On the Schatten-von Neumann properties of some pseudo-differential operators

We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten-von Neumann classes $S_q$ with $0<q\le 1$. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.


Introduction
When working with compact pseudo-differential operators it is often important to know how fast their singular values (or eigenvalues) decay. These properties are conveniently stated in terms of the classical Schatten-von Neumann classes S p , p > 0, or even more general ideals S p,q , p, q > 0. We refer to [2], [3], [7] and [19] for information on compact operator ideals.
Not surprisingly, the Schatten-von Neumann properties of pseudo-differential operators are determined by smoothness of their symbols. The first bound in the trace class S 1 was obtained in [17], and later reproduced in [18], Proposition 27.3, and [13], Theorem II-49, see also [10]. Some useful S 1 -bounds were obtained in the much more recent paper [16]. The other ideals S p , S p,q were studied e.g. in [1], [5], [8], [14], [22], and there one can find further references. The fundamental paper [2] contains S p,q -estimates for integral operators in terms of smoothness of their kernels.
In spite of a relatively large number of available results, they are not always practically useful since in applications one often needs more detailed information. In this paper we obtain some explicit bounds for Schatten-von Neumann norms of various pseudodifferential operators aiming at applications in semi-classical analysis. Let p = p(x, y, ξ), x, y, ξ ∈ R d , d ≥ 1, be a smooth amplitude. For any α > 0 introduce the standard notation for the pseudo-differential operator with amplitude p: (1.1) (Op a α (p))u(x) = α 2π d e iα(x−y)ξ p(x, y, ξ)u(y)dydξ, for any Schwartz class function u. In the literature one uses more often the reciprocal value α −1 which is interpreted as the Planck's constant. It is natural for us to study a somewhat more general variant of the operator (1.1). Let T = {t jk } be a non-degenerate (2 × 2)-matrix with real-valued entries. We concentrate on the operators (1.2) Op a α (p T ), p T (x, y, ξ) = p(w, z, ξ), with w = t 11 x + t 12 y, z = t 21 x + t 22 y.
In this formula the choice of the second row in the matrix T is unimportant as long as T remains non-degenerate. Note also that formally Op α,t (a) * = Op α,1−t (a). The values t = 0 and t = 1 give the standard "left" and "right" quantizations. In these cases the operator (1.3) has the symbol a(x, ξ) (for t = 0) or a(y, ξ) (for t = 1). In the literature one sometimes uses for them the notation Op l α (a) and Op r α (a) respectively. Another important example is the Weyl quantization: If the functions p and a above are sufficiently smooth and decay sufficiently fast at infinity then the operators (1.2),(1.3) belong to S q with a suitable q > 0. The aim of the paper is to study this property for q ∈ (0, 1]. Our results are divided in three groups. First in Section 2 we obtain general estimates in S q for α = 1, see Theorems 2.5, 2.6. The S q -bounds for the operators (1.2) seem to be quite useful from the practical point of view. In particular they allow us to study the operators of the form h 1 Op 1,t (a)h 2 , t ∈ [0, 1] with the weights h 1 , h 2 whose supports are disjoint, and to control explicitly the dependence on the distance between the supports, see Theorem 2.6(2). Our approach stems from a simple idea suggested in the paper [16] where trace class properties of pseudo-differential operators were studied. In fact, our results can be viewed as quantitative variants of Proposition 3.2 and Theorem 3.5. from [16], extended to the ideals S q , q ≤ 1. As the classes S q with q < 1 are not normed, the obtained S q -estimates for the operators (1.2) and (1.3) involve the so-called lattice quasi-norms(see (2.3)) for the amplitudes/symbols and their derivatives (for q = 1 these quasi-norms are simply L 1 -integral norms). The estimates in S q with q > 1 are also of great interest, but they are likely to be stated in different terms, cf. [1], [5], [22], and thus they are not discussed here.
Sections 3 and 4 are devoted to applications. In Section 3 we use Theorems 2.5 and 2.6 to derive estimates for large values of the parameter α, which can be interpreted as the semi-classical regime. These results are stated in terms of the scaling properties of the symbols which makes them flexible and convenient for applications. Section 4 is concerned with semi-classical bounds for operators with discontinuous symbols. The discontinuities are introduced as characteristic functions χ Λ (x) and χ Ω (ξ) of some Lipschitz domains Λ and Ω. We derive S q -semi-classical estimates for the Hankel-type operators χ Λ Op α,t (a)(I − χ Λ ) and χ Λ P Ω,α (I − χ Λ ), P Ω,α = Op α (χ Ω ). This study is motivated by the trace asymptotics for Wiener-Hopf and Hankel operators with discontinuous symbols, both classical, see e.g. [12], [23], and multi-dimensional, see [20], [21].
A number of estimates similar to the ones in Sections 3 and 4 have been established in [20] for the trace class S 1 . However some applications in Mathematical Physics, and in particular in Quantum Information Theory, call for estimates in the classes of compact operators with a faster decay of the singular values, see [6], [9]. This was the main incentive for the current paper.
To conclude the Introduction we make some notational conventions. Throughout the paper we denote by C or c with or without indices various positive constant whose value is unimportant. The notation B(u, r) is used for the open ball in R d , d ≥ 1, of radius r > 0 centred at the point u ∈ R d . The characteristic function of the ball B(u, r) is denoted by χ u,r .
Acknowledgements. The author is grateful to H. Leschke and W. Spitzer for introducing him to problems in Quantum Information Theory involving pseudo-differential operators with discontinuous symbols, and for useful remarks on the paper. This work was supported by EPSRC grant EP/J016829/1.

