Clark Measures on Polydiscs Associated to Product Functions and Multiplicative Embeddings

Abstract

We study Clark measures on the unit polydisc, giving an overview of recent research and investigating the Clark measures of some new examples of multivariate inner functions. In particular, we study the relationship between Clark measures and multiplication; first by introducing compositions of inner functions and multiplicative embeddings, and then by studying products of one-variable inner functions.

1 Introduction

Let

$$\begin{aligned} \mathbb {D}^d:= \{ (z_1, z_2, \ldots , z_d) \in \mathbb {C}^d: |z_j| < 1, \quad j = 1,2,\ldots , d \} \end{aligned}$$

denote the unit polydisc in d variables, and

$$\begin{aligned} \mathbb {T}^d:= \{ (\zeta _1, \zeta _2, \ldots , \zeta _d) \in \mathbb {C}^d: |\zeta _j| = 1, \quad j = 1,2,\ldots , d \} \end{aligned}$$

its distinguished boundary. Note that this is only a subset of the boundary $\partial \mathbb {D}^d$. For $d=2$, the set $\mathbb {T}^2$ defines a two-dimensional torus.

For $z \in \mathbb {D}^d$ and $\zeta \in \mathbb {T}^d$, we introduce the Poisson kernel on $\mathbb {D}^d$ as a product of one-variable Poisson kernels:

$$\begin{aligned} P_z( \zeta ) = P(z, \zeta ) := \prod _{j=1}^d P_{z_j}(\zeta _j) \quad \text {where} \quad P_{z_j}( \zeta _j) := \frac{1-|z_j|^2}{|\zeta _j-z_j|^2}. \end{aligned}$$

Given a complex Borel measure $\mu$ on $\mathbb {T}^d$, we define its Poisson integral as

$$\begin{aligned} P[d \mu ](z):= \int _{\mathbb {T}^d} P(z,\zeta ) d \mu (\zeta ), \quad z \in \mathbb {D}^d. \end{aligned}$$

If $\phi : \mathbb {D}^d \rightarrow \mathbb {D}$ is a bounded holomorphic function, then

$$\begin{aligned} \Re \bigg ( \frac{\alpha + \phi (z)}{\alpha - \phi (z)} \bigg ) = \frac{1-|\phi (z)|^2}{|\alpha -\phi (z)|^2} \end{aligned}$$

is positive and pluriharmonic (i.e. locally the real part of an analytic function) on $\mathbb {D}^d$. Hence, by Herglotz’ theorem, there exists a unique positive Borel measure $\sigma _\alpha$ on $\mathbb {T}^d$ such that

$$\begin{aligned} \frac{1-|\phi (z)|^2}{|\alpha -\phi (z)|^2} = P[d \sigma _\alpha ](z) = \int _{\mathbb {T}^d} P(z,\zeta ) d \sigma _\alpha (\zeta ). \end{aligned}$$

We call $\{ \sigma _\alpha \}_{\alpha \in \mathbb {T}}$ the Aleksandrov-Clark measures associated to $\phi$. A measure whose Poisson integral is the real part of an analytic function is generally called an RP-measure or pluriharmonic measure.

Recall that for a function $f: \mathbb {D} \rightarrow \mathbb {C}$ and some point $\zeta \in \mathbb {T}$, we say that f(z) approaches $L \in \mathbb {C}$ non-tangentially, denoted

$$\begin{aligned} L:= \angle \lim _{z \rightarrow \zeta } f(z), \end{aligned}$$

if $f(z) \rightarrow L$ whenever $z \rightarrow \zeta$ in every fixed Stolz domain

$$\begin{aligned} \Gamma _\alpha ( \zeta ) := \{ z \in \mathbb {D}: |z- \zeta | < \alpha ( 1-|z|) \}, \quad \alpha > 1. \end{aligned}$$

This notion extends to multivariate functions: let $f: \mathbb {D}^d \rightarrow \mathbb {C}$ and $\zeta \in \mathbb {T}^d$. We say that f has a non-tangential limit L at $\zeta$ if $f(z) \rightarrow L$ as $z \rightarrow \zeta$, where each $z_j \rightarrow \zeta _j$ non-tangentially in the one-variable sense.

Let $m_d$ denote the normalized Lebesgue measure on $\mathbb {T}^d$, where we write m instead of $m_1$ when $d=1$. For any bounded holomorphic function $\phi : \mathbb {D}^d \rightarrow \mathbb {C}$, Fatou’s theorem for polydiscs (see Chapter XVII, Theorem 4.8 in [20]) ensures that the non-tangential limits

$$\begin{aligned} \phi ^* ( \zeta ) = \angle \lim _{\mathbb {D}^d \ni z \rightarrow \zeta } \phi ( z) \end{aligned}$$

exist for $m_d$-almost every $\zeta \in \mathbb {T}^d$. We say that $\phi : \mathbb {D}^d \rightarrow \mathbb {D}$ is an inner function if it is bounded, holomorphic and $|\phi ^*(\zeta )| = 1$ for $m_d$-almost every $\zeta \in \mathbb {T}^d$. If $\phi$ is inner, we call $\{ \sigma _\alpha \}_{\alpha \in \mathbb {T}}$ the Clark measures of $\phi$ instead.

In one variable, there are detailed descriptions of Clark measures and their behavior, see e.g. Chapter 11 of [13]. For instance, it is known that to each inner function on the unit disc, one can associate not only a family of Clark measures, but also a family of unitary operators. In [11], Doubtsov studies the families of Clark measures and operators associated to any given inner function on $\mathbb {D}^d$, and successfully extends some classical Clark theory to the multivariate setting. For an inner function $\phi$ on $\mathbb {D}^d$, we define the model space associated to $\phi$ as

$$\begin{aligned} K_\phi := H^2(\mathbb {T}^d) \ominus \phi H^2(\mathbb {T}^d) \end{aligned}$$

and operators

$$\begin{aligned} T_\alpha : K_\phi \rightarrow L^2(\sigma _\alpha ), \quad \alpha \in \mathbb {T}, \end{aligned}$$

In [9], Clark proves these operators to be unitary when $d=1$, and in [11], Doubtsov investigates what happens in higher dimensions. Let

$$\begin{aligned} C_w(z):= \prod _{j=1}^d \frac{1}{1-z_j \overline{w_j}}, \quad z_j \in \overline{\mathbb {D}}^d, w_j \in \mathbb {D}^d \end{aligned}$$

denote the Cauchy kernel in d variables. Doubtsov then defines the operator $T_\alpha$ on reproducing kernels $K_w(z)$ as

$$\begin{aligned} T_\alpha [K_w](\zeta ):= (1- \alpha \overline{\phi (w)}) C_w(\zeta ), \quad \zeta \in \mathbb {T}^d, w \in \mathbb {D}^d, \end{aligned}$$

and later extends this to all functions in $K_\phi$ using density. In his main result (Theorem 3.2, [11]), he shows that $T_\alpha$ being a unitary operator is equivalent to the polydisc algebra $A(\mathbb {D}^d)$ being dense in $L^2(\sigma _\alpha )$. This work will focus on the measure-theoretic aspects of Clark theory, but the reader can find notes on Clark embedding operators for rational inner functions in particular in Section 4 of [3].

The case of rational inner functions is thoroughly investigated in both [3, 5], where the authors explicitly characterize the Clark measures of general rational inner functions of bidegree (m, n). This work aims to extend this theory to other classes of inner functions, constructed from one-variable functions. In one variable, any inner function can be expressed as the product of a Blaschke product, a monomial and a singular inner function (Theorem 2.14, [13]), but there is no such simple general structure known for inner functions in higher dimensions. Hence, it is much harder to say anything about the general case for $d>1$. By studying multivariate functions constructed from one-variable inner functions, we can draw on the one-dimensional properties.

As an aside, we note that there are two natural settings for multivariate Clark theory; one could either study Clark measures on the unit polydisc $\mathbb {D}^d$ or the unit d-ball. Clark theory on the unit d-ball has been explored in detail in e.g. [2]. We restrict ourselves to $\mathbb {D}^d$ in this work, and mainly dimension $d=2$.

1.1 Overview

First, in Sect. 2, we introduce some basic theory concerning Clark measures in $\mathbb {T}^d$. Next, we survey some notable properties of Clark measures in one variable. In Sect. 3, we give an overview of recent progress concerning rational inner functions (RIFs). In particular, we present results from [3]—for example, we will see that the Clark measures of bivariate RIFs are supported on finite unions of analytic curves, and that one can characterize their behavior along these curves.

In Sect. 4, we consider the multiplicative embedding

$$\begin{aligned} \Phi (z):= \phi (z_1 z_2 \cdots z_d), \quad z \in \mathbb {D}^d \end{aligned}$$

which produces a multivariate inner function given an inner function $\phi$ in one variable. First we investigate the case $d=2$, where we characterize the unimodular level sets of $\Phi$ and see that these can always be parameterized by—potentially infinitely many—antidiagonals in $\mathbb {T}^2$. We conclude this section by presenting a concrete structure formula for the Clark measures associated to $\Phi$ in d variables: 4.6

Theorem 4.6

Let $\phi (z): \mathbb {D} \rightarrow \mathbb {C}$ be an inner function with Clark measure $\sigma _\alpha$ for some unimodular constant $\alpha$, and let $\tau _\alpha$ be the Clark measure of $\Phi (z_1, \ldots , z_d):= \phi (z_1 z_2 \cdots z_d)$. Then

$$\begin{aligned} \int _{\mathbb {T}^d} f(\xi ) d\tau _\alpha (\xi )&= \int _{\mathbb {T}} \int _{\mathbb {T}}\cdots \int _{\mathbb {T}} f(\zeta _1, \ldots , \zeta _{d-1}, x\overline{\zeta _1 \cdots \zeta _{d-1}} ) d\sigma _\alpha (x) dm(\zeta _{d-1})\\&\quad \cdots dm(\zeta _1). \end{aligned}$$

In Sect. 5, we turn to product functions

$$\begin{aligned} \Psi (z):= \phi (z_1) \psi (z_2), \quad z \in \mathbb {D}^2 \end{aligned}$$

for inner functions $\phi$ and $\psi$. We then prove the following structure formula for the Clark measures of $\Psi$: 5.1

Theorem 5.1

Let $\Psi (z_1,z_2) = \phi (z_1) \psi (z_2)$ for one-variable inner functions $\phi$ and $\psi$, such that

A.:

$\psi$ extends to be continuously differentiable on $\mathbb {T}$ except at a finite set of points,

B.:

the solutions to $\psi ^* = \beta$ for $\beta \in \mathbb {T}$ can be parameterized by functions $\{ g_k( \beta ) \}_{k \ge 1}$ which are continuous in $\beta$ on $\mathbb {T}$ except at a finite set of points,

C.:

for every $\beta \in \mathbb {T}$, there are no solutions to $\psi ^* = \beta$ with infinite multiplicity, and

D.:

the Clark measures of $\psi$ are all discrete.

Then the Clark measures of $\Psi$ satisfy

$$\begin{aligned} \int _{\mathbb {T}^2} f(\xi ) d\sigma _\alpha ( \xi ) = \sum _{k \ge 1} \int _{\mathbb {T}} f(\zeta , g_k ( \alpha \overline{\phi ^* ( \zeta )}) ) \frac{dm(\zeta )}{|\psi '(g_k ( \alpha \overline{\phi ^* ( \zeta )}))|} \end{aligned}$$

for $f \in C(\mathbb {T}^2)$.

