Nehari’s theorem for convex domain Hankel and Toeplitz operators in several variables

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Several Variables,”

International Mathematics Research Notices, Vol. 2021, No. 5, pp. 3331–3361 Advance Access Publication October 31, 2019

doi:10.1093/imrn/rnz193

Nehari’s Theorem for Convex Domain Hankel and Toeplitz Operators in Several Variables

Marcus Carlsson

^1,∗

and Karl-Mikael Perfekt

²

1

Centre for Mathematical Sciences, Lund University Box 118, SE-22100, Lund, Sweden and

²

Department of Mathematics and Statistics,

University of Reading, Reading RG6 6AX, United Kingdom

∗Correspondence to be sent to: e-mail: [email protected]

We prove Nehari’s theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley–Wiener space, reads as follows. Let = (0, 1)^d be ad-dimensional cube, and for a distributionf on 2, consider the Hankel operator

_f(g)(x)=

f(x+y)g(y)dy, x∈.

Then _f extends to a bounded operator on L²() if and only if there is a bounded function b on R^d whose Fourier transform coincides with f on 2. This special case has an immediate application in matrix extension theory: every finite multilevel block Toeplitz matrix can be boundedly extended to an infinite multilevel block Toeplitz matrix. In particular, block Toeplitz operators with blocks that are themselves Toeplitz can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.

1 Introduction

For an open connected set⊂R^d,d≥1, let

=+= {x+y : x∈,y∈},

Received December 7, 2018; Revised July 8, 2019; Accepted July 10, 2019

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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and consider a distribution f defined on . The associated general domain Hankel operator_f =_f_,is the (densely defined) operator_f: L²()→L²(), given by

_f(g)(x)=

f(x+y)g(y)dy, x∈,

where dyis the Lebesgue measure onR^d.

The case = R₊ = (0,∞)ford = 1 corresponds to the class of usual Hankel operators; when represented in the appropriate basis of L²(R₊), the operator _f_,_R

+ is realized as an infinite Hankel matrix {a_n₊_m}^∞_n,m=0 [31, Ch. 1.8]. Nehari’s theorem[25]

characterizes the bounded Hankel matrices of this type, but it has an equivalent version for operators of the type_f: L²(R₊)→ L²(R₊), which reads as follows (we again refer to [31, Ch. 1.8], Theorem 8.1). For a functiong onR^d, we letgˆ =Fgdenote its Fourier transform,

ˆ

g(ξ )=Fg(ξ )=

R^dg(x)e⁻^2πix^·^ξdx, ξ ∈R^d.

Theorem. Suppose that f is a distribution in R₊, f ∈ D(R₊). Then _f: L²(R₊) → L²(R₊)is bounded if and only if there exists a functionb ∈ L^∞(R)such that b|ˆ _R₊ =f. Moreover, it is possible to choosebso that

_f = b _L_∞. (1.1)

Nehari’s theorem is canonical in operator theory. The two most common proofs proceed either by factorization in the single variable Hardy space or by making use of the commutant lifting theorem.

Ford>1, the operators_f_,_Rd

+, =R^d₊, correspond to (small) Hankel operators on the product domain multi-variable Hardy space H_d². In this case, the analogue of Nehari’s theorem remains true, apart from (1.1), but it is significantly more difficult to prove. It was established by Ferguson and Lacey (d= 2) and Lacey and Terwilleger (d>2) [18,23]. A precise statement is given in Theorem2.1.

The main purpose of this article is to prove Nehari’s theorem when⊂R^dis a simple convex polytope. Whenis convex note that+=2.

Theorem 1.1. Let be a simple convex polytope, and letf ∈ D()where = 2.

Then_f: L²() → L²()is bounded if and only if there is a functionb ∈ L^∞(R^d)such thatbˆ| =f. There exists a constant c>0, depending on, such thatbcan be chosen

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to satisfy

c b _L∞ ≤ _f ≤ b _L∞.

When d = 1, the only open connected sets ⊂ Rare the intervals = I. In this case, Theorem1.1is due to Rochberg [35], who called the corresponding operators _f_,I Hankel/Toeplitz operators on the Paley–Wiener space. They have also been called Wiener–Hopf operators on a finite interval [30]. These operators have inspired a wealth of theory in the single variable setting—see Section2.5, where we shall interpret Theorem1.1in the context of Paley–Wiener spaces.

Even ford=1, our proof of Theorem1.1appears to be new. However, in several variables our proof relies on the Nehari theorem of Ferguson–Lacey–Terwilleger and can therefore not be used to give a new proof of their results.