2.
General estimates in S q -ideals with q ∈ (0, 1]: smooth symbols 2.1. Ideals S q . The notation S q , q > 0, is standard for the set of all compact operators A on a Hilbert space with singular values s k (A), k = 1, 2, . . . , for which the functional is finite. For q ≥ 1 this functional defines a natural norm on S q , whereas for q < 1 it defines a quasi-norm. Nevertheless one has the triangle inequality of the form Sq , 0 < q ≤ 1, see [15] and [3], p.262, and the following Hölder-type inequality: [3], p. 262.
A crucial technical point in the study of the operators (1.2) is to estimate suitable S q -(quasi)-norms for the operators h Op 1 (a), h = h(x), a = a(ξ), which have been studied quite extensively. We need the following estimate which is a slight generalization of the bound found in [2], Theorem 11.1 (see also [4], Section 5.8), and quoted in [19], Theorem 4.5 for s ∈ [1,2].
Let C u ⊂ R m be a cube centred at u ∈ R m with the edge of unit length. For a function h ∈ L r loc (R m ), r ∈ (0, ∞), denote These functionals are sometimes called lattice quasi-norms (norms for r, δ ≥ 1). If h r,δ < ∞ we say that h ∈ l δ (L r )(R m ).
where S : R m → R n is a linear map. Then with a constant C q = C q (S) depending only on the number s 0 in the bound max jk |s jk | ≤ s 0 for the entries s jk , j = 1, 2, . . . , n; k = 1, 2, . . . m, of the matrix S.
We do not give the proof as it repeats that of [2], Theorem 11.1 almost word to word.

2.2.
Estimates for the operators (1.2). Now we need to specify the conditions on the matrix T = {t jk }, j, k = 1, 2. The end results require T to be non-degenerate, i.e. T ∈ GL(2, R). For convenience we sometimes assume that (2.4) t 11 + t 12 = 1, and denote Using the inverse of T, we can recover x and y from the vectors w and z defined in (1.2): We assume that with some fixed positive numbers t 0 , δ 0 . In the estimates below the constants may be dependent on t 0 and δ 0 . We provide appropriate comments in every instance.
, with n satisfying (2.11), and some m = 0, 1, . . . . Then x e iξ·x = e iξ·x , so integrating by parts m times, we get the following formula for the kernel of the operator Op a 1 (p T ): Now it is straightforward to see that

