Interpolation and duality in spaces of pseudocontinuable functions

Given an inner function $\theta$ on the unit disk, let $K^p_\theta:=H^p\cap\theta\bar z\bar{H^p}$ be the associated star-invariant subspace of the Hardy space $H^p$. Also, we put $K_{*\theta}:=K^2_\theta\cap{\rm BMO}$. Assuming that $B=B_{\mathcal Z}$ is an interpolating Blaschke product with zeros $\mathcal Z=\{z_j\}$, we characterize, for a number of smoothness classes $X$, the sequences of values $\mathcal W=\{w_j\}$ such that the interpolation problem $f\big|_{\mathcal Z}=\mathcal W$ has a solution $f$ in $K^2_B\cap X$. Turning to the case of a general inner function $\theta$, we further establish a non-duality relation between $K^1_\theta$ and $K_{*\theta}$. Namely, we prove that the latter space is properly contained in the dual of the former, unless $\theta$ is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in $K_{*B}$, with $B=B_{\mathcal Z}$ as above.


Introduction and results
We write T for the unit circle {ζ ∈ C : |ζ| = 1} and m for the normalized arc length measure on T; thus, dm(ζ) = |dζ|/(2π). We then define the spaces L p := L p (T, m) in the usual way and let · p denote the standard norm on L p . Also, for 1 ≤ p ≤ ∞, we introduce the Hardy space H p by putting The Poisson integral (i.e., harmonic extension) of an H p function being holomorphic on the disk D := {z ∈ C : |z| < 1} (see [15, Chapter II]), we may use this extension to view elements of H p as holomorphic functions on D when convenient.
Furthermore, we write P + (resp., P − ) for the orthogonal projection from L 2 onto H 2 (resp., onto zH 2 = L 2 ⊖H 2 ). By a classical theorem of M. Riesz (see [15,Chapter III]), each of these projections admits a bounded extension-or restriction-to L p , with 1 < p < ∞, and maps L p onto H p (resp., onto zH p ). Now suppose θ is an inner function, meaning that θ ∈ H ∞ and |θ| = 1 a.e. on T. The corresponding star-invariant (or model) subspace K p θ is then defined by (1.1) K p θ := {f ∈ H p : zf θ ∈ H p }, 1 ≤ p ≤ ∞, so that K p θ = H p ∩ θzH p . (When p = 2, yet another equivalent definition is K 2 θ = H 2 ⊖ θH 2 .) It is clear from (1.1) that the antilinear isometry (1.2) f → zf θ =: f leaves K p θ invariant. Also, it is well known (see [6,20]) that each K p θ is invariant under the backward shift operator and conversely, that every closed and nontrivial B-invariant subspace of H p , with 1 ≤ p < ∞, arises in this way.
The functions belonging to some K p θ space (i.e., noncyclic vectors of B) are known as pseudocontinuable functions. In fact, they are characterized by the property of having a meromorphic pseudocontinuation to D − := C\(D∪T); that is, the function in question should agree a.e. on T with the boundary values of some meromorphic function of bounded characteristic in D − (see [6] for details).
The orthogonal projection from H 2 onto K 2 θ is given by f → θP − (θf ), and the M. Riesz theorem shows that the same formula provides, for 1 < p < ∞, a bounded projection from H p onto K p θ parallel to θH p . This yields the direct sum decomposition with orthogonality for p = 2. Among the inner functions θ, of special relevance to us are Blaschke products. Recall that, for a sequence Z = {z j } ⊂ D with the associated Blaschke product is given by . The product converges uniformly on compact subsets of D and defines an inner function that vanishes precisely at the z j 's; see [15,Chapter II]. If, in addition, then we say that B is an interpolating Blaschke product. Accordingly, the sequences Z = {z j } in D that satisfy (1.4) and (1.5), with B = B Z , are called interpolating (or H ∞ -interpolating) sequences. By a celebrated theorem of Carleson (see [3] or [15, Chapter VII]), these are precisely the sequences Z with the property that Here and below, the following standard notation (and terminology) is used. Given a sequence Z = {z j } of pairwise distinct points in D, the trace f Z of a function f : D → C is defined to be the sequence {f (z j )}; and if X is a certain function space on D, then the corresponding trace space is We shall be concerned with interpolation problems for functions in star-invariant subspaces-specifically, for those in K p B , where B is an interpolating Blaschke product. Some of the earlier results in this area can be found in [1,7,16,18], while others, more relevant to our current topic, will be recalled presently.
