TWO DISTINGUISHED SUBSPACES OF PRODUCT BMO AND THE NEHARI–AAK THEORY FOR HANKEL OPERATORS ON THE TORUS

MISCHA COTLAR AND CORA SADOSKY

Abstract. In this paper we show that the theory of Hankel operators in the torus 𝕋^{d}, for d > 1, presents striking differences with that on the circle 𝕋, starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbols L^{∞}(𝕋) by BMOr(𝕋^{d}), a subspace of product BMO, and the singular numbers of Hankel operators by so-called sigma numbers. This leads to versions of the Nehari–AAK and Kronecker theorems, and provides conditions for the existence of solutions of product Pick problems through finite Pick-type matrices. We give geometric and duality characterizations of BMOr, and of a subspace of it, bmo, closely linked with A_{2} weights. This completes some aspects of the theory of BMO in product spaces.

Introduction

This paper deals with the extension of the classical theory of bounded Hankel operators in the circle 𝕋 to (big) Hankel operators in the torus 𝕋^{d}, for d > 1. Some crucial results in the one-variable theory, involving the notions of L^{∞} symbols and the singular numbers of the operators, cannot have, as stated, meaningful extensions to the torus. This difficulty can be overcome by introducing so-called BMOr symbols and sigma numbers of Hankel operators. To explain what changes are to be made in dimension d > 1, we recall some basic features of the theory in 𝕋.

Each function ϕ ∈ L^{2}(𝕋^{d}) gives rise to a Hankel operator Γ_{ϕ}, and ϕ is called a symbol for the operator. In the case d = 1, these operators are closely related to the space BMO, since, by the Nehari theorem [N], a Hankel operator Γ is bounded if and only if Γ1 ∈ BMO, and if and only if Γ = Γ_{φ} with φ ∈ L^{∞}, while ϕ ∈ BMO implies Γ_{ϕ} bounded with ∥Γ_{ϕ}∥ = ∥ϕ∥_{BMO}. In turn, the Helson–Szegő theorem [HS] relates BMO to the boundedness of the Hilbert transform in L^{2}(μ), for μ a given measure on the circle 𝕋.

The Nehari theorem gives the distance of a bounded function φ to the space H^{∞}(𝕋) as the norm of the Hankel operator Γ_{φ}, and the theorem of Adamjan, Arov and Krein (AAK) refines this by giving its distance to H^{∞}(𝕋) + R_{n} (where R_{n} is the space of rational functions with n poles in the disk) as the singular number s_{n} of the operator, or, equivalently, as the distance of the operator to those Hankel operators of finite rank n [AAK].

1991 Mathematics Subject Classification. 47B35, 42B20.

Sadosky was partially supported by NSF grants DMS-9205926, INT-9204043 and GER-9550373, and her visit to MSRI is supported by NSF grant DMS-9022140 to MSRI.

From the Beurling characterization of the invariant subspaces of H^{2}(𝕋) of finite codimension, it follows that a Hankel operator Γ is of finite rank n if and only if Γ = Γ_{ϕ} with ϕ = b̄h, where h ∈ H^{∞} and b is a Blaschke product with n zeros at z_{1}, …, z_{n}. If this is the case, the operator Γ_{ϕ} is closely related to a model operator in a finite subspace of H^{2}, so that its norm ∥Γ_{ϕ}∥ equals that of a finite n × n matrix explicitly given in terms of the z_{k}’s and ϕ(z_{k})’s: the Pick matrix. One of the main applications of the Nehari theorem is that it provides a condition for the existence of solutions of the Pick interpolation problems in terms of the norm of an associated Hankel operator Γ_{ϕ} of finite rank, thus yielding the classical Pick condition in terms of Pick matrices.

The basic properties of BMO(𝕋) can be deduced in a unified way [ACS] through a generalized Bochner theorem, which includes also the results of Nehari and Helson– Szegő. The extension of this theorem to several dimensions led in [CS2] and [CS3] to an extension of the Nehari theorem to 𝕋^{d}, for d > 1, in terms of a class of symbols that we called BMOr (for “restricted” BMO). The extension of the Helson–Szegő theorem to several dimensions was given in [CS1], in terms of a subspace of product BMO = BMO(𝕋^{d}) (defined in [ChF1]), that here we call bmo (for “small” BMO).

Section 1 gives some basic properties of these subspaces of product BMO, starting with the continuous proper inclusions

L^{∞}(𝕋^{d}) ⊂ bmo ⊂ BMOr ⊂ BMO(𝕋^{d}).

The preduals of bmo and BMOr are determined, providing counterparts of the duality result of Chang and Fefferman in product domains [ChF2].

As a corollary of the duality result for BMOr, in Section 2 it is shown that, when d > 1, there are bounded Hankel operators without bounded symbols (Theorem 2.1). This indicates that L^{∞} symbols are not enough to characterize bounded Hankel operators, and that BMOr is the right class of symbols in product domains [CS3].

For d > 1 it is known [Am] that the positivity of the Pick matrix is necessary but not sufficient for the existence of a solution of the Pick problem. Necessary and sufficient conditions involving Pick matrices have been given by Agler for d = 2 [Ag], and by Cole, Lewis and Wermer for all d > 1 [CLW]. However, their conditions are not verifiable in practice, and the relation with Hankel operators is lost in their approach. In Section 3 we return to the consideration of analogues to the Pick problem with BMOr-norm control initiated in [CS3], and give necessary and sufficient conditions for the existence of solutions of a coordinate-wise Pick problem in terms of either the boundedness of a Hankel operator with symbol specified by the data, or the positiveness of d associated n × n Pick matrices.

In the case d > 1, all singular numbers of a Hankel operator are bounded below by d^{−1/2} times its norm (Theorem B), so that all Hankel operators of finite rank are zero [CS2]. This abrupt change from the one-dimensional case is closely related to the failure of the Beurling characterization of invariant subspaces to hold in the polydisk [AhC], and shows that an AAK theory cannot be meaningful in 𝕋^{d}, for d > 1. To recover the main features of the Nehari–AAK theory we need to introduce, not only BMOr symbols, but sigma numbers to replace the singular numbers, and a notion of operators of finite type, to replace that of finite rank.

In Section 4 we rely on a version of Beurling’s characterization in the polydisk given in [CS4] to characterize the symbols of Hankel operators of finite type in terms of tensor products of finite Blaschke products, and to extend the AAK result mentioned above in terms of the sigma numbers of the Hankel operators.

In Section 1 it is shown that, when passing from 𝕋 to 𝕋^{d}, for d > 1, the different equivalent characterizations of BMO(𝕋) give rise to distinct spaces. Similarly, the different characterizations of Carleson measures in 𝔻 give rise to different notions in 𝔻^{d}, for d > 1. One such characterization is that a measure in 𝔻 is Carleson if and only if a canonically associated function is in BMO. In Section 5 we extend this canonical association to d > 1, by defining Carleson–Nikolskii measures, and proving that a measure is of this type if and only if a canonically associated function is in BMOr.

In the circle, the norms of Hankel operators of finite rank coincide with the norms of multipliers acting in finite-dimensional model subspaces, which in turn are determined by finite Pick matrices. In Section 6 we prove that the norms of Hankel operators of finite type coincide with those of multipliers acting in corresponding model subspaces, which now are not finite-dimensional but of bi-finite type, like those appearing in Sections 4 and 5. This significantly reduces the number of steps required to verify norm boundedness.

Acknowledgements. We want to thank Chandler Davis for extensive discussions with the second author on duality, and Nikolai Nikolskii for helpful comments. The last version of this paper was written while the second author was a Research Professor of the Mathematical Sciences Research Institute at Berkeley, and we are happy to acknowledge the hospitality received there by both of us.

Basic Notations. The following notations will be used throughout the paper. For d ≥ 1, 𝒫 = 𝒫(𝕋^{d}) is the class of trigonometric polynomials; f̂ represents the Fourier transform of f;

H^{2}(𝕋^{d}) = {f ∈ L^{2}(𝕋^{d}) : f̂(n_{1}, …, n_{d}) = 0 if n_{k} < 0 for some k = 1, …, d};
H_{x_{k}}^{2} = {f ∈ L^{2}(𝕋^{d}) = f̂(n) = 0 for n_{k} < 0};
H^{2}(𝕋^{d})^{⊥} = L^{2}(𝕋^{d}) ⊖ H^{2}(𝕋^{d});

and the orthogonal projector P : L^{2} → H^{2}(𝕋^{d}) is called the analytic projector. The d shifts S_{k} = S_{x_{k}}, in L^{2}(𝕋^{d}), where k = 1, …, d, are defined by

S_{k}f(x) = S_{k}f(x_{1}, …, x_{d}) := exp(ix_{k})f(x).

In the case d = 2, we write (x, y) for (x_{1}, x_{2}) and (m, n) for (n_{1}, n_{2}), and consider the subspaces of H^{2}(𝕋^{2}) given by

H_{x}^{2} = {f ∈ L^{2}(𝕋^{2}) : f̂(m, n) = 0 for m < 0}, H_{−x}^{2} = L^{2} ⊖ H_{x}^{2},

and

H_{y}^{2} = {f ∈ L^{2}(𝕋^{2}) : f̂(m, n) = 0 for n < 0}, H_{−y}^{2} = L^{2} ⊖ H_{y}^{2},

as well as the projectors

P_{x} : L^{2} → H_{x}^{2}, P_{−x} := (I − P_{x}) : L^{2} → H_{−x}^{2}

and

P_{y} : L^{2} → H_{y}^{2}, P_{−y} := (I − P_{y}).

The two shifts S_{1} and S_{2} in L^{2}(𝕋^{2}) satisfy

S_{1}f(x, y) = e^{ix}f(x), S_{2}f(x, y) = e^{iy}f(y).

Observe that

H^{2}(𝕋^{2})^{⊥} = H_{−x}^{2} + H_{−y}^{2} = H_{−x}^{2} ∔ (H_{−y}^{2} ∩ H_{x}^{2}) = H_{−y}^{2} ∔ (H_{−x}^{2} ∩ H_{y}^{2}).

1. Two Distinguished Subspaces of Product BMO

An integrable function in 𝕋 is of bounded mean oscillation if

(1.1)  \frac{1}{|I|} ∫_{I} |f(x) − f_{I}| dx ≤ C  for all intervals I,

where f_{I} = |I|^{−1} ∫_{I} f(x) dx. The class BMO of functions of bounded mean oscillation is important in analysis. It is closely related to the Carleson measures and to the A_{p} weights, as well as to bounded Hankel operators.

A function ϕ is in BMO = BMO(𝕋) if and only if a canonically associated measure μ in 𝔻 is Carleson, and a measure μ in 𝔻 is Carleson if and only if Γ1 ∈ BMO for a canonically associated Hankel operator Γ. (See definitions below.) As there are different characterizations for the elements of BMO in 𝕋, the same is true for Carleson measures in 𝔻.

BMO coincides with the space L^{∞} +HL^{∞}, where H is the Hilbert transform. This characterization follows from Charles Fefferman’s famous duality result, asserting that BMO is the (real) dual of the Hardy space H^{1}. Another way to prove

BMO = L^{∞} + HL^{∞} (1.2)

is through the characterizations of the weights w for which H is bounded in L^{2}(w) given by the A_{2} condition and by the Helson–Szegő theorem [HS].

In passing from 𝕋 to 𝕋^{d}, for d > 1, the extension of the BMO theory to product domains presents various difficulties [ChF2]. S.-Y. Alice Chang and Robert Fefferman were able to introduce a notion of product BMO = BMO(𝕋^{d}), dual to the space H^{1}(𝔻^{d}), and for which an analogue of (1.2) is retained [ChF1]. In fact,

\begin{aligned}ϕ & ∈ BMO(𝕋^{d}) ⟺ \\ ϕ & = f_{1} + H_{x_{1}}f_{2} + ⋯ + H_{x_{d}}f_{d+1} + H_{x_{1}}H_{x_{2}}f_{d+2} + ⋯ + H_{x_{1}}H_{x_{2}} … H_{x_{d}}f_{2^{d}} \\ & for f_{1}, …, f_{2^{d}} ∈ L^{∞}(𝕋^{d}), \end{aligned} (1.3)

where H_{x_{j}} is the Hilbert transform with respect to the variable x_{j}, for j = 1, …, d, and BMO is a complete normed space with respect to

∥ϕ∥_{BMO} := inf{max_{j} ∥f_{j}∥_{∞} : all decompositions (1.3)}.

But for product BMO the geometric characterizations by mean oscillation and by associated Carleson measures become considerably more complicated (they do not correspond to bounded mean oscillation with respect to rectangles), and, furthermore, the connections with weights and Hankel operators are lost.

In previous work ([CS1], [CS3]), we gave results in product spaces analogous to those linking BMO to weights and to Hankel operators in one variable, in terms of classes of functions that are properly contained in product BMO. In this section we clarify the relation of these classes with product BMO, give some of their basic properties, and characterize their preduals.

Definition 1 (small BMO). A function ϕ ∈ L^{2}(𝕋^{d}), for d ≥ 1, is in bmo(𝕋^{d}) if there exist f_{1}, …, f_{d}, g_{1}, …, g_{d} ∈ L^{∞}(𝕋^{d}) such that

ϕ = f_{1} + H_{x_{1}}g_{1} = ⋯ = f_{d} + H_{x_{d}}g_{d} (1.4)

and

∥ϕ∥_{bmo} := inf{ max_{1≤j≤d}{∥f_{j}∥_{∞}, ∥g_{j}∥_{∞}} : all decompositions (1.4)}.

Observe that ∥ϕ∥_{bmo} = 0 if and only if ϕ is constant, and bmo/ℂ is a complete normed space with respect to ∥ · ∥_{bmo}.

