Pseudo-spectrum for a class of semi-classical operators

Transcription

1 Pseudo-spectrum for a class of semi-classical operators Karel Pravda-Starov To cite this version: Karel Pravda-Starov. Pseudo-spectrum for a class of semi-classical operators. 34 pages <hal > HAL Id: hal Submitted on 9 ov 006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 PSEUDO-SPECTRUM FOR A CLASS OF SEMI-CLASSICAL OPERATORS Karel Pravda-Starov University of California, Berkeley Abstract. We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some almost eigenvalues in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate. We prove in this paper a sharp global subelliptic a priori estimate for a class of pseudodifferential operators with respect to the regularity of their symbols. Our main result extends the a priori estimate of Dencker, Sjöstrand and Zworski Theorem.4 in [5] for a class of pseudo-differential operators with symbols of limited smoothness violating the condition P. hal , version - 9 ov 006. Introduction.. Miscellaneous facts about pseudo-spectrum. In recent years, there has been a lot of interest in studying the pseudo-spectrum of non-self-adjoint operators. We first recall some classical and known facts about pseudo-spectrum. The study of pseudo-spectrum has been initiated by noticing that for certain problems of science and engineering involving non-self-adjoint operators, the predictions suggested by spectral analysis do not match with the numerical simulations. To supplement the lack of information given by the spectrum, some new subsets of the complex plane called pseudo-spectra have been defined. The main idea is to study not only points where the resolvent is not defined, i.e. the spectrum, but also where the resolvent is large in norm. We refer the reader to Trefethen s article [] for the definition of the ε-pseudospectrum Λ ε A of a matrix or an operator A, Λ ε A = {z C, zi A ε }. By convention, we write zi A = + if z belongs to the spectrum of A. The ε-pseudo-spectrum of A is non-decreasing with ε. All these subsets contain the spectrum of the operator. In an equivalent way, pseudo-spectra can be defined in term of the spectra of perturbations. Indeed, for any matrix we have Λ ε A = {z C, z σa + B for some B with B ε}. It follows that a number z belongs to the ε-pseudo-spectrum of A if and only if it belongs to the spectrum of some perturbed operator A + B with B ε. From this second description, we understand the interest in studying such subsets. Indeed, if we want to compute numerically some eigenvalues, we start by discretizing the operator. This discretization and inevitable round-off errors will generate some perturbations of the initial operator. Eventually, algorithms for eigenvalues computing determine the eigenvalues of a perturbation of the initial operator, i.e. a value in some ε-pseudospectrum and not necessarily a spectral value. In the self-adjoint case, the spectrum is The reader will find in this paper more details about interest, history and general properties of pseudo-spectra.

3 stable under small perturbations. In fact, this stability is a consequence of the spectral theorem. The spectral theorem implies that Λ ε A is exactly the set of points in C at distance less than or equal to ε from the spectrum of A. However this property of stability is not true in the non-self-adjoint case in which the spectrum could be very unstable under small perturbations. To illustrate this fact, we recall a suggestive example pointed out by Davies in [3] and Zworski in []. Example. Let us consider the rotated harmonic oscillator in one dimension P α = d dx + eiα x where π < α < π. This operator P α is self-adjoint only for α = 0. Its spectrum is composed of the Figure : Computation of ε-pseudo-spectra for the rotated harmonic oscillator P α with α = 0. The right column gives the corresponding values of log 0 ε dim = following eigenvalues see Theorem 3.3 in [6], e i α n +, n. We can try to compute numerically the spectrum and some ε-pseudo-spectra for some small values of the parameter ε. Computations are performed on the discretization Pα Ψ i, Ψ j L R i,j, where is an integer taken equal to 50 and Ψ j j stands for the basis of L R of Hermite functions. umerical results illustrate the spectral stability in the selfadjoint case. We also notice a strong instability in the non-self-adjoint case, which leads to the computation of false eigenvalues for high energies. In this last case, the resolvent may be very large in norm far from the spectrum... Definition of the pseudo-spectra and injectivity pseudo-spectra. Our interest in this article is to study some notion of semi-classical pseudo-spectrum. In order to justify the definition in the semi-classical setting, we start again with the last example of the rotated harmonic oscillator. Following Zworski in [], we rephrase the problem of finding eigenvalues for operator P α by a change of scaling. Setting y = h / x where h is a positive parameter, one has P α λ = d dx + eiα x λ = h h d dy + eiα y hλ = Pα h z h

4 3 Figure : Computation of ε-pseudo-spectra for the rotated harmonic oscillator P α with α = π 4. The right column gives the corresponding values of log 0 ε dim = where z = hλ and P α h = h d dy + eiα y. Since we are interested in the behaviour of the resolvent for large values of λ, we can work in the semi-classical limit, i.e. h 0, with z fixed. We can now extend in a natural way the definition of pseudo-spectrum in the semi-classical setting as follows Definition... Let P h 0<h be a semi-classical family of operators on L R n defined on a domain D, for all µ 0 the set Λ sc µ P h = {z C : C > 0, h 0 > 0, 0 < h < h 0, P h z Ch µ }, is called the pseudo-spectrum of index µ of the family P h 0<h we write by convention P h z = + if z belongs to the spectrum of P h. The pseudo-spectrum of infinite index is defined by Λ sc P h = µ 0 Λ sc µ P h. With this definition, the points in the complement of Λ sc µ P h are the points of the complex plane where we have the following control of the resolvent s norm for h sufficiently small C > 0, h 0 > 0, 0 < h < h 0, P h z < Ch µ. We are interested in this paper by the study of slightly different sets from these pseudospectra, which are made of points where we can find some quasi-modes with a precise decay in the semi-classical limit. We call these sets the injectivity pseudo-spectra. Definition... Let P h 0<h be a semi-classical family of operators on L R n defined on a domain D, for all µ 0 the set λ sc µ P h = {z C : C > 0, h 0 > 0, 0 < h < h 0, u D, u L =, P h zu L Ch µ },

