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A Schatten -von Neumann class criterion for the magnetic Weyl calculus Nassim Athmouni∗, Radu Purice† January 19, 2016 Abstract We prove criteria for a ’magnetic’ Weyl operator (see [15, 11]) to be in a Schatten-von Neuman class by extendinga method developped by H.Cordes [5], T. Kato [9] and G. Arsu [1]. 6 MSC: 35S05, 47B10, 81Q10,81S10, 81S30, 81V99 1 Keywords: quantization, magnetic fields, quantum Hamiltonians, pseudodifferential operators, Schatten-von 0 Neuman classes. 2 n a 1 Introduction J 8 The’magnetic’ Weyl quantization[15]isprovenin[16]tobe astrict deformation quantizationinthe senseofRiffel 1 [19, 20, 14] and its associated ’magnetic’ Weyl calculus is developped in [17, 11, 13] where a magnetic version of the Calderon-Vaillancourt Theorem is proven. In this paper we prove criteria for a ’magnetic’ Weyl operator to ] h be in a Schatten-von Neuman class by extending a method developped by H. Cordes [5], T. Kato [9] and G. Arsu p [1]. Suchcriteriamay then be usedin connectionwith the analysisof quantumHamiltonians with magnetic fields. - h Our main result is formulated in Theorem 1.2. t Let us fix some general notations. Recall that for any a > 0 we denote by [a] N its integer part (i.e. the a ∈ m largestnatural number less then or equal to a). For any finite dimensional realvector space , we shall denote by V BC( ) (resp. BC ( )) the space of bounded continuous (resp. of bounded uniformly continuous) functions with [ V u V thek·k∞ norm,byC∞(V)thespaceofsmoothfunctionsonV,byCp∞ol(V)itssubspaceofsmoothfunctionsthatare 1 polynomially bounded together with all their derivatives and by BC∞( ) the subspace of smooth functions that v areboundedtogetherwithalltheirderivatives;weconsiderallthesespacVesendowedwiththeirusuallocalyconvex 3 topologies (see [21]). For any m R and any Banach space we shall consider the function spaces Sm( ; ) of 1 ∈ B V B -valued smooth functions on such that 6 B V 4 0 sup<v >−m+M ∂vβF (v) B < ∞, ∀M ∈N. (1.1) 1. v∈V |βX|=M(cid:13)(cid:0) (cid:1) (cid:13) 0 (cid:13) (cid:13) We shall denote by Sm( ) := Sm( ;C). We shall consider the space of Schwartz test functions S( ) endowed 16 withitsFréchettopologyVanditsduValS′( )anddenoteby , V theassociateddualitymap. WedenoVtebyτv the V h· ·i : translation with v (acting on the space of tempered distributions). We shall also consider the usual Sobolev v spaces m( ) of an∈yVorder m R, on . For any vector v we denote by < v >:= 1+ v 2. We denote the Xi convoluHtionVoperation by ∈ V ∈ V | | p r a (f g)(v) := f(v u)g(u)du, (f,g) S( ) S( ) (1.2) ∗ − ∀ ∈ V × V ZV and also its possible extensions to larger spaces of distributions on . For two linear topological spaces and 1 we shall denote by B( , ) the linear space of continuous linearVoperators from to , endowed wLith the 2 1 2 1 2 L L L L L bounded convergence topology [4]. We shall work on the configuration space := Rd and consider its dual ∗ with the duality map denoted by < , >: ∗ R. Let us also considerXthe phase space Ξ := ∗ wXith the canonical symplectic map · · X ×X → X ×X σ(X,Y):=<ξ,y > <η,x> for X :=(x,ξ) and Y :=(y,η) two arbitrary points of Ξ. − TheWeyl quantization (see [6, 7, 8]) defines a linear topological isomorphism Op:S′(Ξ) B S( );S′( ) (1.3) → X X ∗FacultédesSciences deGafsa,Gafsa,Tunisie (cid:0) (cid:1) †Institute of Mathematics SimionStoilow of the RomanianAcademy,Research unit nr. 1; P.O.Box 1-764, Bucharest, RO-014700, RomaniaandLaboratoireEuropéenAssociéCNRSMathématique et Modélisation. 