Definition 2 (restricted BMO). A function ϕ ∈ L^{2}(𝕋^{d}), for d ≥ 1, is in BMOr if there exist φ_{0}, φ_{1}, …, φ_{d} ∈ L^{∞}(𝕋^{d}) such that

\{ \begin{matrix}(I − P_{x_{j}})ϕ = (I − P_{x_{j}})φ_{j} for j = 1, …, d, and \\ P_{x_{1}}P_{x_{2}} … P_{x_{d}}ϕ = P_{x_{1}}P_{x_{2}} … P_{x_{d}}φ_{0}, \end{matrix} (1.5)

where P_{x_{j}} : L^{2} → H_{x_{j}}^{2} is the analytic projector in x_{j}, for j = 1, …, d. Moreover,

∥ϕ∥_{BMOr} := inf{ max_{0≤j≤d} ∥φ_{j}∥_{∞} : all decompositions (1.5)}

= max{ max_{1≤j≤d}{inf{∥ϕ − h_{x_{j}}∥_{∞} : h_{x_{j}} ∈ H_{x_{j}}^{2}}}, inf{∥ϕ − h^{⊥}∥_{∞} : h^{⊥} ∈ H^{2⊥}}}.

Observe that BMOr is a complete normed space with respect to ∥ · ∥_{BMOr}, and coincides with the space restricted BMO introduced in [CS3].

The two definitions given above are justified by the following results.

Theorem (Helson–Szegő theorem in 𝕋^{d}, for d ≥ 1). [CS1] A weight 0 ≤ w ∈ L^{1}(𝕋^{d}) satisfies

∫_{𝕋^{d}} |Hf|^{2}w ≤ M^{2} ∫_{𝕋^{d}} |f|^{2}w for all f ∈ 𝒫,

where H = H_{x_{1}} … H_{x_{d}} is the product Hilbert transform, if and only if ϕ = log w ∈ bmo(𝕋^{d}), with

ϕ = u_{1} + H_{x_{1}}v_{1} = ⋯ = u_{d} + H_{x_{d}}v_{d}

for u_{1}, …, u_{d}, v_{1}, …, v_{d} real-valued bounded functions in 𝕋^{d} satisfying ∥u_{j}∥_{∞} ≤ C_{M} and ∥v_{j}∥_{∞} ≤ π/2 − ε_{M} for j = 1, …, d.

Theorem (Nehari theorem in 𝕋^{d}, for d ≥ 1). [CS3] Let Γ : H^{2}(𝕋^{d}) ∩ 𝒫 → H^{2}(𝕋^{2})^{⊥} be a Hankel operator. (The definition of a Hankel operator is given in Section 2.) Γ is bounded if and only if there exists ϕ ∈ BMOr satisfying Γf = (I − P)(ϕf), for all f ∈ H^{2}(𝕋^{d}), where P : L^{2} → H^{2} is the orthogonal projector, and ∥ϕ∥_{BMOr} ≤ ∥Γ∥ ≤ \sqrt{d}∥ϕ∥_{BMOr}.

Definitions 1 and 2 impose constraints on the functions in small and restricted BMO, which follow immediately from the relation between the analytic projector and the Hilbert transforms,

P_{x_{j}} = \frac{1}{2}(I + H_{x_{j}}) for j = 1, …, d, (1.6)

and can be summarized as follows:

Lemma 1.1. (i) For ϕ ∈ bmo given by (1.4), and for j = 1, …, d, we have

\begin{aligned}P_{x_{j}}ϕ & = P_{x_{j}}(f_{j} − ig_{j}) \\ (I − P_{x_{j}})ϕ & = (I − P_{x_{j}})(f_{j} + ig_{j}). \end{aligned} (1.7)

In particular,

P_{x_{1}} … P_{x_{d}}(f_{1} − ig_{1}) = ⋯ = P_{x_{1}} … P_{x_{d}}(f_{d} − ig_{d}) (1.8)

and

(I − P_{x_{1}}) … (I − P_{x_{d}})(f_{1} + ig_{1}) = ⋯ = (I − P_{x_{1}}) … (I − P_{x_{d}})(f_{d} + ig_{d}). (1.9)

(ii) For ϕ ∈ BMOr given by (1.5), we have

(I − P_{x_{1}}) … (I − P_{x_{d}})φ_{1} = ⋯ = (I − P_{x_{1}}) … (I − P_{x_{d}})φ_{d}. (1.10)

Lemma 1.1 implies that in order to define functions in bmo or BMOr by d or d + 1 bounded functions, respectively, those bounded functions have to satisfy the constraints (1.8), (1.9) and (1.10).

The relation between small, restricted and product BMOs is the following:

Proposition 1.2. The inclusions

L^{∞}(𝕋^{d}) ⊂ bmo(𝕋^{d}) ⊂ BMOr(𝕋^{d}) ⊂ BMO(𝕋^{d})

are topological, and proper for d > 1. For d = 1 we have bmo(𝕋) = BMOr(𝕋) = BMO(𝕋).

Proof. The topological inclusion L^{∞} ⊂ bmo is immediate from Definition 1. If ϕ ∈ bmo, by Lemma 1.1(i), for j = 1, …, d, we have (I − P_{x_{j}})ϕ = (I − P_{x_{j}})(f_{j} + ig_{j}), and P_{x_{1}} … P_{x_{d}}ϕ = P_{x_{1}} … P_{x_{d}}(f_{1} − ig_{1}) = ⋯ = P_{x_{1}} … P_{x_{d}}(f_{d} − ig_{d}), with f_{j} ± ig_{j} ∈ L^{∞}(𝕋^{d}), which means that condition (1.5) is satisfied and ϕ ∈ BMOr with ∥ϕ∥_{bmo} ≥ ∥ϕ∥_{BMOr}. It follows from (1.5) and (1.6) that ϕ ∈ BMOr implies ϕ ∈ BMO, with ∥ϕ∥_{BMOr} ≥ ∥ϕ∥_{BMO}. To show that the inclusions are proper it is enough to consider d = 2.

(a) Example of ϕ ∈ bmo\L^{∞}. Fix v ∈ L^{∞}(𝕋) such that Hv ∉ L^{∞}(𝕋), and let ϕ(x, y) = Hv(x − y). Then ϕ = H_{x}g_{1} = H_{y}g_{2}, for g_{1}, g_{2} ∈ L^{∞}(𝕋^{2}), defined by g_{1}(x, y) = v(x − y), g_{2}(x, y) = −v(x − y), and ϕ ∈ bmo\L^{∞}(𝕋^{2}).

(b) Example of ϕ ∈ BMOr\bmo. Let φ_{0}(x, y) ≡ 0, φ_{1}(x, y) = v(x)h(y), φ_{2}(x, y) = h(x)v(y), where v ∈ L^{∞}(𝕋) is as in (a) and h ∈ H^{∞}(𝕋) is not a constant. Define ϕ ∈ BMOr(𝕋^{2}) by condition (1.5), that is, P_{x}P_{y}ϕ = P_{x}P_{y}φ_{0}, (I − P_{x})ϕ = (I − P_{x})φ_{1}, (I − P_{y})ϕ = (I − P_{y})φ_{2}, which can be done since (1.10) is satisfied:

(I − P_{x})(I − P_{y})φ_{1} = 0 = (I − P_{x})(I − P_{y})φ_{2}.

If ϕ were in bmo(𝕋^{2}), by (1.7), we would have P_{x}ϕ = P_{x}(f_{1} − ig_{1}) and P_{y}ϕ = P_{y}(f_{2} − ig_{2}), with f_{1} − ig_{1}, f_{2} − ig_{2} ∈ L^{∞}(𝕋^{2}). But, in our case, P_{x}ϕ = P_{x}P_{y}ϕ + P_{x}(I − P_{y})ϕ = P_{x}P_{y}φ_{0} + P_{x}(I − P_{y})φ_{2} = h(x)(I − P_{y})v(y), with (I − P_{y})v(y) ∉ L^{∞}(𝕋) by assumption. Since P_{x}ϕ = P_{x}φ for some φ ∈ L^{∞}(𝕋^{2}), the function h(x)(I − P_{y}))v(y) = P_{x}φ(x, y) should be, for all y fixed, a function in BMO(𝕋) satisfying |(I − P_{y})v(y)|∥h∥_{BMO(𝕋)} = ∥P_{x}φ(·, y)∥_{BMO(𝕋)} ≤ c∥φ∥_{∞}. Since ∥h∥_{BMO} ≠ 0 and (I − P_{y})v(y) ∉ L^{∞}(𝕋), this is a contradiction.

(c) Example of ϕ ∈ BMO\BMOr. Given ψ_{1}, ψ_{2} ∈ L^{∞}(𝕋^{2}), take ϕ = H_{x}ψ_{1} +H_{y}ψ_{2} ∈ BMO(𝕋^{2}). In this case, (I − P_{x})ϕ = (I − P_{x})(iψ_{1} + iψ_{2}) − (I − P_{x})P_{y}(2iψ_{2}), so, for ϕ to be in BMOr, by (1.5), (I − P_{x})P_{y}ψ_{2} should equal (I − P_{x})φ, for some φ ∈ L^{∞}(𝕋^{2}). Taking ψ_{2}(x, y) = f(x)v(y), for f, v ∈ L^{∞}(𝕋), with v as in example (a), this means that, for all y fixed, |P_{y}v(y)|∥(I − P_{x})f∥_{BMO} = ∥(I − P_{x})P_{y}φ(·, y)∥_{BMO(𝕋)} ≤ C∥φ∥_{∞}, which is a contradiction.

□

In what follows we limit the statements, as well as their proofs, to the case d = 2, in order to simplify notations. All results remain valid, with obvious modifications, for d > 1.

We will write ϕ ∈ BMO_{x}(𝕋^{2}) if ϕ( ·, y) ∈ BMO(𝕋) for every y. If, in addition, we have sup_{y} ∥ϕ( ·, y)∥_{BMO} ≤ C for some constant C, we say that ϕ ∈ BMO_{x}(𝕋^{2}) with uniformly bounded norm. We define BMO_{y}(𝕋^{2}) similarly.

Proposition 1.3 (Bounded mean oscillation on rectangles). The following conditions on a function ϕ are equivalent:

(a) ϕ ∈ bmo(𝕋^{2}).

(b) For a constant C > 0 we have

\frac{1}{|R|} ∫∫ _{R} |ϕ(x, y) − ϕ_{R}| dx dy ≤ C for all R = I × J, (1.11)

where I, J ⊂ 𝕋 are intervals and

ϕ_{R} = \frac{1}{|R|} ∫∫ _{R} ϕ(x, y) dx dy.

(c) ϕ ∈ BMO_{x}(𝕋^{2}) with uniformly bounded norm and ϕ ∈ BMO_{y}(𝕋^{2}) with uniformly bounded norm.

Proof. (b) ⟹ (c). Condition (1.11), of bounded mean oscillation on rectangles, can be written as

\frac{1}{|I|} \frac{I}{|J|} ∫_{I} ∫_{J} |ϕ(x, y) − ϕ_{R}| dx dy = \frac{1}{|I|} ∫_{I} F (x, J) dx ≤ C

for all intervals I, J. This implies, for almost every x ∈ I,

F (x, J) = \frac{1}{|J|} ∫_{J} |ϕ(x, y) − ϕ_{R}| dy ≤ C for allJ, (1.12)

which is to say that ϕ ∈ BMO_{y} with uniformly bounded norm. Similarly, (1.11) implies ϕ ∈ BMO_{x} with uniformly bounded norm.

(c) ⟺ (a). Obviously, ϕ ∈ bmo(𝕋^{2}) implies ϕ ∈ BMO_{x}(𝕋^{2}) and ϕ ∈ BMO_{y}(𝕋^{2}), both with uniformly bounded norm. Conversely, from (c) we have ϕ = f_{1} + H_{x}g_{1} = f_{2}+H_{y}g_{2}, for f_{1} and g_{1} bounded functions in x, uniformly in y, and f_{2} and g_{2} bounded functions in y, uniformly in x. This means that f_{1}, f_{2}, g_{1} and g_{2} are bounded functions of both x and in y, which is (a).

(c) ⟹ (b). By (1.1), the condition that ϕ ∈ BMO_{x} with uniformly bounded norm is equivalent to

(1.13) \frac{1}{|I|} ∫_{I} |ϕ(x, y) − ϕ_{I}(y)| dx ≤ c for all I, uniformly in y,

and the condition that ϕ ∈ BMO_{y} with uniformly bounded norm is equivalent to

(1.14) \frac{1}{|J|} ∫_{J} |ϕ(x, y) − ϕ_{J}(x)| dy ≤ c′, for all J, uniformly in x,

where

ϕ_{I}(y) = \frac{1}{|I|} ∫_{I} ϕ(x, y) dx, ϕ_{J}(x) = \frac{1}{|J|} ∫_{J} ϕ(x, y) dy,

and c, c′ are positive constants. From (1.14) it follows, for almost all x ∈ I, and y ∈ J, that

|ϕ(x, y) − ϕ_{J}(x)| ≤ c′ for all J,

so that, for R = I × J,

|ϕ_{I}(y) − ϕ_{R}| = |\frac{1}{|I|} ∫_{I} ϕ(x, y) dx − \frac{1}{|I|} \frac{1}{|J|} ∫_{I} ∫_{J} ϕ(x, y) dx dy|

= |\frac{1}{|I|} ∫_{I} ϕ(x, y) dx − \frac{1}{|I|} ∫_{I} ϕ_{J}(x) dx|

≤ \frac{1}{|I|} ∫_{I} |ϕ(x, y) − ϕ_{J}(x)| dx ≤ c′.

Then, by (1.13),

\frac{1}{|R|} ∫∫ _{R} |ϕ(x, y) − ϕ_{R}| dx dy

= \frac{1}{|J|} ∫_{J}(\frac{1}{|I|} ∫_{I} |ϕ(x, y) − ϕ_{R}| dx) dy

≤ \frac{1}{|J|} ∫_{J}(\frac{1}{|I|} ∫_{I} |ϕ(x, y) − ϕ_{I}(y)| dx + \frac{1}{|I|} ∫_{I} |ϕ_{I}(y) − ϕ_{R}| dx) dy

≤ c + c′ = C,

which is (b). □

The relation between BMOr, bounded Hankel operators in 𝕋^{d}, for d > 1, and Carleson measures will be treated in Section 4. Now we consider duality results.

