INTEGRAL REPRESENTATIONS AND UNIQUENESS SETS FOR STAR-INVARIANT SUBSPACES.

ALEXEI POLTORATSKI

Abstract. We discuss properties of pseudocontinuable functions in the unit disk of the complex plane in the context of operator theory.

We start with a review of recent developments in the study of the boundary behavior of the Cauchy integrals and their role in rank-one perturbation problems. After that we obtain new results on integral representations, uniqueness sets and separating measures for the model spaces K_{θ}^{p}.

1. Introduction

This note is devoted to pseudocontinuable functions in the unit disk 𝔻 = {|z| < 1} and their applications in operator theory. We say that an analytic function f in 𝔻 has a pseudocontinuation if there exists a certain meromorphic function f^{#} from the Nevanlinna class in ℂ \ 𝔻̄ satisfying

lim_{r→1−} f(rξ) = lim_{r→1+}(f^{#})(rξ)

for almost every ξ on the unit circle 𝕋 = {|z| = 1}. Here we assume that f has boundary values a. e. on the circle, which is a common feature for all ”reasonable” analytic functions in 𝔻.

The importance of pseudocontinuable functions was established by the famous theorem of R. Douglas, H. Shapiro and A. Shields [9]. It says that a function from the Hardy space H^{2} in the unit disk is a cyclic vector for the backward shift operator S^{∗} : f ↦ \frac{f−f(0)}{z} if and only if it does not posess a pseudocontinuation. Hence, all proper invariant subspaces of S^{∗} are filled with pseudocontinuable functions. Moreover, as was shown by S. Nagy and C. Foias, ”most” of the contractive operators in Hilbert spaces can be realized as compressions of the shift operator S : f ↦ zf on such subspaces of H^{2}. This fact lies in the foundation of the operator model theory, see [11].

To give a precise definition of these subspaces of pseudocontinuable functions let us first consider the case p = 2. For any inner function θ we denote by K_{θ}^{2} the orthogonal difference H^{2} ⊖ θH^{2}. The space K_{θ}^{2} is a Hilbert space. It possesses a system of reproducing kernels k_{λ} with the property that f(λ) =< f, k_{λ} > for any λ ∈ 𝔻 and any f ∈ K_{θ}^{2}. The kernels are given by the formula

k_{λ}(z) = \frac{1 − \overline{θ(λ)}θ(z)}{1 − λ̄z}.  (1)

1991 Mathematics Subject Classification. 30E20.

Key words and phrases. Cauchy integrals, pseudocontinuable functions, rank-one perturbations.

The author is supported in part by N.S.F. grant DMS 9970151 .

The space K_{θ}^{p}, p > 0 can be defined as the closed span of k_{λ}’s in the topology of the Hardy space H^{p}. When p ≥ 1 the spaces K_{θ}^{p} are the only closed invariant subspaces of the backward shift operator S^{∗}, see [11] for p > 1 and [6] for p = 1. (When p < 1 the spaces K_{θ}^{p} are not the only invariant subspaces of S^{∗}, see [6].)

A necessary and sufficient condition for a function f to belong to K_{θ}^{p}, p ≥ 1 is that the function f̃ defined on 𝕋 as f̄θ belongs to H^{p} and f̃(0) = 0. Now the property that any f ∈ K_{θ}^{p} admits a pseudocontinuation can be easily seen from the fact that \overline{f̃}θ is meromorphic outside 𝔻 and equals f a. e. on 𝕋.

In this paper we will focus on the following aspect of the operator model theory discovered by D. Clark in [7]. Let P_{θ} denote the orthogonal projector from H^{2} onto K_{θ}^{2}. Then the classical model operator S_{θ} is defined as a compression of S onto K_{θ}^{2}: S_{θ} : K_{θ}^{2} → K_{θ}^{2}, S_{θ} = P_{θ}S. The operator S_{θ} is a contraction with deficiency indices (1,1). Hence it can be made unitary by a rank-one perturbation. Namely, if α ∈ 𝕋 = {|z| = 1} and θ(0) = 0 (this restriction is not important but simplifies most formulas) one could define the operator U_{α} on K_{θ}^{2} as

U_{α}f = z (f − (f, \frac{θ}{z})\frac{θ}{z} ) + α(f, \frac{θ}{z}).  (2)

It is not difficult to check that U_{α} is unitary for any α ∈ 𝕋. The relation

U_{α} = S_{θ} + α(·, \frac{θ}{z})  (3)

shows that it is a rank-one perturbation of S_{θ}. Moreover, the family {U_{α}}_{α∈𝕋} contains all possible rank one unitary perturbations of the model contraction S_{θ}. The resulting operators U_{α} bear all the information about the model contraction S_{θ} but are much easier to study, as they are unitary.

Perhaps the most remarkable part of Clark’s construction is the explicit formula for the spectral measures of the operators U_{α}. To unveil these measures one simply has to notice that, since θ is inner, the function (α + θ)(α − θ)^{−1} has positive real part in 𝔻 for any α ∈ 𝕋. Hence, there is a unique finite positive Borel measure σ_{α} on 𝕋 satisfying

Re \frac{α + θ}{α − θ} = P σ_{α}  (4)

where P σ_{α} denotes the Poisson integral

P σ_{α} = ∫_{𝕋} \frac{1 − |z|^{2}}{|z − ξ|^{2}} dσ_{α}(ξ).  (5)

It turns out that σ_{α} is the spectral measure for U_{α}. We will denote the family of measures satisfying (4) by M_{θ} = {σ_{α}}_{α∈𝕋}.

Since (α + θ)(α − θ)^{−1} is imaginary a. e. on 𝕋, the measure σ_{α} is singular. It is not difficult to see from (4) that σ_{α} is concentrated on a set where θ = α. Thus one can say that the spectral measures of rank one unitary perturbations of S_{θ} are weighted level sets of θ. In particular, σ_{α} has a point mass at ξ ∈ 𝕋 iff θ(ξ) = α and θ has a non-tangential derivative at ξ, i. e. \frac{θ(z)−α}{z−ξ} has a finite non-tangential limit at ξ (see [17]).

It follows that the measures σ_{α} are not only singular with respect to the Lebesgue measure, but also pairwise singular. Moreover, using Rokhlin’s terminology, σ_{α}’s constitute a ”partition of unity”, i. e. if put together they produce the normalized Lebesgue measure m on 𝕋:

∫_{𝕋} σ_{α}dm(α) = m,  (6)

see [1].

