phi(z)|^{2}}=\frac{|p(z)|^{2}-|\tilde{p}(z)|^{2}}{16|1-z_{1}|^{2}|1-z_{1}z_{2}|^{2}}.$

By the sums of squares formula above, we then obtain

$\begin{aligned}
\frac{1-|\phi(z)|^{2}}{|\alpha-\phi(z)|^{2}}=&\frac{1}{4}|1-z_{1}|^{2}\frac{1-|z_{2}|^{2}}{|1-z_{1}z_{2}|^{2}}+\frac{1}{4}|1-z_{2}|^{2}\frac{1-|z_{1}|^{2}}{|1-z_{1}z_{2}|^{2}}+\frac{1}{2}\frac{1-|z_{1}|^{2}}{|1-z_{1}|^{2}}\\\
=&\frac{1}{4}\frac{(1-|z_{1}|^{2})|1-z_{2}|^{2}+(1-|z_{2}|^{2})|1-z_{1}|^{2}}{|1-z_{1}z_{2}|^{2}}\\\
&+\frac{1}{2}\frac{1-|z_{1}|^{2}}{|1-z_{1}|^{2}}\end{aligned}.$

The second term is evidently the Poisson integral of the measure
$\sigma_{-1}^{(2)}=\frac{1}{2}(\delta_{1}(\zeta_{1})\otimes
m_{1}(\zeta_{2}))$, which matches what we get from computing
$\frac{\partial\phi}{\partial z_{1}}(1,z_{2})=-2$ and taking the reciprocal of
its absolute value.

The first term arises from the measure $\sigma_{-1}^{(1)}$ on $\mathbb{T}^{2}$
having

$\int_{\mathbb{T}^{2}}f(\zeta)d\sigma_{-1}^{(1)}(\zeta)=\tfrac{1}{4}\int_{\mathbb{T}}f(\zeta,\overline{\zeta})|1-\zeta|^{2}dm(\zeta),$

as can be seen by examining the Fourier coefficients

$\widehat{\sigma_{-1}^{(1)}}(k,l)=\left\\{\begin{array}[]{cc}\frac{1}{2},&k=l\\\
-\frac{1}{4},&k=l+1\\\ -\frac{1}{4},&l=k+1\\\
0&\mathrm{otherwise}\end{array}\right.$

and computing the Poisson integral of $\sigma_{-1}^{(1)}$ explicitly. The
specific form of $\sigma_{-1}^{(1)}$ of course agrees with Theorem 1.2 once we
set $\alpha=-1$ to get
$W_{-1}(\zeta)=\frac{4|\zeta-1|^{2}}{|4\zeta^{2}-4\zeta|^{2}}=\frac{1}{4}|\zeta-1|^{2}$.

Level curves $\mathcal{C}_{\alpha}$ for several values of $\alpha$ are
displayed in Figure 2(b).$\hfill\blacklozenge$

######  Remark 5.3.

The RIF $\phi=\frac{\tilde{p}}{p}$ with

$p(z)=2-z_{1}z_{2}-z_{1}^{2}z_{2}\quad\textrm{and}\quad\tilde{p}(z)=2z_{1}^{2}z_{2}-z_{1}-1$

has a singularity at $(1,1)$, and $\phi^{*}(1,1)=-1$ so that $\alpha=-1$ is an
exceptional value.

One verifies that the associated $B_{-1}(z_{1})=z_{1}$ so that $\sigma_{-1}$
for this example is supported on the same set as the exceptional Clark measure
in Example 5.2. However, we have

$W_{\alpha}(\zeta)=\frac{|\zeta-1|^{2}}{|(2+\alpha)\zeta+\alpha|^{2}},$

which collapses to $W_{-1}(\zeta)=1$ at the exceptional value, meaning that
the two Clark measures do not coincide.

Our next example is a degree $(3,1)$ RIF with two different singularities on
$\mathbb{T}^{2}$. Here, we are able to observe qualitative differences in
$W_{\alpha}$ for the two corresponding exceptional values of $\alpha$ that
reflect the finer distinctions between the two singularities.

######  Example 5.4.

Let

$p(z)=4-z_{2}+z_{1}z_{2}-3z_{1}^{2}z_{2}-z_{1}^{3}z_{2}\quad\textrm{and}\quad\tilde{p}(z)=4z_{1}^{3}z_{2}-z_{1}^{3}+z_{1}^{2}-3z_{1}-1$

and set $\phi=\frac{\tilde{p}}{p}$. This function has singularities at $(1,1)$
and $(-1,1)$, and the associated exceptional $\alpha$-values are
$\phi^{*}(1,1)=-1$ and $\phi^{*}(-1,1)=1$. Level sets for this example are
displayed in Figure 3; see also [9, Example 7.4].