In the one-dimensional case, BMO(𝕋) is the dual of the (real) Hardy space

H^{1}(𝕋) := {f ∈ L^{1}(𝕋) : Hf ∈ L^{1}(𝕋)},

or, equivalently, is the space of functions f such that P f and (I −P)f ∈ L^{1}(𝕋), where f = P + (I − P)f is a canonical decomposition of f given by the analytic projector P. In the two-dimensional case, for each trigonometric polynomial f ∈ 𝒫(𝕋^{2}), we consider three canonical decompositions of f, given in terms of the analytic projectors P_{x} : L^{2} → H_{x}^{2} and P_{y} : L^{2} → H_{y}^{2}, as well as of P_{−x} := (I − P_{x}) and P_{−y} := (I − P_{y}):

(1.15)  f = P_{x}f + P_{−x}f = P_{y}f + P_{−y}f;
(1.16)  f = P_{x}f + P_{−x}P_{y}f + P_{−x}P_{−y}f = P_{y}f + P_{−y}P_{x}f + P_{−y}P_{−x}f;
(1.17)  f = P_{x}P_{y}f + P_{x}P_{−y}f + P_{−x}P_{y}f + P_{−x}P_{−y}f;

and norm 𝒫(𝕋^{2}) with three different norms, all stronger than the L^{1} norm. The completion of 𝒫(𝕋^{2}) with respect to these three norms gives rise to Banach spaces, denoted as follows:

(A) The space

H_{x}^{1}(𝕋^{2}) + H_{y}^{1}(𝕋^{2}) := L^{1}(𝕋_{y}; H^{1}(𝕋_{x})) + L^{1}(𝕋_{x}; H^{1}(𝕋_{y})), (1.18)

whose elements are functions f = f(x, y) = f_{x}(y), integrable in y, with values in H^{1}(𝕋_{x}), and f = f_{y}(x), integrable in x, with values in H^{1}(𝕋_{y}); that is, the closure of 𝒫(𝕋^{2}) in the norm

[f] := inf{∥g∥_{H_{x}^{1}} + ∥h∥_{H_{y}^{1}} : f = g + h},

where

∥f∥_{H_{x}^{1}} := ∥P_{x}f∥_{1} + ∥P_{−x}f∥_{1} and ∥f∥_{H_{y}^{1}} := ∥P_{y}f∥_{1} + ∥P_{−y}f∥_{1}

correspond to partition (1.15).

(B) The space

ℋ^{1}(𝕋^{2}) := ℋ_{x}^{1}(𝕋^{2}) + ℋ_{y}^{2}(𝕋^{2}), (1.19)

where ℋ_{x}^{1} and ℋ_{y}^{1} are the closures of 𝒫(𝕋^{2}) under the norms corresponding to partition (1.16), namely

\begin{aligned}∥|f∥|_{(x)} & := ∥P_{x}f∥_{1} + (∥P_{−x}P_{y}f∥_{1} + ∥P_{−x}P_{−y}f∥_{1}), \\ ∥|f∥|_{(y)} & := ∥P_{y}f∥_{1} + (∥P_{−y}P_{x}f∥_{1} + ∥P_{−y}P_{−x}f∥_{1}), \end{aligned}

and ℋ^{1} is normed by

∥|f∥| := inf{∥|g∥|_{(x)} + ∥|h∥|_{(y)} : f = g + h}. (1.20)

Observe that, in particular,

∥|f∥| ≥ ∥P_{−x}P_{−y}f∥_{1} for all f. (1.21)

(C) The space

H^{1}(𝕋^{2}) = H^{1}(𝕋_{x}; H^{1}(𝕋_{y})) = H^{1}(𝕋_{y}; H^{1}(𝕋_{x})), (1.22)

normed by

∥f∥_{H^{1}} := ∥P_{x}P_{y}f∥_{1} + ∥P_{x}P_{−y}f∥_{1} + ∥P_{−x}P_{y}f∥_{1} + ∥P_{−x}P_{−y}f∥_{1}, (1.23)

corresponding to partition (1.17).

Observe that 𝒫(𝕋^{2}) can be partitioned in more ways than those in (1.15)–(1.17). For instance, a function f can be written as f = P_{x}P_{y}f + (I − P_{x}P_{y})f, giving rise to the norm ∥f∥ := ∥P_{x}P_{y}f∥_{1} + ∥(I − P_{x}P_{y})f∥_{1}. Since the Hilbert transform, as well as the analytic projection, is unbounded in L^{1}(𝕋), this norm ∥ · ∥ is not comparable to those above, and in particular to ∥| · ∥|.

Proposition 1.4. For each ε > 0, there is an f satisfying

ε∥|f∥| ≥ ∥f∥ := ∥P_{x}P_{y}f∥_{1} + ∥(I − P_{x}P_{y})f∥_{1},

where ∥| · ∥| is defined in (1.20).

Proof. Consider f(x, y) = \overline{u(x)} v(y) + v(x) \overline{u(y)}, with u an inner function and v ∈ L^{1} such that ∥v∥_{1} = 1, while ∥w∥_{1} > ε^{−1}, for w = (I − P)v. Since this f satisfies P_{x}P_{y}f = 0 and P_{−x}P_{−y}f = \overline{u(x)}w(y) + w(x)\overline{u(y)}, we have ∥f∥ = ∥f∥_{1} ≥ 2, and, by (1.21), ∥|f∥| ≥ ∥P_{−x}P_{−y}f∥_{1}, which, after multiplying by u(x)u(y), is equal to

∫∫ |u(x)w(x) + u(y)w(y)| dx dy > ∫ |w(x)| dx > ε^{−1}.

To justify the last inequality, it is enough to choose a test function G in the predual of L^{1}(𝕋), such that ∫ u(y)w(y)G(y) dy = 0, since then

\begin{aligned}∫∫ |u(x)w(x) + u(y)w(y)| dx dy & ≥ sup_{F} ∫∫ (u(x)w(x) + u(y)w(y))F (x)G(y) dx dy \\ & = sup_{F} ∫ u(x)w(x)F (x) dx = ∫ |u(x)w(x)| dx. \end{aligned}□

Theorem 1.5 (Duality). The spaces defined in (1.18), (1.19) and (1.22) are the preduals of the BMOs in 𝕋^{2}. More precisely:

(a) bmo(𝕋^{2}) is the dual of H_{x}^{1}(𝕋^{2}) + H_{y}^{1}(𝕋^{2}).

(b) BMOr(𝕋^{2}) is the dual of ℋ^{1}(𝕋^{2}) = ℋ_{x}^{1}(𝕋^{2}) + ℋ_{y}^{1}(𝕋^{2}).

(c) BMO(𝕋^{2}) is the dual of H^{1}(𝕋^{2}) [ChF1].

Proof. Note that, for any pair of functions f and ϕ in the variables x and y for which the integrals make sense,

∫ (P_{±x}f)(P_{±x}ϕ) dx = ∫ (P_{±y}f)(P_{±y}ϕ) dy = 0,

so that

∫ (P_{±x}f)(P_{∓x}ϕ) dx = ∫ (P_{±x}f)ϕ dx = ∫ f(P_{∓x}ϕ) dx, (1.24)

and similarly for P_{±y}.

(a) Let ϕ ∈ bmo(𝕋^{2}) and f ∈ H_{x}^{1} + H_{y}^{1}. Then, by Lemma 1.1(i) and (1.24), we have ∫ fϕ dx dy = ∫ gϕ dx dy + ∫ hϕ dx dy, where

∫ gϕ dx dy = ∫ (P_{x}g)ϕ + ∫ (P_{−x}g)ϕ = ∫ (P_{x}g)(P_{−x}ϕ) + ∫ (P_{−x}g)(P_{x}ϕ)
= ∫ (P_{x}g)(P_{−x}ψ_{1}) + ∫ (P_{−x}g)(P_{x}φ_{1})
= ∫ (P_{x}g)ψ_{1} + ∫ (P_{−x}g)φ_{1},

for ψ_{1}, φ_{1} ∈ L^{∞}(𝕋^{2}). Similarly,

∫ hϕ = ∫ (P_{y}h)ψ_{2} + ∫ (P_{−y}h)φ_{2},

for ψ_{2}, φ_{2} ∈ L^{∞}(𝕋^{2}). Thus,

|∫ gϕ| ≤ ∥P_{x}g∥_{1}∥ψ_{1}∥_{∞} + ∥P_{−x}g∥_{1}∥φ_{1}∥_{∞}

and

|∫ hϕ| ≤ ∥P_{y}h∥_{1}∥ψ_{2}∥_{∞} + ∥P_{−y}h∥_{1}∥φ_{2}∥_{∞},

which imply

|∫ gϕ| ≤ ∥g∥_{H_{x}^{1}}∥ϕ∥_{bmo} and |∫ hϕ| ≤ ∥h∥_{H_{y}^{1}}∥ϕ∥_{bmo},

and hence

|∫ fϕ| ≤ [f]∥ϕ∥_{bmo}.

Conversely, if l ∈ (H_{x}^{1} + H_{y}^{1})^{∗}, then for every f ∈ 𝒫(𝕋^{2}), we have |l(f)| ≤ C∥f∥_{H_{x}^{1}}, independently of y, and |l(f)| ≤ C∥f∥_{H_{y}^{1}}, independently of x. Since (H_{x}^{1})^{∗} = BMO_{x} and (H_{y}^{1})^{∗} = BMO_{y}, this implies, by Proposition 1.3, that l can be given by a function in bmo(𝕋^{2}).

(b) Let ϕ ∈ BMOr(𝕋^{2}) be given by

ϕ = P_{x}P_{y}φ_{0} + P_{x}P_{−y}φ_{2} + P_{−x}φ_{1} = P_{x}P_{y}φ_{0} + P_{−x}P_{y}φ_{1} + P_{−y}φ_{2}

for φ_{0}, φ_{1}, φ_{2} ∈ L^{∞}(𝕋^{2}), and let f ∈ ℋ_{x}^{1} + ℋ_{y}^{1}. Then, for f = g + h, g ∈ ℋ_{x}^{1}, h ∈ ℋ_{y}^{1}, we have

∫ gϕ = ∫ (P_{−x}P_{−y}g)φ_{0} + ∫ (P_{−x}P_{y}g)φ_{2} + ∫ (P_{x}g)φ_{1}

and | ∫ gϕ| ≤ ∥|g∥|_{(x)}∥ϕ∥_{BMOr}. Similarly, | ∫ hϕ| ≤ ∥|h∥|_{(y)}∥ϕ∥_{BMOr}; hence

|∫ fϕ| ≤ ∥|f∥| ∥ϕ∥_{BMOr}.

Conversely, if l is continuous on ℋ^{1}(𝕋^{2}), it is continuous on ℋ_{x}^{1} and on ℋ_{y}^{1}. In particular, for all f ∈ 𝒫(𝕋^{2}),

|l(f)| ≤ C∥|f∥|_{(x)} = C∥P_{x}f∥_{1} + C(∥P_{y}P_{−x}f∥_{1} + ∥P_{−y}P_{−x}f∥_{1}).

In particular, if f = P_{x}f, we have |l(f)| ≤ C∥f∥_{1}, and there exists F ∈ L^{∞} such that

l(P_{x}f) = ∫ F (P_{x}f) dx dy = ∫ (P_{−x}F)f dx dy,

by (1.24). Similarly, if f = P_{−x}f,

|l(f)| ≤ C(∥P_{y}f∥_{1} + ∥P_{−y}f∥_{1}) = C∥f∥_{H_{y}^{1}},

and there exists G ∈ BMO_{y} such that

l(P_{−x}f) = ∫ G(P_{−x}f) dx dy = ∫ (P_{x}G)f dx dy.

Then

l(f) = l(P_{x}f) + l(P_{−x}f) = ∫ (P_{−x}F + P_{x}G)f dx dy (1.25)

for F ∈ L^{∞} and G ∈ BMO_{y}. Similarly,

l(f) = l(P_{y}f) + l(P_{−y}f) = ∫ (P_{−y}F′ + P_{y}G′)f dx dy (1.26)

for F′ ∈ L^{∞} and G′ ∈ BMO_{x}.

Since, by (1.25) and (1.26), the two functions representing l coincide as functionals on all f ∈ 𝒫(𝕋^{2}), we conclude that

P_{−x}F + P_{x}G = P_{−y}F′ + P_{y}G′ = ϕ.

Now observe that G ∈ BMO_{y} and G′ ∈ BMO_{x} imply that P_{x}G = P_{x}φ and P_{y}G′ = P_{y}φ′ for some φ, φ′ ∈ L^{∞}(𝕋^{2}). Therefore, the function ϕ satisfies

P_{x}P_{y}ϕ = P_{x}P_{y}φ = P_{x}P_{y}φ′ and P_{−x}ϕ = P_{−x}F, P_{−y}ϕ = P_{−y}F′

for φ, φ′, F, F′ ∈ L^{∞}(𝕋^{2}), which means, by definition, that ϕ ∈ BMOr(𝕋^{2}).

(c) Let ϕ ∈ BMO(𝕋^{2}) = L^{∞} + H_{x}L^{∞} + H_{y}L^{∞} + H_{x}H_{y}L^{∞} (see [ChF1]) and f ∈ H^{1}. Writing

ϕ = P_{x}P_{y}ψ_{1} + P_{x}P_{−y}ψ_{2} + P_{−x}P_{y}ψ_{3} + P_{−x}P_{−y}ψ_{4},

for ψ_{1}, ψ_{2}, ψ_{3}, ψ_{4} ∈ L^{∞}, we get

|∫ fϕ| = |∫ (P_{x}P_{y}f)(P_{−x}P_{−y}ψ_{4}) + ∫ (P_{x}P_{−y}f)(P_{−x}P_{y}ψ_{3})

+ ∫ (P_{−x}P_{y}f)(P_{x}P_{−y}ψ_{2}) + ∫ (P_{−x}P_{−y}f)(P_{x}P_{y}ψ_{1})|

≤ ∥P_{x}P_{y}f∥_{1}∥ψ_{4}∥_{∞} + ∥P_{x}P_{−y}f∥_{1}∥ψ_{3}∥_{∞}
+ ∥P_{−x}P_{y}f∥_{1}∥ψ_{2}∥_{∞} + ∥P_{−x}P_{−y}f∥_{1}∥ψ_{1}∥_{∞},

which implies |∫ fϕ| ≤ ∥f∥_{H^{1}}∥ϕ∥_{BMO}.

