Clark measures on polydiscs associated to product functions and multiplicative embeddings

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.

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 D d , and successfully extends some classical Clark theory to the multivariate setting.For an inner function ϕ on D d , we define the model space associated to ϕ as and operators In [9], Clark proves these operators to be unitary when d = 1, and in [11], Doubtsov investigates what happens in higher dimensions.Let and later extends this to all functions in K ϕ using density.In his main result (Theorem 3.2, [11]), he shows that T α being a unitary operator is equivalent to the polydisc algebra A(D d ) being dense in L 2 (σ α ).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] and [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 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 D d in this work, and mainly dimension d = 2.
1.1.Overview.First, in Section 2, we introduce some basic theory concerning Clark measures in T d .Next, we survey some notable properties of Clark measures in one variable.In Section 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 Section 4, we consider the multiplicative embedding which produces a multivariate inner function given an inner function ϕ in one variable.First we investigate the case d = 2, where we characterize the unimodular level sets of Φ and see that these can always be parameterized by -potentially infinitely many -antidiagonals in T 2 .We conclude this section by presenting a concrete structure formula for the Clark measures associated to Φ in d variables.In Section 5, we turn to product functions for inner functions ϕ and ψ.We then prove a structure formula for the Clark measures of Ψ under certain assumptions on ϕ and ψ.
Throughout the text, we present examples of bivariate inner functions with explicit characterizations of their Clark measures.Finally, in Section 6, we discuss possible further research tied to our results and raise some open questions.

Elementary Clark theory in polydiscs.
If ϕ is an inner function with associated Clark measure σ α , then Clearly, the numerator goes to zero m d -almost everywhere.As ϕ − α is bounded, it lies in the Hardy space H 2 (T d ) ⊂ N (T d ); then Theorem 3.3.5 in [18] states that log(ϕ * − α) lies in L 1 (T d ).This in turn implies that ϕ * − α must be non-zero m d -almost everywhere on T d .Hence, P [dσ α ](z) = 0 m d -almost everywhere on T d , as asserted.
A notable consequence of this result is that if ϕ is an inner function, then its Clark measures {σ α } α∈T must be singular with respect to the Lebesgue measure.To see this, we decompose σ α into an absolutely continuous measure , and a m d -singular measure τ 2 α (see Theorem 6.10, [19]).Then Theorem 2.3.1 in [18] states that the function u(z) However, we saw already that P [dσ α ] = 0 m d -almost everywhere on T d ; hence f α = 0 m d -almost everywhere on T d .We can thus conclude that σ α is a m d -singular measure for each α ∈ T.Moreover, as asserted in [11], two Clark measures σ α and σ β associated to an inner function ϕ are mutually singular whenever α ̸ = β.
Observe that since In particular, σ α is a probability measure if the associated inner function ϕ satisfies ϕ(0) = 0.
For an inner function ϕ and a constant α ∈ T, we define the so called unimodular level set where the closure is taken with respect to T d .The following proposition is a generalization of Lemma 2.1 in [3], where it is proven for rational inner functions.With the exception of some details, the proof uses the same arguments as in [3].We include it for the interested reader.Secondly, since ϕ is bounded on the unit polydisc and ϕ(rζ) ̸ → α on B, we have that lim sup for all ζ ∈ B. Here, we take the limit superior instead of the limit, as the limit of the right-hand side need not exist for every point in B. Now define For η ∈ D r (ζ), one can show that (see details on p. 4, [3]), and so which in turn implies that and, moreover, that the limit superior of this quotient is finite for all ζ ∈ B by (1).
In polar coordinates, we can express D r (ζ) as (details on p. 5, [3]).We observe that this is, as a subset of T d , a product of d copies of the same interval.Hence, as r → 1 − , we may estimate the Lebesgue measure of this set as To prove this, suppose there exists some z ∈ B such that the limit superior in (3) is nonzero.Since σ α is a finite measure, we have that σ α (B ∩ D r (z)) < ∞.Together with the fact that the denominator tends to zero, this would imply that the limit, and hence limit superior, is infinite for z ∈ B, which is a contradiction by our previous arguments.Hence (3) holds.
Since |D r (ζ)| → 0 as r → 1 − , the limit in (3) implies that the n-dimensional upper density of the restriction measure (σ α ) |B , defined as (σ α ) |B (A) := σ α (B ∩ A), is zero for every point in T d (see e.g.Proposition 2.2.2 in [16]).Thus, (σ α ) |B is equal to zero, which in turn implies that σ α (B) = 0. □ 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 supp(σ α ) = C α .
