Interpolation and duality in spaces of pseudocontinuable functions

Abstract

Given an inner function $\theta$ on the unit disk, let $K^p_\theta :=H^p\cap \theta {\overline{z}}\overline{H^p}$ be the associated star-invariant subspace of the Hardy space $H^p$. Also, we put $K_{*\theta }:=K^2_\theta \cap \mathrm{BMO}$. 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.

1 Introduction and results

We write ${{\mathbb {T}}}$ for the unit circle $\{\zeta \in {{\mathbb {C}}}:|\zeta |=1\}$ and m for the normalized arc length measure on ${{\mathbb {T}}}$; thus, $dm(\zeta )=|d\zeta |/(2\pi )$. We then define the spaces $L^p:=L^p({{\mathbb {T}}},m)$ in the usual way and let $\Vert \cdot \Vert _p$ denote the standard norm on $L^p$. Also, for $1\le p\le \infty$, we introduce the Hardy space $H^p$ by putting

$$\begin{aligned} H^p:=\{f\in L^p:\,{\widehat{f}}(n)=0\,\,\,{\mathrm{for}}\,n=-1,-2,\dots \}, \end{aligned}$$

where ${\widehat{f}}(n)$ is the nth Fourier coefficient of f given by

$$\begin{aligned} {\widehat{f}}(n):=\int _{{\mathbb {T}}}\overline{\zeta }^nf(\zeta )\,dm(\zeta ),\quad n\in {{\mathbb {Z}}}. \end{aligned}$$

The Poisson integral (i.e., harmonic extension) of an $H^p$ function being holomorphic on the disk

$$\begin{aligned} {{\mathbb {D}}}:=\{z\in {{\mathbb {C}}}:\,|z|<1\} \end{aligned}$$

(see [15, Chapter II]), we may use this extension to view elements of $H^p$ as holomorphic functions on ${{\mathbb {D}}}$ when convenient.

Furthermore, we write $P_+$ (resp., $P_-$) for the orthogonal projection from $L^2$ onto $H^2$ (resp., onto $\overline{z}\overline{H^2}=L^2\ominus H^2$). 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 $1<p<\infty$, and maps $L^p$ onto $H^p$ (resp., onto $\overline{z}\overline{H^p}$).

Now suppose $\theta$ is an inner function, meaning that $\theta \in H^\infty$ and $|\theta |=1$ a.e. on ${{\mathbb {T}}}$. The corresponding star-invariant (or model) subspace $K^p_\theta$ is then defined by

$$\begin{aligned} K^p_\theta :=\{f\in H^p:\,\overline{z}\overline{f}\theta \in H^p\},\quad 1\le p\le \infty , \end{aligned}$$

(1.1)

so that $K^p_\theta =H^p\cap \theta \overline{z}\overline{H^p}$. (When $p=2$, yet another equivalent definition is $K^2_\theta =H^2\ominus \theta H^2$.) It is clear from (1.1) that the antilinear isometry

$$\begin{aligned} f\mapsto \overline{z}\overline{f}\theta =:{\widetilde{f}} \end{aligned}$$

(1.2)

leaves $K^p_\theta$ invariant. Also, it is well known (see [6, 20]) that each $K^p_\theta$ is invariant under the backward shift operator

$$\begin{aligned} {\mathfrak {B}}:f\mapsto \frac{f-f(0)}{z},\quad f\in H^p, \end{aligned}$$

and conversely, that every closed and nontrivial ${\mathfrak {B}}$-invariant subspace of $H^p$, with $1\le p<\infty$, arises in this way.

The functions belonging to some $K^p_\theta$ space (i.e., noncyclic vectors of ${\mathfrak {B}}$) are known as pseudocontinuable functions. In fact, they are characterized by the property of having a meromorphic pseudocontinuation to ${{\mathbb {D}}}_-:={{\mathbb {C}}}{\setminus }({{\mathbb {D}}}\cup {{\mathbb {T}}})$; that is, the function in question should agree a.e. on ${{\mathbb {T}}}$ with the boundary values of some meromorphic function of bounded characteristic in ${{\mathbb {D}}}_-$ (see [6] for details).

The orthogonal projection from $H^2$ onto $K^2_\theta$ is given by $f\mapsto \theta P_-(\overline{\theta }f)$, and the M. Riesz theorem shows that the same formula provides, for $1<p<\infty$, a bounded projection from $H^p$ onto $K^p_\theta$ parallel to $\theta H^p$. This yields the direct sum decomposition

$$\begin{aligned} H^p=K^p_\theta \oplus \theta H^p,\quad 1<p<\infty , \end{aligned}$$

(1.3)

with orthogonality for $p=2$.

