Functions of self-adjoint operators in ideals of compact operators

For self-adjoint operators $A, B$, a bounded operator $J$, and a function $f:\mathbb R\to\mathbb C$ we obtain bounds in quasi-normed ideals of compact operators for the difference $f(A)J-Jf(B)$ in terms of the operator $AJ-JB$. The focus is on functions $f$ that are smooth everywhere except for finitely many points. A typical example is the function $f(t) = |t|^\gamma$ with $\gamma \in (0, 1)$. The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace $tr f(S)$ with $S$ being a Wiener-Hopf operator with a discontinuous symbol.


Introduction
In this paper we study a pair of self-adjoint operators A, B on Hilbert spaces H and G respectively. We are interested in estimates in various quasi-normed ideals of compact operators for the "quasi-commutators" of the form f (A)J − Jf (B) in terms of the "perturbation" AJ −JB, where J : G → H is a bounded operator and f : R → C is a suitable function. There is a vast literature concerned with problems of this type, with a large number of deep results. Our intention is to improve some of the existing estimates for a very specific class of functions f . The focus will be on continuous functions f that are smooth everywhere except possibly for finitely many points. One example of such function is f (t) = |t| γ with γ > 0. In this introduction we do not provide a detailed survey of the known results but concentrate on the directly relevant ones only, further references can be found e.g. in [1] and [23]. By S we denote a (quasi)-normed two-sided ideal of compact operators, and by S p , 0 < p < ∞ -the classical Schatten-von Neumann ideals.
In [19,21] it was found that if f belongs to the Besov class B 1 ∞1 (R) then the function f is S 1 -operator-Lipschitz, i.e.
for arbitrary self-adjoint operators A, B such that A − B ∈ S 1 . Conversely, as shown in [19], the estimate (1.1) implies that f ∈ B 1 11 (R) locally. Paper [8] identifies a meaningful class of self-adjoint operators, for which the condition f ∈ B 1 11 (R) is also sufficient for (1.1).
For the Schatten-von Neumann classes S p , 1 < p < ∞ conditions on the function f look simpler. Precisely, for arbitrary uniformly Lipschitz functions f it was shown in [23] that The classes S p with p ∈ (0, 1) were studied in [20] for unitary operators A and B. We discuss this in more detail in Remark 2. 5. The function f (t) = |t| γ , γ ∈ (0, 1) was studied in [2]. Let S be a normed ideal with the majorization property, see [10] for the definition. This assumption is not too restrictive as any separable ideal (e.g. S p , 0 < p < ∞) possesses this property. As shown in [2] (see also [4]), for γ ∈ (0, 1), if A ≥ 0 and B ≥ 0 are such that |A − B| γ ∈ S, then Observe that the function |t| γ , γ ∈ (0, 1) belongs to the the Hölder-Zygmund class Λ γ (R) = B γ ∞,∞ (R) locally. Among other functional spaces, this space was considered in the recent article [1]. In fact, [1] brings us closer to the objects studied in the current paper as it contains results on the quasi-commutators f (A)J − Jf (B). Precisely, for any function f ∈ Λ γ (R), γ ∈ (0, 1) it was shown in [1] that (1.4) |f under the assumption that the Boyd index β(S) of the quasi-normed ideal S is strictly less than 1, see [1], Theorem 11.5. The definition of the Boyd index can be found e.g. in [1], Section 3. For the Schatten-von Neumann ideals S p , 0 < p < ∞, the index is found by the simple formula β(S p ) = p −1 . None of the results quoted above generalizes the others but some of them have nonempty intersections. Let us compare, for instance (1.3) and (1.4) for the Schatten -von Neumann classes. Then (1.3) gives Spγ , for any p ≥ 1 and γ ∈ (0, 1), and (1.4) gives (see [1], Theorem 11.7) Spγ , under the condition pγ > 1. On the one hand (1.6) is valid for the entire class Λ γ (R), and it allows J = I, but on the other hand, (1.6) holds under the more restrictive assumption pγ > 1.
