Riesz transforms on generalized Heisenberg groups and Riesz transforms associated to the CCR heat flow

Publ. Mat. 48 (2004), 309–333 RIESZ TRANSFORMS ON GENERALIZED HEISENBERG GROUPS AND RIESZ TRANSFORMS ASSOCIATED TO THE CCR HEAT FLOW Franc¸oise Lust-Piquard Abstract Let 1 <q<∞. We prove that the Riesz transforms Rk=XkL−1 2on a generalized Heisenberg group Gsatisfy PK k=1 |Rk(f)|21 2Lq(G) ≤C(q, J)kfkLq(G)where K,Jare respectively the dimensions of the first and second layer of the Lie algebra of G. We prove similar inequalities on Schatten spaces Sq(H), with dimension free constants, for Riesz transforms associated to commuting inner ∗-derivations Dkand a suitable substitute of the square function. An example is given by the derivations associated to ncommuting pairs of operators (Pj, Qj) on a Hilbert space Hsatisfying the canonical commutation relations [Pj, Qj] = iIH. General introduction This paper is divided in two parts: we solve similar problems in two different settings, using similar methods inspired by the first part of [P1], which contains a proof of the following classical inequalities [S]: for 1< q < ∞and f∈ D(Rn), DqkfkLq(Rn)≤ n X k=1 |Rk(f)|2!1 2Lq(Rn) ≤CqkfkLq(Rn) where Rk(f) = ∂ ∂xk L−1 2(f), L =− n X k=1 ∂2 ∂x2 k and the constants do not depend on n. 2000 Mathematics Subject Classification. 43A80, 46L50, 46L57. Key words. Heat operator, Riesz transforms, H-groups, commuting ∗-inner derivations. 310 F. Lust-Piquard The first part deals with Riesz transforms acting on Lq(G),where G is a generalized Heisenberg group, and owes a lot to [CMZ]. The second part deals with Riesz transforms acting on the Schatten space Sq(H), associated to commuting ∗-inner derivations on the algebra K(H) of compact operators on a Hilbert space H; an example is given by the inner derivations defined by (Pj, Qj)n j=1, where Pj,Qjsatisfy the canonical commutation relation [Qj, Pj] = iIHand the other commutators are zero. We already used Pisier’s method in other settings, see e.g. [LP2], [LP3]. However the difficulties arise at different steps in different applications. Since the two settings we consider are very different, we present more precise introductions in each part. Acknowledgement. We thank V. Georgescu for fruitful discussions on the setting of the second part. We also thank the referee for detecting a gap in the proof of Theorem 1.b in an earlier version of the paper. 1. Riesz transforms on generalized Heisenberg groups 1.1. Introduction. For any stratified Lie group G, we denote by X1,...,XKa basis of the top layer of the Lie algebra Gof G, by L=− K X j=1 X2 j the subelliptic Kohn Laplacian on G, by Rk=XkL−1 2,1≤k≤K, the Riesz transforms. The boundedness of each Rkon Lq(G), 1 < q < ∞, is known: the classical proof uses the homogeneity of its kernel and the singular integral results of [FS, Chapter 6]; see also Lemma 2 below. Our interest in this paper is to look for dimension free inequalities involving PK k=1 |Rk(f)|21 2Lq(G) . We first consider the case where Gis a Heisenberg group Hnand give a simpler proof of the main result of [CMZ]. We extend this result to the generalized Heisenberg groups HK,J defined by Kaplan [K]; they are particular step two stratified Lie