5 4 is called the injectivity pseudo-spectrum of index µ of the family P h 0<h. The injectivity pseudo-spectrum of infinite index is defined by λ sc P h = µ 0 λ sc µ P h. The injectivity pseudo-spectrum of index µ is by definition the set of points in the complex plane which are some almost eigenvalues with a decay in Oh µ when h 0. We notice that the injectivity pseudo-spectra as the pseudo-spectra are non-increasing with the index. The absence of injectivity pseudo-spectrum at a given point is easily characterized by an a priori estimate on the operator. Indeed, there is no injectivity pseudo-spectrum of index µ at z if and only if we have.. C > 0, h 0 > 0, 0 < h < h 0, u D, P h zu L Ch µ u L. We say that there is no loss of any power of h, respectively a loss of at most h µ for the points in the complement of the injectivity pseudo-spectrum of index 0, respectively of index µ when µ is positive. We have the following inclusions µ 0, λ sc µ P h Λ sc µ P h, but to obtain the equality, we need an additional property of surjectivity for the operators, which is fulfilled for instance if we deal with Fredholm operators of index 0. We can also notice that if P h z is a closed operator with a dense domain and that z λ sc µ Ph, the estimate.. for the operator P h z implies the surjectivity for the operator P h z if h sufficiently small. Under these previous assumptions, z Λ sc µ P h implies that z λ sc µ P h. In fact, if we suppose that P h z is a closed operator, the absence of injectivity pseudo-spectrum in z for the operator P h gives a control for the norm of the left inverse P h z : RanP h z D since the estimate.. induces that the range RanP h z is closed in L R n. These above definitions differ from one given in [5] for a semi-classical pseudo-differential operator. We prefer here to give a definition, which depends on the properties of the operator rather than on its symbol in order to study some geometrical conditions on the symbol giving the existence or the absence of pseudo-spectrum or injectivity pseudo-spectrum..3. Remark. The spectrum of a self-adjoint operator is purely real, this property is also true for the injectivity pseudo-spectrum of a family of self-adjoint operators. Proposition.3.. Let P h 0<h be a semi-classical family of self-adjoint operators on L R n defined on a dense domain D then.3. z C, u D, P h u zu L Im z u L. Thus, there is no loss of any power of h in C \ R and in particular for all µ 0 the injectivity pseudo-spectrum of index µ of P h 0<h is contained in R. Proof. Let z be in C \ R, the estimate follows from the Cauchy-Schwarz inequality Im z u L Re P h u zu, isgnim zu L P h u zu L u L, where sgnx denotes the sign of x. In fact, under the assumptions of the previous proposition we have for all µ non negative that λ sc µ P h = Λ sc µ P h. Indeed, we have on one hand by.3. that z C \ R, P h z Im z

6 5 since the spectrum of P h is real by self-adjointness. On the other hand, if z is in λ sc µ P h c R, a previous remark shows that z Λ sc µ P h since we have in this case z = z λ sc µ P h = λsc µ P h..4. Examples. The first example is the harmonic oscillator h d dx + x. The injectivity pseudo-spectrum of infinite index is in this case R + and one has in the complement in C of R + the following estimates h > 0, u C 0 R, h d u dx + x u L R h u L R, z R +, C > 0, h > 0, u C 0 R, h d u dx + x u zu L R C u L R. For the example of the rotated harmonic oscillator P α h = h d dx + eiα x, with 0 < α < π, Davies has proved in [4] Theorem that λ sc Pα h {z C : 0 < argz < α}. In fact, the injectivity pseudo-spectrum of infinite index is exactly {z C : 0 < argz < α}. Indeed for all µ 0, λ sc µ Pα h is contained in the numerical range ΣP α = {z C : 0 argz α} {0}. We can prove the a priori estimates see [0] h > 0, u C0 R, h d u dx + eiα x u + cosα L h u R L R, and for all z in C such that argz {0, α}, there exist some positive constants C z and h 0 such that 0 < h < h 0, u C 0 R, h d u dx + eiα x u zu L R C zh 3 u L R.. Statement of the main result.. The estimate. In the following statement, we give an a priori estimate, which characterizes the absence of injectivity pseudo-spectrum in 0 for the semi-classical operator, hd t + iqt, x, hξ w, where the function q is a real-valued function such that.. qt, x, ξ C [n/]+4 b R t R n x R n ξ, R. The notation C k b stands for the space of Ck functions, which are bounded as well as their derivatives of order lower than or equal to k, [m] stands for the integer part of m and qt, x, hξ w denotes the Weyl quantization of the symbol qt, x, hξ. We assume that for all X = x, ξ R n,.. qt, X > 0 and s > t qs, X 0. This hypothesis means that for all X R n, the function t qt, X can only change sign from negative values to positive ones. We also assume that for all X R n, the function t qt, X only vanishes in a fixed compact set [ A, A], A > 0, exactly

7 6 times,, and that these roots are some Lipschitz functions with respect to the variable X. More precisely, we assume that..3 A > 0, inf qt, X > 0 t A,X Rn and..4, t [ A, A], X R n, qt, X = et, X where e is a positive function on R n+ such that..5 M 0 = inf et, X > 0 t A,X Rn t αj X, and α j, j =,...,, are some real-valued Lipschitz functions on R n such that..6 α j L R n A, j =,...,. j= Theorem... Under these assumptions, there exist some constants C > 0 and 0 < h 0 such that for all u C 0 R t R n x and 0 < h < h 0,..7 hd t u + iqt, x, hξ w u L R n+ Ch + u L R n+. Thus, there is no injectivity pseudo-spectrum of infinite index in 0. More precisely, there is a loss of at most h /+ in 0... Remarks. We can first notice that under the assumptions of Theorem.., there is also no injectivity pseudo-spectrum of infinite index in z for all z in R, and that the loss in these points is also of at most h /+. This fact is a direct consequence of Theorem.. and of the following identity T h hdt + iqt, x, hξ w z T h = hd t + iqt, x, hξ w, where T h is the unitary operator on L R n+ defined by T h ut, x = e iπ h zt ut, x. Let us now make some comments about the class of pseudo-differential operators we study. The interest of studying such a class is that given a pseudo-differential operator with a principal symbol satisfying the principal-type condition ry, η = 0 dry, η 0, we can by multiplication to left and right with some elliptic Fourier integral operators obtain a pseudo-differential operator with a principal symbol, which is microlocally of the type hd t +iqt, x, hξ w. To interpret the assumptions of Theorem.. in a more general setting, we can notice that the assumption.. means that the principal symbol satisfies the condition Ψ see Definition in [7]. If we make more assumptions of smoothness on the function e, the assumption..4 implies in terms of iterated Poisson brackets that for all t, x, τ, ξ R n+, there exists an integer 0 l such that HRep l Im pt, x, τ, ξ 0, if pt, x, τ, ξ = τ + iqt, x, ξ. This means that every point in pr n+ are of finite type with an order bounded above by the fixed integer. The assumption..3 of ellipticity outside of the set [ A, A] R n allows us to obtain a global subelliptic a priori estimate without conditions on the supports size for the functions u in C0 R n+. Dencker, Sjöstrand and Zworski have proved in Theorem.4 in [5] an absence s result