Throughout the text, we present examples of bivariate inner functions with explicit characterizations of their Clark measures. Finally, in Sect. 6, we discuss possible further research tied to our results and raise some open questions.

2 Preliminaries

2.1 Elementary Clark Theory in Polydiscs

If $\phi$ is an inner function with associated Clark measure $\sigma _\alpha$, then

$$\begin{aligned} P[d \sigma _\alpha ](z) = \frac{1-|\phi (z)|^2}{|\alpha -\phi (z)|^2} = 0 \quad m_d\hbox {-}\text { almost everywhere on}\,\,\mathbb {T}^d. \end{aligned}$$

Clearly, the numerator goes to zero $m_d$-almost everywhere. As $\phi - \alpha$ is bounded, it lies in the Hardy space $H^2( \mathbb {T}^d) \subset N(\mathbb {T}^d)$; then Theorem 3.3.5 in [18] states that $\log ( \phi ^* - \alpha )$ lies in $L^1(\mathbb {T}^d)$. This in turn implies that $\phi ^* - \alpha$ must be non-zero $m_d$-almost everywhere on $\mathbb {T}^d$. Hence, $P[d \sigma _\alpha ](z) = 0$ $m_d$-almost everywhere on $\mathbb {T}^d$, as asserted.

A notable consequence of this result is that if $\phi$ is an inner function, then its Clark measures $\{ \sigma _\alpha \}_{\alpha \in \mathbb {T}}$ must be singular with respect to the Lebesgue measure. To see this, we decompose $\sigma _\alpha$ into an absolutely continuous measure $\tau ^1_\alpha = f_\alpha m_d$, $f_\alpha \in L^1(\mathbb {T}^d)$, and a $m_d$-singular measure $\tau ^2_\alpha$ (see Theorem 6.10, [19]). Then Theorem 2.3.1 in [18] states that the function

$$\begin{aligned} u(z):= P[d\sigma _\alpha ](z) = P[f_\alpha dm_d + d\tau ^2_\alpha ](z) \end{aligned}$$

satisfies $u^*(\zeta ) = f_\alpha (\zeta )$ for $m_d$-almost every $\zeta \in \mathbb {T}^d$. However, we saw already that $P[d\sigma _\alpha ] = 0$ $m_d$-almost everywhere on $\mathbb {T}^d$; hence $f_\alpha = 0$ $m_d$-almost everywhere on $\mathbb {T}^d$. We can thus conclude that $\sigma _\alpha$ is a $m_d$-singular measure for each $\alpha \in \mathbb {T}$. Moreover, as asserted in [11], two Clark measures $\sigma _\alpha$ and $\sigma _\beta$ associated to an inner function $\phi$ are mutually singular whenever $\alpha \ne \beta$.

Observe that since

$$\begin{aligned} \int _{\mathbb {T}^d} d \sigma _\alpha = \int _{\mathbb {T}^d} P (0,\zeta ) d \sigma _\alpha = \frac{1-|\phi (0)|^2}{|\alpha -\phi (0)|^2} < \infty , \end{aligned}$$

the measure $\sigma _\alpha$ is finite for each $\alpha \in \mathbb {T}$. In particular, $\sigma _\alpha$ is a probability measure if the associated inner function $\phi$ satisfies $\phi (0)=0$.

For an inner function $\phi$ and a constant $\alpha \in \mathbb {T}$, we define the so called unimodular level set

$$\begin{aligned} \mathcal {C}_\alpha ( \phi ) := \text {Clos} \Bigl \{ \zeta \in \mathbb {T}^d : \lim _{r \rightarrow 1^-} \phi ( r \zeta ) = \alpha \Bigl \}, \end{aligned}$$

where the closure is taken with respect to $\mathbb {T}^d$. The following proposition is a generalization of Lemma 2.1 in [3], where it is proven for rational inner functions.

Proposition 2.1

Let $\phi : \mathbb {D}^d \rightarrow \mathbb {C}$ be an inner function, and let $\alpha$ be a unimodular constant. Then ${{\,\textrm{supp}\,}}(\sigma _\alpha ) \subset \mathcal {C}_\alpha ( \phi )$.

With the exception of some minor details, the proof uses the same arguments as in [3], and is therefore omitted here. Note that the inclusion in Proposition 2.1 is not necessarily strict. In fact, for the classes of functions studied in this text, it follows from our structure formulas (Theorem 3.4, 3.5 and Corollary 4.4.1) that ${{\,\textrm{supp}\,}}(\sigma _\alpha ) = \mathcal {C}_\alpha$.

The definition and basic properties of d-dimensional Clark measures discussed so far are quite analogous to the classical one-variable case. However, some less elementary results also carry over. For instance, Doubtsov established the following d-dimensional generalization of the well-known Aleksandrov Disintegration Theorem:

Proposition 2.2

Let $\phi : \mathbb {D}^d \rightarrow \mathbb {C}$ be an inner function with associated Clark measures $\{ \sigma _\alpha : \alpha \in \mathbb {T} \}$. Then

$$\begin{aligned} \int _\mathbb {T} \int _{\mathbb {T}^d} f d \sigma _\alpha dm(\alpha ) = \int _{\mathbb {T}^d} f dm_d \end{aligned}$$

for all $f \in C(\mathbb {T}^d)$.

For a proof, see Proposition 2.4 in [11].

Finally, we record a fact which will be used in several proofs down the line:

Lemma 2.3

The linear span of Poisson kernels $\mathcal {M}:= \text {span}\{ P_z: z \in \mathbb {D}^d \}$ is dense in $C ( \mathbb {T}^d )$.

The proof is a straight-forward generalization of the proof of Proposition 1.17 in [13].

2.2 Clark Measures in One Variable

Before getting into examples of Clark measures in higher dimensions, we quote some results from Clark theory in one variable. To formulate the main result, we must first introduce the concept of angular derivatives.

Theorem 2.4

For an analytic function f on $\mathbb {D}$ and $\zeta _0 \in \mathbb {T}$, the following are equivalent:

• (i) The non-tangential limits

  $$\begin{aligned} f ( \zeta _0 ) = \angle \lim _{z \rightarrow \zeta _0} f(z) \quad \text {and} \quad \angle \lim _{z \rightarrow \zeta _0} \frac{f(z)- f(\zeta _0)}{z-\zeta _0} \end{aligned}$$

  exist;

• (ii) The derivative function $f'$ has a non-tangential limit at $\zeta _0$.

Under the equivalent conditions above,

$$\begin{aligned} \angle \lim _{z \rightarrow \zeta _0 } \frac{f(z) - f(\zeta _0) }{z- \zeta _0} = \angle \lim _{z \rightarrow \zeta _0 } f'(z). \end{aligned}$$

Proof

See Theorem 2.19 in [13]. $\square$

Definition 2.5

Assuming the conditions of the theorem, we call

$$\begin{aligned} f'( \zeta _0):= \angle \lim _{z \rightarrow \zeta _0 } \frac{f(z) - f(\zeta _0) }{z- \zeta _0} = \angle \lim _{ z \rightarrow \zeta _0} f'(z) \end{aligned}$$

the angular derivative of f at $\zeta _0$. Furthermore, if f maps $\mathbb {D}$ to itself, we say that f has an angular derivative in the sense of Carathéodory at $\zeta _0 \in \mathbb {T}$ if f has an angular derivative at $\zeta _0$ and $f( \zeta _0 ) \in \mathbb {T}$.

We now have the machinery needed to state the following proposition, which will be extremely useful to us in later sections:

Proposition 2.6

Let $\phi : \mathbb {D} \rightarrow \mathbb {C}$ be an inner function and let $\alpha \in \mathbb {T}$. Then the associated Clark measure $\sigma _\alpha$ has a point mass at $\zeta \in \mathbb {T}$ if and only if

$$\begin{aligned} \phi ^* ( \zeta ) = \lim _{r \rightarrow 1^-} \phi (r \zeta ) = \alpha \end{aligned}$$

and $\phi$ has a finite angular derivative in the sense of Carathéodory at $\zeta$. In this case,

$$\begin{aligned} \sigma _\alpha ( \{ \zeta \} ) = \frac{1}{| \phi '(\zeta )|} < \infty \quad \text {and} \quad \phi '(\zeta ) = \frac{\alpha \overline{\zeta }}{\sigma _\alpha ( \{ \zeta \} )}. \end{aligned}$$

Proof

See Proposition 11.2 in [13]. $\square$

Example 2.7

Let B(z) be a non-constant finite Blaschke product of order n, and let $\alpha \in \mathbb {T}$. Then B(z) is an inner function and analytic on $\mathbb {T}$, and $B(\zeta ) = \alpha$ has precisely n distinct solutions; denote these by $\eta ^\alpha _1, \ldots , \eta ^\alpha _n$. Moreover, from properties of finite Blaschke products, its derivative is non-zero on $\mathbb {T}$. By Proposition 2.6, the associated Clark measure then satisfies

$$\begin{aligned} \sigma _\alpha = \sum _{k=1}^n \frac{1}{|B'(\eta ^\alpha _k)|}\delta _{\eta ^\alpha _k}. \end{aligned}$$

Example 2.8

The function

$$\begin{aligned}\phi (z) := \exp \biggl (-\frac{1+z}{1-z}\biggr ), \quad z \in \mathbb {D},\end{aligned}$$

is inner, and $\phi ^*(\zeta )$ exists everywhere on $\mathbb {T}$; this is because

$$\begin{aligned} \bigg | \exp \biggl (-\frac{1+z}{1-z}\biggr ) \bigg | = \exp \biggl (\Re \biggl ( {-\frac{1+z}{1-z}} \biggr ) \biggr ) = \exp \biggl (- \frac{1-|z|^2}{|1-z|^2} \biggr ), \end{aligned}$$

from which we can see that $\phi ^*( 1) = 0$. Observe that every point $\zeta \ne 1$ on the unit circle solves the equation $\phi ^* = \alpha$ for some $\alpha \in \mathbb {T}$, so $\phi ^*(\mathbb {T} {\setminus } \{ 1\} ) = \mathbb {T}$. Moreover, the solutions accumulate in the limit point $\zeta = 1$ for every $\alpha$-value. Since the unimodular level sets are closed by definition, this implies that $1 \in \mathcal {C}_\alpha ( \phi )$ for all $\alpha \in \mathbb {T}$.

Now let $\alpha = 1$ for simplicity. As seen in Example 11.3(ii) in [13], the solutions to $\phi ^*( \zeta ) = 1$ are given by

$$\begin{aligned} \eta _k = \frac{2 \pi k - i}{2 \pi k + i}, \quad k \in \mathbb {Z}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{|\phi '(\eta _k)|} = \frac{8}{1+4 \pi ^2 k^2}. \end{aligned}$$

By Proposition 2.6, the Clark measure of $\phi$ associated to $\alpha = 1$ may thus be expressed as

$$\begin{aligned} \sigma _1 = \sum _{k \in \mathbb {Z}} \frac{8}{1+4 \pi ^2 k^2} \delta _{\eta _k}. \end{aligned}$$

We will revisit variations of this example in later sections.