Conversely, if l ∈ (H^{1})^{∗}, the usual duality argument shows that l is given by a function in L^{∞} + H_{x}L^{∞} + H_{y}L^{∞} + H_{x}H_{y}L^{∞} = BMO(𝕋^{2}), as in [ChF1]. □

A more detailed study of bmo and BMOr in 𝕋^{d}, for d > 1, including their atomic decompositions and their associated Carleson measures, will be the object of a future paper.

2. Big Hankel Operators and Their BMOr Symbols

We consider operators Γ : 𝒫 ∩ H^{2}(𝕋^{d}) → H^{2}(𝕋^{d})^{⊥}, for d ≥ 1. Such operators Γ are called bounded if sup ∥Γf∥_{2}/∥f∥_{2} =: ∥Γ∥ < ∞. A bounded Γ has a unique bounded extension, Γ : H^{2} → H^{2⊥}. It is easy to check that, for every Γ : 𝒫 ∩ H^{2} → H^{2⊥}, the following conditions are equivalent:

(a) ⟨ΓS_{k}f, g⟩ = ⟨Γf, S_{−k}g⟩ for k = 1, …, d and all f ∈ 𝒫 ∩ H^{2} and g ∈ H^{2⊥};

(b) ΓS_{k}f = (I − P)S_{k}Γf for k = 1, …, d, where P : L^{2} → H^{2} is the orthoprojector;

(c) There exists ϕ ∈ L^{2}(𝕋^{2}) ∋ Γ = Γ_{ϕ}, that is, Γf = (I − P)ϕf for all f ∈ H^{2}(𝕋^{2});

(d) Γ = Γ_{ϕ_{−}} for ϕ_{−} = Γ1 ∈ H^{2⊥}.

If (a)–(d) are verified, Γ is called a big Hankel operator, and ϕ as in (c) is called a symbol of Γ. Since

Γ_{ϕ} = Γ_{ψ} ⟺ ϕ − ψ = h ∈ H^{2},

we see that if ϕ is a symbol for Γ so are all ϕ + h, for h ∈ H^{2}. Moreover, among all symbols, there is a unique one in H^{2⊥}, which is Γ1. In what follows, (big) Hankel operators will be referred to as Hankel.

If φ ∈ L^{∞}, then Γ_{φ} is a bounded operator, with ∥Γ_{φ} ≤ ∥φ∥_{∞}. In the onedimensional case, the Nehari Theorem gives the converse: A Hankel operator Γ is bounded if and only if ∃φ ∈ L^{∞} with Γ_{φ} = Γ, if and only if ∃φ ∈ L^{∞} with Γ_{φ} = Γ and ∥φ∥_{∞} = ∥Γ∥, and if and only if Γ1 ∈ BMO. Also, ∥Γ_{φ}∥ = dist_{L^{∞}}(φ, H^{∞}). Since ϕ ∈ BMO(𝕋) implies ϕ = φ + h, for φ ∈ L^{∞} and, h ∈ H^{2}, we have

ϕ ∈ BMO ⟹ Γ_{ϕ} = Γ_{φ} is bounded, and ∥Γ_{ϕ}∥ ≤ ∥ϕ∥_{BMO}.

Thus, in the one-dimensional case, BMO(𝕋) appears as an essential feature both in the weighted norm inequalities for the Hilbert transform, and in the boundedness of the Hankel operators. In [ACS] it was shown that the basic properties of BMO(𝕋) can be deduced in a unified way from a Generalized Bochner Theorem (GBT) that is equivalent to the Nehari theorem in H^{2}(𝕋; μ), and which unifies the results of Nehari and Helson–Szegő. An abstract version of this GBT led to a version of the Nehari theorem in 𝕋^{d} in terms of BMOr, and to an extension of the Helson–Szegő theorem in terms of bmo(𝕋^{d}). Since bmo ≠ L^{∞} and BMOr ≠ BMO for d > 1, this underlines the importance of these two subspaces of product BMO.

Here we will base our considerations on the two-dimensional version of Nehari theorem:

Theorem A. [CS1], [CS2] For every (big) Hankel operator Γ : 𝒫 ∩ H^{2}(𝕋^{2}) → H^{2}(𝕋^{2})^{⊥}, the following conditions are equivalent:

(a) Γ is bounded.

(b) There exist φ_{1}, φ_{2} ∈ L^{∞}(𝕋^{2}), with max{∥φ_{1}∥_{∞}, ∥φ_{2}∥_{∞}} ≤ ∥Γ∥, and such that

P_{−x}Γ = P_{−x}Γ_{φ_{1}}, P_{−y}Γ = P_{−y}Γ_{φ_{2}}. (2.1)

(c) There exists ϕ ∈ BMOr ∩ H^{2⊥} with Γ = Γ_{ϕ} and ∥ϕ∥_{BMOr} ≤ ∥Γ∥ ≤ \sqrt{2}∥ϕ∥_{BMOr}.

(d) Γ1 ∈ BMOr.

Remark that (2.1) implies, for φ_{1} and φ_{2} as in (b), that

Γ = P_{−x}Γ_{φ_{1}} + P_{x}P_{−y}Γ_{φ_{2}} = P_{−y}Γ_{φ_{2}} + P_{−x}P_{y}Γ_{φ_{1}}. (2.2)

Corollary. For every ϕ ∈ BMOr, Γ_{ϕ} is bounded and

(2.3) ∥Γ_{ϕ}∥ = max{dist_{BMOr}(ϕ, BMOr ∩ H_{x}^{2}), dist_{BMOr}(ϕ, BMOr ∩ H_{y}^{2})}.

From Theorem A and the fact that ϕ ∈ BMOr implies ϕ_{−} = (I − P)ϕ ∈ BMOr ∩ H^{2⊥}, with ∥ϕ_{−}∥_{BMOr} ≤ ∥ϕ∥_{BMOr}, it follows that, for all ϕ ∈ BMOr, the operator Γ_{ϕ} is bounded and satisfies ∥Γ_{ϕ}∥ ≤ \sqrt{2}∥ϕ∥_{BMOr}.

Then, ϕ ↦ Γ_{ϕ} is a surjective map from BMOr onto the space 𝒢 of the bounded Hankel operators, whose restriction to BMOr ∩ H^{2⊥} is a bijection. The symbols ϕ ∈ BMOr, a proper subspace of product BMO, are thus enough for the theory of big Hankel operators. The duality theorem for BMOr leads to the following theorem, which highlights that the symbols in L^{∞} are not enough, so that equivalence (c) in Theorem A can be considered sharp.

The map φ ↦ Γ_{φ} from L^{∞}(𝕋^{2}) to the space 𝒢 has kernel H^{∞}(𝕋^{2}) and induces an injective map from L^{∞}/H^{∞} into 𝒢 . If this map were also surjective, by the Banach open mapping theorem, there would be a constant K > 0 such that, for each Γ ∈ 𝒢, there would be a φ ∈ L^{∞} with

Γ_{φ} = Γ and ∥φ∥_{∞} ≤ K∥Γ∥. (2.4)

If (2.4) held, the space L^{∞} + P L^{∞}, the “natural” extension of BMO(𝕋) to d > 1, would indeed coincide with BMOr. We see that this is not the case by showing that the map φ ↦ Γ is not surjective from L^{∞}/H^{∞} to 𝒢 .

Theorem 2.1. There are bounded big Hankel operators from H^{2}(𝕋^{2}) to H^{2}(𝕋^{2})^{⊥} that have no bounded symbol.

Proof. If the map φ ↦ Γ from L^{∞}/H^{∞} to 𝒢 were surjective, there would exist a K > 0 for which (2.4) would be satisfied. If a pair φ_{1}, φ_{2} ∈ L^{∞}(𝕋^{2}), with ∥φ_{1}∥_{∞} ≤ 1 and ∥φ_{2}∥_{∞} ≤ 1, coincide as functionals on H^{2}(𝕋^{2}), by (2.1) it defines a bounded Hankel operator Γ, with ∥Γ∥ ≤ \sqrt{2}, so that there would be a φ ∈ L^{∞} with Γ = Γ_{φ} and ∥φ∥_{∞} ≤ \sqrt{2}K. Now given any trigonometric polynomials p_{0} ∈ H_{−x,−y}^{2} := H_{−x}^{2} ∩ H_{−y}^{2}, p_{1} ∈ H_{−x,y}^{2} := H_{−x}^{2} ∩ H_{y}^{2}, and p_{2} ∈ H_{x,−y}^{2} := H_{x}^{2} ∩ H_{−y}^{2}, we have by (2.2):

\begin{aligned}⟨Γ1, p_{0} + p_{1} + p_{2}⟩ & = ⟨P_{−x}Γ_{φ_{1}}1, p_{0} + p_{1}⟩ + ⟨P_{−y}P_{x}Γ_{φ_{2}}1, p_{2}⟩ \\ & = ⟨Γ_{φ_{1}}1, p_{0} + p_{1}⟩ + ⟨Γ_{φ_{2}}1, p_{2}⟩ = ∫ φ_{1}(p̄_{0} + p̄_{1}) + ∫ φ_{2}p̄_{2}. \end{aligned}

On the other hand, if (2.4) holds, we have

\begin{aligned}|⟨Γ1, p_{0} + p_{1} + p_{2}⟩| & = |⟨Γ_{φ}1, p_{0} + p_{1} + p_{2}⟩| = | ∫ φ(p̄_{0} + p̄_{1} + p̄_{2})| \\ & ≤ ∥φ∥_{∞}∥p_{0} + p_{1} + p_{2}∥_{1} ≤ \sqrt{2}K∥p_{0} + p_{1} + p_{2}∥_{1}, \end{aligned}

so that

|∫ φ_{1}(p̄_{0} + p̄_{1}) + ∫ φ_{2}p̄_{2}| ≤ \sqrt{2}K∥p_{0} + p_{1} + p_{2}∥_{1}. (2.5)

We will now show that (2.5) leads to a contradiction. In fact, to give any pair φ_{1}, φ_{2} as above is the same as to give a ϕ_{−} ∈ BMOr ∩ H^{2⊥}, with P_{−x}ϕ_{−} = P_{−x}φ_{1}, P_{−y}ϕ_{−} = P_{−y}φ_{2}, and P_{x}P_{y}ϕ_{−} = 0. Then (2.5) can be rewritten as

|∫ ϕ_{−}(p̄_{0} + p̄_{1} + p̄_{2})| ≤ \sqrt{2}K∥p_{0} + p_{1} + p_{2}∥_{1} (2.6)

for all trigonometric polynomials p_{0} + p_{1} + p_{2} ∈ H_{−x,−y}^{2} + H_{−x,y}^{2} + H_{x,−y}^{2} = H^{2⊥}.

Now, any ϕ ∈ BMOr can be written as ϕ = P_{x}P_{y}ϕ + (I − P_{x}P_{y})ϕ = P_{x}P_{y}φ_{0} + ϕ_{−} for some φ_{0} ∈ L^{∞} and ϕ_{−} ∈ BMOr ∩ H^{2⊥}. Thus, for every trigonometric polynomial f = p̄ + p̄_{0} + p̄_{1} + p̄_{2}, with p ∈ H^{2}(𝕋^{2}), and ϕ ∈ BMOr, (2.6) yields

\begin{aligned}|∫ ϕf| & = |∫ φ_{0}p̄+∫ ϕ_{−}(p̄_{0} +p̄_{1} +p̄_{2})| ≤ ∥φ_{0}∥_{∞}∥p∥_{1} +\sqrt{2}K∥p_{0} +p_{1} +p_{2}∥_{1} \\ & ≤ (∥ϕ∥_{BMOr} +\sqrt{2}K)(∥P_{x}P_{y}f∥_{1} +∥(I −P_{x}P_{y})f∥_{1}), \end{aligned} (2.7)

where p = P_{x}P_{y}f and p_{0} + p_{1} + p_{2} = (I − P_{x}P_{y})f.

But since the Hilbert transforms, as well as the analytic projections, are unbounded in L^{1}, the norms ∥P_{x}P_{y}f∥_{1} + ∥(I − P_{x}P_{y})f∥_{1} and ∥|f∥| are not comparable (see Proposition 1.4), and there exists for every ε > 0 an f ∈ 𝒫(𝕋^{2}) such that

∥P_{x}P_{y}f∥_{1} + ∥(I − P_{x}P_{y})f∥_{1} < \frac{ε}{1 + \sqrt{2}K} ∥|f∥|.

By Hahn–Banach and the duality of BMOr, there exists ϕ ∈ BMOr such that ∥ϕ∥_{BMOr} ≤ 1 and ∫ ϕf = ∥|f∥|, so that (2.7) implies ∫ ϕf < ε ∫ ϕf, which is a contradiction. □

An important open question is whether for every φ ∈ L^{∞}(𝕋^{2}) there is another ψ ∈ L^{∞}(𝕋^{2}) such that Γ_{ψ} = Γ_{φ} and ∥ψ∥_{∞} ≤ K∥Γ_{φ}∥, where K is a universal constant and Γ_{φ} is the big Hankel operator defined by 𝕋_{φ}f = (I − P)φf. Some geometric properties of H^{∞}(𝕋^{2}) and BMOr make highly improbable a positive answer to this question, which will be considered elsewhere.

3. Interpolation Problems in the Polydisk, Hankel Operators and Pick Matrices

A basic interpolation problem in 𝔻^{d}, for d ≥ 1, is the Pick problem: Given z_{1}, …, z_{n} ∈ 𝔻^{d} and λ_{1}, …, λ_{n} ∈ ℂ, find a necessary and sufficient condition for the existence of an analytic function F on 𝔻^{d} satisfying F (z_{k}) = λ_{k}, for k = 1, …, n, with ∥F ∥_{∞} ≤ 1. This problem can be reformulated in a way that is slightly more general only for d > 1, as follows: Given z_{1}, …, z_{n} ∈ 𝔻^{d} and G ∈ H^{∞}(𝔻^{d}), find an analytic F satisfying F (z_{k}) = G(z_{k}) for k = 1, …, n, and ∥F ∥_{∞} ≤ 1.