8 7 of pseudo-spectrum of infinite index for a general class of pseudo-differential operators. Under the assumptions of Theorem.4 in [5], they reduce their study by a symplectic change of variables to the study of the local model hd t + iqt, x, hξ w and prove for this model the a priori estimate 5.9 in [5]. This a priori estimate 5.9 is sufficient to obtain the resolvent s estimate. in Theorem.4 because the assumptions of this theorem.4 for getting the a priori estimate 5.9 are also fulfilled for the formal adjoint hd t iqt, x, hξ w, which shows the surjectivity of the operator hd t + iqt, x, hξ w if the domains are suitably chosen and h sufficiently small. In this case, there is no pseudo-spectrum and no injectivity pseudo-spectrum of infinite index in 0. The loss is of at most h /+. In the case studied by Dencker, Sjöstrand and Zworski, the condition P is fulfilled see Definition 6.5. in [7]. More precisely, in this case the function q does not change sign on R n+. Our result shows that if we are only interested in obtaining the a priori estimate characterizing the absence of injectivity pseudo-spectrum with a loss of at most h /+, we can obtain a similar a priori estimate as 5.9 for a particular class of pseudo-differential operators violating the condition P since we only assume in Theorem.. that the condition Ψ is fulfilled. Another main difference between our result and the result of Dencker, Sjöstrand and Zworski is that we consider here some symbols with limited smoothness. Although our result does not deal with the general subelliptic case see Proposition 7.6. in [7] in a setting of limited smoothness - indeed, we make a strong assumption of Lipschitz regularity for the roots α j in..4 - we feel that our sharp estimate of the regularity needed for symbols to obtain result of subellipticity is worth noticing and also that it is interesting to have a proof of subelliptic a priori estimates for a class of operators violating the condition P, which is quite simple in comparison with the proof of the general case given by Hörmander in [7] Proposition 7.6., even if this class is particular and that our result does not deal with the general subelliptic case..3. The structure of the proof. The first step in the proof of Theorem.. is to change the quantization and to prove a similar a priori estimate in another quantization. As pointed out before, our assumption on the symbol s sign is essential. To take advantage of this assumption, we use the Wick quantization, which has the main property of being a positive quantization. The next section recalls some results in the Wick quantization, in particular its link with the Weyl quantization. To prove the a priori estimate in the Wick quantization, we need first to use a phase space cut-off to study separately different regions depending on the size of the function q t,.3. h + u L = h + H Wick u, u L + h + H Wick u, u L, where H + H =, supp H {q t ε 0} and supp H {q t ε 0}, ε 0 > 0. The estimate of the first term in the right-hand-side of the previous equality is easier than the second one. We only use for this first term an expansion of a L -norm square and some results of symbolic calculus in the Wick quantization to take advantage of the size of the function q t in Lemma 4... For the second term, the proof s core of its estimate is the following L -norm splitting h + H Wick u, u L = h + H Wu dtdx R n+ = h + { q <h /+ } H Wu dtdx + h + { q h /+ } H Wu dtdx,

9 8 where Wu stands for the wave packets transform of u defined in the next section; and we estimate the two terms of the right-hand-side of the following inequality.3. h + H Wick u, u L h + H Wu dtdx { q <h /+ } + H q Wu dtdx, R n+ in Lemma 4.. and Lemma To estimate the second term of the right-hand-side of.3., we use some techniques developed by Lerner in [8]. 3. Preliminaries 3.. otations and a few facts about the Weyl quantization. We give in this paragraph the notations and normalizations used in this paper. The scalar product on L R n is denoted by u, v L R n = uxvxdx, R n stands for the Euclidean norm and D x = x /iπ. The definition of the Fourier transform chosen here is, for u in the Schwartz space SR n, ûξ = uxe iπx.ξ dx, R n where x.ξ denotes the canonical scalar product on R n of x and ξ. For a classical Hamiltonian ax, ξ defined on R n x Rn ξ, the Weyl quantization defines the operator a w by the following formula x + y a w ux = e iπx y.ξ a, ξ uydydξ. R n In the statement of Theorem.., the variable t is seen as a parameter in the symbol of the operator qt, x, hξ w, i.e. qt, x, hξ w ut, x = e iπx y.ξ q t, x + y, hξ ut, ydydξ. R n A nice feature of the Weyl quantization is the fact that real Hamiltonians get quantized by formally self-adjoint operators. The composition formula in the Weyl quantization, a w b w = a b w, is given by 3.. a bx = n R 4n e 4iπσX Y,X Z ay bzdy dz, where σ, stands for the symplectic form on R n R n defined for all X = x, ξ and Y = y, η by σx, Y = ξ.y η.x. 3.. Wick calculus. The purpose of this second paragraph is to recall the definition and some basic properties of the Wick quantization following [9]. We also prove here some results of symbolic calculus we need in the proof of Theorem... The main reason to introduce this new quantization is its property of positivity, i.e. that non-negative Hamiltonians define non-negative operators q 0 q Wick 0. This property of positivity is not satisfied in the case of the Weyl quantization see [9] for an example of non-negative Hamiltonian defining an operator, which is not non-negative. This property is essential in our approach and permits us to use some