Among the inner functions $\theta$, of special relevance to us are Blaschke products. Recall that, for a sequence ${\mathcal {Z}}=\{z_j\}\subset {{\mathbb {D}}}$ with

$$\begin{aligned} \sum _j(1-|z_j|)<\infty , \end{aligned}$$

(1.4)

the associated Blaschke product is given by

$$\begin{aligned} B(z)=B_{{\mathcal {Z}}}(z):=\prod _j\frac{|z_j|}{z_j}\frac{z_j-z}{1-\overline{z}_jz} \end{aligned}$$

(if $z_j=0$, then we set $|z_j|/z_j=-1$). The product converges uniformly on compact subsets of ${{\mathbb {D}}}$ and defines an inner function that vanishes precisely at the $z_j$’s; see [15, Chapter II]. If, in addition,

$$\begin{aligned} \inf _j|B'(z_j)|\,(1-|z_j|)>0, \end{aligned}$$

(1.5)

then we say that B is an interpolating Blaschke product. Accordingly, the sequences ${\mathcal {Z}}=\{z_j\}$ in ${{\mathbb {D}}}$ that satisfy (1.4) and (1.5), with $B=B_{{\mathcal {Z}}}$, are called interpolating (or $H^\infty$-interpolating) sequences. By a celebrated theorem of Carleson (see [3] or [15, Chapter VII]), these are precisely the sequences ${\mathcal {Z}}$ with the property that

$$\begin{aligned} H^\infty \big |_{{\mathcal {Z}}}=\ell ^\infty . \end{aligned}$$

Here and below, the following standard notation (and terminology) is used. Given a sequence ${\mathcal {Z}}=\{z_j\}$ of pairwise distinct points in ${{\mathbb {D}}}$, the trace $f\big |_{{\mathcal {Z}}}$ of a function $f:{{\mathbb {D}}}\rightarrow {{\mathbb {C}}}$ is defined to be the sequence $\{f(z_j)\}$; and if ${\mathcal {X}}$ is a certain function space on ${{\mathbb {D}}}$, then the corresponding trace space is

$$\begin{aligned} {\mathcal {X}}\big |_{{\mathcal {Z}}}:=\left\{ f\big |_{{\mathcal {Z}}}:\,f\in {\mathcal {X}}\right\} . \end{aligned}$$

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 $p>0$, $\gamma \in {{\mathbb {R}}}$ and a sequence ${\mathcal {Z}}=\{z_j\}\subset {{\mathbb {D}}}$, we write $\ell ^p_\gamma ({\mathcal {Z}})$ for the set of all sequences $\{w_j\}\subset {{\mathbb {C}}}$ satisfying

$$\begin{aligned} \sum _j|w_j|^p(1-|z_j|)^{\gamma }<\infty . \end{aligned}$$

Now, if $1<p<\infty$ and if $B=B_{{\mathcal {Z}}}$ is an interpolating Blaschke product with zero sequence ${\mathcal {Z}}$, then we have

$$\begin{aligned} K^p_B\big |_{{\mathcal {Z}}}=H^p\big |_{{\mathcal {Z}}}=\ell ^p_1({\mathcal {Z}}). \end{aligned}$$

Indeed, the left-hand equality follows from (1.3) with $\theta =B$, while the other holds by a well-known theorem of Shapiro and Shields [21]. In addition, for each sequence ${\mathcal {W}}=\{w_j\}$ in $\ell ^p_1({\mathcal {Z}})$, there is a unique function $f\in K^p_B$ with $f\big |_{{\mathcal {Z}}}={\mathcal {W}}$; the uniqueness is due to the fact that $K^p_B\cap BH^p=\{0\}$.

The case of $K^\infty _B$ is subtler, as the next result shows.

Theorem A

Suppose that ${\mathcal {Z}}=\{z_j\}$ is an interpolating sequence in ${{\mathbb {D}}}$ and $B=B_{{\mathcal {Z}}}$ is the associated Blaschke product. Then we have

$$\begin{aligned} K^\infty _B|_{{\mathcal {Z}}}=\ell ^\infty \end{aligned}$$

(1.6)

if and only if

$$\begin{aligned} \sup \left\{ \sum _j\frac{1-|z_j|}{|\zeta -z_j|}:\,\zeta \in {{\mathbb {T}}}\right\} <\infty . \end{aligned}$$

(1.7)

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 ${\mathcal {Z}}=\{z_j\}$ in ${{\mathbb {D}}}$ 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

$$\begin{aligned} K^\infty _{B^2}|_{{\mathcal {Z}}}=\ell ^\infty \end{aligned}$$

is valid whenever ${\mathcal {Z}}$ is an interpolating sequence and $B=B_{{\mathcal {Z}}}$. It should be noted, however, that $K^\infty _{B^2}$ is strictly larger than $K^\infty _B$.