By Theorem 4 in [4], RP-measures on $\mathbb {T}^d$, $d \ge 2$, cannot be supported on sets of Hausdorff dimension less than one, and in particular, they cannot possess any point masses. One can therefore not hope for an analogous result to Proposition 2.6 for $d \ge 2$. However, the proposition will still be useful in determining the density of certain Clark measures.

3 Rational Inner Functions

There has been significant progress in Clark theory for multivariate rational inner functions, see [3, 5]. We already saw a one-variable RIF in Example 2.7, where we could describe the Clark measures in a straight-forward manner—largely because Blaschke products are in fact analytic on $\mathbb {T}$. The main issue when studying RIFs in higher dimensions arises from dealing with potential singularities. However, it turns out that in two variables, the support of any associated Clark measure is actually a finite union of graphs, and that we can explicitly calculate its weights along these. We aim to give an overview of these results in this section.

We will first need some terminology specific to rational inner functions. We say that a polynomial $p \in \mathbb {C}[z_1,\ldots , z_d]$ is stable if it has no zeros in $\mathbb {D}^d$, and that it has polydegree $(n_1,\ldots , n_d) \in \mathbb {N}^d$ if p has degree $n_j$ when viewed as a polynomial in $z_j$. By Theorem 5.2.5 in [18], any rational inner function in $\mathbb {D}^d$ can be written as

$$\begin{aligned} \phi (z) = e^{ia}\prod _{j=1}^d z_j^{k_j} \frac{\tilde{p}(z)}{p(z)} \end{aligned}$$

where $a \in \mathbb {R}$, $k_1, \ldots , k_d \in \mathbb {N}$, p is a stable polynomial of polydegree $(n_1, \ldots , n_d)$, and

$$\begin{aligned} \tilde{p}(z) := z_1^{n_1}\cdots z_d^{n_d}\overline{p} \biggl ( \frac{1}{\overline{z_1}}, \ldots , \frac{1}{\overline{z_d}} \biggr ) \end{aligned}$$

is its reflection. Note that any zero of p on $\mathbb {T}^d$ will be a zero of $\tilde{p}$ and vice versa, and that p and $\tilde{p}$ have the same polydegree. For simplicity, we will always assume that $\phi (z) = \frac{\tilde{p}}{p}$, where p and $\tilde{p}$ are so called atoral—a concept explored in [1] and [6]. In the context of this text, atoral simply means that p and $\tilde{p}$ share no common factors, and that in two dimensions in particular, p and $\tilde{p}$ have finitely many common zeros on $\mathbb {T}^2$ (see Section 2.1, [6]). Hence, a rational inner function $\phi$ in two variables will have at most finitely many singularities on $\mathbb {T}^2$.

Moreover, we define the polydegree of a rational function $\phi = q/p$ as $(n_1, \ldots , n_d)$, where p and q have no common factors, and $n_j$ is the maximum of the degrees of p and q when viewed as polynomials in variable $z_j$. Thus, the polydegree of $\phi = \tilde{p}/p$ as defined above agrees with the polydegrees of both its numerator and denominator.

A notable fact about RIFs is that their non-tangential limits exist and are unimodular for every $\zeta \in \mathbb {T}^d$ (Theorem C, [15]). In [8], the authors prove the following result, which gives us a straight-forward expression for the level sets of RIFs:

Theorem 3.1

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

$$\begin{aligned} \mathcal {L}_{\alpha }( \phi ) := \{ \zeta \in \mathbb {T}^d: \tilde{p}( \zeta ) - \alpha p(\zeta ) = 0 \}. \end{aligned}$$

Then $\mathcal {C}_\alpha ( \phi ) = \mathcal {L}_{\alpha }( \phi ).$

Proof

See Theorem 2.6 in [8]. $\square$

Note that for any zero of p on $\mathbb {T}^d$, the equation $\tilde{p}- \alpha p = 0$ is trivially satisfied. This implies that all singularities of $\phi$ on $\mathbb {T}^d$ are contained in $\mathcal {C}_\alpha ( \phi )$. Moreover, observe that $\{ \phi ^* = \alpha \}$ is in general not a closed set. However, by the theorem above, we may characterize $\mathcal {C}_\alpha ( \phi )$ as the zeros of a polynomial when $\phi$ is a RIF.

When $d=2$, one can find an even nicer characterization of the unimodular level sets:

Lemma 3.2

Let $\phi = \frac{\tilde{p}}{p}$ be a RIF of bidegree (m, n), and fix $\alpha \in \mathbb {T}$. For any choice of $\tau _0 \in \mathbb {T}$, there exists a finite number of functions $g^{\alpha }_1, \ldots , g^{\alpha }_n$ defined on $\mathbb {T}$ and analytic on $\mathbb {T} \setminus \{ \tau _0 \}$ such that $\mathcal {C}_\alpha ( \phi )$ can be written as a union of curves

$$\begin{aligned} \{ ( \zeta , g^{\alpha }_j ( \zeta )): \zeta \in \mathbb {T}\}, \quad j= 1, \ldots , n, \end{aligned}$$

potentially together with a finite number of vertical lines $\zeta _1 = \tau _1, \ldots , \zeta _1= \tau _k$, where each $\tau _j \in \mathbb {T}$.

Proof

The proof is quite technical and will only be outlined here; for a complete proof, see Lemma 2.5 in [3]. In [3], the authors fix a point $\tau \in \mathbb {T}$ and construct a parameterization of $\mathcal {C}_\alpha (\phi ) \cap (I_\tau \times \mathbb {T})$ for a small interval $I_\tau \ni \tau$ in $\mathbb {T}$. When $\tau$ is not the $z_1$-coordinate of a singularity of $\phi$, one can use properties of RIFs to show that $\phi (\tau ,\cdot )$ satisfies the conditions of the Implicit Function Theorem. Hence the solutions to $\phi ^* = \alpha$ can be parameterized by smooth curves on the strip $I_\tau \times \mathbb {T}$.

The main issue arises from the fact that the curves $g_j^\alpha$ might intersect at singularities of $\phi$, in which case analyticity is not obvious. One must thus ensure that we can in some sense “pull apart” any crossed curves in $\mathcal {C}_\alpha (\phi )$ and prove that they are each analytic when viewed separately. However, it is shown in [6] that near each singularity of $\phi$, the level sets actually do consist of smooth curves. Hence, even in the case where $\tau$ is the $z_1$-coordinate of a singularity, one obtains a smooth parameterization of $I_\tau \times \mathbb {T}$. In the last step of the proof, the authors glue together the local parameterizations, which yields the final result. $\square$

The analysis of Clark measures of RIFs must now be divided into two cases; when the unimodular constant $\alpha$ is generic versus exceptional as defined below.

Definition 3.3

We say that $\alpha \in \mathbb {T}$ is an exceptional value if $\phi (\tau , \zeta _2) \equiv \alpha$ or $\phi (\zeta _1, \tau ) \equiv \alpha$ for some $\tau \in \mathbb {T}$. If $\alpha$ is not exceptional, we say that it is generic.

The different cases stem from the characterization of $\mathcal {C}_\alpha ( \phi )$ in Lemma 3.2; if $\alpha$ is an exceptional value, by the definition above, the level sets will contain lines of the form $\{ \zeta _1 = \tau \}$ or $\{ \zeta _2 = \tau \}$. If $\alpha$ is generic, $\mathcal {C}_\alpha ( \phi )$ can be fully described by the graphs of the functions $g^{\alpha }_1, \ldots , g^{\alpha }_n$.

Theorem 3.4

Let $\phi = \frac{\tilde{p}}{p}$ be a RIF of bidegree (m, n) and $\alpha \in \mathbb {T}$ a generic value for $\phi$. Then the associated Clark measure $\sigma _\alpha$ satisfies

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d \sigma _\alpha ( \xi ) = \sum _{j=1}^n \int _\mathbb {T} f( \zeta , g^{\alpha }_j ( \zeta )) \frac{dm(\zeta )}{|\frac{\partial \phi }{\partial z_2} ( \zeta , g^{\alpha }_j ( \zeta ) ) |} \end{aligned}$$

for all $f \in C ( \mathbb {T}^2)$, where $g^{\alpha }_1, \ldots , g^{\alpha }_n$ are the parameterizing functions from Lemma 3.2.

Proof

See Theorem 3.3 in [3]. $\square$

Theorem 3.5

Let $\phi = \frac{\tilde{p}}{p}$ be a RIF of bidegree (m, n) and $\alpha \in \mathbb {T}$ an exceptional value for $\phi$. Then, for $f \in C(\mathbb {T}^2)$, the associated Clark measure $\sigma _\alpha$ satisfies

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d \sigma _\alpha ( \xi ) = \sum _{j=1}^n \int _\mathbb {T} f( \zeta , g^{\alpha }_j ( \zeta )) \frac{dm(\zeta )}{|\frac{\partial \phi }{\partial z_2} ( \zeta , g^{\alpha }_j ( \zeta ) ) |} + \sum _{k=1}^\ell c^\alpha _k \int _\mathbb {T}f( \tau _k, \zeta ) dm(\zeta ), \end{aligned}$$

where $g_1^\alpha , \ldots , g_n^\alpha$ are the parameterizing functions and $\zeta _1 = \tau _1, \ldots , \zeta _1 = \tau _\ell$ the vertical lines in $\mathcal {C}_\alpha ( \phi )$ from Lemma 3.2, and $c_k^\alpha := 1/|\frac{\partial \phi }{\partial z_1}(\tau _k, z_2)| > 0$ are constants.

Proof

See Theorem 3.8 in [3]. $\square$

Note that in both the generic and exceptional case, the weights of the Clark measures along level curve components are generally given by one-variable functions. The fine structure of RIF weights is thoroughly analyzed in [3]—we will only briefly touch upon this here. For $\phi = \tilde{p} / p$, let $W_j^\alpha (\zeta ):= |\frac{\partial \phi }{\partial z_2} ( \zeta , g^{\alpha }_j ( \zeta ) ) |^{-1}$ denote the weights from Theorems 3.4 and 3.5. Then, by Lemma 5.1 in [3], these functions are in $L^1(\mathbb {T})$ and may be expressed as

$$\begin{aligned} W_j^\alpha (\zeta ) = \frac{|p(\zeta , g_j^\alpha ( \zeta )|}{| \frac{\partial \tilde{p}}{\partial z_2}(\zeta , g_j^\alpha (\zeta )) - \alpha \frac{\partial p}{\partial z_2} (\zeta , g_j^\alpha (\zeta )) |}. \end{aligned}$$

As we established earlier, if $(\tau , \gamma ) \in \mathbb {T}^2$ is a singularity of $\phi$, then $(\tau , \gamma ) \in \mathcal {C}_\alpha (\phi )$ for every $\alpha \in \mathbb {T}$. If $(\tau , \gamma ) = (\tau , g_j^\alpha (\tau ))$ for some level curve $g_j^\alpha$, then $p(\tau , g_j^\alpha (\tau )) = 0$ and one might expect $W_j^\alpha$ to be zero at this point—at least if there is no cancellation from the denominator. However, it could be that there are curve components $g_j^\alpha$ in $\mathcal {C}_\alpha (\phi )$ which do not satisfy $g_j^\alpha (\tau ) = \gamma$. In [3], the authors introduce the notion of contact order and prove the following statement:

For all but finitely many $\alpha$, if a branch $(\zeta , g_j^\alpha (\zeta ))$ of $\mathcal {C}_\alpha (\phi )$ passes through the singularity $(\tau , \gamma )$, the corresponding weight function $W_j^\alpha$ has order of vanishing at $\tau$ that corresponds to the contact order of the corresponding branch of $\mathcal {Z}(\tilde{p})$ at $(\tau , \gamma )$.