We shall also consider general domain Toeplitz operators_f = _f_,:L²()→ L²(). In this context,f is a distribution defined on=−, and_f is densely defined via

_f(g)(x)=

f(x−y)g(y)dy, x∈.

If after a translation is invariant under the ref lection x → −x, then the classes of Hankel operators_f_, and Toeplitz operators_f_, are essentially the same, and Theorem1.1immediately yields a boundedness result. This reasoning is applicable to the cube=(0, 1)^d, for example.

Corollary 1.2. Let be a simple convex polytope such that for somez ∈ R^d it holds that+z= −−z. Letf ∈D(),=−=2+2z. Then_f is bounded if and only if there exists a functionb ∈ L^∞(R^d)such thatbˆ| = f. There exists a constantc >0, depending on, such thatbcan be chosen to satisfy

c b _L∞ ≤ _f ≤ b _L∞.

On the other hand, when is a proper convex unbounded set, containing an open cone say, it is clear that the boundedness characterizations of _f_, and _f_, may be completely different; plainly explained by the fact that = − = R^d in the Toeplitz case, while = + = 2 R^d for Hankel operators. In this setting, identifying the boundedness of _f carries none of the subtleties of Nehari- type theorems. In Theorem6.1we obtain the expected boundedness result for a class

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of “cone-like” domains. Rather than giving a precise statement here, let us record the following corollary of Theorem6.1.

Corollary 6.2. Let⊂R^dbe any open connected domain such that (1,∞)^d⊂⊂(0,∞)^d,

and letf ∈ D(R^d). Then_f:L²() → L²()is bounded if and only iff is a tempered distribution and ˆf _L_∞₍_Rd) <∞, and in this case

_f = ˆf _L_∞.

In the final part of the paper we shall give an application of Theorem1.1 to matrix completion theory, essentially obtained by discretizing Corollary1.2whenis a cube. To avoid introducing further notation, we shall only state the result in words for now. Recall that a Toeplitz matrix is one whose diagonals are constant. AnN×N d-multilevel block Toeplitz matrix is an N×N Toeplitz matrix whose entries are N× N (d−1)-multilevel block Toeplitz matrices. Here N could be finite or infinite. A 1- multilevel block Toeplitz matrix is simply an ordinary Toeplitz matrix. A 2-multilevel block Toeplitz matrix is what is usually considered a block Toeplitz matrix where each block itself is Toeplitz.

Theorem 7.1. Every finiteN×N d-multilevel block Toeplitz matrix can be extended to an infinited-multilevel block Toeplitz matrix bounded on², with a constant that only depends on the dimensiond.

For scalar Toeplitz matrices (d = 1) this result is well known [5, 26, 36, 38], although not as firmly cemented in the literature as the Nehari theorem itself; see [28, Ch. V.2, V.8] for a proof based on Parrot’s lemma and a discussion of the result’s history. Ford = 1, the converse deduction of Theorem1.1starting from Theorem 7.1 can be found in [13].

The paper is laid out as follows. In Section2 we will give a more formal background and introduce necessary notation. We will also discuss the relationship between _f_,, Paley–Wiener spaces, and co-invariant subspaces of the Hardy spaces.

In Section3we will prove approximation results for distribution symbols with respect to Hankel and Toeplitz operators, allowing us to reduce to smooth symbols. Section4 brief ly outlines what we need to know about convex sets and polytopes. In Section5

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we prove Theorem1.1, our Nehari theorem for Hankel operators. We also indicate how the proof extends to certain unbounded polyhedral domains. In Section6 our main result on Toeplitz operators is shown, Theorem6.1. Finally, Section7gives the proof of Theorem7.1.

2 Further background and related results 2.1 Hankel operators on multi-variable Hardy spaces

Let us begin by placing Hankel operators _f into the context of classical Hankel operators on Hardy spaces. As before, for g ∈ L²(R^d), let gˆ = Fg denote its Fourier transform,

ˆ

g(ξ )=Fg(ξ )=

R^dg(x)e^−2πix·^ξdx, ξ ∈R^d.

For the inverse transform we writeF⁻¹(g)= ˇg. The product domain Hardy spaceH_d² is the proper subspace ofL²(R^d)of functions whose Fourier transforms are supported in the coneR^d₊,R₊=(0,∞),

H_d²=

G∈L²(R^d) : suppGˆ ⊂R^d₊ .

We letP_d:L²(R^d)→H_d² denote the orthogonal projection and letJ:L²(R^d)→L²(R^d)be the involution defined byJG(x)=G(−x),x∈R.