First, we need yet another piece of notation. Given numbers p > 0, γ ∈ R and a sequence Z = {z j } ⊂ D, we write ℓ p γ (Z) for the set of all sequences {w j } ⊂ C satisfying Now, if 1 < p < ∞ and if B = B Z is an interpolating Blaschke product with zero sequence Z, then we have . Indeed, the left-hand equality follows from (1.3) with θ = B, while the other holds by a well-known theorem of Shapiro and Shields [21]. In addition, for each sequence W = {w j } in ℓ p 1 (Z), there is a unique function f ∈ K p B with f Z = W; the uniqueness is due to the fact that K p B ∩ BH p = {0}. The case of K ∞ B is subtler, as the next result shows. Theorem A. Suppose that Z = {z j } is an interpolating sequence in D and B = B Z is the associated Blaschke product. Then we have This theorem is essentially a consequence of Hruščev and Vinogradov's work in [19]; see also [5,Section 3] for details.
Condition (1.7) above is known as the uniform Frostman condition, and the sequences Z = {z j } in D that obey it are called Frostman sequences. While a Frostman sequence need not be interpolating (in fact, its points are not even supposed to be pairwise distinct), it does necessarily split into finitely many interpolating sequences; see [19] for a proof. Finally, a Blaschke product whose zeros form a Frostman sequence will be referred to as a Frostman Blaschke product.
We mention in passing that, by a theorem of Vinogradov [22], the identity K ∞ B 2 | Z = ℓ ∞ is valid whenever Z is an interpolating sequence and B = B Z . It should be noted, however, that K ∞ B 2 is strictly larger than K ∞ B . To describe the trace class K ∞ B | Z in the general case (i.e., when (1.7) no longer holds), we first introduce a bit of notation. Once the interpolating sequence Z = {z j } is fixed, we associate with each sequence W = {w j } from ℓ 1 1 (Z) the conjugate sequence W = { w k } whose elements are The absolute convergence of the series in (1.8) is ensured, for any k ∈ N, by the fact that W ∈ ℓ 1 1 (Z) in conjunction with (1.5). Because ℓ 1 1 (Z) contains ℓ ∞ , as well as every ℓ p 1 (Z) with 1 < p < ∞, the sequence W is well defined whenever W belongs to one of these spaces.
The following result was established in [13].
It was further conjectured in [13,14] that the trace space K 1 B Z is describable in similar terms, i.e., that the necessary conditions W ∈ ℓ 1 1 (Z) and W ∈ ℓ 1 1 (Z) are also sufficient for W to be in K 1 B Z . To the best of our knowledge, the conjecture is still open.
Here, our purpose is to supplement Theorem B by characterizing the values of smooth, not just bounded, functions in K 2 B on the (interpolating) sequence Z = B −1 (0). To be more precise, of concern are trace spaces of the form (K 2 B ∩ X) Z , where X is a certain smoothness class on T. Specifically, X will be one of the following spaces.
• The Lipschitz-Zygmund space Λ α = Λ α (T) with α > 0. This is the set of where n is some (any) integer with n > α, and ∆ n h denotes the nth order difference operator with step h. (As usual, the difference operators ∆ k h are defined inductively: where f I := m(I) −1 I f dm; the supremum is taken over the open arcs I ⊂ T. Even though BMO contains discontinuous and unbounded functions, there are reasons for viewing it as a smoothness class. In a sense, it corresponds to the endpoint as α → 0 of the Λ α scale. We also need the analytic subspace BMOA := BMO ∩ H 2 .