groups, and K,Jdenote respectively the dimensions of the first and second layer in the Lie algebra of HK,J . The Heisenberg group Hnis the same as H2n,1. Riesz Transforms 311 Theorem 1. Let 1< q < ∞,1 q+1 q0= 1. a) [CMZ]There exist constants Cqsuch that, for every n∈N∗and every f∈ D(Hn), C−1 q0kfkLq(Hn)≤ 2n X k=1 |Rk(f)|2!1 2Lq(Hn) ≤CqkfkLq(Hn). b) The same holds for generalized Heisenberg groups HK,J , with constants C(q, J)which depend on qand Jbut not on K. It is a standard fact (see e.g. [CMZ]) that, in the above formula, for any Lie group G, the lefthand side inequality for qis an easy consequence of the righthand side one for q0,1 q+1 q0= 1. Theorem 1 relies on Christ’s study [C] of Hilbert transforms along curves for homogeneous nilpotent Lie groups, and on the use of dilations δtin order to get an expression of the convolution operator e−1 2t2L involving the (heat) kernel pof e−1 2L, namely (see Lemma 2 a)) (1) e−1 2t2L(f)(γ) = ZG f(γδtg−1)p(g)dg. Theorem 1 also uses a formula (Lemma 2 b) below) (2) √2πXL−1 2(f)(γ) = ZG F(γ, g)(Xp)(g)dg which holds on every stratified group G,Xlying in the first layer of G,Fbeing a Hilbert transform of f. These ingredients are already in [CMZ]; they are reminiscent of the method of [P1]. In the proof of Theorem 1 a), our improvement upon [CMZ] is that we do not use the explicit formula of pand avoid computations. Denoting p=p(n)when G=Hn, we use only the following properties: (i) [FS]pis a positive function lying in S(G), and RGp dg = 1, (ii) p(n)(x1,...,yn, u) is radial with respect to (x1,...,yn), i.e. depends on (r, u), (iii) p(n)(x1,...,yn, u) = p(1)(x1, y1, u)∗u...∗up(1)(xn, yn, u), where ∗udenotes convolution in Rwith respect to the variable u. Property (ii) is used through the observation that Xjp(n)=xj 1 r ∂p(n) ∂r + 2yj ∂p(n) ∂u . In the proof of Theorem 1 b) we use the analogue for HK,J . 312 F. Lust-Piquard We recall that the heat kernels (i.e. the kernels of e−tL) on Hnand the non isotropic Heisenberg groups, or rather their Fourier transform with respect to u, were first computed in [G] and [H], then in several subsequent papers, by rather complicated methods; the heat kernels on general step two stratified groups were computed in [CY]. A more explicit formula for generalized Heisenberg groups HK,J is given in [R], using the result for Hn. Let us mention our computation of the heat kernels for isotropic or non isotropic Heisenberg groups and the free step two stratified groups Nn,2[LP1], which is simpler than the previous ones and relies on the common starting point of [CMZ] and the present paper, namely formula (1). 1.2. Notation. For a background on stratified groups (which are particular homogeneous groups) we refer to [FS, Chapter I]. We consider stratified Lie groups Gequipped with their Haar measure, denoted by dg or dγ, which is Lebesgue measure on the underlying space Rd.D(G) denotes the space of C∞compactly supported functions on G,S(G) denotes the Schwartz class. The convolution of two functions f,hlying in S(G) is defined by f∗h(γ) = ZG f(γg−1)h(g)dg. The Lie algebra of left invariant vector fields on Gis denoted by G. For X∈ G, (3) X(f∗p) = f∗Xp. The first layer of Gis the linear subspace which spans Gas a