Clark’s construction builds a bridge between the comlex analytic operator theory, in the style of the Havin-Nikolski seminar or Nikolski’s book [11], on one side and Gohberg-Krein-Livshits perturbation theory and mathematial physics on the other. To see this connection let us look at the same construction from the other direction: If U_{1} is an arbitrary unitary operator with simple spectrum and φ, ||φ|| = 1 is its cyclic vector then one can consider the standard family of rank one unitary perturbations of U_{1} generated by φ:

U_{α} = U_{1} + (1 − α)(·, U_{1}^{−1}φ)φ, α ∈ 𝕋.  (7)

Let θ be the characteristic function for the contraction U_{1} − (·, U_{1}^{−1}φ)φ defined as in the operator model theory. Then the spectral measures of U_{α} are immediately produced by (4). If we replace the unitary U_{1} with a self-adjoint A_{0} and consider the family of self-adjoint rank one perturbations

A_{λ} = A_{0} + λ(·, φ)φ λ ∈ ℝ,

which is a more traditional setting for the perturbation theory, then the same scheme will give the spectral measures of A_{λ}: one will just have to write down the upper half-plane analogue of (4), see [14, 15]. From the rank one situation one can, with some additional effort, pass to the case of finite rank and trace class perturbations.

Now let us consider the family of unitary operators U_{α} defined in K_{θ}^{2} by the formula (3). If σ_{α} is the spectral measure of U_{α} then by the spectral theorem U_{α} is unitarily equivalent to M_{α}^{z} , the operator of multiplication by z in L^{2}(σ_{α}). Thus there exists the generalized Fourier transform Φ_{α} – the unitary operator from K_{θ}^{2} to L^{2}(σ_{α}) satisfying

U_{α} = Φ_{α}^{∗}M_{α}^{z}Φ_{α}.  (8)

As was shown in [12], the Fourier transform Φ_{α} acts in the most natural way: it sends every function into its boundary values. Namely, for each α ∈ 𝕋 every function f from K_{θ}^{2} has non-tangential boundary values almost everywhere with respect to the (singular) measure σ_{α} and these boundary values are equal to Φ_{α}f almost everywhere with respect to σ_{α} , see Section 2.

To write down the formula for the inverse operator Φ_{α}^{∗} we need to introduce the so-called normalized Cauchy transform. Denote by M the space of all finite complex Borel measures on 𝕋, by M_{+} the subset consisting of positive measures. If μ ∈ M we can consider its Cauchy integral Kμ in the unit disk:

Kμ(z) = ∫_{𝕋} \frac{dμ(ξ)}{1 − ξ̄z} .

The normalized Cauchy transform C_{μ} is defined for functions f ∈ L^{1}(|μ|) as

C_{μ}f(z) = \frac{Kfμ(z)}{Kμ(z)}.

It turns out that the inverse Fourier transform Φ_{α}^{∗} can be defined as

Φ_{α}^{∗}f = C_{σ_{α}}f.

This paper is devoted to the discussion of the transforms Φ_{α}, Φ_{α}^{∗} and the related results on pseudocontinuable functions. Both the direct operator Φ_{α} and its inverse Φ_{α}^{∗} have many intriguing function theoretic properties. These properties were studied in a number of papers, see for instance [2], [5], [12] and [16]. In Section 2 we give a brief survey of this area.

Sections 3 and 4 contain new results on integral representaions and uniqueness in K_{θ}^{p}. In Section 4 we study the following question: When does the generalized Fourier transform Φ_{α} have a non-trivilal kernel in K_{θ}^{p}, p ≠ 2? This question can be considered within the general framework of the uncertainity principle in harmonic analysis as formulated in [10].

In order to prove our uniqueness results, in Section 3 we have to solve the following elementary problem that may be of independent interest to some readers: When does a ratio of two Cauchy integrals of singular measures belong to H^{p}, p ≥ 1? As we find out in Section 4, all the pseudocontinuable functions from K_{θ}^{p} can be represented as such ratios.

Acknowledgement. The author would like to thank A. B. Aleksandrov for useful discussions and A. Borichev for important comments and corrections.

2. The Fourier transforms Φ_{α} and Φ_{α}^{∗}

Let θ be an inner function. Consider the operators U_{α} on K_{θ}^{2} defined by (3). Then, as we mentioned in the introduction, their spectral measures σ_{α} are given by (4). Recall that Φ_{α} is defined as the unitary operator intertwining U_{α} and M_{α}^{z} , the operator of multiplicaton by z in L^{2}(σ_{α}), see (8).

As was shown by D. Clark in [7], Φ_{α} can be defined in the following way. Recall that θ has non-tangential boundary limits equal to α σ_{α}-a. e. Hence any reproducing kernel

k_{λ}(z) = \frac{1 − \overline{θ(λ)}θ(z)}{1 − λ̄z}, λ ∈ 𝔻,

has non-tangential boundary limits equal to \frac{1−\overline{θ(λ)}α}{1−λ̄z} σ_{α}-a. e. Put

Φ_{α}k_{λ} = \frac{1 − \overline{θ(λ)}α}{1 − λ̄z} ∈ L^{2}(σ_{α}).  (9)

It turns out that Φ_{α} preserves the norms of finite linear combinations of reproducing kernels. Since linear combinations of k_{λ}’s are dense in K_{θ}^{2} as well as their images are in L^{2}(σ_{α}), Φ_{α} can be extended to a unitary operator from K_{θ}^{2} to L^{2}(σ_{α}). It is not difficult to check that (8) is satisfied on reproducing kernels and therefore on the whole K_{θ}^{2}.

After Clark’s paper appeared, D. Sarason rose the natural question about the action of Φ_{α} on functions in K_{θ}^{2} other than finite linear combinations of reproducing kernels. The problem here is that, unlike k_{λ}, an arbitrary H^{p}-function does not have to have non-tangential boundary limits σ_{α}-a. e. This question was answered by the following

Theorem 2.1. [12]

Let θ be an inner function, M_{θ} = {σ_{α}}_{α∈𝕋} be the associated family of measures on 𝕋 defined by (4). For any f ∈ K_{θ}^{2} and any α ∈ 𝕋 the non-tangential boundary limit f^{∗}(ξ) exists for σ_{α}-almost every ξ ∈ 𝕋. The unitary operator Φ_{α} : K_{θ}^{2} → L^{2}(σ_{α}) satisfying (8) can be defined as

(Φ_{α}f)(ξ) = f^{∗}(ξ) for σ_{α} − a.e. ξ.

In particular, the boundary values of f belong to L^{2}(σ_{α}) for any α and have the same norm as f (an analog of Plancherel’s formula).

The following result on convergence in average is much easier to prove than Theorem 2.1. We are able to supply the proof here. We keep the notations of the last statement.

Theorem 2.2. [12]

Let f ∈ K_{θ}^{2}, f(z) = ∑_{n=0}^{∞} a_{n}z^{n}. Then for any α ∈ 𝕋 the partial sums ∑_{n=0}^{k} a_{n}z^{n} converge to f^{∗} (i. e. to Φ_{α}f) in L^{2}(σ_{α}) as k → ∞.

Proof. It is not difficult to show that

∑_{n=0}^{k} a_{n}z^{n} = Φ_{α}f − z^{k}Φ_{α}(S^{∗})^{k}f for σ_{α} − a.e. ξ.

The equality can first be verified for f = k_{λ} and then extended to the whole space in the standard way.

