Pseudo-differential Operators and Schatten-von Neumann Classes

Transcription

1 Pseudo-differential Operators and Schatten-von Neumann Classes Ernesto Buzano and Fabio Nicola Abstract. For a Hörmander s symbol class S(m, g), it is proved that the weight m is in L p (R 2n ), with 1 p <, if and only if all pseudo-differential operators with Weyl symbol in S(m, g) are in the Schatten-von Neumann class S p (L 2 ). 1. Introduction and statement of main result In this paper we study the decay of the singular values of a class of compact pseudo-differential operators on L 2 (R n ). We deal with linear operators of the form (1.1) W a u(x) := e i(x y) ξ a ( 1 2 (x + y), ξ) u(y) dy d ξ, where as usual d ξ = (2π) n dξ. The function a(x, ξ), with (x, ξ) R 2n, is called the Weyl symbol of W a, and the mapping (1.2) a W a is the so-called Weyl quantization. At first one considers symbols, say, in the Schwartz space S(R 2n ); then one sees that the mapping (1.2) extends to the temperate distributions S (R 2n ) and taes values in the space of all linear continuous mappings S(R n ) S (R n ). Moreover, when the symbols are smooth functions satisfying convenient growth estimates at infinity, a complete symbolic calculus is available for the corresponding operators, see e.g. Hörmander s boo [9], Chapter XVIII. We are interested in conditions on the Weyl symbol a which ensure that W a is a compact operator in L 2 (R n ) with a prescribed asymptotics for its singular values, i.e. it belongs to a given Schatten-von Neumann class. Recall that the singular values s j (A), j = 1, 2,... of a compact operator A on a Hilbert space H are the eigenvalues of A := (A A) 1/2. Received by the editors December 23, Mathematics Subject Classification. Primary 47B10; Secondary 35S05. Key words and phrases. Schatten-von Neumann Classes, Hypoellipticity, Psudo-differential Operators, Weyl-Hörmander Calculus.

2 2 Ernesto Buzano and Fabio Nicola A compact operator A on H is in the Schatten-von Neumann class S p (H), with 1 p <, if the sequence of its singular values is in l p. S p (H) is a Banach algebra with respect to the norm { A Sp := s j (A) p} 1/p. j=1 The elements of S 2 (X) are the Hilbert-Schmidt operators, while S 1 (X) is the algebra of trace class operators. One also set S (H) = K(H), the ideal of compact operators on H in the algebra of bounded operators B(H), with the usual norm. We refer to Simon [14], Schatten [12], for the basic theory of these Banach algebras. As it is well-nown (see for example Folland [3]), we have and a L 1 (R 2n ) = W a is compact, a L 2 (R 2n ) W a is Hilbert-Schmidt. More in general, by complex interpolation, we have a well defined continuous mapping (see [10]) L p (R 2n ) a W a S p (L 2 ), 1 p 2, 1 p + 1 p = 1, which shows, in particular, that W a is compact if a L p (R 2n ), for 1 p 2. Without additional conditions one cannot expect that symbols in L p (R 2n ), with p > 2, define bounded operators on L 2 (R n ), let alone in S p (L 2 ). In fact, already in the case p =, it is easily seen that for the bounded symbol a(x, ξ) = e ixξ it turns out that W a (L 2 ) L 2. More in general, Simon showed in [15] that there are no estimates of the form W a B(L 2 ) C a L p when p > 2. If we instead require L -bounds for a and its derivatives, then W a becomes bounded on L 2 (R n ), as established by the celebrated Calderón-Vaillancourt Theorem. Similarly, if the symbol with a certain number of its derivatives are in L 1 (R 2n ) L (R 2n ), the corresponding operator is shown to be trace class, see Robert [13], Theorem (II-53), and Toft [17], where more general properties of Schatten-von Neumann operators in the point of view of the Weyl calculus are given. A variety of other sufficient conditions can be found in the papers by Hörmander [7, 8], Daubechies [2], Heil, Ramanathan and Topiwala [6], Gröchenig and Heil [5], where techniques from Gabor Analysis and modulation spaces are employed, as well. In this connection, Gröchenig s boo [4] is considered a reference wor concerning analysing pseudo-differential operators in terms of time-frequency analysis. Related problems in [5] and [6] have also been considered by Sjöstrand [16], using methods which are more close to traditional approaches in pseudo-differential calculus.