To describe the trace class $K^\infty _B|_{{\mathcal {Z}}}$ in the general case (i.e., when (1.7) no longer holds), we first introduce a bit of notation. Once the interpolating sequence ${\mathcal {Z}}=\{z_j\}$ is fixed, we associate with each sequence ${\mathcal {W}}=\{w_j\}$ from $\ell ^1_1({\mathcal {Z}})$ the conjugate sequence $\widetilde{{\mathcal {W}}}=\{\widetilde{w}_k\}$ whose elements are

$$\begin{aligned} {\widetilde{w}}_k:=\sum _j\frac{w_j}{B'(z_j)\cdot (1-z_j\overline{z}_k)}\quad (k=1,2,\dots ). \end{aligned}$$

(1.8)

The absolute convergence of the series in (1.8) is ensured, for any $k\in {{\mathbb {N}}}$, by the fact that ${\mathcal {W}}\in \ell ^1_1({\mathcal {Z}})$ in conjunction with (1.5). Because $\ell ^1_1({\mathcal {Z}})$ contains $\ell ^\infty$, as well as every $\ell ^p_1({\mathcal {Z}})$ with $1<p<\infty$, the sequence $\widetilde{{\mathcal {W}}}$ is well defined whenever ${\mathcal {W}}$ belongs to one of these spaces.

The following result was established in [13].

Theorem B

Suppose that ${\mathcal {Z}}=\{z_j\}$ is an interpolating sequence in ${{\mathbb {D}}}$ and $B=B_{{\mathcal {Z}}}$ is the associated Blaschke product. Given a sequence ${\mathcal {W}}\in \ell ^\infty$, one has ${\mathcal {W}}\in K^\infty _B\big |_{{\mathcal {Z}}}$ if and only if $\widetilde{{\mathcal {W}}}\in \ell ^\infty$.

It was further conjectured in [13, 14] that the trace space $K^1_B\big |_{{\mathcal {Z}}}$ is describable in similar terms, i.e., that the necessary conditions ${\mathcal {W}}\in \ell ^1_1({\mathcal {Z}})$ and $\widetilde{{\mathcal {W}}}\in \ell ^1_1({\mathcal {Z}})$ are also sufficient for ${\mathcal {W}}$ to be in $K^1_B\big |_{{\mathcal {Z}}}$. 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 ${\mathcal {Z}}=B^{-1}(0)$. To be more precise, of concern are trace spaces of the form $\left( K^2_B\cap X\right) \big |_{{\mathcal {Z}}}$, where X is a certain smoothness class on ${{\mathbb {T}}}$. Specifically, X will be one of the following spaces.

$\bullet$ The Lipschitz–Zygmund space $\Lambda ^\alpha =\Lambda ^\alpha ({{\mathbb {T}}})$ with $\alpha >0$. This is the set of functions $f\in C({{\mathbb {T}}})$ satisfying

$$\begin{aligned} \Vert \Delta _h^nf\Vert _\infty =O(|h|^\alpha ),\quad h\in {{\mathbb {R}}}, \end{aligned}$$

where n is some (any) integer with $n>\alpha$, and $\Delta _h^n$ denotes the nth order difference operator with step h. (As usual, the difference operators $\Delta _h^k$ are defined inductively: we put

$$\begin{aligned} (\Delta _h^1f)(\zeta ):=f(e^{ih}\zeta )-f(\zeta ),\quad \zeta \in {{\mathbb {T}}}, \end{aligned}$$

and $\Delta _h^kf:=\Delta _h^1\Delta _h^{k-1}f$ for $k\ge 2$.)

$\bullet$ $\mathrm{BMO}=\mathrm{BMO}({{\mathbb {T}}})$, the space of functions of bounded mean oscillation on ${{\mathbb {T}}}$. Recall that an integrable function f on ${{\mathbb {T}}}$ belongs to $\mathrm{BMO}$ if and only if

$$\begin{aligned} \Vert f\Vert _*:=\left| \int _{{\mathbb {T}}}f\,dm\right| +\sup _I\frac{1}{m(I)}\int _I|f-f_I|\,dm<\infty , \end{aligned}$$

where $f_I:=m(I)^{-1}\int _I f\,dm$; the supremum is taken over the open arcs $I\subset {{\mathbb {T}}}$. Even though $\mathrm{BMO}$ contains discontinuous and unbounded functions, there are reasons for viewing it as a smoothness class. In a sense, it corresponds to the endpoint as $\alpha \rightarrow 0$ of the $\Lambda ^\alpha$ scale. We also need the analytic subspace $\mathrm{BMOA}:=\mathrm{BMO}\cap H^2$.

$\bullet$ The Gevrey class $G_\alpha =G_\alpha ({{\mathbb {T}}})$ with $\alpha >0$. This is the set of functions $f\in C^\infty ({{\mathbb {T}}})$ satisfying

$$\begin{aligned} \Vert f^{(n)}\Vert _\infty \le Q_f^{n+1}(n!)^{1+1/\alpha },\quad n=0,1,2,\dots , \end{aligned}$$

with some constant $Q_f>0$. Here, we write $f^{(n)}(e^{it})$ for the nth order derivative of the function $t\mapsto f(e^{it})$, which is assumed to be $C^\infty$-smooth on ${{\mathbb {R}}}$.