10 9 sign s hypothesis made on the symbol of studied pseudo-differential operator. Setting for x, y and η in R n, ϕ y,η x = n/4 e πx y e iπx y.η, where x = x x n, the following lemma introduces the wave packets transform. Lemma 3... Let u SR n, we define Wuy, η = u, ϕ y,η L R n = n/4 uxe πx y R n e iπx y.η dx, y, η R n. The mapping u Wu is continuous from SR n to SR n and isometric from L R n to L R n. Moreover, we have the reconstruction formula u SR n, x R n, ux = Wuy, ηϕ y,η xdydη. R n See Lemma. in [9] for a proof. Let Y = y, η R n, we denote by Σ Y the operator defined in the Weyl quantization by the symbol 3.. p Y X = n e π X Y. This operator is a rank-one orthogonal projection. Indeed, a direct computation gives 3.. ΣY u x = WuY ϕ Y x = u, ϕ Y L R n ϕ Y x. Definition 3... Let a L R n, the Wick quantization of a is defined as a Wick = ay Σ Y dy. R n Remark. More generally if a belongs to S R n, the operator a Wick can be defined for all u and v in SR n by < a Wick u, v > S R n,sr n =< ay, Σ Y u, v L R n > S R n,sr n, where the notation <, > S,S denotes the duality bracket between the spaces S and S. Proposition 3... Let a L R n, then a Wick = W a µ W, Wick = id L R n, where W is the isometric mapping from L R n to L R n defined in Lemma 3.. and a µ denotes the operator of multiplication by a in L R n. Moreover, one has a Wick LL R n a L R n and a 0 a Wick 0. See Proposition 3. in [9] for a proof. Remark. We can notice that the previous proposition implies that real Hamiltonians get quantized in the Wick quantization by formally self-adjoint operators.

11 0 Definition 3... Let k and l R, the symbol class S k Λ l, Λ dx denotes the set of C k functions a defined on R n [, + [ such that 0 j k, γ l,j a = sup Λ l+ j a j X, ΛT,..., T j < + X R n,λ Tp R n, Tp = and the symbol class SΛ l, dx denotes the set of C functions a such that j, γ l,j a = sup Λ l a j X, ΛT,..., T j < +. X R n,λ Tp R n, Tp = Proposition 3... Let a L R n, b S Λ, Λ dx, be some real-valued functions then [ a Wick b Wick = ab 4π a.b + ] Wick 4iπ {a, b} + S, [ Rea Wick b Wick = ab 4π a.b ] Wick + S, with S j LL R n d n a L γ, b, j =,, where the derivatives of a are taken in the distribution sense, {a, b} stands for the Poisson bracket of a and b. Here γ, b denotes the semi-norm of b defined in Definition 3.., d n is a positive constant depending only on the dimension n and the distribution Xj a Xl b is defined as Xj a Xl b := Xj a Xl b a X j,x l b S R n. See Proposition 3.4, its proof and the remark following this proposition in [9] for a proof. Lemma 3... If a S Λ l, Λ dx, then the Weyl symbol ã of a Wick, a Wick = ã w, is equal to the function a Γ where ΓX = n e π X, and verifies ã SΛ l, dx and X ã SΛ l, dx. Proof. From Definition 3.. and 3.., one has a Wick = ã w with 3..3 ãx = ay n e π X Y dy = a ΓX. R n Since for all α n, α ã = a α Γ, α ã L a L α Γ L and α Γ L < +, we first get that ã SΛ l, dx because from Definition 3.., one has a L γ l,0 aλ l. Then, since from Definition 3.., X a S 0 Λ l,λ dx, we deduce from the identity, α X Xã = X a α Γ, using the same estimate as before that X ã SΛ l, dx. Lemma If a SΛ l, dx and b SΛ l, dx then the symbol RX = e 4iπ θ σx Y,X Z ay bzdy dz dθ 0 R θ n, 4n belongs to SΛ l+l, dx.

12 Proof. Setting for θ ]0, ], R θ a, bx = R 4n e 4iπ θ σx Y,X Z ay bz dy dz θ n, we deduce from the theorem in [7] that the bilinear map a, b R θ a, b from SΛ l, dx SΛ l, dx to SΛ l+l, dx is weakly continuous. Let us assume that a and b belong to the Schwartz space SR n. Using a change of variables, we obtain that R θ a, bx = e 4iπσY,Z a θy + Xb θz + XdY dz. R 4n Since an explicit computation gives + Y 4n+ + Z 4n+ + D Y 4n+ 4n+ + D Z 4n+ 4n+ e 4iπσY,Z = e 4iπσY,Z, we obtain that R θ a, bx = R 4n + D Y 4n+ 4n+ + D Z 4n+ 4n+ [ e 4iπσY,Z] and we can make some integrations by parts to obtain that R θ a, bx = R 4n e 4iπσY,Z It follows that we can write R θ a, bx = a θy + Xb θz + X + Y 4n+ + Z 4n+ dy dz + D Y 4n+ 4n+ + D Z 4n+ [ a θy + Xb θz + X ] 4n+ + Y 4n+ + Z 4n+ dy dz. R4n e 4iπσY,Z + Y 4n+ + Z 4n+ f α,β Y, Z θ α + β α a θy + X β b θz + XdY dz, α,β n α, β 4n+ where f α,β are some Cb R4n functions. Since a SΛ l, dx, b SΛ l, dx and dy dz + Y 4n+ < +, + Z 4n+ R 4n we can differentiate with respect to X the integral of the previous identity and we obtain after these differentiations that for all γ n, X R n, Λ and θ ]0, ], γ X R θa, bx 4n + 4n R 4n dy dz + Y 4n+ + Z 4n+ sup γ l,ja j 4n++ γ sup f α,β L α,β n α, β 4n+ sup j 4n++ γ γ l,jb Λ l+l. Using the weakly continuity of the map a, b R θ a, b, we deduce from these estimates that the symbol R θ a, b belongs uniformly to the class SΛ l+l, dx with respect to θ ]0, ] if a SΛ l, dx and b SΛ l, dx. It follows that R SΛ l+l, dx.

