Wiener-Hopf operators in higher dimensions: the Widom conjecture for piece-wise smooth domains

We prove a two-term quasi-classical trace asymptotic formula for the functions of multi-dimensional Wiener-Hopf operators with discontinuous symbols. The discontinuities occur on the surfaces which are assumed to be piece-wise smooth. Such a two-term formula was conjectured by H. Widom in 1982, and proved by A. V Sobolev for smooth surfaces in 2009.


Introduction
The quasi-classical functional calculus for smooth pseudo-differential operators was developed more than three decades ago (see e.g. [5] ) and now it is considered a standard tool of microlocal analysis and spectral theory. On the contrary, for pseudo-differential operators with discontinuous symbols results are sparse and less well known. Various quasi-classical trace type formulas for Wiener-Hopf operators were obtained by H. Widom in the 80's. In this article we shall be concerned with a multi-dimensional generalisation of one such result which has become known as The Widom Conjecture. Let a = a(x, ξ), x, ξ ∈ R d , d ≥ 1 be a smooth symbol. Introduce the standard notation for the left and right pseudo-differential operators with symbol a and a quasi-classical parameter α > 0: for any function u from the Schwartz class on R d . If the function a depends only on ξ then the operators Op l α (a), Op r α (a) coincide with each other, and we simply write Op α (a). Here and below integrals without indication of the domain are assumed to be taken over the entire Euclidean space R d .
(1. 3) tr as α → ∞. The precise formulas for the coefficients W 0 , W 1 are given in Sect. 2. The first term in (1.3) is the standard Weyl asymptotics, whereas the second term is non-standard, and it describes the contribution of the boundaries ∂Λ, ∂Ω. Emphasise that the second term contains a log-factor which makes it different from the familiar asymptotic expansion in powers of α −1 . The formula (1.3) was proved by H. Widom in [8] for d = 1. For d ≥ 2, in the case when one of the domains is a half-space, (1.3) was justified in [10]. For arbitrary bounded smooth domains in R d , d ≥ 2, the conjecture was proved in [6].
The main aim of this paper is to extend (1.3) to piece-wise smooth domains. Apart from the purely mathematical motivation, the interest in such domains is dictated by applications in Mathematical Physics, and in particular in Quantum Information Theory, see [2], [3], [4]. The proof is based on the papers [6], [7]. Using a convenient partition of unity one separates contributions from the smooth and non-smooth parts of the boundaries ∂Λ and ∂Ω. For the smooth part one applies directly the local version of the asymptotic formula of the form (1.3) from [6], whereas for the non-smooth part it suffices to establish appropriate trace bounds. Here a key role is played by inequalities obtained for arbitrary Lipschitz domains in [7]. As a result one checks that the non-smooth portion of the boundaries contributes a term of size o(α d−1 log α), which leads to the global asymptotics (1.3).
The author is grateful to J. Oldfield and W. Spitzer for critical remarks. This work was supported by EPSRC grant EP/J016829/1.