One aim of this paper is to derive the following "hybrid" of (1.3) and (1.4). For the sake of discussion we state the result in a somewhat simplified form, see Theorem 2.4 for the precise statement. Assume that f ∈ C ∞ (R \ {z}), z ∈ R, is a compactly supported function satisfying the condition with some γ > 0. Let S be a quasi-normed ideal. Then for any σ ∈ (0, γ), σ ≤ 1, the bound holds Emphasize that in contrast to (1.4), the value σ = γ is not allowed. On the other hand, there are no restrictions on the ideal S. If γ > 1 then in the formula (1.8) one can take σ = 1. Thus for S = S p , 1 < p < ∞ and J = I the bound (1.8) is in agreement with (1.2). For p ∈ (0, 1) and J = I the bound (1.8) is in line with the results of [20], see Remark 2.5 for details.
Since our choice of the function f is very specific, the proof of (1.8) does not require sophisticated methods employed in [1,19,20,21,23] where various general functional classes were studied. In particular, we do not make use of the Double Operator Integrals techniques. Instead we rely on the representation of f (A) for a self-adjoint operator A in terms of the quasi-analytic extension of the function f , which has become known as the Helffer-Sjöstrand formula, see [11,5]. The convenient quasi-analytic extension is constructed in Lemma 3.3.
In Theorem 2.10 we focus on the following useful special case of the bound (1.8). Let A be a self-adjoint operator and let P be an orthogonal projection. Then, using (1.8) with J = P , B = P AP we obtain the bound A bound of a similar nature was previously derived in [15,16] for arbitrary f ∈ W 2,∞ loc (R): The above two inequalities are helpful in problems involving Szegő-type estimates and/or asymptotics, see e.g. [15,16,14,25]. The last section of the paper, Sect. 4, illustrates the practical use of the bound (1.9) with an example of a multi-dimensional Wiener-Hopf operator with a discontinuous symbol. The operator in question, denoted by S α , is defined by (4.1), where α ≥ 1 is the "quasi-classical parameter". The objective is to obtain a two-term asymptotics of the trace tr g(S α ) with a non-smooth function g, as α → ∞. The function g is allowed to have finitely many singularities of the type described by (1.7). For smooth g the sought two-term asymptotic formula was justified in [25] and [27]. The main result of Sect. 4 is contained in Theorem 4.4, and its proof consists in "closing" the asymptotic formula for smooth g with the help of the bound (1.9). The generalization to non-smooth functions is motivated, in part, by applications in information theory and statistical physics, see e.g. [9], [13], [17]. Further discussion is deferred until Sect. 4.
Acknowledgements. The author is grateful to W. Spitzer for useful remarks. This work was supported by EPSRC grant EP/J016829/1.

Main results
2.1. Quasi-normed ideals of compact operators. We need some information from the theory of ideals of compact operators. Details can be found in [10], [3], [22]. Let S ⊂ S ∞ be a two-sided ideal. Recall that a functional · S defined for T ∈ S is said to be a quasi-norm if there exists a number κ ≥ 1 such that If, in addition, the conditions below are satisfied (4) XT Y S ≤ X Y T S , for any bounded X, Y and A ∈ S, (5) T S = T for any one-dimensional operator T , then the quasi-norm · S is said to be symmetric. The ideal S is said to be a quasi-normed ideal if it is endowed with a (symmetric) quasi-norm, and is complete. We usually omit the term "symmetric" for brevity. If κ = 1 then the quasi-norm becomes a norm.
Note an important property of quasi-norms. Below by s k (T ), k = 1, 2, . . . , we denote singular numbers of the operator T ∈ S ∞ . Lemma 2.1. Let T ∈ S and let S ∈ S ∞ be operators such that s k (S) ≤ Ms k (T ), k = 1, 2, . . . , with some constant M > 0. Then S ∈ S and S S ≤ M T S .
For normed ideals this lemma was proved in [10], and the proof for quasi-normed ideals is the same. It shows that the quasi-norm T S depends only on the singular numbers of the operator T ∈ S. This means in particular that T S = T * S = |T | S , where |T | = √ T * T . We say that a quasi-normed ideal S is a q-normed ideal if there exists an equivalent quasi-norm · S which satisfies the q-triangle inequality: S , for any T 1 , T 2 ∈ S, see e.g. [22]. In fact, any quasi-normed ideal S is a q-normed ideal with the q ∈ (0, 1] found from the equation κ = 2 q −1 −1 (q = 1 refers to a normed ideal).