In the case of d = 1, the problem was solved by G. Pick in 1916, in terms of the positivity of an associated n × n matrix given by the data. Another solution has been given in terms of the boundedness of an associated Hankel operator given by the data.

Theorem 3.1 (Pick). Given z_{1}, …, z_{n} ∈ 𝔻 and G ∈ H^{∞}(𝔻), the following assertions are equivalent:

(i) The Pick matrix

((1 − G(z_{j})\overline{G(z_{k})})(1 − z_{j}\overline{z_{k}})^{−1})_{j,k=1,...,n} (3.1)

is positive definite. (ii) The Hankel operator Γ_{ϕ} with symbol ϕ = b̄G, where b is the Blaschke product

with simple zeros at z_{1}, …, z_{n}, is bounded, and ∥Γ_{ϕ}∥ ≤ 1. (iii) The Pick problem has a solution.

The equivalence of (i) and (iii) was proved in [P], and that of (ii) and (iii) can be obtained as a corollary of the Nehari theorem (see, for instance, [Ni]).

For d > 1 it is known [Am] that the positivity of the Pick matrix analogous to (3.1) is necessary but not sufficient for the existence of a solution to the Pick problem. Necessary and sufficient conditions involving Pick matrices have been given by Agler for d = 2 [Ag], and by Cole, Lewis and Wermer for all d > 1 [CLW]. However, their conditions are not verifiable in practice. Moreover, in their approach the relation with Hankel operators is lost.

As the Nehari theorem for d > 1 can be recovered by replacing the L^{∞} norm by the BMOr norm, considering the Pick problem with BMOr-norm control allowed us in [CS3] to retain the relation with Hankel operators (within a constant \sqrt{d}), but not a Pick condition.

The following result, which also reduces to the Pick theorem when d = 1, gives necessary and sufficient conditions for the existence of solutions of a coordinate-wise Pick problem in terms of either the boundedness of a Hankel operator with symbol specified by the data, or the positivity of d associated n × n Pick matrices. We state it here only for d = 2, but it holds for all d > 1, with obvious changes.

Theorem 3.2. Given (z_{1}, w_{1}), …, (z_{n}, w_{n}) ∈ 𝔻^{2} and G ∈ H^{∞}(𝔻^{2}), let b = b_{1} ⊗ b_{2}, where b_{1} and b_{2} are finite one-dimensional Blaschke products with simple zeros at z_{1}, …, z_{n} and w_{1}, …, w_{n}, respectively. The following assertions are equivalent (up to a constant \sqrt{2}):

(i) The Pick matrices

((1 − G(z_{j}, y)\overline{G(z_{k}, y)})(1 − z_{j}\overline{z_{k}})^{−1})_{j,k=1,...,n} (3.2)

and

((1 − G(x, w_{j})\overline{G(x, w_{k})})(1 − w_{j}\overline{w_{k}})^{−1})_{j,k=1,...,n} (3.3)

are positive definite for every y ∈ 𝕋 and every x ∈ 𝕋, respectively. (ii) The Hankel operator Γ_{ϕ}, for ϕ = b̄G, is bounded, with ∥Γ_{ϕ}∥ ≤ 1. (iii) There is a function F ∈ H^{2}(𝔻^{2}) satisfying F (z_{k}, w) = G(z_{k}, w) and F (z, w_{k}) =

G(z, w_{k}), for k = 1, …, n, with ∥b̄F ∥_{BMOr} ≤ 1.

(iv) There exist two bounded functions on 𝔻^{2}, F_{1} analytic in z and F_{2} analytic in w, satisfying F_{1}(z_{k}, y) = G(z_{k}, y) and F_{2}(x, w_{k}) = G(x, w_{k}), for k = 1, …, n, with ∥F_{1}∥_{∞} ≤ 1 and ∥F_{2}∥_{∞} ≤ 1.

More precisely, (ii) implies (iii) and (iv), and either (iii) or (iv) imply (ii) with ∥Γ_{ϕ}∥ ≤ \sqrt{2}, while (i) is equivalent to (iv). (Compare [BH].)

Remark. Observe that the loss of the L^{∞} norm in condition (iii) is compensated by the strengthening of the interpolation condition to each variable independently.

Proof. (ii) ⟹ (iii). If ∥Γ_{ϕ}∥ ≤ 1, by Theorem A there exists ψ ∈ BMOr with ∥ψ∥_{BMOr} ≤ 1, such that Γ_{ϕ} = Γ_{ψ}. Therefore ψ − ϕ = ψ − b̄G ∈ H^{2}(𝔻^{2}). Setting F = bψ = G + b_{1}b_{2}h ∈ H^{2}(𝔻^{2}) it is immediate that F satisfies all the conditions of (iii).

(iii) ⟹ (ii). If F satisfies the interpolation conditions (iii), then

F − G = b_{1}(z)h_{1}(z, w) = b_{2}(w)h_{2}(z, w)

for h_{1}, h_{2} ∈ H^{2}(𝔻^{2}). This implies that there is an h ∈ H^{2}(𝔻^{2}) such that F − G = b_{1}(z)b_{2}(w)h(z, w). Thus, setting ψ = b̄F, we have ∥ψ∥_{BMOr} ≤ 1, and, by Theorem A, ∥Γ_{ψ}∥ ≤ \sqrt{2}. But Γ_{ψ} = Γ_{ϕ}, since ψ − ϕ = b̄(F − G) = h, so ∥Γ_{ϕ}∥ ≤ \sqrt{2}.

(ii) ⟹ (iv). By Theorem A(b), the condition ∥Γ_{ϕ}∥ ≤ 1 implies that there exist φ_{1}, φ_{2} ∈ L^{∞}(𝕋^{2}) with ∥φ_{j}∥_{∞} ≤ 1 for j = 1, 2, such that ϕ = b̄G = ϕ_{1} + h_{x} = ϕ_{2} + h_{y} for h_{x} ∈ H_{x}^{2} and h_{y} ∈ H_{y}^{2}. The functions F_{1} = b_{1}φ_{1} and F_{2} = b_{2}φ_{2} satisfy ∥F_{j}∥_{∞} ≤ 1 for j = 1, 2, as well as F_{1}(z_{k}, w) = G(z_{k}, w) and F_{2}(z, w_{k}) = G(z, w_{k}). Moreover, F_{1} = \overline{b_{2}(w)}G(z, w) + b_{1}(z)h_{x}(z, w) is analytic in z, and F_{2} = \overline{b_{1}(z)}G(z, w) + b_{2}(w)h_{y}(z, w) is analytic in w.

(iv) ⟹ (ii). By the interpolation conditions satisfied by F_{1} and F_{2}, for each y ∈ 𝕋 we have G(z, y) − F_{1}(z, y) = b_{1}(z)h_{x}(z, y) for h_{x} ∈ H_{x}^{2}, and, for each x ∈ 𝕋, we have G(x, w)−F_{2}(x, w) = b_{2}(w)h_{y}(x, w) for h_{y} ∈ H_{y}^{2}. Hence, setting φ_{j} = b̄F_{j} for j = 1, 2, we have ∥φ_{j}∥_{∞} ≤ 1, and

\begin{aligned}ϕ − φ_{1} & = b̄(G − F_{1}) = \overline{b_{2}}h_{x} ∈ H_{x}^{2}, \\ ϕ − φ_{2} & = b̄(G − F_{2}) = \overline{b_{1}}h_{y} ∈ H_{y}^{2}. \end{aligned}

Again by Theorem A, this implies ∥Γ_{ϕ}∥ ≤ \sqrt{2}.

(i) ⟺ (iv). Apply Pick’s Theorem to the one-variable functions G(z, ·) and G(·, w) separately. Then the two solutions F_{1}(z, ·) and F_{2}(·, w) satisfy ∥F_{1}(·, w)∥_{∞} ≤ 1 for all w ∈ 𝔻, and ∥F_{2}(z, ·)∥_{∞} ≤ 1 for all z ∈ 𝔻, so that ∥F_{1}∥_{∞} ≤ 1 and ∥F_{2}∥_{∞} ≤ 1. Conversely, observe that (iv) implies the analogues of (3.2) and (3.3) with w ∈ 𝔻 instead of y ∈ 𝕋, and z ∈ 𝔻 instead of x ∈ 𝕋, respectively, which is equivalent to (3.2) and (3.3) by the analyticity of G in both variables. □

Given (z_{1}, w_{1}), …, (z_{n}, w_{n}) ∈ 𝔻^{2} and λ_{1}, …, λ_{n} in ℂ, let b_{1} and b_{2} be the corresponding one-dimensional Blaschke products. Writing λ_{k} = λ′_{k}λ′′_{k}, for k = 1, …, n, set

G_{1}(z) = ∑_{k=1}^{n} \frac{b_{1}(z)}{b′_{1}(z_{k})} \frac{λ′_{k}}{z − z_{k}} and G_{2}(w) = ∑_{k=1}^{n} \frac{b_{2}(w)}{b′_{2}(w_{k})} \frac{λ′′_{k}}{w − w_{k}} ,

so that G_{1}(z_{k}) = λ′_{k}, G_{2}(w_{k}) = λ′′_{k}, and G(z_{k}, w_{k}) = λ_{k} for G(z, w) = G_{1}(z)G_{2}(w). For such G_{1} and G_{2}, or any others satisfying the interpolating conditions, we have:

Corollary 3.3. Given (z_{1}, w_{1}), …, (z_{n}, w_{n}) ∈ 𝔻^{2} and λ_{1}, …, λ_{n} in ℂ, there exists a function F ∈ H^{2}(𝔻^{2}) satisfying F (z_{k}, w) = G(z_{k}, w) and F (z, w_{k}) = G(z, w_{k}), for k = 1, …, n, with ∥b̄F ∥_{BMOr} ≤ 1, as well as two bounded functions on 𝔻^{2}, F_{1} analytic in z and F_{2} analytic in w, satisfying F_{1}(z_{k}, y) = G(z_{k}, y) and F_{2}(x, w_{k}) = G(x, w_{k}), for k = 1, …, n, with ∥F_{1}∥_{∞} ≤ 1 and ∥F_{2}∥_{∞} ≤ 1, whenever the two numerical n × n matrices

((1 − ∥G_{2}∥_{∞}^{2}G_{1}(z_{j})\overline{G_{1}(z_{k})})(1 − z_{j}\overline{z_{k}})^{−1})_{j,k=1,...,n}

and

((1 − ∥G_{1}∥_{∞}^{2}G_{2}(w_{j})\overline{G_{2}(w_{k})})(1 − w_{j}\overline{w_{k}})^{−1})_{j,k=1,...,n}

are positive definite.

4. Hankel Operators of Finite Type and Versions of the Kronecker and AAK Theorems

In the one-dimensional case, once the relation between the bounded Hankel operators and their symbols was established, it was important to characterize the symbols of operators of finite rank. The characterization is given by the Kronecker theorem: A bounded Hankel operator Γ is of finite rank if and only if Γ = Γ_{φ} for φ = b̄h, where b is a finite Blaschke product and h ∈ H^{∞}, so b̄h ∈ H^{∞} +R_{n} (where R_{n} is the class of rational functions with n poles in the disk). Since the range of Γ is finite-dimensional if and only if its kernel has finite codimension, and since this kernel is a subspace of H^{2}(𝕋) invariant under the shift S, the Kronecker theorem can be deduced from the Beurling theorem, asserting that a subspace ℐ ⊂ H^{2}(𝕋) is invariant if and only if ℐ = θH^{2}(𝕋), where θ is an inner function with |θ| ≡ 1, and that an invariant subspace 𝕋 is of finite codimension if and only if θ = b, a finite Blaschke product. The S^{∗}- invariant subspaces, called the model spaces, are of the form K_{θ} = H^{2}(𝕋) ⊖ θH^{2}(𝕋), and K_{θ} is finite-dimensional if and only if θ = b.

Recall that for an operator T and for n ∈ ℕ, the singular numbers of T are defined as

(4.1) s_{n}(T) := inf{∥T − T_{n}∥ : T_{n} of finite rank ≤ n}, which is equivalent to

s_{n}(T) = inf{∥T |_{E}∥ : E of codimension ≤ n}.

Here s_{0}(T) = ∥T ∥ ≥ s_{1} ≥ ⋯ ≥ s_{n} ≥ ⋯, and T is of finite rank if there is an m ∈ ℕ such that s_{n}(T) = 0 for n > m.

A theorem of Adamjan–Arov–Krein [AAK] asserts that for every Hankel operator Γ : H^{2}(𝕋) → H^{2}(𝕋)^{⊥} we have, for n ∈ ℕ,

(4.2) s_{n}(Γ) = inf{∥Γ − Γ_{n}∥ : Γ_{n} Hankel and of finite rank ≤ n}.

This, combined with the Kronecker theorem, gives, for all φ ∈ L^{∞}(𝕋),

s_{n}(Γ_{φ}) = dist(φ, H^{∞} + R_{n}).

An equivalent form of (4.2) was given by S. Treil [T1] as

(4.3) s_{n}(Γ) = inf{∥Γ|_{ℐ}∥ : ℐ invariant under S and codim ℐ ≤ n},

where, by Beurling’s theorem, the subspace ℐ is of the form bH^{2}(𝕋), for b a Blaschke product with n factors.

Through an abstract version of the AAK theorem, in [CS1] it was shown that the situation is radically different for (big) Hankel operators:

Theorem B. For every bounded Γ : H^{2}(𝕋^{2}) → H^{2}(𝕋^{2})^{⊥} and for all n ∈ ℕ, we have

s_{n}(Γ) ≥ 1/\sqrt{2}∥Γ∥.