Main results
We begin with describing the classes of domains with which we work. In what follows we always assume that d ≥ 2. (1) We say that Λ ⊂ R d is a basic Lipschitz domain (resp. basic C m -domain, m = 1, 2, . . . ) if there exists a Lipschitz (resp. C m -) function Φ = Φ(x),x ∈ R d−1 , such that with a suitable choice of the Cartesian coordinates For a basic Lipschitz domain the function Φ is assumed to be uniformly Lipschitz, i.e. the constant For a basic C m -domain all the derivatives ∇ n Φ, n = 1, 2, . . . , m, are assumed to be uniformly bounded on R d−1 . For a basic domain we use the notation Λ = Γ(Φ).
locally it can be represented by basic Lipschitz (resp. C m -) domains, i.e. for any z ∈ Λ there is a radius r > 0 such that In this case the boundary ∂Λ is said to be a (d − 1)-dimensional Lipschitz (resp. C m -) surface. (3) A basic Lipschitz domain Λ = Γ(Φ) is said to be piece-wise C m with some m = 1, 2, . . . , if the function Φ is C m -smooth away from a collection of finitely many (d − 2)-dimensional Lipschitz surfaces L 1 , L 2 , . . . , L n ⊂ R d−1 . More precisely, if Λ is given by (2.1) then for any open ball B ⊂ R d−1 such that B is disjoint with all surfaces L j , j = 1, 2, . . . , n, we have Φ ∈ C m (B). Note that the derivatives of Φ are not required to be bounded uniformly in the choice of the ball B. We denote i.e. (∂Λ) s ⊂ ∂Λ is the set of points where the C m -smoothness of the surface ∂Λ may break down. (4) A Lipschitz domain Λ is said to be piece-wise C m , m = 1, 2, . . . , if locally it can be represented by piece-wise C m basic domains. As for the basic domains, by (∂Λ) s ⊂ ∂Λ we denote the set of points where the C m -smoothness of ∂Λ may break down.
Let us define the asymptotic coefficients entering the main asymptotic formulas. For a symbol b = b(x, ξ) let For any (d − 1)-dimensional Lipschitz surfaces L, P denote where n L (x) and n P (ξ) denote the exterior unit normals to L and P defined for a.a. x and ξ respectively. For any continuous function g on C such that g(0) = 0, and any number s ∈ C, we also define The next theorem contains the main result of the paper.
and Ω is piece-wise C 3 . Let a = a(x, ξ) be a symbol whose distributional derivatives satisfy the bounds Let g be a function on C such that g(0) = 0, analytic in a disk of sufficiently large radius. Then tr g(T α (a)) = α d W 0 (g(a); Λ, Ω) as α → ∞.
For the self-adjoint operator S α (a) we have a wider choice of functions g: Theorem 2.3. Let the domains Λ, Ω ⊂ R d , d ≥ 2, and the symbol a be as in Theorem 2.2. Then for any function g ∈ C ∞ (R), such that g(0) = 0, one has tr g(S α (a)) = α d W 0 (g(Re a); Λ, Ω) As in [6] the crucial step of the proof is to prove the formula (2.7) for polynomial functions.
Theorem 2.4. Let the domains Λ, Ω ⊂ R d , d ≥ 2, and the symbol a be as in Theorem 2.2. Then for g p (t) = t p , p = 1, 2, . . . , we have as α → ∞. If T α (a) is replaced with S α (a), then the same formula holds with the symbol a replaced by Re a on the right-hand side.
In the next theorem the domain Λ is allowed to be unbounded, in which case we replace formula (2.9) with its regularized variant.
(1) Ω is bounded and piece-wise C 3 , (2) Λ or R d \ Λ is bounded, and Λ is piece-wise C 1 . Let the symbol a be as in Theorem 2.2. Then , then the same formula holds with the symbol a replaced by Re a on the right-hand side.
Note that for bounded domains Λ formula (2.10) is just another way to write the asymptotics (2.9), see Proof of Theorem 2.5. On the other hand, for unbounded Λ formula (2.10) is an independent result. Theorems 2.2 and 2.3 are derived from Theorem 2.4 in the same way as in [6] for smooth domains, and we do not provide details. However the methods of [6] do not allow one to derive from Theorem 2.5 analogues of Theorems 2.2 or 2.3 for unbounded domains Λ. This generalization will be done in another publication.
The main focus of the rest of this paper is on the proof of Theorems 2.4 and 2.5.