Lie algebra. We denote by σthe automorphism of G, corresponding to the automorphism σof Gwhose action on the first layer is σ:X−→ −X. Gis equipped with a dilation δt,t > 0, corresponding to the automorphism of Gwhose action on the first layer is (4) δtX=tX. The induced action on functions f:G→Ris denoted by (δtf)(g) = f(δtg), and [FS, I C] (5) Xδtf=tδt(Xf). A stratified Lie group Gis said to be step two if the central layer of Gis the second one; denoting by X1,...,XKa basis of the first layer, and by U1,...,UJa basis of the second layer, it means that all Riesz Transforms 313 commutators [Xj, Xk] belong to the linear span of the Uj’s and the other commutators are zero. Every g∈Gis defined in a unique way by g= exp PK k=1 xkXk+PJ j=1 ujUjand we denote g= (x, u) = (x1,...,xK, u1,...,uJ). In this setting, the Haar measure on G, i.e. the Lebesgue measure on RK+J, is also denoted by dx du. In particular, σg =σ(x, u) = (−x, u), hence σg−1= (x, −u), and δt(x, u) = (tx, t2u). We will use the following easy, but crucial, result which is standard when J= 0 and ψis the gaussian density on RK. Lemma 2. Let ψbe a measurable function: RK+J→Rsuch that RRJ|ψ(x, u)|du only depends on |x|,x= (xk)K k=1 ∈RK. Then a) if 1≤q < ∞,ak∈C,  K X k=1 akxkLq(|ψ|dx du) = K X k=1 |ak|2!1 2 kx1kLq(|ψ|dx du); b) if 1<q<∞,1 q+1 q0= 1, for every h∈Lq(|ψ|dx du), K X k=1 ZRK+J xkhψ dx du 2!1 2 ≤ kx1kLq0(|ψ|dx du)khkLq(|ψ|dx du). Proof: For short, we write Lq(|ψ|) instead of Lq(|ψ|dx du). We use polar coordinates in RK, namely a= (a1,...,aK) = |a|w,x=|x|v=rv, with w, v ∈ΣK, v = (v1,...,vK), here ΣKdenotes the unit sphere of RK and dσKis the uniform measure on it, with total mass the area of ΣK. a) Since RRJ|ψ|du only depends on r,  K X k=1 akxk q Lq(|ψ|) =|a|qZ∞ 0 rqZRJ|ψ|durK−1drZΣK|hw, vi|qdσK(v) =|a|qZ∞ 0 rqZRJ|ψ|durK−1drZΣK|v1|qdσK(v) =|a|qkx1kq Lq(|ψ|). 314 F. Lust-Piquard b) This follows from a): indeed, by H¨older inequality, K X k=1 Zxkhψ dx du 2!1 2 = sup |a|=1 Z K X k=1 akxk!hψ dx du ≤ khkLq(|ψ|)sup |a|=1  K X k=1 akxkLq0(|ψ|) =khkLq(|ψ|)kx1kLq0(|ψ|). Let Gbe a stratified Lie group; to g∈Gwe associate the curve: R→G g(t) = δtgfor t≥0, g(t) = δ|t|σg for t < 0. In particular, if Gis step two, the curve associated to g= (x, u) is g(t) = (tx, t2u), t ∈R. The Hilbert transform of falong the curve g(t) is F(γ, g) = pv Z∞ −∞ f(γg(t)−1)dt t = lim ε→0+Zε<|t|<ε−1 f(γg(t)−1)dt t = lim ε→0+F(γ, g, ε). The function F(γ, g) is well defined on G×Gbecause f∈ D(G). F(γ, g, ε) is called the truncated Hilbert transform. The map hg t:f→f(.g(t)−1), t∈R,L∞(G)→L∞(G), is a ∗-homomorphism of the ∗-algebra L∞(G), but {hg t}t∈Ris not a one parameter group. This makes an important difference with the settings of [LP1], [LP2], [LP3] and the second part of this paper. The next result comes from [CMZ, Proof of Lemmas 1 and 5]. For the sake of completeness we give a more precise proof. Riesz Transforms 315 Lemma 3. Let Gbe a stratified Lie group and f∈ D(G). Then (1) e−1 2t2L(f)(γ) = ZG f(γδtg−1)p(g)dg, t > 0, and, for Xin the first layer of G,dγ a.s., (2) √2πXL−1 2(f)(γ) = ZG F(γ, g)(Xp)(g)dg, where F(γ, g)is the Hilbert transform of falong the curve g(t),g∈G. For 1< q < ∞, the Riesz transforms satisfy √2πXL−1 2(f)Lq(G)≤hqkXpkL1(G)kfkLq(G). Proof: Formula (1) holds true for t= 1 by definition of p. By (4) (see also [LP1]), e−1 2t2L=δt−1e−1 2Lδt, hence, by (5), (1) e−1 2t2L(f)(γ) = δt−1[(f◦δt)∗p(γ)] = ZG f(γδtg−1)p(g)dg. By (5), (1) and (3), Xe−1 2t2L(f) = t−1δt−1X[(f◦δt)∗p)] = t−1δt−1[(f◦δt)∗(Xp)]. The automorphism σmaps Lto L, so p=p◦σand Xp =X(p◦σ) = −(Xp)◦σ. For h∈ D(G), since σ2g=gand σis measure preserving, h∗(Xp)(γ) = ZG h(γg−1)(Xp)(g)dg =−ZG h(γg−1)(Xp)(σg)dg =−ZG h(γσg−1)(Xp)(g)dg. In particular, for f∈ D(G), 2Xe−1 2t2L(f)(γ) = t−1ZG [f(γδtg−1)−f(γδtσg−1)](Xp)(g)dg. Since rπ 2XL−1 2(f) = Z∞ 0 Xe−1 2t2L(f)dt 316 F. Lust-Piquard we get √2πXL−1 2(f)(γ) = Z∞ 0ZG [f(γδtg−1)−f(γδtσg−1)](Xp)(g)dgdt t. Since p∈ S(G), Xp ∈L1(G), and formula (2) now comes from the subsequent Lemma 4 b). By (2) and H¨older inequality with 1 q+1 q0= 1, dγ a.s., √2πXL−1 2(f)(γ)≤ kF(γ, .)kLq(|Xp|dg)kXpk 1 q0 L1(G). By the subsequent Lemma 4 a), kF(γ, g)kLq(dγ,Lq(|Xp|dg)) ≤hqkfkLq(G)kXpk 1 q L1(G), which ends the proof. The next lemma comes from [C]; it is already used in this setting in [CMZ], see [CMZ, Lemma 5]. Lemma 4. Let Gbe a stratified Lie group and f∈ D(G). For g∈G let F(γ, g)be the Hilbert transform of falong the curve g(t)and let ψ∈L1(G). Then a) for 1<q<∞, there exists a constant hqsuch that kFkLq(dγ⊗|ψ|dg)≤hqkfkLq(dγ)kψk 1 q 1. b) ZG F(γ, g)ψ(g)dg =Z∞ 0ZG (f(γδtg−1)−f(γδtσg−1))ψ(g)dgdt t, dγ a.s. Proof: a) When gruns through G, the corresponding family of curves g(t) satisfies the assumptions of [C, pp. 579 and 594]. By the main result of [C], if 1 < q < ∞, there exists a constant hqsuch that, for every g∈G, kF(., g)kLq(G)≤hqkfkLq(G). This proves a) by integration with respect to gand Fubini theorem. b) For every g∈G, the truncated Hilbert transform F(γ, g, ε) along the curve g(t) satisfies [C, Lemma 6.3] kF(., g)−F(., g, ε)kL2(G)−→ε→0+0 Riesz Transforms 317 and sup ε>0kF(., g, ε)kL2(G)≤h2kfkL2(G). Hence, since ψ∈L1(G), the dominated convergence theorem implies lim ε→0+ZG (F(γ, g)−F(γ, g, ε))ψ(g)dgL2(G) ≤lim ε→0+ZGkF(., g)−F(., g, ε)kL2(G)|ψ(g)|dg = 0. This implies b), because, by Fubini theorem, dγ a.s., ZG F(γ, g, ε)ψ(g)dg =Zε<t<ε−1ZG (f(γδtg−1)−f(γδtσg−1))ψ(g)dgdt t. Proof of Theorem 1: We treat Heisenberg groups first because the proof in this case is simpler and the idea is more apparent. a) When G=Hn, we denote p=p(n). The group law on Hnis γg = (x1, y1,...,xn, yn, u)(x0 1, y0 1,...,x0 n, y0 n, u0) = x1+x0 1, y1+y0 1,...,xn+x0 n, yn+y0 n, u+u0+2 n X j=1 (yjx0 j−xjy0 j) . By definition, for g= (x1,...,yn, u), (Xjp(n))(g) = ∂p(n) ∂xj (g) + 2yj ∂p(n) ∂u (g) (Yjp(n))(g) = ∂p(n) ∂yj (g)−2xj ∂p(n) ∂u (g). The Laplacian is given by −L= n X j=1 "∂2 ∂x2 j +∂2 ∂y2 j + 4(x2 j+y2 j)∂2 ∂u2+ 4 xj ∂2 ∂yj∂u −yj ∂2 ∂xj∂u# hence commutes with rotations on (x1,...,yn). It follows that p(n)is radial with respect to (x1,...,yn),hence so are ∂p(n) ∂u and the function 1 r ∂p(n) ∂r =1 xj ∂p(n) ∂xj =1 yj ∂p(n) ∂yj , 324 F. Lust-Piquard where, since PK i=1 aiXi= 1, Ba(x) = "K X k=1 xkXk, K X i=1 aiXi#=adPK i=1 aiXi K X k=1 xkXk! =PEa K X k=1 xkXk! with Ea= (ker adPK i=1 aiXi)⊥. By rotation on the xvariables, we may suppose that Eais the span of X1,...,XJ. Since ∂p(K,J) ∂ρ is radial with respect to x Aq0 q0= sup |a|=1 ZRK+J|Ba(x)|q0 |u1|q01 ρ ∂p(K,J) ∂ρ (x, u)dx du =ZRK+J J X k=1 |xk|2!q0 2 |u1|q0−1 ∂p(K,J) ∂u1 (x, u)dx du ≤ZRKJ+J J X k=1 |xk|2!q0 2 |u1|q0−1 ∂p(KJ,J) ∂u1 (x, u)dx du where the inequality is verified as in a), replacing p(1) by p(KJ,J), since J < KJ. Finally C2(q, J)=hq J X k=1|xk|2 !1 2 |u1|Lq0(|1 u1 ∂p(KJ,J) ∂u1|dx du)  1 u1 ∂p(KJ,J) ∂u1 1 q L1 (RKJ+J) . This constant is finite because p(KJ,J)∈ S(HKJ,J ) and we get  K X i=1 |Ri(f)(γ)|2 !1 2Lq(HK,J ) ≤1 √2πC1(q, J)+ 1 2C2(q, J)kfkLq(HK,J ). This implies as in a) the lefthand side inequality. Riesz Transforms 325 Remark. The above proof of a) does not seem to extend to the setting of non isotropic Heisenberg groups, because it uses in a crucial way the radiality of p; so does the proof in [CMZ]. However, it is stated in [BDJ, p. 59], without explanation, that the proof in [CMZ] can be extended to the non isotropic case. 2. Riesz transforms associated to commuting inner ∗-derivations 2.1. Introduction and notation. We now show how the classical result on Riesz transforms on Lq(Rn , dx) can be extended to Riesz transforms on Schatten spaces Sq(H), defined by inner (bounded or unbounded) commuting ∗-derivations acting on the C∗algebra K(H) of compact operators on the Hilbert space H. We denote by γnthe standard gaussian density on Rnand by Π the orthogonal projection onto the linear span of the coordinates yj in L2(γn). As well known, Π extends as a bounded operator: Lq(γn)→ Lq(γn), 1 ≤q < ∞, which we still denote by Π. The Fourier transform on Ris defined by b f(u) = RRe−iyuf(y)dy. We recall that Sq(H) is the space of compact operators Xon Hsuch that tr(X∗X)q 2<∞, 1 ≤q < ∞.B(H) is the dual space of S1(H) which is itself the dual space of K(H). S2(H) is the Hilbert space of Hilbert Schmidt operators on H. From now on, we assume His separable. We denote by |X|2 s=X∗X+XX∗the symmetrized modulus of X∈ K(H). Let hbe a ∗-automorphism of K(H),hence h∗∗ is a ∗-automorphism of B(H). By [Pe, Theorem 8.9.2] there exists a unitary operator Uon H such that h(X) = UXU∗,X∈K(H). Let (hy)y∈Rbe a strongly continuous one parameter group of ∗-automorphisms of K(H); then there exists a one parameter group of unitaries (Uy)y∈Rsuch that hy(X) = UyXU∗ y, X∈K(H) (see e.g. [V, Theorem 11.1] for an actually stronger result, or [Par, pp. 86–87]); by Stone’s theorem Uy=eyA, where iA is selfadjoint on H. Hence the generator Dof (hy)y∈Ris an inner ∗-derivation, in general unbounded, defined by D(X) = [A, X]. In particular (hy)y∈Ris a group of isometries of S2(H) (and a group of isometries of every Sq(H), 1 ≤q < ∞). By [RS, Theorem VIII.9], (hy)y∈Ris strongly continuous on S2(H). Its generator is naturally induced by D, and iD is self-adjoint on S2(H) by Stone’s theorem. Let us now recall some facts about joint functional calculus for “commuting” ∗-inner derivations. We consider nstrongly continuous one 326 F. Lust-Piquard parameter groups of ∗-automorphisms of K(H), with respective generators Dj, 1 ≤j≤n, which we denote respectively by (eyjDj)yj∈R. We assume that (i) their restrictions commute on S2(H) (ii) the only X∈ S2(H) which is invariant under every eyjDjis X= 0. In particular, by (i), y−→ U(y) = ePn j=1 yjDj is a strongly continuous map of Rninto the unitary