$\bullet$ The Sobolev space ${\mathcal {L}}^p_s={\mathcal {L}}^p_s({{\mathbb {T}}})$ with $1<p<\infty$ and $s>0$, defined by

$$\begin{aligned} {\mathcal {L}}^p_s:=\{f\in L^p:\,f^{(s)}\in L^p\}, \end{aligned}$$

with the appropriate interpretation of the (possibly fractional) derivative $f^{(s)}$. Precisely speaking, we write $f^{(s)}\in L^p$ to mean that there is a function $g\in L^p$ satisfying ${\widehat{g}}(n)=(in)^s{\widehat{f}}(n)$ for all $n\in {{\mathbb {Z}}}$.

For each of these choices of X, we now characterize the sequences ${\mathcal {W}}$ from the trace space $\left( K^2_B\cap X\right) \big |_{{\mathcal {Z}}}$ in terms of the conjugate sequence $\widetilde{{\mathcal {W}}}$, as defined by (1.8) above. The description always involves a certain decay condition (or growth restriction) on $\widetilde{{\mathcal {W}}}$, as we shall presently see.

Theorem 1.1

Let $\alpha >0$, $1<p<\infty$ and $s>0$. Also, let X be one of the following spaces: $\Lambda ^\alpha$, $\mathrm{BMO}$, $G_\alpha$ or ${\mathcal {L}}^p_s$. Given an interpolating Blaschke product $B=B_{{\mathcal {Z}}}$ with zeros ${\mathcal {Z}}=\{z_k\}$ and a sequence ${\mathcal {W}}=\{w_k\}\in \ell ^2_1({\mathcal {Z}})$, we have

$$\begin{aligned} {\mathcal {W}}\in \left( K^2_B\cap X\right) \big |_{{\mathcal {Z}}} \end{aligned}$$

if and only if

(a) $|\widetilde{w}_k|=O\left( (1-|z_k|)^\alpha \right)$ when $X=\Lambda ^\alpha$,

(b) $\widetilde{{\mathcal {W}}}\in \ell ^\infty$ when $X=\mathrm{BMO}$,

(c) there is a constant $c>0$ such that

$$\begin{aligned} |{\widetilde{w}}_k|=O\left( \exp \left( -\frac{c}{(1-|z_k|)^\alpha }\right) \right) \end{aligned}$$

when $X=G_\alpha$,

(d) $\widetilde{{\mathcal {W}}}\in \ell ^p_{1-sp}({\mathcal {Z}})$ when $X={\mathcal {L}}^p_s$.

The intersection $K^2_B\cap \mathrm{BMO}$, which corresponds to case (b) above, will be henceforth denoted by $K_{*B}$. Similarly, for a general inner function $\theta$, we define

$$\begin{aligned} K_{*\theta }:=K^2_\theta \cap \mathrm{BMO}. \end{aligned}$$

Comparing Theorem B with the $\mathrm{BMO}$ part of Theorem 1.1, we see that the structure of the trace space $K^\infty _B\big |_{{\mathcal {Z}}}$ is remarkably similar to that of $K_{*B}\big |_{{\mathcal {Z}}}$. In light of this observation, we may wonder what the $\mathrm{BMO}$ counterpart of Theorem A could look like. Specifically, we may ask if there exist infinite Blaschke products $B=B_{{\mathcal {Z}}}$ for which the trace space $K_{*B}\big |_{{\mathcal {Z}}}$ is completely determined by the natural (and necessary) logarithmic growth condition on the values.

To be more precise, suppose that ${\mathcal {Z}}=\{z_k\}$ is a sequence of pairwise distinct points in ${{\mathbb {D}}}$, and write $\ell ^\infty _{\log }({\mathcal {Z}})$ for the space of sequences ${\mathcal {W}}=\{w_k\}\subset {{\mathbb {C}}}$ with

$$\begin{aligned} |w_k|=O\left( \log \frac{2}{1-|z_k|}\right) . \end{aligned}$$

It is well known (and easily shown) that every $f\in \mathrm{BMOA}$ satisfies

$$\begin{aligned} |f(z)|=O\left( \log \frac{2}{1-|z|}\right) ,\quad z\in {{\mathbb {D}}}, \end{aligned}$$

so $\mathrm{BMOA}\big |_{{\mathcal {Z}}}$ is always contained in $\ell ^\infty _{\log }({\mathcal {Z}})$. The equality

$$\begin{aligned} \mathrm{BMOA}\big |_{{\mathcal {Z}}}=\ell ^\infty _{\log }({\mathcal {Z}}) \end{aligned}$$

(1.9)

obviously need not hold in general, but it does actually occur for some infinite sequences ${\mathcal {Z}}=\{z_k\}$ (which form a tiny subfamily among the $H^\infty$-interpolating sequences). For instance, (1.9) will be valid provided that

$$\begin{aligned} |z_j-z_k|\ge c(1-|z_j|)^s,\quad j\ne k, \end{aligned}$$

for some constants $c>0$ and $s\in (0,\frac{1}{2})$; see [10, Theorem 11].