Consider_f =_f_,for=R^d₊withf ∈L²(R^d₊). For a dense set ofg,h∈ L²(R^d₊) we have that

_fg,h_L2(R^d₊)= ˇf Jg,ˇ hˇ _H²

d. (2.1)

It follows that the (possibly unbounded) operator _f: L²(R^d₊) → L²(R^d₊) is unitarily equivalent to the small Hankel operatorZ_f_ˇ:H_d²→H_d²,

Z_f_ˇG=P_d(fˇ·JG).

Note that anybsuch thatb|ˆ _R^d

+=f generates the same Hankel operator asfˇ,Z_b=Z_f_ˇ. To justify the above computation easily we assumed that f ∈ L²(R^d₊). An approximation argument is needed to consider general symbols f, which may only be distributions inR^d₊. We provide this later in Proposition3.2. We can then read off the boundedness of_f from the boundedness of the corresponding Hankel operator onH_d². When d = 1 and = = R₊, the analogue of Theorem1.1 is exactly the classical

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Nehari theorem. In higher dimensions the corresponding theorem is due to Ferguson–

Lacey–Terwilleger [18,23]. In our notation, their results read as follows.

Theorem 2.1. Suppose==R^d₊and thatf is a distribution inR^d₊,f ∈D(R^d₊). Then _f:L²(R^d₊)→L²(R^d₊)is bounded if and only if there exists a functionb∈ L^∞(R^d)such thatbˆ|_R^d

+ =f. Moreover, there exists a constantc>0, depending ond, such thatbcan be chosen to satisfy

c b _L_∞ ≤ _f ≤ b _L_∞. (2.2) For d > 1 it is not possible to take c = 1 in (2.2), see for example [29]. This result, as stated in [18,23], requires thatf ∈L²(R^d₊). The extension to the more general situation considered here is a technicality, but for completeness the details are provided in Section3.

2.2 Hankel operators on bounded domains

We now discuss bounded domains, the setting of our main result. The only convex bounded domains inRare the intervalsI ⊂ R. Translations, dilations, and ref lections carry the operator_f_,I onto_f_˜_,J, whereJ ⊂R is any other interval andf˜ arises from transforming f appropriately. In one variable it thus suffices to consider operators _f_,(0,1) where = (0, 1). Rochberg [35] called these operators Hankel operators on the Paley–Wiener space and proved Theorem1.1in the one-dimensional case.

In the same article [35], it is posed as an open problem to characterize the bounded Hankel operators _f_, when is a disc inR². We are not able to settle this question, but Theorem1.1does provide the answer when = (0, 1)^d is a cube inR^d. As we will see, the Hankel operators_f_,(0,1)d constitute a natural generalization of the Hankel operators on the Paley–Wiener space. On a technical level, the reason that we are able to prove Theorem1.1whenis a simple convex polytope, but not whenis a ball, is that we rely on Theorem2.1. In applying Theorem2.1 to our situation, the corners of the boundary ofare actually of help rather than hindrance. We consider the case of a ball to be an interesting open problem for which we do not dare to make a firm conjecture. In view of Fefferman’s disproof of the disc conjecture [17], Nehari theorems might turn out to be quite different for balls and polytopes.

2.3 Toeplitz operators

When d = 1 and = R₊, = R, the operators _f are known as Wiener–Hopf operators [11, Ch. 9]. Analogously with Hankel operators, these can be shown to be

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unitarily equivalent to Toeplitz matrix operators on²(N). In this case the boundedness characterization is easy to both state and prove

_f = ˆf _L∞. (2.3)

In Theorem6.1 we extend (2.3) to Toeplitz operators _f_, for a class of “cone-like”

domains⊂R^d, for which=−=R^d. 2.4 Truncated correlation operators

For open connected sets,ϒ ⊂ R^d it is also convenient to introduce the more general

“truncated correlation operators”_f_,ϒ,:L²(ϒ)→L²(), defined by

_f(g)(x)=

ϒ

f(x+y)g(y)dy, x∈,

whereflives on=+ϒ. This class of operators includes both general domain Hankel and Toeplitz operators, by lettingϒ=andϒ = −, respectively.

For our purposes, general truncated correlation operators will only appear in intermediate steps toward proving the main results, but they also carry independent interest. They were introduced in [1], where their finite rank structure was investigated.

