BOUNDED BEREZIN-TOEPLITZ OPERATORS ON THE SEGAL-BARGMANN SPACE HIROYUKICHIHARA Abstract. We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex 9 0 n-space. ThisspaceischaracterizedbytheimageofaglobalBargmann- 0 typetransformintroducedbySj¨ostrand. Wealsoobtainthedeformation 2 estimates of the composition of Berezin-Toeplitz operators whose sym- bols and their derivatives up to order three are in the Wiener algebra n of Sj¨ostrand. Our method of proofs is based on the pseudodifferen- a J tial calculus and theheat flow determined by thephase function of the Bargmann transform. 3 2 ] A 1. Introduction F . We study the boundedness and the deformation estimates of Berezin- h t Toeplitz operators on a generalized Segal-Bargmann space (Fock space) in- a m troduced by Sj¨ostrand in [15]. This space is a reproducing kernel Hilbert space of square-integrable holomorphic functions on the complex n-space, [ and is characterized by the image of a global Bargmann-type transform. 3 We begin with a review of Sj¨ostrand’s “linear” theory in [15] to intro- v 3 duce the setting of the present paper. Let φ(X,Y) be a quadratic form 9 of (X,Y) Cn Cn of the form 1 ∈ × 1 1 0 φ(X,Y) = X,AX + X,BY + Y,CY , . 2h i h i 2h i 6 0 where A, B and C are complex n n matrices, tA = A, tC = C and 8 × X,Y = X Y + +X Y for X = (X ,...,X ) and Y = (Y ,...,Y ). :0 hSet i =i √ 11, C1R =···(C +nC¯)n/2 and CI = (1C C¯)/n2i. We denote1by In tnhe v − − n n identity matrix. Assume that i X × (1) detφ′′ = detB = 0, r XY 6 a (2) Imφ′′ = C > 0. YY I −1/2 −1/2 −1/2 −1/2 WeremarkthatdetC = detC det(C C C +iI )= 0sinceC C C I I R I n 6 I R I is a real symmetric matrix. Let h (0,1] be a semiclassical parameter, and let S(Rn) be the Schwartz class o∈n Rn. A global Bargmann-type transfor- mation of u S(Rn) is defined by ∈ Tu(X) = C h−3n/4 eiφ(X,y)/hu(y)dy, φ Rn Z 1991 Mathematics Subject Classification. Primary 47B35; Secondary 47B32, 47G30. Keywordsandphrases. Bargmanntransform,Segal-Bargmannspace,Berezin-Toeplitz operator, pseudodifferential operator. Supportedby theJSPS Grant-in-Aidfor ScientificResearch #20540151. 1 2 H.CHIHARA where C is a normalizing constant as φ C = 2−n/2π−3n/4 detB (detC )−1/4. φ I | | The assumption (2) guarantees the existence of a function Φ(X) = max Imφ(X,y) y∈Rn{− } 1 1 = Im(tBX),C−1Im(tBX) Im X,AX 2h I i− 2 h i = X,Φ′′ X¯ +Re X,Φ′′ X , h XX¯ i h XX i BC−1tB¯ BC−1tB A Φ′′ = I > 0, Φ′′ = I . XX¯ 4 XX − 4 − 2i We denote the Lebesgue measure on Cn by L. Set X = X,X¯ for X Cn. Let L2 be the set of all square-integrable