The question is what happens to (1.9) when $\mathrm{BMOA}$ gets replaced by its subspace $K_{*B}$, with $B=B_{{\mathcal {Z}}}$. The property that arises is thus

$$\begin{aligned} K_{*B}\big |_{{\mathcal {Z}}}=\ell ^\infty _{\log }({\mathcal {Z}}), \end{aligned}$$

(1.10)

and we regard it as an analogue of (1.6) in the $\mathrm{BMO}$ 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 ${\mathcal {Z}}=B^{-1}(0)$ is a finite set.

Proposition 1.2

Whenever $B=B_{{\mathcal {Z}}}$ is an infinite Blaschke product with simple zeros, the trace space $K_{*B}\big |_{{\mathcal {Z}}}$ is properly contained in $\ell ^\infty _{\log }({\mathcal {Z}})$.

This will be deduced from another result, which deals with the case of a general inner function $\theta$ and asserts an amusing lack of duality between the star-invariant subspaces $K^1_\theta$ and $K_{*\theta }$.

It is well known that, for $1<p<\infty$, the dual of the Hardy space $H^p$ (under the pairing $\langle f,g\rangle =\int _{{\mathbb {T}}}f\overline{g}\,dm$) is $H^q$ with $q=p/(p-1)$, while the dual of $H^1$ is $\mathrm{BMOA}$; see, e.g., [15, Chapter VI]. The former duality relation has a natural counterpart in the $K^p_\theta$ setting, namely $(K^p_\theta )^*=K^q_\theta$ for p and q as above (see [4, Lemma 4.2]), and one may wonder if the identity $(K^1_\theta )^*=K_{*\theta }$ has any chance of being true, at least for some inner functions $\theta$. Our last theorem says that this is never the case, except when $\theta$ is a finite Blaschke product.

Theorem 1.3

Given an inner function $\theta$, other than a finite Blaschke product, there exists a non-$\mathrm{BMO}$ function $g\in K^2_\theta$ such that the functional

$$\begin{aligned} f\mapsto \int _{{\mathbb {T}}}f\overline{g}\,dm, \end{aligned}$$

defined initially for $f\in K^2_\theta$, extends continuously to $K^1_\theta$.

In other words, whenever $\theta$ is an “interesting” (i.e., nonrational) inner function, $K_{*\theta }$ is properly contained in $(K^1_\theta )^*$. One might compare this non-duality result with Bessonov’s duality theorem for $K^1_\theta$ that appears in [2]. There, $\theta$ was assumed to be a one-component inner function, meaning that the set $\{z\in {{\mathbb {D}}}:|\theta (z)|<\varepsilon \}$ is connected for some $\varepsilon \in (0,1)$, and the dual of $K^1_\theta \cap zH^1$ was identified with a certain discrete $\mathrm{BMO}$ space on ${{\mathbb {T}}}$.

In the remaining part of the paper, we first list a number of auxiliary facts (these are collected in Sect. 2) and then use them to prove our current results. The proofs are in Sects. 3 and 4.

2 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, $\Lambda ^\alpha$, $\mathrm{BMO}$, $G_\alpha$ or ${\mathcal {L}}^p_s$), the admissible values of the parameters $\alpha$, p and s being as above.

Lemma 2.1

Let $f\in H^2$ and let $B=B_{{\mathcal {Z}}}$ be an interpolating Blaschke product with zeros ${\mathcal {Z}}=\{z_k\}$. In order that $P_-(\overline{B}f)\in X$, it is necessary and sufficient that

(a) $|f(z_k)|=O\left( (1-|z_k|)^\alpha \right)$ when $X=\Lambda ^\alpha$,

(b) $\{f(z_k)\}\in \ell ^\infty$ when $X=\mathrm{BMO}$,

(c) for some $c>0$,

$$\begin{aligned} |f(z_k)|=O\left( \exp \left( -\frac{c}{(1-|z_k|)^\alpha }\right) \right) \end{aligned}$$

when $X=G_\alpha$,

(d) $\{f(z_k)\}\in \ell ^p_{1-sp}({\mathcal {Z}})$ when $X={\mathcal {L}}^p_s$.

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 $\mathrm{BMOA}$ enjoys the so-called K-property of Havin, as defined in [17]. The precise meaning of this assertion is as follows.

Lemma 2.2

For every $\psi \in H^\infty$, the Toeplitz operator $T_{\overline{\psi }}$ given by

$$\begin{aligned} T_{\overline{\psi }}f:=P_+(\overline{\psi }f),\quad f\in \mathrm{BMOA}, \end{aligned}$$

maps $\mathrm{BMOA}$ boundedly into itself.

To prove this, it suffices to observe (in the spirit of [17]) that $T_{\overline{\psi }}$ is the adjoint of the multiplication operator $g\mapsto \psi g$, which is obviously bounded on $H^1$.