In [2] it was shown that they have a fundamental connection with frequency estimation on general domains, motivating the practical need for understanding such operators, not only on domains of simple geometrical structure. In [3] it is explained how one may infer certain results for the integral operators _f from their discretized matrix counterparts. We warn the reader that in naming the operators _f, _f, and _f we have slightly departed from previous work, reserving the term (general domain) Hankel operator for truncated correlation operators of the form_f_,,.

2.5 Hankel operators on multi-variable Paley–Wiener spaces

Another viewpoint is offered through co-invariant subspaces of the Hardy spacesH_d². For a domain⊂R^d, let PWdenote the subspace ofL²(R^d)of functions with Fourier transforms supported in,

PW= {G∈L²(R^d) : suppGˆ ⊂}. In the classical case=(0, 1)⊂R, note that

PW_(0,1)=H₁² {G∈H₁² : suppGˆ ⊂[1,∞)} =H₁²θH₁²,

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where

θ (x)=e^i2πx, x∈R.

Hence PW_(0,1)is the ortho-complement (inH₁²) ofθH₁², the shift-invariant subspace ofH₁² with inner factorθ. This space is usually denotedK_θ,

PW_(0,1)=K_θ :=(θH₁²)^⊥.

By a calculation similar to (2.1) we see that _f_,(0,1) is unitarily equivalent to the compression of the Hankel operatorZ_f_ˇto PW_(0,1),

_f_,(0,1)P_PW_(0,1)Z_f_ˇ|_PW_(0,1),

where P_PW_(0,1): H₁² → PW_(0,1) denotes the orthogonal projection onto PW_(0,1). Such truncated Toeplitz and Hankel operators are now very well studied on general K_θ- spaces [6,7,9,10,14,20,27,30,36].

In the case of the cube=(0, 1)^d⊂R^d,d>1, the Hankel operator_f_,may, just as ford =1, be understood as the compression of a Hankel operator to a co-invariant subspace ofH_d². Namely,

PW_(0,1)d = {G∈H_d² : suppGˆ ⊂[0, 1]^d} = {G∈H_d² : suppGˆ ⊂R^d₊\(0, 1)^d}^⊥. IfG∈H_d²∩L^∞(R^d), it is clear thatGPW^⊥

(0,1)^d ⊂PW^⊥

(0,1)^d, since F(GH)(ξ )=

R^d₊

G(y)ˆ H(ξˆ −y)dy=0, H∈PW^⊥_(0,1)_d, ξ ∈[0, 1]^d.

Hence PW^⊥

(0,1)^d ⊂ H_d² is an invariant subspace (under multiplication by bounded holomorphic functions), and as before we have that

_f_,(0,1)d P_PW

(0,1)dZ_f_ˇ|_PW

(0,1)d, whereP_PW

(0,1)d:H_d²→PW_(0,1)d denotes the orthogonal projection onto PW_(0,1)d.

Finally, let us brief ly discuss the viewpoint of weak factorization. The Hardy spaceH_d¹is defined as the closure ofF⁻¹(C_c^∞(R^d₊))inL¹(R^d). Similarly, we define PW¹as the closure ofF⁻¹(C_c^∞())inL¹(R^d). As is well known, see for example [24, Theorem 6.4], Theorem2.1 is equivalent to the fact that H_d¹ is the projective tensor product of two

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copies ofH_d²,

H_d¹ =H_d²H_d², (2.4)

with equivalence of norms. Here the projective tensor product norm onXX,Xa Banach space of functions, is given by

G _X_X=inf

⎧⎨

⎩ j

G_j _X H_j _X : G=

j

G_jH_j, G_j,H_j∈X

⎫⎬

⎭,

XXbeing defined as the completion of finite sums

jG_jH_jin this norm.

The reason that Theorem2.1is equivalent to (2.4) is the following: by (2.1),_f_,_Rd +

is bounded if and only if

|ˇf,GH_H²

d| ≤C G _H2 d H _H2

d,

which means precisely thatfˇ induces a bounded functional onH_d²H_d²,fˇ ∈(H_d²H_d²)^∗. On the other hand, the existence ofb∈L^∞(R^d)such thatbˆ|_R^d

+ =f|_R^d

+, so thatˇf,GH_H²

d = b,GH_H²

d,G,H∈H_d², means, by the Hahn–Banach theorem, precisely thatfˇ∈(H_d¹)^∗. Theorem1.1 yields a similar weak factorization theorem for Paley–Wiener spaces. We postpone the proof to Section5, but mention now that corresponding weak factorization forK_θspaces plays an important role in [6] and [9]. Corollary 5.3 might also be compared to the results in [37], where weak factorization for multivariate analytic polynomials is deduced as a consequence of Theorem2.1.