Since ||(S^{∗})^{k}f||_{K_{θ}^{2}} → 0 and Φ_{α} is unitary, ||Φ_{α}(S^{∗})^{k}f||_{L^{2}(σ_{α})} → 0. □

If one applies the Abel-Poisson summation method to the series of f one immediately obtaines the following

Corollary 2.3. [12] The functions f_{r}^{∗}, f_{r}(z) = f(rz) converge to f^{∗} (Φ_{α}f) in L^{2}(σ_{α}) as r → 1−.

We will return to Φ_{α} and discuss its uniqueness properties in Section 4. As for now, let us move on to the inverse transform Φ_{α}^{∗}.

To write down an explicit formula for Φ_{α}^{∗} : L^{2}(σ_{α}) → K_{θ}^{2} let us start with the

reproducing kernels. As usual we assume that θ(0) = 0. Denote l_{λ}^{α} = \frac{1−\overline{θ(λ)}α}{1−λ̄z} ∈ L^{2}(σ_{α}). Let us recall that by (9)

Φ_{α}^{∗}l_{λ}^{α} = k_{λ}.

Since Kσ_{α} = 1/(1 − ᾱθ), it is not difficult to verify that

k_{λ} = \frac{Kl_{λ}^{α}σ_{α}}{Kσ_{α}} = C_{σ_{α}}l_{λ}^{α}

(Kμ and C_{μ} are as defined in the introduction). Now, if f is an arbitrary function from L^{2}(σ_{α}) we can consider a sequence f_{n} of finite linear combinations of l_{λ}^{α}’s tending to f in L^{2}(σ_{α}). Since Φ^{∗} is unitary, Φ^{∗}f_{n} will tend to Φ^{∗}f in H^{2}. At the same C_{σ_{α}}f_{n} will tend to C_{σ_{α}}f pointwise in the disk. Hence, we can extend from l_{λ}^{α}’s to the whole L^{2}(σ_{α}) to conclude that Φ_{α}^{∗} is the normalized Cauchy transform associated with σ_{α}:

Φ_{α}^{∗} : f ↦ C_{σ_{α}}f.  (10)

Let μ ∈ M_{+} be a singular measure. Then there exists an inner function satisfying the equation

P μ = Re \frac{1 − θ}{1 + θ} .

By our definitions this means that μ = σ_{1} ∈ M_{θ} and C_{μ} = Φ_{1}^{∗}. Hence, any transform C_{μ} associated with singular positive μ can be represented as Φ_{α}^{∗} : L^{2}(μ) → K_{θ}^{2} for a suitably chosen θ. This allows us to switch to the discussion of the normalized Cauchy transform C_{μ}. Some of the results below do not require μ to be singular or real.

Together with Theorem 2.1, (10) gives the following result:

Theorem 2.4. [12]

Let μ ∈ M and f ∈ L^{1}(|μ|). Then the function C_{μ}f has finite non-tangential boundary values μ-a. e. These values are equal to f μ^{s}-a. e., where μ^{s} is the singular component of μ.

The case of singular positive μ follows directly from Theorem 2.1. The general case requires a little additional effort, see [12]. One can also formulate this statement in a slightly more general form:

Theorem 2.5. Let μ, ν ∈ M. Then the meromorphic function \frac{Kμ}{Kν} has non-tangential boundary limits ν-a. e. on 𝕋.

If μ = fν + γ where f ∈ L^{1}(|ν|) and γ ∈ M, γ ⊥ ν then these non-tangential limits are equal to f ν^{s}-a. e.

We suggest it to the reader as an exercise to deduce Theorem 2.5 from Theorem 2.4.

In Section 3 we will need the following simple corollary of the last result:

Corollary 2.6. Let μ, ν ∈ M. Suppose that ν is singular and μ ⊥ ν. Then \frac{Kμ}{Kν} has non-tangential boundary limits equal to 0 ν-a. e. on 𝕋.

Remark 2.7. If ϕ is an arbitrary (not necessarily inner) function from the unit ball of H^{∞} one can still define the family of measures M_{ϕ} on 𝕋 satisfying (4). In the case of non-inner ϕ the measures will not be singular. Nevertheless, the whole construction works in this case as well. The role of K_{θ}^{2} will now be played by the de Branges-Rovnyak space N_{ϕ} (see [17], where it was denoted by M_{ϕ}). The measures σ_{α} will, once again, present a family of spectral measures of rank-one perturbations of a unitary operator (whose spectral measure is σ_{1}), see [14]. The operator C_{σ_{α}} will map L^{2}(σ_{α}) onto N_{ϕ}. In Theorems 2.1, 2.2 and Corollary 2.3 K_{θ}^{2} can be replaced by N_{ϕ}. Theorem 2.4 says that any f ∈ N_{ϕ} has boundary values σ_{α}-a. e. for any α ∈ 𝕋.

This extension allows us to include not only singular but all cyclic unitary families of rank-one perturbations in our model. It also provides motivation to study the operator C_{μ} for arbitrary positive μ.

The rest of this section is devoted to the L^{p}-estimates for C_{μ} and the associated maximal non-tangential operator.

For any μ ∈ M_{+} C_{μ} is bounded as an operator from L^{2}(μ) to H^{2} (or to L^{2}(m)). The situation becomes more interesting for p ≠ 2:

Theorem 2.8. [2]

For any μ ∈ M_{+} the operator C_{μ} is bounded as an operator from L^{p}(μ) to H^{p} (to L^{p}(m)) for 1 < p ≤ 2. The operator C_{μ} is unbounded, in general, for p > 2. In particular, if μ is singular and C_{μ} : L^{p}(μ) ↦ H^{p} (L^{p}(m)) is bounded for some p > 2 then μ is discrete.

Theorem 2.4 shows that for any f ∈ L^{p}(μ), p ≥ 1 the function C_{μ}f is defined μ-a. e. on 𝕋. Hence it makes sense to ask if C_{μ}f ∈ L^{p}(μ) i. e. to study the boundedness of C_{μ} as a transform in L^{p}(μ). The question appears to be even more interesting (and more difficult) if one considers the associated non-tangential maximal operator. The results in this direction can be used to estimate the asymptotics of the resolvents of our operators inside the disk (upper half-plane in the case of self-adjoint operators). These problems can also be viewed as a generalisation of the Hunt-Muckenhoupt-Weeden theory to the case of arbitrary weights, see [16].

Let us start with C_{μ}. The problem of its boundedness as a transform in L^{p}(μ) is solved by Theorem 2.4 for all p ≥ 1 in the most difficult case of singular μ. For such measures C_{μ} is not only bounded, it is identical. This reflects the fact that for singular μ the measure fμ is never “antianalytic.” For arbitrary μ the situation is not as nice. The same “forces” that make Theorem 2.8 fail for p > 2 play a role here. All in all, we have the following

Theorem 2.9. [16] For any μ ∈ M_{+}(𝕋) the operator C_{μ} is bounded in L^{p}(μ) for 1 < p ≤ 2.

The operator is unbounded (in general) for p > 2, see Example 1 in [16]. It is not necessarily bounded when p = 1 either: the simplest counterexample is μ = m.