Here, $\mathcal {Z}(\tilde{p})$ denotes the zero set of the polynomial $\tilde{p}$. The precise statement can be found in Theorem 5.6 in [3]; the gist is that there are constants c, C such that one can bound

$$\begin{aligned} 0 < c \le \frac{W_j^\alpha (\zeta )}{|\zeta -\tau |^{K_j}} \le C \end{aligned}$$

for all $\zeta$ in a neighborhood of $\tau$, where $K_j$ is the contact order of $\phi$ at $(\tau , \gamma ) = (\tau , g_j^\alpha (\tau ))$ associated with the branch $g_j^\alpha$. Consequently, under these conditions, $W_j^\alpha$ is a bounded function.

The case of RIFs of bidegree (n, 1) specifically has been studied in great detail in [5]. For these functions, we obtain a more explicit version of Theorem 3.5. If $\phi = \tilde{p}/p$ has bidegree (n, 1), we may write

$$\begin{aligned} p(z) = p_1(z_1) + z_2 p_2(z_1) \quad \text {and} \quad \tilde{p}(z) = z_2 \tilde{p}_1(z_1)+ \tilde{p}_2(z_1) \end{aligned}$$

for reflections $\tilde{p}_i = z_1^n \overline{p}_i(1/\overline{z}_i)$. In this case, solving $\phi ^* = \alpha$ for $z_2$ yields $z_2 = \frac{1}{B_\alpha (z_1)}$, where

$$\begin{aligned} B_\alpha (z) := \frac{\tilde{p}_1(z) - \alpha p_2(z)}{\alpha p_1(z) - \tilde{p}_2(z)}. \end{aligned}$$

Moreover, define

$$\begin{aligned} W_\alpha (\zeta ) := \frac{|p_1(\zeta )|^2-|p_2(\zeta )|^2}{|\tilde{p}_1(\zeta ) - \alpha p_2(\zeta )|^2}. \end{aligned}$$

Then, by Theorem 1.2 in [5], we have

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d \sigma _\alpha ( \xi ) = \int _\mathbb {T} f( \zeta , \overline{B_\alpha ( \zeta )}) W_\alpha ( \zeta ) dm(\zeta ) + \sum _{k=1}^\ell c^\alpha _k \int _\mathbb {T}f( \tau _k, \zeta ) dm(\zeta ) \end{aligned}$$

with $c_k^\alpha = 1/|\frac{\partial \phi }{\partial z_1}(\tau _k, z_2)|$ is non-zero if and only if $\alpha$ is an exceptional value. It is worth noting that for any RIF $\phi$ of bidegree (n, 1), a value $\alpha \in \mathbb {T}$ is exceptional if and only if it is the non-tangential value of $\phi$ at some singularity (see Section 3 of [5]).

Example 3.6

For an explicit example, we use Example 5.2 from [5]: let $\phi = \frac{\tilde{p}}{p}$ for

$$\begin{aligned} p(z) = 4-z_2-3z_1-z_1z_2+z_1^2 \quad \text {and} \quad \tilde{p}(z) = 4z_1^2z_2 - z_1^2- 3z_1 z_2- z_1 + z_2. \end{aligned}$$

Observe that $\phi$ has only one singularity, which occurs at (1, 1). For each $\alpha \in \mathbb {T}$, the formulas above yield

$$\begin{aligned} B_\alpha (z ) = \frac{4z_1^2 - 3z_1 + 1 +\alpha + \alpha z_1}{4 \alpha - 3z_1 \alpha + z_1^2 \alpha +z_1^2 + z_1} \end{aligned}$$

and

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

We see that $\alpha = -1$ is an exceptional value, as $\phi = -1$ is solved by $(1, z_2)$ as well as $\big (z_1, \frac{1}{B_{-1}(z_1)}\big ) = (z_1, 1/z_1)$. Since $\phi$ only has one singularity, this point gives rise to the only exceptional value and $\phi ^*(1,1) = -1$. Hence, for $\alpha \ne -1$, we have

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d \sigma _\alpha ( \xi ) = \int _\mathbb {T} f( \zeta , \overline{B_\alpha ( \zeta )}) \frac{4|\zeta -1|^4}{|4 \zeta ^2 - 3 \zeta + 1 + \alpha + \alpha \zeta |^2} dm(\zeta ). \end{aligned}$$

Moreover, we see that $W_{-1} ( \zeta ) = \frac{1}{4}|\zeta -1|^2$ and $\frac{\partial \phi }{\partial z_1}(1,z_2) = -2$, which yields

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d \sigma _{-1} ( \xi ) = \frac{1}{4} \int _\mathbb {T} f( \zeta , \overline{\zeta })|\zeta -1|^2 dm(\zeta ) + \frac{1}{2}\int _{\mathbb {T}} f( 1, \zeta ) dm (\zeta ) \end{aligned}$$

for $\alpha = -1$. In Fig. 1, we have plotted the level curves corresponding to different $\alpha$-values.

Fig. 1

Level curves in Example 3.6 for $\alpha = 1$ (black), $\alpha = e^{i\pi /4}$ (gray), $\alpha = e^{i \pi /2}$ (orange), and exceptional value $\alpha = - 1$ (red)

4 Multiplicative Embeddings

Given an inner function $\phi$ in one complex variable, we define the multiplicative embedding

$$\begin{aligned} \Phi (z) = \Phi (z_1, z_2):= \phi (z_1 z_2), \quad z \in \mathbb {D}^2. \end{aligned}$$

The function defined by $(z_1, z_2) \mapsto z_1 z_2$ maps $\mathbb {D}^2$ to $\mathbb {D}$, and so $\phi$ being an inner function implies that $\Phi$ is inner as well. In the following proposition, we characterize the support set of $\Phi$ with the help of the original function $\phi$.

Proposition 4.1

Let $\phi (z)$ be an inner function in one variable, and $\alpha \in \mathbb {T}$. Define $\Phi (z_1,z_2):= \phi (z_1 z_2)$. Then

$$\begin{aligned} \mathcal {C}_\alpha ( \Phi ) = \bigcup _{\zeta \in \mathcal {C}_\alpha ( \phi ) } \{ (z, \zeta \overline{z} ): z \in \mathbb {T} \}. \end{aligned}$$

Proof

First, for ease of notation, define

$$\begin{aligned} \mathcal {C}_\alpha ' (f):= \Bigl \{ \zeta \in \mathbb {T}^d: \lim _{r \rightarrow 1^-} f(r \zeta ) = \alpha \Bigl \} \end{aligned}$$

for any inner function f, so that $\mathcal {C}_\alpha (f) = \text {Clos} (\mathcal {C}_\alpha ' (f))$.

Let $\zeta \in \mathcal {C}_\alpha ' ( \phi )$. Then we know that $\lim _{r \rightarrow 1^-} \phi (r \zeta ) = \alpha .$ For every $z \in \mathbb {T}$,

$$\begin{aligned} \lim _{r \rightarrow 1^-} \Phi (r (z, \zeta \overline{z})) = \lim _{r \rightarrow 1^-} \phi (r^2 \zeta z \overline{z} ) = \lim _{r \rightarrow 1^-} \phi (r \zeta ) = \alpha , \end{aligned}$$

implying that $(z, \zeta \overline{z} ) \in \mathcal {C}_\alpha ( \Phi )$. Thus

$$\begin{aligned} \bigcup _{\zeta \in \mathcal {C}_\alpha '( \phi ) } \{ (z, \zeta \overline{z} ): z \in \mathbb {T} \} \subset \mathcal {C}_\alpha ( \Phi ). \end{aligned}$$

To extend this to a union over $\mathcal {C}_\alpha ( \phi )$, let $\zeta \in \mathcal {C}_\alpha ( \phi )$. Then there exists some sequence $( \zeta _n)_{n\ge 1}$ in $\mathcal {C}_\alpha '( \phi )$ that converges to $\zeta$ as n tends to infinity. This also implies that for any $z \in \mathbb {T}$, $(z, \zeta _n \overline{z}) \rightarrow (z, \zeta \overline{z}) \in \mathcal {C}_\alpha (\Phi )$ as $n \rightarrow \infty$. Hence,

$$\begin{aligned} \bigcup _{\zeta \in \mathcal {C}_\alpha ( \phi ) } \{ (z, \zeta \overline{z} ): z \in \mathbb {T} \} \subset \mathcal {C}_\alpha ( \Phi ). \end{aligned}$$

Conversely, let $(z_1, z_2) \in \mathcal {C}_\alpha '( \Phi )$, so

$$\begin{aligned} \lim _{r \rightarrow 1^-} \Phi (r (z_1, z_2)) = \lim _{r \rightarrow 1^-} \phi (r^2 z_1z_2 ) = \alpha . \end{aligned}$$

Then $\zeta := z_1 z_2 \in \mathcal {C}_\alpha ( \phi )$. Since $z_1, z_2 \in \mathbb {T}$, we may write

$$\begin{aligned} z_2 = \frac{\zeta }{z_1} = \zeta \overline{z_1}, \end{aligned}$$

so $(z_1, z_2) = (z_1, \zeta \overline{z_1} ) \in \{ (z, \zeta \overline{z}): z \in \mathbb {T} \}$. Hence,

$$\begin{aligned} \mathcal {C}_\alpha ' ( \Phi ) \subset \bigcup _{\zeta \in \mathcal {C}_\alpha ( \phi ) } \{ (z, \zeta \overline{z} ): z \in \mathbb {T} \}. \end{aligned}$$

Now let $(z_1, z_2) \in \mathcal {C}_\alpha ( \Phi )$. Then there is some sequence of $(z_{1,n}, z_{2,n} )$ in $\mathcal {C}_\alpha ' ( \Phi )$ converging to $(z_1, z_2)$ as $n \rightarrow \infty$. But this implies that $z_{1,n} z_{2,n} \rightarrow z_1 z_2 \in \mathcal {C}_\alpha ( \phi )$, and the same argument as above then yields

$$\begin{aligned} \mathcal {C}_\alpha ( \Phi ) \subset \bigcup _{\zeta \in \mathcal {C}_\alpha ( \phi ) } \{ (z, \zeta \overline{z} ): z \in \mathbb {T} \}. \end{aligned}$$

$\square$

As in the RIF case, the unimodular level sets of this class of functions may be expressed as unions of curves. However, as opposed to in Lemma 3.2, the unions need not be finite—or even countable—here.

Remark 4.2

Observe that by Lemma 2.2 in [3], any positive, pluriharmonic, $m_d$-singular probability measure defines the Clark measure of some inner function. Hence, there exist Clark measures with significantly more intricate supports than what we have seen so far.

