Laplace--Carleson embeddings on model spaces and boundedness of truncated Hankel and Toeplitz operators

A characterisation is given of bounded embeddings from weighted $L^2$ spaces on bounded intervals into $L^2$ spaces on the half-plane, induced by isomorphisms given by the Laplace transform onto weighted Hardy and Bergman spaces (Zen spaces). As an application necessary and sufficient conditions are given for the boundedness of truncated Hankel and Toeplitz integral operators, including the weighted case.


Introduction
By a Laplace-Carleson embedding, we mean the composition of two mappings: first, the Laplace transform from a space Z of functions defined on (0, ∞) to a space A of analytic functions defined on the right-half plane C + , then the natural embedding from A into L 2 (C + , µ) where µ is a positive regular Borel measure. As we shall make more precise in Section 2, the boundedness of such embeddings (which has applications in the study of weighted Hankel operators [2], and linear systems [9,10]) can be tested by considering the measure of Carleson squares.
The simplest example of the above situation is the case Z = L 2 (0, ∞), which, by the Paley-Wiener theorem is isomorphic to the Hardy space H 2 (C + ) by means of the Laplace transform. The boundedness of the Laplace-Carleson embedding is then described by means of the classical Carleson embedding theorem (see Section 2).
Finite-interval embeddings, on the other hand, form a special case of an embedding of a model space (an invariant subspace for the backwards shift), for which a theory of Carleson measures in the unweighted case has been studied by many authors, see e.g. [4,1,11].
Here we prove a very general theorem, Theorem 3.1, for a class of spaces (Zen spaces), including the Hardy and Bergman spaces, for finite-interval embeddings. We also prove embedding theorems for a wider class of model spaces in case that µ is supported on a sector or on the real line (Thm. 3.7, 3.8).
One application of these embedding results lies in the theory of boundedness of weighted Hankel integral operators, including the Glover operator studied in rational L 2 approximation [8]. This can even be used to characterize the boundedness of certain truncated Hankel and Toeplitz operators acting on general model spaces (Thm. 4.4,Rem. 4.6). We also discuss further applications in the theory of admissibility of control operators for linear semigroup systems (see, e.g. [19]).
Theorem 2.1. Let µ be a positive regular Borel measure on C + . Then the following are equivalent.
2. There is a constant C 1 > 0 such that for each w ∈ C + 3. There is a constant C 2 > 0 such that µ(Q a,h ) ≤ C 2 h for each Carleson square in C + .
The above theorem was generalized considerably in [9], with applications to problems in control theory given in [10]. The context used was that of Zen spaces, defined using a measure ν =ν ⊗ λ, whereν is a positive regular measure on [0, ∞) with 0 ∈ suppν, satisfying the ∆ 2 condition and λ denotes Lebesgue measure. Defining there is an isometric map L : [9,Prop. 2.3] and the references therein). Note thatν = δ 0 , a Dirac measure, corresponds to the Hardy space and w(t) = 2π for all t, whilẽ ν = λ corresponds to the Bergman space with w(t) = π/t. The following result, which we shall extend to finite-time embeddings, is given in [9, Thm. 2.1].
Theorem 2.2. With ν and w as above, the Laplace transform L defines a bounded linear map from L 2 (0, ∞; w(t) dt) into L 2 (C + , µ) if and only if there is a constant k > 0 such that for each Carleson square Q a,h we have

Model spaces on the disc and half-plane
For an inner function φ ∈ H ∞ (D), with D the unit disc, we write K D φ = H 2 (D) ⊖ φH 2 (D), the associated model space. These are the nontrivial subspaces invariant under the backward shift operator, and have been much studied. A key reference is [6].
We will be particularly interested in the case that θ T (s) = exp(−T s) for some T > 0, and thus φ T (z) = exp(T (z−1)/(z+1)). This is because L 2 (0, T ) is mapped isomorphically by the Laplace transform onto the Paley-Wiener In the context of Carleson embeddings from model spaces we may construct the following commutative diagram: where the relation between the measures ρ and µ is since to within a constant the mapping from F to f is an isometry from H 2 (C + ) onto H 2 (D). It takes θH 2 (C + ) onto φH 2 (D), and preserves orthogonal complements. Using the mappings above, one can translate the results of Cohn [4], Lacey et al [11] and others into the half-plane for a wide class of inner functions. This is discussed in Section 3.2.

