Continuity and Schatten–Von Neumann Properties for Pseudo–Diﬀerential Operators and Toeplitz Operators on Modulation Spaces

The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria

Continuity and Schatten–von Neumann Properties for Pseudo–Diﬀerential Operators and Toeplitz operators on Modulation Spaces

Joachim Toft

Vienna, Preprint ESI 1732 (2005) November 2, 2005

Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at CONTINUITY AND SCHATTEN-VON NEUMANN PROPERTIES FOR PSEUDO-DIFFERENTIAL OPERATORS AND TOEPLITZ OPERATORS ON MODULATION SPACES

JOACHIM TOFT

p,q Abstract. Let M(ω) be the modulation space with parameters p,q and weight function ω. We prove that if p1 = p2, q1 = q2, α ∞ ω1 = ω0ω and ω2 = ω0, and ∂ a/ω0 ∈ L for all α, then the Ψdo p1,q1 p2,q2 p,q at(x, D): M(ω1) → M(ω2 ) is continuous. If instead a ∈ M(ω) for appropriate p, q and ω, then we prove that the map here above is continuous, and if in addition pj = qj = 2, then we prove that at(x, D) is a Schatten-von Neumann operator of order p. We use these results to discuss continuity for Toeplitz operators.

Mathematics Subject Classiﬁcations (2000): Primary: 47B10, 35S05, 47B35, 47B37; Secondary: 42B35, 46E35.

Key words: Schatten-von Neumann, pseudo-diﬀerential operators, Toeplitz operators, modulation spaces, embeddings.

0. Introduction In [G2] and [GH1], Gröchenig and Heil present an alternative method, based on time-frequency analysis when investigating pseudo-differential operators with non-smooth symbols belonging to non-weighted modulation spaces. Here they make suitable Gabor expansions of the symbols, which in certain extent essentially reduce the problems in such way that the symbols are translations and modulations of a fix and well-known function. In that end, they are able to make a somewhat detailed study of compactness, and prove embedding properties between Schatten-von Neumann classes in pseudo-differential operators acting on L2, and modulation spaces. Furthermore, they also prove that any pseudo-differential operator ∞,1 with symbol in the modulation space M (denoted by Sw in [Sj2] by Sjöstrand) is continuous on any non-weighted modulation space M p,q. Since L2 = M 2,2, it follows that in particular that such operators are continuous on L2, a property which was proved by Sjöstrand in [Sj1], where modulation spaces were used as symbol classes for the first time. 0 Furthermore, since S0 , the set of functions which are bounded together 1 with all their derivatives, is contained in M ∞,1, it follows from these 0 investigations that any pseudo-differential operator with symbol in S0 is continuous on M p,q. The latter result was remarked in the L2-case in [Sj1], and for general p and q, the result is a special case of Theorem 2.1 in [Ta] by Tachizawa. Some further generalizations can also be found in the independent papers [To5] and [GH2], where pseudo-differential operators with symbol class M p,q are considered. In [To6] these results were further extended for Weyl operators where the symbols belong to weighted modulation spaces. Important parts of the investigations here above concern modulation spaces, introduced by Feichtinger in Fe2 and [Fe4] during the period 1980–1983 as an appropriate family of function and distribution spaces to have in background when discussing certain problems within time-frequency analysis. The basic theory of such spaces were thereafter established by H. Feichtinger and K. Gröchenig in the papers [Fe3], [FG1]–[FG2] and [G1]. Roughly speaking, for an appropriate p,q weight function ω, the modulation space M(ω) is obtained by imposing p,q a mixed L(ω)-norm on the short-time Fourier transform of a tempered distribution. The non-weighted modulation space M p,q is then obtained by choosing ω = 1. In terms of modulation spaces it is sometimes easy to obtain information concerning growth and decay properties, as well as certain localization and regularity properties for distributions. In this paper we continue the discussions in [To5] and [To6] concerning continuity for pseudo-differential operators in background of mod- ulaiton space theory. More precisely, we consider pseudo-differential operators with smooth symbols or with symbols belonging appropriate modulation spaces, and discuss continuity for such operators when acting on modulation spaces. Especially we are concerned with a somewhat detailed study of continuity and compactness for pseudo-differential operators acting between modulation spaces of Hilbert type in terms of Schatten-von Neumann classes. In particular we investigate trace-class and Hilbert-Schmidt properties. Except for the Hilbert-Schmidt case it is in general a hard task to find complete characterizations of Schatten-von Neumann classes. One is therefore forced to find embeddings between such classes and other spaces which are more convenient. In Section 5 we discuss embeddings between such classes and modulation spaces, and generalize certain results in [GH1], [To2] and [To5]. The situation in Section 5 is more complicated depending on the fact that we consider operators acting on modulation spaces which involve weight functions of general types, instead of operators acting on L2 which is the case in the latter papers. In particular, by choosing the involved weight functions in appropriate

2 ways, we may use our results to discuss Schatten-von Neumann properties for pseudo-differential operators acting between weighted Lebesgue spaces and/or Sobolev spaces of Hilbert type. The general types of modulation spaces which are involved in the continuity investigations cause new problems comparing to [GH1] and [To5]–[To6]. These problems are overcome by using a related Gabor technique as in [GH1], leading to a convenient way to expand the symbols, and discretization of certain parts of the problems. The requested results are thereafter obtained by using techniques in modulation space theory, which are well-known within time-frequency analysis, in combination with certain results in harmonic analysis and pseudodifferential calculus. In the last part we apply these results to Toeplitz operators (in the litterature, the terms localization operators and anti-Wick operators also occur), which are convenient to study in background of pseudo- differential calculus. In particular we extend certain continuity and Schatten-von Neumann results in [Bg], [CG] and [To5]. Some attention is also paid to discuss pseudo-differential operators with smooth symbols acting on modulation spaces. These investigations are presented in Section 3, where we extend some results by Tachizawa in [Ta] and in [To6]–[To7]. In order to describe our results in more detail we recall the definition of modulation spaces. Assume that χ ∈ S (Rm) \ 0, p, q ∈ [1, ∞] and 2m that ω is an appropriate function on R , and let τxχ(y) = χ(y − x) when x,y ∈ Rm. (We use the same notation for the usual functions and distribution spaces as in e. g. [Hö].) Then the modulation space p,q m S ′ m M(ω)(R ) consists of all f ∈ (R ) such that

kfk p,q = kfk p,q,χ M(ω) M(ω)

(0.1) q/p 1/q p ≡ |F (f τxχ)(ξ)ω(x, ξ)| dx dξ < ∞, Z Z with the obvious modiﬁcations when p = ∞ and/or q = ∞. Here F denotes the Fourier transform on S ′(Rm), which takes the form

F f(ξ)= f(ξ)=(2π)−m/2 f(x)e−ihx,ξi dx Z b when f ∈ S (Rm). Next assume that t ∈ R is fixed and that a ∈ S (R2m). Then the pseudo-differential operator at(x, D) is the continuous operator on S (Rm) which is defined by the formula

(at(x, D)f)(x) = (Opt(a)f)(x) (0.2) = (2π)−m a((1 − t)x + ty,ξ)f(y)eihx−y,ξi dydξ. ZZ 3 ′ 2m The definition of at(x, D) extends to any a ∈ S (R ), and then m ′ m at(x, D) is continuous from S (R ) to S (R ). (See e.g. [Hö].) If w t = 1/2, then at(x, D) is equal to the Weyl operator a (x, D) for a. If instead t = 0, then the standard (Kohn-Nirenberg) representation a(x, D) is obtained. In Section 5 we discuss continuity for pseudo-differential operators acting on modulation spaces when the operator symbols belong to modulation spaces. Some of these questions were discussed for Weyl operators (i. e. when t = 1/2 in (0.2)) already in Section 5 in [To6]. In the first part of Section 5 we extend the results in [To6] to arbitrary t ∈ R. In particular we find appropriate conditions on ω, ω1, ω2 and p, q, pj ,qj for j = 1, 2, in order for a (x, D): M p1,q1 → M p2,q2 t (ω1) (ω2) p,q to be continuous when a ∈ M(ω). The second part of Section 5 is devoted to the case when the M pj,qj (ωj) above are Hilbert spaces, i.e. pj = qj = 2 for j = 1, 2. In this case a more detailed continuity study in terms of Schatten-von Neumann properties is performed. These considerations are dependent on some preparations which are made in Section 2 and in Section 4. Recall that an operator T from M 2,2 to M 2,2 belongs to I , the set of Schatten- (ω1) (ω2) p von Neumann operators of order p ∈ [1, ∞], if and only if 1/p p sup |(T fj, gj )M 2 | < ∞, X (ω2) where the supremum is taken over all orthonormal sequences (fj) in M 2,2 and (g ) in M 2,2 . In particular, this implies that I is the (ω1) j (ω2) ∞ set of linear and continuous operators, and that T is compact when T ∈ Ip and p < ∞. (Cf. [Si] and [ST].) We are then concerned with classification and embedding properties for the set st,p(ω1, ω2) which ′ 2m consists of all a ∈ S (R ) such that at(x, D) ∈ Ip. In Section 5 we prove that p,q1 p,q2 M(ω) ⊆ st,p(ω1, ω2) ⊆ M(ω) , for appropriate choices of q1, q2 and ω. In particular, our investigations concern Schatten-von Neumann properties for pseudo-differential op- 2 erators which map the Sobolev space Hs1 or the weighted Lebesgue 2 2 2 2 space Ls1 to Hs2 or Ls2 , since each of these spaces agrees with M(ω) if p m ω is chosen in an appropriate way. Here Hs (R ) is the Sobolev space of distributions with s derivatives in Lp, i.e. it consists of all f ∈ S ′ such that (1 − ∆)s/2f ∈ Lp(Rm). In Section 6 we discuss continuity and Schatten-von Neumann properties for Toeplitz operators in the context of modulation space theory. Recall that the Weyl symbol for a Toeplitz operator is obtained by a convolution between the Toeplitz symbol and a Wigner distribution (see e. g. [To1]–[To7] or [W1]). Consequently, any property which hold 4 for Weyl operators imposes certain properties for Toeplitz operators. In particular we may apply the continuity and Schatten-von Neumann properties in Section 5 to the theory of Toeplitz operators. To that end we obtains some extensions and improvements of certain results in [BCG], [CG] and Section 5 in [To6]. In Section 3 we discuss continuity for pseudo-differential operators 2m 2m with symbols in S(ω)(R ), the set of all smooth functions a on R such that ∂αa/ω ∈ L∞(R2m). Here ω is an appropriate weight function 2m on R . We generalize Theorem 2.2 in [To6], and prove that if ω0 on R2m is an appropriate weight function on R2m, t ∈ R and a ∈ 2m S(ω)(R ), then the pseudo-differential operator at(x, D) is continuous from M p,q to M p,q . For example, we may choose ω(x, ξ) = hxithξis, (ω0ω) (ω0) which is considered in Theorem 2.1 in [Ta]. Here and in what follows we let hxi =(1+ |x|2)1/2.

1. Preliminaries In this section we discuss basic properties for modulation spaces. The proofs are in many cases omitted since they can be found in [BT], [Fe2]–[Fe4], [FG1]–[FG3], [G2] or [To1]–[To6]. We start by recalling some properties of the weight functions which ∞ m are involved. We say that the function ω ∈ Lloc(R ) is v-moderate for ∞ m some appropriate function v ∈ Lloc(R ), if there is a constant C > 0 such that

m (1.1) ω(x1 + x2) ≤ Cω(x1)v(x2), x1,x2 ∈ R . The function v is then said to moderate ω. If in addition (1.1) holds for ω = v, then v is said to be a moderate or submultiplicative function. As in [To6] we let P(Rm) denote the cone which consists of all ∞ m 0 < ω ∈ Lloc(R ) such that ω is v-moderate, for some polynomial v on m m m R . Moreover, we let P0(R ) be the set of all smooth ω ∈ P(R ) such that (∂αω)/ω is bounded for every α. Note that if ω ∈ P(Rm), then ω(x)+ ω(x)−1 ≤ P (x), x ∈ Rm for some polynomial P on Rm. Let ω1 and ω2 be positive functions such that ω2/ω1 is a bounded. Then we write ω2 ≺ ω1. If ω1 ≺ ω2 ≺ ω1, then ω1 and ω2 are called equivalent and we write ω1 ∼ ω2. In most of the applications, it is no restriction to assume that the weight functions belong to P0, which is a consequence of the following lemma. (See also [To6].)

m Lemma 1.1. Assume that ω ∈ P(R ). Then there is a function ω0 ∈ m P0(R ) such that ω0 ∼ ω.