Corollary 5.3. Letbe a simple convex polytope, and let=2. Then PW¹ =PWPW.

The norms of these Banach spaces are equivalent.

2.6 Brief historical overview

Z. Nehari published his famous theorem in 1957 [25], inspiring the search for analogous statements in other contexts; positive results are themselves often referred to as Nehari theorems. The most natural inquiries are perhaps those related to Hankel operators on Hardy spaces of several variables. Nehari’s theorem for the Hardy space of the unit ball was proven by Coifman, Rochberg, and Weiss in 1976 [15, Thm. VII], but this setting is rather different from the one considered in this paper.

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For the product domain Hardy spaceH_d², Hankel operators can be defined by either projecting onH_d² or on the larger spaceL²(R^d)H_d². The 1st option leads to the

“small” Hankel operators considered in Section2.1, while the 2nd type of operator is commonly referred to as a “big” Hankel operator. In the notation of Section2.4, a small Hankel operator is an operator _f_,Rd

+,R^d₊ = _f_,Rd

+, whereas big Hankel operators are of the form

f,R^d₊,R^d\R^d₊. When transferred to operators on the Hardy space of the polydisc, small Hankel operators correspond, in the standard basis, to infinite matrices with a certain block Hankel structure (cf. Section7).

The big Hankel operators were extensively studied by Cotlar and Sadosky. In particular, boundedness of the big Hankel operators was characterized in terms of certain BMO type estimates in [16]. Small Hankel operators were investigated by Janson and Peetre [22] in 1988. They introduced “generalized Hankel and Toeplitz operators”

as particular cases of a more general class of pseudo-differential operators called paracommutators. In their terminology, an operator of the form_f_,,ϒ is a generalized Hankel operator if andϒ are open cones and∩(−ϒ) = {0}, whereas it is called Toeplitz if∩(−ϒ)= ∅. Hence the general domain Hankel operators_f_,are generalized Hankel operators a lá Janson–Peetre wheneveris a cone with mild restrictions, while _f_,is a generalized Toeplitz operator a lá Janson–Peetre for every open cone. In the Toeplitz case, a full boundedness characterization is given in [22, p. 482]. In the Hankel case, only sufficient conditions for boundedness and Schatten class membership are provided, in terms of BMO and Besov spaces, respectively.

As previously mentioned, R. Rochberg considered Hankel operators for bounded domains in 1987 [35], studying the case of a finite interval in one dimension. Further- more, he posed as an open problem to understand the case when⊂R²is a disc. In this latter setting, L. Peng [32] characterized when_f_,belongs to the Schatten classS_p, for 1≤p≤2, in terms of certain Besov spaces adapted to the disc. L. Peng also carried out a similar study [33] for the case of the multidimensional cube,=(−1, 1)^d, describing membership in S_p for all p, 0 < p < ∞, as well as giving a sufficient condition for boundedness.

Since then it seems that the field did not see progress until the results of Ferguson–Lacey–Terwilleger [18, 23] settled the issue of boundedness of small Hankel operators.

3 Distribution symbols

Let ,ϒ ⊂ R^d be any open connected sets and letf ∈ D() be a distribution on , =+ϒ. We follow the notation of [21] in our use of distributions. We then define

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the truncated correlation operator _f as an operator _f_,ϒ_, : C^∞_c (ϒ) → C^∞()by the formula

_f(ϕ)(x)=(f,T_xϕ), x∈, where(f,ϕ)denotes the action off onϕand

T_xϕ(·)=ϕ(· −x).

We reserve the notationf,ϕfor scalar products that are anti-linear in the 2nd entry.

Since T_xϕ is compactly supported in forx ∈ , it follows that_f(ϕ) this is well defined and smooth in (see, e.g., [21, Theorem 4.1.1]). Since C^∞_c (ϒ)is dense in L²(ϒ),_f gives rise to a densely defined operator on the latter space, which extends to a bounded operator_f:L²(ϒ)→L²()if and only if

_f =sup

_f(ϕ) _L2()

ϕ _L2(ϒ )

:ϕ∈C^∞_c (ϒ),ϕ=0

<∞.

It is clear that _f(ϕ)(x) =

f(x+y)ϕ(y)dy whenever f ∈ L¹_loc(). By slight abuse of notation, we write the action of_f in this way even whenf is not locally integrable.