By Corollary 2.3 this implies the proclaimed result.
In the next Theorem we replace the (2, 2q)-quasi-norms of functions h 1 , h 2 by much weaker ones.
and let P n,m ∈ l q (L 1 )(R 3d ) with some q ∈ (0, 1], with n satisfying (2.11) and some m = 0, 1, . . . . Then Proof. Let us define a convenient partition of unity. The open balls B(j, 2 √ d), j ∈ Z d , form a covering of R d . Let {ψ j } be an associated partition of unity such that The first two factors are estimated by C h 1 2,∞ and C h 2 2,∞ respectively, with some constant C = C(t 0 , δ 0 ). Thus by the triangle inequality (2.1) Remembering that the number of intersecting balls B(j, 2 √ d) is uniformly bounded, we can estimate the sum on the right-hand side byC P n,m (p) q 1,q . This completes the proof.

2.3.
Estimates for the operators (1.3). Theorem 2.5 allows amplitudes independent of z, e.g. it allows one to consider t-pseudo-differential operators (1.3). We isolate this observation in a separate theorem. For a symbol a = a(x, ξ) denote  The constants in the next theorem are independent of t ∈ [0, 1].
, let n be as in (2.11), and q ∈ (0, 1] be some number. (1) Suppose that F n,n (a) ∈ l q (L 1 )(R 2d ). Then for any t ∈ [0, 1] we have (2) Suppose that the distance between the supports of the functions h 1 , h 2 is at least so that τ = 0, see (2.5). By definitions (2.9) and (2.16), To estimate P n,m (p) 1,q write for any k, s, j ∈ Z d : Consequently, for m ≥ n, Now Theorem 2.5 with m = n implies (2.17). Proof of (2.18). Let ζ ∈ C ∞ (R) be a function such that , where the matrix T is defined as in (2.19). We use Theorem 2.5 again but in a slightly different way than above -first we implement integration by parts similar to the one done in the proof of Lemma 2.4. Let P z e −iξ·z = e −iξ·z , so, integrating by parts m times, we get the following formula for the kernel of the operator Op a 1 (g T ): It is straightforward to see that with a constant independent of r. Arguing as in the first part of the proof we get the bound Theorem 2.5 with m ≥ n leads to (2.18).
As the next Theorem shows, in the case d = 1, when h 1 and h 2 have disjoint supports, one can sometimes allow symbols a depending only on ξ. Here and below we use x and ξ for one-dimensional variables.
with some r ≥ 1. Let q ∈ (0, 1] be some number, and let h = [q −1 ] + 1. Suppose that a = a(ξ) satisfies the condition ∂ m a ∈ l q (L 1 )(R), for some m ≥ n. Then we have Proof. As in the proof of the previous theorem, (2.20). Furthermore, integrating by parts m times we get the following formula for the kernel: By definition of h 1 , h 2 we obtain P n,0 (x, y, ξ; g (m) ) ≤ C |∂ m a(ξ)| |x| m + |y| m + r m .
Since m ≥ n = [q −1 ] + 1, the right-hand side belongs to l q (L 1 )(R 3 ), and the quasi-norm is bounded from above by ∂ m a 1,q . Now the estimate (2.21) follows from Theorem 2.5.