Now let us discuss the action of the non-tangential maximal operator MC_{μ}f in L^{p}(μ). For any function g in 𝔻 we define its non-tangential maximal function Mg as

Mg(ξ) = sup_{z∈Γ_{ξ}} |g(z)|

where ξ ∈ 𝕋 and Γ_{ξ} is the sector \{|z − ξ| ≤ \sqrt{2} Re(1 − ξ̄z), |z| > \frac{1}{\sqrt{2}}\}. If μ ∈ M_{+} then MC_{μ}f is finite μ-a.e. on 𝕋 for any f ∈ L^{1}(μ) by Theorem 2.4. That allows us to ask if the maximal operator MC_{μ} : f ↦ MC_{μ}f is bounded in L^{p}(μ).

As usual, we say that an operator has the weak type (p, p) if it acts from L^{p}(μ) into the “weak L^{p}(μ)”, defined as

L^{p,∞}(μ) = \{f| μ(\{|f| > t\}) < \frac{C}{t^{p}} , for some C < ∞\}

(with the inf of such C raised into the power 1/p viewed as a norm).

One of the results of [12] implies that MC_{μ} has the weak type (2,2). For p ≠ 2 we have the following:

Theorem 2.10. [16]

For any μ ∈ M_{+}(𝕋) the maximal operator MC_{μ} is bounded in L^{p}(μ) for 1 < p < 2.

The operator MC_{μ} is, generally, unbounded for p > 2. An example from [16] presents a singular μ and f ∈ L^{∞}(μ) such that MC_{μ}f ∉ L^{p}(μ) for any p > 2. The operator is obviously unbounded for p = 1 as well. Hence, the remaining question is weather the weak type (2,2) can be improved to the strong type. It seems that the answer is negative, but a counterexample is yet to be found. Apart from the boundedness, it is also unknown if the operator has the weak type (1,1).

In the classical theory of Cauchy transforms in L^{p} spaces with absolutely continuous weights the maximal operator always follows the pattern of the Cauchy transform itself: they are bounded under the same conditions. For arbitrary weights and the normalized Cauchy transform the situation is different. As we saw above, if we restrict our attention to the case of singular measures μ, then the boundedness of C_{μ} will follow trivially from Theorem 2.4 for all p ≥ 1. The maximal operator however will still be, generally, unbounded for p > 2 as follows from the example from [16] mentioned above.

3. Ratios of Cauchy integrals belonging to H^{p}

In this section we first answer the following basic question: Let μ, ν ∈ M_{+} be two singular measures. Suppose that μ ⊥ ν. Can Kμ/Kν belong to H^{p}?

The answer is, trivially, ”yes” for p < 1, since |Kν| ≥ 1/2||ν|| and Kμ ∈ H^{p}. The following theorem shows that it is also possible for 1 ≤ p < 2, but not for p ≥ 2.

Theorem 3.1.

I) Let μ ∈ M_{+} be a singular measure. Then there exists singular ν ∈ M_{+} such that μ ⊥ ν and Kμ/Kν ∈ H^{p} for every 0 < p < 2.

II) Let ν ∈ M_{+} be a singular measure with non-trivial continuous part. Then there exists a singular μ ∈ M_{+} such that μ ⊥ ν and Kμ/Kν ∈ H^{p} for every 0 < p < 2.

III) For any singular measures μ ∈ M and σ ∈ M_{+} such that μ ⊥ σ the ratio Kμ/Kσ is not in H^{2}.

Remark 3.2. The measure μ in part I) does not have to be positive. The proof would require an additional technical step. Here we only discuss the positive case for the sake of brevity.

Proof.

I) We first consider an important particular case when μ is δ_{1} (the unit point mass at 1). Then Kμ = (1 − z)^{−1}. Choose singular η ∈ M_{+} such that (1 − z)^{−1} belongs to any L^{p}(η), p < 2 but does not belong to L^{2}(η). Let θ be the inner function defined by Kη = (1+θ)^{−1}. Consider the corresponding family M_{θ} = {η_{α}} (then η = η_{−1} ∈ M_{θ}). Now we can put ν = η_{1}.

Indeed, it is well known (see for instance [17], pages 51-53) that since (1−z)^{−1} ∉ L^{2}(η), no other measure from M_{θ} can have a point mass at 1. Hence ν ⊥ μ. Also, since (1 − z)^{−1} ∈ L^{p}(η) for any p < 2 the function C_{η}(1 − z)^{−1} belongs to H^{p} for any p < 2 by Theorem 2.8. But, since 1 − θ = 2 η-a. e., the boundary values of \frac{1−θ}{1−z} are equal to 2/(1 − z) η-a. e. and therefore Φ^{∗}(1 − z)^{−1} = C_{η}(1 − z)^{−1} = \frac{1}{2} \frac{1−θ}{1−z} by Theorem 2.1. Hence \frac{1−θ}{1−z} belongs to H^{p} for any p < 2. It is left to recall that \frac{1−θ}{1−z} = Kμ/Kν.

If μ is an arbitrary positive singular measure, consider once again the corresponding inner function I, Kμ = (1−I)^{−1}. WLOG ||μ|| = 1. Then Kμ(z) = Kδ_{1}(I(z)). By the previous argument Kδ_{1}/Kσ ∈ H^{p}, p < 2 for some singular σ ∈ M_{+}, σ ⊥ δ_{1}. Therefore, Kδ_{1}(I(z))/Kσ(I(z)) ∈ H^{p}, p < 2. It is left to notice that Kσ(I(z)) is a Cauchy integral of a positive singular measure ν, μ ⊥ ν. Namely, if we put ν = ∫ μ_{α}dσ(α) where μ_{α} ∈ M_{I} then

\frac{Kμ}{Kν} = \frac{Kδ_{1}(I(z))}{Kσ(I(z))} ∈ H^{p}, p < 2.

It is not difficult to show that such ν and μ are mutually singular.

II) Let Σ ⊂ 𝕋 be a closed set such that |Σ| = 0, ν_{p.p.}(Σ) = 0 and ν_{c}(Σ) > 0 (ν_{p.p.} and ν_{c} are the pure point and continuous parts of ν respectively). Denote by η the restriction of ν on Σ. Then η is continuous. Let E_{1}, E_{2}, ... be a sequence of open sets on 𝕋 containing Σ such that E_{1} ⊃ E_{2} ⊃ ... and |E_{n}| = 2^{−n}. Denote by d_{n} the distance between Σ and Ē_{n} (Ē_{n} = 𝕋 \ E_{n}).

Construction of μ. We construct μ, μ ⊥ ν using a Cantor-type process.

Step 1. Let E_{1} = ∪I^{1,n} where I^{1,n}, n = 1, 2, ... are disjoint open arcs. Inside each arc I^{1,n} such that η(I^{1,n}) > 0 cover η-almost all Σ with dijoint open arcs J_{1}^{1,n}, J_{2}^{1,n}, ..., J_{2k}^{1,n}, k ≥ 1 satisfying η(J_{1}^{1,n}) = η(J_{2}^{1,n}) = ... = η(J_{2k}^{1,n}) and |J_{i}^{1,n}| < (d_{1})^{2}.