Proof. The assertion follows by letting ω0 = ω ∗ ϕ for some 0 ≤ ϕ ∈ S (Rm) \ 0. 5 If H is a Hilbert space, then its scalar product is denoted by ( · , · )H , or ( · , · ) when there are no confusions about the Hilbert space structure. The duality between a topological vector space and its dual is denoted by h· , · i. For admissible a and b in S ′(Rm),we set(a,b)= ha, bi, and it is obvious that ( · , · ) on L2 is the usual scalar product. Next assume that B1 and B2 are topological spaces. Then B1 ֒→ B2 means that B1 is continuously embedded in B2. In the case that B1 and B2 are Banach spaces, B1 ֒→ B2 is equivalent to B1 ⊆ B2 and kxkB2 ≤ CkxkB1, for some constant C > 0 which is independent of x ∈ B1. m Next let V1 and V2 be vector spaces such that V1 ⊕V2 = R and V2 = ⊥ S ′ V1 , and assume that v0 ∈ (V1) and that v(x1,x2)=(v0 ⊗1)(x1,x2), where xj ∈ Vj for j = 1, 2. Then v(x1,x2) is identiﬁed with v0(x1), and we set v(x1,x2)= v(x1). Assume that ω ∈ P(R2m), p, q ∈ [1, ∞], and that χ ∈ S (Rm) \ 0. p,q m Then recall that the modulation space M(ω)(R ) is the set of all f ∈ S ′ m p,q (R ) such that (0.1) holds. We note that the deﬁnition of M(ω) is independent of the choice of χ. (See Proposition 1.3 below.) p,q p,q If ω = 1, then the notation M is used instead of M(ω). Moreover p p,p p p,p we set M(ω) = M(ω) and M = M . Remark 1.2. We are also concerned with the following family of function and distribution spaces which are related to the Wiener amalgam spaces. Assume that p, q ∈ [1, ∞] and that ω ∈ P(R2m). Then the p,q m S ′ m space W(ω)(R ) consists of all a ∈ (R ) such that

q/p 1/q p,q F p kakW = | (a τxχ)(ξ)ω(x, ξ)| dξ dx (ω) Z Z is ﬁnite. (Cf. Deﬁnition 4 in [FG3].) We recall that W p,q = F M p,q when ω (x, ξ)= ω(−ξ,x) ∈ P(R2m). (ω) (ω0) 0 In fact, letχ ˇ(x) = χ(−x) as usual. Then Parseval’s formula and a change of the order of integration shows that

−1 (1.2) |F (a τξχ)(x)| = |F (a τxχˇ)(ξ)|, and the assertion follows.b Web refer to [Fe4] and [FG3] for more facts p,q about the W(ω)-spaces. The convention of indicating weight functions with parenthesis is P m p m used also in other situations. For example, if ω ∈ (R ), then L(ω)(R ) is the set of all measurable functions f on Rm such that fω ∈ Lp(Rm), i. e. such that kfk p ≡kfωkLp is ﬁnite. L(ω) Next we consider the Fourier transform of functions and distributions deﬁned on R2m. By interpreting R2m as the phase space with dual 6 variables (y, η), it follows that the Fourier transform takes the form

(1.3) (Ff)(y, η)= f(y, η) ≡ (2π)−m f(x, ξ)e−i(hx,ηi+hy,ξi) dxdξ, ZZ f b when f ∈ L1(R2m). Then it follows that F is a homeomorphism on f ∈ S (R2m) which extends to a homeomorphism on f ∈ S ′(R2m) f and to a unitary map on L2(R2m), since similar facts hold for F . For conveniency we call the Fourier transform in (1.3) as the phase space Fourier transform. p,q p p,q p p,q p We use the notation M(ω), M(ω), M and M instead of M(ω), M(ω), M p,q and M p respectively,f whenf F fis used insteadf of F , in the deﬁni- tion of modulation spaces of distributions on R2m. f The following proposition is a consequence of well-known facts in [Fe4] or [G2]. Here and in what follows, we let p′ denotes the conjugate exponent of p, i.e. 1/p + 1/p′ = 1.

Proposition 1.3. Assume that p, q, pj,qj ∈ [1, ∞] for j = 1, 2, and 2m that ω, ω1, ω2,v ∈ P(R ) are such that ω is v-moderate. Then the following are true: 1 m p,q m (1) if χ ∈ M(v)(R ) \ 0, then f ∈ M(ω)(R ) if and only if (0.1) p,q m holds, i.e. M(ω)(R ) is independent of the choice of χ. More- p,q over, M(ω) is a Banach space under the norm in (0.1), and diﬀerent choices of χ give rise to equivalent norms;

(2) if p1 ≤ p2, q1 ≤ q2 and ω2 ≺ ω1, then ;(S (Rm) ֒→ M p1,q1(Rm) ֒→ M p2,q2 (Rm) ֒→ S ′(Rm (ω1) (ω2) (3) the sesqui linear form ( · , · ) on S extends to a continuous ′ ′ p,q m p ,q m sesqui linear form from M(ω)(R ) × M(1/ω)(R ) to C. On the other hand, if kak = sup |(a,b)|, where the supremum is taken p′,q′ m ′ ′ over all b ∈ M(1/ω)(R ) such that kbk p ,q ≤ 1, then k·k and M(1/ω) k·k p,q are equivalent norms; M(ω) S m p,q m (4) if p,q < ∞, then (R ) is dense in M(ω)(R ). The dual space ′ ′ p,q m p ,q m of M(ω)(R ) can be identiﬁed with M(1/ω)(R ), through the S m ∞ m form ( · , · )L2 . Moreover, (R ) is weakly dense in M(ω)(R ). Proposition 1.3 (1) permits us to be rather vague about to the choice 1 of χ ∈ M(v) \ 0 in (0.1). For example, if C > 0 is a constant and Ω ′ is a subset of S , then kak p,q ≤ C for every a ∈ Ω, means that the M(ω) 1 inequality holds for some choice of χ ∈ M(v) \ 0 and every a ∈ Ω. 1 Evidently, for any other choice of χ ∈ M(v) \ 0, a similar inequality is true although C may have to be replaced by a larger constant, if necessary. 7 Next we discuss weight functions which are common in the applications. For any s,t ∈ R, set t s t (1.4) σt(x)= hxi , σs,t(x, ξ)= hξi hxi , m m 2m when x, ξ ∈ R . Then it follows that σt ∈ P0(R ) and σs,t ∈ P0(R ) for every s,t ∈ R, and σt is σ|t|-moderate and σs,t is σ|s|,|t|-moderate. 2 2 s/2 Obviously, σs(x, ξ)=(1+|x| +|ξ| ) , and σs,t = σt ⊗σs. Moreover, if m ω ∈ P(R ), then ω is σt-moderate provided t is chosen large enough. p p,q p,q For conveniency, we use the notations Ls, Ms and Ms,t instead of Lp , M p,q and M p,q respectively. (σs) (σs) (σs,t) p,q m p,q 2m It is also convenientto let M(ω)(R )(M(ω)(R )) be the completion p,q S m S 2m p,q of (R )( (R )) under the norm k·kM (k·kfp,q ). Then M ⊆ f (ω) M(ω) (ω) p,q M(ω) with equality if and only if p < ∞ and q < ∞. It follows that most of the properties which are valid for the usual modulation spaces p,q m also hold for M(ω)(R ).

Remark 1.4. Assume that p,q,q1,q2 ∈ [1, ∞]. Then the following properties for modulation spaces hold:

′ ′ p,q1 p p,q2 (1) if q1 ≤ min(p, p ) and q2 ≥ max(p, p ), then M ⊆ L ⊆ M . In particular, M 2 = L2; 0 ∞,1 (2) S0 = ∩s∈RMs,0 ; P 2m p,q m →֒ ( if ω ∈ (R ) is such that ω(x, ξ)= ω(x), then M(ω)(R (3) C(Rm) if and only if q = 1; (4) M 1,∞ is a convolution algebra which contains all measures on Rm with bounded mass; (5) if Ω is a subset of P(R2m) such that for any polynomial P on R2m, there is an element ω ∈ Ω such that P/ω is bounded, then S m p,q m S ′ m p,q m (R )= ∩ω∈ΩM(ω)(R ), (R )= ∪ω∈ΩM(1/ω)(R ); (6) if s,t ∈ R are such that t ≥ 0, then 2 2 2 2 2 2 2 Ms,0 = Hs , M0,s = Ls, and Mt = Lt ∩ Ht . (See e. g. [G2], [Fe2]–[Fe4], [FGb], [FG1]–[FG3], [To4], [To5] or [To6].)

Remark 1.5. Assume that pj ,qj ∈ [1, ∞] for 0 ≤ j ≤ N, satisfy 1 1 1 1 1 1 1 1 + + ··· + = N − 1+ , + + ··· + = . p1 p2 pN p0 q1 q2 qN q0 2m and that ω0,...,ωN ∈ P(R ) satisfy

ω0(x1 + ··· + xN , ξ) ≤ Cω1(x1, ξ) ··· ωN (xN , ξ), m for some constant C independent on x1,...,xN , ξ ∈ R . Then Sec- tion 5 in [To7] shows that there is a canonical way to extend the map (f1,...,fN ) 7→ f1 ∗···∗ fN from S ×···× S to S to a continuous, 8 symmetric and associative map from M p1,q1 ×···× M pN ,qN to M p0,q0 , (ω1) (ωN ) (ω0) and if f ∈ M pj,qj for j = 1,...,N, then j (ωj)

p ,q p ,q (1.5) kf1 ∗···∗ fN kM 0 0 ≤ C kfjkM j j , (ω0) (ω ) Yj=1 j for some constant C, independent of f1,...,fN .

Remark 1.6. In some situations we consider continuity of Banach function spaces (or solid BF-spaces), which is a natural generalization of modulation spaces. (See. e. g. [Fe4].) We recall the deﬁnition of such spaces. Assume that B1 and B2 are Banach spaces of complex-valued m measurable functions on R such that for j = 1, 2 and f ∈ Bj , then ′ ∞ τxf ∈ Bj and S ֒→ Bj ֒→ S , and Bj · L ⊆ Bj. Assume also that there is a constant C > 0 and v ∈ P(Rm) such that

∞ kτxfkBj ≤ v(x)kfkBj , and kf hkBj ≤ CkfkBj khkL , m ∞ m for every x ∈ R , f ∈ Bj and h ∈ L (R ). For any ω ∈ P(R2m) and χ ∈ S (Rm) \ 0, set

fχ,ω(x, ξ)= |F (f τxχ)(ξ)ω(x, ξ)|,

Φf (ξ)= kfχ,ω(·, ξ)kB1 , Φf (x)= kfχ,ω(x, ·)kB2.

Then let M(ω) = M(ω)(B1, B2) be thee Banach space which consists of ′ m S B all f ∈ (R ) such that kfkM(ω) ≡kΦf k 2 is finite. Also let W(ω) = ′ m W(ω)(B1, B2) be the Banach space which consists of all f ∈ S (R ) B such that kfkW(ω) ≡kΦf k 1 is finite. It follows that many properties which are valid for the modulation e spaces also hold for the spaces M(ω)(B1, B2) and W(ω)(B1, B2). Next we recall some facts from Chapter 12 and Chapter 13 in [G2] concerning discretizations of modulation spaces. Let (xj)j∈I and (ξk)k∈I be lattices in Rm, and assume that χ ∈ S (Rm) is fixed and satisfies

−1 2 m (1.6) C ≤ |χ(x − xj)| ≤ C, x ∈ R , Xj∈I ′ m for some constant C. Also assume that f ∈ S (R ). If (ξk) is suﬃ- ciently dense, then it follows from [G2] and Section 7.3 in [H¨o] that there exists a function ψ ∈ S (Rm) such that

ihx,ξki (1.7) f(x)= cj,k(f)e χ(x − xj) j,kX∈I

′ ihx,ξki (1.7) = dj,k(f)e ψ(x − xj), j,kX∈I 9 where cj,k and dj,k are the ”Fourier coeﬃcients” for f, given by the formulas F (1.8) cj,k(f)= cj,k = (fτxj ψ)(ξk) and ′ F (1.8) dj,k(f)= dj,k = (fτxj χ)(ξk). Here the sums converge in S ′(Rm). In the case of modulation spaces we can say more.

Proposition 1.7. Assume that (xj)j∈I , (ξk)k∈I , χ and ψ are the same as the above and assume that f ∈ S ′(Rm). Then the following conditions are equivalent. p,q m (1) f ∈ M(ω)(R ); q/p 1/q p (2) |cj,k(f)ω(xj, ξk)| < ∞; Xk Xj q/p 1/q p (3) |dj,k(f)ω(xj , ξk)| < ∞. Xk Xj On the other hand, if χ ∈ S and f is given by (1.7), and (2) holds, p,q then f ∈ M(ω). If in addition p,q < ∞, then the sum in (1.7) converges p,q to f in M(ω).

Remark 1.8. Let (xj)j∈I , (ξk)k∈I , χ, ψ and f be as the above, and assume that in addition

cj,k(f)ω(xj, ξk) → 0 or dj,k(f)ω(xj , ξk) → 0 as |xj|+|ξk| tends to inﬁnity. Then Proposition 1.7 holds with condition (1) replaced by ′ p,q m (1) f ∈M(ω)(R ); Next we apply Theorem 1.7 and Remark 1.8 to prove (complex) interpolation properties for modulation spaces. Such properties were carefully investigated in [Fe4] for non-weighted modulation spaces. Here we extend these results to weighted modulation spaces.