As an example, we can take as S any Schatten-von Neumann ideal S p , p ∈ (0, ∞) with the standard (quasi)-norm If p ≥ 1, then this functional is a norm, and if p ∈ (0, 1) then it is a p-norm, see [24] and also [3]. Lemma 2.2. Suppose that the operator V = AJ − JB is such that |V | σ ∈ S with some σ ∈ (0, 1]. Then for all y = Im z = 0 we have By definition of the quasi-norm, where we have used the trivial bound for the left-hand side of (2.2): W ≤ 2|y| −1 J . In order to estimate the quasi-norm on the right-hand side of (2.3) estimate the singular values s k (|W | σ ): Therefore by Lemma 2.1 |W | σ S ≤ |y| −2σ |V | σ S . Substituting this bound into (2.3) we get the required estimate.
We are interested in bounds for the difference where f is a function satisfying the following condition. Below we denote by χ R the characteristic function of the interval (−R, R), R > 0.
For a function f satisfying the above condition the bound holds: One can immediately deduce from (2.5) that for any t 1 , t 2 ∈ R, so that g ∈ C 0,κ (R). Here by C 0,κ (R n ), n ≥ 1, we denote the standard class of Hölder-continuous functions f with the finite norm The next theorem constitutes the main result of the paper.
Theorem 2.4. Suppose that f satisfies Condition 2.3 with some γ > 0, n ≥ 2 and R > 0. Let S be a q-normed ideal where (n − σ) −1 < q ≤ 1 with some number σ ∈ (0, 1], σ < γ. Let A, B be two self-adjoint operators as described above such that V = AJ − JB is a bounded operator. Suppose that |V | σ ∈ S. Then One should observe that the parameters n and σ in Theorem 2.4 are not entirely independent. Indeed, the condition (n − σ) −1 < q ≤ 1 does not allow n = 2 and σ = 1 at the same time. We'll need to remember this fact in the proof of Theorem 4.4 later on.
Remark 2.5. It is appropriate to compare Theorem 2.4 with the results of the paper [20] mentioned in the introduction. In [20] it was shown for a pair of unitary operators U 1 and U 2 that . These conditions can certainly be appropriately rephrased for self-adjoint operators with the help of the Cayley transform.
For the sake of comparison, in Theorem 2.4 assume for simplicity that f ∈ C ∞ (R\{0}) is a function such that f (t) = |t| γ , γ > 0, for all |t| ≤ 1 and f (t) = 0 for |t| ≥ 2. Then using (2.7) with S = S p , J = I, R = 2 and γ > 1 we get that On the other hand, the necessary condition f ∈ B 1 p pp (R) is satisfied for any γ > 0. Note the following scaling property: Remark 2.6. Theorem 2.4 for arbitrary R > 0 follows from Theorem 2.4 for R = 1. Indeed, without loss of generality one may assume that x 0 = 0. Note that the function g(t) = R −γ f (Rt) satisfies (2.5) with R = 1 and that g n = f n . Now use bound (2.7) for the function g and the operators It is also convenient to have a separately stated result for smooth functions f .
Let A, B be two self-adjoint operators as in Theorem 2.4. Then with a constant C independent of the operators A, B, J, function g and parameter R.
Proof. Suppose first that ρ = 1 and without loss of generality set Then the function clearly satisfies (2.5) with γ = 2, x 0 = 2, R = 3 and f n ≤ C. Therefore by Theorem 2.10, If ρ > 0 is arbitrary, then use the first part of the proof for the function f (t) = g(ρt) and operators A ′ = Aρ −1 , B ′ = Bρ −1 . Now we use Corollary 2.7 to obtain bounds similar to (2.7) for functions with unbounded supports. We concentrate on smooth functions g satisfying the bound Corollary 2.8. Suppose that g satisfies (2.10) with some n ≥ 2 and β > 0. Let the ideal S and operators A, B be as in Corollary 2.7. If qβ > 1, then , with a positive constant C n independent of the operators A, B, J and function g.
with a constant C independent of m and g. Now use the q-triangle inequality (2.1): Since qβ > 1, the above bound leads to (2.11). 2.