operators on S2(H), satisfying U(y+z) = U(y)U(z), y, z ∈Rnand U(0) = I. Then [RS, Theorem VIII.12], there is a projection valued measure Eon Rnsuch that (8) ePn j=1 yjDj(X), Y =ZRn eihy,λidhEλ(X), Y i, X, Y ∈ S2(H), y ∈Rn. By bounded functional calculus and Fubini theorem, for every F∈ S(Rn), as bounded operators on S2(H) [RS, p. 272], (9) b F(iD1,...,iDn) = ZRn ePn k=1 ykDkF(y)dy. Moreover, let h:Rn→Rbe a Borel measurable function and let X∈ S2(H) be such that RRn|h(λ)|2dhEλ(X), Xi<∞; then the formula hh(iD1,...,iDn)(X), Y i=ZRn h(λ)dhEλ(X), Y i, Y ∈ S2(H) defines an operator h(iD1,...,iDn) which is densely defined and selfadjoint on S2(H). This holds in particular for L=− n X k=1 D2 k. By assumption (ii), the projection E{0}is null. Indeed, let Z∈ S2(H) lying in its range. Since the measure E−δ0⊗E{0}is valued in the set of orthogonal projections Pon S2(H) such that P E{0}= 0, hence P(Z) = 0, the scalar measure 1{Rn\{0}}(λ)dhEλ(Z), Y iis zero for every Y∈ S2(H); then, by (8), Y, ePn j=1 yjDj(Z)=Deihy,λi, δ0EY, E{0}Z=hY, Zi, which, by (ii), implies Z= 0. Riesz Transforms 327 The Riesz transforms are defined by Rj=DjL−1 2,1≤j≤n, so they are contractions on their domain in S2(H). Since E{0}= 0, Rjis actually defined on S2(H) by (10) hRj(X), Y i=ZRn\{0} iλj |λ|dhEλ(X), Y i. The main result of this part is the following theorem: Theorem 5. Let Hbe a separable Hilbert space and let (eyjDj)yj∈R,1≤ j≤nbe strongly continuous one parameter groups of ∗-automorphisms of K(H)satisfying the above conditions (i), (ii). Let 1<q<∞. a) Then, for every X∈K(H)and t∈R, e−1 2t2L(X) = ZRn etPn j=1 yjDj(X)γn(y)dy; e−1 2t2Lis a completely positive contraction: K(H)→K(H)and a contraction of every Sq(H). b) The operator R=Pn j=1 yjRjis a (completely) bounded operator: Sq(H)→Lq(γn(y)dy, Sq)which satisfies √2πR(X) = (Π ⊗ISq)pv Z∞ −∞ etPn j=1 yjDAj(X)dt t. c) For X∈ Sq,kXkSqis respectively equivalent, with constants which depend only on q, to (i) Pn j=1 |Rj(X)|2 s1 2Sq ,2≤q < ∞, and to (ii) infPn j=1 BjB∗ j1 2Sq +Pn j=1 C∗ jCj1 2Sq,1<q<2, where the infimum is taken over all decompositions Rj(X) = Bj+ Cjin Sq(H). Note that, on H, the operators Rj(X), 1 ≤j≤n, do not commute in general, and Rj(X)∗=Rj(X∗) does not commute in general with Rj(X). 328 F. Lust-Piquard Proof of Theorem 5: The strategy is similar to Pisier’s in the first part of [P1] and to the one we used in the first part of this paper. On one hand, things are much easier than for Heisenberg groups because we deal with a one parameter group; on the other hand, some difficulties arise from the setting of non commutative Lq’s. a) Formula (9) defines bounded operators on K(H) and Sq(H), 1 ≤ q < ∞, with norm less than kFkL1(dy), because ePn k=1 ykDkis an isometry of K(H) and Sq(H) for every y. This formula gives a Stinespring factorization (see e.g. [Pa]) of b F(iD1,...,iDn) acting on K(H) because X−→ ePn j=1 yjDj(X) is a ∗-homomorphism: K(H)→L∞(dγn, B(H)) ⊂B(L2(dγn, H)). Applying this to F(y) = γn(y) and tD1,...,tDnproves assertion a). b) α) By (10) and Fubini theorem, for X∈ S2(H), rπ 2hRj(X), Xi=rπ 2ZRn iλj |λ|dhEλ(X), Xi =ZRn iλjZ∞ 0 e−1 2t2|λ|2dtdhEλ(X), Xi =Z∞ 0ZRn iλje−1 2t2|λ|2dhEλ(X), Xidt =Z∞ 0DDje−1 2t2L(X), XEdt. β) Let F(y) = yjγn(y),hence