Before proceeding, we need to introduce a bit of notation. Namely, with an inner function $\theta$ and a number $\varepsilon \in (0,1)$ we associate the sublevel set

$$\begin{aligned} \Omega (\theta ,\varepsilon ):=\{z\in {{\mathbb {D}}}:\,|\theta (z)|<\varepsilon \}. \end{aligned}$$

The following result is a restricted version of [9, Theorem 1].

Lemma 2.3

Suppose that $f\in \mathrm{BMOA}$ and $\theta$ is an inner function. Then $f\overline{\theta }\in \mathrm{BMO}$ if and only if

$$\begin{aligned} \sup \{|f(z)|:\,z\in \Omega (\theta ,\varepsilon )\}<\infty \end{aligned}$$

(2.1)

for some (or every) $\varepsilon$ with $0<\varepsilon <1$.

Next, we recall a remarkable maximum principle for $K^2_\theta$ functions that was established by Cohn in [5].

Lemma 2.4

Let $\theta$ be inner, and suppose $f\in K^2_\theta$ is a function that satisfies (2.1) for some $\varepsilon \in (0,1)$. Then $f\in H^\infty$.

Our last lemma reproduces yet another result of Cohn (see [4, p. 737]), which characterizes the inner functions $\theta$ with the property that $K_{*\theta }$ 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 $\theta$ be an inner function. Then $K_{*\theta }=K^\infty _\theta$ if and only if $\theta$ is a Frostman Blaschke product.

We also refer to [12, Theorem 1.7] for a refinement of this result in terms of ${\mathrm{inn}}(K_{*\theta })$, the set of inner factors for functions from $K_{*\theta }$.

3 Proof of Theorem 1.1

We shall only give a detailed proof of part (a), the other cases being similar. Since ${\mathcal {W}}=\{w_k\}\in \ell ^2_1({\mathcal {Z}})$, we know that there exists a unique $f\in K^2_B$ such that $f\big |_{{\mathcal {Z}}}={\mathcal {W}}$. Therefore, in order that

$$\begin{aligned} {\mathcal {W}}\in \left( K^2_B\cap \Lambda ^\alpha \right) \big |_{{\mathcal {Z}}} \end{aligned}$$

(3.1)

it is necessary and sufficient that

$$\begin{aligned} f\in \Lambda ^\alpha . \end{aligned}$$

(3.2)

To find out when the latter condition holds, we apply Lemma 2.1, part (a), to the function $g:=\overline{z}\overline{f}B$ in place of f. (Note that $g\in H^2$ because $f\in K^2_B$.) This tells us that $P_-(\overline{B}g)\in \Lambda ^\alpha$ if and only if

$$\begin{aligned} |g(z_k)|=O\left( (1-|z_k|)^\alpha \right) ,\quad k\in {{\mathbb {N}}}. \end{aligned}$$

(3.3)

On the other hand,

$$\begin{aligned} P_-(\overline{B}g)=P_-(\overline{z}\overline{f})=\overline{z}\overline{f}, \end{aligned}$$

and it is clear that the function $\overline{z}\overline{f}$ belongs to $\Lambda ^\alpha$ 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 ${\mathcal {W}}$. For $z\in {{\mathbb {D}}}$, Cauchy’s formula yields

$$\begin{aligned} g(z)=\int _{{{\mathbb {T}}}}\frac{g(\zeta )}{1-\overline{\zeta }z}\,dm(\zeta ). \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{aligned} \overline{g(z_k)}&=\int _{{{\mathbb {T}}}}\frac{\overline{g(\zeta )}}{1-\zeta \overline{z}_k}\,dm(\zeta ) =\int _{{{\mathbb {T}}}}\frac{\zeta f(\zeta )\overline{B(\zeta )}}{1-\zeta \overline{z}_k}\,dm(\zeta )\\&=\frac{1}{2\pi i}\int _{{{\mathbb {T}}}}\frac{f(\zeta )}{B(\zeta )\cdot (1-\zeta \overline{z}_k)}\,d\zeta . \end{aligned} \end{aligned}$$

Computing the last integral by residues, while recalling that $f(z_j)=w_j$, we find that

$$\begin{aligned} \overline{g(z_k)}=\sum _j\frac{w_j}{B'(z_j)\cdot (1-z_j\overline{z}_k)}={\widetilde{w}}_k \end{aligned}$$

(3.4)

for each $k\in {{\mathbb {N}}}$. (To justify the application of the residue theorem, one may begin by evaluating the integral over the circle $r_n{{\mathbb {T}}}$, where $\{r_n\}\subset (0,1)$ is a suitable sequence tending to 1, and then pass to the limit as $n\rightarrow \infty$.)

Finally, we use (3.4) to rewrite (3.3) in the form

$$\begin{aligned} |{\widetilde{w}}_k|=O\left( (1-|z_k|)^\alpha \right) ,\quad k\in {{\mathbb {N}}}. \end{aligned}$$

(3.5)

The equivalence of (3.1) and (3.5) is thereby established, proving the $\Lambda ^\alpha$ part of the theorem.