After that we define the function f_{1} to equal 2 on J_{1}^{1,n}, J_{3}^{1,n}, ..., J_{2k−1}^{1,n} for n = 1, 2, ... and 0 on the rest of 𝕋. Then we put μ_{1} = f_{1}η.

During this step we obtained a countable collection of arcs J_{i}^{1,n}. To simplify further notations we reenumerate them as J^{1,k}, k = 1, 2, .... The particular order of the new enumeration is not important.

Step m, m ≥ 2. Inside each arc J^{m−1,n} such that μ_{m−1}(J^{m−1,n}) > 0 cover μ_{m−1}-almost all Σ with disjoint open arcs J_{1}^{m,n}, J_{2}^{m,n}, ..., J_{(m+1)k}^{m,n}, k ≥ 1 satisfying μ_{m−1}(J_{1}^{m,n}) = μ_{m−1}(J_{2}^{m,n}) = ... = μ_{m−1}(J_{(m+1)k}^{m,n}) and |J_{i}^{m,n}| < (d_{m})^{2}. (Note that the last condition provides that all J_{i}^{m,n} ⊂ E_{m}.)

After that we define function f_{m} to equal 0 on J_{m+1}^{m,n}, J_{2m+2}^{m,n}, ..., J_{km+k}^{m,n}, to (m + 1)/m on other arcs J_{i}^{m,n} for n = 1, 2, ... and to 0 outside of ∪_{i=1}^{(m+1)k} J_{i}^{m,n}. Then we put μ_{m} = f_{m}μ_{m−1}. Before proceeding to step m + 1 we again redenote J_{i}^{m,n} as J^{m,k}, k ∈ ℕ.

The obtained sequence μ_{m} converges in the ∗-weak topology of M. We denote its limit by μ. It is not difficult to see that μ is singular. By the standard Cantor argument, μ ⊥ η: it is concentrated on a set Σ′ ⊂ Σ such that η(Σ′) = ∏(1− \frac{1}{n}) = 0. Therefore μ ⊥ ν_{c}. Since ν_{p.p.}(Σ) = 0, μ ⊥ ν.

Estimate of Kμ/Kν. Note that after the first step of the above construction we only change our measure inside the arcs J_{i}^{1,n} ⊂ E_{1} leaving the total mass of each arc unchanged. The condition |J_{i}^{1,n}| < (d_{1})^{2} means that each arc is rather far from Ē_{1}. Hence the Cauchy integral Kμ can be estimated through Kμ_{1} = Kf_{1}η on Ē_{1}. More precisely, it is not difficult to see that, since d_{1} ≤ 1/2, the condition |J_{i}^{1,n}| < (d_{1})^{2} implies

|Kμ| < |Kf_{1}η| + ||μ|| ≤ |Kf_{1}η| + ||ν_{c}||

on Ē_{1}. WLOG ||ν|| = 1. Since |Kν| ≥ 1/2 and f_{1}η = f_{1}ν the last inequality gives

||Kμ/Kν||_{L^{2}(Ē_{1})} ≤ C_{1}||Kf_{1}ν/Kν||_{H^{2}} + C_{2}

for some C_{1}, C_{2} > 0. By our construction ||f_{1}||_{L^{2}(ν)} ≤ \sqrt{2}. Note that ||Kf_{1}ν/Kν||_{H^{2}} = ||f_{1}||_{L^{2}(ν)} because C_{ν} = Φ_{1}^{∗} where Φ_{1} : K_{θ}^{2} → L^{2}(ν) is the Fourier transform discussed in the introduction corresponding to θ satisfying Kν = (1 − θ)^{−1}. Hence,

||Kμ/Kν||_{L^{2}(Ē_{1})} ≤ C_{1}||Kf_{1}ν/Kν||_{H^{2}} + C_{2} ≤ \sqrt{2}C_{1} + C_{2}.

In a similar way, on E_{n} \ E_{n+1}

||Kμ/Kν||_{L^{2}(E_{n}\E_{n+1})} ≤ C_{1}||\frac{K(∏_{1}^{n+1} f_{k})ν}{Kν} ||_{H^{2}} + C_{2} ≤ C_{1}|| ∏_{1}^{n+1} f_{k}||_{L^{2}(ν)} + C_{2}.

By our construction the product ∏_{1}^{n+1} f_{k} is equal to n + 1 on the set of measure 1/(n + 1) and to 0 elsewhere. Hence

||Kμ/Kν||_{L^{2}(E_{n}\E_{n+1})} < C_{3}\sqrt{n + 1}

for some C_{3} > 0.

If 0 < p < 2 then by Hölder’s inequality

||Kμ/Kν||_{L^{p}(E_{n}\E_{n+1})}^{p} ≤ [||Kμ/Kν||_{L^{2}(E_{n}\E_{n+1})}^{2}]^{p/2} [m(E_{n} \ E_{n+1})]^{\frac{2−p}{2}}

≤ C_{3}^{p}(n + 1)^{p/2}[2^{−n−1}]^{\frac{2−p}{2}} .

Hence, ||Kμ/Kν||_{L^{p}(𝕋)}^{p} < ∞.

III) Let μ ∈ M, σ ∈ M_{+}, μ ⊥ σ be two singular measures such that f = Kμ/Kσ ∈ H^{2}. Then f ∈ K_{θ}^{2} where inner θ is defined by (1 − θ)^{−1} = Kσ (see Lemma 4.2 below). Hence by Clark’s Theorem f = KF σ/Kσ for some F ∈ L^{2}(σ). Therefore, μ = F σ which contradicts the condition μ ⊥ σ. □

Remark 3.3. Another way to construct an inner θ such that \frac{1−θ}{1−z} belongs to any H^{p}, p < 2 but not to H^{2} (as in part I above) is suggested by A. B. Aleksandrov. Theorem 3.3 from [2] says that any Carleson Blaschke product with zeros a_{n} satisfying ∑ \frac{1−|a_{n}|^{2}}{|1−a_{n}|^{p}} < ∞ for all p < 2 but ∑ \frac{1−|a_{n}|^{2}}{|1−a_{n}|^{2}} = ∞ will have the desired properties.

In the rest of this section we will try to understand how ”close” the measures μ and ν should be to each other in order for Kμ/Kν to be in H^{p}. If μ ∈ M is singular we denote by Hol μ the holomorphic support of μ:

Hol μ = {ξ ∈ 𝕋| lim_{z⟶_{∢}ξ} Kμ(z) = ∞}.

The notation Hol^{#} μ stands for a slightly larger set:

Hol^{#} μ = {ξ ∈ 𝕋| lim sup_{z⟶_{∢}ξ} |Kμ(z)| = ∞}.

We define the space H^{1,∞} (”weak” H^{1}) as the subspace of H^{p}, p < 1 consisting of functions f whose boundary values f^{∗} satisfy

sup_{t} t m({|f^{∗}| > t}) < ∞.