• The Gevrey class G α = G α (T) with α > 0. This is the set of functions f ∈ C ∞ (T) satisfying For each of these choices of X, we now characterize the sequences W from the trace space (K 2 B ∩ X) Z in terms of the conjugate sequence W, as defined by (1.8) above. The description always involves a certain decay condition (or growth restriction) on W, as we shall presently see.
B ∩ BMO, which corresponds to case (b) above, will be henceforth denoted by K * B . Similarly, for a general inner function θ, we define K * θ := K 2 θ ∩ BMO. Comparing Theorem B with the BMO part of Theorem 1.1, we see that the structure of the trace space K ∞ B Z is remarkably similar to that of K * B Z . In light of this observation, we may wonder what the BMO counterpart of Theorem A could look like. Specifically, we may ask if there exist infinite Blaschke products B = B Z for which the trace space K * B Z is completely determined by the natural (and necessary) logarithmic growth condition on the values.
To be more precise, suppose that Z = {z k } is a sequence of pairwise distinct points in D, and write ℓ ∞ log (Z) for the space of sequences W = {w k } ⊂ C with It is well known (and easily shown) that every f ∈ BMOA satisfies so BMOA Z is always contained in ℓ ∞ log (Z). The equality (1.9) BMOA Z = ℓ ∞ log (Z) obviously need not hold in general, but it does actually occur for some infinite sequences Z = {z k } (which form a tiny subfamily among the H ∞ -interpolating sequences). For instance, (1.9) will be valid provided that for some constants c > 0 and s ∈ (0, 1 2 ); see [10,Theorem 11]. The question is what happens to (1.9) when BMOA gets replaced by its subspace K * B , with B = B Z . The property that arises is thus , and we regard it as an analogue of (1.6) in the BMO setting. In contrast to (1.6), however, (1.10) does not lead to any nontrivial class of sequences. Indeed, our next result shows that (1.10) is only possible when Z = B −1 (0) is a finite set. Proposition 1.2. Whenever B = B Z is an infinite Blaschke product with simple zeros, the trace space K * B Z is properly contained in ℓ ∞ log (Z). This will be deduced from another result, which deals with the case of a general inner function θ and asserts an amusing lack of duality between the star-invariant subspaces K 1 θ and K * θ . It is well known that, for 1 < p < ∞, the dual of the Hardy space H p (under the pairing f, g = T f g dm) is H q with q = p/(p − 1), while the dual of H 1 is BMOA; see, e.g., [15,Chapter VI]. The former duality relation has a natural counterpart in the K p θ setting, namely (K p θ ) * = K q θ for p and q as above (see [4,Lemma 4.2]), and one may wonder if the identity (K 1 θ ) * = K * θ has any chance of being true, at least for some inner functions θ. Our last theorem says that this is never the case, except when θ is a finite Blaschke product. Theorem 1.3. Given an inner function θ, other than a finite Blaschke product, there exists a non-BMO function g ∈ K 2 θ such that the functional defined initially for f ∈ K 2 θ , extends continuously to K 1 θ .
In other words, whenever θ is an "interesting" (i.e., nonrational) inner function, K * θ is properly contained in (K 1 θ ) * . One might compare this non-duality result with Bessonov's duality theorem for K 1 θ that appears in [2]. There, θ was assumed to be a one-component inner function, meaning that the set {z ∈ D : |θ(z)| < ε} is connected for some ε ∈ (0, 1), and the dual of K 1 θ ∩ zH 1 was identified with a certain discrete BMO space on T.
In the remaining part of the paper, we first list a number of auxiliary facts (these are collected in Section 2) and then use them to prove our current results. The proofs are in Sections 3 and 4.
Acknowledgement. I thank Carlo Bellavita for a helpful conversation. In particular, Theorem 1.3 of this paper arose in response to a question he asked me.