3 Schatten-von Neumann Classes 3 In this paper we find necessary and sufficient conditions so that W a S p (L 2 ), when a is a symbol in Hörmander classes S(m, g), we describe below: refer to [9], Chapter XVIII and [8] for further details. We employ the following notation. Given two functions f, g : X R, f(x) g(x), x X, (or simply f g) means that there exists a constant C > 0 such that f(x) Cg(x) for all x X. An admissible metric is a measurable function g : (x, ξ) g x,ξ of R 2n into the set of positive definite quadratic forms on R 2n, which is slowly varying, σ- temperate, and satisfies the uncertainty principle. A g-weight is a positive measurable function m : R 2n R +, which is g- continuous and (σ, g)-temperate. We say that a smooth function a : R 2n C is a symbol if for some g-weight m sup (x,ξ) a g (x, ξ) <, for all N, m(x, ξ) where a g 0 (x, ξ) := a(x, ξ) and a ()( a g (x, ξ); (t 1, τ 1 ),..., (t, τ ) ) (x, ξ) := sup, for 1. (t j,τ j) g x,ξ (t 1, τ 1 ) 1/2 g x,ξ (t, τ ) 1/2 We denote by S(m, g) the class of all symbols of weight m and metric g. S(m, g) is a Frechét space with respect to the seminorms: a ;S(m,g) := sup (x,ξ) a g (x, ξ), ( N), m(x, ξ) where a g (x, ξ) := sup a g j (x, ξ). j Whenever a S(m, g), the operator W a is continuous on the Schwartz class S(R n ), and has a continuous extension to the tempered distributions S (R n ). Moreover W a is closable in L 2 (R n ): we denote by W a its closure. Let h(x, ξ) := where g σ is the dual quadratic form: g σ x,ξ(t, τ) := ( sup (t,τ) ) 1/2 g x,ξ (t, τ) gx,ξ σ, (t, τ) sup σ ( (t, τ); (y, η) ) 2, g x,ξ (y,η)=1 with respect to the standard symplectic form σ = n i=1 dξ i dx i in R 2n. We say that the metric g satisfies the strong uncertainty principle if there exists a positive constant δ such that h(x, ξ) (1 + x + ξ ) δ, (x, ξ).

4 4 Ernesto Buzano and Fabio Nicola Our result is the following Theorem 1.1. Consider an admissible metric g satisfying the strong uncertainty principle. For any g-weight m and all 1 p <, we have (1.3) m L p (R 2n ) Ψ(m, g) S p (L 2 ), where Ψ(m, g) = { W a : a S(m, g) }. We prove this theorem in section 3. When p = 1 the implication = in (1.3) has been already proved by Hörmander ([7], Theorem 3.9) under more general assumptions. After submission of the manuscript for publication in the ISAAC Proceedings, Toft [18] informed us that he obtained the equivalence (1.3) for every p by interpolation, without assuming the strong uncertainty principle. However, we thin that our approach has still some interest because it is simple and natural. 2. Hypoelliptic symbols and complex powers We present a definition of hypoelliptic symbol which generalizes the one given by Tulovsiǐ and Shubin in [19]: Definition 2.1. A symbol a S(m, g) is called g-hypoelliptic if there exists a positive constant R such that (i) for all N we have 1 (2.1) a g (x, ξ) a(x, ξ), for (x, ξ) R. (ii) there exists a g-weight m 0 such that (2.2) a(x, ξ) m 0 (x, ξ), for (x, ξ) R, When m 0 = m we say that the symbol a is g-elliptic. We denote by HS(m, m 0 ; g) the class of g-hypoelliptic symbols belonging to S(m, g) and satisfying (2.2). It is easily seen that the sums and (pointwise) products of hypoelliptic symbols are in turn hypoelliptic. We emphasize that in Definition 2.1 we do not require a to be slowly varying or temperate. In the sequel, we need the following composition theorem which is proved in [9], (Theorem ): Theorem 2.2. Given two symbols a S(m 1, g) and b S(m 2, g), we have that W a W b is a pseudo-differential operator with symbol a#b S(m 1 m 2, g) 1 (x, ξ) 2 = x 2 + ξ 2 = n i=1 x2 i + n i=1 ξ2 i.