b F(u) = −iuje−1 2|u|2. By (9) applied to Fand tD1,...,tDn, by symmetry with respect to t, for X∈ Sq(H), tDje−1 2t2L(X) = ZRn etPn k=1 ykDk(X)yjγn(y)dy =1 2ZRn (etPn k=1 ykDk(X))−e−tPn k=1 ykDk(X))yjγn(y)dy. Riesz Transforms 329 Hence, by α), for X∈ S2(H), √2πRj(X) = Z∞ 0ZRn (etPn k=1 ykDk(X) −e−tPn k=1 ykDk(X))yjγn(y)dydt t =Z∞ −∞ ZRn etPn k=1 ykDk(X)yjγn(y)dydt t. (11) γ) We claim that, for every X∈ Sq(H), y∈Rn, 1 <q<∞, F(y, X) = pv Z∞ −∞ etPn k=1 ykDk(X)dt t is well defined as the norm limit in Sq(H) of F(y, X, ε) = Zε<|t|<ε−1 etPn k=1 ykDk(X)dt t and satisfies (12) kF(y, X)kSq≤hqkXkSq. Indeed, Sq(H) is UMD for 1 <q<∞[BGM, Theorem 6.1] and, for fixed y, (etPn k=1 ykDk)t∈Ris a strongly continuous one parameter group of isometries of Sq(H). Hence [BGM, Theorems 5.12 and 5.16] prove the claim. δ) It follows that, for the S2(H) norm, by Fubini theorem and (11), ZRn F(y, X)yjγn(y)dy = lim εZRn F(y, X, ε)yjγn(y)dy = lim εZε<|t|<ε−1ZRn etPn k=1ykDk(X)yjγn(y)dydt t =√2πRj(X). Hence, for X∈ Sq(H)∩S2(H), √2πR(X) = n X j=1 yjZRn F(y, X)yjγn(y)dy = (Π ⊗ISq)(F(y, X)), and, by (12), kF(y, X)kLq(γndy,Sq)≤hqkXkSq. 330 F. Lust-Piquard ε) By [P2, Remark 8.4.6], Π ⊗ISqis bounded on Lq(γndy, Sq), 1<q<∞, and its norm depends only on q. By δ) this proves the boundedness of Ron Sq(H) and ends the proof of b). Actually, Π is completely bounded, hence so is Π ⊗ISq, and the same argument as above, applied to (etPn j=1 yjDAj⊗ISq)t∈Rshows that, for fixed y, pv R∞ −∞ etPn j=1 yjDAjdt tis completely bounded: Sq(H)→ Sq(H). c) By [P2, Theorem 8.4.1], the norms in the statement are equivalent, with constants which depend only on q, to  n X j=1 yjRj(X)Lq(dγn,Sq) =kR(X)kLq(dγn,Sq). Hence, we have to show that kXkSqis equivalent to kR(X)kLq(dγn,Sq). One inequality has been proved in b). For the other one, we notice that R∗R= Id on S2(H) because hR(X),R(Y)i=ZRn*n X j=1 yjRj(X), n X k=1 ykRk(Y)+γn(y)dy = n X j=1 hRj(X), Rj(Y)i= n X j=1 R∗ jRj(X), Y =hX, Y i. Hence, for 1 q+1 q0= 1, kXkSq≤kR∗kq→qkR(X)kLq(γndy,Sq)=kRkq0→q0 kR(X)kLq(γndy,Sq). Example 6. The CCR heat flow on K(L2(Rn)). The assumptions of Theorem 5 are satisfied by the following example, taken from [A], where assertion a) is proved in this special case. The operator P=−id dx and the operator Qof multiplication by x are well defined: S(R)→ S(R) and satisfy the Canonical Commutation Relation [Q, P ] = iI. P,Qare formally selfadjoint unbounded operators on L2(R). They generate two one parameter unitary groups of operators on L2(R), respectively eisP (translation by −s) and eitQ (multiplication by eitx), satisfying (13) eisP eitQ =eisteitQeisP , s, t ∈R. Riesz Transforms 331 We may define the inner ∗-derivations DP(X) = i[P, X], DQ(X) = i[Q, X] for operators Xon L2(R) which are defined by kernels k(x, y)∈ S(R2). Note that the kernels of DP(X), DQ(X) are respectively ∂ ∂x +∂ ∂y k and (x−y)k.DP,DQare generators of groups of ∗-automorphisms of K(L2(R)), namely esDP(X) = eisP Xe−isP , etDQ(X) = eitQXe−itQ and, owing to (13), esDPand etDQcommute on K(L2(R)). Obviously, only X= 0 is stable under these groups. On S(Rn) we consider in the same way Pj=−i∂ ∂xj,Qjthe operator of multiplication by xj. 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