A natural next step is to investigate whether we can characterize the density of a given Clark measure $\tau _\alpha$ on the antidiagonals in $\mathcal {C}_\alpha ( \Phi )$. We do this in the next result and its subsequent corollary:

Theorem 4.3

Let $\phi (z)$ be an inner function in one variable, with Clark measure $\sigma _\alpha$ for some unimodular constant $\alpha$. Let $\tau _\alpha$ be the corresponding Clark measure of $\Phi (z_1, z_2):= \phi (z_1 z_2)$. Then, for any function $f \in C(\mathbb {T}^2)$,

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d\tau _\alpha ( \xi ) = \int _\mathbb {T} \bigg ( \int _\mathbb {T} f(\zeta , x \overline{\zeta }) d \sigma _\alpha (x) \bigg ) dm(\zeta ). \end{aligned}$$

Proof

We first prove this in the case when f is the product of one-variable Poisson kernels. Fixing $z_2 \in \mathbb {D}$, let

$$\begin{aligned} u_{z_2}( z_1) := \frac{1-|\phi (z_1 z_2)|^2}{|\alpha -\phi (z_1 z_2)|^2} = \int _{\mathbb {T}^2} P_{z_1}( \xi _1) P_{z_2}( \xi _2) d \tau _\alpha (\xi ), \quad z_1 \in \mathbb {D}. \end{aligned}$$

As the middle expression is pluriharmonic, $u_{z_2}$ must be harmonic on $\mathbb {D}$. Since $z_1 z_2 \in \mathbb {D}$ for any $z_1 \in \overline{\mathbb {D}}$, and $\phi$ is analytic (and hence continuous) on $\mathbb {D}$, we see that $\Phi (z_1, z_2) = \phi (z_1 z_2)$ as a function of $z_1$ is continuous on $\overline{\mathbb {D}}$. Moreover, by the maximum principle, $|\phi | < 1$ on the unit disc, which implies that the denominator will always be non-zero. We conclude that $u_{z_2}$ is continuous in $\overline{\mathbb {D}}$, and we may thus apply the Poisson integral formula:

$$\begin{aligned} u_{z_2}( z_1 ) = \int _\mathbb {T} \frac{1-|\phi (\zeta z_2)|^2}{|\alpha -\phi (\zeta z_2)|^2} P_{z_1}( \zeta ) dm(\zeta ). \end{aligned}$$

(1)

Moreover, for $\zeta \in \mathbb {T}$, we see that

$$\begin{aligned} \int _{\mathbb {T}} P_{z}( \zeta , x \overline{\zeta } ) d \sigma _\alpha (x)&= \int _{\mathbb {T}} P_{z_1}( \zeta ) P_{z_2} (x \overline{\zeta } ) d \sigma _\alpha (x)\\&= P_{z_1}( \zeta ) \int _{\mathbb {T}} \frac{1-|z_2|^2}{|x \overline{\zeta }- z_2|^2} d\sigma _\alpha (x) \\&= P_{z_1}( \zeta ) \int _{\mathbb {T}} \frac{1-|\zeta z_2|^2}{|x- \zeta z_2|^2} d\sigma _\alpha (x) \\&= P_{z_1}( \zeta ) \int _{\mathbb {T}} P_{\zeta z_2}(x) d\sigma _\alpha (x) \\&= P_{z_1} ( \zeta ) \frac{1-|\phi ( \zeta z_2)|^2 }{| \alpha - \phi (\zeta z_2)|^2}, \end{aligned}$$

where we use the definition of the Clark measure $\sigma _\alpha$ in the last step. By integrating the above and applying (1), we get

$$\begin{aligned} \int _\mathbb {T} \bigg ( \int _{\mathbb {T}} P_{z} ( \zeta , x \overline{\zeta } ) d \sigma _\alpha (x) \bigg ) dm(\zeta )&= \int _\mathbb {T} \frac{1-|z_1|^2}{|\zeta -z_1|^2} \frac{1-|\phi (\zeta z_2)|^2}{|\alpha -\phi (\zeta z_2)|^2} dm(\zeta ) \\&= \frac{1-|\phi (z_1 z_2)|^2}{|\alpha -\phi (z_1 z_2)|^2} \\&= \int _{\mathbb {T}^2} P_{z_1}( \xi _1) P_{z_2}( \xi _2) d \tau _\alpha (\xi ). \end{aligned}$$

Now apply Lemma 2.3 to obtain the final result. $\square$

Remark 4.4

It is a priori not obvious that $f( \zeta , x \overline{\zeta })$ is integrable with respect to $\sigma _\alpha$. Integrability is ensured by the fact that $f( \zeta , x \overline{\zeta })$ is continuous on $\mathbb {T}$, as it is composed by two functions f and $g_x(z):= ( z, x \overline{z})$ which are continuous there. Since $\sigma _\alpha$ is a finite, positive Borel measure on a compact space, all continuous functions on said space are integrable with respect to $\sigma _\alpha$.

In particular, when the Clark measures associated to $\phi$ are discrete, one gets the following result:

Corollary 4.4.1

Let $\phi : \mathbb {D} \rightarrow \mathbb {C}$ be an inner function with Clark measure $\sigma _\alpha$ for some unimodular constant $\alpha$, and let $\tau _\alpha$ be the Clark measure of $\Phi (z_1, z_2):= \phi (z_1 z_2)$. If $\sigma _\alpha$ is supported on a countable collection of points $\{ \eta _k \}_{k \ge 1} \subset \mathcal {C}_\alpha ( \phi )$, then

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d\tau _\alpha ( \xi ) = \sum _{k\ge 1} \int _\mathbb {T} f(\zeta , \eta _k \overline{\zeta }) \frac{ dm(\zeta )}{|\phi '(\eta _k)|} \end{aligned}$$

for all $f \in C(\mathbb {T}^2)$.

Proof

By Proposition 2.6, $\sigma _\alpha$ having a point mass at some $\eta _k$ implies that $\sigma _\alpha ( \{ \eta _k \}) =1 / |\phi '(\eta _k)|$. Then, following the steps in the proof of Theorem 4.3,

$$\begin{aligned} \int _{\mathbb {T}} P_{z}( \zeta , x \overline{\zeta } ) d \sigma _\alpha (x)&= P_{z_1}( \zeta ) \int _{\mathbb {T}} \Bigg ( \sum _{k\ge 1} \frac{1}{|\phi '(\eta _k)|} P_{\zeta z_2}(x) \Bigg ) d \delta _{\eta _k}(x). \end{aligned}$$

This then reduces to

$$\begin{aligned} P_{z_1}( \zeta ) \sum _{k\ge 1} \frac{1}{|\phi '(\eta _k)|} P_{\zeta z_2}(\eta _k) = \sum _{k\ge 1} \frac{1}{|\phi '(\eta _k)|} P_{z_1}( \zeta ) P_{z_2}( \eta _k \overline{\zeta }). \end{aligned}$$

Hence,

$$\begin{aligned} \int _{\mathbb {T}} P_{z}( \zeta , x \overline{\zeta } ) d \sigma _\alpha (x) = \sum _{k\ge 1}\frac{1}{|\phi '(\eta _k)|} P_{z}( \zeta , \eta _k \overline{\zeta }). \end{aligned}$$

Integrating over this and applying Theorem 4.3 then shows that

$$\begin{aligned} \int _{\mathbb {T}^2} P_{z}( \xi ) d \tau _\alpha (\xi )&= \int _\mathbb {T} \Bigg ( \sum _{k\ge 1} \frac{1}{|\phi '(\eta _k)|} P_{z}( \zeta , \eta _k \overline{\zeta }) \Bigg ) d m (\zeta ) \\ {}&= \sum _{k\ge 1} \int _\mathbb {T} P_{z}( \zeta , \eta _k \overline{\zeta }) \frac{d m (\zeta )}{|\phi '(\eta _k)|}, \end{aligned}$$

where we have used positivity of the summands in the last step. The result now follows from Lemma 2.3. $\square$

It is interesting to compare the above result to the corresponding theorems, Theorems 3.4 and 3.5, for rational inner functions. In the RIF case, we saw that the weights of Clark measures along the curves in the unimodular level sets were one-variable functions. Corollary 4.4.1 shows that for the multiplicative embeddings, the weights are simpler than their RIF counterparts—they are constant along each curve in the level sets. This implies that given any univariate inner function $\phi$, regardless of its complextiy, the associated Clark measures of $\phi (z_1 z_2)$ will always be very “well-behaved”, in the sense that they are supported on straight lines and—when the Clark measures of $\phi$ are discrete—have constant density along each such line.

Example 4.5

Recall the function

$$\begin{aligned}\phi (z) := \exp \biggl (-\frac{1+z}{1-z}\biggr ), \quad z \in \mathbb {D},\end{aligned}$$

from Example 2.8. We saw there that the solutions to $\phi ^*( \zeta ) = 1$ are given by

$$\begin{aligned} \eta _k = \frac{2 \pi k - i}{2 \pi k + i}, \quad k \in \mathbb {Z}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{|\phi '(\eta _k)|} = \frac{8}{1+4 \pi ^2 k^2}. \end{aligned}$$

Now consider $\Phi (z_1, z_2):= \phi (z_1 z_2)$, which appears in e.g. Example 13.1 in [7]. Applying Corollary 4.4.1 for the Clark measure $\tau _1$ of $\Phi$ then results in

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d \tau _1 ( \xi ) = \sum _{k \in \mathbb {Z}}\frac{8}{1+4 \pi ^2 k^2} \int _\mathbb {T} f( \zeta , \eta _k \overline{\zeta } )dm(\zeta ) \end{aligned}$$

for $f \in C(\mathbb {T}^2)$. This marks our first example of a non-rational bivariate function, for which we can explicitly characterize the Clark measures. Moreover, this is our first example of an inner function whose unimodular level sets consist of infinitely many curves, as opposed to the RIF case.

We now extend this theory to d variables. For an inner function $\phi$ in one variable, define the multiplicative embedding

$$\begin{aligned} \Phi (z):= \phi (z_1 z_2 \cdots z_d), \quad z \in \mathbb {D}^d. \end{aligned}$$

By the same argument as for two variables, this is an inner function. In the next result, we prove a d-dimensional version of Theorem 4.3:

Theorem 4.6

Let $\phi (z): \mathbb {D} \rightarrow \mathbb {C}$ be an inner function with Clark measure $\sigma _\alpha$ for some unimodular constant $\alpha$, and let $\tau _\alpha$ be the Clark measure of $\Phi (z_1, \ldots , z_d):= \phi (z_1 z_2 \cdots z_d)$. Then

$$\begin{aligned} \int _{\mathbb {T}^d} f(\xi ) d\tau _\alpha (\xi )&= \int _{\mathbb {T}} \int _{\mathbb {T}}\cdots \int _{\mathbb {T}} f(\zeta _1, \ldots , \zeta _{d-1}, x\overline{\zeta _1 \cdots \zeta _{d-1}} ) d\sigma _\alpha (x) dm(\zeta _{d-1}) \\&\quad \cdots dm(\zeta _1). \end{aligned}$$

Proof

We prove the result by induction, where Theorem 4.3 marks our base case. As usual, we prove the formula for Poisson kernels first.