This theorem implies that all (big) Hankel operators of finite rank are zero, and no satisfactory extension of the AAK theorem can be expected in terms of their singular numbers. This is linked to the fact that, by a theorem of Ahern and Clark [AhC], the subspaces of the form bH^{2}(𝕋^{2}) for b = b_{1} ⊗ b_{2}, with b_{1} and b_{2} finite one-dimensional Blaschke products, are not of finite codimension in H^{2}(𝕋^{2}). However, as shown in [CS4], for these subspaces bH^{2}(𝕋^{2}), it is still true that

{f_{y_{0}}(x) = f(x, y_{0}) : f(x, y) ∈ bH^{2}(𝕋^{2}), y_{0} ∈ 𝕋 fixed}

and

{f_{x_{0}}(y) = f(x_{0}, y) : f(x, y) ∈ bH^{2}(𝕋^{2}), x_{0} ∈ 𝕋fixed}

are finite-codimensional subspaces of H^{2}(𝕋), leading to a notion of subspaces of finite bi-codimension.

A decomposable subspace V of H^{2}(𝕋^{2}), with V = V_{1} ⊗ V_{2}, where V_{1} ⊂ H^{2}(𝕋) and V_{2} ⊂ H^{2}(𝕋), is called of finite bi-codimension (m, n) if and only if codim V_{1} = m and codim V_{2} = n. The orthogonal complement of such V is

V^{⊥} = H^{2}(𝕋^{2}) ⊖ V = V_{1}^{⊥} ⊗ H^{2}(𝕋) + H^{2}(𝕋) ⊗ V_{2}^{⊥},

with V_{k}^{⊥} = H^{2}(𝕋) ⊖ V_{k} for k = 1, 2. Since orthogonal complements of this form will appear again in Section 5 and in other contexts, we give them a name. Given a subspace ℒ ⊂ L^{2}(𝕋) and two positive integers m and n, a subspace W ⊂ L^{2}(𝕋^{2}) is said to be of bi-finite type (ℒ; m, n) if and only if there exist two subspaces W_{1} and W_{2} of H^{2}(𝕋), with dim W_{1} = m and dim W_{2} = n, such that W = W_{1} ⊗ ℒ + ℒ ⊗ W_{2}. With this notation, the orthogonal complement V^{⊥} of a subspace V ⊂ H^{2}(𝕋^{2}) of finite bi-codimension (m, n) is a subspace of bi-finite type (H^{2}(𝕋); m, n).

For these notions we have the following analogue to the Beurling theorem for invariant subspaces of finite codimension in the disk:

Theorem C. [CS4] For a subspace ℐ ⊂ H^{2}(𝕋^{2}), invariant under both shifts, S_{1} and S_{2}, of H^{2}(𝕋^{2}), the following conditions are equivalent:

(a) ℐ is of finite bi-codimension (m, n), that is, ℐ = V_{1} ⊗ V_{2};

(b) ℐ^{⊥} = H^{2}(𝕋^{2}) ⊖ ℐ is of bi-finite type (H^{2}(𝕋); m, n), that is, ℐ^{⊥} = W_{1} ⊗ H^{2}(𝕋) + H^{2}(T) ⊗ W_{2} with dim W_{1} = m and dim W_{2} = n.

(c) ℐ = bH^{2}(𝕋^{2}) = b_{1}H^{2}(𝕋) ⊗ b_{2}H^{2}(𝕋), for b = b_{1} ⊗ b_{2}, that is, b(x, y) = b_{1}(x)b_{2}(y), where b_{1} and b_{2} are one-dimensional Blaschke products with m and n factors, respectively.

Furthermore, W_{k} = V_{k}^{⊥} = H^{2}(𝕋) ⊖ b_{k}H^{2}(𝕋) for k = 1, 2.

A Hankel operator Γ : H^{2}(𝕋^{2}) → H^{2}(𝕋^{2})^{⊥} is called of finite type (m, n) ∈ ℕ^{2} if the kernel of Γ is of finite bi-codimension (m, n).

Theorem 4.1. (Kronecker-type characterization of Hankel operators of finite type) Let Γ : H^{2}(𝕋^{2}) → H^{2}(𝕋^{2})^{⊥} be a bounded Hankel operator. The following conditions are equivalent:

(a) Γ is of finite type (m, n).

(b) Γ = Γ_{ϕ} for ϕ ∈ BMOr ∩ H^{2⊥}, and ϕ = b̄h, where b = b_{1} ⊗ b_{2} for b_{1} and b_{2} one-dimensional Blaschke products with m and n factors, respectively, and h ∈ H^{2}(𝕋^{2}). Moreover, h = bh_{x} + φ_{1} = bh_{y} + φ_{2} for φ_{1}, φ_{2} ∈ L^{∞}(𝕋^{2}), h_{x} ∈ H_{x}^{2}(𝕋^{2}), h_{y} ∈ H_{y}^{2}(𝕋^{2}), so that

ϕ = h_{x} + b̄φ_{1} = h_{y} + b̄φ_{2}.

Proof. The kernel K of Γ is invariant under both shifts S_{1} and S_{2}, since Γ is Hankel and, for f ∈ K, ⟨ΓS_{k}f, g⟩ = ⟨Γf, S_{k}^{−1}g⟩ = 0, for k = 1, 2. Then, by Theorem C, there is b = b_{1} ⊗ b_{2} such that K = bH^{2}(𝕋^{2}). For ϕ ∈ BMOr the H^{2⊥} symbol of Γ, we have ⟨Γf, g⟩ = ∫ fḡϕ, and thus ∫ bfḡϕ = 0 for all f ∈ H^{2} and g ∈ H^{2⊥}, that is, bϕ = h ∈ H^{2} and ϕ = b̄h, and the converse holds. Moreover, since b̄h = ϕ ∈ BMOr, the function h must satisfy h = bh_{x} + φ_{1} = bh_{y} + φ_{2} for φ_{1}, φ_{2} ∈ L^{∞}(𝕋^{2}), h_{x} ∈ H_{x}^{2}, h_{y} ∈ H_{y}^{2}, and the conclusion follows. □

Theorem C suggests, in order to develop a version of the AAK theorem, to replace the ordinary singular numbers of Hankel operators by some σ-numbers defined in analogy with (4.3). For Γ : H^{2}(𝕋^{2}) → H^{2}(𝕋^{2})^{⊥} a Hankel operator and (m, n) ∈ ℕ^{2}, let

(4.4) σ_{mn}(Γ) := inf{∥Γ|_{ℐ}∥ : ℐ ⊂ H^{2}(𝕋^{2}) invariant under S_{1} and S_{2} and of finite bi-codimension (m, n)}.

Equivalently, for ϕ ∈ BMOr,

σ_{mn}(Γ_{ϕ}) = inf{∥Γ_{bϕ}∥: b = b_{1} ⊗ b_{2}, with b_{1} and b_{2} one-dimensional Blaschke products having at most m and n factors, respectively}, (4.5)

since, by Theorem C, the subspaces ℐ in (4.4) can be written as ℐ = bH^{2}(𝕋^{2}) with ∥bh∥_{2} = ∥h∥_{2}, so that

∥Γ_{ϕ}|_{ℐ}∥ = sup_{h} \frac{∥Γ_{ϕ}bh∥_{2}}{∥bh∥_{2}} = sup_{h} \frac{∥Γ_{bϕ}h∥_{2}}{∥h∥_{2}} = ∥Γ_{bϕ}∥.

It is easy to check that the infima are attained in (4.4) and (4.5). Clearly, we have again

σ_{00}(Γ) = ∥Γ∥,
σ_{mn}(Γ) ≥ σ_{(m+1)n}(Γ),
σ_{mn}(Γ) ≥ σ_{m(n+1)}(Γ),
σ_{mn}(Γ) ≤ ∥Γ∥

for all m, n ∈ ℕ.

Corollary 4.2. For every (m, n) ∈ ℕ^{2} there exists a non-zero Hankel operator

Γ : H^{2}(𝕋^{2}) → H^{2}(𝕋^{2})^{⊥}

of finite type (m, n), such that

σ_{pq}(Γ) = 0 for p > m, q > n.

Proof. Take h(x, y) = b_{1}(x)h_{1}(y)+b_{2}(y)h_{2}(y) for b_{1} and b_{2} one-dimensional Blaschke products with at most m and n factors, respectively, and h_{1}, h_{2} ∈ H^{∞}(𝕋). Further take ϕ = b̄h, for b = b_{1} ⊗ b_{2}. Then h ∈ H^{∞}(𝕋^{2}), and

ϕ = \overline{b_{1}(x)}h_{2}(x) + \overline{b_{2}(y)}h_{1}(y) ∈ BMOr ∩ H^{2}(𝕋^{2})^{⊥},

since

P_{−x}ϕ = P_{−x}\overline{b_{1}(x)}h_{2}(x), P_{−y}ϕ = P_{−y}\overline{b_{2}(x)}h_{1}(y),

with b̄_{1}h_{2} and b̄_{2}h_{1} in L^{∞}(𝕋^{2}), and P_{x}P_{y}ϕ = 0 for the right choice of h_{1} and h_{2}. By Theorem (4.1), the Hankel operator Γ = Γ_{ϕ} satisfies the conclusion. □

From Theorem B follows that there are no nonzero compact big Hankel operators, that is, Hankel operators whose sequence of singular numbers tend to zero. Since this corollary says that there are big Hankel operators Γ ≠ 0 with σ_{mn}(Γ) → 0 as m, n → ∞, it is interesting to study the class of such operators, and this will be done elsewhere.

For bounded Hankel operators in the one-dimensional case the AAK theorem asserts that s_{n}(Γ_{φ}) = dist_{L^{∞}}(φ, H^{∞} + R_{n}).

This precise statement does not hold for all bounded (big) Hankel operators Γ = Γ_{ϕ′} given by a symbol ϕ ∈ BMOr, but we still have a substitute by replacing the distance

dist(ϕ, H^{∞} + R_{n}) = inf{∥ϕ − b^{−1}h∥_{∞}: h ∈ H^{∞} = L^{∞} ∩ H^{2},
b = b_{1} ⊗ ⋯ ⊗ b_{d}, with b_{k} a one-dimensional Blaschke product of at most n_{k} factors, for k = 1, …, d}
= inf{∥bϕ − h∥_{∞}: h ∈ H^{∞} = L^{∞} ∩ H^{2}, b}

by

δ(ϕ, BMOAr + R_{n}) := inf{∥bϕ − h∥_{BMOr} : h ∈ BMOAr = BMOr ∩ H^{2},
b = b_{1} ⊗ ⋯ ⊗ b_{d}, with b_{k} a one-dimensional Blaschke product of at most n_{k} factors, for k = 1, …, d}.

Observe that BMOAr = BMOr ∩ H^{2} = BMO ∩ H^{2} = BMOA.

Theorem 4.3. For every ϕ ∈ BMOr(𝕋^{d}) and n ∈ ℕ^{d}, with d > 1, we have

1/\sqrt{d}σ_{n}(Γ_{ϕ}) ≤ δ(ϕ, BMOAr + R_{n}) ≤ σ_{n}(Γ_{ϕ}),

where Γ_{ϕ} is the Hankel operator with symbol ϕ.

Proof. By (4.5), for every ε > 0, there are b_{1} and b_{2}, and b = b_{1} ⊗ b_{2}, such that

σ_{mn}(Γ_{ϕ}) ≤ ∥Γ_{bϕ}∥ ≤ σ_{mn}(Γ_{ϕ}) + ε.

The operator Γ_{bϕ} : H^{2} → H^{2⊥} is also Hankel, and, by Theorem A, Γ_{bϕ} = Γ_{ψ} for some ψ ∈ BMOr, with ψ = bϕ − h, h ∈ H^{2}, and ∥Γ_{bϕ}∥ ≤ \sqrt{2}∥ψ∥_{BMOr}, ∥ψ∥_{BMOr} ≤ σ_{mn}(Γ_{ϕ}) + ε. Hence, 1/\sqrt{2}σ_{mn}(Γ_{ϕ}) ≤ ∥bϕ − h∥_{BMOr} ≤ σ_{mn}(Γ_{ϕ}) + ε, for all ε > 0, which is the conclusion. □

5. Carleson Measures, Model Subspaces of Finite Type, and BMOr Symbols

In the one-dimensional case, there is a close relation linking Hankel operators, BMO functions and Carleson measures. Carleson measures in the disk are those positive measures μ satisfying the Carleson imbedding condition

∫_{𝔻} |f(z)|^{2} dμ(z) ≤ C^{2} ∫_{𝕋} |f(t)|^{2} dt, for all f ∈ H^{2}(𝕋), (5.1)

where f(z) stands for the analytic extension of f to 𝔻.

Carleson characterized those measures as satisfying the tent condition for intervals, that is, μ(S(I)) ≤ C|I| for every interval I, where S(I) is a tent in 𝔻 with base I. Moreover, the H^{2}-imbedding condition (5.1) is equivalent to the H^{p}-imbedding condition being valid for all p such that 1 ≤ p < ∞.

Following Nikolskii and Treil [Ni], [T2], condition (5.1) can be expressed in terms of projectors on one-dimensional model subspaces

K_{z} = K_{b_{z}} = H^{2} ⊖ b_{z}H^{2}

defined by single-factor Blaschke products

b_{z}(ζ) = \frac{|z|}{z} \frac{z − ζ}{1 − ζz̄}.

It is well known that such a subspace K_{z} is spanned by the normalized function

ϕ_{z}(ξ) = \frac{(1 − |z|^{2})^{1/2}}{1 − z̄ξ} for z ∈ 𝔻 and ξ ∈ 𝕋, (5.2)

which has the reproducing property

⟨f, ϕ_{z}⟩ = (1 − |z|^{2})^{1/2}f(z) for all f ∈ H^{2}. (5.3)

Thus, for P_{z} : H^{2}(𝕋) → K_{z} the orthogonal projector, the identity

∥P_{z}f∥_{2}^{2} = (1 − |z|^{2})|f(z)|^{2} (5.4)

holds for all f ∈ H^{2}(𝕋). The Carleson imbedding condition (5.1) can, therefore, be rewritten as

∫_{𝔻} ∥P_{z}f∥_{2}^{2} dν(z) ≤ C^{2}∥f∥_{2}^{2}, for all f ∈ H^{2}(𝕋), (5.5)

with dν(z) = (1 − |z|^{2})^{−1} dμ(z).