1 for the strong topologies. Explicitely, for F S( ) we have the formula ∈ X Op(F)=(2π)−d/2 (2π)−d/2 eiσ(X,Y)F(Y)dY W(X)dX (2π)−d/2 −[F](X)W(X)dX (1.4) ≡ FΞ ZΞ(cid:18) ZΞ (cid:19) ZΞ W((x,ξ))φ (z):=e(i/2)<ξ,x>e−i<ξ,z>φ(z+x), φ S( ). (1.5) ∀ ∈ X We shall use some clas(cid:0)ses of Ho¨rm(cid:1)ander type symbols. For m R and ρ [0,1] let us define: ∈ ∈ νm (F):= sup <ξ >−m ∂α∂βF (x,ξ) , (N,M) N N, F C∞(Ξ), (1.6) N,M x ξ ∀ ∈ × ∀ ∈ (x,ξ)∈Ξ |αX|=N|βX|=M(cid:12)(cid:12)(cid:0) (cid:1) (cid:12)(cid:12) (cid:12) (cid:12) Sm(Ξ):= F C∞(Ξ) νm−Mρ(F)< , (N,M) N N . (1.7) ρ ∈ N,M ∞ ∀ ∈ × n (cid:12) o Evidently Sm( ∗) may be considered as the subs(cid:12)pace of Sm(Ξ) of functions constant in the directions in . X (cid:12) 1 X We shall usualy work in the Hilbert space L2( ) (defined with respect to the Lebesgue measure). In general X for a complex Hilbert space we shall denote by (, ) its scalar product (supposed to be anti-linear in the first K variable). For any Hilbert spKace we denote by B(· ·) the C∗-algebra of bounded operators on and by B ( ) ∞ K K K K its ideal of compact operators. Definition 1.1. Givena Hilbert space , forany p [1, ) we considerthe linear subspaceof compactoperators A B ( ) with the property that K ∈ ∞ ∞ ∈ K lim µ (A) < , (1.8) n ∃Nր∞ ∞ n≤N X where µn(A) n∈N are the singular values of the operator A B∞( ), [3]. This subspace, denoted by Bp( ) and { } ∈ K K called the Schatten-von Neumann class of order p, is a Banach space for the norm 1/p kAkBp(K) := Nlրim∞  µn(A)p . (1.9) n≤N X   We recall that B ( ) is the space of trace-class operators and B ( ) the space of Hilbert-Scmidt operators that is 1 2 a Hilbert space forKthe scalar product (A,B)B2(K) :=Tr(A∗B). K 1.1 The magnetic Weyl calculus. The magnetic fields are closed 2-forms on that we shall suppose to have components of class BC∞( ). To any X X such magnetic field B one can associate in a highly non-unique way a vector potential A, i.e. a 1-form such that B =dA; different choices for the vector potential are related by a change of gauge (i.e. dA=B =dA′ if and only if ϕ, A′ = A+dϕ). We shall always suppose the vector potential to have components of class C∞( ) because ∃ pol X such a choice always exists for magnetic fields of class BC∞( ). We use two important ’phase factors’ defined in X terms of these exterior forms: ΛA(x,z) := exp i A (1.10) (− Z[x,z] ) ΩB(x,y,z) := exp i B (1.11) − (cid:26) Z<x,y,z> (cid:27) where [x,z] is the oriented line segment from x to z and < x,y,z > is the oriented triangle of vertices ∈ X ∈ X x,y,z . From Stoke’s Theorem we deduce that ΩB(x,y,z) = ΛA(x,y)ΛA(y,z)ΛA(z,x). { }⊂X Let us recall from [15] the magnetic Weyl system defined as the family of unitary operators in L2( ): X WA(X) , WA((x,ξ))u (z):=ΛA(z,z+x) W((x,ξ))u (z), u . (1.12) X∈Ξ ∀ ∈H (cid:8) (cid:9) (cid:0) (cid:1) (cid:0) (cid:1) As explained in [15] they are defined as unitary groups associated to the canonical observables in the minimal coupling formalism for the vector potential A. With the help of this magnetic Weyl system one can define a magnetic Weyl calculus (i.e. a magnetic quantization) as in [15, 11] OpA(F)=(2π)−d/2 −[F](X)WA(X)dX. (1.13) FΞ ZΞ 2 Let us make the connection with the ‘twisted integral kernels’ formalism in [18]. For any integral kernel K S′( ) one can associate its ’magnetic’ twisted integral kernel ∈ X ×X KA(x,y) := ΛA(x,y)K(x,y). (1.14) Let us denote by ntK the corresponding linear operator on S( ); i.e. v,( ntK)u = K,v u for any I X I L2(X) h ⊗ iX (u,v) S( ) 2. Letusrecallthe linearbijectionW:S′(Ξ) S′( (cid:0) )associat(cid:1)edtotheusualWeylcalculus (1.3) b∈y theXequality Op(F)= nt(WF): → X×X (cid:2) (cid:3) I x+y WF (x,y) := (2π)−d ei<ξ,x−y>F ,ξ dξ. (1.15) ZX∗ 2 (cid:0) (cid:1) (cid:0) (cid:1) Then we have the equality OpA(F) = nt(ΛAWF). (1.16) I This functionalcalculusinduces amagnetic Moyal product♯B :S(Ξ) S(Ξ) S(Ξ)suchthatOpA(f♯Bg)= OpA(f)OpA(g). Explicitely we have × → f♯Bg = π−2d e−2iσ(Y,Z)ΩB(x y z,x+y z,x y+z)f(X Y)g(X Z)dY dZ (1.17) − − − − − − ZΞZΞ (cid:0) (cid:1) as oscillating integrals (see [8]). We shall use the notation ωB(x,y,z):=ΩB(x y z,x+y z,x y+z). (1.18) − − − − In[11]onegivestheextensionofthismagneticMoyalproducttotheusualHo¨rmandertypesymbolsandin[11,13] it is proven that this calculus has similar properties with the usual Moyal product. If a symbol F S′(Ξ) is invertible for this magnetic Moyal product we shall denote by F− its inverse. ∈ B In[11]itisproventhatforanysymbolF S0( )the operatornormofOpA(F)isboundedbysomeseminorm ∈ 0 X definingtheFréchettopologyonS0( )andthisseminormonlydependsonthedimensiondof andsomeFréchet 0 X X seminorm of the components of the magnetic field in BC∞( ) (this second fact, although not explicitely stated X there, easily follows when looking at the detailed proof of Theorem 3.1 in [12]). We shall define the following associated norm on the S0( ) symbols: 0 X F B := OpA(F) B(H). (1.19) k k k k In [15] it is proven that OpA(F) is Hilbert-Scmidt if and only if F ∈L2(Ξ) and kFkL2(Ξ) =kOpA(F)kB2(H). 