Proposition 1.9. Assume that 0 <θ< 1 and that p, q, pj,qj ∈ [1, ∞] satisfy 1 1 − θ θ 1 1 − θ θ = + and = + . p p1 p2 q q1 q2 P 2m 1−θ θ Also assume that ω1, ω2 ∈ (R ) and let ω = ω1 ω2. Then (Mp1,q1 (Rm), Mp2,q2 (Rm)) = Mp,q (Rm). (ω1) (ω2) [θ] (ω) p,q Proof. For any sequence λ =(λj,k)j,k≥0 of positive numbers, let ℓ(λ) be the set of all sequences (cj,k)j,k≥0 of complex numbers such that q/p 1/q p |cj,kλj,k| < ∞ Xk Xj 10 and cj,kλj,k turns to zero as j + k →∞. Then if follows from Remark 1.8 that it suﬃces to prove that

p1,q1 p2,q2 p,q (1.9) (ℓ(λ1) , ℓ(λ2) )[θ] = ℓ(λ), i i where λ = (λj,k)j,k≥0 for i = 1, 2 are sequences of positive numbers, 1 1−θ 2 θ and λj,k =(λj,k) (λj,k) . Now let T (z) for 0 ≤ Re z ≤ 1 be the operator which is defined by the formula 1 1−z 2 z T (z) (cj,k)j,k≥0 =(cj,k(λj,k) (λj,k) )j,k≥0 p,q It follows that T (z) is an isometric and bijective map from ℓ(λ) to ℓp,q when Re z = θ. An application of Stein’s interpolation theorem 1 (see [RS]) now shows that it suffices to prove (1.9) in the case λj,k = 2 λj,k = λj,k = 1. The result is then a consequence of Theorem 5.1.1 and Theorem 5.1.2 in [BL]. The proof is complete. Next we recall some facts in Chapter XVIII in [Hö] concerning pseudo- differential operators. As remarked in the introduction, it follows that 2m if a ∈ S (R ), then the pseudo-differential operator at(x, D) in (0.2) is a linear and continuous operator on S (Rm) with operator kernel −m/2 F −1 Kt,a(x,y)=(2π) ( 2 a)((1 − t)x + ty,y − x), for every fixed t ∈ R. Here F F is the partial Fourier transform of F (x,y) with respect to the y-variable. Since F2 and the map F (x,y) 7→ F ((1 − t)x + ty,y − x) are homeomorphisms on S ′(R2m), it follows ′ 2m that the definition of Kt,a extends to any a ∈ S (R ), and that the ′ ′ 2m map a 7→ Kt,a is a homeomorphism on S . For every a ∈ S (R ), m at(x, D) is defined as the linear and continuous operator from S (R ) ′ m to S (R ) with distribution kernel Kt,a. Furthermore, any linear and continuous operator from S (Rm) to S ′(Rm) has a distribution kernel in S ′(R2m) in view of kernel theorem of Schwartz. Hence, for every fixed t ∈ R, it follows that there is a one to one correspondance between such operators, and pseudo-differential operators of the form at(x, D). Consequently, for every a ∈ S ′(R2m) and s,t ∈ R, there is a unique ′ 2m b ∈ S (R ) such that as(x, D)= bt(x, D). By straight-forward applications of Fourier’s inversion formula, it follows that

i(t−s)hDx,Dξi (1.10) as(x, D)= bt(x, D) ⇔ b(x, ξ)= e a(x, ξ). (Cf. [H¨o].) By Fourier’s inversion formula it follows that if t =6 0 and Φ(x, ξ)= ihx, ξi/t, then there is a constant c such that eithDx,Dξia = ceiΦ ∗a. This motivates us to consider continuity properties for operators of the form iΦ (1.11) f 7→ SΦf ≡ (e ⊗ δV2 ) ∗ f, m where δV2 is the delta function on the vector space V2 ⊆ R and Φ ⊥ is a real-valued and non-degenerate quadratic form on V1 = V2 . By 11 Fourier’s inversion formula it follows that SΦ is a homeomorphism on S (Rm) which extends to a homeomorphism on S ′(Rm). In a similar way as in [To2]–[To6], we are concerned with continuity properties of SΦ when acting on modulation spaces. It was proved in [To2] that such operators are continuous on any M p,q. In [To6] the lat- p,q ter result was extended to modulation spaces of the form M(ω), where ω(x, ξ) = ω(ξ). In the following proposition we show that the arguments in [To6] can be used in a larger context. Proposition 1.10. Assume that ω ∈ P(R2m), p, q ∈ [1, ∞], and that m ⊥ V1, V2 ⊆ R are vector spaces such that V2 = V1 . Assume also that Φ is a real-valued and non-degenerate quadratic form on V1, and let AΦ/2 be the corresponding matrix. If ξ = (ξ1, ξ2) where ξj ∈ Vj for j = 1, 2, then ′ m p,q,χ S kSΦfkM = kfk p,q,ψ , where f ∈ (R ), (ω ) M(ω) (1.12) Φ −1 ωΦ(x, ξ)= ω(x − AΦ ξ1, ξ) and ψ = SΦχ. In particular, the following are true: (1) the map (1.11) on S ′(Rm) restricts to a homeomorphism from M p,q(Rm) to M p,q (Rm); (ω) (ωΦ) 4m (2) if t ∈ R, ω0 ∈ P(R ), and

ωt(x,ξ,y,η)= ω0(x − ty,ξ − tη,y,η),

then the map eithDx,Dξi on S ′(Rm) restricts to a homeomorphism from M p,q (R2m) to M p,q (R2m). (ω0) (ωt) Proof. We only prove the ﬁrst part of the proposition in the case m V1 = R , leaving the modiﬁcations to the general case for the reader. Assume that f ∈ S ′(Rm) and χ ∈ S (Rm). It follows easilly that the Hilbert adjoint for SΦ is given by S−Φ. This gives

ihξ,·i ihξ,·i (1.13) |F (SΦf τxχ)(ξ)| = |(SΦf, τxχe )| = |(f,S−Φ(τxχe ))|.

ihξ,·i We have to analyze S−Φ(τxχe ). By straight-forward computations it follows that

S (τ χeihξ,·i)(y)= e−iΦ(y−z)χ(z − x)eihz,ξi dz −Φ x Z

= eihx,ξi e−iΦ(y−x−z)χ(z)eihz,ξi dz Z

− ihy,ξi iΨ(x,ξ) −iΦ(y−(x−A 1 ξ)−z) = e e e Φ χ(z) dz Z

ihy,ξi iΨ(x,ξ) = e e (τ −1 (SΦχ))(y), x−AΦ ξ 12 for some real quadratic form Ψ on R2m. By inserting this into (1.13) we get ih· ,ξi |F (SΦf τxχ)(ξ)| = |(f,e (τ −1 (SΦχ)))| = |F (f τ −1 ψ)(ξ)|. x−AΦ ξ x−AΦ ξ The assertion (1) and (1.12) now follows by applying the Lp,q -norm (ωΦ) on the latter equalities, and using Proposition 1.3 (1). The assertion (2) follows now by replacing Rm with R2m, and letting 2m 2m V1 = R and Φ(x, ξ)= hx, ξi/t when t =6 0, and V2 = R when t = 0. The proof is complete. We ﬁnish the section by giving some remarks on Wigner distributions and Weyl operators of rank one. The Wigner distribution for f ∈ S (Rm) and g ∈ S (Rm) is deﬁned by the formula

(1.14) W (x, ξ)=(2π)−m/2 f(x − y/2)g(x + y/2)eihy,ξi dy. f,g Z m m 2m The map (f, g) 7→ Wf,g is continuous from S (R )×S (R ) to S (R ) and extends uniquely to a continuous mapping from S ′(Rm)×S ′(Rm) to S ′(R2m), and from L2(Rm) × L2(Rm) to L2(R2m). (See [Fo], [To1] or [To3].) Next assume that a ∈ S (R2m) and that f, g ∈ S (Rm). Then it follows from (0.2) with t = 1/2 and straight-forward computations that w −m/2 (1.15) (a (x, D)f, g)=(2π) (a, Wg,f ), In particular, (1.15) and Fourier’s inversion formula imply that if m f1, f2 ∈ S (R ), then w −m/2 (1.16) a (x, D)f(x)=(2π) (f, f2)f1(x) ⇔ a = Wf1,f2 . Consequently, a Weyl operator is a rank one operator if and only if its symbol is a Wigner distribution.

2. Schatten-von Neumann classes for operators acting on Hilbert spaces In this section we discuss Schatten-von Neumann classes of linear operators from a Hilbert space H1 to another Hilbert space H2. Such operator classes were introduced by R. Schatten in [Su] in the case H1 = H2. (See also [Si]). The general situation when H1 is not necessarily equal to H2 have thereafter been discussed in [ST]. Here we give an alternative approach based on an argument where the situation is reduced to the case H1 = H2. Let ON(Hj ), j = 1, 2, denote the family of orthonormal sequences in Hj , and assume that T : H1 → H2 is linear, and that p ∈ [1, ∞]. Then set 1/p p kT kIp = kT kIp(H1,H2) ≡ sup |(T fj, gj )H2 | 13 X (with obvious modifications when p = ∞). Here the supremum is taken over all (fj) ∈ ON(H1) and (gj ) ∈ ON(H2). Then recall that Ip = Ip(H1, H2), the Schatten-von Neumann class of order p, consists of H H all linear operators T from 1 to 2 such that kT kIp(H1,H2) is finite. Obviously, I∞(H1, H2) consists of all continuous operators from H1 to H2. If H1 = H2, then the shorter notation Ip(H1) is used instead of Ip(H1, H2). We also let I♯(H1, H2) be the set of all linear and compact operators from H1 to H2, and equip this space with the norm k·kI∞ as usual. The spaces I1(H1, H2) and I2(H1, H2) are called the sets of trace-class operators and Hilbert-Schmidt operators respectively. These definitions agree with the old ones when H1 = H2, and in this case the norms k·kI1 and k·kI2 agree with the trace-class norm and Hilbert-Schmidt norm respectively. Another describtion of the Schatten-von Neumann classes can be obtained in terms of singular numbers. Assume first that T here above is compact. Then by the spectral theorem it follows that ∞

T f = λj (f, gj)fj, Xj=1 ∞ H ∞ H for some sequences (fj)j=1 ∈ ON( 2) and (gj)j=1 ∈ ON( 1), and some sequence λ1 ≥ λ2 ≥ ··· ≥ 0. Here the numbers λj are called the singular numbers for T , and we use the notation σj(T ) for these numbers, i. e. σj(T )= λj . There is a canonical way to extend the deﬁnition of singular values to non-compact operators. More precisely, for any closed subspace V of H1, set

µV (T ) = sup kT fkH2 , H f∈V, kfk 1≤1

Then let σj(T ) be deﬁned by the formula

σj(T )= σj,H1,H2 (T ) ≡ inf µV (T ). dim V ⊥=j−1

It is straight-forward to control that σj(T ) agrees with the earlier deﬁ- p nition when T is compact. Moreover, T ∈ Ip if and only if (σj(T )) ∈ l , and

p (2.1) kT kIp = k(σj(T ))kl .

From now on we assume that the Hilbert spaces Hj are separable. Then without loss of generality we may in many questions concerning Ip(H1, H2) reduce ourself to the case H1 = H2. 0 0 In fact, assume that (fj ) and (gj ) are fixed orthonormal basis for H1 and H2 respectively, and let T0 be the linear map, defined by the formula 0 0 T0 αjfj = αjgj . X 14 X 2 Here (αj) ∈ l is arbitrary. Then T0 is an isometric bijection from H1 H ∗ to 2, and T0 ◦ T0 = IdH1 . Consequently, (fj) 7→ (T0fj) is a bijection from ON(H1) to ON(H2). This in turn implies that T 7→ T0 ◦ T is an isometric homeomorphism from Ip(H1) to Ip(H1, H2), i.e. there is a canonical identification between Ip(H1) and Ip(H1, H2). In Proposition 2.1–2.7 below we have listed some properties for spaces of the type Ip(H1, H2), and which are well-known in the case H1 = H2. (See [Si].) The general case, when H1 is not necessarily equal to H2 is now a consequence of the identification here above and the results in [Si]. Recall that p′ ∈ [1, ∞] denotes the conjugate exponent for p ∈ [1, ∞], i.e. 1/p + 1/p′ = 1.

Proposition 2.1. Assume that p, pj ∈ [1, ∞] for 1 ≤ j ≤ 2 such that p1 < p2 < ∞. Also assume that H1 and H2 are separable Hilbert spaces. Then Ip is a Banach space, I H H I H H I H H I H H p1 ( 1, 2) ⊆ p2 ( 1, 2) ⊆ ♯( 1, 2) ⊆ ∞( 1, 2), and

I∞ I I I H H (2.2) kT k ≤kT k p2 ≤kT k p1 ,T ∈ ∞( 1, 2). Moreover, equalities in (2.2) occur if and only if T is a rank one H H operator, i. e. T f = (f, g1)H1 g2 for some g1 ∈ 1 and g2 ∈ 2, and then kT kIp = kg1kH1 kg2kH2 for every p ∈ [1, ∞]. Proposition 2.2. Assume that p,q,r ∈ [1, ∞] such that 1/p + 1/q = 1/r. Also assume that Hj for 1 ≤ j ≤ 3 are separable Hilbert spaces. If T1 ∈ Ip(H1, H2) and T2 ∈ Iq(H2, H3), then T = T2 ◦ T1 ∈ Ir(H1, H3), and

(2.3) kT2 ◦ T1kIr ≤kT1kIp kT2kIq .

On the other hand, for any T ∈ Ir(H1, H3), there are operators T1 ∈ Ip(H1, H2) and T2 ∈ Iq(H2, H3) such that T = T2 ◦ T1 and equality is attained in (2.3).

Remark 2.3. Assume that p ∈ [1, ∞] and that Hj for j = 1,... 4 are Hilbert spaces such that H1 ֒→ H2, and H3 ֒→ H4. Then it follows from Proposition 2.2 that Ip(H2, H3) ֒→ Ip(H1, H4). This is also a consequence of (2.1), since it follows from the assumptions that it exists a constant C such that if T is linear from H2 to H3 and j ≥ 1, then

σj,H1,H4 (T ) ≤ Cσj,H2,H3 (T ).

∗ We note that T ∈ Ip(H1, H2) if and only if T ∈ Ip(H2, H1).

Proposition 2.4. Assume that p ∈ [1, ∞], and that H1 and H2 are separable Hilbert spaces. Then

∗ (T1,T2)=(T1,T2)H1,H2 ≡ trH1 (T2 ◦ T1) 15 deﬁnes a bilinear and continuous form from Ip(H1, H2)×Ip′ (H1, H2) to C, and for every T1 ∈ Ip and T2 ∈ Ip′ , then

(T1,T2)H1,H2 = (T2,T1)H2,H1

H H I I ′ I H H |(T1,T2) 1, 2 |≤kT1k p kT2k p , kT1k p = sup |(T1,S) 1, 2 |,

I ′ I ′ where the supremum is taken over all S ∈ p such that kSk p ≤ 1. If in addition p< ∞, then the dual space for Ip(H1, H2) can be identiﬁed with Ip′ (H1, H2) through this form.