Embedding theorems 3.1 Carleson embeddings for Zen spaces
We now consider finite-interval embeddings induced by the Laplace transform. It will be seen that the results do not depend on T (although the implied norm bounds do). We write . w for the norm in L 2 (0, ∞; w(t) dt).
Theorem 3.1. With ν and w related as in (2) and T > 0, the Laplace transform L defines a bounded linear map from L 2 ([0, T ], w(t) dt) into L 2 (C + , µ) if and only if there is a constant k > 0 such that for each Carleson square Q a,h with h ≥ 1 we have Proof. We define a new measure on C + by µ 1 (E) = µ(E − 1), so that µ 1 is a shifted version of µ, supported on the half-plane {z ∈ C : Re z > 1}. For , which is well-defined as Lf is an entire function. By (6), we have that µ 1 satisfies the Carleson condition (relative to ν), and so for some absolute constant c, and so the Laplace-Carleson embedding is bounded.

Embeddings of Model spaces
Recall the result of Cohn [4] (a version of his Theorem 3.2): let ρ be a measure on D and write The spectrum of an inner function φ consists of the closure of the union of the zero set of φ and the support of the singular measure associated with φ, if they exist. Equivalently it can be defined as Suppose φ is an inner function such that D ε = {z ∈ D : |φ(z)| < ε} is connected for some ε > 0, and suppose that ρ is a measure that assigns zero mass to the spectrum of φ. Then the following are equivalent: for all ξ ∈ D.
The question of the equivalence of 1. and 2. is known as the Cohn conjecture. This was shown to fail for general model spaces by Nazarov and Volberg in 2002 [12]; however, it does hold for special cases. The advantage of the Cohn criterion (7) is that it is fairly easy to check, compared to the test function conditions for the associate Clark measure used in [11], [3]. We will therefore be interested in settings in which versions of the Cohn conjecture hold. The following result by Cohn [5] for the case of radial embeddings will be very useful: 1. ρ is a Carleson measure for K D φ ; that is, the natural mapping Proof. The equivalence of (8) and (9) is just Theorem 3 in [5]. The consideration of the case 1 / ∈ σ(φ) etc. is not part of Cohn's original statement, but it can easily be extracted from his proof: As in (16), (17) of [5], reformulate (9) for 0 ≤ r < 1 as Suppose that 1 / ∈ σ(φ) and let R = 1 − dist(1, σ(φ))/2. Then φ extends analytically through 1, and whereC is a constant only depending on φ. Hence for R ≤ r < 1, where the last expression is the testing condition (10) for r = R. The proof for −1 / ∈ σ(φ) and −1 < r ≤ 0 follows similarly. In the context of the equivalence between disc and half-plane given in (4) and (5), we have where ξ = (1 − w)/(1 + w) and z = (1 − s)/(1 + s).
It is therefore possible to derive results on embeddings of spaces K C + θ into L 2 (C + , µ) from those in the setting of the unit disc. Let us consider the case of the singular inner function φ(z) = exp(T (z − 1)/(z+1)) for some fixed T > 0, which does satisfy the connectivity condition above, and whose spectrum is simply {−1}. Then, using Cohn's result we deduce the following.
Note that 1 − exp(−T Re w) tends to 1 as Re w → ∞ and is asymptotic to T Re w as Re w → 0. It is clear that the condition does not depend on T , which is as one would expect. We therefore have a simpler condition for a bounded Laplace-Carleson embedding.
This is the same as saying that the norm of the reproducing kernel s → 1/(s + w) in L 2 (C + , µ) is dominated by its norm in the mixed Hardy space From this we may obtain the following result, which is a special case of Theorem 3.1, and was first proved using the methods of this section before the more general theorem was derived. We omit the details of the proof.
This is a special case of the Carleson embedding theorem for model spaces by Treil and Volberg in [18]. In the case of general model spaces, Theorem 3.3 translates via the correspondences (4) and (11) into Theorem 3.7. Suppose θ is an inner function on C + , and suppose that µ is a measure supported on [0, ∞). Then the following are equivalent: 1. µ is a Carleson measure for K C + θ ; that is, the natural mapping 2. there is a constant c > 0 such that for all w ∈ (0, ∞).