The central question in this paper is the following:for which domainsϒ and is the boundedness of_f equivalent to the existence of a function b ∈ L^∞(R^d)such thatbˆ| =f ?Some care must be taken in interpreting this question. For example, the prototypical example of a bounded Hankel operator is the Carleman operator

_1/x,R

+ =_1/x,R

+,R+. The symbol f(x) = ¹_xχ_R

+(x) is in this case not a tempered distribution on R (so the meaning of fˇ is unclear)—it is, however, the restriction of the tempered distribution p.v.¹_x toR₊. An example with a delta function makes it clear that it is not necessary for f to be locally integrable ineither.

We first record the answer to our question in the trivial direction.

Proposition 3.1. Consider any connected open domains , ϒ ⊂ R^d, with associated domain=ϒ+. Letb∈L^∞(R^d)be given and supposef = ˆb|. Then_f:L²(ϒ)→L²() is bounded and

_f ≤ b _L∞. (3.1)

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Proof. Forϕ∈C^∞_c (ϒ)we have that

_f(ϕ)=FM_bJF⁻¹ϕ|,

whereM_b is the operator of multiplication byb. The statement is obvious from here.

Next we establish two technical results on the approximation of distribution symbols by smooth compactly supported functions, Propositions3.2and3.3. They will help us to overcome the technical issues mentioned earlier, in particular allowing us to deduce Theorem2.1from the corresponding statements in [18,23].

Given open connected domains,ϒ ⊂R^d, let ϒ_n_∞

n=1be an increasing sequence of connected open subdomainsϒ_n⊂ϒsuch that

dist(ϒ_n,∂ϒ) >1/n, ∪^∞_n₌₁ϒ_n=ϒ.

Note that_n=ϒ_n+is also increasing and satisfies

dist(_n,∂) >1/n, ∪^∞_n₌₁_n=.

Letψ ∈ C^∞_c (R^d)be a fixed non-negative function with compact support in the ballB(0, 1/2)such that

R^dψ(x)dx=1. Forn≥1 let ψ_n(x)=n^dψ(nx),

so that (ψ_n)^∞_n₌₁ is an approximation of the identity. Since f ∈ D() and suppψ_n ⊂ B(0, 1/2n), the convolutionf ∗ψ_nis well defined as a function inC^∞(_2n). Letρ_n be a smooth cut-off function that is 1 in a neighborhood of_nbut zero in a neighborhood of ^c_2n, and note thatρ_n(f ∗ψ_n)then naturally defines a function inC^∞(Rⁿ). Finally, for a non-negative functionη∈C^∞_c (R^d)with η _L2 =1, letω=η∗ ˜η, whereη(x)˜ =η(−x). Then ω∈C^∞_c (R^d)and

ω(0)= ˆω _L1=1.

Letω_n(x)=ω(x/n). We introduce

f_n=ω_nρ_n(f∗ψ_n)

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as an approximant off, where the role ofω_nis to enforce compact support in caseis unbounded. By construction,f_n∈C^∞_c ()and it is straightforward to check thatf_n→f inD(). As for_f

n,ϒn,, we have the following result.

Proposition 3.2. Let , ϒ be connected open domains, = ϒ+, and suppose f ∈ D(). Forn≥1, let_n=ϒ_n+andf_nbe constructed as above. Then

_f

n,ϒn, ≤ _f_,ϒ, .

Proof. We can assume that _f_,ϒ, <∞, since otherwise there is nothing to prove.

First note that

ω_n(x)=

R^dn^dω(nξ )eˆ ^2πix·^ξdξ,

the integrand on the right havingL¹-norm equal to ˆω _L1(R^d). Lettingg_n=ρ_n(f∗ψ_n), we have forϕ∈C^∞_c (ϒ_n)andx∈that

_f

n(ϕ)(x)=

ϒn

R^dn^dω(nξ )eˆ ^2πi(x⁺^y)^·^ξdξg_n(x+y)ϕ(y)dy=

R^dn^dω(nξ )eˆ ^2πiξ^·^x_g

n(ϕ_ξ)(x)dξ, whereϕ_ξ(y)=e^2πiy^·^ξϕ(y). Since ϕ_ξ _L2 = ϕ _L2 it follows by the triangle inequality (for L²-valued Bochner integrals) that

_f_n_,ϒ_n_, ≤ ˆω _L1 _g_n_,ϒ_n_, = _g_n_,ϒ_n_, . This reduces our task to proving that the operators

_g

n,ϒn,=_ρ

n(f∗ψn),ϒn,=_f_∗_ψ

n,ϒn,

are uniformly bounded inn. We have forϕ∈C^∞_c (ϒ_n)andx∈that _f_∗_ψ

n(ϕ)(x)=

R^d

R^df((x+y)−z)ψ_n(z)dzϕ(y)dy

=

R^df(x+z)

R^dψ_n(y−z)ϕ(y)dydz=_f(ψ_n∗ϕ)(x), whereψ_n(x)=ψ_n(−x). Since

ψ_n∗ϕ _L2(ϒ )≤ ψ_n _L1 ϕ _L2(ϒn)= ψ _L1 ϕ _L2(ϒn) = ϕ _L2(ϒn),

this completes the proof.