Moreover, for P : L^{2} → H^{2} the analytic projector, we have

P_{z}f = b_{z}(I − P)b̄_{z}f = b_{z}Γb̄_{z}f (5.6)

and

∥P_{z}f∥_{2}^{2} = ∥Γb̄_{z}f∥_{2}^{2}. (5.7)

From (5.7) it can be deduced (see the development leading to (5.10) below) that μ ≥ 0 is Carleson if and only if a canonically associated (vector-valued) Hankel operator Γ is bounded, and (through the Nehari theorem) if and only if its antianalytic symbol Γ1 ∈ BMO.

In Section 1 we observed that the different definitions of BMO, which coincide for d = 1, give rise to different classes in 𝕋^{d}, for d > 1. In fact, Chang and Fefferman defined product BMO to circumvent Carleson’s counterexample showing that the class of measures in 𝔻^{2} characterized by the tent condition on rectangles R = I × J does not necessarily satisfy the H^{1}-imbedding condition. Enlarging the class of tents in their definition of product Carleson measures, Chang and Fefferman proved that a function is in product BMO if and only if a canonically associated measure is product Carleson. Here we adopt in 𝔻^{d} the Nikolskii formulation (5.5) and show that a measure is Carleson–Nikolskii if and only if a canonically associated function is BMOr.

In 𝕋^{2} the one-dimensional subspace K_{b_{z}}, where b_{z} is a Blaschke factor, is replaced by K_{b_{z}⊗b_{ζ}} =: K_{zζ}, where b_{z} and b_{ζ} are one-variable Blaschke factors, and K_{zζ} is not a one-dimensional subspace of H^{2}(𝕋^{2}). But now, according to Theorem C in Section 4, K_{zζ} is of bi-finite type (H^{2}(𝕋); 1, 1), and its elements are of the form A(y)ϕ_{z}(x) + B(x)ϕ_{ζ}(y), for A(y) and B(x) varying in H^{2}(𝕋). For the orthogonal projector P_{zζ} from H^{2}(𝕋^{2}) onto K_{zζ} we have the following result.

Lemma 5.1 (Lemma on the projection). If P_{zζ} : H^{2}(𝕋^{2}) → K_{zζ} is the orthogonal projector, we have, for all f ∈ H^{2}(𝕋^{2}),

P_{zζ}f(x, y) = c_{z}f(z, y)ϕ_{z}(x) + c_{ζ}f(x, ζ)ϕ_{ζ}(y) − c_{z}c_{ζ}f(z, ζ)ϕ_{z}(x)ϕ_{ζ}(y), (5.8)

where c_{ν} = (1 − |ν|^{2})^{1/2}, ν ∈ 𝔻. Equivalently,

P_{zζ}f(x, y) = ⟨f, ϕ_{z}⟩_{L^{2}(𝕋_{x})}ϕ_{z}(x) + ⟨f, ϕ_{ζ}⟩_{L^{2}(𝕋_{y})}ϕ_{ζ}(y) − ⟨f, ϕ_{z} ⊗ ϕ_{ζ}⟩_{L^{2}(𝕋^{2})}ϕ_{z}(x)ϕ_{ζ}(y).

Proof. Denoting the right-hand side of (5.8) by g(x, y), for g ∈ K_{zζ}, it remains to check that, for arbitrary A, B ∈ H^{2}(𝕋), we have

\begin{aligned}⟨g(x, y), A(y)ϕ_{z}(x)⟩ & = ⟨f(x, y), A(y)ϕ_{z}(x)⟩, \\ ⟨g(x, y), B(x)ϕ_{ζ}(y)⟩ & = ⟨f(x, y), B(x)ϕ_{ζ}(y)⟩. \end{aligned}

Since, by (5.3), for every F ∈ H^{2}(𝕋^{2}) we have

⟨F (x, y), ϕ_{z}(x)⟩ = c_{z}F (z, y) and ⟨F (x, y), ϕ_{ζ}(y)⟩ = c_{ζ}F (x, ζ), (5.9)

NEHARI–AAK THEORY FOR HANKEL OPERATORS ON THE TORUS 27

and, by (5.2), ∫ |ϕ_{z}(x)|^{2} dx = ∫ |ϕ_{ζ}(y)|^{2} dy = 1, we obtain, as desired,

⟨g(x, y), A(y)ϕ_{z}(x)⟩ = c_{z} ∫ f(z, y)\overline{A(y)} dy + c_{ζ} ∫∫ f(x, ζ)ϕ_{ζ}(y)\overline{A(y)}\overline{ϕ_{z}(x)} dx dy
− c_{z}c_{ζ}f(z, ζ) ∫∫ ϕ_{z}(x)ϕ_{ζ}(y)\overline{A(y)}\overline{ϕ_{z}(x)} dx dy

= c_{z} ∫ f(z, y)\overline{A(y)} dy + c_{ζ}c_{z}f(z, ζ)c_{ζ} \overline{A(ζ)} − c_{z}c_{ζ}f(z, ζ)c_{ζ} \overline{A(ζ)}

= ⟨f(x, y), A(y)ϕ_{z}(x)⟩,

and similarly for the other term. □ Lemma 5.2. For P_{zζ} : H^{2}(𝕋^{2}) → K_{zζ}, where (z, ζ) ∈ 𝔻^{2}, and f ∈ H^{2}(𝕋^{2}), we have

\begin{aligned}∥P_{zζ}f∥_{2}^{2}(1 − |z|^{2})^{−1}(1 − |ζ|^{2})^{−1} & = (1 − |ζ|^{2})^{−1} ∫_{𝕋} |f(z, y)|^{2} dy \\ & + (1 − |z|^{2})^{−1} ∫_{𝕋} |f(x, ζ)|^{2} dx − |f(z, ζ)|^{2}, \end{aligned}

where f(z, y), f(x, ζ) and f(z, ζ) are the analytic extensions of f to z ∈ 𝔻, ζ ∈ 𝔻, and (z, ζ) ∈ 𝔻^{2}, respectively. Proof. From the expression of P_{zζ}f given in (5.8) it follows, using (5.2), (5.3), (5.4) and (5.9), that |P_{zζ}f|^{2} = c_{z}^{2}|f(z, y)|^{2}|ϕ_{z}(x)|^{2} + c_{ζ}^{2}|f(x, ζ)|^{2}|ϕ_{ζ}(y)|^{2} + c_{z}^{2}c_{ζ}^{2}|f(z, ζ)|^{2}|ϕ_{z}(x)|^{2}|ϕ_{ζ}(y)|^{2}

+ 2 Re(c_{z}c_{ζ}f(z, y)\overline{f(x, ζ)}ϕ_{z}(x)\overline{ϕ_{ζ}(y)} + c_{z}^{2}c_{ζ} \overline{f(z, ζ)}f(z, y)|ϕ_{z}(x)|^{2}ϕ_{ζ}(y)

−c_{z}c_{ζ}^{2}f(z, ζ)\overline{f(x, ζ)}ϕ_{z}(x)|\overline{ϕ_{ζ}(y)}|^{2}).

Then,

\begin{aligned}∥P_{zζ}f∥_{2}^{2} & = ∫∫ |P_{zζ}f(x, y)|^{2} dx dy \\ & = c_{z}^{2} ∫ |f(z, y)|^{2} dy +c_{ζ}^{2} ∫ |f(x, ζ)|^{2} dx+c_{z}^{2}c_{ζ}^{2}|f(z, ζ)|^{2} −2 Re(c_{z}^{2}c_{ζ}^{2}|f(z, ζ)|^{2}), \end{aligned}

so

(c_{z}^{2}c_{ζ}^{2})^{−1}∥P_{zζ}f∥_{2}^{2} = c_{ζ}^{−2} ∫ |f(z, y)|^{2} dy + c_{z}^{−2} ∫ |f(x, ζ)|^{2} dx − |f(z, ζ)|^{2},

which is the conclusion. □

Following Nikolskii’s approach, we say that a measure μ ≥ 0 defined in 𝔻^{2} is Carleson–Nikolskii if

dν(z, ζ) = (1 − |z|^{2})^{−1}(1 − |ζ|^{2})^{−1}dμ(z, ζ)

satisfies

∫∫ _{𝔻^{2}} ∥P_{zζ}f∥_{2}^{2} dν(z, ζ) ≤ C^{2}∥f∥_{2}^{2} for all f ∈ H^{2}(𝕋^{2}).

Formula (5.6) is still valid in H^{2}(𝕋^{2}), that is, for all f ∈ H^{2}(𝕋^{2}) we have

P_{zζ}f = (b_{z} ⊗ b_{ζ})(I − P)(b̄_{z} ⊗ b̄_{ζ})f = (b_{z} ⊗ b_{ζ})Γ_{b̄_{z}⊗b̄_{ζ}}f.

Thus, again we have

∥P_{zζ}f∥_{2}^{2} = ∥Γ_{b̄_{z}⊗b̄_{ζ}}f∥_{2}^{2}.

Therefore, a measure μ is Carleson–Nikolskii if and only if

∫∫ _{𝔻^{2}} ∥Γ_{b̄_{z}⊗b̄_{ζ}}f∥_{2}^{2} dν ≤ C^{2}∥f∥_{2}^{2} for all f ∈ H^{2}(𝕋^{2}).

Let Γ : H^{2}(𝕋^{2}) → L^{2}(𝔻^{2}, ν; H^{2}(𝕋^{2})^{⊥}) be the operator assigning to each f ∈ H^{2}(𝕋^{2}) the function

(z, ζ) ↦ Γ_{b̄_{z}⊗b̄_{ζ}}f ∈ H^{2}(𝕋^{2})^{⊥},

so that μ is of Carleson type if and only if Γ is bounded in L^{2}(𝔻^{2}, ν), with ∥Γ∥ ≤ C. By Fubini’s theorem, the space L^{2}(𝔻^{2}, ν; H^{2}(𝕋^{2})^{⊥}) of square integrable functions in the bidisk, with values in H^{2}(𝕋^{2})^{⊥}, is isometrically isomorphic to the space H^{2}(𝕋^{2})^{⊥}(L^{2}(𝔻^{2}, ν)) of antianalytic functions with values in L^{2}(𝔻^{2}, ν). Under this isomorphism the operator Γ corresponds to the operator

Γ⃗ : H^{2}(𝕋^{2}) → H^{2}(𝕋^{2})^{⊥}(L^{2}(𝔻^{2}, ν)). (5.10)

This Γ⃗ is a (vector-valued) big Hankel operator, since, for k = 1, 2, we have

Γ⃗S_{k}f = F (x, y; z, ζ) = Γ_{b̄_{z}⊗b̄_{ζ}}(S_{k}f)(x, y) = (I − P)S_{k}Γ_{b̄_{z}⊗b̄_{ζ}}f = (I − P)S_{k}Γ⃗f.

The operator Γ⃗ is called the Hankel operator canonically associated to μ. Theorem A (which can be used since its proof through abstract liftings extends to Hankel operators from the scalar spaces H^{2}(𝕋^{2}) to a vector-valued H^{2}(𝕋^{2})^{⊥}(ℋ), where ℋ is a Hilbert space) applied to Γ⃗ yields:

Theorem 5.3. A measure μ ≥ 0 in 𝔻^{2} is of Carleson type, with constant C, if and only if the canonically associated operator Γ⃗ is bounded with norm ∥Γ⃗∥ = C, and if and only if Γ⃗1 ∈ BMOr(𝕋^{2}; L^{2}(𝔻^{2}, ν)), with norm ≅ C.

Thus the connection between measures satisfying the Carleson imbedding condition, Hankel operators and BMO(𝕋), is recovered in 𝕋^{2} in terms of BMOr.

6. Estimates for the Norm of the Hankel Operators of Finite Type

Let us recall some basic properties of the finite-dimensional model subspaces K_{b} ⊂ H^{2}(𝕋), where b is a finite Blaschke product, which include as a special case the properties of the K_{z} considered in Section 5. For a finite Blaschke product b with simple zeros in 𝔻, we again denote by P_{b} the orthogonal projector from H^{2}(𝕋) onto K_{b} = H^{2}(𝕋)⊖bH^{2}(𝕋), and define the model operator T_{b} : K_{b} → K_{b} by T_{b} := P_{b}S|K_{b}, so that T_{b}^{∗} = S^{∗}|K_{b}. Similarly, for each G ∈ H^{∞}(𝕋), G(T_{b}) is defined by G(T_{b})f := P_{b}Gf.

If ϕ_{z} is given by (5.2), then, for each z ∈ 𝔻, ϕ_{z} is an eigenfunction of S^{∗}, and if z_{1}, …, z_{m} ∈ 𝔻 are the zeros of b, then {ϕ_{z_{1}}, …, ϕ_{z_{m}}} is a basis of K_{b} composed of eigenfunctions of T_{b}^{∗}. Similarly, K_{b} has a basis {ψ_{z_{1}}, …, ψ_{z_{m}}}, of eigenfunctions of T_{b}, where

ψ_{z}(ξ) = b(ξ − z)^{−1}, T_{b}ψ_{z} = zψ_{z}. (6.1)

Thus T_{b} and T_{b}^{∗} are multiplier operators, that is, they are given by diagonal finite matrices in the corresponding bases, so that, for each G ∈ H^{∞}(𝕋), the condition

∥G(T_{b})∥ ≤ 1

is equivalent to the positive definiteness of the associated Pick matrix

((1 − G(z_{j})\overline{G(z_{k})})(1 − z_{j}\overline{z_{k}})^{−1})_{j,k=1,...,m}.

The Kronecker theorem characterizes the symbols of the Hankel operators Γ : H^{2}(𝕋) → H^{2}(𝕋)^{⊥} of finite rank n as those of the form b̄G, for b a Blaschke product with n factors and G ∈ H^{∞}(𝕋). Since the projector P_{b} is related to the analytic projector P : L^{2} → H^{2} by

P_{b}f = b(I − P)b̄f = bΓb̄f, (6.2)

we derive the identities

|Γ_{b̄G}f| = |Γb̄Gf| = |P_{b}Gf| = |G(T_{b}f)|,

and thus

∥Γ_{b̄G}∥ = ∥G(T_{b})∥.