13 Lemma If a S Λ, Λ dx and b S, Λ dx then there exists a positive constant C such that for all Λ, [a Wick, b Wick ] LL C, where [a Wick, b Wick ] denotes the commutator of the operators a Wick and b Wick. Proof. We get from Lemma 3.. that 3..4 [a Wick, b Wick ] = [ã w, b w ] = ã w bw b w ã w = ã b b ã w, where ã SΛ, dx and b S, dx are some symbols verifying 3..5 X ã SΛ, dx and X b SΛ, dx. The composition formula 3.. and some results of calculus in the Weyl quantization see the formula following 5 in [] show that in the normalization chosen here, one has 3..6 ã bx = n R 4n e 4iπσX Y,X Z ãy bzdy dz = ãx bx + R X, where 3..7 R X = n 0 e 4iπ θ σx Y,X Z iπσd Y, D Z [ ãy bz ] dy dz dθ R θ n. 4n We deduce from the lemma 3..3, 3..5 and 3..7 that R S, dx. In the same way, one has 3..8 b ã = ã b + R, where R is a symbol in the class S, dx. It follows from 3..4, 3..6 and 3..8 that [a Wick, b Wick ] = R w with R = R R S, dx, which by using the Calderón-Vaillancourt theorem proves the lemma Lemma If a S, Λ dx and b S Λ, Λ dx are some real-valued functions then there exists a positive constant C such that for all Λ, a Wick b Wick a Wick a b Wick LL C. Proof. We can apply Proposition 3.. to obtain that [ 3..9 a Wick b Wick = ab 4π a.b + ] Wick 4iπ {a, b} + S, where S LL d n γ 0,0 aγ, b and d n is a positive constant depending only on the dimension n. Let us denote 3..0 c = 4π a.b + {a, b}. 4iπ Since a S, Λ dx and b S Λ, Λ dx, we get that c S, Λ dx. It follows from Proposition 3.., 3..9 and the use of the triangular inequality that c Wick a Wick + S a Wick LL c Wick LL a Wick LL + S LL a Wick LL 3.. c L a L + d n γ 0,0 aγ, b a L γ 0,0 c + d n γ 0,0 aγ, b γ 0,0 a < +.

14 3 Since Λ ab S, Λ dx and Λa S Λ, Λ dx, another use of Proposition 3.. gives 3.. ab Wick a Wick = Λ ab Wick Λa Wick [ = a b 4π ab.a + ] Wick 4iπ {ab, a} + S, where S LL d n γ 0,0 Λ abγ, Λa < +. According to our assumptions, the symbol 4π ab.a + {ab, a}, 4iπ belongs to the class S, Λ dx and it follows from Proposition 3.. that 3..3 [ 4π ab.a + 4iπ {ab, a}] Wick LL Since we have from 3..9, 3..0 and 3.., γ 0,0 4π ab.a + 4iπ {ab, a}. a Wick b Wick a Wick a b Wick = ab Wick a Wick + c Wick a Wick + S a Wick a b Wick [ = 4π ab.a + ] Wick 4iπ {ab, a} + S + c Wick a Wick + S a Wick, we deduce from 3.., 3.., 3..3 and the use of the triangular inequality that there exists a positive constant C such that for all Λ, a Wick b Wick a Wick a b Wick LL C. Lemma If a S [n/]+4 Λ, Λ dx where [n/] stands for the integer part of n/ then there exists a positive constant C such that for all Λ, a Wick a w LL C. Proof. As in 3..3, one has a Wick = ã w where 3..4 ãx = ay n e π X Y dy. R n Using a change of variables and a Taylor formula at the second order, we obtain from 3..4 that 3..5 ãx = ay + X n e π Y dy = ax + a X.Y n e π Y dy R n R n + θa X + θy Y n e π Y dy dθ. 0 R n Since Y e π Y dy = 0, R n it follows from 3..5 that 3..6 a Wick a w = ã w a w = R w, where 3..7 RX = 0 θa X + θy Y n e π Y dy dθ. R n

15 4 Since a S [n/]+4 Λ, Λ dx and Y e π Y dy < +, R n we deduce from 3..7 that R S [n/]+, Λ dx and we can apply the L estimate for Weyl quantization of Boulkhemair Theorem. in [] to obtain the existence of a positive constant C such that for all Λ, R w LL C. In view of 3..6, this ends the proof of Lemma Another preliminary lemma. Lemma If F C R, C, α is a real-valued Lipschitz function and G = F α then one has 3.3. G x = F αx α x, for a.e. x in R if G, resp. α, stands for the derivative of G, resp. α, in the distribution sense. Proof. To prove this lemma, it is sufficient to show that for all R > 0, the identity 3.3. is fulfilled a.e. on ] R, R[. Let us consider R > 0. Since α is a L R function because α is a Lipschitz function, we can find a sequence of C 0 R, R functions u n n such that for all n, 3.3. u n L [ R,R] α L [ R,R] and lim n + u nx = α x, for a.e. x in [ R, R]. We define for all n, α n x = α0 + x 0 u n tdt, x R, and we obtain from 3.3. that α n is a C R, R function, which verifies for all n, α n L [ R,R] α L [ R,R] and lim n + α nx = α x, for a.e. x in [ R, R]. Since we can easily check that the derivative in the distribution sense of the function x 0 α tdt, is equal to α, we deduce from the continuity of the function α that for all x R, αx = α0 + x 0 α tdt. Using now 3.3., and 3.3.5, we obtain from the Lebesgue convergence theorem that for all x [ R, R], lim n + α nx = αx. If ϕ is a C0 ] R, R[, C function and <, > D,D denotes the duality bracket between the distribution space D and the space of test functions D, one has < G, ϕ > D,D= < G, ϕ > D,D = F αx ϕ xdx R = lim F α n x ϕ xdx, n + R