By a theorem of Kolmogorov all Cauchy integrals belong to H^{1,∞} (it also follows, for instance, from Theorem 3.5 below). Any quotient Kν/Kμ where μ ∈ M_{+} will also belong to H^{1,∞} since |Kμ| ≥ 1/2||μ||.

We denote by H^{o(1,∞)} the subset of H^{1,∞} consisting of functions satisfying

lim_{t→∞} t m({|f^{∗}| > t}) = 0.

We have the following result:

Theorem 3.4. Let μ ∈ M, ν ∈ M_{+}, μ ⊥ ν be two singular measures. I. If Kμ/Kν ∈ H^{1} then Hol ν supports μ. II. If Kμ/Kν ∈ H^{o(1,∞)} then Hol^{#} ν supports μ.

Before we prove Theorem 3.4 we need some prepapation. To simplify notations, in what follows we write Kμ instead of Kμ^{∗} to denote the non-tangential boundary values of Kμ and χ_{{|Kμ|>t}} denotes the characteristic function of the set {|Kμ| > t} ⊂ 𝕋. As usual, μ_{s} stands for the singular component of μ.

Theorem 3.5. [13] Let μ ∈ M. Then

πtχ_{{|Kμ|>t}}m ⟶^{∗−weakly}_{t→∞} |μ_{s}|.

We will also need the following

Lemma 3.6. Let μ ∈ M, ν ∈ M_{+}, μ ⊥ ν be two singular measures. The property Kμ/Kν ∈ H^{o(1,∞)} is spectral for μ, i. e. if Kμ/Kν ∈ H^{o(1,∞)} then Kfμ/Kν ∈ H^{o(1,∞)} for any f ∈ L^{1}(|μ|).

Proof. First, let μ be positive.

To show that Kfμ/Kν ∈ H^{o(1,∞)} approximate f in L^{1}(μ) by a smooth function g. It is not difficult to check that for any smooth g the function Kgμ/Kμ is bounded. Hence Kgμ/Kν ∈ H^{o(1,∞)}. Since the H^{1,∞}-norm of K(f − g)μ/Kν can be made arbitrarily small, we obtain the statement.

Now, let μ be an arbitrary real measure, μ = μ_{+} − μ_{−}, where μ_{±} are mutually singular positive measures. To reduce this case to the previous one, we need to demonstrate that if Kμ/Kν ∈ H^{o(1,∞)} then Kμ_{±}/Kν ∈ H^{o(1,∞)}.

Let E, F be disjoint open sets such that μ_{+}(E) > ||μ_{+}||−ϵ and μ_{−}(F) > ||μ_{−}||−ϵ. By Theorem 3.5

πtm({|Kμ_{+}| > t} ∩ E) → μ_{+}(E) > ||μ_{+}|| − ϵ as t → ∞.

Hence

πtm({|Kμ_{+}| > t} ∩ E) > ||μ_{+}|| − ϵ − o(1) as t → ∞.  (11)

In the same way

πtm({|Kμ_{−}| > t} ∩ E) < ϵ + o(1) as t → ∞.  (12)

Since |Kν| ≥ 1/2, {|Kμ_{+}/Kν| > t} ⊂ {|Kμ_{+}| > t/2}. Hence by (11)

m(\{|Kμ_{+}/Kν| > t\} \ E) < \frac{2ϵ}{πt} + o(1/t)

and, similarly, by (12)

m(\{|Kμ_{−}/Kν| > t\} ∩ E) < \frac{2ϵ}{πt} + o(1/t).

Since Kμ/Kν ∈ H^{o(1,∞)}, this implies

m(\{|Kμ_{+}/Kν| > t\}) < m(\{|Kμ_{+}/Kν| > t\} ∩ E) + \frac{2ϵ}{πt} + o(1/t)

≤ m ([\{|Kμ/Kν| > t/2\} ∪ \{|Kμ_{−}/Kν| > t/2\}] ∩ E) + \frac{2ϵ}{πt} + o(1/t)

≤ m(\{|Kμ/Kν| > t/2\} ∩ E) + m(\{|Kμ_{−}/Kν| > t/2\} ∩ E) + \frac{2ϵ}{πt} + o(1/t)

< \frac{6ϵ}{πt} + o(1/t) as t → ∞.

Therefore Kμ_{+}/Kν ∈ H^{o(1,∞)}. The measure μ_{−} can be treated in the same way.

The case of arbitrary complex μ can be proved by separating its real and imaginary parts. □

We say that a point ξ ∈ 𝕋 is a density point of a set E ⊂ 𝕋 if

\frac{|I_{ε} ∩ E|}{ε} → 1 as ε → 0,

where I_{ε} is the arc of size ε centered at ξ.

Proof of Theorem 3.4. Denote f = Kμ/Kν.

I. Since f is summable on 𝕋, f = Kf. Suppose that there exists a set E, E ∩ Hol ν = ∅ such that |μ|(E) > 0. Then 1/Kν = Kf/Kμ does not tend to 0 as z ⟶_{∢} ξ for μ-a. e. ξ in E. But this contradicts Corollary 2.6 since fm ⊥ μ.

II. By Lemma 3.6 we can assume that there exists C > 0 such that for μ-a. e. ξ

sup_{z=rξ, 0<r<1} |Kν(z)| < C < ∞.  (13)

(Otherwise just multiply μ by

ϕ_{C} = χ_{{ξ| sup_{z=rξ, 0<r<1} |Kν(z)|<C}},

prove the statement for ϕ_{C}μ instead of μ and then consider larger and larger C’c.)

Denote L_{n} = {ξ ∈ 𝕋| |Kμ(ξ)| > n}. If ξ ∈ L_{n} is a density point for L_{n}, consider I_{ξ} – an open arc containing ξ satisfying

|L_{n} ∩ I_{ξ}| > 3/4|I_{ξ}|.  (14)

Such an I_{ξ} can always be chosen to satisfy μ(I_{ξ}) > 0. Indeed, since I_{ξ} ∩ L_{n} ≠ ∅, if both endpoints of I_{ξ} are not in L_{n} then automatically μ(I_{ξ}) > 0. If one or both endpoints are in L_{n} one can consider a slightly larger arc such that (14) still holds but the endpoints are no longer in L_{n}. By the standard argument, we can choose a collection I_{1}^{n}, I_{2}^{n}, ... of such open arcs I_{ξ} that covers almost all L_{n}. Moreover, I_{i}^{n} can be chosen to form a covering of rank 2 (each point is covered by at most 2 arcs).

Recall that, since μ is a singular measure, Theorem 3.5 implies |L_{n}| > D \frac{||μ||}{n} for some D > 0. Notice that since f ∈ H^{o(1,∞)} there exists a sequence A_{n} → ∞ such that

|L_{n} ∩ {|Kν| > A_{n}}| > 3/4|L_{n}|.