In view of Proposition 2.4 we note that I2(H1, H2) is a Hilbert space with scalar product (·, ·)H1,H2 , and that the corresponding norm agrees with the Hilbert-Schmidt norm k·kI2 .

Proposition 2.5. Assume that p ∈ [1, ∞), and that H1 and H2 are separable Hilbert spaces. Then I1(H1, H2) is dense in I♯(H1, H2) and in Ip(H1, H2) when p < ∞. It is dense in I∞(H1, H2) with respect to the weak∗ topology. The next proposition deals with spectral properties in context of Schatten-von Neumann classes.

Proposition 2.6. Assume that T ∈ I♯(H1, H2). Then for some choice ∞ H ∞ H ∞ of sequences (fj)j=1 ∈ ON( 1), (gj)j=1 ∈ ON( 2) and λ = (λj)j=1 ∈ ∞ l0 it holds ∞

(2.4) T f = λj (f, fj)H1 gj Xj=1 where the sum on the right-hand side convergences with respect to the operatornorm. Moreover, if 1 ≤ p < ∞ then T ∈ Ip(H1, H2), if and p p only if λ ∈ l , and then kT kIp = kλkl and the sum on the right-hand side of (2.4) converges with respect to the norm k·kIp. Next we consider interpolation properties.

Proposition 2.7. Assume that p, p1, p2 ∈ [1, ∞] and 0 ≤ θ ≤ 1 such that 1/p = (1−θ)/p1 +θ/p2. Assume also that H1 and H2 are separable I I Hilbert spaces. Then the (complex) interpolation space ( p1 , p2 )[θ] is equal to Ip with equality in norms. 3. Continuity for pseudo-differential operators with symbols in S(ω) In this section we continue the discussions in [To6] concerning continuity for operators in Op(S(ω0)) in context of modulation spaces. In the first part we prove in Theorem 3.2 below that if ω, ω0 ∈ P, t ∈ R and a ∈ S , then a (x, D) is continuous from M p,q to M p,q. In particu- (ω0) t (ω0ω) (ω) lar, Theorem 2.1 in [Ta] as well as Theorem 2.2 in [To6] are covered. Later on in Section 5 we prove that Theorem 3.2 is a special case of 16 a more general result. (See Remark 5.3.) The reason why we have in- cluded a separate proof of Theorem 3.2 is that in contrast to Section 5, the proof in the present section also hold in a more general context where the continuity assertions for the involving pseudo-differential operators in terms of modulation spaces are replaced by solid BF-spaces. (See Remark 3.5.) In the second part we present some applications and prove that certain properties which are valid for Sobolev spaces carry over to modulation spaces. 2m 2m We start to give some remarks on S(ω)(R ) when ω ∈ P(R ). By 2m straight-forward computations it follows that S(ω)(R ) agrees with 2 2 S(ω, g) when g(x,ξ)(y, η) = |y| + |η| is the (constant) standard euclidean metric on R2m. (See Section 18.4–18.6 in [Hö].) Since the metric g is constant it follows that it is trivially slowly varying and σ- temperate, where σ denotes the standard symplectic form on R2m. Moreover, from the fact that ω is σt-moderate when t is large enough, it follows by straight-forward computations that ω is σ,g-temperate. The following lemma is now a consequence of Theorem 18.5.10 in [Hö].

Lemma 3.1. Assume that ω ∈ P(R2m), s,t ∈ R, and that a,b ∈ ′ 2m S (R ) such that as(x, D)= bt(x, D). Then

2m 2m a ∈ S(ω)(R ) ⇔ b ∈ S(ω)(R ). We have now the following result.

2m 2m Theorem 3.2. Assume that ω, ω0 ∈ P(R ), a ∈ S(ω)(R ) and that t ∈ R. Then a (x, D) is continuous from M p,q (Rm) to M p,q(Rm). t (ω0ω) (ω) For the proof we recall Minkowski’s inequality in a somewhat general form. Assume that p ∈ [1, ∞], dµ and dν are positive measures, and that f is measureable with respect to the product measure dµ ⊗ dν. Then Minkowski’s inequality asserts that

p 1/p 1/p f(x,y) dν(y) dµ(x) ≤ |f(x,y)|p dµ(x) dν(y). Z Z Z Z

Proof of Theorem 3.2. By Lemma 3.1 it follows that it is no restriction to assume that t = 0. We may also assume that a is real-valued, and m that ω0 ∈ P0 in view of Lemma 1.1. Let χ1 = χ ∈ S (R ). Then it follows that

ih·,ξi F (a(·, D)f τxχ)(ξ)=(a(·, D)f, τxχ1 e )

∗ ih·,ξi (3.1) =(f,a(·, D) (τxχ1 e ))

−m ih· ,ξi = (2π) (f,e H(x,ξ, · ))ω0(x, ξ), 17 where

m −ihy,ξi ∗ ih·,ξi H(x,ξ,y)=(2π) e (a(·, D) (τxχ1 e ))(y)/ω0(x, ξ)

a(z, ζ) ihy−z,ζ−ξi = χ1(z − x)e dzdζ. ZZ ω0(x, ξ)

We have to analyze H(x,ξ,y). Let N1 and N2 be even and large N1 integers, and set χ2 = h·i χ1 ∈ S and a(x + z, ξ + ζ) Φ(x,ξ,z,ζ)= N N . ω0(x, ξ)hzi 1 hζi 1 P N Since a ∈ S(ω0) and ω0 ∈ 0 is h·i -moderate for some large integer N, it follows that for any multi-index α, there is a constant Cα such that

α −N2 −N2 (3.2) |(∂ Φ)(x,ξ,z,ζ)|≤ Cαhzi hζi , provided N1 was chosen large enough. By partial integrations we obtain for some constants Cβ,γ that

H(x,ξ,y)

= Φ(x,ξ,z − x, ζ − ξ)χ (z − x)((1 − ∆ )N1/2eihy−z,ζ−ξi) dzdζ ZZ 2 z

= Cβ,γHβ,γ(x,ξ,y),

|β+Xγ|≤N1 where

H (x,ξ,y)= Φ (x,ξ,z − x, ζ − ξ)χ (z − x)eihy−z,ζ−ξi dzdζ β,γ ZZ β γ

= Φ (x,ξ,y − z − x, ζ)χ (y − z − x)eihz,ζi dzdζ. ZZ β γ

Here Φβ and χγ are some derivatives of Φ and χ2 of order β and γ respectively. In particular, (3.2) is fulﬁlled after Φ has been replaced by Φβ, and χγ ∈ S . An integration with respect to the ζ-variable in the last integral now gives

(3.3) H (x,ξ,y)= Ψ (x,ξ,y − z − x,z)χ (y − z − x) dz, β,γ Z β γ

m/2 where (2π) Ψβ(x,ξ,y,z) is the inverse partial Fourier transform of Φβ(x,ξ,y,η) with respect to the η-variable. Assume next that N3 is a large and ﬁxed integer. Then it follows from (3.2) that there is a constant C such that

α −N3 −N3 (3.4) |∂ Ψβ(x,ξ,y − z,z)|≤ Chyi hzi 18 for every multi-index α such that |α| ≤ N3, provided N2 was chosen large enough. Hence, if N4 and M are ﬁxed integers, then it follows from (3.3) that

M hyi Hβ,γ(x,ξ,x + y)= ϕβ,γ(x,ξ,y),

where ϕβ,γ satisﬁes

α −N4 (3.5) |∂ ϕβ,γ(x,ξ,y)|≤ Chyi , |α|≤ N4,

for some constant C, provided N3 was chosen large enough. By inserting these expressions into (3.1) we get

|F (a(·, D)f τxχ)(ξ)|≤ κβ,γ(x, ξ)ω0(x, ξ), X where

ih·,ξi κβ,γ(x, ξ)= |(f,e χ0(y − x)ϕβ,γ(x,ξ, ·))| = |F (f τ χ )τ ϕ(x,ξ, · ))(ξ)| (3.6) x 0 x

= (2π)−m/2 |F (f τ χ )(ξ − η)||F (ϕ(x,ξ, · ))(η)| dη, Z x 0

−M and χ0 = h·i . Now let N5 ≥ 0 be a ﬁxed integer. Then it follows from (3.5) that

|F (ϕ(x,ξ, · ))(η)|≤ Chηi−N5 ,

for some constant C, provided N4 was chosen large enough. A combination of these estimates now gives

|F (a(x, D)f τxχ)(ξ)|≤ (F (x, ·) ∗ h)(ξ) ω0(x, ξ) (3.7) where F (x, ξ)= |F (fτxχ0)(ξ)|, and h(ξ) = hξi−N5 . Now we choose a polynomial v such that ω and −N5 2 ω0 are v-moderate. If we set G(ξ) = hξi v(0, ξ) and M and N5 −M 1 are chosen large enough, then it follows that χ0 = h · i ∈ M(v) and 1 m p,q G ∈ L (R ). Hence if ω1 = ω0ω and the L(ω)-norm is applied on (3.7), 19 then Minkowski’s inequality gives

ka(x, D)fk p,q M(ω)

q/p 1/q p = |F (a(x, D)f τxχ)(ξ)ω(x, ξ)| dx dξ Z Z p q/p 1/q −N5 ≤ C1 F (x, ξ − η)hηi dη ω1(x, ξ) dx dξ Z Z Z p q/p 1/q ≤ C2 F (x, ξ − η)ω1(x, ξ − η)G(η) dη dx dξ Z Z Z q/p 1/q p ≤ C2 (F (x, ξ − η)ω1(x, ξ − η)) dx dξ G(η) dη Z Z Z

= C2kF kLp,q kGkL1 = C3kfkM p,q . (ω1) (ω1) for some constants C1, C2 and C3. This proves that

ka(x, D)fkM p,q ≤ CkfkM p,q (ω) (ω0ω) for some constant C, and the assertion follows. 2

We note that if t = 0 and ω0 = σs1,s2 where s1,s2 ∈ R, then Theorem 3.2 agrees with Theorem 1.1 in [Ta]. Next we show that Theorem 2.2 in [To6] is essentially a consequence of Theorem 3.2.

2m Corollary 3.3. Assume that p, q ∈ [1, ∞], ω ∈ P(R ) and ω0 ∈ 2m P0(R ) such that ω0(x, ξ)= ω0(x) or ω0(x, ξ)= ω0(ξ). Then ω0(x, D) is a homeomorphism from M p,q to M p,q. (ω0ω) (ω)

Proof. It follows from Theorem 3.2 that ω0(x, D) is continuous from M p,q to M p,q. On the other hand, since ω (x, ξ)= ω (x) or ω (x, ξ)= (ω0ω) (ω) 0 0 0 ′ m ω0(ξ), it follows that the inverse of ω0(x, D) on S (R ) is equal to (1/ω0)(x, D). Hence Theorem 3.2 together with the obvious fact that 1/ω0 ∈ P0 give

kfkM p,q = k(1/ω0)(x, D)(ω0(x, D)f)kM p,q ≤ Ckω0(x, D)fkM p,q (ω0ω) (ω0 ω) (ω) for some constant C. This proves that ω (x, D) is bijective from M p,q 0 (ω0ω) p,q to M(ω) with continuous inverse (1/ω0)(x, D). The proof is complete. As in [To6] we remark that an immediate consequence of Corollary m 3.3 is that if p, q ∈ [1, ∞], ω0(x, ξ) = ω1(x)ω2(ξ) where ωj ∈ P0(R ) for j = 1, 2, and ω ∈ P(R2m), then M p,q (Rm)= { f ∈ S ′(Rm); ω (x)ω (D)f ∈ M p,q(Rm) }. (ω0ω) 1 2 (ω) s In particular, if s ∈ R and ω0(x, ξ)= σs,0(x, ξ)= hξi , then M p,q (Rm)= { f ∈ S ′(Rm); σ (D)f ∈ M p,q(Rm) }. (σs,0ω) s (ω) 20 Note here that σs(D), s ∈ R, appears frequently in harmonic analysis and in the pseudo-diﬀerential calculus. For example, if p ∈ [1, ∞], then S ′ m p m recall that f ∈ (R ) belongs to the Sobolev space Hs (R ) if and p p only if kfkHs ≡kσs(D)fkL is ﬁnite. It is well-known that if s = N is p a positive integer and 1

Proposition 3.4. Assume that N1, N2 ≥ 0 are integers, p, q ∈ [1, ∞], and that ω ∈ P(R2m). Then

p,q m p,q m β α p,q M (R )= {f ∈ M (R ); x ∂ f ∈ M , |α|≤ N1, |β|≤ N2} (σN1,N2 ω) (ω) (ω)

p,q m N2 N1 N2 N1 p,q = {f ∈ M(ω)(R ); xj f, ∂k f, xj ∂k f ∈ M(ω), j, k = 1,...,m}.