Auxiliary results
Here we collect some trace estimates and asymptotic formulas from [6] and [7] used in the proofs. The trace estimates established in [6] required that Λ and Ω be C 1 -smooth domains. In [7] most of those estimates are proved under the Lipschitz assumption only. On the other hand, the article [7] does not duplicate [6], and thus in the current article some of the estimates from [6] are re-proved for Lipschitz domains.
3.1. Notation. Smooth symbols. In order to allow consideration of symbols b = b(x, ξ) with different scaling properties, we define for any ℓ, ρ > 0 the norms with n, m = 0, 1, . . . . If the norm (3.1) is finite for some (and hence for all) ℓ, ρ > 0 then we say that the symbol b belongs to the class S (n,m) . Below we often assume that various symbols b = b(x, ξ) are compactly supported, and the choice of the parameters ℓ, ρ in (3.1) is coordinated with the size of support. Precisely, we suppose that In what follows most of the bounds are obtained under the assumption that αℓρ ≥ ℓ 0 with some fixed positive number ℓ 0 . The constants featuring in all the estimates below are independent of the symbols involved as well as of the parameters z, µ, α, ℓ, ρ but may depend on the constant ℓ 0 .
We begin with some natural estimates for smooth symbols. The notation S 1 is used for the trace class, and · S 1 -for the trace class norm.

3.2.
Bounds for basic domains. Theorems 2.4 and 2.5 will be deduced from the asymptotics of "local" traces of the form tr Op l α (b)g p (T α (a)) , g p (t) = t p , p = 1, 2, . . . , with a compactly supported symbol b. In this section we concentrate on such "localized" operators. In fact, due to the bound (3.3) it will be unimportant which of the operators Op l α (b) or Op r α (b) is used for this localization. Thus we often use the notation Op α (b) to denote any of these two operators.
First we obtain some bounds for the case when both domains Λ and Ω are basic Lipschitz, i.e. Λ = Γ(Φ) and Ω = Γ(Ψ) with some uniformly Lipschitz functions Φ and Ψ. The choice of Cartesian coordinates for which Λ or Ω have the form (2.1) is not assumed to be the same for both domains. The constants in the estimates below depend only on the Lipschitz constants M Φ , M Ψ for the functions Φ and Ψ, and not on any other properties of the domains.
First one needs the following commutator estimates.
Using these commutator estimates we can now reduce the problem to the operator T α (1). Lemma 3.3. Let each of the domains Λ and Ω be either a basic Lipschitz domain, or R d . Let a, b ∈ S (d+2,d+2) , and assume that b satisfies (3.2). Let αℓρ ≥ ℓ 0 . Then