Preliminaries
Several background results will be needed. When stating the first of these, we shall assume that X is one of our smoothness spaces (namely, Λ α , BMO, G α or L p s ), the admissible values of the parameters α, p and s being as above.
Lemma 2.1. Let f ∈ H 2 and let B = B Z be an interpolating Blaschke product with zeros Z = {z k }. In order that P − (Bf ) ∈ X, it is necessary and sufficient that The statements corresponding to parts (a) and (b) were proved in [8] as Theorems 4.1 and 5.2. For parts (c) and (d), we refer to [11]; specifically, see Theorems 1 and 7 therein.
Another (well-known) fact to be used below is that the space BMOA enjoys the socalled K-property of Havin, as defined in [17]. The precise meaning of this assertion is as follows.
Lemma 2.2. For every ψ ∈ H ∞ , the Toeplitz operator T ψ given by maps BMOA boundedly into itself.
To prove this, it suffices to observe (in the spirit of [17]) that T ψ is the adjoint of the multiplication operator g → ψg, which is obviously bounded on H 1 .
Next, we recall a remarkable maximum principle for K 2 θ functions that was established by Cohn in [5].
Our last lemma reproduces yet another result of Cohn (see [4, p. 737]), which characterizes the inner functions θ with the property that K * θ contains only bounded functions. This characterization is, in turn, a consequence of Hruščev and Vinogradov's earlier work from [19] on the multipliers of Cauchy type integrals.
Lemma 2.5. Let θ be an inner function. Then K * θ = K ∞ θ if and only if θ is a Frostman Blaschke product.
We also refer to [12, Theorem 1.7] for a refinement of this result in terms of inn(K * θ ), the set of inner factors for functions from K * θ .

Proof of Theorem 1.1
We shall only give a detailed proof of part (a), the other cases being similar. Since W = {w k } ∈ ℓ 2 1 (Z), we know that there exists a unique f ∈ K 2 B such that f Z = W. Therefore, in order that it is necessary and sufficient that To find out when the latter condition holds, we apply Lemma 2.1, part (a), to the function g := zf B in place of f . (Note that g ∈ H 2 because f ∈ K 2 B .) This tells us that P − (Bg) ∈ Λ α if and only if On the other hand, and it is clear that the function zf belongs to Λ α if and only if f does. Thus, we may rephrase (3.2) as (3.3).
To arrive at a further-and definitive-restatement of (3.3), we need to express the numbers g(z k ) in terms of W. For z ∈ D, Cauchy's formula yields Consequently, Computing the last integral by residues, while recalling that f (z j ) = w j , we find that (To justify the application of the residue theorem, one may begin by evaluating the integral over the circle r n T, where {r n } ⊂ (0, 1) is a suitable sequence tending to 1, and then pass to the limit as n → ∞.) Finally, we use (3.4) to rewrite (3.3) in the form The equivalence of (3.1) and (3.5) is thereby established, proving the Λ α part of the theorem.
The remaining statements (i.e., those involving BMO, G α and L p s ) are proved similarly, by combining the appropriate parts of Lemma 2.1 with identity (3.4).

Proofs of Proposition 1.2 and Theorem 1.3
We begin by proving Theorem 1.3. Once this is done, Proposition 1.2 will be derived as a corollary.
Proof of Theorem 1.3. Given an inner function θ distinct from a finite Blaschke product, we want to find a function g ∈ K 2 θ \ BMO that induces a bounded linear functional on K 1 θ . We shall distinguish two cases. Case 1. Assume that θ is an infinite Frostman Blaschke product. Its zero sequence, say Z = {z j }, must then have a limit point on T. Of course, nothing is lost by assuming that Z clusters at 1. Now let where "log" stands for the holomorphic branch of the logarithm that lives on the right half-plane and satisfies log 1 = 0. We have ϕ ∈ BMOA (because Im ϕ ∈ L ∞ ), so the corresponding linear functional acts boundedly on H 1 and hence on K 1 θ . Clearly, the same functional on K 1 θ is also induced, in a similar manner, by the function g := θP − (θϕ), which is the orthogonal projection (in H 2 ) of ϕ onto K 2 θ . Precisely speaking, the functional extends continuously to K 1 θ .