Since I_{i}^{n} form a covering of rank 2, there exists I_{k}^{n} such that

|I_{k}^{n} ∩ {|Kν| > A_{n}}| > 3/8|I_{k}^{n}|.  (15)

Since Kν is an outer function, we can consider the function G = \sqrt{Kν}. The function Re G is positive and is at least \sqrt{A_{n}/2} on \{|Kν| > A_{n}\}. Together with (15), the properties of the Poisson kernel now imply that sup_{0<r<1} Re G(rξ) > c\sqrt{A_{n}} for any ξ ∈ I_{k}^{n} for some c > 0. But since μ(I_{k}^{n}) > 0 and A_{n} can be arbitrarily large (recall that A_{n} → ∞) this contradicts (13). □

4. Uniqueness and integral representations of pseudocontinuable functions

In this section we first discuss for what θ and p the generalized Fourier transform Φ_{α} (defined as in the introduction) may have a non-trivial kernel in K_{θ}^{p}. One can also look at this problem from the following function theoretic viewpoint.

If X is a linear space of functions in 𝔻 we say that Σ ⊂ 𝕋 is a uniqueness set for X if the only f ∈ X such that

lim_{z⟶_{∢}ξ} f(z) = 0 for all ξ ∈ Σ

is the zero function.

Note that the existence of boundary values on Σ for all functions in X is not required. If Σ is a uniqueness set for X and f, g ∈ X are two different functions, our definition means that f and g behave differently near Σ. Namely, the limit

lim_{z⟶_{∢}ξ} f − g

either does not exist or is non-zero for at least one point ξ ∈ Σ.

Also, if μ is a finite Borel measure on 𝕋 we say that μ separates points in X (μ is a separating measure for X) if the only f ∈ X satisfying

lim_{z⟶_{∢}ξ} f(z) = 0 for μ-a. e. ξ ∈ 𝕋

is the zero function.

The classical theory implies that Σ is a uniqueness set for H^{p} iff m(Σ) > 0. Similarly, μ ∈ M, μ ≠ 0 is a separating measure for H^{p} iff μ has a nontrivial absolutely continuous part. The situation is very different for the spaces K_{θ}^{p}.

It is hardly possible to discribe all uniqueness sets for all K_{θ}^{p} in any reasonable terms. The main difficulty here is that the structure of uniqueness sets depends too much on the concrete properties of θ, e. g. [8]. For instance, if θ is a finite Blaschke product of degree n then K_{θ}^{p} is n-dimensional and any n points on 𝕋 constitute a uniqueness set. Nothing like this takes place if we consider an infinite Blaschke product or a singular inner function. In that case, however, other ”uncommon” uniqueness sets may appear if the spectrum of θ contains an arc or if θ is a socalled single-component function (see [2] for the definition).

As was announced before, in this note we are only going to discuss for what θ and p the measures σ_{α} (defined by (4)) are separating and when their natural supports, the level sets {θ = α}, are uniqueness sets. (We denote by {θ = α} the subset of 𝕋 consisting of points where the non-tangential boundary limit of θ is equal to α.)

If θ is inner we denote by σ(θ) the spectrum of θ:

σ(θ) = {ξ ∈ 𝕋| lim inf_{z→ξ} |θ(z)| = 0}.

Theorem 4.1.

I. If p ≥ 2 then every σ_{α} ∈ M_{θ} is a separating measure for K_{θ}^{p} for any inner θ (and therefore any set {θ = α}, α ∈ 𝕋 is a uniqueness set);

II. If 1 ≤ p < 2 then there exists θ such that none of the measures σ_{α} ∈ M_{θ} is a separating measure for K_{θ}^{p};

III. If p < 1 then there may not be any uniqueness sets of Lebesgue measure 0 (singular separating measures) for the space K_{θ}^{p}. In particular, if σ(θ) ⊃ 𝕋 there are no uniqueness sets of Lebesgue measure 0 (singular separating measures).

For 1 ≤ p < 2 it is unknown wether the set {θ = α} is a uniqueness set for K_{θ}^{p}. Although this seems highly unlikely, we were unable to find a counterexample. Instead we prove that {θ = α} is a uniqueness set for a large subset of K_{θ}^{p} consisting of functions admitting certain integral representations, see Theorem 4.5 below.

To prove Theorem 4.1 we now turn to the discussion of integral representations for pseudocontinuable functions.

For μ ∈ M the Cauchy integral Kμ is naturally defined in ℂ \ 𝕋. Moreover, if μ is singular then by Privalov’s theorem the two parts of Kμ defined inside and outside the disk have equal boundary values on the circle at a. e. point. Hence Kμ presents an example of a pseudocontinuable function in 𝔻. Now, let μ ∈ M and ν ∈ M_{+} be two singular measures. Then, once again, \frac{Kμ}{Kν} is an example of a pseudocontinuable function in 𝔻.

For any singular μ Kμ belongs to any H^{p}, p < 1 but not to H^{1}. However, as we will demonstrate shortly, our second example \frac{Kμ}{Kν} covers all pseudocontinuable functions in H^{1}: for any function f ∈ K_{θ}^{p}, p ≥ 1 there exist μ ∈ M and ν ∈ M_{+} such that f = \frac{Kμ}{Kν}. We will also show how these measures are related to θ.

Lemma 4.2. Let f = \frac{Kμ}{Kν} where μ ∈ M, ν ∈ M_{+} are singular. If f ∈ H^{p}, p ≥ 1 then f ∈ K_{θ}^{p}, where the inner function θ is uniquely defined by the equation

Kν = \frac{1}{1 − θ} .  (16)

Proof. It is easy to check that for any singular positive ν there is a unique θ satisfying (16). Let us notice that θ̄f = (θ̄ − 1)Kμ. Since Kμ(∞) = 0, the antianalytic function g(z) = Kμ(1/z̄) satisfies g(0) = 0. Hence θ̄f = (θ̄−1)g ∈ H̄_{0}^{p}.

□

As we mentioned before, the relation (16) between ν and θ means simply that ν = σ_{1} ∈ M_{θ}. Along with the representaion f = \frac{Kμ}{Kσ_{1}} our function always posesses representations involving other σ_{α}’s. In fact, the definition of the inverse Fourier transform Φ_{α}^{∗} (see formula (10)) implies that if f ∈ K_{θ}^{2} then for any α ∈ 𝕋 there is a unique measure μ ∈ M such that f = \frac{Kμ}{Kσ_{α}}. By Theorem 2.1 the measure μ satisfies μ = f^{∗}σ_{α} where f^{∗} ∈ L^{2}(σ_{α}) is the non-tangential boundary values of f. For p < 2 we have

Lemma 4.3. Let f ∈ K_{θ}^{p}, p ≥ 1. Then for almost any α ∈ 𝕋 f = \frac{Kf^{∗}σ_{α}}{Kσ_{α}}, where f^{∗} ∈ L^{p}(σ_{α}) is the non-tangential boundary values of f.

Proof. Consider a sequence f_{n} of K_{θ}^{2}-functions tending to f in H^{p}. By (6) the boundary values f_{n}^{∗} → f^{∗} in L^{p}(σ_{α}) for almost every α. If f_{n}^{∗} → f^{∗} in L^{p}(σ_{α}), then Kf_{n}^{∗}σ_{α} → Kf^{∗}σ_{α} pointwise in the disk. Hence, pointwise in the disk,

f = lim_{n→∞} f_{n} = lim_{n→∞} \frac{Kf_{n}^{∗}σ_{α}}{Kσ_{α}} = \frac{Kf^{∗}σ_{α}}{Kσ_{α}} .