funct|ion|s on Chn witih respec∈t Φ p to e−2Φ(X)/hL(dX), and let H be the set of all holomorphic functions in Φ L2. We remark that Φ 1 Re iφ(X,y) = Φ(X) C1/2(y+C−1Im(tBX))2. { } − 2| I I | The Bargmann transform T is well-defined for any tempered distribution u S′(Rn). Moreover Tu satisfies e−Φ(X)/hTu(X) S′(Cn), and is holo- mo∈rphiconCn. Inparticular,T givesaHilbertspace∈isomorphismofL2(Rn) onto H , whereL2(Rn)isthesetof allLebesguesquare-integrable functions Φ onRn. Wehereremarkthate−Φ(X)/hT(S(Rn)) S(Cn),andT(S(Rn))is denselyembeddedinH andT(S′(Rn))respecti⊂velysinceS(Rn)isdensely Φ embedded in L2(Rn) and S′(Rn) respectively. The Bargmann transform T is interpreted as a Fourier integral operator associated with a linear canon- ical transform κ :Cn Cn (Y, φ′ (X,Y)) (X,φ′ (X,Y)) Cn Cn, T × ∋ − Y 7→ X ∈ × κ (x,ξ) = ( tB−1(Cx+ξ),Bx AtB−1(Cx+ξ)). T − − If we set 2 ∂Φ Λ = X, (X) X Cn , Φ ( i ∂X (cid:12) ∈ ) (cid:18) (cid:19) (cid:12) then Λ = κ (R2n). This means that th(cid:12)e singularities of u S′(Rn) Φ T (cid:12) described in the phase space R2n are transla(cid:12)ted into those of Tu∈described in the Lagrangian submanifold Λ . Φ Let Ψ(X,Y) be a holomorphic quadratic function on Cn Cn defined by × the critical value of φ(X,Z) φ(Y¯,Z¯) /2i for Z Cn, that is, −{ − } ∈ 1 1 Ψ(X,Y)= X,Φ′′ Y + X,Φ′′ X + Y,Φ′′ Y . h XX¯ i 2h XX i 2h XX i Note thatΨ(X,X¯)= Φ(X). TT∗ is an orthogonal projector ofL2 onto H , Φ Φ and given by C (3) TT∗u(X) = Φ e[2Ψ(X,Y¯)−2Φ(Y)]/hu(Y)L(dY), hn Cn Z n 2 C = det(Φ′′ )= (2π)−n detB 2(detC )−1. Φ π XX¯ | | I (cid:18) (cid:19) BEREZIN-TOEPLITZ OPERATORS 3 Here we state the definition of Berezin-Toeplitz operators on H . If we Φ set R = C−1/2tB/2, then R∗R = Φ′′ . Let T be a class of symbols defined I X¯X by T = b(X) e−2|R(X−Y)|2/h b(Y) 2L(dY) < for any X Cn . ( (cid:12) Cn | | ∞ ∈ ) (cid:12) Z (cid:12) A Berezin-Toep(cid:12)litz operator T˜ associated with a symbol b T is defined by T˜u= TT∗(b(cid:12)u) for u H . Sbince ∈ b Φ ∈ Re 2Ψ(X,Y¯) 2Φ(Y) = Φ(X) Φ(Y) R(X Y)2, { − } − −| − | e−Φ(X)/hT˜u(X) takes a finite value for each X Cn provided that u L2 and b Tb. Historically, Berezin introduced thi∈s type of operators ac∈tinΦg ∈ on a class of holomorphic functions over some complex spaces or manifolds, and established the foundation of geometric quantization in his celebrated paper [1]. Properties of such operators and related problems on the usual Segal-Bargmann space have been investigated in several papers. See [2], [3], [5], [6], [7], [17] and references therein. Here