1.2 The main result. Inthepapers[1,2],G.ArsuusessomeideasandresultsofH.O.Cordes[5]andT.Kato[9]andthecharacterization of Schatten-von Neumann classes of operators coming from J.W. Calkin and R. Schatten (see [3, 23]) in order to obtain an interesting criterion for a Weyl operator to be in a given Schatten-von Neumann class. Our aim in this paper is to replace the usual Weyl system with the magnetic Weyl system (1.12) and prove a criterion for a magnetic Weyl operator (1.13) to be in a given Schatten-von Neumann class. We prove the following Theorem. Theorem 1.2. Suppose that B is a magnetic field with components of class BC∞( ) and we choose some vector potentialAfor B withcomponents ofclass C∞( ). Supposethat F S′(Ξ) andletXusdenotebys(d):=2[d/2]+2 pol X ∈ and t(d):=d+[d/2]+1. 1. If ∂α∂βF L∞(Ξ) for α s(d) and β t(d), then OpA(F) B L2( ) and there exists some finite x ξ ∈ | | ≤ | | ≤ ∈ X constant C >0 such that (cid:0) (cid:1) kOpA(F)kB(H) ≤C ∂xα∂ξβF L∞(Ξ). |α|X≤s(d) |β|X≤t(d)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 2. For p [1, ) if ∂α∂βF Lp(Ξ) for α s(d) and β t(d), then OpA(F) B L2( ) and there exists ∈ ∞ x ξ ∈ | | ≤ | | ≤ ∈ p X some finite constant C >0 such that p (cid:0) (cid:1) kOpA(F)kBp(H) ≤Cp ∂xα∂ξβF Lp(Ξ). |α|X≤s(d) |β|X≤t(d)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 3 3. If∂α∂βF L∞(Ξ)for α s(d)and β t(d), and lim ∂α∂βF (x,ξ)=0for α s(d)and β t(d), x ξ ∈ | |≤ | |≤ (x,ξ)→∞ x ξ | |≤ | |≤ then OpA(F) B L2( ) . (cid:0) (cid:1) ∞ ∈ X Remark 1.3. We note(cid:0)that po(cid:1)ints (1) and (2) of the Theorem are the ’magnetic’ version of Theorem 6.4 in [1]. First, let us consider the value of s(d) N (the number of derivatives with respect to the -variables) that we obtain. For d N odd, we have s(d) =∈d+1 exactly as in [1], while for d N even we hXave s(d) = d+2 ∈ ∈ that is larger by one unit with respect to the value in [1]; this is just the consequence of our choice to work withoutfractionaryderivatives,thatwouldlargelycomplicatethetechnicalargumentswithoutarealimprovement. Concerning t(d) N, it is interesting to note that it is larger then its value given in [1] for the zero magnetic field ∈ situation and that reflects the fact that the presence of a magnetic field that does not vanish at infinity obliges us to control severalderivatives of the symbol. Moreover,if we go into the details of our proof of Theorem 1.2 (more precisely the proof of Proposition 2.8) we easily see that in the absence of the magnetic field (i.e. of the factor ωτzB) we can take t(d) = d+1 as in [1]. Let us also note that points (1) and (3) in our Theorem are similar to Theorem 1 in [10] but with assumptions on fewer derivatives of the symbol. 2 Proof of Theorem 1.2. While the idea of the prooffollowsclosely the argumentsandsome results from[1, 5, 9], severalessentialtechnical steps have to be completely reconsidered in order to control the ’magnetic phase factors’ present in the magnetic Weyl calculus. Letusrecallthatin[1,9]onebeginsbynoticingthatthefundamentalsolutionsofsomesimpleellipticdifferential operatorsaresymbolsoftrace-classoperators(asimplied byCordesLemma [5,2])andstartingfromthe following formula, valid for two symbols f and g of class S(Ξ), Op(f g)= f(X)Op(τ g)dX = f(X) W( X)Op(g)W(X) dX, (2.20) −X ∗ − ZΞ ZΞ (cid:16) (cid:17) a procedure elaborated by G. Arsu [1] using the results of J.W. Calkin and R. Schatten (see [3, 22, 23]) and some ideasofT.Kato[9]allowstoobtainthedesiredresult. Letusdeveloptheseideasandadaptthemtooursituation. For (s,t) R R let us consider the following ΨDO on Ξ: + + ∈ × s/2 t/2 Ls,t := 1l ∆X 1l ∆X∗ (2.21) − − where (cid:0) (cid:1) (cid:0) (cid:1) ∆X := ∂x2j, ∆X∗ := ∂ξ2j. (2.22) 1≤j≤d 1≤j≤d X X Let us denote by ψ S′( ) the unique fundamental solution of 1l ∆ s/2 and by ψ˙ S′( ∗) the unique s X t ∈ X − ∈ X t/2 fundamental solution of 1l ∆X∗ . Let us recall the following w(cid:0)ell know(cid:1)n result (see for example section 5 in − [1] and Corollary 2.6 in [2] for the last statement). Proposition 2.4. For (cid:0)any s > 0(cid:1) the distribution ψ S′( ) is in fact a function of class L1( ) that is in s S( 0 ). For x 0 we have that ∈ X X X \{ } | |ց ∂αψ O 1+ xs−d−|α| , s d α =0, (2.23) x s ∼ | | − −| |6 ∂αψ O(cid:0) 1+lnx−1 ,(cid:1) s d α =0. (2.24) x s ∼ | | − −| | For s>d we have that ψ p( ) for any p<(cid:0) (s/2). We(cid:1)have evidently a similar behaviour for ψ˙ S′( ∗). s t This resultandthe Cor∈deHsLeXmma [5,2])allowto provethat ψ ψ˙ is anintegralkerneldefining∈a tracXe-class s t operator. Then, using 2.20 and the trivial fact that for any f S⊗′(Ξ), if we denote by δ the Dirac measure of 0 ∈ mass 1 at 0 Ξ we have that ∈ f = f δ = f L (ψ ψ˙ ) = L f (ψ ψ˙ ) 0 s,t s t s,t s t ∗ ∗ ⊗ ∗ ⊗ andKato’s operator calculusin[9]andLemma4(cid:0).3in[1]givet(cid:1)hede(cid:0)siredr(cid:1)esultintheabsenceofthemagneticfield. An importantdifficulty for the case of the ’magnetic’ Weyl calculus comes from the fact that equation (2.20) is no longer valid for the magnetic Weyl calculus; more precisely we have OpA(τ g) = WA(X)∗OpA(g)WA(X). (2.25) −X 6 The following subsection is devoted to the control of this difficulty. 4 2.1 Magnetic translations of symbols. In Proposition 3.4 in [13] one defines the action of Ξ on the symbols in S′(Ξ) by ’magnetic translations’: Ξ Z TB B S′(Ξ);S′(Ξ) (2.26) ∋ 7→ Z ∈ as the conjugate action associated to the magnetic Weyl s(cid:0)ystem: (cid:1) OpA TB g := WA(Z)∗OpA(g)WA(Z). (2.27) −Z Explicitely we have the formula: (cid:0) (cid:1) TBZg = (1l⊗FX−)e−iSzB ⋆ τZg , ∀g ∈S′(Ξ) (2.28) where we have used (as in [13]) the the in(cid:2)verse Fourier tra(cid:3)ns(cid:2)form(cid:3)on X −φ (ξ) := (2π)−d/2 ei<ξ,y>φ(y)dy, φ S( ), (2.29) FX ∀ ∈ X ZX (cid:0) (cid:1) the notation 1/2 1 SB(x,y) := y z ds dtB (x+sy+tz), (2.30) z − j k jk Xj6=k −1Z/2 Z0 and the following ’mixed’ product: f ⋆g (x,ξ) := f(x,η)g(x,ξ η)dη. (2.31) ZX∗ − (cid:0) (cid:1) We note that SB(x,y) as defined in (2.30) is in fact the flux of the magnetic field B through the orientedparallel- z ogram of vertices x+(y/2),x (y/2),x (y/2)+z,x+(y/2)+z . We shall also use the notation { − − } ΘB := (1l −)e−iSzB S′(Ξ) (2.32) z ⊗FX ∈ and remark that ΘB⋆Θ−B = 1 δ , the ident(cid:2)ity element for t(cid:3)he ’mixed’ product ⋆. We also remark that z z ⊗ 0 TBf := ΘB⋆ τ g = τ Θτ−zB ⋆f . (2.33) Z z Z Z z These arguments allow us to write (cid:2) (cid:3) (cid:2) (cid:3) OpA(f g) = f(Z)OpA(τ g)dZ = f(Z)OpA τ (ΘτzB ⋆Θ−τzB ⋆g) dZ = ∗ −Z −Z −z −z ZΞ ZΞ (cid:0) (cid:1) = f(Z)OpA TB (Θ−τzB ⋆g) dZ = f(Z)WA(Z)∗OpA Θ−τzB ⋆g WA(Z)dZ. (2.34) −Z −z −z ZΞ ZΞ (cid:0) (cid:1) (cid:0) (cid:1) This last formula replaces (2.20) in the case of the ’magnetic’ Weyl calculus. 2.2 Kato’s operator calculus. We recall here one of the main results in [1] using the operator calculus elaborated by T. Kato in [9]. Suppose given a measurable map W : Ξ B( ) for the weak operator topology on B( ). For any trace-class operator T B ( ) and any ϕ S(Ξ) we→canHdefine the following integral (with respectHto the weak operator topology): 1 ∈ H ∈ ϕ T := ϕ(X) W(X)∗TW(X) dX. (2.35) { } ZΞ (cid:0) (cid:1) Proposition 2.5. (Lemma 4.3 in [1]) 1. If there exists a finite C >0 such that u,W(X)v 2dX C u 2 v 2 , (u,v) , (2.36) H ≤ k kHk kH ∀ ∈H×H ZΞ (cid:12)(cid:0) (cid:1) (cid:12) then for any ϕ L∞(Ξ) t(cid:12)he integral (2.