□

We will say that the function f ∈ K_{θ}^{p} admits an α-representation if f = \frac{Kμ}{Kν} for some μ ∈ M and ν = σ_{α} ∈ M_{θ}. We will call a representation f = \frac{Kμ}{Kν} regular if μ << ν.

Using this terminology we can say that any f ∈ K_{θ}^{2} has regular α-representations for all α. Lemma 4.3 says that for 1 ≤ p < 2 f has regular α-representations for almost all α. Let us also make the following observations:

Remark 4.4.

I) Each f ∈ K_{θ}^{p}, 0 < p ≤ ∞ admits at most one α-representaion.

II) By Theorem 3.1 and Lemma 4.2 functions from K_{θ}^{2} admit only regular representaions, whereas functions from K_{θ}^{p}, 1 ≤ p < 2 may have both regular and irregular ones. The same function f ∈ K_{θ}^{p}, 1 ≤ p < 2 may have regular αrepresentations for some α and irregular ones for other α. By Lemma 4.3 any f admits almost all α-representations and they are regular. Hence, f can only have irregular α-representaions for a zero-set of α’s.

III) By Theorem 3.4 if f ∈ K_{θ}^{p}, 1 ≤ p < 2, f = \frac{Kμ}{Kσ_{α}} then μ is supported on the

set {θ = α}. If f ∈ K_{θ}^{o(1,∞)} (= K_{θ}^{q} ∩ H^{o(1,∞)}, 0 < q < 1) then μ is supported on the set

{ξ ∈ 𝕋 | lim θ(z_{n}) = α for some sequence z_{n} ⟶_{∢} ξ}.

IV) In general, a K_{θ}^{p}-function may not admit an α-representaion for some α if 1 ≤ p < 2. In [2] it was shown that a function f ∈ K_{θ}^{p}, 1 ≤ p < 2 may have no non-tangential boundary values on a set E, σ_{α}(E) > 0. At the same time, it follows from Theorem 2.5 that if a function has an α-representaion then it must have non-tangential boundary values σ_{α}-a. e. Thus, such an f does not have an α-representaion (neither a regular nor an irregular one).

V) Denote

(K_{θ}^{p})_{α} = {f ∈ K_{θ}^{p} | f admits an α-representation }.

Note that (K_{θ}^{p})_{α} , 1 ≤ p, p ≠ 2 contains Φ_{α}^{∗}(L^{p}(σ_{α})) but is rarely equal to that set. The set Φ_{α}^{∗}(L^{p}(σ_{α})) consists of functions admitting regular α-representations Kfσ_{α}/Kσ_{α} with f ∈ L^{p}(σ_{α}). If p < 2 an α-representaion may not be regular. If p > 2 the representation is always regular by Theorem 3.1, but f may not be in L^{p}(σ_{α}), as follows from the results of [2].

VI) We have (K_{θ}^{2})_{α} = K_{θ}^{2} for any α. For 1 ≤ p < 2 Lemma 4.3 implies

⋃_{α∈Ω} (K_{θ}^{p})_{α} = K_{θ}^{p}

where Ω is any subset of 𝕋 of non-zero Lebesgue measure.

VII) As was shown in [5] for any 1 ≤ p and any α, β ∈ 𝕋, α ≠ β

Φ_{α}^{∗}(L^{p}(σ_{α})) + Φ_{β}^{∗}(L^{p}(σ_{β})) = K_{θ}^{p}

i. e. for any f ∈ K_{θ}^{p} there exist h ∈ L^{p}(σ_{α}) and g ∈ L^{p}(σ_{β}) such that f = C_{σ_{α}}h + C_{σ_{β}}g.

Proof of Theorem 4.1. I) Follows from the fact that any f ∈ K_{θ}^{p}, p ≥ 2 has the regular α-representation f = Kf^{∗}σ_{α}/Kσ_{α} and from Theorem 2.5.

II) If θ is such that any σ_{α} ∈ M_{θ} has a non-trivial continuous part then by Theorem 3.1 for any α there exists a non-zero μ ∈ M, μ ⊥ σ_{α} such that \frac{Kμ}{Kσ_{α}} ∈ H^{p}. By Lemma 4.2 f = \frac{Kμ}{Kσ_{α}} ∈ K_{θ}^{p}. By Theorem 2.5 the boundary values of f are 0 σ_{α}-a.e.

To find such a θ take, for instance, any inner function that does not have angular derivatives at any point on 𝕋. Then all σ_{α} will be continuous (see [17], pages 51-53).

III) It is well-known (see [6]) that the necessary and sufficient condition for a function f to belong to K_{θ}^{p}, 0 < p < 1 is to satisfy f̄θ ∈ H^{p} and to admit an analytic continuation through 𝕋 into the domain of θ in ℂ. If σ(θ) = 𝕋 then the second requirement dissapears. Also if f = Kμ for some singular μ ∈ M then f − ||μ||/2 is imaginary on 𝕋 and therefore f̄θ = (−f + ||μ||)θ ∈ H^{p}. Hence, K_{θ}^{p}, p < 1 contains all Cauchy integrals of singular measures. Let Σ ⊂ 𝕋 be a uniqueness set of Lebesgue measure 0. Then there exists a singular measure μ such that P μ(ξ) = ∞ at every ξ ∈ Σ. Consider the Herglotz integral of μ

Hμ = ∫_{𝕋} \frac{1 + ξ̄z}{1 − ξ̄z} dμ(ξ).

Then the function (Hμ)^{−1} has positive real part in 𝔻 and is real at 0. Hence it is a Herglotz integral of another measure ν. Since (Hμ)^{−1} is imaginary a.e. on 𝕋, ν is singular. To obtain a contradiction it is left to notice that Kν(ξ) ∈ K_{θ}^{p} but Kν(ξ) = 0 at any ξ ∈ Σ. □

As we mentioned before, it is unclear whether or not {θ = α} is a uniqueness set for K_{θ}^{p} for every θ when 1 ≤ p < 2. We can prove, however, that (K_{θ}^{p})_{α}, the set of functions admitting an α-representation, satisfies the following

Theorem 4.5. For any inner θ, α ∈ 𝕋 and p ≥ 1 the set {θ = α} is the uniqueness set for (K_{θ}^{p})_{α}.

Proof. Let f ∈ (K_{θ}^{p})_{α}, f = \frac{Kμ}{Kσ_{α}} , μ ≠ 0. By Theorem 2.5 the ratio \frac{Kσ_{α}}{Kμ} has finite non-tangential boundary limits μ-a. e. Hence f has non-zero limits μ-a. e. But by Theorem 3.4 μ is supported on {θ = α}. □

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Dept. of Math., Texas A&M University, College Station, TX 77843

E-mail address: alexei@math.tamu.edu