Figure 3. Generic level curves for
$(4z_{1}^{3}z_{2}-z_{1}^{3}+z_{1}^{2}-3z_{1}-1)/(4-z_{2}+z_{1}z_{2}-3z_{1}^{2}z_{2}-z_{1}^{3}z_{2})$
corresponding to several values of $\alpha$ (black, gray, orange, pink). Level
sets corresponding to exceptional values $\alpha=-1$ and $\alpha=1$ marked in
green and red, respectively.

For $\alpha\neq 1,-1$, we have

$B_{\alpha}(z_{1})=\frac{\alpha-\alpha z_{1}+3\alpha
z_{1}^{2}+4z_{1}^{3}+\alpha z_{1}^{3}}{1+4\alpha+3z_{1}-z_{1}^{2}+z_{1}^{3}}.$

Note that for $\alpha=-1$, we get

$B_{-1}(z_{1})=\frac{z_{1}-1}{z_{1}-1}\frac{3z_{1}^{2}+1}{3+z_{1}^{2}}=\frac{3z_{1}^{2}+1}{3+z_{1}^{2}},$

a Blaschke product of degree $2$, while for $\alpha=1$,

$B_{1}(z_{1})=\frac{z_{1}+1}{z_{1}+1}\frac{5z_{1}^{2}-2z_{1}+1}{z_{1}^{2}-2z_{1}+5}=\frac{5z_{1}^{2}-2z_{1}+1}{z_{1}^{2}-2z_{1}+5},$

another degree $2$ Blaschke product. The graphs
$\\{(\zeta,\overline{B_{-1}(\zeta)})$ and
$\\{(\zeta,\overline{B_{1}(\zeta)})\\}$ together with vertical lines at
$\zeta_{1}=1$ and $\zeta_{1}=-1$ constitute $\mathrm{supp}(\sigma_{-1})$ and
$\mathrm{supp}(\sigma_{1})$, respectively.

We further read off that

$p_{1}(z_{1})=4\quad\mathrm{and}\quad
p_{2}(z_{1})=-1+z_{1}-3z_{1}^{2}-z_{1}^{3}$

so that, with $W_{\alpha}$ as in Remark 3.1,

$W_{\alpha}(\zeta)=\frac{16-|1-\zeta+3\zeta^{2}+\zeta^{3}|^{2}}{|4\zeta^{3}+\alpha\zeta^{3}+3\alpha\zeta^{2}-\alpha\zeta+\alpha|^{2}}.$

After some simplifications, we find that

$W_{\alpha}(z)=\frac{|\zeta-1|^{2}|\zeta+1|^{4}}{|4\zeta^{3}+\alpha\zeta^{3}+3\alpha\zeta^{2}-\alpha\zeta+\alpha|^{2}}.$

For the exceptional values $\alpha=\pm 1$, the weights in the point mass parts
of $\sigma_{\pm 1}$ can be obtained by computing

$\frac{\partial\phi}{\partial
z_{1}}(1,z_{2})=-1\quad\textrm{and}\quad\frac{\partial\phi}{\partial
z_{1}}(-1,z_{2})=-2,$

which imply

$c^{-1}_{1}=\frac{1}{|\frac{\partial\phi}{\partial
z_{1}}(1,z_{2})|}=1\quad\textrm{and}\quad
c^{1}_{1}=\frac{1}{|\frac{\partial\phi}{\partial
z_{1}}(-1,z_{2})|}=\frac{1}{2}.$

(Note that $\phi(0,0)=-\frac{1}{4}$ so that the Clark measures $\sigma_{\pm
1}$ are not probability measures in this example.) Putting $\alpha=\pm 1$ in
$W_{\alpha}$, we have cancellation in numerator and denominator, and we obtain

$W_{-1}(\zeta)=\frac{|\zeta+1|^{4}}{|3\zeta^{2}+1|^{2}}$

and

$W_{1}(\zeta)=\frac{|\zeta-1|^{2}|\zeta+1|^{2}}{|5\zeta^{2}-2\zeta^{2}+1|^{2}}.$

This gives us a complete description of the exceptional Clark measures.