Finally, we record a fact which will be used in several proofs down the line: Lemma 2.2.The linear span of Poisson kernels The proof is a straight-forward generalization of the proof of Proposition 1.17 in [13].
(ii) The derivative function f ′ has a non-tangential limit at ζ 0 .Under the equivalent conditions above, Proof.See Theorem 2.19 in [13].□ Definition 2.4.Assuming the conditions of the theorem, we call the angular derivative of f at ζ 0 .Furthermore, if f maps D to itself, we say that f has an angular derivative in the sense of Carathéodory at ζ 0 ∈ T if f has an angular derivative at ζ 0 and f (ζ 0 ) ∈ T.
We now have the machinery needed to state the following proposition, which will be extremely useful to us in later sections: .
Proof.See Proposition 11.2 in [13].□ Example 2.6.Let B(z) be a non-constant finite Blaschke product of order n, and let α ∈ T.
Then B(z) is an inner function and analytic on T, and B(ζ) = α has precisely n distinct solutions; denote these by η α 1 , . . ., η α n .Moreover, from properties of finite Blaschke products, its derivative is non-zero on T. By Proposition 2.5, the associated Clark measure then satisfies Example 2.7.The function is inner, and ϕ * (ζ) exists everywhere on T; this is because from which we can see that ϕ * (1) = 0. Observe that every point ζ ̸ = 1 on the unit circle solves the equation ϕ * = α for some α ∈ T, so ϕ * (T \ {1}) = T.Moreover, the solutions accumulate in the limit point ζ = 1 for every α-value.Since the unimodular level sets are closed by definition, this implies that 1 ∈ C α (ϕ) for all α ∈ T. Now let α = 1 for simplicity.As seen in Example 11.3(ii) in [13], the solutions to ϕ * (ζ) = 1 are given by By Proposition 2.5, the Clark measure of ϕ associated to α = 1 may thus be expressed as We will revisit variations of this example in later sections.By Theorem 4 in [4], RP-measures on T d , d ≥ 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 2.5 for d ≥ 2. However, the proposition will still be useful in determining the density of certain Clark measures.

Rational inner functions
There has been significant progress in Clark theory for multivariate rational inner functions, see [3] and [5].We already saw a one-variable RIF in Example 2.6, where we could describe the Clark measures in a straight-forward manner -largely because Blaschke products are in fact analytic on 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 ∈ C[z 1 , . . ., z d ] is stable if it has no zeros in D d , and that it has polydegree (n 1 , . . ., n d ) ∈ 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 D d can be written as where a ∈ R, k 1 , . . ., k d ∈ N, p is a stable polynomial of polydegree (n 1 , . . ., n d ), and z 2 is its reflection.Note that any zero of p will be a zero of p and vice versa, and that p and p have the same polydegree.For simplicity, we will always assume that ϕ(z) = p p , where p and p are so called atoral -a concept explored in [1] and [6].In the context of this text, atoral simply means that p and p share no common factors, and that in two dimensions in particular, p and p have finitely many common zeros on T 2 (see Section 2.1, [6]).Hence, a rational inner function ϕ in two variables will have at most finitely many singularities on T 2 .
Moreover, we define the polydegree of a rational function ϕ = q/p as (n 1 , . . ., 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 ϕ = 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 ζ ∈ 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: Proof.See Theorem 2.6 in [8].□ Note that for any zero of p, the equation p − αp = 0 is trivially satisfied.This implies that all singularities of ϕ on T d are contained in C α (ϕ).Moreover, observe that {ϕ * = α} is in general not a closed set.However, by the theorem above, we may characterize C α (ϕ) as the zeros of a polynomial when ϕ is a RIF.
When d = 2, one can find an even nicer characterization of the unimodular level sets: Lemma 3.2.Let ϕ = p p be a RIF of bidegree (m, n), and fix α ∈ T. For any choice of τ 0 ∈ T, there exists a finite number of functions g α 1 , . . ., g α n defined on T and analytic on T \ {τ 0 } such that C α (ϕ) can be written as a union of curves {(ζ, g α j (ζ)) : ζ ∈ T}, j = 1, . . ., n, potentially together with a finite number of vertical lines Proof.See Lemma 2.5 in [3].