p,q β α p,q Proof. Let M0 be the set of all f ∈ M(ω) such that x ∂ f ∈ M(ω) p,q when |α| ≤ N1 and |β| ≤ N2, and let M0 be the set of all f ∈ M(ω) N2 N1 p,q such that xj ∂k f ∈ M(ω) for j, k = 1f,...,N. We shall prove that p,q α M0 = M0 = M . Obviously, M0 ⊆ M0. Since the symbol ξ of (σN1,N2 ω) the operator Dα belongs to S when |α| ≤ N, it follows from f (σN1,N2 ) f p,q Theorem 3.2 that M ⊆ M0. The result therefore follows if it is (σN1,N2 ω) p,q proved that M0 ⊆ M . (σN1,N2 ω) In order tof prove this, assume ﬁrst that N1 = N, N2 = 0, f ∈ M0, and choose open sets f m m Ω0 = { ξ ∈ R ; |ξ| < 2 }, and Ωj = { ξ ∈ R ; 1 < |ξ|

m m Then ∪j=0Ωj = R , and there are non-negative functions ϕ0,...,ϕm in 0 m m S0 such that supp ϕj ⊆ Ωj and j=0 ϕj = 1. In particular, f = j=0 fj p,q when fj = ϕj(D)f. The resultP follows if we prove that fj ∈ MP (σN,0ω) for every j. −N Now set ψ0(ξ) = σN (ξ)ϕ0(ξ) and ψj(ξ) = ξj σN (ξ)ϕj (ξ) when j = 0 1,...,m. Then ψj ∈ S0 for every j. Hence Corollary 3.3 gives

kfj kM p,q ≤ C1kσN (D)fj kM p,q (σN,0ω) (ω)

N N = C1kψj(D)∂ fk p,q ≤ C2k∂ fk p,q < ∞ j M(ω) j M(ω) and kf0kM p,q ≤ C1kσN (D)f0kM p,q = C1kψ0(D)fkM p,q ≤ C2kfkM p,q < ∞ (σN,0ω) (ω) (ω) (ω) 21 for some constants C1 and C2. This proves that

N N (3.8) kfkM p,q ≤ C kfkM p,q + k∂j fkM p,q , (σN,0ω) (ω) (ω) Xj=1 and the result follows in this case. If we instead split up f into ϕjf, then similar arguments show that P

p,q p,q N p,q (3.9) kfkM ≤ C kfkM + kxk fkM , (σ0,N ω) (ω) (ω) Xk=1 and the result follows in the case N1 = 0 and N2 = N from this estimate. The general case follows now if combine (3.8) with (3.9). The proof is complete.

P 2m 2m Remark 3.5. Assume that ω, ω0 ∈ (R ), t ∈ R, a ∈ S(ω0)(R ), and that B1 and B2 are the same as in Remark 1.6. Then the proof B B of Theorem 3.2 shows that at(x, D) is continuous from M(ω0ω)( 1, 2) B B B B B B to M(ω)( 1, 2), and from W(ω0ω)( 1, 2) to W(ω)( 1, 2). In particular, if p, q ∈ [1, ∞] then it follows that at(x, D) is continuous from W p,q (Rm) to W p,q(Rm). (ω0ω) (ω) Moreover, from the proof of Proposition 3.4 it follows that if N1, N2 ≥ 0 are integers, p, q ∈ [1, ∞], then

M (B , B ) (σN1,N2 ω) 1 2 β α = { f ∈ M(ω)(B1, B2); x ∂ f ∈ M(ω)(B1, B2), |α|≤ N1, |β|≤ N2 }

m B B N2 N1 N2 N1 B B = ∩j,k=1{ f ∈ M(ω)( 1, 2); xj f, ∂k f, xj ∂k f ∈ M(ω)( 1, 2) }. and

W (B , B ) (σN1,N2 ω) 1 2 β α = { f ∈ W(ω)(B1, B2); x ∂ f ∈ W(ω)(B1, B2), |α|≤ N1, |β|≤ N2 }

m B B N2 N1 N2 N1 B B = ∩j,k=1{ f ∈ W(ω)( 1, 2); xj f, ∂k f, xj ∂k f ∈ W(ω)( 1, 2) }.

Remark 3.6. By using techniques of ultra-distributions, Pilipovićand Teofanov prove in [PT1] and [Te] parallel results comparing to Theo- rem 3.2. Here they consider generalized modulation spaces, where less growth restrictions are assumed on the weight function ω. It is for example not necessary that ω should be bounded by polynomials. 22 4. Schatten-von Neumann classes for operators acting on modulation spaces In this section we present some properties for Schatten-von Neu- mann classes of linear operators acting from M 2 (Rm) to M 2 (Rm), (ω1) (ω2) 2m where ω1, ω2 ∈ P(R ). We are especially concerned of finding appropriate identifications of the dual of st,p(ω1, ω2) when p< ∞. In the first part we prove that there is a canonical way to identify the dual for st,p(ω1, ω2) with st,p′(ω1, ω2). In the second part we use this property to prove that the dual for st,p(ω1, ω2) can also be identified with st,p′(1/ω1, 1/ω2) through a canonical extension of the scalar product on L2. We start to consider Schatten-von Neumann classes in background of pseudo-differential calculus. Let t ∈ R, p ∈ [1, ∞] and ω1, ω2 ∈ 2m P(R ) be fixed. Then recall that st,p(ω1, ω2) consists of all a ∈ S ′(R2m) such that a (x, D) ∈ I (M 2 , M 2 ). Also let s (ω , ω ) t p (ω1) (ω2) t,♯ 1 2 be the set of all a ∈ S ′(R2m) such that a (x, D) ∈ I (M 2 , M 2 ), t ♯ (ω1) (ω2) We let st,p(ω1, ω2) and st,♯(ω1, ω2) be equipped by the norms

2 2 kakst,p = kakst,p(ω1,ω2) ≡kat(x, D)kIp(M ,M ) (ω1) (ω2) and k·kst,∞ respectively. Since the Weyl quantization is important in w w our investigations we also use the notations sp and s♯ instead of st,p and st,♯ respectively when t = 1/2 and p ∈ [1, ∞]. From the fact that any linear and continuous operator from S (Rm) ′ m ′ 2m to S (R ) is equal to at(x, D), for a unique a ∈ S (R ), it follows that the map a 7→ at(x, D) is an isometric homeomorphism from s (ω , ω ) to I (M 2 , M 2 ) when p ∈ [1, ∞], and from s (ω , ω ) to t,p 1 2 p (ω1) (ω2) t,♯ 1 2 I (M 2 , M 2 ). Consequently, most of the properties which are listed ♯ (ω1) (ω2) in Section 2 carry over to the st,p-spaces. Hence Proposition 2.1 shows that st,p(ω1, ω2) is a Banach space which increases with the parameter p ∈ [1, ∞]. Moreover, Proposition 2.4 shows that the norm k·kst,2 (ω1,ω2) in st,2(ω1, ω2) induces a scalar product

(·, ·)st,2 =(·, ·)st,2(ω1,ω2) and that the dual for st,p(ω1, ω2) can be identified with st,p′(ω1, ω2) through this scalar product. A problem in this context is the somewhat complicated structure 2 of the form (·, ·)st,2(ω1,ω2), comparing to e. g. the scalar product on L , which in general fit pseudo-differential calculus well. In the remaining part of the section we therefore focus on a possible replacement of 2 the form (·, ·)st,2 with L when discussing the dual space. From our investigations it turns out that indeed the dual space of st,p(ω1, ω2) may be identified with st,p′(1/ω1, 1/ω2), by a unique and continuous extention of the definition of scalar product on L2. 23 We start to make some necessary preparations. Assume that ω ∈ P 2m 2 m (R ). By Theorem 1.3 it follows that the dual for M(ω)(R ) can 2 m 2 m be identified with M(1/ω)(R ) through the scalar product on L (R ). 2 On the other hand, since M(ω) is a Hilbert space, its dual can also be 2 2 identified with M(ω) through the scalar product on M(ω). Consequently, there exist unique homeomorphisms 2 ∗ 2 2 2 Sω : M(ω) → M(ω) and Tω : M(ω) → M(1/ω) 2 ∗ 2 such that if ℓ ∈ M(ω) and g = Sωℓ ∈ M(ω), then 2 (4.1) ℓ(f)=(f, g) 2 =(f,Tωg)L2 , f ∈ M M(ω) (ω) and −1 (4.2) C kTωgk 2 ≤kℓk = kgk 2 ≤ CkTωgk 2 M(1/ω) M(ω) M(1/ω) for some constant C which is independent of g. We observe that (4.1) ∞ 2 and (4.2) imply that if (fj)j=1 ∈ ON(M(ω)) and gk = Tωfk, then −1 (4.3) (fj, gk)L2 = δj,k and C ≤kgkk 2 ≤ C. M(1/ω) 2 ∗ For conveniency we set ONω = ON(M(ω)), and we let ONω be the ∞ 2 set of all sequences (gj )j=1 in M(1/ω) such that (4.3) gj = Tωfj for ∞ 2 some (fj)j=1 ∈ ON(M(ω)). The following characterization of st,p(ω1, ω2) immediately from (1.15), Proposition 2.6 and the homoeomorphism property of Tω. 2m Proposition 4.1. Assume that t ∈ R and ω1, ω2 ∈ P(R ), Also ∞ ∗ ∞ assume that a ∈ st,♯(ω1, ω2). Then for some (gj)j=1 ∈ ONω1 , (hj)j=1 ∈ ∞ ∞ ONω2 and λ =(λj )j=1 ∈ l0 it holds ∞

(4.4) a = λjWhj ,gj Xj=1

(with convergence with respect to the norm k·kst,∞ ). Moreover, if 1 ≤ p p< ∞ then a ∈ st,p(ω1, ω2), if and only if λ ∈ l , and then

p kakst,p(ω1,ω2) = kλkl for some constant C which is independent of a. ∞ ∗ On the other hand, if a is given by (4.4) for some (gj )j=1 ∈ ONω1 , ∞ ∞ p (hj)j=1 ∈ ONω2 and λ =(λj )j=1 ∈ l , then a ∈ st,p(ω1, ω2).

w Next we prove that the dual space of s1 (ω1, ω2) may be identiﬁed w with s∞(1/ω1, 1/ω2) through a continuous extension of the scalar prod- 2 2m w ∗ uct on L (R ). Assume that ℓ ∈ s1 (ω1, ω2) . It follows from the above that an arbitrary rank one operator from M 2 to M 2 can be written (ω1) (ω2) as w T f =(f, g)L2 h = b (x, D)f, 24 where b = W , g ∈ M 2 and h ∈ M 2 . Moreover h,g (1/ω1) (ω2) −1 C kgk 2 khk 2 ≤kbksw(ω ,ω ) M(1/ω ) M(ω ) 1 1 2 (4.5) 1 2 2 2 = kT kI1 ≤ CkgkM khkM , (1/ω1) (ω2) for some constant C. Hence

|ℓ(Wh,g)|≤ CkgkM 2 khkM 2 . (1/ω1) (ω2) In particular it follows that the mappings

f 7→ ℓ(W 0 ) and f 7→ ℓ(W 0 ) 1 f2 ,f1 2 f2,f1 S m 0 are continuous and linear mappings from (R ) to C for every f1 0 S m and f2 in (R ). Hence Schwartz kernel theorem and its proof (see Theorem 5.2.1 in [H¨o]) show that there is a unique distribution K ∈ S (Rm ⊕ Rm) such that

ℓ(Wf2,f1 )=(K, f2 ⊗ f 1) m for every f1, f2 ∈ S (R ). By letting a be the Weyl symbol for the operator with kernel K, it follows that w 2 (4.6) ℓ(Wf2,f1 )=(a (x, D)f1, f2)=(a, Wf2,f1 )L . w Next we prove that a ∈ s∞(1/ω1, 1/ω2). By the positive linearity in the deﬁnition of the trace norm we have kℓk = sup |ℓ(b)| = sup |ℓ(b)| kbk w ≤1 s1 w where the last supremum is taken over all b ∈ s1 (ω1, ω2) such that w w kbks1 ≤ 1 and b (x, D) is an operator of rank one. Hence (4.5) and (4.6) give w (4.7) |(a (x, D)f1, f2)|≤ Ckf1kM 2 kf2kM 2 , (1/ω1) (ω2) and it follows from Proposition 1.3 (4) that aw(x, D) is continuous from M 2 to M 2 , i.e. a ∈ s∞(1/ω , 1/ω ). (1/ω1) (1/ω2) w 1 2 w Assume next that b ∈ s1 (ω1, ω2) is arbitrary. Then b = λj Whj,gj ∗ 1 for some (gj ) ∈ ONω1 , (hj) ∈ ONω2 , and (λj ) ∈ l . HenceP (4.2), (4.5) and (4.6) give

2 w |(a,b)L |≤ |λj||(a, Whj,gj )|≤ C |λj | = Ckbks1 X X for some constant C. By combining the latter estimate with (4.5) it follows that ℓ(b)=(a,b)L2, and that b 7→ (a,b)L2 is continuous w w from s1 (ω1, ω2) to C. Consequently, the dual space of s1 (ω1, ω2) can w be identiﬁed with s∞(1/ω1, 1/ω2) through the form (·, ·)L2 . We also w note that the extension of (·, ·)L2 to a duality between s1 (ω1, ω2) and sw (1/ω , 1/ω ) is unique since S is dense in M 2 for j = 1, 2, and ∞ 1 2 (ωj) 25 that ﬁnite rank operators are dense in I (M 2 , M 2 ). In particular 1 (ω1) (ω2) we have proved the following result in the case p = 1 and t = 1/2.

Theorem 4.2. Assume that t ∈ R, p ∈ [1, ∞) and that ω1, ω2 ∈ P(R2m). Then the scalar product on L2(R2m) extends uniquely to a w duality between st,p(ω1, ω2) and st,p′(1/ω1, 1/ω2), and the dual space for w st,p(ω1, ω2) can be identiﬁed with st,p′ (1/ω1, 1/ω2) through this form. ∗ Moreover, if ℓ ∈ st,p(ω1, ω2) and a ∈ st,p′ (1/ω1, 1/ω2) such that ℓ(b)= (a,b)L2 when b ∈ st,p(ω1, ω2), then −1 ′ ′ C kakst,p (ω1,ω2) ≤kℓk≤ Ckakst,p (ω1,ω2) for some constant C which only depends on ω1 and ω2.