The remaining statements (i.e., those involving $\mathrm{BMO}$, $G_\alpha$ and ${\mathcal {L}}^p_s$) are proved similarly, by combining the appropriate parts of Lemma 2.1 with identity (3.4).

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 $\theta$ distinct from a finite Blaschke product, we want to find a function $g\in K^2_\theta {\setminus }\mathrm{BMO}$ that induces a bounded linear functional on $K^1_\theta$. We shall distinguish two cases.

Case 1

Assume that $\theta$ is an infinite Frostman Blaschke product. Its zero sequence, say ${\mathcal {Z}}=\{z_j\}$, must then have a limit point on ${{\mathbb {T}}}$. Of course, nothing is lost by assuming that ${\mathcal {Z}}$ clusters at 1. Now let

$$\begin{aligned} \varphi (z):=\log (1-z), \end{aligned}$$

where “log” stands for the holomorphic branch of the logarithm that lives on the right half-plane and satisfies $\log 1=0$. We have $\varphi \in \mathrm{BMOA}$ (because ${\mathrm{Im}}\,\varphi \in L^\infty$), so the corresponding linear functional acts boundedly on $H^1$ and hence on $K^1_\theta$. Clearly, the same functional on $K^1_\theta$ is also induced, in a similar manner, by the function

$$\begin{aligned} g:=\theta P_-(\overline{\theta }\varphi ), \end{aligned}$$

which is the orthogonal projection (in $H^2$) of $\varphi$ onto $K^2_\theta$. Precisely speaking, the functional

$$\begin{aligned} f\mapsto \int _{{\mathbb {T}}}f\overline{g}\,dm\left( =\int _{{\mathbb {T}}}f\overline{\varphi }\,dm\right) ,\quad f\in K^2_\theta , \end{aligned}$$

extends continuously to $K^1_\theta$.

We know that $g\in K^2_\theta$, and to conclude that g does the job, we only need to check that

$$\begin{aligned} g\notin \mathrm{BMO}. \end{aligned}$$

(4.1)

To this end, observe first that $\sup _j|\varphi (z_j)|=\infty$ and hence, a fortiori,

$$\begin{aligned} \sup \{|\varphi (z)|:\,z\in \Omega (\theta ,\varepsilon )\}=\infty \end{aligned}$$

for every $\varepsilon \in (0,1)$. By Lemma 2.3, this implies that $\overline{\theta }\varphi \notin \mathrm{BMO}$. On the other hand,

$$\begin{aligned} \overline{\theta }\varphi =P_-(\overline{\theta }\varphi )+P_+(\overline{\theta }\varphi ), \end{aligned}$$

where the last term, $P_+(\overline{\theta }\varphi )$, is in $\mathrm{BMOA}(\subset \mathrm{BMO})$ thanks to Lemma 2.2. It follows readily that $P_-(\overline{\theta }\varphi )\notin \mathrm{BMO}$. In particular, $P_-(\overline{\theta }\varphi )\notin L^\infty$ (just note that $L^\infty \subset \mathrm{BMO}$). Equivalently, the function $\theta P_-(\overline{\theta }\varphi )=g$ is not in $L^\infty$.

Now, if g were in $\mathrm{BMO}$, then we would have $g\in K_{*\theta }$; and since our current assumption on $\theta$ yields $K_{*\theta }=K^\infty _\theta$ (in accordance with Lemma 2.5), g would have to be bounded, which it is not. This proves (4.1).

Case 2

Assume that $\theta$ is not a Frostman Blaschke product. This time, using Lemma 2.5 again, we can find an unbounded function $h\in K_{*\theta }$. We have then ${\widetilde{h}}:=\overline{z}\overline{h}\theta \in K^2_\theta$, and we go on to claim that ${\widetilde{h}}\notin \mathrm{BMO}$. (Here and below, the “tilde operation” (1.2) is being used repeatedly.) Indeed, if ${\widetilde{h}}$ were in $\mathrm{BMO}$, then so would be $h\overline{\theta }$, and Lemma 2.3 would tell us that

$$\begin{aligned} \sup \{|h(z)|:\,z\in \Omega (\theta ,\varepsilon )\}<\infty \end{aligned}$$

for some (any) $\varepsilon \in (0,1)$. This, however, would imply that $h\in H^\infty$ by virtue of Lemma 2.4, whereas h is actually unbounded by assumption.

Now we know that ${\widetilde{h}}\in K^2_\theta {\setminus }\mathrm{BMO}$, and we proceed by showing that ${\widetilde{h}}$ generates a continuous linear functional on $K^1_\theta$. This will allow us to conclude that ${\widetilde{h}}$ is eligible as g (the function we are looking for), and the proof will be complete.