The proof is quite technical and will only be outlined here.Specifically, the authors fix a point τ ∈ T and construct a parameterization of C α (ϕ) ∩ (I τ × T) for a small interval I τ ∋ τ in T. When τ is not the z 1 -coordinate of a singularity of ϕ, one can use properties of RIFs to show that ϕ(τ, •) satisfies the conditions of the Implicit Function Theorem.Hence the solutions to ϕ * = α can be parameterized by smooth curves on the strip I τ × T.
The main issue arises from the fact that the curves g α j might intersect at singularities of ϕ, in which case analyticity is not obvious.One must thus ensure that we can in some sense "pull apart" any crossed curves in C α (ϕ) and prove that they are each analytic when viewed separately.However, it is shown in [6] that near each singularity of ϕ, the level sets actually do consist of smooth curves.Hence, even in the case where τ is the z 1 -coordinate of a singularity, one obtains a smooth parameterization of I τ × T. In the last step of the proof, the authors glue together the local parameterizations, which yields the final result.
The analysis of Clark measures of RIFs must now be divided into two cases; when the unimodular constant α is generic versus exceptional as defined below.Definition 3.3.We say that α ∈ T is an exceptional value if ϕ(τ, ζ 2 ) ≡ α or ϕ(ζ 1 , τ ) ≡ α for some τ ∈ T. If α is not exceptional, we say that it is generic.
The different cases stem from the characterization of C α (ϕ) in Lemma 3.2; if α is an exceptional value, by the definition above, the level sets will contain lines of the form {ζ 1 = τ } or {ζ 2 = τ }.If α is generic, C α (ϕ) can be fully described by the graphs of the functions g α 1 , . . ., g α n .
Theorem 3.4.Let ϕ = p p be a RIF of bidegree (m, n) and α ∈ T a generic value for ϕ.Then the associated Clark measure σ α satisfies , where g α 1 , . . ., g α n are the parameterizing functions from Lemma 3.2.Proof.See Theorem 3.3 in [3].□ Theorem 3.5.Let ϕ = p p be a RIF of bidegree (m, n) and α ∈ T an exceptional value for ϕ.Then, for f ∈ C(T 2 ), the associated Clark measure σ α satisfies where g α 1 , . . ., g α n are the parameterizing functions and and c [3]. □ 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 ϕ = p/p, let W α j (ζ) := | ∂ϕ ∂z 2 (ζ, g α j (ζ))| −1 denote the weights from Theorem 3.4 and Theorem 3.5.Then, by Lemma 5.1 in [3], these functions are in L 1 (T) and may be expressed as .
If (τ, γ) = (τ, g α j (τ )) for some level curve g α j , then p(τ, g α j (τ )) = 0 and one might expect W α j 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 α j in C α (ϕ) which do not satisfy g α j (τ ) = γ.In [3], the authors introduce the notion of contact order and prove the following statement: For all but finitely many α, if a branch (ζ, g α j (ζ)) of C α (ϕ) passes through the singularity (τ, γ), the corresponding weight function W α j has order of vanishing at τ that corresponds to the contact order of the corresponding branch of Z(p) at (τ, γ).Here, Z(p) denotes the zero set of the polynomial 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 for all ζ in a neighborhood of τ , where K j is the contact order of ϕ at (τ, γ) = (τ, g α j (τ )) associated with the branch g α j .Consequently, under these conditions, W α j 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 ϕ = p/p has bidegree (n, 1), we may write p(z) = p 1 (z 1 ) + z 2 p 2 (z 1 ) and p(z) = z 2 p1 (z 1 ) + p2 (z 1 ) for reflections pi = z n 1 p i (1/z i ).In this case, solving ϕ * = α for z 2 yields z 2 = 1 Bα(z 1 ) , where Moreover, define Then, by Theorem 1.2 in [5], we have | is non-zero if and only if α is an exceptional value.It is worth noting that for any RIF ϕ of bidegree (n, 1), a value α ∈ T is exceptional if and only if it is the non-tangential value of ϕ at some singularity (see Section 3 of [5]).