(cid:12)35) is well defined in the weak operator topology on B( ) and we ∈ H have the estimation kϕ{T}kB(H) ≤ CkϕkL∞(X)kTkB1(H). (2.37) 5 2. If there exists a finite C >0 such that W(X) B(H) √C almost everywhere on Ξ, then for any ϕ L1(Ξ) the integral (2.35) is well defined in thekweak okperato≤r topology on B( ), belongs to B ( ) and we h∈ave the 1 H H estimation kϕ{T}kB1(H) ≤ CkϕkL1(X)kTkB1(H). (2.38) 3. If the map W :Ξ B( ) satisfies both conditions above for some finite C >0, then for any ϕ Lp(Ξ), for some p (1, ), t→he inHtegral (2.35) is well defined in the weak operator topology on B( ), belon∈gs to B ( ) p ∈ ∞ H H and we have the estimation kϕ{T}kBp(H) ≤ CkϕkLp(X)kTkB1(H). (2.39) Remark 2.6. Wenotethatforanyvectorpotential,ourmagneticWeylsystemΞ X WA(X) B( )satisfies ∋ 7→ ∈ H both conditions in Theorem 2.5 with a constant C = 1, the first one as proven in Proposition 3.8 (a) in [15] and the second one due to their unitarity. Using (2.34)andtheaboveRemarkweobtainthefollowingCorollaryofProposition2.5(the’magnetic version’ of Theorem 4.5 in [1]): Corollary2.7. Supposegiven amagneticfieldB withcomponentsof class BC∞( ) andsupposefixedsomevector potential A for B with components of class C∞( ); if a symbol F S′(Ξ) hasXthe property OpA(F) B ( ), pol X ∈ ∈ 1 H then 1. For any f ∈L∞(Ξ) we have that OpA(f ∗F)∈B(H) and kOpA(f ∗F)kB(H) ≤kfkL∞(X)kOpA(F)kB1(H). 2. For any f Lp(Ξ), for some p [1, ), we have that OpA(f F) B ( ) and p ∈ ∈ ∞ ∗ ∈ H kOpA(f ∗F)kBp(H) ≤kfkLp(X)kOpA(F)kB1(H). 2.3 Estimations for OpA Θ−τzB ⋆Ψ . −z s,t Thus in order to finish the proof o(cid:0)f our Theorem(cid:1)1.2, we only have to prove that OpA Θ−τzB ⋆Ψ B ( ) for −z s,t ∈ 1 H Ψ :=ψ ψ˙ with s>0 and t>0 large enough (with the notations introduced at the begining of Section 2). s,t s⊗ t (cid:0) (cid:1) We use (1.16) in connection with (2.32) and (2.30), denote by Ψ :=WΨ S′( ) and write s,t s,t ∈ X ×X OpA Θ−−τzzB ⋆Ψs,t = Int ΛAW(eΘ−−τzzB ⋆Ψs,t) , (2.40) (cid:0) (cid:1) (cid:0) (cid:1) W(Θ−τzB⋆Ψ ) (x,y) = exp iS−τzB (x+y)/2,y x Ψ (x,y) (2.41) −z s,t − −z − s,t (cid:0) (cid:1) = expniS−τzzB (x(cid:0)+y)/2,y−x Ψ(cid:1)os,te(x,y), n (cid:0) (cid:1)o 1/2 1 e SτzB (x+y)/2,y x = (y x )z ds dtB (x+y)/2+s(y x)+tz , (2.42) −z − − j − j k jk − (cid:0) (cid:1) Xj6=k −1Z/2 Z0 (cid:0) (cid:1) andthis is the flux ofthe magneticfieldB throughthe orientedparallelogramofvertices y,x,x+z,y+z . Using { } the relation B =dA and Stoke’s Theorem we conclude that ΛA(x,y)exp iSτzB (x+y)/2,y x = ΛA(x,y)ΛA(y,x)ΛA(x,x+z)ΛA(x+z,y+z)ΛA(y+z,y) (2.43) −z − n (cid:0) (cid:1)o = ΛA(x,x+z)ΛA(x+z,y+z)ΛA(y+z,y) = ΛA(x,x+z)ΛτzA(x,y)ΛA(y+z,y). Thus,ifwedenotebyUA theunitaryoperatorinL2( )definedbymultiplicationwiththefunctionx ΛA(, +z), z X 7→ · · we can write that OpA Θ−τzB ⋆Ψ = UAOpτzA(Ψ ) UA −1and it is enough to prove the following ’magnetic −z s,t z s,t z version’ of Lemma 1 in [5]. (cid:0) (cid:1) (cid:2) (cid:3) Proposition 2.8. Suppose given a magnetic field B = dA with components of class BC∞( ); for t > 3d/2 and s>2[d/2]+2 we have that OpτzA(Ψ ) B ( ) uniformly for z . X s,t 1 ∈ H ∈X Proof. We shall proceed as in [5, 2] but we shall work with the magnetic Moyal product (1.17). The idea is to write Ψ as a magnetic Moyal product of two symbols of class L2(Ξ): s,t Ψ = Φ(1)♯τzBΦ(2), Φ(j) L2(Ξ), j =1,2. (2.44) s,t ∈ 6 Let us consider the symbols p (X) :=< ξ >m +λ for any m > 0 and some λ > 0 large enough; they are m,λ evidently elliptic symbols of class Sm( ∗) that, for λ > 0 large enough, are invertible for the magnetic Moyal X product due to Theorem 1.8 in [17]. More precisely, looking at the proof of this cited Theorem we see that r := p − = <ξ >m +λ −1♯τzB s (λ)♯τzB...♯τzBs (λ) (2.45) m,λ m,λ τzB  z,m z,m  k∈N (cid:0) (cid:1) (cid:0) (cid:1) X k    with s (λ) S−κ( ) for some κ (0,1), having the operator nor|m strictly le{szs then 1 for}λ > 0 large enough z,m ∈ X ∈ and the defining Fréchet seminorms bounded by some seminorm of the components of τ B in BC∞( ); as these z X seminorms are translation invariant, we have uniform bounds for z . Thus, using Proposition 3.10 in the ∈ X − AppendixandProposition6.2in[13]weconcludethatforλ>0largeenough,thesymbolseminormsof p m,λ τzB ∈ S−m( ) are bounded by some constants that do not depend on z . 