5 Schatten-von Neumann Classes 5 such that (2.3) R N (a, b) := a#b N j=0 {a, b} j (2i) j j! S(m 1m 2 h N+1, g), for all N N, where {a, b} 0 = ab, and ( n ( {a, b} j = ) ) j a(x, ξ)b(y, η) ξ i y i x i η i i=0 y=x η=ξ for j > 0. More precisely, for each N, N there exists an integer l N, such that R N (a, b) ;S(m1 m 2 h N+1,g) a l N, ;S(m 1,g) b l N, ;S(m 2,g) for all a S(m 1, g) and all b S(m 2, g). Furthermore we have the following technical result, see [1] for the proof. Lemma 2.3. Given two smooth functions we have for all, j Z +. {a, b} j g (2n) j a, b : R 2n C, l=1 ( ) a g j+l l b g j+ l hj, An important property of hypoelliptic symbols is that they possess an approximate parametrix, when a big power of h is small compared to m 0 /m: Proposition 2.4. Consider a g-hypoelliptic symbol a HS(m, m 0 ; g) and assume there exists N 0 N such that (2.4) h N0 m 0 m. Then for all N N there exists a symbol q = q N S(m 1 0, g) such that (2.5) q#a 1 S(h N+1, g). Moreover for all N we have (2.6) q g (x, ξ) 1, for (x, ξ) 2R, a(x, ξ) where R is a positive constant such that the estimates (2.1) and (2.2) are satisfied. Proof. The proof is a slight generalization of an argument due to Hörmander ([7], Lemma 3.1). Let χ : R 2n R be a smooth non negative function such that { 0, if (x, ξ) R, (2.7) χ(x, ξ) = 1, if (x, ξ) 2R,

6 6 Ernesto Buzano and Fabio Nicola and define 2 (2.8) r := 1 ( a 1 χ ) #a. From the estimate (18.4.4) of [9], for all Z + we have ( ) g 1/j a 1 g (x, ξ) 1 a j (x, ξ), for (x, ξ) R, a(x, ξ) a(x, ξ) j=1 which, by hypoellipticity, implies for all N: a 1 g (x, ξ) 1, for (x, ξ) R. a(x, ξ) Because N 0 r = j=1 { a 1 χ, a } j (2i) j j! R N0 ( a 1 χ, a ), and h is a g-weight, from Lemma 2.3, Theorem 2.2, and (2.4) we obtain ) (2.9) r S (h + mhn0+1, g S(h, g). Now let and m 0 r #0 := 1, (2.10) r #j := r#r# #r, (j 1). }{{} j times Then from Theorem 2.2 and (2.9) we have r #j S(h j, g), j N. Therefore, from Lemma 2.3 and Theorem 2.2 for all, j N we obtain r #j #(a 1 χ) g (x, ξ) N 0 { r #j, a 1 χ } g i (x, ξ) + ( R N0 r #j, a 1 χ ) g (x, ξ) (2.11) i=0 h(x, ξ)j a(x, ξ) h(x, ξ)j a(x, ξ) + h(x, ξ)j+n0+1 m 0 (x, ξ) + h(x, ξ)j m(x, ξ) h(x, ξ)j, for (x, ξ) 2R. a(x, ξ) 2 We mean that a(x, ξ) 1 χ(x, ξ) = 0 when a(x, ξ) = 0 and χ(x, ξ) = 0.

7 Schatten-von Neumann Classes 7 For all N N define q := N r #j # ( a 1 χ ). j=0 From the estimate (2.11) we have (2.6) and q S(m 1 0, g). Finally, from (2.8) we have N q#a = r #j # ( a 1 χ ) #a = j=0 N r #j # (1 r) = 1 r #(N+1), j=0 which implies q#a 1 S(h N+1, g). When h is small enough, we can show that hypoelliptic operators have essentially only one closed extension: Theorem 2.5. Consider a g-hypoelliptic symbol a HS(m, m 0 ; g) and assume there exists N 0 N such that (2.12) h N0 inf {1, m 0}. m Then the minimal and the maximal extension of W a to L 2 (R n ) coincide. Proof. The proof is a slight generalization of an argument due to Hörmander. Denote by W a the extension of W a to S (R n ). We have to show that for any u S (R n ) such that W a u L 2 (R n ), there exists a sequence u j in S(R n ) such that u j u and Wa u j W a u in L 2 (R n ). Thans to Theorem 2.4 there exists q S(m 1 0, g) such that r := 1 q#a S(h N 0+1, g). Using the partition of unity associated to the metric g in Lemma of [9], one constructs a sequence of symbols χ j S(R 2n ) which is bounded in S(1, g) and converges to 1 in the C topology. Then we set u j = W χj u. We have that u j S(R n ), because χ j S(R 2n ). From the definition of r we have W a u j = W a Wχj Wq Wa u + W a Wχj Wr u = W a#χj #q W a u + W a#χj #ru.