Proof. By (1.10) and the fact that eihDx,Dξi is unitary on L2, it suffices to prove the result in the case t = 1/2. We have already proved the result in the case p = 1. Therefore assume that p > 1. We prove w first that ℓ(b)= (a,b)L2 defines a continuous linear form on sp (ω1, ω2) w I I I ′ for every element sp′ (1/ω1, 1/ω2). Since 1 is dense in p and in p , Proposition 4.1 shows that it suffices to prove

′ |(a,b)L2|≤ Ck(λj)klp k(µj)klp , for some constant C, where

0 0 a = λj Whj ,gj and b = µj Wh ,g X X j j 0 0 are ﬁnite sums. Here gj = T1/ω1fj , gj = Tω1 fj for some (fj) ∈ ON1/ω1 , 0 0 (fj ) ∈ ONω1 , (hj) ∈ ON1/ω2 and (hj ) ∈ ONω2 . It is also no restriction ′ to assume that k(λj)klp = k(µj )klp = 1, By straight-forward computations we get

2 0 0 2 |(a,b)L | = λj µk(Whj,gj , Wh ,g )L k k Xj,k

0 0 ≤ |λj µk(hj , hk)L2 (gj , gk)L2 | Xj,k Hence the inequality between aritmetic and geometric mean-values gives (4.8) 1 p′ 1 p 1 0 2 1 0 2 |(a,b) 2|≤ |λ | + |µ | |(h , h ) 2 | + |(g , g ) 2 | . L p′ j p k 2 j k L 2 j k L Xj,k

0 0 By letting κj = T1/ω2hj and κj = Tω2 hj it follows from (4.3) that 0 0 0 0 (gj, gk)L2 =(fj, gk)M 2 =(gj , fk )M 2 =(fj, fk )L2 , and (1/ω1) (ω1) 0 0 0 0 (hj , hk)L2 =(hj, κk)M 2 =(κj, hk)M 2 =(κj, κk)L2 . (1/ω2) (ω2) 26 Hence the orthogonality assumptions imply

0 2 0 0 2 |(gj , gk)L2 | ≤kgkkM 2 ≤ C, |(gj , gk)L2 | ≤kgjkM 2 ≤ C, (1/ω1) (ω1) Xj Xk 0 0 and similarily when gj and gk are replaced by hj and hk respectively. By using these estimates in (4.8), the assertion follows. w ∗ Assume next that ℓ ∈ sp (ω1, ω2) . By Proposition 2.4 there is a unique operator T ∈ I ′ (M 2 , M 2 ) such that p (ω1) (ω2) w ℓ(b)=(b (x, D),T )M 2 ,M 2 (ω1) (ω2) w for any b ∈ sp (ω1, ω2). Then

w 0 0 T f = λj (f, fj)M 2 hj , and b (x, D)= µj (f, fj )M 2 hj X (ω1) X (ω1) 0 0 p′ for some (fj), (fj ) ∈ ONω1 and (hj), (hj ) ∈ ONω2 ,(λj ) ∈ l and (µj) ∈ lp. This implies that

0 0 ℓ(b)= λj µk(hk, hj)M 2 (fj, fk )M 2 . (ω2) (ω1) Xj,k 0 0 Now let gj = Tω1 fj , κj = Tω2 hj and a = λj Wκj,fj . Then it follows by straight-forward computations that b =P µ W 0 0 , and that j hj ,gj P (a,b) 2 = λj µk(Wκ ,f , W k k ) 2 L j j h0 ,g0 L Xj,k

k k = λj µk(κj, h0)L2 (g0 , fj)L2 = ℓ(b). Xj,k From the ﬁrst part of the proof it also follows that

w ′ ka (x, D)k I ′ (M 2 ,M 2 ) ≤ Ck(λj )klp < ∞, p (1/ω1) (1/ω2) for some constant C. The proof is complete. Remark 4.3. Theorem 4.2 appeared after fruitful discussions with Paolo Boggiatto.

5. Continuity for Pseudo-differential operators with symbols in modulation spaces In this section, we discuss continuity and Schatten-von Neumann properties for pseudo-differential operators, when the operator symbols belong to appropriate classes of modulation spaces. In particular we extend some of the continuity properties in Section 5 in [To6]. An important ingredient in these investigations concern continuity properties for the Wigner distributions in context of modulation spaces, as well as the central role for Wigner distributions in the Weyl calculus of pseudo-differential operators. 27 We start to recall some basic facts about Wigner distributions and Weyl operators. When discussing Weyl calculus in context of modulation spaces, it is convenient to use the following convention. Assume that Bj for j = 1, 2, 3 are Frechét spaces such that

2m ′ 2m m ′ m ,( S (R ) ֒→ B1 ֒→ S (R ), S (R ) ֒→ B2, B3 ֒→ S (R and that (a,f,g) 7→ (a, Wg,f ) is well-defined and sequently continuous from B1 × B2 × B3 to C. Then (1.15) is taken as the definition of w B′ B a (x, D)f as an element in 3 when f ∈ 2, and it follows that w B B′ a (x, D) is a continuous operator from 2 to 3. Next we discuss continuity properties for pseudo-differential operator, and prove in a moment that if t ∈ R,

(5.1) 1/p1 − 1/p2 = 1/q1 − 1/q2 = 1 − 1/p − 1/q, q ≤ p2,q2 ≤ p, p,q ω1, ω2 and ω are appropriate weight functions and a ∈ M(ω), then a (x, D) is continuous from M p1,q1 to M p2,q2 . As a ﬁrst step we recall t (ω1) (ω2) in Proposition 5.1 below continuity properties for Wigner distributions in background of modulation space theory. The proof of the following result is omitted, since the result is a restatement of Theorem 5.1 in [To6], and the fact that

Wf,g = Wg,f when f, g ∈ S ′. (See also [CG], [G2], [To4] and [To5].)

Proposition 5.1. Assume that pj ,qj,p,q ∈ [1, ∞] such that p ≤ pj ,qj ≤ q, for j = 1, 2, and that

(5.2) 1/p1 + 1/p2 = 1/q1 + 1/q2 = 1/p + 1/q.

2m 2m 2m Assume also that ω1, ω2 ∈ P(R ), and that ω, ω0 ∈ P(R ⊕ R ) satisfy ω ≺ T (ω1 ⊗ ω2) and ω0(X, Y )= ω(X, −Y ), where T is the map on P(R2m ⊕ R2m), given by (T ω)(x,ξ,y,η)= ω(x − y/2, ξ + η/2,x + y/2, ξ − η/2). S ′ m S ′ m S ′ 2m Then the map (f1, f2) 7→ Wf1,f2 from (R ) × (R ) to (R ) restricts to a continuous mapping from M p1,q1 (Rm) × M p2,q2 (Rm) to (ω1) (ω2) p,q 2m M(ω)(R ), and for some constant C,

p,q p,q p ,q p ,q (5.3) kWf1,f2 kM = kWf2,f1 kM ≤ Ckf1kM 1 1 kf2kM 2 2 (ω) (ω0) (ω1) (ω2) ′ m when f1, f2 ∈ S (R ). We shall next apply Proposition 5.1 to pseudo-diﬀerential calculus. The following result extends Theorem 14.5.2 in [G2], Theorem 7.1 in [GH2] and Theorem 4.3 in [To6]. 28 Theorem 5.2. Assume that t ∈ R and p, q, pj,qj ∈ [1, ∞] for j = 2m 2m 1, 2, satisfy (5.1). Also assume that ω ∈ P(R ⊕ R ) and ω1, ω2 ∈ P(R2m) satisfy ω (x − ty,ξ + (1 − t)η) (5.4) 2 = ω(x,ξ,y,η) ω1(x + (1 − t)y, ξ − tη) p,q 2m S m for some constant C. If a ∈ M(ω)(R ), then at(x, D) from (R ) to S ′(Rm) extends uniquely to a continuous mapping from M p1,q1 (Rm) (ω1) to M p2,q2 (Rm). (ω2) Moreover, if in addition a ∈Mp,q , then a (x, D): M p1,q1 → M p2,q2 (ω) t (ω1) (ω2) is compact.

Proof. It suﬃces to prove the theorem in the case t = 1/2 in view of Proposition 1.10. The conditions on pj and qj implies that ′ ′ ′ ′ ′ ′ ′ ′ p ≤ p1,q1, p2,q2 ≤ q , 1/p1 + 1/p2 = 1/q1 + 1/q2 = 1/p + 1/q . Hence Proposition 5.1, and (5.4) show that for some constant C > 0,

′ ′ ′ ′ p ,q p1,q1 kWg,f k ≤ CkfkM kgk p2,q2 M(1/ω) (ω1) M (1/ω2) ′ ′ when f ∈ M p1,q1 (Rm) and g ∈ M p2,q2 (Rm). (ω1) (1/ω2) p,q 2m Now assume that a ∈ M(ω)(R ). Then for some constants C > 0 and C′ > 0 we get

′ ′ |(a, Wg,f )|≤ CkakM p,q kWg,f k p ,q (ω) M(1/ω) (5.5) ′ p,q p1,q1 ′ ′ ≤ C kakM kfkM kgk p2,q2 . (ω) (ω1) M (1/ω2) The result now follows by combining (5.5) and Proposition 1.3(3). It remains to prove that if a ∈ Mp,q , then aw(x, D): M p1,q1 → (ω) (ω1) M p2,q2 is compact. By straight-forward computations it follows that (ω2) any linear operator with Schwartz kernel, which acts from one Banach space to an other must be compact, provided that the space of the Schwartz functions are continuously embedded in the involved Banach spaces. This implies that aw(x, D) is compact when a ∈ S . p,q For general a ∈M(ω), it follows now from the ﬁrst part of the proof that aw(x, D) may be approximated in operator norm by compact operators. Hence aw(x, D) is compact and the proof is complete. 2

2m Remark 5.3. Assume that t ∈ R, p, q ∈ [1, ∞], ω1, ω2 ∈ P(R ) and

ω(x,ξ,y,η)= σN (y, η)ω2(x, ξ)/ω1(x, ξ) ∞,1 2m where N ≥ 0 is an integer. If a ∈ M(ω) (R ), then Theorem 5.2 shows that a (x, D) is continuous from M p,q (Rm) to M p,q (Rm). Since t (ω1) (ω2) ∞,1 S(ω2/ω1) ⊆ M(ω) , we obtain Theorem 3.2 as a special case of Theorem 5.2.

29 Remark 5.4. If p = p1 = p2 = ∞ and q = q1 = q2 = 1, or p = q1 = q2 = ∞ and q = p1 = p2 = 1 in Theorem 5.2, then S is not dense in any of the involving spaces. However, by applying (5.5) it follows that aw(x, D)f makes sense as an element in S ′ when f ∈ M p1,q1 . Then (ω1) (5.5) in combination with Proposition 1.3 (3) shows that aw(x, D) is continuous from M p1,q1 to M p2,q2 . (ω1) (ω2)

Corollary 5.5. Assume that p, q, pj,qj ∈ [1, ∞] for j = 1, 2, satisfy ′ q ≤ p2,q2 ≤ p and (5.2) . Assume also that t ∈ R \{0, 1}, sj,tj ∈ R p,q 2m for j = 0, 1, 2 such that s1,s2,t1,t2 ≥ 0, and that a ∈ M(ω)(R ), for some ω ∈ P(R2m ⊕ R2m). Then the following are true:

(1) if |s0|≤ s1 + s2, and

ω(x,ξ,y,η)= σs1 (x, ξ)σs2 (y, η), p1,q1 Rm p2,q2 Rm then at(x, D) is continuous from Ms0 ( ) to Ms0 ( ); (2) if |s0|≤ s1 + s2, |t0|≤ t1 + t2, and

ω(x,ξ,y,η)= σs1,t1 (x, ξ)σs2,t2 (y, η), p1,q1 m p2,q2 m then at(x, D) is continuous from Ms0,t0 (R ) to Ms0,t0 (R ).

Proof. The assertion (1) follows by letting ω1 = ω2 = σs0 in Theorem 5.2, and using that h(y/2 − x, ξ − η/2)i−1h(y/2+ x, ξ + η/2)i ≤ C min(hxi, hyi)min(hξi, hηi) for some constant C. The assertion (2) is proved in a similar way. The proof is complete. In the case p = q = ∞ in Theorem 5.2, the converse is also true, i. e. we have the following result. Theorem 5.6. Assume that t ∈ R, a ∈ S ′(R2m), ω ∈ P(R2m ⊕ 2m 2m R ), and ω1, ω2 ∈ P(R ) such that (5.4) holds. Then the operator m ′ m at(x, D) from S (R ) to S (R ) extends to a continuous mapping from M 1 (Rm) to M ∞ (Rm), if and only if a ∈ M ∞ (R2m). (ω1) (ω2) (ω) For the proof we need the following two propositions of independent interest, where the ﬁrst one is a slight generalization of Schwartz- Gr¨ochenig’s kernel theorem. (See Theorem 14.4.1 in [G2] or Theorem 4.1 in [To6].)