We begin by introducing some notation: for $n < m$, let

$$\begin{aligned} {\textbf {z}}_n^m:= z_n \cdots z_m \end{aligned}$$

for $z_j \in \mathbb {D}, n \le j \le m$. Note that $\Phi (z) = \phi ({\textbf {z}}_1^d)$ per definition.

Suppose the formula holds for $\phi (z_1 \cdots z_{d-1})= \phi ({\textbf {z}}_1^{d-1})$, in which case

$$\begin{aligned} \frac{1-|\phi ({\textbf {z}}_1^{d-1})|^2}{|\alpha - \phi ({\textbf {z}}_1^{d-1})|^2}&= \int _{\mathbb {T}} \int _{\mathbb {T}} \cdots \int _{\mathbb {T}} P_{z_1}(\zeta _1) \cdots P_{z_{d-1}}(x\overline{\zeta _1 \zeta _2 \cdots \zeta _{d-2}} ) d\sigma _\alpha (x) dm(\zeta _{d-2}) \\&\quad \cdots dm(\zeta _1). \end{aligned}$$

We now want to show the result for d variables. Fix $z_2, \ldots , z_d$ and define the one-variable function

$$\begin{aligned} u(z_1):= \frac{1-|\phi ({\textbf {z}}_1^{d})|^2}{|\alpha - \phi ({\textbf {z}}_1^{d})|^2} = \frac{1-|\phi (z_1 \cdot {\textbf {z}}_2^{d})|^2}{|\alpha - \phi (z_1 \cdot {\textbf {z}}_2^{d})|^2}, \quad z_1 \in \mathbb {D}. \end{aligned}$$

By the same argument as in the proof of Theorem 4.3, this function is harmonic on the unit disc and continuous on its closure. Hence, we may apply the Poisson integral formula:

$$\begin{aligned} \frac{1-|\phi ({\textbf {z}}_1^{d})|^2}{|\alpha - \phi ({\textbf {z}}_1^{d})|^2} = \int _{\mathbb {T}^d} P_{z_1}(\zeta _1) \frac{1-|\phi ({\textbf {z}}_2^d \cdot \zeta _1)|^2}{|\alpha - \phi ({\textbf {z}}_2^d \cdot \zeta _1)|^2} \end{aligned}$$

(2)

For fixed $\zeta _1 \in \mathbb {T}$, define $\phi _{\zeta _1}({\textbf {z}}_2^d):= \phi ({\textbf {z}}_2^d \cdot \zeta _1)$. Then, by our induction assumption, it holds that

$$\begin{aligned} \frac{1-|\phi _{\zeta _1}({\textbf {z}}_2^d)|^2}{|\alpha - \phi _{\zeta _1}({\textbf {z}}_2^d)|^2}&= \int _{\mathbb {T}} \int _{\mathbb {T}} \cdots \int _{\mathbb {T}} P_{z_2}(\zeta _2) \cdots P_{\zeta _1 z_{d-1}}(x\overline{\zeta _2 \cdots \zeta _{d-1}} ) d\sigma _\alpha (x) dm(\zeta _{d-1})\\&\quad \cdots dm(\zeta _2) \\&= \int _{\mathbb {T}} \int _{\mathbb {T}} \cdots \int _{\mathbb {T}} P_{z_2}(\zeta _2) \cdots P_{z_{d-1}}(x\overline{\zeta _1 \zeta _2 \cdots \zeta _{d-1}} ) d\sigma _\alpha (x) dm(\zeta _{d-1}) \\&\quad \cdots dm(\zeta _2). \end{aligned}$$

Finally, by (2), we arrive at

$$\begin{aligned} \frac{1-|\phi ({\textbf {z}}_1^{d})|^2}{|\alpha - \phi ({\textbf {z}}_1^{d})|^2}&= \int _{\mathbb {T}} \int _{\mathbb {T}} \cdots \int _{\mathbb {T}} P_{z_1}(\zeta _1) \cdots P_{z_{d-1}}(x\overline{\zeta _1 \cdots \zeta _{d-1}}) d\sigma _\alpha (x) dm(\zeta _{d-1}) \\&\quad \cdots dm(\zeta _1), \end{aligned}$$

as desired. Application of Lemma 2.3 yields the final result. $\square$

Similarly to in the two-variable case, this gives us a sense of the geometry of ${{\,\textrm{supp}\,}}\{ \tau _\alpha \}$. For example, for $d=3$ and $x = e^{i \nu } \in \mathbb {T}$, the set

$$\begin{aligned} \{ (\zeta _1, \zeta _2, x \overline{\zeta _1 \zeta _2}): \zeta _1, \zeta _2 \in \mathbb {T} \} = \{ (e^{is}, e^{it}, e^{i(\nu -s-t)} ): -\pi \le s,t \le \pi \} \end{aligned}$$

has logarithmic coordinates $(s, t, \nu -s-t)$, which defines a plane in $\mathbb {T}^3$.

As in the case of $d=2$, we get the following consequence when $\sigma _\alpha$ is discrete:

Corollary 4.6.1

Let $\phi (z): \mathbb {D} \rightarrow \mathbb {C}$ be an inner function with Clark measure $\sigma _\alpha$ for some unimodular constant $\alpha$, and let $\tau _\alpha$ be the Clark measure of $\Phi (z_1, \ldots , z_d):= \phi (z_1 z_2 \cdots z_d)$. If $\sigma _\alpha$ is supported on a countable collection of points $\{\eta _k \}_{k \ge 1} \subset \mathcal {C}_\alpha (\phi )$, then

$$\begin{aligned} \int _{\mathbb {T}^d} f(\xi ) d\tau _\alpha (\xi )&= \sum _{k \ge 1} \int _{\mathbb {T}} \cdots \int _{\mathbb {T}} f(\zeta _1, \ldots , \zeta _{d-1}, \overline{\zeta _1 \cdots \zeta _{d-1}} \eta _k) dm(\zeta _{d-1}) \\&\quad \cdots \frac{dm(\zeta _1)}{|\phi '(\eta _k)|}. \end{aligned}$$

5 Product Functions

Given one-variable inner functions $\phi$ and $\psi$, define the product function

$$\begin{aligned} \Psi (z) := \phi (z_1) \psi (z_2), \quad z \in \mathbb {D}^2. \end{aligned}$$

Then $\Psi$ is an inner function in $\mathbb {D}^2$. The analysis of the Clark measures of $\Psi$ is not as straight-forward as for the multiplicative embeddings. A key argument in the proofs of Theorems 3.4, 3.5 and 4.3 is the Poisson integral formula. To use this for $\Psi (z_1, z_2)$, we require that for fixed $z_2 \in \mathbb {D}$, the function

$$\begin{aligned} u_{z_2}(z_1) := \frac{1- |\phi (z_1)\psi (z_2)|^2}{|\alpha - \phi (z_1) \psi (z_2)|^2} \end{aligned}$$

is continuous on the closed unit disc. However, for a general inner function $\phi$, its non-tangential limits need only exist m-almost everywhere on $\mathbb {T}$. Even if they do exist on the entire unit circle, $\phi ^*$ need not be continuous. For this reason, we introduce the function $\Psi _r(z):= \phi (rz_1)\psi (z_2)$ for $0<r<1$. This is not an inner function, as $|\phi (r z_1)|<1$ on the unit circle. However, since $\Psi _r \rightarrow \Psi$ as $r \rightarrow 1^-$, we can investigate the Clark measures of $\Psi$ via $\Psi _r$.

Theorem 5.1

Let $\Psi (z_1,z_2) = \phi (z_1) \psi (z_2)$ for one-variable inner functions $\phi$ and $\psi$, such that

• A. $\psi$ extends to be continuously differentiable on $\mathbb {T}$ except at a finite set of points,

• B. the solutions to $\psi ^* = \beta$ for $\beta \in \mathbb {T}$ can be parameterized by functions $\{ g_k( \beta ) \}_{k \ge 1}$ which are continuous in $\beta$ on $\mathbb {T}$ except at a finite set of points,

• C. for every $\beta \in \mathbb {T}$, there are no solutions to $\psi ^* = \beta$ with infinite multiplicity, and

• D. the Clark measures of $\psi$ are all discrete.

Then the Clark measures of $\Psi$ satisfy

$$\begin{aligned} \int _{\mathbb {T}^2} f(\xi ) d\sigma _\alpha ( \xi ) = \sum _{k \ge 1} \int _{\mathbb {T}} f(\zeta , g_k ( \alpha \overline{\phi ^* ( \zeta )}) ) \frac{dm(\zeta )}{|\psi '(g_k ( \alpha \overline{\phi ^* ( \zeta )}))|} \end{aligned}$$

for $f \in C(\mathbb {T}^2)$.

Remark 5.2

The assumptions A-D are most likely excessive, but we impose them here to get an easy guarantee that the right-hand side is finite and integrable. Nevertheless, we will see some interesting examples of product functions and their Clark measures for which Theorem 5.1 can be applied, e.g. when $\phi (z_1) = \exp \bigl (-\frac{1+z_1}{1-z_1} \bigr )$.

Remark 5.3

Observe that there exist examples of inner functions where the Clark measure $\sigma _\alpha$ is discrete for one specific $\alpha$-value but $\sigma _\beta$ is singular continuous for $\beta \in \mathbb {T} {\setminus } \{ \alpha \}$, and vice versa. See Example 1 and 2 in [10].

Proof

Define $\Psi _r(z_1,z_2):= \phi (r z_1)\psi (z_2)$ for $0<r<1$, and note that for each $z \in \mathbb {D}^2$,

$$\begin{aligned} \frac{1-|\Psi _r(z_1,z_2)|^2}{|\alpha - \Psi _r(z_1,z_2)|^2} \rightarrow \frac{1-|\Psi (z_1,z_2)|^2}{|\alpha - \Psi (z_1,z_2)|^2} = \int _{\mathbb {T}^2} P_z(\xi ) d\sigma _\alpha ( \xi ) \end{aligned}$$

as $r \rightarrow 1^-$. Define, for fixed $z_2 \in \mathbb {D}$ and fixed $0<r<1$,

$$\begin{aligned} u_{z_2}^r(z_1) := \frac{1- |\psi (z_2)\phi (rz_1)|^2}{|\alpha - \psi (z_2) \phi (rz_1)|^2}, \quad z_1 \in \mathbb {D}. \end{aligned}$$

As $\phi (rz_1)$ is continuous and satisfies $|\phi (rz_1)|<1$ on the unit circle, $u^r_{z_2}$ is continuous on $\overline{\mathbb {D}}$. Moreover, even though $\Psi _r$ is not an inner function, it holds that

$$\begin{aligned} \frac{1-|\Psi _r|^2}{|\alpha - \Psi _r|^2} = \Re \bigg ( \frac{\alpha + \Psi _r}{\alpha - \Psi _r} \bigg ) \end{aligned}$$

where $( \alpha + \Psi _r)/(\alpha - \Psi _r)$ is analytic on $\mathbb {D}^2$. Hence, the left-hand side is pluriharmonic in $\mathbb {D}^2$, which in turn implies that $u^r_{z_2}$ is harmonic in $\mathbb {D}$. By the Poisson integral formula,

$$\begin{aligned} \frac{1- |\psi (z_2)\phi (z_1)|^2}{|\alpha - \psi (z_2) \phi (z_1)|^2} = \lim _{r \rightarrow 1^-} u_{z_2}^r(z_1) = \lim _{r \rightarrow 1^-}\int _{\mathbb {T}} u^r_{z_2}(\zeta ) P_{z_1}(\zeta ) dm(\zeta ). \end{aligned}$$