3. An important special case. As explained in the Introduction, it is of particular interest for us to consider the case when the operator J is an orthogonal projection. Theorem 2. 10. Suppose that f satisfies Condition 2.3 with some γ > 0, n ≥ 2 and R > 0. Let S be a q-normed ideal where (n − σ) −1 < q ≤ 1 with some number σ ∈ (0, 1], σ < γ. Let A, P be a self-adjoint operator and an orthogonal projection satisfying Condition 2.9. Suppose that |P A(I − P )| σ ∈ S. Then S . with a positive constant C independent of the operators A, P , function f and parameter R. We also state the following consequence of Corollary 2.7: Corollary 2.11. Suppose that g ∈ C n 0 (−ρ, ρ), with some ρ > 0 and n ≥ 2. Let S be a q-normed ideal with (n − σ) −1 < q ≤ 1, where σ ∈ (0, 1]. Let the operator A and orthogonal projection P be as in Theorem 2.10. Then with a constant C independent of the operator A and projection P .

Proof of Theorem 2.4
3.1. A quasi-analytic extension. In order to study functions of self-adjoint operators we use the formula known as the Helffer-Sjöstrand formula, see [11], [5]. It requires the notion of a quasi-analytic extension of f . We use a somewhat more complicated definition than that in [11] since we are working with non-smooth functions. For the sake of simplicity we concentrate on compactly supported functions, although all the definitions with appropriate modifications can be given for more general functions. Let Thenf is said to be a quasi-analytic extension of f .

Proposition 3.2. Let
A be a self-adjoint operator on a Hilbert space H. Let f : R → R be a function as in Definition 3.1, and letf be its quasi-analytic extension. Then Proof. It suffices to show that For a δ > 0 split the plane into two regions: First estimate the contribution from D 2 : By Definition 3.1 the integral tends to zero as δ → 0. Using the property we simplify the remaining integral: Rewrite: The last two terms converge to zero for all and hence the last two terms on the right-hand side do not exceed C|t − x| κ−1 . Thus their integral over x converges to zero as δ → 0. Consequently, Since f is continuous, the integral converges to f (t), as claimed.
Versions of the formula (3.1) have been known well before the paper [11]. In [6] E.M. Dyn'kin developed functional calculus for operators in Banach spaces, based on a formula in the spirit of (3.1). Similar functional constructions can be found in L. Hörmander's book [12], Section 3.1, so (3.1) must have been known to him earlier. In [7] E.M. Dyn'kin found a characterization of the classical Besov and Sobolev classes in terms of quasi-analytic extensions. These results were used in [8].
In order to establish (3.5) we use (3.6), so that for all x = 0 we have The first term on the right-hand side already satisfies (3.5). Using the formula we conclude that in the second term on the right-hand side of (3.8) we have |x|/2 ≤ |y| ≤ |x|. Hence this term satisfies (3.5) as well.

Proof of Theorem 2.4.
Without loss of generality we may assume that x 0 = 0, and that f n = 1. Also, in view of Remark 2.6, it suffices to obtain (2.7) for R = 1 only.
Letf be the quasi-analytic extension constructed in Lemma 3.3. By the formula (3.1) and resolvent identity (2.2) we can write Let ρ(x, y) = 8 −1 |y|, and let {D j }, j = 1, 2, . . . , be a family of open discs with finite intersection property centred at some points z j = (x j , y j ) ∈ Π, of radius ρ(x j , y j ) such that ∪ j D j = Π.
Let φ j ∈ C ∞ 0 (Π) be an associated partition of unity such that |∂ l x ∂ k y φ j (x, y)| ≤ C l,k ρ(x, y) −l−k . By the "finite intersection property" we mean that the number of discs having non-empty common intersection is uniformly bounded. The existence of such a covering and such a partition of unity follows from [12], Theorem 1.4.10. Estimate the quasi-norm of For z ∈ D j expand R(z; A) and R(z; B) in the uniformly norm-convergent series The uniformity of convergence is guaranteed by the bound |z − z j | R(z j ; K) ≤ 1/8. Denote k = (k 1 , k 2 ), k 1 ≥ 0, k 2 ≥ 0. Therefore we can now expand T j in the norm-convergent series:

By Lemma 2.2,
A straightforward calculation shows that if D j ∩ F 1 = ∅ then (x j , y j ) ∈ F 2 and D j ⊂ F 4 , see (3.2) for the definition of F b , b > 0. Thus for all (x, y) ∈ D j we have Consequently, Use the q-triangle inequality again to sum over j: where we have used the finite intersection property. By assumption, we have q(n−σ)−2 > −1, and hence the integral on the right-hand side is bounded by This proves (2.7) for R = 1. As explained in Remark 2.6, this immediately leads to (2.7) for general R > 0.