we give two examples of H . Φ Example 1: If φ(X,Y) = iβ(X2/2 2XY +Y2), β > 0 and XY = − X,Y ,thenH istheusualSegal-Bargmannspace(theFockspace), Φ h i and β i i Ψ(X,Y¯)= XY¯, κ (x,ξ) = x ξ, iβ x+ ξ . T 2 − 2β − 2β (cid:18) (cid:18) (cid:19)(cid:19) It is remarkable that Φ(X) = β X 2/2 is strictly convex and Φ′′ = | | XX 0 in this case. The strict convexity justifies the change of quantiza- tion parameter. See [15, Proposition 1.3]. These facts are effectively used in the analysis on the usual Segal-Bargmann space. See e.g., [8] for the detail. Example 2: If we set φ(X,Y) = i(X Y)2/2, then T is the heat − kernel transform, and (X Y¯)2 (ImX)2 Ψ(X,Y¯) = − , Φ(X) = , κ (x,ξ) = (x iξ,ξ). T − 8 2 − In this case, the global FBI transform e−Φ(X)/hT and the space H Φ are used as strong tools for microlocal and semiclassical analysis of linear differential operators on Rn. See [12] for the detail. Thepurposeof thepresentpaperis to studytheboundednessandthede- formation estimates of Berezin-Toeplitz operators on the generalized Segal- BargmannspaceH . Tostateourresults, weintroducenotation andreview Φ pseudodifferential calculus on H developed in [15]. Φ We denote by L(H ) the set of all bounded linear operators of H to Φ Φ H , and set Φ ∂a ∂b Q(a,b) = ,(Φ′′ )−1 , a,b = iQ(a,b) iQ(b,a) ∂X X¯X ∂X¯ { } − (cid:28) (cid:29) for a,b C1(Cn). Pick up χ S(Cn) such that χ(X)L(dX) = 0. Cn Sj¨ostran∈d’s Wiener algebra S ∈(Cn) is the set of all tempered distribu6tions W R 4 H.CHIHARA on Cn satisfying (4) U(ζ;b) = sup F[uτ χ](ζ) L1(Cn), Z∈Cn| Z | ∈ ζ where F is the usual (not semiclassical) Fourier transform on Cn R2n, τ χ(X) = χ(X Z), and L1(Cn) is the set of all Lebesgue integrabl≃e func- Z tions on Cn. Se−t kbkSW = kU(·;b)kL1(Cn). We also denote by L∞(Cn) the set of all essentially bounded functions on Cn. The definition of S (Cn) W is independent of the choice of χ, and S (Cn) is invariant under linear W transforms on Cn. It is remarkable that B2n+1(Cn) S (Cn) B0(Cn), W ⊂ ⊂ andtheWeylquantizationofanyelementofS isaboundedlinearoperator. W Set N = 0,1,2,... for short. Bk(Cn), k N is the set of all bounded 0 0 Ck-functio{ns on Cn w}hose derivatives of any o∈rder up to k are also bounded on Cn. Next we introduce the Weyl quantization on H . For fixed X Cn, set Φ ∈ 2 ∂Φ X +Y Γ(X) = (Y,θ) Y Cn, θ = , ( (cid:12) ∈ i ∂X 2 ) (cid:12) (cid:18) (cid:19) (cid:12) and a volume of Γ(X) is defi(cid:12)ned by dΩ = dY1 dYn dθ1 dθn. For u H , the reproducing f(cid:12)ormula u= TT∗u∧h·a·s·a∧nothe∧r expr∧es·s·io·n∧ Φ ∈ 1 (5) u(X) = eihX−Y,θi/hu(Y)dΩ. (2πh)n ZΓ(X) The right hand sides of (3) and (5) coincide to each other via the change of variables called the Kuranishi trick. The Weyl quantization of a symbol a(X,θ) S (Λ ) = (κ−1)∗S (R2n) is defined by ∈ W Φ T W 1 X +Y OpW(a)u(X) = eihX−Y,θi/ha ,θ u(Y)dΩ h (2πh)n 2 ZΓ(X) (cid:18) (cid:19) for u T(S(Rn)). OpW(a)u is holomorphic in Cn since ∈ h ∂ X +Y ∂ X +Y eihX−Y,θi/ha ,θ = eihX−Y,θi/ha ,θ ∂X¯ 2 ∂Y¯ 2 (cid:18) (cid:19) (cid:18) (cid:19) in the sense of distribution. The Weyl quantization of a κ is defined by T ◦ 1 x+y OpW(a κ )u(x) = eihx−y,ξi/ha κ ,ξ u(y)dydξ h ◦ T (2πh)n R2n ◦ 2 Z (cid:18) (cid:19) for u S(Rn). It is remarkable that OpW(S (Λ )) is extended on H and a sub∈algebra of L(H ), and the exact EhgoroWv thΦeorem Φ Φ (6) OpW(a) T = T OpW(a κ ) h ◦ ◦ h ◦ T holds for a S (Λ ). Moreover, Guillemin discovered in [9] that T˜ = W Φ b OpW(b′ ) f∈or b(X) = b(X,X¯), where h 1/2 i b′ (X,θ) = b X,(Φ′′ )−1 θ Φ′′ X , 1/2 1/2 XX¯ 2 − XX (cid:18) (cid:18) (cid:19)(cid:19) and b is the heat flow of b defined by t t>0 { } b (X) = eth∆b(X) t BEREZIN-TOEPLITZ OPERATORS 5 = CΦ e−2|R(X−Y)|2/thb(Y)L(dY), (th)n Cn Z 1 ∂ ∂ ∆ = ,(Φ′′ )−1 . 2 ∂X X¯X ∂X¯ (cid:28) (cid:29) b makes sense for b T and t (0,2). We use only t [0,1] as a t ∈ ∈ ∈ quantization parameter. b is said to be the Berezin symbol of a Berezin- 1 Toeplitz operator T˜. These facts show that pseudodifferential calculus (See b e.g., [10], [12] and [16]) and the heat flow determined by the phase function play essential roles in the analysis of Berezin-Toeplitz operators. Here we state our results. Theorem 1. Suppose that b T . We have (i) If T˜ L(H ), then for a∈ny t (1/2,1], b Φ ∈ ∈ (7) kbtkL∞(Cn) 6 k(T˜2btkL(1H)Φn). − (ii) If b L∞(Cn) for some t [0,1/2), then T˜ L(H ). t b Φ ∈ ∈ ∈ (iii) Suppose that b S′(Cn) in addition. Set bλ(X) = eiRehX,λib(X) for λ Cn. Then, b1/2 ∈SW(Cn) if and only if ∈ ∈ (8) k(bλ)1(·)kL∞(Cn)e−h|tR−1λ|2/8 ∈ L1(Cnλ). In this case, T˜ L(H ). b Φ ∈ Theorem 2. Suppose that ∂Xα∂Xβ¯a,∂Xα∂Xβ¯b ∈ SW(Cn) for any multi-indices satisfying α +β 6 3. Then, there exists a positive constant C which is 0 | | independent of a, b and h, such that h ih T˜ T˜ T˜ + T˜ , [T˜ ,T˜] T˜ a◦ b− ab 2 Q(a,b) a b − 2 {a,b} (cid:13) (cid:13)L(HΦ) (cid:13) (cid:13)L(HΦ) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 6 C h2 ∂α∂β(cid:13)a (cid:13) ∂µ∂ν b . (cid:13) (cid:13) 0 k X X(cid:13)¯ kSW (cid:13) k X X¯ kSW (cid:13) |α+β|63 |µ+ν|63 X X Here we explain the known results and the detail of our results. Theo- rem 1-(i) is