3.3.
Bounds and asymptotics for more general domains. The next group of results expresses the fact that the local asymptotics are determined by local properties of the boundaries ∂Λ, ∂Ω. This is the key idea in the proof of Theorem 2.4. Let Λ, Ω and Λ 0 , Ω 0 be two pairs of domains such that each of Λ 0 , Ω 0 is either (1) a basic Lipschitz domain, or (2) the entire space R d , or (3) the empty set. Suppose that The next localization result is crucial.
For C 1 -domains Λ, Ω estimates of this type were established in [6], Section 7. Generalization to the Lipschitz domains is quite straightforward and we present a proof here for the sake of completeness.
Proof of Lemma 3.6. Without loss of generality assume that the both N-norms on the right-hand sides of (3.8) and (3.9) equal 1. For any two operators A 1 and A 2 we write Assume that Λ 0 , Ω 0 are basic Lipschitz domains. The following relations are consequences of (3.3) and Proposition 3.2: . Taking the adjoints we also get χ Λ Op α (b) ∼ Op α (b)χ Λ 0 . In the same way one obtains similar relations for P Ω,α : Thus by Proposition 3.2, If Λ 0 or Ω 0 are either R d or ∅, then the above relations hold for trivial reasons. This proves (3.8).
Applying repeatedly the relations (3.10) and (3.11) in combination with Proposition 3.1 we arrive at This relation coincides with (3.9). The same argument leads to the bound of the form (3.9) for the operator S α .
Lemma 3.7. Let a, b ∈ S (d+2,d+2) , and assume that b satisfies (3.2). Let αℓρ ≥ ℓ 0 . Suppose that Λ and Ω satisfy (3.7), and one of the following two conditions is satisfied: Then Proof. By Lemmas 3.6 and 3.3 we may assume that Λ = Λ 0 , Ω = Ω 0 and a ≡ 1. Under any of the conditions of the lemma we have either T α (1; Λ 0 , Ω 0 ) = 0 or χ Λ 0 or P Ω 0 ,α . In the first case the left-hand side of (3.12) equals zero, and there is nothing to prove. If T α (1) = χ Λ 0 , then the sought trace has the form tr(Op α (b)χ Λ 0 ). This trace is easily found by integrating the kernel of the operator over the diagonal, and it does not depend on the choice of quantization. This immediately leads to (3.12). If T α (1) = P Ω 0 ,α , then computing the trace tr(Op l α (bχ Ω 0 )) we obtain (3.12) again. Note that in this case it is convenient to choose the l-quantization for Op α (b).
The next result is also useful. Lemma 3.8. Let the symbols a, b be as in Lemma 3.7, and let αℓρ ≥ ℓ 0 . Suppose that Λ and Ω satisfy (3.7). Then Proof. Due to (3.8), the problem reduces to finding the trace of the operator As in the proof of Lemma 3.7, by virtue of Lemmas 3.6 and 3.3 we may assume that Ω = Ω 0 and a ≡ 1. Thus T α (1; R d , Ω 0 ) = P Ω 0 ,α . Again, the trace of the operator χ Λ Op l α (b)P Ω 0 ,α χ Λ is easily seen to be equal to α d W 0 (b; Λ, Ω). So far it was enough to assume that the domains were Lipschitz. To state the asymptotic result we need more restrictive conditions. Proposition 3.9. Let a, b ∈ S (d+2,d+2) , and let b satisfy (3.2). Assume that (3.7) holds with some basic domains Λ 0 , Ω 0 such that Λ 0 is C 1 and Ω 0 is C 3 . Then tr Op l α (b)g p (T α (a)) = α d W 0 (bg p (a); Λ, Ω) as α → ∞.
This proposition follows from [6], Theorem 11.1 upon application of Lemma 3.6.

Proof of the main theorems
Here we concentrate on proving Theorems 2.4 and 2.5. As explained earlier, Theorem 2.4 implies the main results -Theorems 2.2 and 2.3.

An intermediate local asymptotics.
We begin with the following local result: is a symbol with compact support in both variables, and that Λ is a piece-wise C 1 basic domain, and Ω a piece-wise C 3 basic domain. Then Without loss of generality assume that the symbol b is supported on B(0, 1) × B(0, 1). If B(0, 1) ∩ ∂Λ = ∅ or B(0, 1) ∩ ∂Ω = ∅, then the required asymptotics immediately follow from Lemma 3.7. Assume that neither of the above intersections is empty. Cover the boundaries ∂Λ ∩ B(0, 1) and ∂Ω ∩ B(0, 1) with finitely many open balls of radius ε > 0. Denote the number of such balls by J = J ε and K = K ε respectively. Since ∂Λ and ∂Ω are Lipschitz, one can construct these coverings in such a way that the number of intersections of each ball with the other ones is bounded from above uniformly in ε and Let Σ Λ (resp. Σ Ω ) be the set of indices j (resp. k) such that the ball from the constructed covering indexed j (resp. k) has a non-empty intersection with the set (∂Λ) s (resp. (∂Ω) s ). Since the sets (∂Λ) s , (∂Ω) s are built out of Lipschitz surfaces, by construction of the covering we have We may assume that the covering balls with indices j / ∈ Σ Λ (resp. k / ∈ Σ Ω ) are separated from (∂Λ) s (resp. (∂Ω) s ). Thus in each of these balls the boundary ∂Λ (resp. ∂Ω) is C 1 (resp. C 3 ).
Denote by φ j , j = 1, 2, . . . , J, and ψ k , k = 1, 2, . . . , K the associated smooth partitions of unity, so that the functions equal 1 on a neighbourhood of ∂Λ ∩ B(0, 1) and ∂Ω ∩ B(0, 1) respectively, and uniformly in x and ξ. The symbol b(1 − φψ) is supported away from ∂Λ ∩ B(0, 1) × ∂Ω ∩ B(0, 1) . Thus Lemma 3.7 implies that The constant C ε on the right-hand side depends on the symbol b, and on ε, but the latter fact does not matter for the rest of the proof. It remains to study the trace tr Op l α (bφψ)g p (T α (1)) . Let us separate contributions from the smooth and singular parts of the boundaries ∂Λ and ∂Ω. Denotẽ The support ofb contains only smooth parts of the boundaries ∂Λ and ∂Ω, so by Proposition 3.9 we have It remains to handle the cases when j ∈ Σ Λ or k ∈ Σ Ω . Let Lemma 4.2. Let b jk be as defined above, and let p ≥ 1. Then as α → ∞.
Proof. It is enough to establish the estimate lim sup Indeed, in view of (4.2) and (4.3), the number of summands on the left-hand side of (4.7) does not exceed Cε 3−2d , and hence summing (4.8) up over (j, k) ∈ Σ we obtain (4.7). If p = 1, then the trace asymptotics of the operator Op l α (b jk )T α (1) are easy to find. Indeed, by Lemma 3.8, we have . Thus it remains to study the trace of the operator with a polynomialg, and estimate using (3.6): for sufficiently large α. Together with (4.9) this implies that (4.10) lim sup 1 as α → ∞. It follows straight from the definition (2.4) that so (4.10) entails (4.8), as claimed.