Given $f\in K^2_\theta$, we have the elementary identity $\overline{f}{\widetilde{h}}={\widetilde{f}}\overline{h}$. Recalling also the facts that ${\widetilde{f}}\in H^2(\subset H^1)$ and $h\in \mathrm{BMOA}$, we use the duality relation $(H^1)^*=\mathrm{BMOA}$ to infer that

$$\begin{aligned} \begin{aligned} \left| \int _{{{\mathbb {T}}}}f\,\overline{{\widetilde{h}}}\,dm\right|&= \left| \int _{{{\mathbb {T}}}}\overline{f}\,{\widetilde{h}}\,dm\right| =\left| \int _{{{\mathbb {T}}}}{\widetilde{f}}\,\overline{h}\,dm\right| \\&\le C\Vert {\widetilde{f}}\Vert _1\Vert h\Vert _*=C\Vert f\Vert _1\Vert h\Vert _* \end{aligned} \end{aligned}$$

with some absolute constant $C>0$. Consequently, for $g={\widetilde{h}}$, the (densely defined) functional

$$\begin{aligned} f\mapsto \int _{{{\mathbb {T}}}}f\overline{g}\,dm \end{aligned}$$

(4.2)

is indeed continuous on $K^1_\theta$, and we are done. $\square$

Proof of Proposition 1.2

Assuming that $B=B_{\mathcal {Z}}$ is an infinite Blaschke product with zeros ${\mathcal {Z}}=\{z_j\}$, where the $z_j$’s are pairwise distinct, we want to find a sequence of values ${\mathcal {W}}=\{w_j\}$ in $\ell ^\infty _{\log }({\mathcal {Z}})$ that is not the trace of any $K_{*B}$ function on ${\mathcal {Z}}$. Consider, for each $j\in {{\mathbb {N}}}$, the function

$$\begin{aligned} f_j(z):=\frac{1}{1-\overline{z}_jz} \end{aligned}$$

and note that $f_j\in K^2_B$. Observe also that

$$\begin{aligned} \Vert f_j\Vert _1\le M\log \frac{2}{1-|z_j|},\quad j\in {{\mathbb {N}}}, \end{aligned}$$

(4.3)

for some fixed constant $M>0$.

Now, Theorem 1.3 provides us with a function $g\in K^2_B{\setminus }\mathrm{BMO}$ such that the associated functional (4.2), defined initially for $f\in K^2_B$, acts boundedly on $K^1_B$, say with norm $N_g$. When applied to $f=f_j$, this functional takes the value $\overline{g(z_j)}$; indeed, Cauchy’s formula gives

$$\begin{aligned} \int _{{\mathbb {T}}}\overline{f}_jg\,dm=g(z_j) \end{aligned}$$

for each j. In conjunction with (4.3), this yields

$$\begin{aligned} |g(z_j)|\le N_g\Vert f_j\Vert _1\le MN_g\log \frac{2}{1-|z_j|},\quad j\in {{\mathbb {N}}}. \end{aligned}$$

(4.4)

Finally, we put

$$\begin{aligned} w_j:=g(z_j),\quad j\in {{\mathbb {N}}}. \end{aligned}$$

The sequence ${\mathcal {W}}=\{w_j\}$ is then in $\ell ^\infty _{\log }({\mathcal {Z}})$, as (4.4) shows, while

$$\begin{aligned} {\mathcal {W}}\notin K_{*B}\big |_{\mathcal {Z}} \end{aligned}$$

(4.5)

as required. To verify (4.5), it suffices to note that g is the only function in $K^2_B$ that interpolates ${\mathcal {W}}$ on ${\mathcal {Z}}$ (indeed, a $K^2_B$ function is uniquely determined by its trace on ${\mathcal {Z}}=B^{-1}(0)$), whereas $g\notin \mathrm{BMO}$. The proof is complete. $\square$

Remark

We have seen above that if $g\in K^2_B$, with $B=B_{{\mathcal {Z}}}$, and if the functional (4.2) is continuous on $K^1_B$, then $g\big |_{{\mathcal {Z}}}\in \ell ^\infty _{\log }({\mathcal {Z}})$. Now, if ${\mathcal {Z}}$ has the $\mathrm{BMOA}$-interpolating property (1.9), then every sequence ${\mathcal {W}}$ in $\ell ^\infty _{\log }({\mathcal {Z}})$ is actually writable as $g\big |_{{\mathcal {Z}}}$ for some $g\in K^2_B$ that induces a continuous linear functional on $K^1_B$. (To see why, take $G\in \mathrm{BMOA}$ with $G\big |_{{\mathcal {Z}}}={\mathcal {W}}$ and then put $g=BP_-(\overline{B}G)$, 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 ${\mathcal {Z}}$ for which $\ell ^\infty _{\log }({\mathcal {Z}})\not \subset \ell ^2_1({\mathcal {Z}})$; and if this is the case, then no sequence in $\ell ^\infty _{\log }({\mathcal {Z}}){\setminus }\ell ^2_1({\mathcal {Z}})$ is the trace of any $H^2$ function on ${\mathcal {Z}}$.