1 LeXt us also consider the function q (X) :=< x >r with r R, ∈deXfining a symbol of class Sr( ) fo(cid:0)r any(cid:1)r R r ∈ X ∈ and also of class S0(Ξ) for any r 0. Formally we can write 1 ≤ Ψ = q ♯τzBr ♯τzB p ♯τzBq ♯τzBΨ (2.46) s,t −r m,λ m,λ r s,t (cid:0) (cid:1) (cid:0) (cid:1) Using once again Proposition 3.10 in the Appendix and the fact that the seminorms of the components of the magnetic field that control the magnetic Moyal products are translation invariant, we easily conclude that for r >0, m>0 and λ>0 large enough q ♯τzBr S−m( ), (2.47) −r m,λ ∈ 1 X uniformly for z . Moreover,for any a (cid:0)0 and b 0(cid:1)we can write: ∈X ≥ ≥ <x>a<ξ >b q ♯τzBr (x,ξ) = (2.48) −r m,λ (cid:0) (cid:1) = π−2d <x>a<ξ >b e−2iσ(Y,Y′)ωτzB(x,y,y′)<x y >−r r (x y′,ξ η′)dY dY′ = m,λ − − − Z Ξ×Ξ <x>a = π−dC <x y >−(r−a) a − <x y >a<y >a × ZX (cid:18) − (cid:19) <y >a<η′ >b e2i<η′,y> <ξ >b <ξ η′ >b r (x,ξ η′) dη′ dy. ×(cid:20)ZX∗ (cid:18)<ξ−η′ >b<η′ >b(cid:19) − m,λ − (cid:21) (cid:0) (cid:1) We use the identities: <y >2N1 e2i<η′,y> =(1 4−1∆η′)N1e2i<η′,y>, <η′ >2N2 e2i<η′,y> =(1 4−1∆y)N2e2i<η′,y> (2.49) − − and after some integrations by parts as in the proof of Proposition 3.10 in the Appendix, taking 0 a r, ≤ ≤ 0 b m and 2N [a]+d+1, 2N [b]+d+1 we get that 1 2 ≤ ≤ ≥ ≥ <x>a<ξ >b q ♯τzBr (x,ξ) C sup <ξ >b ∂αr (x,ξ) C(a,b)νm (r ). (2.50) −r m,λ ≤ a,d ξ m,λ ≤ 0,2N2 m,λ (x,ξ)∈Ξ (cid:12)(cid:0) (cid:1) (cid:12) |α|X≤2N2(cid:12)(cid:0) (cid:1) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) A similar computation can be made for any derivative ∂xα∂ξβ q−r♯τzBrm,λ so that we conclude that (cid:0) (cid:1) q p q ♯τzBr S0( ), (a,b) [0,r] [0,m] (2.51) a b,0 −r m,λ ∈ 1 X ∀ ∈ × and taking r >d/2 and m>d/2 w(cid:0)e note that Φ(cid:1)(1) :=q−r♯τzBrm,λ L2(Ξ) so that OpA(Φ(1)) B2 L2( ) . ∈ ∈ X Now let us study the second factor in (2.46). We note that for m > 0 and r > 0 the first two functions of (cid:0) (cid:1) this second magnetic Moyal product, namely p and q , are in fact C∞(Ξ) functions with polynomial growth m,λ r at infinity uniformly for all their derivatives, and thus Proposition 4.23 in [15] shows that their magnetic Moyal product may be well defined in the sense of tempered distributions and moreover this product (as a tempered distribution) may be further composed by magnetic Moyal product with any tempered distribution on Ξ. Thus Φ(2) is well defined as a tempered distribution on Ξ and we can also use the associativity of the magnetic Moyal product. Let us note that this tempered distribution depends in fact on z due to the translated magnetic ∈ X field appearing in the two ’magnetic’ Moyal products in the definition of Φ(2) as the second paranthesis in (2.46); thus we shall use the notation Φ(2) and notice that this dependence is uniformly smooth with respect to the weak z distribution topology. 7 We begin by computing qr♯τzBΨs,t =qr♯τzB ψs ψ˙t for r >d/2>0: ⊗ q ♯(cid:0)τzB ψ (cid:1)ψ˙ (x,ξ) = r s t ⊗ (cid:2) (cid:0) (cid:1)(cid:3) = π−2d e−2iσ(Y,Y′)ωτzB(x,y,y′)<x y >r ψ (x y′)ψ˙ (ξ η′)dY dY′ = s t − − − Z Ξ×Ξ = π−dψ (x) e2i<η′,y> <x y >r ψ˙ (ξ η′)dydη′ = (2.52) s t − − X×ZX∗ = π−dψ (x) e2i<ξ,y> <x y >r e−2i<ξ−η′,y>ψ˙ (ξ η′)dη′ dy = (2.53) s t Z − (cid:18)ZX∗ − (cid:19) X <x y >r <x y >r = 2dψ (x) e2i<ξ,y> − dy = 2d q ψ (x) e2i<ξ,y> − dy = (2.54) s <2y >t r s <x>r<2y >t Z Z X (cid:0) (cid:1) X = (2π)d/2 q ψ (x) (1l −)f (x,ξ) (2.55) r s ⊗FX where: (cid:0) (cid:1) (cid:0) (cid:1) <x (y/2)>r f(x,y) := − . (2.56) <x>r<y >t It is easy to chek that f C∞( ) and satisfies the estimations: ∈ pol X ×X ∂α∂βf (x,y) C <x>−|α|<y >r−t−|β| . (2.57) x y ≤ αβ Now let us consider some m(cid:12)(cid:12)(cid:0)> d/2,(cid:1)and u(cid:12)(cid:12)se the notations: f := (2π)d/2(1l −)f and for any r 0 the ⊗FX ≥ function ψ (x) :=< x >r ψ (x). We notice that for any r 0 the function ψ has exactly the same properties s,r s s,r ≥ as those of ψ given in Proposition 2.4. e s We waent to show that: e Φ(2) := p ♯τzBq ♯τzBΨ = (2π)d/2 p ♯τzB q ψ 1 (1l −)f (2.58) z m,λ r