8 8 Ernesto Buzano and Fabio Nicola Thans to Theorem 2.2 and Lemma 2.3, for all N there exists l, l N, l l, such that we have the estimates: N 0 a#χ j #q g 1 {a#χ 2 l j, q} g l! l + R N 0 (a#χ j, q) g l=0 a#χ j g l q g l + mhn 0+1 m 0 ( ) a g l χ j g l + mh N 0+1 q g l + mhn 0+1 m 0 ( a + mh N0+1) a 1 + mhn0+1 m mhn 0+1 m 0 1, for all (x, ξ) R and j Z +. It follows that sup j Z + a#χ j #q ;S(1,g) <. Moreover from (2.12) and Theorem 2.2 we have also a#χ j #r ;S(1,g) a#χ j #r ;S(mh N 0 +1,g) a l ;S(m,g) χ l ;S(1,g) r l ;S(h N 0 +1,g), for all j Z +. This means that the sequences a#χ j #q and a#χ j #r belong to a bounded set in S(1, g). Finally, from Theorem of [9] we have that a#χ j #q and a#χ j #r converge pointwise to a#q and a#r. The last part of the proof is dependent of the following lemma, due to Hörmander ([7], Lemma 3.3): Lemma 2.6. If φ j is a bounded sequence in S(1, g), converging pointwise to φ S(1, g), then W φj u W φ u in L 2 (R n ) for all u L 2 (R n ). End of the proof of Theorem 2.5. From Lemma 2.6 we obtain that u j = W χj u W 1 u = u, W a u j = W a#χj #q W a u + W a#χj #ru W a#q Wa u + W a#r u with convergence in L 2 (R n ). = W a#q#a+a#r u = W a#(q#a+r) u = W a u, Corollary 2.7. Under the same hypotheses of Theorem 2.5 we have that Wā is the adjoint of W a. In particular W a is self-adjoint iff a is real-valued. Now we recall from [1] and [11] some results on complex powers of a nonnegative operator.

9 Schatten-von Neumann Classes 9 Definition 2.8 (Komatsu). A closed operator A on a Banach space X is called non-negative if (i) (, 0) is contained in the resolvent set of A; (ii) sup (A + λi) 1 B(X) <. λ R + λ For example, when X is a Hilbert space, A is non-negative if (Au, u) X 0, Set C + := { z C : Re z > 0 } and (2.13) γ (z) := for all u D(A). Γ() Γ(z)Γ( z) = ( 1)! sin πz ( 1 z) (1 z)π, for Z + and z C \ Z. The proof of Proposition 2.9 and Theorem 2.11 below are given in [11]. Proposition 2.9. Consider a non-negative operator A on a Banach space X. Given z C +, and u D(A [Re z]+1 ) we have that the integral 3 I z A,u := γ (z) 0 λ z 1 [ A (A + λi) 1] u dλ, is convergent for all integers > Re z. Moreover these integrals are independent of : I z A,+1u = I z A,u, > Re z. Definition 2.10 (Balahrishnan). Given a non-negative operator A on a Banach space X and a complex number z C +, define a new operator JA z on X as { D (JA z ) := D ( A [Re z]+1), JA z u := Iz A,u, for any > Re z. Theorem Assume that A is a non-negative, densely defined operator on a Banach space X, then A z := JA z, z C + is the unique family of operators which enjoys the following set of properties: (i) D(JA z ) D(Az ); (ii) A z is closed; (iii) A = AA }{{ A } ; times (iv) A z A w = A z+w ; (v) (Spectral Mapping Theorem) the spectrum of A z is given by σ(a z ) = {λ z : λ σ(a)}. 3 The complex power λ z is the principal branch λ z = exp z (log λ + i arg λ), with π < arg λ π.