4.2.
Proof of Theorems 2.4 and 2.5. The proofs amount to putting together local asymptotic formulas and estimates obtained above. The argument is based on partition of unity, and is rather standard. We present it for the sake of completeness. Also, all the proofs are conducted for the operator T α only -the argument for S α is essentially the same. The next two lemmas are the last building blocks in the proofs of Theorems 2.4 and 2.5. If T α (a) is replaced with S α (a), then the same formulas hold with the symbol a replaced by Re a in W 0 and W 1 .
Proof. Let R > 0 be such that supp h ∈ B(0, R), and either Λ or R d \ Λ is contained in B(0, R). Since the domains Λ ∩ B(0, R) and Ω are bounded, we can cover their closures by finitely many open balls such that in each of them each domain Λ or Ω is represented by a basic domain or by R d . Denote by {φ j } and {ψ k } the partitions of unity subordinate to these coverings. Represent Consequently, in order to prove (4.11) it suffices to find the sought asymptotics for the operator χ Λ Op l α (b jk )P Ω,α Op l α (a)P Ω,α (T α (a; Λ, Ω)) p−1 , for each j and k. By virtue of (3.8) this is equivalent to studying the operator Now, due to (3.9), we can replace each domain Λ or Ω by the appropriate basic domain or by R d . Furthermore, Lemma 3.3 ensures that the symbol a can be replaced by the constant symbol a ≡ 1. Now Theorem 4.1 implies that as α → ∞. Summing over j and k we obtain formula (4.11).
The following lemma concentrates on the case of unbounded Λ.
Proof. For brevity we write T α (Λ) = T α (a; Λ, Ω). For any two operators A 1 and A 2 we write A 1 ∼ A 2 if A 1 − A 2 S 1 ≤ Cα d−1 , with a constant C independent of α.
Suppose now that (4.15) holds for p = k, and let us prove it for p = k + 1. Write: The sought bound follows from (4.15) for p = 1 and p = k.
To conclude the proof write so that (4.14) follows from (4.15) used twice: for the domain Λ itself, and for Λ = R d .
Now we can proceed to the proof of Theorems 2.4 and 2.5. As explained earlier, the proofs are conducted only for the operators T α .
Proof of Theorem 2.4. Since Λ is bounded, use formula (4.11) with a function h ∈ C ∞ 0 (R d ) such that hχ Λ = χ Λ . This completes the proof.