Furthermore, observe that, $W_{-1}(1)\neq 0$ and so, $W_{-1}$ does not vanish
at the $z_{1}$-coordinate of the singularity with non-tangential value $-1.$
In contrast, $W_{1}(-1)=0$, so $W_{1}$ does vanish at the $z_{1}$-coordinate
of the singularity with non-tangential value $1.$ This mirrors the singular
behavior in Example 5.2, where function $W_{-1}$ vanishes at $\zeta=1$, the
$z_{1}$-coordinate of the singularity where $\phi^{*}(1,1)=-1$. This pattern
suggests a connection with contact order, which was studied in [8] and governs
the integrability of partial derivatives of a RIF $\phi$; in that sense,
higher contact order indicates a stronger singularity. In our computations,
the singularities at $(1,1)$ in Example 5.2 and at $(-1,1)$ in this example
(where the exceptional $W_{\alpha}$ vanish at the $z_{1}$-coordinate of the
associated singularity) are instances of singularities where $\phi$ exhibits
contact order $4$; the singularities in Example 5.1 and at $(1,1)$ in this
example (where the exceptional $W_{\alpha}$ do not vanish at the
$z_{1}$-coordinate of the associated singularity) are singularities where
$\phi$ exhibits contact order $2$, the lowest possible contact order.
$\hfill\blacklozenge$

######  Remark 5.5.

It would interesting to investigate how the exact nature of a singularity
$\tau\in\mathbb{T}^{2}$ (contact order, number of branches of $p$ coming
together at $\tau$, etc) of a RIF is reflected in the associated exceptional
Clark measure. For example, if $\phi=\frac{\tilde{p}}{p}$ is a general degree
$(m,n)$ RIF having contact order at least $4$, does the corresponding
exceptional Clark measure have a density along $\mathcal{C}_{\alpha}$ that
vanishes at $\tau$?

Our final example is a rational inner function having bidegree $(3,3)$, and is
not covered by our general results. It serves as a counterexample showing that
Theorem 3.11 fails for higher-degree RIFs, and illustrates some complexities
that arise from the fact that for RIFs of bidegree $(m,n)$ with $m,n\geq 2$, a
general $\alpha$-level set is not necessarily parametrized by a single
function.

######  Example 5.6.

Let $\phi(z)=\frac{\tilde{p}}{p}(z)$ where

$p(z)=2-z_{1}^{2}z_{2}-z_{1}z_{2}^{2}\quad\textrm{and}\quad\tilde{p}(z)=z_{1}z_{2}(2z_{1}^{2}z_{2}^{2}-z_{1}-z_{2}).$

This example is obtained by applying the level line embedding construction
described in [9, Section 6.1] to the essentially $\mathbb{T}^{2}$-symmetric
polynomial

$r(z)=(1-z_{1}^{2}z_{2})(1-z_{1}z_{2}^{2}).$

As is guaranteed by the embedding construction, we have
$p(1,1)=0=\tilde{p}(1,1)$ and $\phi^{*}(1,1)=-1$, as well as

$\tilde{p}+p=2(1-z_{1}^{2}z_{2})(1-z_{1}z_{2}^{2}).$

These facts can also be checked directly. We also note that $p$ and
$\tilde{p}$, and hence $\phi$, are invariant under the simultaneous
coordinatewise rotations $z_{j}\mapsto e^{2i\pi/3}z_{j}$ and $z_{j}\mapsto
e^{-2i\pi/3}z_{j}$. Some level sets of $\phi$ are displayed in Figure 4.

Recall from Example 5.1 that $|2-x-y|^{2}-|2xy-
x-y|^{2}=(1-|x|^{2})2|1-y|^{2}+(1-|y|^{2})2|1-x|^{2}$. Substituting
$x=z_{1}^{2}z_{2}$ and $y=z_{1}z_{2}^{2}$ into this formula, we get the
decomposition

$|2-z_{1}^{2}z_{2}-z_{1}z_{2}^{2}|^{2}-|2z_{1}^{3}z_{2}^{3}-z_{1}^{2}z_{2}-z_{1}z_{2}^{2}|^{2}=(1-|z_{1}^{2}z_{2}|^{2})2|1-z_{1}z_{2}^{2}|^{2}+(1-|z_{1}z_{2}^{2}|^{2})2|1-z_{1}^{2}z_{2}|.$

It follows that

$\frac{1-|\phi(z)|^{2}}{|1+\phi(z)|^{2}}=\frac{1}{2}\frac{1-|z_{1}^{2}z_{2}|^{2}}{|1-z_{1}^{2}z_{2}|^{2}}+\frac{1}{2}\frac{1-|z_{1}z_{2}^{2}|^{2}}{|1-z_{1}z_{2}^{2}|^{2}},$

and by computing Fourier coefficients, one can show that the two expressions
on the right are the Poisson integrals of the measures having

$\int_{\mathbb{T}^{2}}f(\zeta)d\sigma_{-1}^{(1)}=\int_{\mathbb{T}}f(\zeta,\overline{\zeta}^{2})dm(\zeta)\quad\textrm{and}\quad\int_{\mathbb{T}^{2}}f(\zeta)d\sigma_{-1}^{(2)}=\int_{\mathbb{T}}f(\overline{\zeta}^{2},\zeta)dm(\zeta)$

respectively. By Doubtsov’s Theorem $3.2$ in [12], which applies to general
RIFs, $J_{-1}$ is unitary if and only if the bidisk algebra is dense in
$L^{2}(\sigma_{-1})$.