a refinement and a generalization of the results of Berger and Coburnin[3]. TheyprovedthatkbtkL∞(Cn) 6 C(t)kT˜bkL(HΦ) fort ∈ (1/2,1] withsomefunction C(t)in casethat H is theusualSegal-Bargmann space. Φ For a general H , we need some ideas to avoid difficulties coming from Φ Φ′′ = 0. Theorem 1-(ii) is obvious by the L2-boundedness theorem of XX 6 pseudodifferential operators of order zero with smooth symbols. The con- dition (8) is a special form of the condition for which b S (Cn). This 1/2 W ∈ is given by a special choice of a Schwartz function χ appearing in the def- inition of S (Cn). Theorem 1-(iii) seems to extend the known results by W Berger and Coburn in [3, Theorem 13], that is, if b > 0 and b L∞(Cn), 1 ∈ then T˜ L(H ). b Φ ∈ Theorem 2 reminds us of the recent interesting results of Lerner and Morimoto in [11] on the Fefferman-Phong inequality. Coburn proved in [5] the deformation estimates on the usual Segal-Bargmann space under the assumption a,b the set of all trigonometric polynomials+C2n+6(Cn), ∈ 0 6 H.CHIHARA where C2n+6(Cn) is the set of all compactly supported C2n+6-functions on 0 Cn. Roughly speaking, Theorem 2 asserts that the deformation estimates hold for a,b B2n+4(Cn). The relationship between Berezin-Toeplitz oper- ∈ ators and Weyl pseudodifferential operators on H gives a formal identity Φ T˜ T˜ = T˜, c = e−h∆/2(a′ #b′ ), a◦ b c 1/2 1/2 where # is the product of S (Λ ) in the sense of the Weyl calculus intro- W Φ duced later. Unfortunately, however, the backward heat kernel e−h∆/2 can act only on a class of real-analytic symbols, and it is very hard to obtain the symbol c. We apply the forward heat kernel eth∆ to the construction of the asymptotic expansion of the backward heat kernel h e−h∆/2 = 1 ∆+ (h2), − 2 O and give an elementary proof of Theorem 2. The organization of the present paper is as follows. In Section 2 we prove (i) and (iii) of Theorem 1. In Section 3 we prove Theorem 2. 2. Boundedness of Berezin-Toeplitz operators In this section we prove (i) and (iii) of Theorem 1. On one hand, to prove (i), we express the boundedness of T˜ in terms of a complete orthonormal b system of H . We introduce a trace class operator defined by T˜ and the Φ b complete orthonornal system, and take its trace which becomes b (X) for t any fixed X Cn. This idea is basically due to Berger and Coburn in [3]. ∈ In our case, however, Φ(X) is not supposed to be strictly convex, nor Φ′′ XX is not supposed to vanish. We need to be careful about these obstructions. On the other hand, the