Multiplicative embeddings
Given an inner function ϕ in one complex variable, we define the multiplicative embedding The function defined by (z 1 , z 2 ) → z 1 z 2 maps D 2 to D, and so ϕ being an inner function implies that Φ is inner as well.In the following proposition, we characterize the support set of Φ with the help of the original function ϕ.Proof.First, for ease of notation, define Then we know that lim r→1 − ϕ(rζ) = α.For every z ∈ T, lim To extend this to a union over C α (ϕ), let ζ ∈ C α (ϕ).Then there exists some sequence (ζ n ) n≥1 in C ′ α (ϕ) that converges to ζ as n tends to infinity.This also implies that for any z ∈ T, (z, Then there is some sequence of (z 1,n , z 2,n ) in C ′ α (Φ) converging to (z 1 , z 2 ) as n → ∞.But this implies that z 1,n z 2,n → z 1 z 2 ∈ C α (ϕ), and the same argument as above then yields

□
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 τ α on the antidiagonals in C α (Φ).We do this in the next result and its subsequent corollary: Theorem 4.3.Let ϕ(z) be an inner function in one variable, with Clark measure σ α for some unimodular constant α.Let τ α be the corresponding Clark measure of Φ(z 1 , z 2 ) := ϕ(z 1 z 2 ).Then, for any function f ∈ C(T 2 ), Proof.We first prove this in the case when f is the product of one-variable Poisson kernels.Fixing z 2 ∈ D, let As the middle expression is pluriharmonic, u z 2 must be harmonic on D. Since z 1 z 2 ∈ D for any z 1 ∈ D, and ϕ is analytic (and hence continuous) on D, we see that as a function of z 1 is continuous on D.Moreover, by the maximum principle, |ϕ| < 1 on the unit disc, which implies that the denominator will always be non-zero.We conclude that u z 2 is continuous in D, and we may thus apply the Poisson integral formula: Moreover, for ζ ∈ T, we see that , where we use the definition of the Clark measure σ α in the last step.By integrating the above and applying (4), we get In particular, when the Clark measures associated to ϕ are discrete, one gets the following result: Corollary 4.4.1.Let ϕ : D → C be an inner function with Clark measure σ α for some unimodular constant α, and let τ α be the Clark measure of Φ(z 1 , z 2 ) := ϕ(z 1 z 2 ).If σ α is supported on a countable collection of points {η k } k≥1 ⊂ C α (ϕ), then Proof.By Proposition 2.5, σ α having a point mass at some Then, following the steps in the proof of Theorem 4.3, This then reduces to Hence, Integrating over this and applying Theorem 4.3 then shows that where we have used positivity of the summands in the last step.The result now follows from Lemma 2.2.□ It is interesting to compare the above result to the corresponding theorems, Theorem 3.4 and Theorem 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 ϕ, regardless of its complextiy, the associated Clark measures of ϕ(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 ϕ are discrete -have constant density along each such line.
Now consider Φ(z 1 , z 2 ) := ϕ(z 1 z 2 ), which appears in e.g.Example 13.1 in [7].Applying Corollary 4.4.1 for the Clark measure τ 1 of Φ then results in ).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 ϕ in one variable, define the multiplicative embedding 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 ϕ(z) : D → C be an inner function with Clark measure σ α for some unimodular constant α, and let τ α be the Clark measure of 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 We now want to show the result for d variables.Fix z 2 , . . ., z d and define the one-variable function 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: For fixed . Then, by our induction assumption, it holds that Finally, by (5), we arrive at as desired.Application of Lemma 2.2 yields the final result.□ Similarly to in the two-variable case, this gives us a sense of the geometry of supp{τ α }.For example, for d = 3 and x = e iν ∈ T, the set As in the case of d = 2, we get the following consequence when σ α is discrete: Corollary 4.6.1.Let ϕ(z) : D → C be an inner function with Clark measure σ α for some unimodular constant α, and let τ α be the Clark measure of

Product functions
Given one-variable inner functions ϕ and ψ, define the product function Then Ψ is an inner function in D 2 .The analysis of the Clark measures of Ψ is not as straightforward as for the multiplicative embeddings.A key argument in the proofs of Theorem 3.4, Theorem 3.5 and 4.3 is the Poisson integral formula.To use this for Ψ(z 1 , z 2 ), we require that for fixed z 2 ∈ D, the function is continuous on the closed unit disc.However, for a general inner function ϕ, its non-tangential limits need only exist m-almost everywhere on T.Even if they do exist on the entire unit circle, ϕ * need not be continuous.For this reason, we introduce the function Ψ r (z) := ϕ(rz 1 )ψ(z 2 ) for 0 < r < 1.This is not an inner function, as |ϕ(rz 1 )| < 1 on the unit circle.However, since Ψ r → Ψ as r → 1 − , we can investigate the Clark measures of Ψ via Ψ r .