Trace asymptotics for multidimensional Wiener-Hopf operators with discontinuous symbols
4.1. Definitions. Now we derive from the theorems established above estimates for some Wiener-Hopf operators on L 2 (R d ). Let Λ, Ω ⊂ R d , d ≥ 2, be two domains, and let χ Λ , χ Ω be their characteristic functions. For a bounded complex-valued function a = a(x, ξ), called symbol, define the pseudo-differential operator It is a standard fact that under the condition a ∈ W d+1,∞ (R d × R d ) the norm of the operator Op α (a) is bounded uniformly in α ≥ 1, see e.g. [25], Lemma 3.9. Under Wiener-Hopf operators with discontinuous symbols here we understand operators of the form The function a is assumed to be smooth, and it is the presence of the projection P Ω,α that suggests the term "discontinuous symbol". In Sect. 5 we consider somewhat more general discontinuous symbols. We impose the following conditions on the domains Λ and Ω.
The domain Λ is piece-wise C 1 , and Ω is piece-wise C 3 .
By the Lipschitz domain we understand a domain which locally looks like a set of points above the graph of a suitable Lipschitz function. Precise definitions of this property as well as of piece-wise smoothness are given in [27], Definition 2.1.
We are interested in the large α asymptotics of the trace of the operator with a function g : R → C which is smooth except for finitely many points. Let us define the asymptotic coefficients entering the main asymptotic formulas. For 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.e. x and ξ respectively. For any function g ∈ C 0,κ (C), κ > 0, and any number s ∈ C, we also define The next result is found in [27], Theorem 2.5:  Observe that for the polynomials g 0 (t) ≡ 1 and g 1 (t) = t both sides of the above formula equal zero, so the asymptotics hold trivially.
The main focus of of [27] was on non-smooth domains Λ and Ω. As a result the above proposition was formally proved in [27] for the case d ≥ 2 only, although a similar, somewhat simplified argument should give (4.6) for the case d = 1 as well, with an appropriately modified definition of the coefficient W 1 , see e.g. [28]. However we do not pursue this objective in the current paper.
Our aim here is to extend Proposition 4.3 to functions g that have just C 2 local smoothness, and may lose differentiability at finitely many points.  = a(x, ξ) be a globally bounded C ∞ -function. Let X = {z 1 , z 2 , . . . , z N } ⊂ R, N < ∞, be a collection of points on the real line. Suppose that g ∈ C 2 (R \ X) is a function such that in a neighbourhood of each point z ∈ X it satisfies the bound with some γ > 0. Then Let us make some comments on Theorem 4.4.
(1) The assumption a ∈ C ∞ is made for simplicity. In fact, some finite smoothness, depending on the value of the parameter γ, would suffice, but we have chosen to avoid ensuing technicalities.
(2) Suppose that Λ is bounded and that g ∈ C ∞ (R) is a function such that g(0) = 0.
Then both operators on the right-hand side of (4.2) are trace class, and formula (4.7) is just an indirect way to write the asymptotics tr g(S α ) = α d W 0 (g(Re a); Λ, Ω) as α → ∞, established in [27], Theorem 2.3. Indeed, one can show that see e.g. [25], Section 12.3, where a similar calculation was done. Substituting this in (4.7) one obtains (4.8).
If the symbol a depends only on the variable ξ, i.e. a = a(ξ), then the reduction of (4.7) to (4.8) for bounded Λ is more straightforward. Indeed, in this case the second operator on the right-hand side of (4.2) is given by (4.9) χ Λ Op α g(Re aχ Ω ) χ Λ .
This operator is clearly trace class for any continuous g such that g(0) = 0. Integrating its kernel along the diagonal, one easily finds the exact value of its trace: α d W 0 (g(Re a); Λ, Ω). (3) Suppose that a = a(ξ), and that g = 0 on the range of the function Re a. Then the operator (4.9) equals zero, so that Theorem 4.4 implies that g(S α (a, Λ, Ω)) is trace class and its trace satisfies the asymptotic formula It is interesting to point out that this formula holds for both bounded or unbounded domains Λ. Another proof of formula (4.10) in the special case a ≡ 1 and g(0) = g(1) = 0 was given in [17]. It was motivated by the study of the entanglement entropy for free Fermions at zero temperature, see also [9] and [13]. Mathematically speaking, the entropy is found as trace of the operator η β (S α ), β > 0, with the operator S α = S α (1, Λ, Ω) for bounded Λ and Ω, and with the function η β defined by (4.11) η β (t) = if t ∈ [0, 1], end extended by 0 to the rest of the real line. Clearly, for β = 1 the function η β satisfies the conditions of Theorem 4.4 with γ = β, and for β = 1with arbitrary γ < 1.