proof of (iii) is a simple computation. We choose a Schwartz function χ as a heat kernel at the time t = 1/2. Here we give two lemmas used in the proof of (i). For u,v H , the inner Φ ∈ product , is defined by h· ·iHΦ u,v = u(X)v(X)e−2Φ(X)/hL(dX), h iHΦ Cn Z which is the restriction of h·,·iL2Φ on HΦ. Set 1/2 u (X) = CΦ 2|α| (RX)αehX,Φ′X′XXi/h α hn α!h|α| ( ) for a multi-index α Nn. The first lemma is concerned with a complete ∈ 0 orthonormal system of H which is naturally generated by the Taylor ex- Φ pansion of the reproducing kernel e2Ψ(X,Y¯)/h. Lemma 3. {uα}α∈Nn0 is a complete orthonormal system of HΦ. Incase thatH is theusualSegal-Bargmann space, theproofof Lemma3 Φ is given in [8, page 40, (1.63) Theorem]. In this case, {T∗uα}α∈Nn0 is said to be the family of Hermite functions. The general case can be proved in the same way, and we here omit the proof of Lemma 3. Next lemma is concerned with the family of Weyl operators, which is a family of unitary operators on H and acts on symbols of Berezin-Toeplitz Φ BEREZIN-TOEPLITZ OPERATORS 7 operatorsasagroupofshiftsonCn. ThefamilyofWeyloperators Wλ λ∈Cn { } on H is defined by Φ W u(X) = e[2ϕ(X,λ)−ϕ(λ,λ)]/hu(X λ), λ − where ϕ(X,λ) = X,Φ′′ λ¯ + X,Φ′′ λ . h XX¯ i h XX i We remark that ϕ(X,λ) is holomorphic in X, and if u is holomorphic, then W u is also. Properties of Weyl operators are the following. λ Lemma 4. We have (i) W∗ = W on H . λ −λ Φ (ii) W∗ W = I on H . λ◦ λ Φ (iii) W∗ T˜ W = T˜ on H for b T . λ ◦ b◦ λ b(·+λ) Φ ∈ Proof. A direct computation shows that (9) 2ϕ(X +λ,λ) ϕ(λ,λ) 2Φ(X +λ) = 2ϕ(X +λ,λ)+ϕ(λ,λ) 2Φ(X) − − − − (10) = 2ϕ(X, λ) ϕ( λ, λ) 2Φ(X). − − − − − Let u,v H . Using a translation X X +λ and (10), we deduce Φ ∈ 7→ W u,v = e[2ϕ(X,λ)−ϕ(λ,λ)−2Φ(X)]/hu(X λ)v(X)L(dX) h λ iHΦ Cn − Z = e[2ϕ(X+λ,λ)−ϕ(λ,λ)−2Φ(X+λ)]/hu(X)v(X +λ)L(dX) Cn Z = e[2ϕ(X,−λ)−ϕ(−λ,−λ)−2Φ(X)]/hu(X)v(X +λ)L(dX) Cn Z = u,W v , h −λ iHΦ which shows that W∗ = W . λ −λ W∗ W = I is also proved by a direct computation λ◦ λ W∗ W u(X) = W W u λ◦ λ −λ◦ λ = e[2ϕ(X,−λ)−ϕ(−λ,−λ)]/h(W u)(X +λ) λ = e[−2ϕ(X,λ)−ϕ(λ,λ)]/h(W u)(X +λ) λ = e[−2ϕ(X,λ)−ϕ(λ,λ)+2ϕ(X+λ,λ)−ϕ(λ,λ)]u(X) = u(X), since ϕ(X +λ,λ) = ϕ(X,λ)+ϕ(λ,λ). TT∗ is self-adjoint on L2 and TT∗W v = W v for v H . Using this and Φ λ λ ∈ Φ (9), we deduce W∗ T˜ W u,v h λ ◦ b◦ λ iHΦ = T˜ W u,W v h b◦ λ λ iHΦ = TT∗(bW u),W v h λ λ iHΦ =hbWλu,WλviL2Φ = b(X)e[2ϕ(X,λ)+2ϕ(X,λ)−ϕ(λ,λ)−ϕ(λ,λ)−2Φ(X)]/h Cn Z u(X λ)v(X λ)L(dX) × − − 8 H.CHIHARA = b(X +λ)e[2ϕ(X+λ,λ)+2ϕ(X+λ,λ)−ϕ(λ,λ)−ϕ(λ,λ)−2Φ(X+λ)]/h Cn Z u(X)v(X)L(dX) × = b(X +λ)e−2Φ(X)/hu(X)v(X)L(dX) Cn Z = T˜ u,v , h b(·+λ) iHΦ which proves W∗ T˜ W = T˜ . (cid:3) λ ◦ b◦ λ b(·+λ) Here we prove Theorem 1-(i). Proof of Theorem 1-(i). Suppose T˜b ∈ L(HΦ), and set M = kT˜bkL(HΦ) for fsohroratn.yLXemmaCn4.shInowtsertmhastoTf˜bt(·h+eXc)o∈mpLlet(He Φor)thaonndorMma=l skyTs˜tbe(·m+Xg)kivLen(HΦin) Lemma 3, T˜∈ L(H ) implies that T˜u ,u 6 M for any α,β Nn. b ∈ Φ |h b α βiHΦ| ∈ 0 Since Φ(Y) = RY 2+Re Y,Φ′′ Y , we deduce that for any X Cn | | h XX i ∈ T˜ u ,u h b(·+X) α βiHΦ = TT∗(b( +X)u ),u h · α βiHΦ =hb(·+X)uα,uβiL2Φ 1/2 C 1 Φ = b(X +Y) hn α!β! Cn (cid:18) (cid:19) Z α β 1/2 1/2 2 RY 2 RY e−2|RX|2/hL(dY). × h h ( ) ( ) (cid:18) (cid:19) (cid:18) (cid:19) In particular, if we take α= β and sum it up for α = k, then we have | | T˜ u ,u h b(·+X) α αiHΦ |α|=k X (11) =CΦ 1 2|RY|2 ke−2|RY|2/hb(X +Y)L(dY). hn Cn k! h Z (cid:18) (cid:19) Fix(t,X) (1/2,1] Cn. Whenk = 0,(11)showsthat T˜ u ,u = ∈ × h b(·+X) 0 0iHΦ b1(X), and |hT˜b(·+X)u0,u0iHΦ| 6 M implies that kb1kL∞(Cn) 6 M, which is (7) at t = 1. We consider (t,X) (1/2,1) Cn below, and set s = 1/t 1 ∈ × − ∈ (0,1). Here we introduce a trace class operator ∞ H u= ( s)k u,u T˜ u s,X − h αiHΦ b(·+X) α k=0 |α|=k X X for u H . Let K (Y,Z) be the integral kernel of H , that is, Φ s,X s,X ∈ ∞ K (Y,Z)= ( s)k T˜ u (Y)u (Z). s,X b(·+X) α α − k=0 |α|=k X X It is easy to see that K (Y,Y) L1(Cn;e−2Φ(Y)/hL(dY)) since s,X ∈ ∞ sk T˜ u (Y) u (Y)e−2Φ(Y)/hL(dY) b(·+X) α α Cn| || | k=0 |α|=kZ X X BEREZIN-TOEPLITZ OPERATORS 9 ∞ n Mtn 6M s|α| = M sk = M(1 s)−n = . − (2t 1)n αX∈Nn0 Xk=0 ! − Then, the Lebesgue convergence theorem and (11) impliy that N t−n ( s)k T˜ u (Y)u (Y)e−2Φ(Y)/hL(dY) b(·+X) α α Cn − Z k=0 |α|=k X X (12) = CΦ N ( s)k 1 2|RY|2 ke−2|RY|2/hb(X +Y)L(dY) (th)n Cn − k! h Z k=0 (cid:18) (cid:19) X converges as N . Thus we have (7) for t (1/2,1) since the right hand → ∞ ∈ side of (12) converges to b (X). (cid:3) t Next we prove Theorem 1-(iii). Boulkhemair proved in [4] that (4) is equivalent to (13) sup F−1[F[b]τ χ˜](X) L1(Cn) X∈Cn| λ | ∈ λ with some χ˜ S(Cn) satisfying χ˜(X)L(dX) = 0, where F−1 is the Cn usual inverse F∈ourier transform on Cn. 6 R Proof of Theorem 1-(iii). Wecomputethecondition(13). WechooseF[χ](X) = C e−4|RX|2/h whichis theheatkernelatthetimet = 1/2, andexpectacom- 1 prehensive expression coming from the parallelogram law. Let X∗ Cn be ∈ thedualvariable underthe Fourier transform. We choose a constant C > 0 1 so that χ(X∗) = e−h|tR¯−1X∗|2/16. Set χ = τ χ for short. The parallelogram λ λ law implies that F[b ](X∗)χ (X∗)= e−h|tR¯−1X∗|2/16−h|tR¯−1(X∗−2λ¯)|2/16F[b](X∗) 1/2 2λ¯ = e−h|tR−1λ|2/8−h|tR¯−1(X∗−λ¯)|2/8F[b](X∗). Taking the inverse Fourier transformation of the above, we deduce F−1[F[b χ ]](X) 1/2 2λ¯ =e−h|tR−1λ|2/8CΦ eiRehX−Y,λi−2|R(X−Y)|2/hb(Y)L(dY) hn Cn Z =eiRehX,λi−h|tR−1λ|2/8(b−λ) (X). 