Carleson Embedding Theorems on model spaces for sectorial measures
In the theory of infinite-dimensional linear systems, measures supported on a sector of the right half plane appear frequently in a natural way, arising from the spectrum of the underlying semigroup. Embedding theorems for measures supported on such a sector are therefore of interests, and such embeddings behave in many ways better than in the general case, see e.g. [10]. To our knowledge, it is unknown whether the Cohn conjecture holds for general model spaces in case of measures supported on a sector. Thm 1.1. and Section 2.5 in [3], previous to [11], provide a relatively accessible criterion in terms of Clark measures and test functions in this case. The counterexample of Nazarov and Volberg [12] concerns measures supported on the unit circle (in the case of the disc) respectively the imaginary axis (in the case of the right half-plane), thus it does not provide information on the case of sectorial measures. Here, we prove a version of the Cohn conjecture for sectorial measures and model spaces with some extra condition.
Theorem 3.8. Suppose θ is an inner function on C + , and suppose that µ is a measure supported on a sector Γ based at 0. In addition, suppose that 0 / ∈ σ(θ), and that either ∞ / ∈ σ(θ), or that |θ| is bounded away from 1 on the intersection of Γ with some right half plane. Then the following are equivalent: 1. µ is a Carleson measure for K C + θ ; that is, the natural mapping 2. there is a constant c > 0 such that for all w ∈ (0, ∞).
For the proof, we require the following refinement of Lemma 6.1 in [1].
Here, we use that for any w ∈ C + with θ(w) = 0, To prove Theorem 3.8, note that by Lemma 3.9, |θ(z)| will be bounded away from 1 on the set {w ∈ Γ : δ ≤ Re w ≤ N } for any N > 0. That means, for w in this set, condition (17) of Theorem 3.8 will be just the ordinary boundedness condition for reproducing kernels, and the ordinary Carleson condition for Carleson squares of sidelength between δ/2 and N follows. If both 0 and ∞ are not in σ(θ), this implies for suitable N > 0 the boundedness of the embedding (1) by the sufficiency part of the embedding theorem of Treil and Volberg in [18], which holds for any model space. If |θ| is bounded away from 1 on the intersection of Γ with some right half plane {z ∈ C : Re z ≥ N }, then we obtain the ordinary Carleson condition for Carleson squares of sidelength greater or equal to δ/2, and again, the embedding theorem of Treil and Volberg [18] implies the boundedness of the embedding.
The necessity of the Cohn condition (17) holds as always, since it follows from boundedness of the embedding on the reproducing kernels of the model space. This finishes the proof.