s,t m,λ r s⊗ ⊗FX (cid:16) (cid:2)(cid:0) (cid:1)(cid:0) (cid:1)(cid:3)(cid:17) asatempereddistributiononΞisinfactanL2(Ξ)functionuniformlyforz . Inordertodealwiththepossible ∈X singularities of this distribution we shall regularize it by introducing 4 cut-off functions in the oscillatory integrals appearingin the definition (1.17), more precisely we shallapproachΦ(2), in the weak distributiontopology,by the z following continuous functions on Ξ depending also on 4 positive parameters R : j j=1,2,3,4 { } Φ(2) (x,ξ) := e−2i<η,y′>e2i<η′,y>χ (y)χ (y′)χ (η)χ (η′) (2.59) (Rj,z) R1 R2 R3 R4 × Z Z Ξ Ξ g p (ξ η)ψ (x y′)f(x y′,ξ η′)ωτzB(x,y,y′)dydy′dηdη′, m,λ s,r × − − − − where for any R>0 we define χ (v) :=χ(R−1 v ) with χ : R R a smooth decreasing function that satisfies R + + e | e| → χ(t)=1 for 0 t 1 and χ(t)=0 for t 2. ≤ ≤ ≥ We shall first consider the term < ξ η >m in the function p (ξ η) =< ξ η >m +λ and the associated m,λ − − − integral Φ(3) (x,ξ) := e−2i<η,y′>e2i<η′,y>χ (y)χ (y′)χ (η)χ (η′) (2.60) (Rj,z) R1 R2 R3 R4 × Z Z Ξ Ξ g <ξ η >m ψ (x y′)f(x y′,ξ η′)ωτzB(x,y,y′)dydy′dηdη′. s,r × − − − − We make the measure preserving change of variables: e e u:=x y′ − (y,y′,η,η′) (y,u,ζ,ζ′); ζ :=ξ η (2.61) 7→  − ζ′ :=ξ η′,  − so that (2.60) may be written as  Φ(3) (x,ξ) := e−2i<ξ,x−u−y>e2i<ζ,x−u>e−2i<ζ′,y>χ (y)χ (x u)χ (ξ ζ)χ (ξ ζ′) (2.62) (Rj,z) R1 R2 − R3 − R4 − × Z Z Ξ Ξ g 8 <ζ >m ψ (u)f(u,ζ′)ωτzB(x,y,x u)dydudζdζ′ = s,r × − = e2i<ζ,x> <ζe>m χe (ξ ζ) e−2i<ζ,u>ψ (u)χ (x u) (2.63) ZX∗ R3 − (cid:26)ZX s,r R2 − × e−2i<ξ,x−u−y> e−2i<ζ′,y>χ (ξ ζ′)f(u,ζ′)dζ′ ωτzeB(x,y,x u)χ (y)dy du dζ × (cid:20)ZX (cid:18)ZX∗ R4 − (cid:19) − R1 (cid:21) (cid:27) ≡ e e2i<ζ,x> <ζ >m χ (ξ ζ) e−2i<ζ,u>Θ (x,ξ,u)du dζ. (2.64) ≡ ZX∗ R3 − (cid:18)ZX (Rj,z) (cid:19) Let us study closer the continuous function introduced in (2.64): Θ (x,ξ,u) := ψ (u)χ (x u) (2.65) (Rj,z) s,r R2 − × e−2i<ξ,x−u−y> e−2i<ζ′,y>χ (ξ eζ′)f(u,ζ′)dζ′ ωτzB(x,y,x u)χ (y)dy ; × (cid:20)ZX (cid:18)ZX∗ R4 − (cid:19) − R1 (cid:21) we make the change of variable y v :=x u y tehat allow us to write it as: X ∋ 7→ − − ∈X Θ (x,ξ,u) := ψ (u)χ (x u)T (x,ξ,u), (2.66) (Rj,z) s,r R2 − R1,R4,z TRe1,R4,z(x,ξ,u) := (2.67) = e−2i<ξ,v> e−2i<ζ′,x−u−v>χ (ξ ζ′)f(u,ζ′)dζ′ ωτzB(x,x u v,x u)χ (x u v)dv . (cid:20)ZX (cid:18)ZX∗ R4 − (cid:19) − − − R1 − − (cid:21) We recall that e f := (2π)d/2(1l −)f (2.68) ⊗FX and the fact that the distribution f S′( ) defined in (2.57) is in fact a smooth function of class S0( ) in ∈ X ×eX X the first variable and of class Sr−t( ) in the second variable uniformly with respect to the first variable and thus, X for t > r+(d/2) > d, it belongs to S0 ;L2( ) . Thus, using the Fourier inversion Theorem and noticing that X X for any g ∈ S0 X;L2(X) we have tha(cid:0)t k(1l⊗τ−(cid:1)u)g(u,·)kL2(X) = kg(u,·)kL2(X), we conclude that the tempered distributions (cid:0) (cid:1) T (x,ξ,u,v) := e−2i<ζ′,x−u−v>χ (ξ ζ′)f(u,ζ′)dζ′, R [1, ) (2.69) R4 ZX∗ R4 − 4 ∈ ∞ areafamilyoffunctionsofclassS0 ;L2( ) withrespecttothevaeriables(u,v) andbytheDominatewd X X ∈X×X Convergence Theorem (cid:0) (cid:1) (x,ξ) Ξ, lim T (x,ξ,u,v) = (2π)df(u,2(x u v)), in S0 ;L2( ) (2.70) ∀ ∈ ∃R4ր∞ R4 − − X X (cid:0) (cid:1) uniformly with respect to (x,ξ) Ξ. Due to the fact that by definition we have that ωτzB BCu( 3) uniformly ∈ ∈ X and smoothly for z we conclude that ∈X (z,x,ξ) Ξ, lim T (x,ξ,u,v)ωτzB(x,x u v,x u)χ (x u v) = (2.71) ∀ ∈X × ∃R4ր∞ R4 − − − R1 − − = (2π)df(u,2(x u v))ωτzB(x,x u v,x u)χ (x u v)=:θτzB(x,u,v), − − − − − R1 − − in S0 ;L2( ) uniformly with respect to (z,x,ξ) Ξ. Moreover,for any magnetic field B with components X X ∈X × of class BC∞( ) we have that for any (x,u) 2 (cid:0) X(cid:1) ∈X θB(x,u,v) C(B)f(u,2(x u v)=C(B)<2(x u v)>r−t (2.72) ≤ − − − − (cid:12) (cid:12) and thus (cid:12) (cid:12) (x,suu)∈pX2 θB(x,u,·) L2(X) ≤C(B)kqr−tkL2(X), (2.73) (cid:13) (cid:13) with C(B) depending only on some semi(cid:13)norm of BC(cid:13)∞( ) of the components of the magnetic field. We easily X conclude that θB BCu( 3) BCu x u;L2( v) and the map