Figure 4. Level curves for
$(2z_{1}^{3}z_{2}^{3}-z_{1}^{2}z_{2}-z_{1}z_{2}^{2})/(2-z_{1}^{2}z_{2}-z_{1}z_{2}^{2})$
corresponding to $\alpha=1$ (black) and $\alpha=e^{\pi i/2}$ (orange). Level
set corresponding to exceptional value $\alpha=-1$ marked in red.

Let us show that this is indeed the case. By definition, $h_{1}(z)=z_{1}$ and
$h_{2}(z)=z_{2}$ are elements of $A(\mathbb{D}^{2})$. Next consider the
function $g_{1}(z)=\bar{z}_{1}$ and the function

$f_{1}(z)=z_{1}z_{2}+(1-z_{1}^{2}z_{2})z_{2}^{2}\in A(\mathbb{D}^{2}).$

Since

$f_{1}(\zeta,\bar{\zeta}^{2})=\zeta\bar{\zeta}^{2}+(1-\bar{\zeta}^{2}\zeta^{2})\bar{\zeta}^{4}=\bar{\zeta}=g_{1}(\zeta,\bar{\zeta}^{2})$

and

$f_{1}(\bar{\zeta}^{2},\zeta)=\bar{\zeta}^{2}\zeta+(1-\zeta\bar{\zeta}^{4})\zeta^{2}=\zeta^{2}=g_{1}(\bar{\zeta}^{2},\zeta)$

we have $g_{1}=f_{1}$ on the support of $\sigma_{-1}$. A similar computation
shows that the bidisk algebra function

$f_{2}(z)=z_{1}z_{2}+(1-z_{1}z_{2}^{2})z_{1}^{2}$

coincides with $g_{2}(z)=\bar{z}_{2}$ on $\mathrm{supp}(\sigma_{-1})$. Thus,
if $g(\zeta)=\zeta_{1}^{m}\zeta_{2}^{n}$ is any trigonometric polynomial,
then, on the support of $\sigma_{-1}$, $g$ coincides with one of functions
$h_{1}^{|m|}h_{2}^{|n|}$, $h_{1}^{|m|}f_{2}^{|n|}$, $f_{1}^{|m|}h_{2}^{|n|}$,
and $f_{1}^{|m|}f_{2}^{|n|}$, which are all in $A(\mathbb{D}^{2})$. Since the
trigonometric polynomials are dense in $C(\mathbb{T}^{2})$, which in turn is
dense in $L^{2}(\sigma_{-1}$), $A(\mathbb{D}^{2})$ is also dense. Thus,
$J_{-1}$ is unitary even though $\alpha=-1$ is the non-tangential value of
$\phi$ at a singularity. $\hfill\blacklozenge$