Observe that $\Re ( ( \alpha + \Psi _r(z_1,z_2))/(\alpha - \Psi _r(z_1,z_2) ) )$ is bounded for every $(z_1, z_2) \in \overline{\mathbb {D}} \times \mathbb {D}$ and every $0< r < 1$. The dominated convergence theorem then states that we can move the limit into the integral: so, for fixed $z_2 \in \mathbb {D}$,

$$\begin{aligned} \frac{1- |\psi (z_2)\phi (z_1)|^2}{|\alpha - \psi (z_2) \phi (z_1)|^2} = \int _{\mathbb {T}} \lim _{r \rightarrow 1^-} u^r_{z_2}(\zeta ) P_{z_1}(\zeta ) dm(\zeta ). \end{aligned}$$

(3)

Moreover,

$$\begin{aligned} \lim _{r \rightarrow 1^-} u^r_{z_2}(\zeta ) = \lim _{r \rightarrow 1^-} \frac{1- |\psi (z_2)\phi (r \zeta )|^2}{|\alpha - \psi (z_2) \phi (r \zeta )|^2} = \frac{1- |\psi (z_2)\phi ^*(\zeta )|^2}{|\alpha - \psi (z_2) \phi ^*(\zeta )|^2} \end{aligned}$$

for m-almost every $\zeta \in \mathbb {T}$. Let E denote the set of points $\zeta$ such that $|\phi ^*(\zeta )|=1$.

By our assumptions, the solutions to $\psi ^* = \beta$ can be parameterized by functions $g_k( \beta )$ continuous on $\mathbb {T}$ except on a finite collection of points. Since we have also assumed that the Clark measures of $\psi$ consist of point masses, by Proposition 2.6, the measure associated to any $\beta \in \mathbb {T}$ is given by $\sum _{k\ge 1} | \psi '( g_k ( \beta ))|^{-1} \delta _{g_k(\beta )}$ where $|\psi '(g_k(\beta ))| > 0$ for each k. For fixed $\zeta \in E$, this holds for $\beta = \alpha \overline{\phi ^*( \zeta )}$.

Hence, for $\zeta \in E$, we have that

$$\begin{aligned} \frac{1- |\psi (z_2)\phi ^*(\zeta )|^2}{|\alpha - \psi (z_2) \phi ^*(\zeta )|^2} = \sum _{k \ge 1} \frac{1}{| \psi '(g_k(\alpha \overline{\phi ^*(\zeta )}))|} P_{z_2}(g_k(\alpha \overline{\phi ^*(\zeta )})). \end{aligned}$$

(4)

To apply the Poisson integral formula, we must first check that the product of the right-hand side with $P_{z_1}(\zeta )$ is integrable. Recall that by Fatou’s theorem, $\phi (r \zeta )$ converges to $\phi ^*(\zeta )$ as $r \rightarrow 1^-$ m-almost everywhere on $\mathbb {T}$ and in $L^1(\mathbb {T})$. Moreover, the curves $\{ g_k \}_{k \ge 1}$ are assumed to be continuous on the unit circle except at finitely many points. Hence, the composition $P_{z} (\zeta , g_k (\alpha \overline{\phi ^*(\zeta )}))$ must be measurable—indeed, $f(\zeta , g_k (\alpha \overline{\phi ^*(\zeta )}))$ is measurable for any $f\in C(\mathbb {T}^2)$. Similarly, we see that the weights $|\psi '(g_k ( \alpha \overline{\phi ^* ( \zeta )}))|$ are measurable, as $\psi$ is assumed to be continuously differentiable on $\mathbb {T}$ except at finitely many points. Since we are integrating over a compact space, this is enough to ensure integrability.

Moreover, for fixed $\zeta \in E$, the sum $\sum _{k \ge 1} |\psi '(g_k ( \alpha \overline{\phi ^* ( \zeta )}))|^{-1}$ must be finite, since the Clark measure of $\psi$ associated to the parameter value $\alpha \overline{\phi ^* ( \zeta )}$ exists by assumption. As we have excluded the situation where infinitely many of the curves intersect, the weights cannot sum up to infinity as we integrate over $\mathbb {T}$. The curves could still have infinite intersections at limit points of $g_k( \alpha \overline{\phi ^*(\zeta )})$, which per definition do not solve $\psi ^* = \alpha \overline{\phi ^*(\zeta )}$. However, by Proposition 2.6, the weights of the Clark measures must be zero for these points.

Since Eq. (4) holds for m-almost every $\zeta \in \mathbb {T}$, the integrals of the left- and right-hand side will coincide. By combining this with (3), we see that

$$\begin{aligned} \frac{1- |\psi (z_2)\phi (z_1)|^2}{|\alpha - \psi (z_2) \phi (z_1)|^2}&= \int _{\mathbb {T}} \frac{1- |\psi (z_2)\phi ^*(\zeta )|^2}{|\alpha - \psi (z_2) \phi ^*(\zeta )|^2}P_{z_1}(\zeta ) dm(\zeta ) \\&= \int _{\mathbb {T}}\sum _{k \ge 1} \frac{1}{| \psi '(g_k(\alpha \overline{\phi ^*(\zeta )}))|} P_{z}(\zeta , g_k(\alpha \overline{\phi ^*(\zeta )})) dm (\zeta ). \end{aligned}$$

As the summands are all positive, we may interchange summation and integration. Thus,

$$\begin{aligned} \frac{1- |\Psi (z_1,z_2)|^2}{|\alpha - \Psi (z_1,z_2)|^2} = \sum _{k \ge 1} \int _{\mathbb {T}} P_{z}(\zeta ,g_k(\alpha \overline{\phi ^*(\zeta )})) \frac{dm(\zeta )}{| \psi '(g_k(\alpha \overline{\phi ^*(\zeta )}))|}, \end{aligned}$$

i.e.

$$\begin{aligned} \int _{\mathbb {T}^2} P_z(\xi ) d \sigma _\alpha ( \xi ) = \sum _{k \ge 1} \int _{\mathbb {T}} P_{z}(\zeta ,g_k(\alpha \overline{\phi ^*(\zeta )})) \frac{dm(\zeta )}{| \psi '(g_k(\alpha \overline{\phi ^*(\zeta )}))|}. \end{aligned}$$

Since the span of Poisson kernels is dense in $C(\mathbb {T}^2)$, we may conclude that

$$\begin{aligned} \int _{\mathbb {T}^2} f(\xi ) d\sigma _\alpha ( \xi ) = \sum _{k \ge 1} \int _{\mathbb {T}} f(\zeta , g_k ( \alpha \overline{\phi ^* ( \zeta )}) ) \frac{dm(\zeta )}{|\psi '(g_k ( \alpha \overline{\phi ^* ( \zeta )}))|} \end{aligned}$$

for all $f \in C(\mathbb {T}^2)$. $\square$

Note that the weights of these measures strongly resemble their RIF counterparts from Theorem 3.4. Moreover, as in the case of the multiplicative embeddings, Theorem 5.1 allows for infinite collections of parameterizing functions.

Remark 5.4

Let us convince ourselves that there actually exist inner functions $\psi$ that meet the requirements of Theorem 5.1. For example, finite Blaschke products define one such class. Let $\psi$ be a non-constant finite Blaschke product of order n. As in Example 2.7, this implies that $\psi$ is analytic on $\mathbb {T}$ and $\psi ^*(\zeta ) = \beta$ has precisely n distinct solutions for each $\beta \in \mathbb {T}$, and $\psi ' \ne 0$ on $\mathbb {T}$. By the Implicit Function Theorem, we may thus parameterize the solutions with functions $\{g_k( \beta ) \}_{k=1}^n$ analytic on the unit circle. Additionally, we saw in Example 2.7 that the Clark measures of $\psi$ are discrete for every $\beta \in \mathbb {T}$. Hence, Theorem 5.1 works for any product function $\Psi (z) = \phi (z_1) \psi (z_2)$ where $\psi$ is a non-constant finite Blaschke product and $\phi$ is an arbitrary inner function.

In the case where both $\phi$ and $\psi$ are finite Blaschke products, the theorem reproduces what we know about RIFs, as

$$\begin{aligned} \bigg | \frac{\partial \Psi }{\partial z_2}( \zeta , g_k^\alpha (\zeta )) \bigg | = |\phi (\zeta ) \psi '(g_k^\alpha ( \zeta )) | = |\psi '(g_k^\alpha ( \zeta )) | \end{aligned}$$

Then Theorem 5.1 reduces to Theorem 3.4.

Observe that if $\psi \in C(\mathbb {T})$, it must be a finite Blaschke product (Corollary 4.2, [14]). Similarly, if $\psi ' \in H^1( \mathbb {T})$, then $\psi$ is continuous on $\mathbb {T}$ (Theorem 3.11, [12]) and thus a finite Blaschke product. Hence, to be able to construct varied examples, we need $\psi ^*$ to have some discontinuities on the unit circle (see e.g. Example 5.6).

In what comes next, we let $g_k^\alpha (\zeta ):=g_k(\alpha \overline{\phi ^*(\zeta )})$ for ease of notation.

Example 5.5

Let

$$\begin{aligned} \Psi (z_1, z_2) := \psi (z_2) \phi (z_1) = z_2\frac{\lambda -z_2}{1-\overline{\lambda }z_2} \exp \biggl (-\frac{1+z_1}{1-z_1}\biggr ) \end{aligned}$$

for $\phi$ as in Example 2.8 and some constant $\lambda \in \mathbb {D}$. Note that $\Psi ^* = 0$ for $\zeta _1 = 1$. The equation $\Psi ^* = \alpha$ for $\alpha \in \mathbb {T}$ can be rewritten as

$$\begin{aligned} \zeta _2\frac{\lambda -\zeta _2}{1-\overline{\lambda }\zeta _2} = \alpha \exp \biggl (\frac{1+\zeta _1}{1-\zeta _1}\biggr ). \end{aligned}$$

For $\alpha = e^{i \nu }$, the solutions to this are given by $\zeta _2 = g_k^\alpha (\zeta _1)$, $k=1,2$, where

$$\begin{aligned} g^\alpha _k( \zeta _1)&:= \frac{1}{2} \biggl (\lambda + \exp \biggl ( i \nu + \frac{1+\zeta _1}{1-\zeta _1} \biggr ) \overline{\lambda } \\ {}&\quad \pm \sqrt{-4 \exp \biggl ( i \nu + \frac{1+\zeta _1}{1-\zeta _1} \biggr ) + \biggl (- \lambda - \exp \biggl ( i \nu + \frac{1+\zeta _1}{1-\zeta _1} \biggr ) \overline{\lambda } \biggr )^2} \, \biggr ). \end{aligned}$$

In Fig. 2, we have plotted the level curves for certain parameter values.