2.4.
Trace-class estimates. For q = 1 the lattice quasi-norms in Theorems 2.5 and 2.6 coincide with the standard L 1 -norms. Due to the relative simplicity of these bounds it seems appropriate to write them out separately. Moreover making the change αξ = ξ ′ we can immediately extend them to all values α ≥ 1: and P d+1,m ∈ L 1 (R 3d ), with some m = 0, 1, . . . . Then for any α ≥ 1 we have with a constant C m = C m (t 0 , δ 0 ).
(2) Suppose that the distance between the supports of the functions h 1 , h 2 is at least The constants C and C m do not depend on t ∈ [0, 1].
For T = I and t = 0, 1 the above estimates were obtained in [20].
3. Amplitudes from classes S (n 1 ,n 2 ,m) : semi-classical estimates 3.1. Compactly supported amplitudes/symbols. Now we proceed to estimates for arbitrary q ∈ (0, 1] for the operators containing the parameter α > 0. Due to the nature of the bounds derived in the previous section we do not expect the semi-classical bounds to look as simple as in Theorems 2.8 and 2.9. Thus we do not try to find integral bounds but instead we concentrate on the scaling properties of the S q -estimates. For arbitrary numbers ℓ > 0 and ρ > 0 introduce the following norms: We say that p belongs to the class S (n 1 ,n 2 ,m) if the norm (3.1) is finite for some (and hence for all) positive ℓ, ρ. For a symbol a = a(w, ξ) (resp. function a = a(ξ)) we use the notation N (n,m) (a; ℓ, ρ) (resp. N (m) (a; ρ)). Accordingly, we define classes S (n,m) and S (m) . The presence of the parameters ℓ, ρ allows one to consider amplitudes and symbols with different scaling properties. Let U ℓ be the unitary operator on L 2 (R d ) defined by . Then a straightforward calculation gives for any ℓ, ρ > 0 the following unitary equivalence: T ), p (ℓ,ρ) (w, z, ξ) = p(ℓw, ℓz, ρξ), β = αℓρ. The norms (3.1) are also invariant: The operators Op a α (p T ) transform in a standard way under Euclidean isometries ( i.e. orthogonal transformations and shifts), their norms (3.1) remain invariant. We use these facts regularly without introducing formal notation for these transformations.
All the S q -bounds below will be derived under the following conditions on the amplitudes or symbols. For the operator Op a α (p T ) we assume that with some u, µ ∈ R d and some ℓ > 0, ρ > 0. For the t-operators Op α,t (a) we assume 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 u, µ, α, ℓ, ρ but may depend on the constant ℓ 0 .
Theorem 3.1. Let T ∈ GL(2, R) be a matrix satisfying (2.4), and let s, t ∈ [0, 1]. Let q ∈ (0, 1] and αℓρ ≥ ℓ 0 . Let p ∈ S (n,n,n) , with n defined in (2.11), be an amplitude satisfying the condition (3.4), and let a ∈ S (n,n) be a symbol satisfying the condition (3.5). Then Op a α (p T ) ∈ S q , Op α,t (a) ∈ S q , and with a constant C q = C q (t 0 , δ 0 ) (see (2.7)), and with a constant C q independent of t. If, in addition a ∈ S (n,n+1) then with a constant C q independent of s, t.
Proof. The estimate (3.7) is a special case of (3.6) with the matrix T defined in (1.4).
Note that det S = s − t, and assume that |s − t| ≥ 1/4. For all n 1 , n 2 ≤ n, l ≤ n + 1 we have Arguing as in the first part of the proof we arrive at the estimate P n,n+1 (g) 1,q ≤ Cℓ d q −1 , C = C(ℓ 0 ), which implies (3.8) by virtue of Theorem 2.5. As we have assumed that | det S| ≥ 1/4, the constant in (3.8) does not depend on s, t.

3.2.
Symbols with non-compact support. Here we illustrate the use of the obtained estimates and derive a semi-classical bound for the t-pseudo-differential operators whose symbols are not necessarily compactly supported. Suppose that for some constant A > 0, and some number q ∈ (0, 1] the symbol a satisfies the bound where n is as in (2.11).