Theorem 5.1.Let Ψ(z 1 , z 2 ) = ϕ(z 1 )ψ(z 2 ) for one-variable inner functions ϕ and ψ, such that A. ψ extends to be continuously differentiable on T except at a finite set of points, B. the solutions to ψ * = β for β ∈ T can be parameterized by functions {g k (β)} k≥1 which are continuous in β on T except at a finite set of points, C. for every β ∈ T, there are no solutions to ψ * = β with infinite multiplicity, and D. the Clark measures of ψ are all discrete.Then the Clark measures of Ψ satisfy 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 ϕ(z 1 ) = exp − 1+z 1 1−z 1 .
Remark 5.3.Observe that there exist examples of inner functions where the Clark measure σ α is discrete for one specific α-value but σ β is singular continuous for β ∈ T \ {α}, and vice versa.See Example 1 and 2 in [10].
Proof.Define Ψ r (z 1 , z 2 ) := ϕ(rz 1 )ψ(z 2 ) for 0 < r < 1, and note that for each z ∈ D 2 , as r → 1 − .Define, for fixed z 2 ∈ D and fixed 0 < r < 1, As ϕ(rz 1 ) is continuous and satisfies |ϕ(rz 1 )| < 1 on the unit circle, u r z 2 is continuous on D.Moreover, even though Ψ r is not an inner function, it holds that where (α + Ψ r )/(α − Ψ r ) is analytic on D 2 .Hence, the left-hand side is pluriharmonic in D 2 , which in turn implies that u r z 2 is harmonic in D. By the Poisson integral formula, Observe that ℜ((α ) is bounded for every (z 1 , z 2 ) ∈ D × 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 ∈ D, Hence, for ζ ∈ E, we have that To apply the Poisson integral formula, we must first check that the product of the righthand side with P z 1 (ζ) is integrable.Recall that by Fatou's theorem, ϕ(rζ) converges to ϕ * (ζ) as r → 1 − m-almost everywhere on T and in L 1 (T).Moreover, the curves {g k } k≥1 are assumed to be continuous on the unit circle except at finitely many points.Hence, the composition P z (ζ, g k (αϕ * (ζ))) must be measurable -indeed, f (ζ, g k (αϕ * (ζ))) is measurable for any f ∈ C(T 2 ).Similarly, we see that the weights |ψ ′ (g k (αϕ * (ζ)))| are measurable, as ψ is assumed to be continuously differentiable on T except at finitely many points.Since we are integrating over a compact space, this is enough to ensure integrability.
Moreover, for fixed ζ ∈ E, the sum k≥1 |ψ ′ (g k (αϕ * (ζ)))| −1 must be finite, since the Clark measure of ψ associated to the parameter value αϕ * (ζ) 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 T. The curves could still have infinite intersections at limit points of g k (αϕ * (ζ)), which per definition do not solve ψ * = αϕ * (ζ).However, by Proposition 2.5, the weights of the Clark measures must be zero for these points.
Since equation ( 7) holds for m-almost every ζ ∈ T, the integrals of the left-and right-hand side will coincide.By combining this with (6), we see that As the summands are all positive, we may interchange summation and integration.Thus, i.e.
Since the span of Poisson kernels is dense in C(T 2 ), we may conclude that 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 ψ that meet the requirements of Theorem 5.1.For example, finite Blaschke products define one such class.Let ψ be a non-constant finite Blaschke product of order n.As in Example 2.6, this implies that ψ is analytic on T and ψ * (ζ) = β has precisely n distinct solutions for each β ∈ T, and ψ ′ ̸ = 0 on T. By the Implicit Function Theorem, we may thus parameterize the solutions with functions {g k (β)} n k=1 analytic on the unit circle.Additionally, we saw in Example 2.6 that the Clark measures of ψ are discrete for every β ∈ T. Hence, Theorem 5.1 works for any product function Ψ(z) = ϕ(z 1 )ψ(z 2 ) where ψ is a non-constant finite Blaschke product and ϕ is an arbitrary inner function.
In the case where both ϕ and ψ are finite Blaschke products, the theorem reproduces what we know about RIFs, as Then Theorem 5.1 reduces to Theorem 3.4.