2mj Proposition 5.7. Assume that m = m1 + m2, ωj ∈ P(R ) for j = 1, 2 and ω ∈ P(Rm ⊕ Rm) such that

(5.6) ω(x,y,ξ,η)= ω2(x, ξ)/ω1(y, −η). Also assume that T is a linear and continuous map from S (Rm1 ) to S ′(Rm2 ). Then T extends to a continuous mapping from M 1 (Rm1 ) (ω1) 30 to M ∞ (Rm2 ), if and only if it exists an element K ∈ M ∞ (Rm) such (ω2) (ω) that (5.7) (T f)(x)= hK(x, ·), fi. Here the distribution u on the right-hand side in (5.7) is deﬁned by the formula hu, gi = hK, g ⊗ fi, or alternatively by the formula (u, g)=(K, g ⊗ f). Proof. Assume that T extends to continuous map from M 1 to M ∞ . (ω1) (ω2) It follows from the kernel theorem of Schwartz that (5.7) holds for some S ′ m ∞ K ∈ (R ). We shall prove that K belongs to M(ω). From the assumptions and Proposition 1.3 (3) it follows that

(5.8) |(K, g ⊗ f)L2 |≤ CkfkM 1 kgkM 1 (ω1) (1/ω2) holds for some constant C which is independent of f ∈ S (Rm1 ) and g ∈ S (Rm2 ). By letting χ = g ⊗ f be ﬁxed, and replacing f and g with −ih·,ηi ih·,ξi fy,η = e f(·− y) and gx,ξ = e f(·− x), it follows that (5.8) takes the form ′ F (5.8) | (Kτ(x,y)χ)(ξ, η)|≤ Ckfy,ηkM 1 kgx,ξkM 1 . (ω1) (1/ω2) We have to analyze the right-hand side of (5.8)′. If v is chosen such m1 that ω1 is v-moderate, and χ1 ∈ S (R ) \ 0 is a ﬁxed function, then we obtain

F kfy,ηkM 1 = | (fτz−y χ1)(ζ + η)ω1(z, ζ)| dzdζ (ω1) ZZ

= |F (fτ χ )(ζ)ω (z + y, ζ − η)| dzdζ ZZ z 1 1

′ ≤ Cω1(y, −η)kfk 1 = C ω1(y, −η). M(v) In the same way we get −1 kgx,ξkM 1 ≤ Cω2(x, ξ) . (1/ω2) If these estimates are inserted into (5.8)′ we obtain

|F (Kτ(x,y)χ)(ξ, η)ω(x,y,ξ,η)|≤ C, for some constant C which is independent of x, y, ξ and η. By taking the supremum of the left-hand side it follows that kKk ∞ < ∞. Hence M(ω) ∞ K ∈ M(ω), and the necessity follows. The suﬃciency follows by straight-forward computations. The details are left for the reader. (See also the proof of Theorem 4.1 in [To6].) The proof is complete. 31 Proposition 5.8. Assume that a ∈ S ′(R2m), and that K ∈ S ′(R2m) is the distribution kernel for the Weyl operator aw(x, D). Also assume 2m 2m that p ∈ [1, ∞], and that ω, ω0 ∈ P(R ⊕ R ) such that

ω(x,ξ,y,η)= ω0(x − y/2,x + y/2, ξ + η/2, −ξ + η/2). Then a ∈ M p (R2m) if and only if K ∈ M p (R2m). Moreover, if (ω) (ω0) χ ∈ S (R2m) and

ψ(x,y)= χ((x + y)/2, ξ)eihy−x,ξi dξ, Z then kak p,χ = kKk p,ψ . M(ω) M (ω0) Note here that the deﬁnitions of the Fourier transforms are diﬀerrent for K and a respectively in Proposition 5.8. In fact, K should be treated as a distribution on the space RN with N = 2m. On the other hand, for a(x, ξ), the variables x and ξ should be interpreted as dual variables, leading to that the phase space Fourier transform should be applied on a (cf. (1.3)).

Proof. By Fourier’s inversion formula, it follows by straight-forward computations that

|F (Kτ(x−y/2,x+y/2)ψ)(ξ + η/2, −ξ + η/2)| = |F (aτ(x,ξ)χ)(y, η)|, where the Fourier transform on the right-hand side should be interpreted as a Fourier transform on distributions deﬁned on the phase p space. The result now follows by applying the L(ω)-norm on these expressions. The proof is complete.

Next we discuss embeddings between Schatten-von Neumann classes in pseudo-diﬀerential operators and modulation spaces. As a ﬁrst step we consider Hilbert-Schmidt properties for operators acting on modulation spaces of Hilbert type.

2m Proposition 5.9. Assume that ω1, ω2 ∈ P(R ) such that (5.6) holds, and that T is a linear and continuous operator from S (Rm) to S ′(Rm) with distribution kernel K ∈ S ′(R2m). Then T ∈ I (M 2 , M 2 ), if 2 (ω1) (ω2) 2 2m and only if K ∈ M(ω)(R ), and

(5.9) kT kI = kKk 2 . 2 M(ω)

Proof. Let (fj) ∈ ONω1 and (hk) ∈ ONω2 be orthonormal basis for M 2 (Rm) and M 2 (Rm) respectively. Then (ω1) (ω2)

2 2 2 2 2 2 kT kI2 = |(T fj, hk)M )| = |(K, hk ⊗ fj )M ⊗L | (ω2) (ω2) Xj,k Xj,k 32 ′ 2 Now let Tω0 = IM ⊗ T1/ω0, where ω0(x, ξ) = ω1(x, −ξ). Then we (ω2) obtain

2 ′ 2 2 2 kT kI2 = |(Tω0K, hk ⊗ fj)M ⊗M | (ω2) (ω1) Xj,k

′ 2 2 2 2 2 2 2 2 = kTω0 KkM ⊗M = kKkM ⊗M = kKkM (ω2) (ω0) (ω2) (1/ω0) (ω)

Hence (5.9) holds, and the proof is complete.

The following result is now an immediate consequence of Proposition 5.9 and Proposition 5.8 for p = 2.

′ 2m 2m Proposition 5.10. Assume that a ∈ S (R ), ω1, ω2 ∈ P(R ), and assume that (5.4) holds for t = 1/2. Then aw(x, D) ∈ I (M 2 , M 2 ), 2 (ω1) (ω2) 2 2m if and only if a ∈ M(ω)(R ). Moreover, for some constant C > 0 it holds

−1 w C kak 2 ≤ka (x, D)kI ≤ Ckak 2 . M(ω) 2 M(ω)

For general Schatten-von Neumann classes, we have the following generalization of Proposition 5.10.

Theorem 5.11. Assume that t ∈ R and p, q, pj,qj ∈ [1, ∞] for j = 1, 2, satisfy

′ ′ (5.10) p1 ≤ p ≤ p2, q1 ≤ min(p, p ) and q2 ≥ max(p, p ).

2m 2m 2m Also assume that ω ∈ P(R ⊕ R ) and ω1, ω2 ∈ P(R ) satisfy (5.4) for some constant C. Then

p1,q1 2m p2,q2 2m ( M(ω) (R ) ֒→ sp,t(ω1, ω2) ֒→ M(ω) (R (5.11)

For the proof we need the following lemma.

Lemma 5.12. Assume that

(xj1 )j1∈I1 , (ξj2 )j2∈I2 , (yk1 )k1∈I1 and (ηk2 )k2∈I2 are lattices in Rm. Also assume that ϕ(x) = e−|x|2/2 where x ∈ Rm, 2m ω1, ω2 ∈ P (R ), (fl) ∈ ONω1 , (hl) ∈ ONω2 , and set κl = Tω2hl and

θ1 = |F (f τ ϕ)(ξ − η /2)|2ω (x + y /2, ξ − η /2)2, j,k,l l xj1 +yk1 /2 j2 k2 1 j1 k1 j2 k2 θ2 = |F (κ τ ϕ)(ξ + η /2)|2/ω (x − y /2, ξ + η /2)2, j,k,l l xj1 −yk1 /2 j2 k2 2 j1 k1 j2 k2 33 where j = (j1, j2) ∈ I1 × I2 ≡ I, k = (k1, k2) ∈ I. Then for some constant C and integer N ≥ 0 it holds

i 2 (5.12) θ ≤ Ckϕk 2 < ∞, i = 1, 2, j,k,l MN Xl

1 2 (5.13) θj,k,l ≤ CkflkM 2 ≤ C < ∞, (ω1) Xk∈I

2 2 (5.14) θj,k,l ≤ CkhlkM 2 ≤ C < ∞. (ω2) Xk∈I

Proof. Let N ≥ 0 be chosen such that ω1 and ω2 are σN -moderate. Then (5.12) for i = 1 follows if we prove that

2 2 2 |F (flτzϕ)(ζ)| ≤ Ckϕk 2 /ω1(z, ζ) MN Xl for some constant C which is independent of (z, ζ). Since

ih· ,ζi ih· ,ζi F 2 2 (flτzϕ)(ζ)=(fl,e τzϕ)L =(fl,T1/ω1e τzϕ)M (ω1) S and (fl) ∈ ONω1 , we obtain for some χ ∈ \ 0 that

2 ih· ,ζi 2 F 2 | (flτzϕ)(ζ)| = |(fl,T1/ω1e τzϕ)M | (ω1) Xl Xl

ih· ,ζi 2 ih· ,ζi 2 ≤kT1/ω1e τzϕkM 2 ≤ Cke τzϕkM 2 (ω1) (1/ω1)

= C |F (eih· ,ζiτ ϕτ χ)(ξ)/ω (x, ξ)|2 dxdξ ZZ z x 1

= C |F (ϕτ χ)(ξ − ζ)/ω (x, ξ)|2 dxdξ ZZ x−z 1

= C |F (ϕτ χ)(ξ)/ω (x + z, ξ + ζ)|2 dxdξ ZZ x 1

′ F 2 2 ′ 2 2 ≤ C | (ϕτxχ)(ξ)σN (x, ξ)| dxdξ/ω1(z, ζ) = C kϕkM 2 /ω1(z, ζ) . ZZ N This proves the assertion. The case i = 2 in (5.12) follows from similar arguments together with the fact that F ih· ,ζi ih· ,ζi (κlτzϕ)(ζ)=(κl,e τzϕ)L2 =(hl,e τzϕ)M 2 . (ω2)

Next we prove (5.13). For some lattices (zk1 )k1∈I1 and (ζk2 )k2∈I2 we have 1 2 2 2 (θ ) = |F (f τ ϕ)(ζ )ω (z , ζ )| ≤ Ckf k 2 j,k,l l zk1 k2 1 k1 k2 l M (ω1) Xk Xj,k 34 for some constant C, where the last inequality follows from Proposition

1.7. This proves (5.13). By replacing the lattices (zk1 ) and (ζk2 ) by some other ones, if necessary, we obtain

(θ2 )2 = |F (κ τ ϕ)(ζ )ω (z , ζ )|2 j,k,l l zk1 k2 2 k1 k2 Xk Xk

2 ′ 2 ≤ CkκlkM 2 ≤ C khlkM 2 (1/ω2) (ω2) for some constants C and C′, and (5.14) follows. The proof is complete.

Proof of Theorem 5.11. We use the same notations as in Lemma 5.12. In view of Proposition 1.10 it is no restriction to assume that t = 1/2, and that equalities are attained in (5.10). Then the result is an immediate consequence of Proposition 5.10 in the case p = q = 2. Next we consider the case q = 1. Let Xj = (xj1 , ξj2 ) and Yk =(yk1 , ηk2 ). Then it follows 2m p,1 that (Xj)j∈I and (Yk)k∈I are lattices in R . Assume that a ∈ M(ω). By Chapters 5, 6, 12 and 13 in [G2] it follows that if the lattices here above are chosen dense enough, then

ihX,Yki a(X)= cj,ke (τXj Wϕ,ϕ)(X), j,kX∈I for some sequence (cj,k)j,k∈I which satisﬁes

1/p p λj,k ≤ Ckak p,1 < ∞, M(ω) (5.15) Xj∈I Xk∈I

λj,k = |cj,kω(xj1 , ξj2 yk1 , ηk2 )|.