16 5 where the last equality is a consequence of the Lebesgue convergence theorem since on one hand, the continuity of F and prove that for all x R, lim F α n x ϕ x = F αx ϕ x, n + because supp ϕ ] R, R[ and that on the other hand, one has for all n, F α n x ϕ x sup [ M,M] F ϕ x L, because it follows from 3.3., and the use of the triangular inequality that α n L [ R,R] α0 + R u n L [ R,R] M, if M := α0 + R α L R. We can now deduce from an integration by parts and that < G, ϕ > D,D= lim F α n x α n + n xϕxdx = F αx α xϕxdx, R where the last equality is still a consequence of the Lebesgue convergence theorem since on one hand, one has from and 3.3.6, lim n + F α n x α n xϕx = F αx α xϕx, for a.e. x in R because F C 0 R, C and supp ϕ ] R, R[; and that on the other hand, one has from 3.3.4, F α n x α n xϕx sup F α L [ R,R] ϕx L, [ M,M] where M is the constant defined in This proves that G = F αα a.e. on [ R, R] and ends the proof of Lemma Proof of Theorem A preliminary reduction. To prove Theorem.., it is sufficient to prove the following estimate : there exist some constants C > 0 and Λ 0 such that for all u C 0 R n+ and Λ Λ 0, 4.. D t u + iλqt, Λ X Wick u L R n+ CΛ + u L R n+, where the variable t is seen as a parameter in the Wick quantization of the symbol qt, Λ X see Definition 3.., qt, Λ X Wick = qt, Λ XΣX dx. R n Indeed, let us assume that the estimate 4.. holds and set 4.. Qt, X, Λ = Λqt, Λ X. We deduce from.. and 4.. that the function Qt, belongs to the symbol class S [n/]+4 Λ, Λ dx uniformly with respect to the parameter t in R see Definition 3.., 4..3 Q S [n/]+4 Λ, Λ dx. It follows from Lemma 3..6 that there exists a positive constant c 0 such that for all Λ, 4..4 Q Wick Q w LL c 0. R

17 6 We deduce from 4.., 4.., 4..4 and the use of the triangular inequality that for all u C 0 Rn+ and Λ Λ 0, D t u + iq w u L R n+ D t u + iq Wick u L R n+ Q Wick u Q w u L R n+ CΛ + c0 u L R n+. This estimate induces that there exists Λ 0 Λ 0 such that for all u C 0 Rn+ and Λ Λ 0, 4..5 D t u + iλqt, Λ X w u L R n+ C Λ + u L R n+. By setting h 0 = Λ 0 and h = Λ, we get from 4..5 that in the semi-classical setting, one has for all u C0 Rn+ and 0 < h < h 0, 4..6 hd t u + iqt, h X w u L R n+ C h + u L R n+. To obtain an analogous estimate for the operator hd t + iqt, x, hξ w, we use the following symplectic linear mapping χ h x, ξ = h / x, h / ξ, x, ξ R n. Using the symplectic invariance of the Weyl quantization, it follows that 4..7 hd t + iqt, h / x, h / ξ w = U h hdt + iqt, x, hξ w U h, where U h is the unitary operator of L R n+, U h vt, x = h n 4 vt, h / x. Eventually, we deduce from 4..6 and 4..7 that for all u C 0 R n+ and 0 < h < h 0, hd t u + iqt, x, hξ w u L R n+ C h + u L R n+, which proves the estimate of Theorem Proof of the estimate 4... To prove the estimate 4.., we need to use a phase space cut-off to study separately different regions depending on the size of the function Q t. Let us consider χ a C R, [0, ] function such that 4.. χ = 0 on ], ], χ = on [, + [, and let us define the following C R, [0, ] functions 4.. χ = χ and χ = χ, which verify 4..3 χ + χ = on R, χ = 0 on ], ], χ = on [, + [ and χ = on ], ], χ = 0 on [, + [. Since the functions α j, j =,...,, appearing in..4 are supposed to be Lipschitzian, we can choose a positive constant ε 0 such that sup α j L R n ε 0 <, j=,..., where α j stands for the gradient in the distribution sense of the function α j and α j L R n, the L -norm of its Euclidean norm. We define the following symbols 4..5 h t, X, Λ = χ Q tt, X, Λε 0 Λ, H t, X, Λ = χ Q t t, X, Λε 0 Λ It follows from.., 4.. and 4..3 that and H t, X, Λ = χ Q t t, X, Λε 0 Λ H + H = and h, H, H S [n/]+3, Λ dx,

18 7 uniformly with respect to the parameter t R, and we deduce from 4..3 and 4..5 that 4..7 supp H {t, X R n+ : Q tt, X, Λ ε 0 Λ} and supp H {t, X R n+ : Q tt, X, Λ ε 0 Λ}. Step. The following lemma gives an estimate of the first term of the right-hand-side of.3.. Lemma 4... There exists a positive constant c such that for all u C 0 Rn+ and Λ, 4..8 c ΛH Wick u, u L R n+ D t u + iq Wick u L R n+ + u L R n+. Proof of Lemma 4... By expanding the following L -norm, we obtain that 4..9 D t + iq Wick h Wick u L = D th Wick u L + ReD th Wick u, iq Wick h Wick u L + Q Wick h Wick u L ReD th Wick u, iq Wick h Wick u L = [D t, iq Wick ]h Wick u, h Wick u L, since D t and iq Wick are respectively formally self-adjoint and anti-self-adjoint operators because Q is a real-valued function see the remark following the proposition 3... Since according to.. and 4.., Q t S [n/]+3 Λ, Λ dx, it follows from Lemma 3..5 and 4..6 that there exists a positive constant c such that for all Λ, 4..0 h Wick Q t Wick h Wick h Q t Wick LL c. Using that h Wick is a formally self-adjoint operator because h is real-valued function, we deduce from the Cauchy-Schwarz inequality and 4..0 that 4.. [D t, iq Wick ]h Wick u, h Wick u L = π = h Wick Q t π Wick h Wick u, u L π Q t Wick h Wick u, h Wick u L h Q t Wick u, u L c π u L. Since from 4.. and 4..5, h = H, we obtain from Proposition 3.. and 4..7 that h Q t Wick u, u L R n+ = W h Q twu, u L R n+ = h Q twu, Wu L R n+ = H t, X, ΛQ tt, X, Λ Φt, X dtdx R n+ ε 0 Λ H t, X, Λ Φt, X dtdx R n+ and R n+ H t, X, Λ Φt, X dtdx = H Wu, Wu L R n+ = W H Wu, u L R n+ = H Wick u, u L R n+, if Φt, X = W ut, X where W stands for the wave packets transform in the variable x. We deduce from 4..9, 4.. and the two last formulas that 4.. ε 0 Λ H Wick u, u L π D t + iq Wick h Wick u L + c u L.