This means that, in the circle, the norm of a Hankel operator of finite rank is equal to the norm of an associated multiplier operator acting in finite-dimensional K_{b}, which in turn is determined by a finite Pick matrix.

The same result holds for Hankel operators of finite type in the torus (see Theorem 6.2 below), but the association with the multiplier operators acting in K_{b} is not so simple. This is due to the fact that here K_{b}, for b the tensor product of d Blaschke products, is not finite-dimensional but of multiple-finite type. As before, we present here the case d = 2.

In 𝕋^{2}, if we restrict ourselves to the case when b_{1} and b_{2} have the same number of zeros, at z_{1}, …, z_{n} and w_{1}, …, w_{n}, respectively, and when G = G_{1} ⊗ G_{2}, with G_{1}, G_{2} ∈ H^{∞}(𝕋), we have the following equivalences, in terms of

K_{b_{1}b_{2}} = H^{2}(𝕋^{2}) ⊖ (b_{1} ⊗ b_{2})H^{2}(𝕋^{2}) (6.3)

and

K′_{b_{1},b_{2}} = \overline{H^{2}(𝕋^{2})^{⊥}} ⊖ [(b_{1} ⊗ b_{2})H^{2}(𝕋^{2}) ⊕ b_{1}(H_{x}^{2} ∩ H_{−y}^{2}) ⊕ b_{2}(H_{−x}^{2} ∩ H_{y}^{2})] (6.4)

subspaces of finite type (H^{2}(𝕋); n, n) and (L^{2}(𝕋); n, n), respectively, as follows from Theorem C and from Theorem 2 in [CS4] (see Section 4).

Proposition 6.1. Given ϕ = (b̄_{1} ⊗ b̄_{2})(G_{1} ⊗ G_{2}), for b_{1}, b_{2}, G_{1} and G_{2} as above, and K_{b_{1}b_{2}} and K′_{b_{1}b_{2}} defined by (6.3) and (6.4), the following conditions are equivalent:

(a) ∥Γ_{ϕ}∥ ≤ 1, that is, ∥Γ_{ϕ}f∥_{L^{2}} ≤ ∥f∥_{L^{2}}, for all f ∈ H^{2}(𝕋^{2}).

(b) ∥Γ_{ϕ}|K_{b_{1}b_{2}}∥ ≤ 1, that is, ∥Γ_{ϕ}e∥_{L^{2}} ≤ ∥e∥_{L^{2}}, for all e ∈ K_{b_{1}b_{2}}.

(c) For all e ∈ K_{b_{1}b_{2}} and e′ ∈ K′_{b_{1}b_{2}}, the inequality

|∫∫ eēϕ dx dy| ≤ ∥e∥_{L^{2}}∥e′∥_{L^{2}}

holds.

Proof. (a) ⟺ (b). Every f ∈ H^{2}(𝕋^{2}) can be written as an orthogonal sum

f(x, y) = b_{1}(x)b_{2}(y)h(x, y) + e(x, y), with h ∈ H^{2}(𝕋^{2}) and e ∈ K_{b_{1}b_{2}},

and

\begin{aligned}Γ_{ϕ}(b_{1} ⊗ b_{2})h & = (I − P)(b_{1}(x)b_{2}(y)h(x, y)\overline{b_{1}(x)}\overline{b_{2}(y)}G_{1}(x)G_{2}(y)) \\ & = (I − P)(h(x, y)G_{1}(x)G_{2}(y)) = 0, \end{aligned}

since hG_{1}G_{2} ∈ H^{2}(𝕋^{2}). Thus, Γ_{ϕ}f = Γ_{ϕ}e, ∀f ∈ H^{2}(𝕋^{2}) and ∥Γ_{ϕ}f∥_{2} = ∥Γ_{ϕ}e∥_{2} ≤ ∥e∥_{2} ≤ ∥f∥_{2}, so (b) implies (a). The converse follows from K_{b_{1}b_{2}} ⊂ H^{2}(𝕋^{2}).

(a) ⟺ (c). Similar proof, observing the equivalence of (a) with

|∫∫ f(x, y)\overline{g(x, y)}ϕ(x, y) dx dy| ≤ ∥f∥_{2}∥g∥_{2}

for all f ∈ H^{2}(𝕋^{2}) and g ∈ \overline{H^{2}(𝕋^{2})^{⊥}}, and writing in terms of the decomposition of H^{2}(𝕋^{2})^{⊥} in the direct sum of K′_{b_{1}b_{2}} and its orthogonal complement. □

Proposition 6.1 says that ∥Γ_{ϕ}∥ = ∥Γ_{ϕ}|K_{b_{1}b_{2}}∥, and we will prove that ∥Γ_{ϕ}|K_{b_{1}b_{2}}∥ coincides with the norm of a multiplier operator in K_{b_{1}b_{2}} (see Theorem 6.2 below), thus generalizing the one-dimensional results.

The systems of eigenfunctions {ψ_{z_{1}}, …, ψ_{z_{n}}} and {ψ_{w_{1}}, …, ψ_{w_{n}}}, where {z_{1}, …, z_{n}} and {w_{1}, …, w_{n}} are the zeros of b_{1} and b_{2}, are bases for K_{b_{1}} and K_{b_{2}}, respectively. Through Theorem C of Section 4, this allows to write the elements e ∈ K_{b_{1}b_{2}} as

e(x, y) = ∑_{i=1}^{n} A_{i}(y)ψ_{z_{i}}(x) + ∑_{j=1}^{n} B_{j}(x)ψ_{w_{j}}(y) (6.5)

where, for i, j = 1, …, n, we have A_{i}, B_{j} ∈ H^{2}(𝕋). In what follows we write, for simplicity,

ψ_{z_{i}}(x) = ξ_{i}(x) and ψ_{w_{j}}(y) = η_{j}(y) for i, j = 1, …, n,

and remark that the ξ′_{i}s are eigenfunctions of the model operator T_{b_{1}}, for which, by (6.1), T_{b_{1}}ξ_{i} = z_{i}ξ_{i}, and the η′_{j}s are eigenfunctions of T_{b_{2}}, with T_{b_{2}}η_{j} = w_{j}η_{j}. For their part, each A_{i}(y), B_{j}(x) can be written as

\begin{aligned}A_{i}(y) & = b_{2}(y)h′′_{i}(y) + ∑_{k} c_{ik}η_{k}(y), \\ B_{j}(x) & = b_{1}(x)h′_{j}(x) + ∑_{l} d_{jl}ξe(x) \end{aligned} (6.6)

with h′′_{i}, h′_{j} ∈ H^{2}(𝕋), for i, j = 1, …, n. Moreover

Γ_{ϕ}e = (I − P)ϕe = (P_{−y} + P_{−x}P_{y})ϕe = (P_{−x} + P_{x}P_{−y})ϕe,

and, by (6.2) and the definitions of 𝕋_{b} and G(𝕋_{b}), we get

\begin{aligned}P_{−x}(b̄_{1}G_{1}ξ_{i})(x) & = b̄_{1}T_{b_{1}}G_{1}ξ_{i}(x) = b̄_{1}G_{1}(z_{i})ξ_{i}(x), \\ P_{−y}(b̄_{2}G_{2}η_{j})(y) & = b̄_{2}G_{2}(w_{j})η_{j}(y). \end{aligned}

Since P_{x}b̄_{1}G_{1}ξ_{i} = b̄_{1}G_{1}ξ_{i} − P_{−x}b̄_{1}G_{1}ξ_{i}, we have

\begin{aligned}P_{x}\overline{b_{1}(x)}G_{1}(x)ξ_{i}(x) & = \overline{b_{1}(x)}(G_{1}(x) − G_{1}(z_{i}))ξ_{i}(x), \\ P_{y}\overline{b_{2}(y)}G_{2}(y)η_{j}(y) & = \overline{b_{2}(y)}(G_{2}(y) − G_{2}(w_{j}))η_{j}(y). \end{aligned}

Now for every e ∈ K_{b_{1}b_{2}}, we can write Γ_{ϕ}e in terms of functions expressible by b_{1}, b_{2}, G_{1} and G_{2}. By (6.5) and (6.6), every e ∈ K_{b_{1}b_{1}} has the expression

e(x, y) =_{i} ∑_{i,j=1}^{n} c_{ij}ξ_{i}(x)η_{j}(y) + ∑_{i=1}^{n} b_{2}(y)h′′_{i}(y)ξ_{i}(x) + ∑_{j=1}^{n} b_{1}(x)h′_{j}(x)_{j}(y), (6.7)

where, for i, j = 1, …, n, we have c_{ij} ∈ ℂ and h′_{j}, h′′_{j} ∈ H^{2}(𝕋) are one-variable functions. From all the above and the fact that e ∈ K_{b_{1}b_{2}}, we have

\begin{matrix}Γ_{ϕ}e = \overline{b_{1}(x)}\overline{b_{2}(x)}(∑_{i,j=1}^{n}(G_{1}(x)G_{2}(w_{j})+G_{1}(z_{i})G_{2}(y)−G_{1}(z_{i})G_{2}(w_{j}))·ξ_{i}(x)η_{j}(y)c_{ij} \\ + ∑_{i=1}^{n} G_{1}(z_{i})G_{2}(y)ξ_{i}(x)b_{2}(y)h′′_{i}(y)+∑_{j=1}^{n} G_{1}(x)G_{2}(w_{j})b_{1}(x)η_{j}(y)h′_{j}(x)). \end{matrix} (6.8)

Remark. Expressions (6.7)–(6.8) allow us to check that the three equivalent conditions of Proposition 6.1 are also equivalent to the positive-definiteness of a finite Pick matrix, defined in terms of b_{1}, b_{2}, G_{1} and G_{2}, whose elements are bounded operators acting in H^{2}(𝕋), L^{2}(𝕋) or from H^{2}(𝕋) to L^{2}(𝕋).

Expression (6.7) shows that K_{b_{1}b_{2}} = K^{0} ⊕ K^{1} ⊕ K^{2}, where K^{0} is the direct sum of the n^{2} one-dimensional spaces ℂξ_{i}(x)η_{j}(y), K^{1} is the direct sum of the n subspaces b_{2}(y)ξ_{i}(x)H^{2}(𝕋), and K^{2} is the direct sum of the n subspaces b_{1}(x)η_{j}(y)H^{2}(𝕋).

Whenever F^{1}, F^{2}, F^{3}, F^{4} ∈ H^{∞}(𝕋), it is clear that, for i, j = 1, …, n, we have

\begin{aligned}(F^{1}(x) + F^{2}(y))ξ_{i}(x)η_{j}(y) & ∈ K_{b_{1}b_{2}}, \\ F^{3}(y)ξ_{i}(x)b_{2}(y)H^{2}(𝕋) & ⊂ K_{b_{1}b_{2}}, \\ F^{4}(x)η_{j}(y)b_{1}(x)H^{2}(𝕋) & ⊂ K_{b_{1}b_{2}}. \end{aligned}

Accordingly, we say that an operator T : K_{b_{1}b_{2}} → K_{b_{1}b_{2}} is a multiplier in K_{b_{1}b_{2}} if, for i, j = 1, …, n,

(6.9)  T ξ_{i}(x)η_{j}(y) = (F_{ij}^{1}(x) + F_{ij}^{2}(y))ξ_{i}(x)η_{j}(y),
(6.10)  T ξ_{i}(x)b_{2}(y)h′′(y) = F_{i}^{3}(y)ξ_{i}(x)b_{2}(y)h′′(y) for h′′ ∈ H^{∞}(𝕋),
(6.11)  T η_{j}(y)b_{1}(x)h′(x) = F_{j}^{4}(x)ξ_{1}(x)η_{j}(y)h′(x) for h′ ∈ H^{∞}(𝕋).

The development above implies the following result:

Theorem 6.2. Given two one-dimensional Blaschke products b_{1} and b_{2}, with simple zeros at z_{1}, …, z_{n} and w_{1}, …, w_{n}, respectively, and given G_{1}, G_{2} ∈ H^{∞}(𝕋), let ϕ = (b̄_{1}⊗b̄_{2})(G_{1}⊗G_{2}). If Γ_{ϕ} is the Hankel operator defined by symbol ϕ, then ∥Γ^{ϕ}∥ = ∥Γ_{ϕ}∥, where Γ^{ϕ} is the multiplier in K_{b_{1}b_{2}} (in the sense of (6.9)–(6.11)) defined by

\begin{aligned}F_{ij}^{1}(x) + F_{ij}^{2}(y) & = G_{1}(x)G_{2}(w_{j}) + G_{1}(z_{i})G_{2}(y) − G_{1}(z_{i})G_{2}(w_{j}), \\ F_{i}^{3}(y) & = G_{1}(z_{i})G_{2}(y) \\ F_{j}^{4}(x) & = G_{1}(x)G_{2}(w_{j}) \end{aligned}

for i, j = 1, …, n.

Theorem 6.2 has a valid formulation in 𝕋^{d}, for d ≥ 1. For d = 1 it reduces to the Pick formula.

Theorem 6.2 allows us to write the boundedness condition ∥Γ_{ϕ}∥ ≤ 1 as a formula of Pick matrix type, but more complicated than in the one-dimensional case, and we will not go into the details here. Still, remark that the verification of boundedness of the norm of Γ^{ϕ} is not as involved as that for the restriction of Γ_{ϕ} to the model subspace K_{b_{1}b_{2}} (condition (b) of Proposition 6.1), since it is done through the defining properties (6.9)–(6.11) of multipliers.

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Mischa Cotlar, Facultad de Ciencias, Universidad Central de Venezula, Caracas 1040, Venezuela

  E-mail address: cs@@scs.howard.edu

Cora Sadosky, Department of Mathematics, Howard University, Washington, DC 20059, USA

  E-mail address: mcotlar@@dino.conicit.ve