2m Here X = (x, ξ) ∈ R and hX, Yk i = hx, ηk2 i + hyk1 , ξi. Hence if ∗ (fl) ∈ ONω1 and (κl) ∈ ONω2 , then

w ih·,Yki |(a (x, D)fl, κl)|≤ |cj,k(e τXj Wϕ,ϕ, Wκl,fl )|. Xj,k

By straight-forward computations it follows that

ihX,Yki iΦ(Yk,Xj) e (τX Wϕ,ϕ)(X)= e W 1 2 , j ϕj,k,ϕj,k where Φ is a real-valued quadratic form on R2m ⊕ R2m, and

1 ihx,ξj2+ηk2 /2i ϕj,k(x)= e ϕ(x − xj1 + yk1 /2),

2 ihx,ξj2−ηk2 /2i ϕj,k(x)= e ϕ(x − xj1 − yk1 /2). 35 This gives

ih·,Yki |(e τX Wϕ,ϕ, Wκ ,f )| = |(W 1 2 , Wκ ,f )| j l l ϕj,k,ϕj,k l l

1 2 1 2 ω2(xj1 − yk1 /2, ξj2 + ηj/2) = |(ϕj,k, κl)(ϕj,k, fl)| = θj,k,lθj,k,l ω1(xj1 + yk1 /2, ξj2 − ηk2 /2) 1 2 ≤ Cθj,k,lθj,k,lω(xj1 , ξj2 ,yk1 , ηk2 ), for some constant C. Hence

ih·,Yki 1 2 1/2 |(e τXj Wϕ,ϕ, Wκl,fl )|≤ Cλj,k(θj,k,lθj,k,l) . This gives 1/p w p |(a (x, D)fl, κl)| ≤ C(J1 + J2)/2, Xl where p 1/p i 2 Ji = λj,k(θj,k,l) , j = 1, 2. Xl Xj,k

We have to estimate J1 and J2. By Minkowski’s and H¨older’s inequal- ities we get

p 1/p 1 J1 ≤ C λj,kθj,k,l Xj Xl Xk

p 1/p 1 1/p 1 1/p′ = C (λj,k(θj,k,l) )(θj,k,l) Xj Xl Xk

p/p′ 1/p p 1 1 ≤ C λj,kθj,k,l θj,k,l . Xj Xl Xk Xk Now(5.15) and (5.12)–(5.14) gives p/p′ 1/p p 1 1 J1 ≤ C1 λj,kθj,k,l sup θj,k,l l Xj Xk,l Xk

′ 1/p 2/p p 1 ≤ C2 sup kκlkM 2 λj,k θj,k,l l (1/ω2) Xj Xk Xl

1/p 2/p p ≤ C3kϕk 2 λj,k ≤ C4kakM p,1. MN (ω) Xj∈I Xk∈I

In the same way we get J2 ≤ Ckak p,q for some constant C. Hence it M(ω) follows from these estimates that 1/p w p |(a (x, D)fl, κl)| ≤ Ckak p,q . M(ω) Xl 36 For some constant which is independent of the choice of sequences (fl) and (κl). The result now follows by taking the supremum of the left- hand side with respect to all sequences (fl) and (κl). The ﬁrst embedding in (5.11) now follows by interpolation of the case q = 1 and the case p = q = 2, using Proposition 1.9 and Proposition 2.7. The second embedding in (5.11) now follows from the ﬁrst one and duality, using Theorem 4.2. The proof is complete. 2

6. Continuity for Toeplitz operators with symbols in modulation spaces In this section, we extend certain continuity properties in Section 5 in [To6] concerning Toeplitz operators, when the operator symbols belong to appropriate classes of modulation spaces. As in [To1], [To3], [To6] or [W1]–[W3] we reduce ourself to continuity for Weyl operators by using that the Weyl symbol for a Toeplitz operator is equal to a convolution between a Wigner distribution and the corresponding symbol for the Toeplitz operator. We start to recall the definition of Toeplitz operators. Assume that m 2m h1, h2 ∈ S (R ) \ 0 are fixed. For any a ∈ S (R ), the Toeplitz S Rm operator Tph1,h2 (a) on ( ), with respect to h1, h2 and symbol a is defined by the formula S Rm (Tph1,h2 (a)f, g)=(a(2·)Wf,h1 , Wg,h2 ), f,g ∈ ( ).

(Cf. [To1], [To3] or [W1]–[W3].) Obviously, Tph1,h2 (a) extends to a continuous map from S ′(Rm) to S (Rm). Now recall that if t ∈ R, then ′ 2m ′ the map a 7→ at(x, D) is a bijection from S (R ) to Op(S ), the set of all continuous operators from S (Rm) to S ′(Rm). In particu- w S ′ R2m lar, b (x, D) = Tph1,h2 (a) for some b ∈ ( ), and by straight- forward computations it follows that b = a ∗ uh1,h2 , where uh1,h2 = −m/2 (2π) Wh2,h1 , i.e. w Tph1,h2 (a)=(a ∗ uh1,h2 ) (x, D), (6.1) −m/2 where uh1,h2 (X)=(2π) Wh2,h1 (−X). (Cf. [Sh], [To1], [To3] and [W1].) ′ 2m ′ m More generally, assume that a ∈ S (R ) and h1, h2 ∈ S (R ) are S ′ 2m distributions such that a∗uh1,h2 makes sense as an element in (R ).

Then (6.1) is taken as the deﬁnition of the Toeplitz operator Tph1,h2 (a). We refer to [Bg], [CG], [Sh], [To1], [To3], [To4] and [W1]–[W3] for more facts concerning Toeplitz operators.

Next Tph1,h2 (a) is discussed when h1, h2 and a belong to certain types of modulation spaces. By combining Proposition 5.1 with the convolution properties for modulation spaces in Section 1, the following p,q result is obtained. Here Op(M(ω)) is the set of all operators of the type 37 w p,q 2m S ′ a (x, D) such that a ∈ M(ω)(R ), and the topologies for Op( ) and p,q S ′ 2m p,q 2m Op(M(ω)) are inherited from (R ) and M(ω)(R ) respectively. Proposition 6.1. Let T be the same as in Proposition 5.1, and assume that rj,sj, p, p0,q,q0 ∈ [1, ∞] for j = 1, 2 satisfy 1 1 1 1 1 1 1 1 + = + = 1 − − + + r1 r2 s1 s2 p q p0 q0 (6.2) 1 1 1 1 1 1 0 ≤ − ≤ 1 − , 1 − ≤ 1+ − ≤ 1, j = 1, 2. p p0 rj sj q q0 Assume also that ν ∈ P(R2m), h ∈ M rj,sj (Rm), j = 1, 2, and that j j (νj) 2m 2m ω, ω0 ∈ P(R ⊕ R ) satisfy 2m ω0(X1+X2, Y ) ≤ Cω(X1, Y ) T (ν2⊗ν1)(−X2, −Y ), X1, X2, Y ∈ R , S R2m for some constant C > 0. Then a 7→ Tph1,h2 (a) from ( ) to Op(S ′) extends to a continuous map from M p,q(R2m) to Op(M p0,q0 ). (ω) (ω0)

Proof. By (6.2) it is possible to ﬁnd p1,q1 ∈ [1, ∞] such that p1 ≤ rj,sj ≤ q1 for j = 1, 2 and

1/p−1/p0 = 1−1/p1, 1/q−1/q0 = −1/q1, 1/r1 +1/r2 = 1/p1 +1/q1 Hence Proposition 5.1 implies that u ∈ M p1,q1 , whereω ˇ = T (ν ⊗ h1,h2 (ω1) 1 2 p,q 2m ν1). If a ∈ M(ω)(R ), then Remark 1.5 gives a ∗ u ∈ M p,q ∗ M p1,q1 ⊆ M p0,q0 , h1,h2 (ω) (ω1) (ω0) and the result follows from (6.1). The proof is complete. Next we combine Proposition 6.1 with Theorem 5.2 in order to Tp (a) should be continuous from M p1,q1 to M p2,q2 , when a ∈ M p,q h1,h2 (ω1) (ω2) (ω) and h ∈ M rj ,sj for j = 1, 2. More precisely we establish continuity j (νj ) properties of the form

p1,q1 p2,q2 p,q r1,s1 r2,s2 (6.3) k Tph1,h2 (a)kM →M ≤ CkakM kh1kM kh2kM , (ω1) (ω2) (ω) (ν1) (ν2) when the involving weight functions satisfy ω (x − y − z, ξ + η − ζ) (6.4) sup 2 ≤ Cω(x,ξ,y,η), (z,ζ)ω1(x − z, ξ − ζ)ν1(z, ζ)ν2(y + z, ζ − η) and the exponents p, q, pj,qj,rj,sj satisfy (6.2) and 1 1 1 1 1 1 (6.5) − = − = 1 − − , and q0 ≤ p2,q2 ≤ p0, p1 p2 q1 q2 p0 q0 for some p0,q0 ∈ [1, ∞].

Theorem 6.2. Assume that p, q, pj,qj,rj,sj ∈ [1, ∞] for j = 1, 2 satisfy (6.2) and (6.5) for some p0,q0 ∈ [1, ∞], and assume that νj, ωj ∈ P(R2m) for j = 1, 2 and ω ∈ P(R2m ⊕ R2m) satisfy (6.4) for some 38 constant C > 0 independent of x,ξ,y,η ∈ Rm. Also let h ∈ M rj ,sj (Rm) j (νj) p,q 2m for j = 1, 2 and a ∈ M(ω)(R ). Then the following are true: (1) Tp (a) is continuous from M p1,q1 (Rm) to M p2,q2 (Rm), and h1,h2 (ω1) (ω2) p,q (6.3) holds for some constant C independent of a ∈ M(ω) and h ∈ M rj ,sj for j = 1, 2; j (νj ) (2) if in addition a ∈Mp,q (R2m) and h ∈Mrj,sj (Rm) for j = 1, 2, (ω) j (νj) then Tp (a): M p1,q1 → M p2,q2 is compact; h1,h2 (ω1) (ω2) ′ (3) if in addition q0 ≤ min(p0, p0), then I 2 2 Tph1,h2 (a) ∈ p(M(ω1), M(ω2)). Proof. Let

ω0(x,ξ,y,η)= ω2(x − y/2, ξ + η/2)/ω1(x + y/2, ξ − η/2) . Then it follows from (6.4) that

ω0(X1 + X2, Y ) ≤ Cω(X1, Y ) T (ν2 ⊗ ν1)(−X2, −Y ),

2m for some constant C independent of X1, X2, Y ∈ R . Hence Tph1,h2 (a) ∈ Op(M p0,q0 ) by Proposition 6.1. It follows now from the deﬁnition of (ω0)

ω0, Theorem 5.2, (6.2) and (6.5) that Tph1,h2 (a) is continuous from M p1,q1 to M p2,q2 . (ω1) (ω2) ′ If in addition q0 ≤ min(p0, p0), then these arguments together with Theorem 5.11 show that Tp (a) ∈ I (M 2 , M 2 ). Hence (1) and h1,h2 p (ω1) (ω2) (3) follows. The compactness assertions in (2) now follows by similar arguments as in the proof of Theorem 5.2. The proof is complete.

Proposition 6.3. Assume that p, q, pj ,qj,rj ,sj ∈ [1, ∞] for j = 1, 2 satisfy (6.2) and (6.5) for some p0,q0 ∈ [1, ∞]. Assume also that ν , ω ∈ P(R2m) and that h ∈ M rj,sj for j = 1, 2. Then the following j j j (νj) are true: p,q 2m (1) if ω1 is ν1-moderate, ω2 is νˇ2-moderate, and a ∈ M(ω)(R )

where ω(X, Y ) = ω2(X)/ω1(X), then Tph1,h2 (a) is continuous from M p1,q1 (Rm) to M p2,q2 (Rm). Moreover, if q ≤ min(p , p′ ), (ω1) (ω2) 0 0 0 then Tp (a) ∈ I (M 2 , M 2 ); h1,h2 p (ω1) (ω2) p,q 2m (2) if νj is ωj -moderate for j = 1, 2, and a ∈ M(ω)(R ) where −1 ω(X, Y )=(ν1(X)ν2(X)) , then Tph2,h1 (a) is continuous from M p1,q1 (Rm) to M p2,q2 (Rm). Moreover, if q ≤ min(p , p′ ), then (ω1) (1/ω2) 0 0 0 Tp (a) ∈ I (M 2 , M 2 ). h1,h2 p (ω1) (1/ω2) Moreover, if in addition a ∈ Mp,q (R2m) and h ∈ Mrj,sj (Rm) for (ω) j (νj) j = 1, 2 in (1) or (2), then Tph1,h2 (a) is compact. 39 Proof. The assertion (1) is an immediate consequence of Theorem 6.2, since the condition that ω1 is ν1-moderate and ω2 isν ˇ2-moderate immediately gives (6.4). ′ (2) It follows from (1.14) that Wf,g = Wg,f for every f, g ∈ S . ′ 2m ′ m This in turn implies that if a ∈ S (R ) and fj, hj ∈ S (R ), then

(Tph1,h2 (a)f1, f2) is well-deﬁned if and only if (Tpf2,f1 (a)h2, h1) is well- deﬁned, and then

(Tph1,h2 (a)f1, f2) = (Tpf2,f1 (a)h2, h1). The assertion (2) now follows this fact in combination with (1), and the fact that ω is ν-moderate if and only if 1/ω isν ˇ-moderate. The proof is complete. In view of Lemma 1.1 (4) and Theorem 3.2, Corollary 6.3 also can be formulated in the following way. ′ Proposition 6.3 . Assume that p, q, pj ,qj,rj,sj ∈ [1, ∞] for j = 1, 2 satisfy (6.2) and (6.5) for some p0,q0 ∈ [1, ∞]. Assume also that ν , ω ∈ P (R2m) and that h ∈ M rj,sj for j = 1, 2. Then the following j j 0 j (νj) are true:

(1) if ω1 is ν1-moderate, ω2 is νˇ2-moderate, ω = ω2/ω1 and ωa ∈ M p,q(R2m), then Tp (a) is continuous from M p1,q1 (Rm) to h1,h2 (ω1) M p2,q2 (Rm). Moreover, if q ≤ min(p , p′ ), then Tp (a) ∈ (ω2) 0 0 0 h1,h2 I (M 2 , M 2 ); p (ω1) (ω2) p,q 2m (2) if νj is ωj -moderate for j = 1, 2, and ωa ∈ M (R ) where ω =(ν ν )−1, then Tp (a) is continuous from M p1,q1 (Rm) to 2 1 h2,h1 (ω1) M p2,q2 (Rm). Moreover, if q ≤ min(p , p′ ), then Tp (a) ∈ (1/ω2) 0 0 0 h1,h2 I (M 2 , M 2 ). p (ω1) (1/ω2) Moreover, if in addition ωa ∈ Mp,q(R2m) and h ∈ Mrj,sj (Rm) for j (νj) j = 1, 2 in (1) or (2), then Tph1,h2 (a) is compact. Remark 6.4. In [CG], Cordero and Gr¨ochenig present a converse of Corollary 6.3 in the case p = q = ∞ and rj = sj = 1 for j = 1, 2, and the weight functions satisfy some extra conditions. (Cf. Theorem 4.3 in [CG].)

Acknowledgements It is a pleasure to gratitude professors P. Boggiatto, N. Kruglyak and I. Asekritova for fruitful discussions.

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Department of Mathematics and Systems Engineering, Vaxj¨ o¨ Uni- versity, Sweden E-mail address: [email protected]