## References

  * [1] J. Agler, J.E. Mc Carthy, and M. Stankus, Toral algebraic sets and function theory on polydisks, J. Geom. Anal. 16 (2006), no. 4, 551–562.
  * [2] J. Agler, J.E. McCarthy, and N.J. Young, A Carathéodory theorem for the bidisk via Hilbert space methods, Math. Ann. 352 (2012), 581-624.
  * [3] A.B. Aleksandrov and E. Doubtsov, Clark measures on the complex sphere, J. Funct. Anal. 278 (2020), 108314.
  * [4] J.A. Ball, C. Sadosky, and V. Vinnikov, Scattering systems with several evolutions and multidimensional input/state/output systems. Integral Equations Operator Theory 52 (2005), no. 3, 323–393.
  * [5] K. Bickel, Fundamental Agler decompositions. Integral Equations Operator Theory 74 (2012), no. 2, 233–257.
  * [6] Kelly Bickel and Greg Knese, Inner functions on the bidisk and associated Hilbert spaces, J. Funct. Anal. 265 (2013), no. 11, 2753–2790.
  * [7] K. Bickel and P. Gorkin, Compressions of the shift on the bidisk and their numerical ranges. J. Operator Theory 79 (2018), no. 1, 225–265.
  * [8] K. Bickel, J.E. Pascoe, and A. Sola, Derivatives of rational inner functions: geometry of singularities and integrability at the boundary, Proc. London Math. Soc. 116 (2018), 281-329.
  * [9] K. Bickel, J.E. Pascoe, and A. Sola, Level curve portraits of rational inner functions, Ann. Sc. Norm. Sup. Pisa Cl. Sc. XXI (2020), 451-494.
  * [10] K. Bickel, J.E. Pascoe, and A. Sola, Singularities of rational inner functions in higher dimensions. American J. Math, to appear. Preprint available at https://arxiv.org/pdf/1906.10913.pdf.
  * [11] J.A. Cima, A.L. Matheson, and W.T. Ross, The Cauchy transform, Math. Surveys and Monographs 125, Amer. Math. Soc., Providence, RI, 2006.
  * [12] E. Doubtsov, Clark measures on the torus. Proc. Amer. Math. Soc. 148 (2020), no. 5, 2009–2017.
  * [13] R.M. Dudley, _Real Analysis and Probability._ Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.
  * [14] G. Folland, Real analysis. _Modern techniques and their applications._ Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.
  * [15] F. Forelli, A necessary condition on the extreme points of a class of holomorphic functions, II, Pacific J. Math. 92 (1981), no. 2, 277–281.
  * [16] S.R. Garcia, J. Mashreghi, and W.T. Ross, Introduction to model spaces and their operators, _Cambridge studies in advanced mathematics_ 148, Cambridge University Press, 2016.
  * [17] S.R. Garcia, J. Mashreghi, and W.T. Ross, Finite Blaschke products: a survey, in _Harmonic analysis, function theory, operator theory, and their applications_ , Theta Ser. Adv. Math. 19, Theta, Bucharest, 2017, 133-158.
  * [18] M.T. Jury, Clark theory in the Drury-Arveson space, J. Funct. Anal. 266 (2014), 3855-3893.
  * [19] G. Knese, Polynomials with no zeros on the bidisk, Anal. PDE 3 (2010), no. 2, 109–149.
  * [20] G. Knese, Rational inner functions in the Schur-Agler class of the polydisk. Publ. Mat. 55 (2011), no. 2, 343-357.
  * [21] G. Knese, Integrability and regularity of rational functions, Proc. London. Math. Soc. 111 (2015), 1261-1306.
  * [22] G. Knese, Extreme points and saturated polynomials. Illinois J. Math. 63 (2019), no. 1, 47–74.
  * [23] A. Luger and M. Nedic, Geometric properties of measures related to holomorphic functions having positive imaginary or real part, 2018, to appear in J. Geom. Anal.
  * [24] R.A. Martínez-Avendaño and P. Rosenthal, _An introduction to operators on the Hardy-Hilbert space._ Graduate Texts in Mathematics, 237. Springer, New York, 2007.
  * [25] J.N. McDonald, Measures on the torus which are real parts of holomorphic functions, Michigan Math. J. 29 (1982), 259–265.
  * [26] J.N. McDonald, Examples of RP-measures, Rocky Mountain J. Math. 16 (1986), 191–201.
  * [27] J.N. McDonald, Holomorphic functions on the polydisc having positive real part, Michigan Math. J. 34 (1987), no. 1, 77–84.
  * [28] J.N. McDonald, An extreme absolutely continuous $RP$-measure, Proc. Amer. Math. Soc. 109 (1990), 731–738.
  * [29] J.E. Pascoe, A wedge-of-the-edge theorem: analytic continuation of multivariable Pick functions in and around the boundary, Bull. London Math. Soc. 49, 916-925.
  * [30] A.G. Poltoratski, Boundary behavior of pseudocontinuable functions. (Russian) Algebra i Analiz 5 (1993), no. 2, 189–210; translation in St. Petersburg Math. J. 5 (1994), no. 2, 389–40.
  * [31] W. Rudin, _Function Theory in polydisks_ , W. A. Benjamin, Inc., New York-Amsterdam, 1969.
  * [32] W. Rudin, “Harmonic analysis in polydiscs” in Actes du Congrès International des Mathématiciens (Nice, 1970), Gauthier-Villars, Paris, 1971, 489–493.
  * [33] W. Rudin and E. L. Stout, Bounday properties of functions of several complex variables, J. Math. Mech. 14 (1965), 991–1005.
  * [34] D. Sarason, _Sub-Hardy Hilbert spaces in the unit disk._ University of Arkansas Lecture Notes in the Mathematical Sciences, 10. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1994.
  * [35] A. Zygmund, _Trigonometric series. Vol. I, II_. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002.