We know that g ∈ K 2 θ , and to conclude that g does the job, we only need to check that (4.1) g / ∈ BMO.
Case 2. Assume that θ is not a Frostman Blaschke product. This time, using Lemma 2.5 again, we can find an unbounded function h ∈ K * θ . We have then h := zhθ ∈ K 2 θ , and we go on to claim that h / ∈ BMO. (Here and below, the "tilde operation" (1.2) is being used repeatedly.) Indeed, if h were in BMO, then so would be hθ, and Lemma 2.3 would tell us that sup{|h(z)| : z ∈ Ω(θ, ε)} < ∞ for some (any) ε ∈ (0, 1). This, however, would imply that h ∈ H ∞ by virtue of Lemma 2.4, whereas h is actually unbounded by assumption. Now we know that h ∈ K 2 θ \ BMO, and we proceed by showing that h generates a continuous linear functional on K 1 θ . This will allow us to conclude that h is eligible as g (the function we are looking for), and the proof will be complete.
Given f ∈ K 2 θ , we have the elementary identity f h = f h. Recalling also the facts that f ∈ H 2 (⊂ H 1 ) and h ∈ BMOA, we use the duality relation (H 1 ) * = BMOA to infer that with some absolute constant C > 0. Consequently, for g = h, the (densely defined) functional (4.2) f → T f g dm is indeed continuous on K 1 θ , and we are done.
Proof of Proposition 1.2. Assuming that B = B Z is an infinite Blaschke product with zeros Z = {z j }, where the z j 's are pairwise distinct, we want to find a sequence of values W = {w j } in ℓ ∞ log (Z) that is not the trace of any K * B function on Z. Consider, for each j ∈ N, the function f j (z) := 1 1 − z j z and note that f j ∈ K 2 B . Observe also that for some fixed constant M > 0. Now, Theorem 1.3 provides us with a function g ∈ K 2 B \ BMO such that the associated functional (4.2), defined initially for f ∈ K 2 B , acts boundedly on K 1 B , say with norm N g . When applied to f = f j , this functional takes the value g(z j ); indeed, Cauchy's formula gives T f j g dm = g(z j ) for each j. In conjunction with (4.3), this yields (4.4) |g(z j )| ≤ N g f j 1 ≤ MN g log 2 1 − |z j | , j ∈ N.
Finally, we put w j := g(z j ), j ∈ N. The sequence W = {w j } is then in ℓ ∞ log (Z), as (4.4) shows, while (4.5) W / ∈ K * B Z as required. To verify (4.5), it suffices to note that g is the only function in K 2 B that interpolates W on Z (indeed, a K 2 B function is uniquely determined by its trace on Z = B −1 (0)), whereas g / ∈ BMO. The proof is complete.
Remark. We have seen above that if g ∈ K 2 B , with B = B Z , and if the functional (4.2) is continuous on K 1 B , then g Z ∈ ℓ ∞ log (Z). Now, if Z has the BMOAinterpolating property (1.9), then every sequence W in ℓ ∞ log (Z) is actually writable as g Z for some g ∈ K 2 B that induces a continuous linear functional on K 1 B . (To see why, take G ∈ BMOA with G Z = W and then put g = BP − (BG), so that g is the orthogonal projection of G onto K 2 B .) Of course, things become different if condition (1.9) is dropped. For instance, there are interpolating sequences Z for which ℓ ∞ log (Z) ⊂ ℓ 2 1 (Z); and if this is the case, then no sequence in ℓ ∞ log (Z) \ ℓ 2 1 (Z) is the trace of any H 2 function on Z.