1 Hence, we obtain Xs∈uCpn|F−1[F[b1/2]χ−2λ¯](X)| = e−h|tR−1λ|2/8k(bλ)1kL∞(Cn). This completes the proof. (cid:3) 3. Deformation estimates for compositions Finally, we prove Theorem 2. We first review the composition of pseudo- differential operators on H . Let σ be a canonical symplectic form on C2n, Φ that is, n σ = dΞ dX = dΞ dX j j ∧ ∧ j=1 X 10 H.CHIHARA at (X,Ξ) Cn Cn. Split σ into real and imaginary parts, and denote σ = σR+i∈σI. R2×n and ΛΦ are I-Lagrangian and R-symplectic. Indeed, this is obvious for R2n, and a direct computation shows that σ = 0 and I|ΛΦ n ∂2Φ 2 ∂Φ σR|ΛΦ = 2i ∂X ∂X¯ dXj∧dX¯k for θ = i ∂X(X), j k j,k=1 X which is nondegenerate. We use this fact as κ∗σ = σR on R2n. T Let a′,b′ S (Λ ). It is well-known that W Φ ∈ OpW(a′ κ ) OpW(b′ κ ) = OpW(a′ κ #b′ κ ), h ◦ T ◦ h ◦ T h ◦ T ◦ T 1 a′ κ #b′ κ (x,ξ) = e−2iσR(y,η;z,ζ)/h ◦ T ◦ T (2πh)2n R4n Z a′ κ (x+y,ξ+η)b′ κ (x+z,ξ +ζ)dydηdzdζ. T T × ◦ ◦ Set θ(X) = 2iΦ′ (X) for X Cn. Using the exact Egorov theorem (6) − X ∈ together with thesymplectic transform κ or a direct computation, we have T OpW(a′) OpW(b′) = OpW(a′#b′), h ◦ h h 2nC 2 a′#b′(X,θ(X)) = Φ e−2iσ(Y,θ(Y);Z,θ(Z))/h hn C2n (cid:18) (cid:19) Z a′(X +Y,θ(X +Y))b′(X +Z,θ(X +Z))L(dY)L(dZ). × Here we begin the proof of Theorem 2. Suppose that ∂α∂βa,∂α∂βb X X¯ X X¯ ∈ S (Cn) for any multi-indices satisfying α+β 6 3. Set a = eth∆a, b = W t t | | eth∆b, i 2 a′ (X,θ) = a X, (Φ′′ )−1 θ Φ′′ X , 1/2 1/2 2 XX¯ − i XX (cid:18) (cid:18) (cid:19)(cid:19) i 2 b′ (X,θ) = b X, (Φ′′ )−1 θ Φ′′ X . 1/2 1/2 2 XX¯ − i XX (cid:18) (cid:18) (cid:19)(cid:19) Then,wehaveT˜ T˜ = OpW(a′ #b′ ). Sincea′ (X,θ(X)) = a (X,X¯), a◦ b h 1/2 1/2 1/2 1/2 if we write a (X) = a (X,X¯) and b (X) = b (X,X¯) simply, then T˜ T˜ = t t t t a b ◦ OpW(a #b ), and h 1/2 1/2 2nC 2 a #b (X) = Φ e−2iσ(Y,θ(Y);Z,θ(Z))/h t t hn C2n (cid:18) (cid:19) Z a (X +Y)b (X +Z)L(dY)L(dZ). t t × To complete the proof of Theorem 2, we have only to show that h (14) a #b eh∆/2(ab) eh∆/2Q(a,b) mod h2S (Cn). 1/2 1/2 W ≡ − 2 Here we remark that 2iσ(Y,θ(Y);Z,θ(Z)) = 4 Y,Φ′′ Z¯ 4 Z,Φ′′ Y¯ = 8iIm Y,Φ′′ Z¯ , − h XX¯ i− h XX¯ i h XX¯ i h ∂ Ye−2iσ(Y,θ(Y);Z,θ(Z))/h = (Φ′′ )−1 e−2iσ(Y,θ(Y);Z,θ(Z))/h, 4 X¯X ∂Z¯ h ∂ Y¯e−2iσ(Y,θ(Y);Z,θ(Z))/h = (Φ′′ )−1 e−2iσ(Y,θ(Y);Z,θ(Z))/h. −4 XX¯ ∂Z