Observe that if ψ ∈ C(T), it must be a finite Blaschke product (Corollary 4.2, [14]).Similarly, if ψ ′ ∈ H 1 (T), then ψ is continuous on T (Theorem 3.11, [12]) and thus a finite Blaschke product.Hence, to be able to construct varied examples, we need ψ * to have some discontinuities on the unit circle (see e.g.Example 5.6).
In what comes next, we let g α k (ζ) := g k (αϕ * (ζ)) for ease of notation.Example 5.5.Let for ϕ as in Example 2.7 and some constant λ ∈ D. Note that Ψ * = 0 for ζ 1 = 1.The equation Ψ * = α for α ∈ T can be rewritten as For α = e iν , the solutions to this are given by In Figure 2, we have plotted the level curves for certain parameter values.
Let us calculate the weights of the Clark measures.Observe that Hence, by Theorem 5.1, for all f ∈ C(T 2 ).In Figure 3, we have plotted the weights where ϕ again is as in Example 2.  Since Ψ * is well-defined and unimodular on T 2 except on the lines {ζ 1 = 1} ∪ {ζ 2 = 1} where Ψ * = 0, we need to solve the equation Ψ = α.We may view this as Solving for ζ 2 yields Note that functions g α k are continuous on the unit circle; their only singularities occur at points ζ 1 = 2πk+ν ν+2πk+2i , which do not have modulus one.Moreover, all g α k pass through the point (1, 1) ∈ T 2 , which does not solve * = α as Ψ * (1, 1) = 0.However, since C α (Ψ) is closed, the point (1, 1) nevertheless lies in the unimodular level set.Hence, where g α k is analytic on T for every k.We have plotted some of these curves in Figure 4. Recall that by Lemma 3.2, the unimodular level sets of RIFs can be parameterized by graphs that are analytic on 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 Ψ.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, Ψ does not seem to meet the requirements of Theorem 5.1; there is a point on T 2 where all g k intersect, as g k (1) = 1 for all k ∈ Z.However, as noted above, this value does not in fact solve the equation Ψ * = α since ϕ * (1) = 0.This point would cause a problem if the Clark measure of ϕ had positive weight there.Fortunately, we are saved by Proposition 2.5; the measure associated to α has a point mass at 1 if and only if ϕ * (1) = α, and so |ϕ ′ (1)| −1 = 0. Let us now calculate the weights of the Clark measures associated to Ψ. First note that  reduce to zero for ζ = 1, as expected.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 Ψ -much like in the case of the rational inner functions in Section 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.

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 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 µ on T d is given by the real part of an analytic function if and only if its Fourier coefficients satisfy μ(k) = 0 for every k outside the set −Z d + ∪ Z d + , where −Z d + denotes the set of points (k 1 , . . ., k d ) where every k j ≤ 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 α are unitary for the simple multiplicative embedding ϕ(z 1 z 2 ) = z 1 z 2 where ϕ(z) = z, for which the Clark measure σ α satisfies For holomorphic monomials f , the functions f (ζ, αζ) are dense in L 2 (m), which in turn implies that A(D 2 ) is dense in L 2 (σ α ), as desired.It seems plausible that a similar argument can be applied to show that given any Φ(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.
for constants c(d), c ′ (d) dependent on d.Together with(2), this shows thatlim r→1 − σ α (B ∩ D r (ζ)) |D r (ζ)| = 0 m d -almost everywhere in B, and that the limit superior must be finite for all ζ ∈ B. Note that per definition, D r (ζ) is a d-dimensional cube with volume tending to zero as r → 1 − for every ζ ∈ B. We now claim that lim sup r→1 − σ α (B ∩ D r (ζ)) |D r (ζ)| = 0 for every ζ ∈ B.

Proposition 2 . 5 .
Let ϕ : D → C be an inner function and let α ∈ T. Then the associated Clark measure σ α has a point mass at ζ ∈ T if and only if ϕ * (ζ) = lim r→1 − ϕ(rζ) = α and ϕ has a finite angular derivative in the sense of Carathéodory at ζ.In this case,
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.
Remark 4.4.It is a priori not obvious that f (ζ, xζ) is integrable with respect to σ α .Integrability is ensured by the fact that f (ζ, xζ) is continuous on T, as it is composed by two functions f and g x (z) := (z, xz) which are continuous there.Since σ α is a finite, positive Borel measure on a compact space, all continuous functions on said space are integrable with respect to σ α .
Now apply Lemma 2.2 to obtain the final result.□