Before proving Theorem 4.4 we list some useful facts.
A crucial ingredient in the proof of Theorem 4.4 is the following lemma.
The above bound can be derived from [26] Theorem 4.6 in the same way as [26] Corollary 4.7.

4.2.
Proof of Theorem 4.4. Throughout the proof we denote for brevity D α (g) = D α (a, Λ, Ω; g), and W 1 (g) = W 1 (A(g; Re a); ∂Λ, ∂Ω). Recall that the norm of the operator Op α (a) is bounded uniformly in α ≥ 1, so without loss of generality we may assume that Op α (a) ≤ 1/2 and that g is supported on the interval [−1, 1], and it is real-valued.
The proof splits into two parts.
Corollary 2.11, Lemma 4.7 and bound (4.15) give that and as a consequence, Now, using Proposition 4.3 for the polynomial g ε we get lim sup Due to (4.12) and (4.15), the asymptotic coefficient W 1 (f ε ) tends to zero as ε → 0, so that This implies that lim sup In the same way one obtains the appropriate lower bound for the lim inf. This completes the proof of (4.7) for g ∈ C 2 (R).
Step 2. Completion of the proof. Let g be a function as specified in the theorem. As before we assume that g is real-valued and that it is supported on the interval [−1, 1]. By choosing an appropriate partition of unity we may assume that the set X consists of one point only, which we denote by z.
Let ζ ∈ C ∞ 0 (R) be a real-valued function, satisfying (3.4). Represent g = g R (t). It is clear that g (2) R ∈ C 2 (R), so one can use the formula (4.7) established in Part 1 of the proof: R ).
For g (1) R we use Theorem 2.10 with S = S 1 , n = 2, an arbitrary σ ∈ (0, 1), σ < γ, and with the operators A, P defined in (4.16). Noticing that g (1) R 2 ≤ C g 2 , we get from Theorem 2.10 and Lemma 4.7 that Using (4.17) we obtain the bound lim sup Due to (4.13) the asymptotic coefficient W 1 (g R ) converges to zero as R → 0. Thus In the same way one obtains the appropriate lower bound for the lim inf. This completes the proof.
Both symbols a and a 1 are assumed to have compact supports in the variable ξ.
The derivation of this fact from (5.2) repeats almost word for word the proof of formula (4.8).

(5.3)
If V α is replaced by H α , then the same formula (5.3) holds with a, a 1 replaced with Re a, Re a 1 .
The derivation of Theorem 5.1 from Theorem 5.2 follows the plan of the proof of Theorem 4.4, and is omitted.
As far as Theorem 5.2 itself is concerned, its proof essentially repeats that of Proposition 4.3 (given in [27]) with some obvious modifications. Below we provide only a sketch of this proof, leaving out some details that can be easily reconstructed.
We'll need the following estimates.
Proof of Theorem 5.2 (sketch). We give the proof only for the operator V α . The version of (5.3) for the self-adjoint operator H α can be obtained following the elementary argument detailed in [25], p. 77. Furthermore, for simplicity we assume that Λ is a bounded domain, so that (5.3) amounts to lim α→∞ 1 α d−1 log α tr g p (V α (a, a 1 ; Λ, Ω)) − α d W 0 ( g p (a 1 ); Λ, Ω) + W 0 (g p (a 1 ); Λ, Ω 1 ) = W 1 (D(g p ; a, a 1 ); ∂Λ, ∂Ω), (5.4) see the remark after Theorem 5.1. Since both Λ and Ω are bounded, we may assume that the symbols a, a 1 are compactly supported in both variables. In what follows we use the following convention: for any two operators A, B depending on the parameter α ≥ 1 we write A ∼ B if A − B S 1 ≤ Cα d−1 with a constant C independent of α.