z θτzB BCu x u;L2( v) is ∈ X ∩ X ×X X X ∋ 7→ ∈ X ×X X smooth and bounded. Using Plancherel Theorem and the Dominatewd Convergence Theorem we conclude that (cid:0) (cid:1) (cid:0) (cid:1) z , lim lim T =: F , in BC ;L2( ∗) , (2.74) ∀ ∈X ∃R1→∞R4→∞ R1,R4,z z u Xx×Xu Xξ (cid:0) (cid:1) 9 uniformly with respect to z . Moreover,taking into account the properties of the function f defined in (2.57) (see also Proposition 2.4), w∈eXmay also conclude that F (x, ,u) S( ∗ 0 ) uniformly with respect to the z · ∈ X \{ } variable z . Finaly,∈noXticing that by Proposition 2.4, ψ L2( ) for any (s,r) R R we conclude that s,r + + ∈ X ∈ × ∀z ∈X, ∃Rl1i→m∞Rl2i→m∞Rl4i→m∞Θ(Rj,z)e= 1⊗1⊗ψs,r Fz, in L2 Xu;BCu Xx;L2(Xξ∗) (2.75) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) uniformly for z . e ∈X In order to control the factor <ζ >m in the first integral in (2.63), that we consider as a Fourier transform of a tempered distribution, let us study now the derivatives of Θ with respect to the variable u : (Rj,z) ∈X ∂αΘ ((x,ξ,u), α =p N∗. (2.76) u (Rj,z) | | ∈ (cid:0) (cid:1) By Proposition 2.4 we know that for s > d we have that ψ p( ) for p < (s/2) and thus all the derivatives s,r ∈ H X ∂αψ are of class L2( ) for s>2α. Let us study the behaviour of the distributions s,r X | | e e ∂uαFz(x,ξ,u), |α|=p∈N∗. (2.77) Whencomputing∂αT ,usingLeibnitzrulewehavetocontrolthederivativesoforderuptop Nwithrespect u R1,R4,z ∈ to u of f(u,2(x u v)), of ωτzB(x,x u v,x u) and of the cut-off functions. Now, ∂αf(u,2(x u v)) ∈X − − − − − u − − is easy to compute and it is clearly a function of class S−|α|( ) with respect to the first variable and of class Sr−t−|α|( )withrespecttothesecondvariableuniformlywithrXespecttothe firstvariable,andthusforanyp N X ∈ these functions have the same properties as the function f in (2.57). Using then Lemma 1.1 in [11] we know that we have the estimations ∂α∂βωτzB (x,y,y′) = θτzB(x,y,y′) <x>+<y >+<y′ > |α|+|β| (2.78) y y′ α,β (cid:0) (cid:1) (cid:0) (cid:1) where θατz,βB ∈BCu(X3) uniformly in z ∈X. In conclusionwe can write ∂uαωτzB(x,x−u−v,x−u) as a finite sum of terms of the form θτzB(x,u,v) < x >p< u >p< x u v >p with θτzB BCu( 3) uniformly in z . We − − ∈ X ∈ X get rid of the growing factor <u>p by replacing ψ by ψ that has the same properties as ψ . The factor s,r s,r+p s,r < x u v >p may be absorbed in the factor f without changing its properties that we used above, as long as − − t > p+r+(d/2). We remain with the factor < xe>p; ineorder to control its growth at infinity wee turn back at formula (2.63) and notice that Φ(3) (x,ξ) = (2.79) Rj,z (1 ∆ )p/2 = − ζ e2i<ζ,x> <ζ >m χ g(ξ ζ) e−2i<ζ,u>ψ (u)χ (x u)Θ (x,ξ,u) . ZX∗(cid:18) <2x>p (cid:19) R3 − (cid:26)ZX s,r R2 − (Rj,z) (cid:27) Consideringthe ζ integralinthe senseofdistributionswe cantransferthee differentialoperator(1 ∆ )p/2 onthe ζ S( ∗) function − − X ∗ ζ <ζ >m χ (ξ ζ) e−2i<ζ,u>ψ (u)χ (x u)Θ (x,ξ,u) C. (2.80) X ∋ 7→ R3 − s,r R2 − (Rj,z) ∈ (cid:26)ZX (cid:27) e Using the well known facts that (1 ∆)−1/2 and (1 ∆)−1/2∂ are bounded operators in L2( ∗) we notice that j for p N we can write: − − X ∈ (1 ∆ )p/2 = X ∂α (2.81) − ζ α ζ |αX|≤p with X B L2( ∗) for any α Nd. Then we only have to notice that ∂α <ζ >m is a symbol of type Sm( ∗) α ∈ X ∈ ζ X for any α Nd and ∈ (cid:0) (cid:1) ∂αe−2i<ζ,u> =( 2i)|α|uαe−2i<ζ,u> ζ − and we can control the factor uα by < u >|α| that can be absorbed in ψ for any α N without changing its s,r | | ∈ properties needed for the arguments above to hold. Finaly we notice that all the terms contaning derivatives of the cut-off functions χ clearly go to 0 when R by the Lebesguee Dominated Convergence Theorem. In Rj j → ∞ conclusion, for s > 2m, all the derivatives ∂αΘ are functions of class L2 ;BC ;L2( ∗) uniformly u (Rj,z) Xu u Xx Xξ for z and choosing m = [d/2] + 1, in the first integral in (2.63) considered as a Fourier transform of a ∈ X (cid:0) (cid:0) (cid:1)(cid:1) tempered distribution, we intertwine the multiplication with < ζ >m with the Fourier transform with respect 10