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Proof of Theorem2.1. Suppose that_f_,_Rd

+=_f_,,ϒ is bounded, where=ϒ =R^d₊. In this case, we letϒ_n=(2/n,∞)^d. By Proposition3.2we then have that

_f

n,ϒn ≤ _f

n,ϒn, ≤ _f_,_Rd

+ , n≥1.

Sinceϒ_n=z_n+R^d₊,z_n=(2/n,. . ., 2/n), we have that

_f_n_,ϒ_n(g)(x)=_f_˜

n,R^d₊(˜g)(x−z_n),

wheref˜_n(x)=f_n(x+2z_n)andg˜_n(x)=g(x+z_n). Sincef˜_n∈L²(R^d₊), the computation that led to (2.1) is justified, and we conclude from [18,23] that there isb_n∈L^∞(R^d)such that

bˆ_n|_2ϒ_n =f_n|_2ϒ_n, b_n _L∞ ≤C _f_,_Rd + .

By Alaoglu’s theorem it follows that there is a weak-star convergent subsequence (b_n

k)^∞_k₌₁ with limit b ∈ L^∞ having norm less than C _f_,_Rd

+ . It remains to prove that f = ˆb|_R^d

+, that is,(f,ϕ)=(b,ϕ)ˆ holds for allϕ∈C^∞_c (R^d₊). However, this is clear from the construction; sinceϕˆ∈L¹we have that

(b,ϕ)ˆ = lim

k→∞(b_n

k,ϕ)ˆ = lim

k→∞(f_n

k,ϕ)=(f,ϕ).

In Section6we will consider Toeplitz operators_f_,for which=−=R^d. In this casef ∗ψ_n is a smooth function defined in all of R^d, and there is no need to multiply withρ_nor to introduce the subdomainsϒ_n. In this case we simply let

f_n=ω_n(f ∗ψ_n).

Clearly,f_n→f inD(R^d)and we have, with the exact same proof as for Proposition3.2, the following approximation result.

Proposition 3.3. Let,ϒ be connected open domains for which=ϒ+=R^d, and supposef ∈D(R^d). Forn≥1, letf_nbe constructed as above. Then

_f

n,ϒ, ≤ _f_,ϒ, .

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4 On convex sets and polytopes

We recall some basic properties of convex sets. Given an unbounded convex set⊂R^d which is either open or closed, its characteristic cone, also known as its recession cone, is the closed set

cc= {x∈R^d : +xR₊⊂}.

The support functionh:R^d→(−∞,∞] is defined by h(θ )=sup

x∈

x·θ.

We refer to [21, Sec. 7.4] for the basic properties ofh. The barrier cone ofis the set bc= {θ∈R^d : h(θ ) <∞}. (4.1) The characteristic cone cc coincides with the polar cone of the barrier cone bc, that is,

cc= {x∈R^d : x·y≤0, ∀y∈bc}.

To give a complete reference for this claim, first note that for closed convex sets , cc coincides with the asymptotic cone of , giving (4.1) by [4, Theorem 2.2.1]. When instead is open and convex we have thatis equal to itsrelative interiorri(), and since cc_ri()=cc[8, Proposition 1.4.2], it follows that cc=ccin this case.

We next recall some standard terminology and facts of polytopes, referring to for example [12, Ch. 7–9]. By an open half-space inR^dwe mean a set

H_ν^r= {x∈R^d : x·ν >r},

whereν ∈R^dis a non-zero vector andr∈R. A closed half-space is the closure of such a set. A finite intersection of half-spaces is called a polyhedral set.

A convex polytope is a bounded polyhedral set. A closed convex polytope is the convex hull of a finite set of points. The minimal set of such points coincides with the extreme points of the polytope, that is, its vertices. If the minimal number of defining hyperspaces of a convex polytope isd+1 (equivalently, if it has preciselyd+1 vertices), the polytope is called a simplex. For a non-closed polytope we define its vertices (and its edges and facets) as those of its closure.