Lipschitz domains.
Here we obtain S q -estimates for operators with symbols having jump discontinuities. The discontinuities are introduced via the projections χ Λ and/or P Ω,α = Op α (χ Ω ) where Λ and Ω are some suitable domains whoce properties are specified in the next definition.
It is assumed that the function Φ is uniformly Lipschitz, i.e. the constant is finite. In this case we use the notation Λ = Γ(Φ). A domain Λ is said to be Lipschitz if locally it can be represented by basic domains, i.e. for any z ∈ R d there is a radius r > 0 such that B(z, r) ∩ Λ = B(z, r) ∩ Λ 0 with some basic domain Λ 0 = Λ 0 (z).
Our results are also applicable in the case d = 1. To state them simultaneously for all dimensions, in the case d = 1 we use the term basic domain for the domain Λ which is either (−∞, 0) or (0, ∞). The role of Lipschits domains will be played by intervals of the form (0, L), L > 0.
Our objective is to obtain semi-classical S q -estimates for the Hankel-type operators χ Λ Op α,t (a)(I − χ Λ ), P Ω,α Op α,t (a)(I − P α,Ω ) and χ Λ P α,Ω (I − χ Λ ), with suitable domains Λ, Ω and suitable symbols a. We work either with t = 0 or t = 1. First we establish the sought estimates for basic domains Λ and Ω, and then extend the result to the general bounded Lipschitz ones using appropriate partitions of unity.
For d ≥ 2 all the S q -estimates obtained for the basic domains are uniform in the Lipschitz constants M Φ and M Ψ satisfying the condition with some constant M. Needless to say, the choice of the coordinates for which Λ or Ω have the form (4.1) does not have to be the same for the domains Λ and Ω. As in the previous section we assume as a rule that the symbols are compactly supported and satisfy the condition (3.5). The constants in the obtained estimates will be independent of the symbols, and of u, µ and ℓ, ρ but may depend on the constant ℓ 0 in the bound αℓρ ≥ ℓ 0 , and, for d ≥ 2, on M. As mentioned in the Introduction some estimates were obtained in [20] for the class S 1 . Note also that for d ≥ 2 the results of [20] require C 1 -smoothness of the domains Λ, Ω whereas in the current paper the Lipschitz property suffices.
Since these operators contain only one characteristic function we refer to this case as the case of discontinuity in one variable. Next we look at the operators of the form χ Λ Op α,t (a)P Ω,α (I − χ Λ ) which is naturally referred to as the case of discontinuity in two variables.
It is useful to remark on the scaling properties of basic domains in d ≥ 2. Applying (3.2) to the characteristic function χ Λ , Λ = Γ(Φ), we observe that under scaling U ℓ the domain Λ transforms into Γ(Φ), whereΦ(x) = ℓΦ(ℓ −1x ). It is obvious that for all x, y ∈ R d , so that In the case d = 1, for a basic domain Λ the same type of bound is obvious:

Discontinuity in one variable.
Here we study the combinations involving an operator with a smooth symbol and one of the operators χ Λ or P Ω,α .  Suppose that the symbol a ∈ S (n,m) satisfies (3.5). Then for t = 0 or 1 we have Proof. The bound (4.8) follows from (4.7) upon exchanging the roles of the variables x and ξ. Thus it suffices to prove (4.7). Proof of (4.7). Assume without loss of generality that N (n,m) (a; ℓ, ρ) = 1. We prove (4.7) for the operator Op α,0 (a) only, the case t = 1 is done in the same way.
Let d ≥ 2. We use the same scaling argument as in the proof of Theorem 3.1, and the fact that the Lipschitz constant of the domain Λ does not change under scaling, see the remark at the end of Subsection 4.1. Thus it suffices to prove (4.7) for α = ρ = 1 and arbitrary ℓ ≥ ℓ 0 with a ℓ 0 > 0. Moreover without loss of generality assume that u = µ = 0.
Choose the coordinates in such a way that Λ is represented as in (4.1). Denote By virtue of (4.4), Cover the closure Λ with open balls of radius 2 √ d centred at the lattice points j ∈ Z d .
These definitions ensure that where ∁Λ = R d \ Λ. Let ψ j , j ∈ Σ, be a smooth partition of unity subordinate to the introduced covering, such that Since N (n,m) (a; 1, 1) ≤ CN (n,m) (a, ℓ, 1) ≤ C, by Theorem 3.2 we obtain By the triangle inequality (2.1), where we have used the fact that qm > d + 1, see (4.6). For j ∈ Σ 1 we use the bound T j Sq ≤ Op 1,0 (ψ j a) ≤ C, which follows from (3.7). As #Σ 1 ≤ Cℓ d−1 , C = C(ℓ 0 ), with the help of the triangle inequality we obtain Together with (4.9) this leads to As explained earlier this bound implies (4.7).
The proof in the case d = 1 is a simplified version of that for d ≥ 2. In particular, instead of (4.4) one uses (4.5). We omit the details.  respectively. Suppose that the symbol a ∈ S (n,m) satisfies (3.5). Then for t = 0 or 1 we have q N (n,m) (a; ℓ, ρ), The constant C q in the above estimates may depend on the domains Λ, Ω.
Proof. In the proof there is no difference between the cases d = 1 and d ≥ 2. As in Theorem 4.2 the bound (4.11) follows from (4.10). Cover Λ with finitely open balls B(z j , r), j = 1, 2, . . . , J where r is chosen in such a way that for each j we have B(z j , 4r)∩ Λ = B(z j , 4r) ∩ Λ 0 with some basic domain Λ 0 = Λ 0 (j). Let {φ j }, j = 1, 2, . . . , J, be a finite partition of unity subordinate to the above covering. Due to the triangle inequality (2.1) it suffices to obtain the bound (4.20) for the operators of the form where b(w, ξ) = φ(w)a(w, ξ), and φ is an element of the partition above supported in the ball B(z, r). Here we have omitted the index j for brevity. If Λ had been a basic domain then the required bound would have followed from (4.16). Let Λ 0 be a basic domain such that By construction, Now we need to show that the estimate (4.10) is preserved if one replaces Λ with Λ 0 in the last bracket on the right-hand side. Let ζ ∈ C ∞ (R d ) be as defined in (2.20), and let Observe that the distance between the supports of φ and h is at least r. Thus by Theorem 3.2 we have Here we have used (4.12). In a similar way we show that the last term on the right-hand side is bounded by Since Λ 0 is a basic domain we can use (4.7) to obtain (4.10) for the symbol b. As explained earlier, this leads to (4.10) for the symbol a.