19 8 Using now the proposition 3.. and the triangular inequality, we obtain that D t + iq Wick h Wick u L h Wick D t + iq Wick u L + [D t + iq Wick, h Wick ]u L h Wick LL D tu + iq Wick u L + 4 [D t, h Wick ]u L + 4 [Q Wick, h Wick ]u L h L D tu + iq Wick u L + 4 [D t, h Wick ]u L [Q Wick, h Wick ]u L, where [P, Q] stands for the commutator of P and Q. Using again Proposition 3.., we get that 4..4 [D t, h Wick ]u L = π th Wick u L π th L u L. Then, we deduce from Lemma 3..4, 4..3 and 4..6 that there exists a positive constant c 3 such that for all Λ, 4..5 [Q Wick, h Wick ] LL c 3. Since from.., 4.., 4.. and 4..5, h and t h S [n/]+, Λ dx, uniformly with respect to the parameter t in R, it follows from 4..3, 4..4 and 4..5 that there exists a positive constant c 4 such that for all u C 0 R n+ and Λ, 4..7 c 4 D t + iq Wick h Wick u L D tu + iq Wick u L + u L. Then, we get 4..8 from 4.. and This ends the proof of Lemma 4... Step. In this second step, we estimate the second term of the right-hand-side of.3.. This part of the proof uses some techniques developed by Lerner in [8]. Lemma 4... There exists a positive constant c 5 such that for all u C 0 R n+ and Λ, 4..8 c 5 R n+ H t, X, Λ Qt, X, Λ Φt, X dtdx if Φt, X = W ut, X. D t u + iq Wick u L u L + u L, Proof of Lemma 4... For X R n and Λ, we define 4..9 θx, Λ = inf{t R : Qt, X, Λ > 0}, if the set {t R : Qt, X, Λ > 0} is not empty, otherwise we set 4..0 θx, Λ = +. According to 4.., the assumption.. of Theorem.. implies that for all X R n and Λ, 4.. Qt, X, Λ > 0 and s > t Qs, X, Λ 0.

20 9 It follows from 4..9, 4..0 and 4.. that for all t, X R n+ and Λ, one has 4.. Qt, X, Λsgn t θx, Λ = Qt, X, Λ, where the function sgn is defined by 4..3 sgnx = x x, x R, sgn0 = 0, sgn = and sgn+ =. We can now introduce the following multiplier 4..4 St, X, Λ = sgn t θx, Λ H t, X, Λ, t, X R n+, Λ. For u in C 0 R n+, we obtain by the Cauchy-Schwarz inequality 4..5 ReD t u + iq Wick u, is Wick u L D t u + iq Wick u L S Wick u L D t u + iq Wick u L u L, because it follows from Proposition 3.., 4.., 4..5, 4..3 and 4..4 that for all t R, 4..6 S Wick t, LL R n St, L R n. Since S is a real-valued function, S Wick is a self-adjoint operator and we have 4..7 ReQ Wick u, S Wick u L = ReS Wick Q Wick u, u L = ReS Wick Q Wick u, u L. Since from 4..3 and 4..6, S L and Q S [n/]+4 Λ, Λ dx, we can apply the proposition 3.. to obtain using 4..6 and 4..7 that for all u C 0 R n+ and Λ, 4..8 ReD t u + iq Wick u, is Wick u L R n+ = ReD t u, is Wick u L R n+ + SQ Wick u, u L R n+ 4π < [Q Xt,.S Xt, ] Wick u, u > S R n+,sr n+ + Rtut,, ut, L R n dt, where for all t R, and Rt LL R n d n St, L R n γ, Q d n γ, Q 4..9 Q X.S X :=. S Q R S TraceQ XX S R n+. Let us now estimate the four terms of the right-hand-side of the last equality. Using the Cauchy-Schwarz inequality and the previous estimate, we get for the last one that for all u C0 Rn+ and Λ, Rtut,, ut, L R n dt d n γ, Q u L R n+. R For the first term, we obtain from Proposition 3.. that 4..3 ReD t u, is Wick u L R n+ = ReD t u, iw S Wu L R n+ = ReD t Wu, is Wu L R n+ = ReD t Φ, is Φ L R n+,

21 0 if Φt, X = W ut, X. A direct computation using 4..4 and an integration by parts gives 4..3 ReD t Φ, is Φ L R n+ = H θx, Λ, X, Λ Φ θx, Λ, X dx π R n + t H t, X, Λsgn t θx, Λ Φt, X dtdx. 4π R n+ Since from.., 4.., 4..3 and 4..5, one has for all t, X R n+ and Λ, t H t, X, Λsgn t θx, Λ = ε 0 q tt t, Λ X χ Q t t, X, Λε 0 Λ ε 0 q tt L χ L < +, we first obtain that there exists a positive constant c 6 such that for all u C0 Rn+ and Λ, t H t, X, Λsgn t θx, Λ Φt, X dtdx 4π R n+ c 6 Φ L R n+ = c 6 Wu L R n+ = c 6 u L R n+, because according to Proposition 3.., W is an isometric mapping from L R n x to L R n X and we deduce from 4..3, 4..3 and that ReD t u, is Wick u L H θx, Λ, X, Λ Φ θx, Λ, X dx c 6 u L π. R n For the second term, we deduce from Proposition 3.. that SQ Wick u, u L R n+ = W SQ Wu, u L R n+ = SQ Wu, Wu L R n+ = H t, X, Λ Qt, X, Λ Φt, X dtdx, R n+ if Φt, X = W ut, X, because according to 4.. and 4..4, SQ = H sgn t θx, Λ Q = H Q. For the third term, we first deduce using the triangular inequality, 4..3 and 4..6 that S TraceQ XX L R n+ nγ, Q. Using the Cauchy-Schwarz inequality, the proposition 3.. and 4..36, we get that for all u C 0 R n+ and Λ, < [S TraceQ XX ]Wick u, u > S,S = [S TraceQ XX ]Wick u, u L [S TraceQ XX ]Wick u L u L [S TraceQ XX ]Wick LL u L S TraceQ XX L u L nγ,q u L. In the following, we need to study the distribution. S Q, to estimate the term < [. S Q ] Wicku, u >S R n+,sr n+.