The boundary of a polytope set is made up of a finite amount of facets (i.e.

d−1 dimensional faces), see Corollary 7.4 and Theorem 8.1 of [12]. For a polytope with vertexx_j, we denote by∂_far,x

j the part of its boundary made up of all facets not containingx_j.

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A vertex of a polytope will be called simple if it is contained in preciselydof its edges. We say that a polytope is simple if all of its vertices are simple, which coincides with the standard terminology. Equivalently, this means that each vertex is contained in preciselydof its facets (cf. [12, Theorem 12.11]).

By an affine linear transformation we mean a map of the formA(x)=x₀+L(x) whereLis a linear map, and we callx₀ the origin of such a map. The following simple lemma gives a 3rd characterization of simple vertices.

Lemma 4.1. Let {x_j}^J_j₌₁ be the vertices of a closed polytope. Then the vertex x_j is simple if and only if it is the origin of an invertible affine transformationA_j such that locally coincides withA_j(R^d₊)aroundx_j, in the sense thatA⁻_j¹()⊂R^d₊and the facets ofA⁻¹_j ()containing 0 are precisely those of the form

A⁻¹()∩ {x∈R^d : x·e_k=0}, 1≤k≤d,

where{e_k}^d_k₌₁denotes the standard basis ofR^d.

Proof. We may assume that x₁ = 0 is simple and that x₂,x₃,. . .,x_d₊₁ are the other endpoints of the edges containing 0. LetA:R^d→R^dbe the linear map such thatA(e_k)= x_k₊₁, 1 ≤ k ≤ d.Ais invertible [12, Corollary 11.7], so that A⁻¹()is a closed convex polytope contained inR^d₊. Since 0 is a vertex ofA⁻¹()with adjacent verticese₁,. . .,e_d, thedfacets containing 0 must be precisely those of the formA⁻¹()∩{x∈R^d : x·e_k =0}.

For the converse, simply note that the property of being a simple vertex is

preserved under affine isomorphisms.

By compactness it is easy to construct a partition of unity adapted to the vertices of.

Lemma 4.2. Given a polytopewith vertices{x_j}^J_j=1there exist functions{μ_j}^J_j=1such thatμ_j∈C^∞_c (R^d),_J

j=1μ_j(x)=1 forx∈, and suppμ_j∩∂_far,x

j= ∅.

Proof. Forε >0 and 1≤j≤J, let

V_j^ε= {x∈R^d|dist(x,∂_far,x

j) > εand dist(x,) <1}.

Since everyx ∈ is contained in some set V_j^ε, there is by compactness a fixedε₀ >0 such that⊂_J

j=1V_j^ε⁰. LetV_J+1=R^d\and choose a smooth partition of unity{μ_j}^J+1_j=1 ofR^dsubordinate toV₁^ε⁰,. . .,V_J^ε⁰,V_J+1. Then{μ_j}^J_j=1is the required partition of unity.

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5 General domain Hankel operators

We now consider general domain Hankel operators_f_,for convex domains. Observe that in this case=+ =2. We begin with a proposition that links the bounded Hankel operators with weak factorization.

Proposition 5.1. Letbe an open convex domain. Then X=

_f_, : _f_, <∞

is a closed subspace of the space of bounded linear operators on L²(). As a Banach space, it is isometrically isomorphic to the dual space(PWPW)^∗. More precisely, bounded functionals μ on the projective tensor product correspond to distributions f on=2,

(f,g)=μ(F⁻¹g), g∈C_c^∞(), for which _f_, = μ .

Proof. The main fact to be proved is that

F⁻¹(C^∞_c ())⊂PWPW.

SinceC^∞_c ()is dense inL²(), it then follows thatF⁻¹(C^∞_c ())is dense in the product PWPW.

We will actually show a little more than the claim. Namely, everyg∈C_c^∞()can be written

g=

k

s_k∗t_k, s_k,t_k ∈L²(), in such a way that the corresponding mapg→

k s_k _L2() t_k _L2() is continuous from C_c^∞(), equipped with the usual test function topology, toR. By employing a partition of unity in which each member is compactly supported in a cube, it is sufficient to prove the claim when=(0, 1/2)^d. For this we employ Fourier series. Letλ(t)=1/2−|t−1/2|, t∈[0, 1], and let

(x)= d i=1

λ(x_i), x∈(0, 1)^d.

Note that λ is in the Wiener algebra A([0, 1]), the space of functions on [0, 1] with absolutely convergent Fourier series, equipped with pointwise multiplication. Therefore