Truncated Hankel and Toeplitz operators
In [2], Carleson-measure techniques were developed to obtain information on the boundedness of certain integral operators, including the Glover operator studied in rational L 2 approximation [8]. Here we take this further, and consider truncated Hankel operators acting on L 2 (0, T ), with symbol h supported on (0, 2T ), given by Note that by using the unitary reversal operator Rf (x) = f (T − x) on L 2 (0, T ), and defining g(x) = h(T − x) so that g is supported on (−T, T ), we obtain which is a truncated Toeplitz operator acting on L 2 (0, T ). Taking Laplace transforms and writing G = Lg and θ T (s) = e −sT , as before, we see also that (if bounded) RΓ h is unitarily equivalent to a "standard" truncated Toeplitz operator A θ T G : f → P K θ T Gf, acting on the model space K θ T ⊂ H 2 (C + ). Clearly questions of boundedness of (18) and (19) are equivalent.
We recall Widom's theorem [20] (see also [16,Chap. 2]) which applies to the case that h is given as the Laplace transform of a positive Borel measure µ defined on R + ; such h are sometimes called diffusive, and correspond to linear systems that can be modelled using the heat equation [13].
In this situation Γ h (with T = ∞) is bounded if and only if there is a constant C > 0 such that µ(0, x) ≤ Cx for all x > 0.
We can now state a finite-interval version of Widom's theorem. Proof. The proof is a modification of an argument in [2]. Defining Z µ : (20) for f, g ∈ L 2 (0, T ), and so Γ h is bounded. The converse follows on taking f = g.
For example, taking dµ(x) = dx/ √ x we see that the function h(t) = 1/ √ t defines a bounded Hankel operator on the space L 2 (0, T ), although not on L 2 (0, ∞), since the Carleson measure condition is not satisfied uniformly for all x > 0.
Remark 4.2. In the case that h is the Laplace transform (sometimes called Fourier-Borel transform) of a complex measure µ defined on C + the condition µ(Q a,h ) ≤ kh for h ≥ 1 will be sufficient for boundedness, although not necessary in general.
Some extensions to weighted Hankel operators were given in [2], and we give a brief account of a finite-time version of these results. Supposing once more that h is the Laplace transform of a positive Borel measure µ on R + , consider the integral operator Γ h,w defined on L 2 (0, T ) by As above, the boundedness of this operator is easily seen to be equivalent to the boundedness of the Laplace transform from L 2 (0, T ; dt/(w(t)) 2 ) into L 2 (R + , µ), and this is a variation on Theorem 3.1.
The case α = 0 is covered by Theorem 4.1.
In the case of truncated Hankel operators on general model spaces, we can proceed in exactly the same way as in Theorem 4.1, using the embedding theorem 3.7. Let θ : C + → C be an inner function and K := L −1 K θ ⊆ L 2 (0, ∞) be the inverse Laplace transform of the corresponding model space. Then we obtain: Theorem 4.4. Let h be given as the Laplace transform of a positive Borel measure µ defined on R + and let θ be an inner function on C + . Then the truncated Hankel operator Γ h on K given by is bounded, if and only if condition (15) of Theorem 3.3 holds, that is, Moreover, in case that 0 / ∈ σ(θ) (resp. ∞ / ∈ σ(θ)), it suffices to check (22) only for w ≥ min{dist(0, σ(θ))/2, 1} (resp. for w ≤ 2 sup{|z| : z ∈ σ(θ)}).
Furthermore, a simple criterion for trace class membership of Γ h is available in this setting. The formal adjoint of the Laplace-Carleson embedding is given by where the integral kernel u(s, t) is given by and (k θ s (t)) s∈C + denotes the reproducing kernel of the model space K. Hence This identity was proved in [1], Prop 5.1, for the case of the unit disc. Using the factorization identity (4.1) again, we obtain Theorem 4.5. Let h be given as the Laplace transform of a positive Borel measure µ defined on R + , let θ be an inner function on C + and K θ the corresponding model space. Then the trace norm of the truncated Hankel operator Γ h on K = L −1 K θ is given by and Γ h is of trace class if and only if the right-hand side is finite.
Remark 4.6. As in the special, case of (18) and (19), the truncated Hankel integral operator Γ h as defined in (21) is equivalent to a truncated Toeplitz operator in the case when θ(s) = θ(s) for s ∈ iR.
For it follows from [15,Cor 4.4] that under these circumstances the Laplace transform of Γ h f equals P K θ (Lh)(RLf ), at least on a dense set of f ∈ K, where R : L 2 (iR) → L 2 (iR) denotes the reversal operator RF (s) = F (−s) (s ∈ iR).
Now it easily verified that if F ∈ K θ , then θ(RF ) also lies in K θ , and has the same norm, at least for inner functions θ such that θ(s) = θ(s). Thus the operator Γ h in (21) is equivalent to a truncated Toeplitz operator on K θ with symbol θ(Lh).
Here x(t) ∈ X is the state of the system at time t, u(t) ∈ C the input and B : C → D(A * ) ′ the control operator. Suppose now that (T (t)) t≥0 has a Riesz basis of eigenvectors (φ k ), that is, T (t)φ k = e λ k t φ k for each k, where the sequence (λ k ) lies in C − .
We take a Hilbert space Z of functions, which in [9,10] is a weighted L 2 (0, ∞) space, but in our case will be a weighted space L 2 (0, T ; w(t) dt) for some T > 0.
Let B be a linear bounded map from C to D(A * ) ′ , corresponding to the sequence (b k ). Then the control operator B is said to be Z-admissible for (T (t)) t≥0 if and only if there is a constant m 0 > 0 such that ∞ 0 T (t)Bu(t) dt X ≤ m 0 u Z , (u ∈ Z).
As discussed in [9,10], B will be admissible if and only if the Laplace transform induces a bounded mapping from Z into L 2 (C + , µ), where µ is the measure |b k | 2 δ −λ k . Finally, the condition we arrive at is that it is necessary and sufficient for Z-admissibility that there exist a constant k > 0 such that −λ k ∈Q a,h |b k | 2 ≤ kν(Q a,h ) for all Carleson squares with h ≥ 1, where ν and w are related is in (2). This is consistent with results produced by other methods, some of which are discussed in [19].