Discontinuity in two variables.
In this subsection we prove analogues of Theorem 4.2 and Corollary 4.4 with the smooth symbol a replaced by the symbol a(x, ξ)χ Ω (ξ). Now we need a partition of unity of a special type which is described in [11], Ch. 1.
Proposition 4.5. Let τ = τ (ξ) > 0 be a Lipschitz function on R d such that for all ξ, η ∈ R d with some κ ∈ [0, 1). Then there exists a set ξ j ∈ R d , j ∈ N such that the balls B(ξ j , τ (ξ j )) form a covering of R d with the finite intersection property, i.e. each ball intersects no more than N = N(κ) < ∞ other balls. Furthermore, there exist non-negative functions ψ j ∈ C ∞ 0 (R d ), j ∈ N, supported in B(ξ j , τ (ξ j )) such that j ψ j (ξ) = 1, for all m uniformly in j.
For d = 1 the proof follows the same line argument and is somewhat simpler. We omit the details.
Just as before, using an appropriate partition of unity one can deduce the following.  The constant C q may depend on the domains Λ, Ω.
Proof. The proof is similar to that of Corollary 4.4. Cover Λ with finitely open balls B(z j , r), j = 1, 2, . . . , J where r is chosen in such a way that for each j, B(z j , 4r) ∩ Λ = B(z j , 4r) ∩ Λ 0 with some basic domain Λ 0 = Λ 0 (j). Let {B(µ k , r)}, k = 1, 2, . . . , K be a covering of Ω with the same properties. Let {φ k } and {ψ j } be finite partitions of unity subordinate to the above coverings. Due to the triangle inequality (2.1) it suffices to obtain the bound (4.20) for the operators of the form where b(x, ξ) = φ(x)ψ(ξ), and φ, ψ are elements of the partitions above supported in the balls B(z, r) and B(µ, r). We omit the indices j, k for brevity. If Λ and Ω had been basic domains then the required bound would have followed from (4.16). Let Λ 0 and Ω 0 be basic domains such that By construction, T α = χ Λ 0 Op α,0 (b)P Ω 0 ,α (I − χ Λ ).
Now we show that the estimate (4.20) is preserved if one replaces Λ with Λ 0 in the last bracket. By (4.8), In order to estimate the last term on the right-hand side let ζ ∈ C ∞ (R d ) be as defined in (2.20), and let h(x) = ζ (|x − z|(4r) −1 ) ,h = 1 − h. Observe that the distance between the supports of φ and h is at least r. Thus by Theorem 3.2, for any m ≥ [dq −1 ] + 1 we have