10 10 Ernesto Buzano and Fabio Nicola (vi) For all u D(A n ), with n Z +, the mapping z A z u is analytic in the strip { z C : 0 < Re z < n }. Assume now that A is the closure in L 2 (R n ) of a pseudo-differential operator W a and is non-negative. Next theorem shows that under suitable hypotheses W a z is pseudo-differential. Theorem Consider an admissible metric g satisfying the strong uncertainty principle and a g-hypoelliptic symbol a HS(m, m 0 ; g) such that Re a(x, ξ) R Im a(x, ξ), for (x, ξ) R, where R is a positive constant such that estimates (2.1) and (2.2) are satisfied. Assume that the closure W a of W a in L 2 (R n ) is non-negative. Let a 0 = ( a χ + 1 χ) e i(arg a)χ, where arg a = 2 arctan Im a Re a+ a and χ C (R 2n ) is a real valued function satisfying 0 χ(x, ξ) 1, (x, ξ) R 2n and (2.7). Then for all z C + there exists a g-hypoelliptic symbol a #z HS ( m Re z, m Re 0 z ; g ) such that (i) for all N and z C + we have (2.14) a #z a z g 0 (x, ξ) a 0(x, ξ) Re z h(x, ξ), z (ii) for all z C + we have W a = Wa #z. (x, ξ); Proof. We give some indications on the proof of this theorem. Details will appear in [1]. First we show that for all λ > 0 the operator W a + λ : S(R n ) S(R n ) has a continuous inverse which is a g-hypoelliptic pseudo-differential operator with Weyl symbol ã λ such that for all, l N we have the estimate ( ) (2.15) (a#ã λ) # a0 g a 0 h a 0 + λ a 0 + λ, (x, ξ), λ > 0, where ã # λ := ã λ # #ã }{{ λ. } -times Then we define a #z (x, ξ) := γ (z) l 0 λ z 1 (a#ã λ ) # (x, ξ) dλ where is an integer greater than Re z > 0 and we show that the left-hand side is independent of and the definiton of a #z is consistent with the definition of a # = a# #a by composition.

11 Schatten-von Neumann Classes 11 Integration of (2.15) yiels (2.14). Because h(x, ξ) 0 as x + ξ, (2.14) implies in particular that a #z is a g-hypoelliptic symbol. In order to complete the proof, it remains to show that (2.16) W a #z = ( W a ) z, z C+. It suffices to prove this identity on S. But then (2.16) becomes (2.17) W a #zu(x) = γ (z) 0 λ z 1 W (a#ãλ )#u(x) dλ, for all > Re z > 0, all u S and all x R n. Now a #z is the (pointwise) limit of a sequence of Riemann sums m γ (z) λ z 1 ( ) # j a#ãλj (x, ξ) γj, j=1 which are bounded in S(h N, g) for a large N. This implies that the Riemann sums m (2.18) γ (z) λ z 1 j W (a#ãλj ) #u(x) γ j, j=1 converge to W a #zu(x). Because we now that (2.18) coverge also to the right-hand side of (2.17), the proof is complete. 3. Proof of Theorem 1.1 First we prove a version of Theorem 1.1 for hypoelliptic operators: Proposition 3.1. Consider an admissible metric g satisfying the strong uncertainty principle. Then for all g-hypoelliptic symbols a we have a L p (R 2n ) W a S p (L 2 ). Proof. From Corollary 2.7 we have that the operator WāW a is essentially selfadjoint and its closure is non-negative. Moreover, the symbol a is hypoelliptic and ā#a = a 2 + b with (3.1) b g (x, ξ) C a(x, ξ) 2 h(x, ξ), for all N, (x, ξ) R. Indeed, if a is, say, in HS(m, m 0 ; g), then by Theorem 2.2 and Lemma 2.3 for all N, N N there exists l,n such that (3.2) b g (x, ξ) N l=1 1 2 l l! {a, a} g (x, ξ) + R N(a, a) g (x, ξ) a g l,n (x, ξ) a g l,n (x, ξ)h(x, ξ) + m(x, ξ) 2 h(x, ξ) N+1 a(x, ξ) 2 h(x, ξ) + m(x, ξ) 2 h(x, ξ) N+1,