Fig. 2

Level curves $g_k^\alpha$ in Example 5.5 for $\alpha = e^{i\pi /4}$ and $\lambda = i/2$

Let us calculate the weights of the Clark measures. Observe that

$$\begin{aligned} \psi '(z_2) = \frac{\lambda -2z_2+z_2^2\overline{\lambda }}{(1-\overline{\lambda }z_2)^2}. \end{aligned}$$

Hence, by Theorem 5.1,

$$\begin{aligned} \int _{\mathbb {T}^2} f(\xi ) d \sigma _\alpha ( \xi ) = \sum _{k=1}^2 \int _{\mathbb {T}} f(\zeta , g_k^\alpha ( \zeta )) \frac{|1-\overline{\lambda }g_k^\alpha (\zeta )|^2}{|\lambda -2g_k^\alpha (\zeta )+g_k^\alpha (\zeta )^2\overline{\lambda }|}dm(\zeta ) \end{aligned}$$

for all $f \in C(\mathbb {T}^2)$. In Fig. 3, we have plotted the weights

$$\begin{aligned} W^{\alpha }_k(\zeta ):= \frac{|1-\overline{\lambda }g_k^\alpha (\zeta )|^2}{|\lambda -2g_k^\alpha (\zeta )+g_k^\alpha (\zeta )^2\overline{\lambda }|}. \end{aligned}$$

Fig. 3

Weight curves $W^\alpha _k$ in Example 5.5 for $\alpha = e^{i\pi /4}$ and $\lambda = i/2$

Example 5.6

Define

$$\begin{aligned} \Psi (z_1, z_2):= \phi (z_1) \phi (z_2) = \exp \biggl (-\frac{1+z_1}{1-z_1}\biggr ) \exp \biggl (-\frac{1+z_2}{1-z_2}\biggr ), \end{aligned}$$

where $\phi$ again is as in Example 2.8. As $\phi ^*$ exists everywhere on $\mathbb {T}$, we have $\Psi ^*(\zeta ) = \phi ^*(\zeta _1) \phi ^*( \zeta _2)$. On the lines $\{ (1,\zeta _2): \zeta _2 \in \mathbb {T} \}$ and $\{ (\zeta _1, 1): \zeta _1 \in \mathbb {T} \}$ in $\mathbb {T}^2$, we see that $\Psi ^* = 0$. Otherwise, $|\Psi ^*| = 1$.

Since $\Psi ^*$ is well-defined and unimodular on $\mathbb {T}^2$ except on the lines $\{ \zeta _1=1 \} \cup \{ \zeta _2= 1 \}$ where $\Psi ^* = 0$, we need to solve the equation $\Psi = \alpha$. We may view this as

$$\begin{aligned} \exp \biggl (-\frac{1+\zeta _1}{1-\zeta _1}-\frac{1+\zeta _2}{1-\zeta _2}\biggr ) = e^{i( \nu + 2 \pi k)}, \quad k \in \mathbb {Z}, \end{aligned}$$

i.e.

$$\begin{aligned} -\frac{1+\zeta _1}{1-\zeta _1}-\frac{1+\zeta _2}{1-\zeta _2} = i(\nu + 2 \pi k), \quad k \in \mathbb {Z}. \end{aligned}$$

Solving for $\zeta _2$ yields

$$\begin{aligned} \zeta _2 = g^\alpha _k( \zeta _1) := \frac{\nu (\zeta _1 - 1) + 2 \pi k( \zeta _1 -1) + 2 i }{\nu ( \zeta _1 - 1) + 2 \pi k (\zeta _1 - 1) + 2 i \zeta _1}, \quad k \in \mathbb {Z}. \end{aligned}$$

Note that functions $g^\alpha _k$ are continuous on the unit circle; their only singularities occur at points $\zeta _1 = \frac{2\pi k+\nu }{\nu + 2 \pi k + 2i }$, which do not have modulus one.

Moreover, all $g^\alpha _k$ pass through the point $(1,1) \in \mathbb {T}^2$, which does not solve $\Psi ^* = \alpha$ as $\Psi ^*(1,1) = 0$. However, since $\mathcal {C}_\alpha ( \Psi )$ is closed, the point (1, 1) nevertheless lies in the unimodular level set. Hence,

$$\begin{aligned} \mathcal {C}_\alpha ( \Psi ) = \bigcup _{k \in \mathbb {Z}} \{( \zeta , g^\alpha _k(\zeta )): \zeta \in \mathbb {T} \} \end{aligned}$$

where $g^\alpha _k$ is analytic on $\mathbb {T}$ for every k. We have plotted some of these curves in Fig. 4.

Fig. 4

Level curves $g^1_k$ in Example 5.6 for $k=-1$ (red), $k=0$ (orange) and $k=2$ (gray)

Recall that by Lemma 3.2, the unimodular level sets of RIFs can be parameterized by graphs that are analytic on $\mathbb {T}^2$ except possibly at a single point. One might then expect that the Clark measures of a product function which is rational in at least one variable, like in Example 5.5, would be supported on smoother curves than this $\Psi$. However, we see that in this case, the unimodular level sets are actually parameterized by much more “well-behaved” curves than in our previous example.

At first sight, $\Psi$ does not seem to meet the requirements of Theorem 5.1; there is a point on $\mathbb {T}^2$ where all $g_k$ intersect, as $g_k(1) = 1$ for all $k \in \mathbb {Z}$. However, as noted above, this value does not in fact solve the equation $\Psi ^* = \alpha$ since $\phi ^*(1) = 0$. This point would cause a problem if the Clark measure of $\phi$ had positive weight there. Fortunately, we are saved by Proposition 2.6; the measure associated to $\alpha$ has a point mass at 1 if and only if $\phi ^* (1) = \alpha$, and so $|\phi '(1)|^{-1} = 0$.

Let us now calculate the weights of the Clark measures associated to $\Psi$. First note that

$$\begin{aligned} \phi '(z_2) = - \frac{2 \exp \bigr (- \frac{1+z_2}{1-z_2} \bigl ) }{(1-z_2)^2} = -\frac{2 \phi (z_2)}{(1-z_2)^2}. \end{aligned}$$

Then

$$\begin{aligned} \phi '(g_k^\alpha (\zeta _1)) = -\frac{2 \alpha }{\phi (\zeta _1) (1-g_k^\alpha (\zeta _1))^2} = \frac{2 \alpha }{\phi (\zeta _1)} \frac{(\nu (\zeta _1 - 1)+ 2 \pi k( \zeta _1 - 1) + 2i\zeta _1)^2}{4(\zeta _1-1)^2} \end{aligned}$$

for $\zeta _1 \in \mathbb {T} \setminus \{ 1 \}$. When taking moduli, we find

$$\begin{aligned} |\phi '(g_k^\alpha (\zeta _1))| = \bigg | \frac{\nu (\zeta _1 - 1) + 2 \pi k (\zeta _1 - 1) + 2i\zeta _1}{2(\zeta _1-1)} \bigg |^2 \end{aligned}$$

for $\zeta _1 \in \mathbb {T} \setminus \{ 1 \}$. Hence, Theorem 5.1 yields

$$\begin{aligned} \int _{\mathbb {T}^2} f( \xi ) d \sigma _\alpha ( \xi ) = \sum _{k \in \mathbb {Z}}\int _{\mathbb {T}} f( \zeta , g_k^\alpha (\zeta )) \bigg | \frac{2(\zeta -1)}{\nu (\zeta - 1) + 2 \pi k (\zeta - 1) + 2i\zeta } \bigg |^2 dm(\zeta ) \end{aligned}$$

for all $f \in C(\mathbb {T}^2)$, where $\alpha = e^{i \nu }$. Since $\sum _{k \in \mathbb {Z}} \frac{1}{k^2}$ converges, we see that the right-hand side is finite.

Note that the weights

$$\begin{aligned} W^\alpha _k(\zeta ):= \bigg | \frac{2(\zeta -1)}{\nu (\zeta - 1) + 2 \pi k (\zeta - 1) + 2i\zeta } \bigg |^2 \end{aligned}$$

reduce to zero for $\zeta = 1$, as expected (see Fig. 5). Moreover, we established earlier that all the level curves pass through the singularity (1, 1). Based on this example, it seems that the weights “detect” the singularities of $\Psi$—much like in the case of the rational inner functions in Sect. 3. Recall our brief discussion on the connection between the order of vanishing of weights at RIF singularities and contact order on page 9. It might be interesting to study if the singularities of general product functions are connected to the density of their Clark measures in some similar way (Fig. 5).

Fig. 5

Weight curves $W_k^1$ in Example 5.6 for $k=-1$ (red), $k=0$ (orange) and $k=2$ (gray)

6 Closing Remarks

It is important to note that Clark measures of general bivariate inner functions still remain unexplored. In one variable, any singular probability measure on $\mathbb {T}$ defines the Clark measure of some inner function (pp. 234-235, [13]). In several variables, we need added requirements on a measure for it to be a Clark measure—as discussed in Remark 4.2, any positive, pluriharmonic, singular probability measure defines the Clark measure of some inner function. The distinction arises from the fact that in several variables, it is not as easy to ensure that a given harmonic function is the real part of an analytic function. By Theorem 2.4.1 in [18], the Poisson integral of a real measure $\mu$ on $\mathbb {T}^d$ is given by the real part of an analytic function if and only if its Fourier coefficients satisfy $\hat{\mu }(k) = 0$ for every k outside the set $-\mathbb {Z}_{+}^d \cup \mathbb {Z}_{+}^d$, where $-\mathbb {Z}_{+}^d$ denotes the set of points $(k_1, \ldots , k_d)$ where every $k_j \le 0$.

Furthermore, the kind of smooth curve-parameterizations that were obtained for the classes of inner functions in this text are certainly not applicable for general inner functions. What we do know is that RP-measures cannot be supported on sets of Hausdorff dimension less than one (Theorem 4, [4]). In two dimensions, we have seen examples of Clark measures supported on curves (i.e. sets of Hausdorff dimension one). In [17], the author constructs an RP-measure whose support has Hausdorff dimension two. However, it is not clear to the author how one would construct an RP-measure with support of Hausdorff dimension $1<d<2$. For an in-depth discussion about the supports of RP-measures, see [4].

We end with a brief note on Clark embedding operators associated to the classes of inner functions introduced here. In Example 4.2 in [11], it is shown that all $T_\alpha$ are unitary for the simple multiplicative embedding $\phi (z_1 z_2) = z_1 z_2$ where $\phi (z) = z$, for which the Clark measure $\sigma _\alpha$ satisfies

$$\begin{aligned} \int _{\mathbb {T}^2} f(\xi ) d \sigma _\alpha ( \xi ) = \int _\mathbb {T} f( \zeta , \alpha \overline{\zeta }) dm(\zeta ), \quad f \in C(\mathbb {T}^2). \end{aligned}$$

For holomorphic monomials f, the functions $f(\zeta , \alpha \overline{\zeta })$ are dense in $L^2(m)$, which in turn implies that $A(\mathbb {D}^2)$ is dense in $L^2(\sigma _\alpha )$, as desired. It seems plausible that a similar argument can be applied to show that given any $\Phi (z)$ satisfying the conditions of Corollary 4.6.1, the associated Clark embedding operators are all unitary. In the case of product functions, however, it is not so clear when the operators would be unitary and further analysis is required.