12 12 Ernesto Buzano and Fabio Nicola for (x, ξ) R. Since the weights m and m 0 are temperate, the functions m ±1, m ±1 0 have a polynomial growth; hence we see that if N is large enough, by the strong uncertainty principle, m(x, ξ) 2 h(x, ξ) N m 0 (x, ξ) 2 a(x, ξ) 2 for (x, ξ) R, and therefore (3.2) implies (3.1). Because h(x, ξ) 0 as (x, ξ), we have that ā#a is g-hypoelliptic. It follows therefore from Theorem 2.12 that the powers of WāW a are pseudo-differential operators. In particular W a p/2 = WāW a p/4 = W(ā#a) #(p/4). On the other hand, by (i) of Theorem 2.12, (ā#a) #(p/4) = a p/2 + b p, with b p (x, ξ) C p a(x, ξ) p/2 h(x, ξ), (x, ξ) R. Because h(x, ξ) 0 as (x, ξ), we obtain that (ā#a) #(p/4) L 2 (R 2n ) a L p (R 2n ). Now it is well-nown that a pseudo-differential operator is Hilbert-Schmidt if and only if its symbol is in L 2 (R 2n ). Thus we have shown that W a p/2 is Hilbert- Schmidt if and only if a L p (R 2n ). On the other hand, by Spectral Mapping Theorem, we have that the singular values of W a p/2 are those of Wa to the power p/2: s j ( Wa p/2 ) = s j ( Wa ) p/2. Thus W a p/2 S2 (L 2 ) W a S p (L 2 ). Now we may prove Theorem 1.1. Assume that Ψ(m, g) S p (L 2 ). It is nown (see for example [9], pg. 143) that there exists a symbol m S(m, g) such that C 1 m m Cm for a suitable constant C > 1. In particular m is g-elliptic and W m belongs to S p (L 2 ). Then by Proposition 3.1 we have m L p (R 2n ), which implies m L p (R 2n ). Assume now that m L p (R 2n ), i.e. m L p (R 2n ) and let a S(m, g). We have to show that W a S p (L 2 ). By linearity we may assume that a is real and non-negative. Moreover, by hypothesis we now that m + a g m + a ;S(m,g) m m + a ;S(m,g) ( m + a)

13 Schatten-von Neumann Classes 13 for all N. Thus m + a is g-elliptic and therefore, by Proposition 3.1, W m+a S p (L 2 ), for m + a L p (R 2n ). Since W m S p, it follows that W a = W a+ m W m S p. This concludes the proof of Theorem 1.1. Notice that in the proof of Theorem 1.1 we did not apply Proposition 3.1 in its full generality, but only for symbols which are g-elliptic. On the other hand, Proposition 3.1 seems interesting in its own right, because the hypoelliptic symbol a L p (R 2n ) is not assumed to belong to a class S(m, g) associated with a weight m in L p (R 2n ). References [1] Buzano E. and Nicola F., Complex powers of hypoelliptic operators, In preparation. [2] Daubechies L., On the distributions corresponding to bounded operators in the Weyl quantization, Comm. Math. Phys. 75 (1980), [3] Folland G. B., Harmonic analysis in phace space, Princeton University Press, [4] Gröchenig K.H., Foundation of Time-Frequency Analysis, Birhäuser, Boston, [5] Gröchenig K.H. and Heil C., Modulation spaces and pseudo-differential operators, Integr. Equ. Op. Theory 34 (1999), no. 4, [6] Ramanathan J., Heil C. and Topiwala P., Singular values of compact pseudodifferential operators, J. Funct. Anal. 150 (1997), [7] Hörmander L., On the asymptotic distribution of the eigenvalues of pseudodifferential operators in R n, Ariv för Mat. 17 (1979), [8], The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979), [9], The analysis of linear partial differential operators III, Grundleheren der mathematischen Wissenschaften, vol. 274, Springer-Verlag, Berlin, [10] Howe, R., Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), [11] Martínez Carracedo C. and Sanz Alix M., The theory of fractional powers of operators, North-Holland Mathematics Studies, vol. 187, Elsevier Science B.V., Amsterdam, [12] Schatten R., Norm ideals of completely continuous operators, Ergebnisse der Mathemati und ihrer Grenzgebiete, Berlin-Göttingen-Heidelberg, Springer, [13] Robert D., Autour de l approximation semi-classique, Progress in Mathematics, vol. 68, Birhäuser, Boston, MA, [14] Simon B., Trace ideals and their applications I, London Math. Soc. Lecture Note Series, Cambridge University Press, Cambridge London New Yor Melbourne, [15], The Weyl transform and L p functions on phace space, Proc. Amer. Math. Soc. 116 (1992), [16] Sjöstrand J., Wiener type algebras of pseudodifferential operators, Séminaire Equations aux Dérivées Partielles, Ecole Polytechnique, 1994/1995, Exposé n o IV.

14 14 Ernesto Buzano and Fabio Nicola [17] Toft J., Continuity properties for non-commutative convolution algebras with applications in pseudo-differential calculus, Bull. Sci. Math. (2) 126 (2002), [18], Personal communication. [19] Tulovsiǐ, V. N. and Shubin, M. A., On the asymptotic distribution of eigenvalues of pseudodifferential operators in R n, Math. USSR Sborni 21 (1973), no. 4, Acnowledgments It is a pleasure for us to express our gratitude to Joachim Toft and Luigi Rodino for helpful discussions. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, Torino, Italy address: buzano@dm.unito.it, nicola@dm.unito.it.