22 Let χ 0 : R [0, ] be a C function such that χ 0 = on [, ] and supp χ 0 [, ]. Setting τ 0 =. χ 0 Q X S Q and τ =. w 0 Q X S Q, where w 0 = χ 0, one has S Q = τ 0 + τ, with τ 0 S R n+ and τ S R n+. Estimate of < τ Wick 0 u, u > S,S. We deduce from and the remark following the definition 3.. that < τ0 Wick u, u > S R n+,sr n+ = < R. χ 0 Q X S Q = χ 0 Q X S Q R n+ To evaluate 4..4, we need to use the following lemma., Σ Y ut,, ut, L R n > S R n Y,SRn Y dt [ ] ΣX ut,, ut, L R n dtdx. Lemma There exists a constant D n depending only on the dimension such that for all α in L R n and for all j =,..., n, αy Σ Y dy R Y n j D n α L R n. LL R n See 6.0 in [8] for a proof of this lemma. Since from 4..39, supp χ 0 [, ], we notice from 4..6 that one has χ 0 Q X S Q X L χ 0 L. Then, using successively the triangular inequality, the Cauchy-Schwarz inequality, Lemma 4..3 and 4..43, we deduce that there exists a positive constant c 7 such that for all u C0 Rn+ and Λ, χ 0 Q X S Q R. [ ] ΣX ut,, ut, L R n dtdx n [ = χ 0 Q X t, St, Q R t,. n n [ j= n j= c 7 u L. ] Σ XdX u, u R n χ 0 Q X t, St, Q j t, j Σ X dx R n χ 0 Q X t, St, Q j t, L R n+ ] u, u L R n+ Σ X dx j LL u L R n+ Eventually, we deduce from 4..4 and that for all u C 0 Rn+ and Λ, < τ Wick 0 u, u > S R n+,sr n+ c 7 u L R n+.

23 Estimate of < τ Wick u, u > S,S. Let us set St, X, Λ = sgn t θx, Λ. We get from 4..4 that S = H S, and we deduce from and the fact that S is a zero order distribution that τ =. w 0 Q Q X H S + w0 Q Q X H. S. Since from and 4..40, the support of the function w 0 is contained in [, ] and 0 w 0, we get using the triangular inequality, 4..3 and 4..6 that for all t, X R n+ and Λ,. w 0 Q X Q H w 0 Q X H Q XXQ X, Q X + w 0 Q X H TraceQ XX + w 0 Q X H. Q 4 w 0 L γ 0,0 H γ, Q + n w 0 L γ 0,0 H γ, Q w 0 L γ 0, H γ, Q < +. It follows from and that there exists a positive constant c 8 such that for all Λ, w 0 Q X Q H S c 8. L R n+ We deduce using the Cauchy-Schwarz inequality, Proposition 3.. and that for all u C0 Rn+ and Λ, [ <. w 0 Q Q ] Wicku, X H S u >S R n+,sr n+ [ =. w 0 Q X Q S] Wicku, H u L R n+ [. w 0 Q X Q ] Wicku H S LRn+ u L R n+ [. w 0 Q X Q ] Wick H S LL u L R n+. w 0 Q X Q H S L R n+ u L R n+ c 8 u L R n+. To study the second term in the right-hand-side of 4..47, we first prove the following lemma. Lemma One has the following inclusions supp S X {t, X Rn+ : Qt, X, Λ = 0} and 4..5 supp w 0 Q X Q H. S K = {t, X R n+ : Qt, X, Λ = 0, Q Xt, X, Λ and Q tt, X, Λ ε 0 Λ}. Moreover, the distribution δqw 0 Q X is well defined on R n+ and one has w 0 Q X H Q. S = δq Q X w 0 Q X H.

24 3 Proof of lemma Since from 4.. and 4..46, S = on {Q > 0} and S = on {Q < 0}, the support of S X is included in {t, X Rn+ : Qt, X, Λ = 0}. Moreover, since from and 4..40, supp w 0 ], ] [, + [, and from 4..7, supp H {t, X R n+ : Q t t, X, Λ ε 0Λ}, we deduce that the support of the distribution w 0 Q Q X H. S, verifies the inclusion Since from..3,..4,..5,..6 and 4.., one has {t, X R n+ { : Qt, X, Λ = 0} = αj Λ X, X : X R n }, it follows that for all Λ the Lebesgue measure in R n+ of the set j= {t, X R n+ : Qt, X, Λ = 0}, is zero and we get from 4.. and that S is equal to the L function Q/ Q and 4..5 w 0 Q X Q H. S = w 0 Q X Q H. Q. Q We denote by δq the pullback of δ 0 the Dirac measure at 0 in D R by Q. This distribution δq = Q δ 0 is well defined on Ω = {t, X R n+ : Q X t, X, Λ > 0} and one has on Ω, which induces from 4..5 that Q = Q Q sgn = Q Q sgn = Q δq, w 0 Q X H Q. S = δq Q X w 0 Q X H, on Ω. Since from 4..39, and 4..53, supp w 0 Q X Ω, we deduce that the distribution δq Q X w 0 Q X H is well defined on R n+ and that the identity is fulfilled on R n+. We deduce now from the remark following the definition 3.. and Lemma 4..4 that [ < w 0 Q Q X H. S ] Wicku, u >S R n+,sr n+ = < w 0 Q Q X H. S, Σ X ut,, ut, L R n > S R n+,sr n = < δq Q X w 0 Q X H, Φ > S R n+,sr n+ because Σ X ut,, ut, L R n = Φt, X according to Lemma 3.. and 3.. if Φt, X = W ut, X. To estimate 4..56, we need to prove the following lemma.

25 4 Lemma For all Λ and t, X {Q = 0 and Q X > /}, we have < Q X t, X, Λ Λ sup α j L R n Q t t, X, Λ. j=,..., Proof of Lemma Let t 0, X 0 R n+ such that Qt 0, X 0, Λ = 0 and Q Xt 0, X 0, Λ >. It follows from..3,..4,..5 and 4.. that there exists j 0 {,..., } such that t 0 = α j0 Λ X0. Since from..4,..6 and 4.., one has for all j {,..., }, X R n and Λ, Q α j Λ X, X, Λ = 0, we obtain from Lemma 3.3. that Q t αj Λ X, X, Λ α j Λ XΛ + Q X αj Λ X, X, Λ = 0, for a.e. X in R n, which induces that 4..6 Q X αj Λ X, X, Λ Λ sup j=,..., α j L R n Q t αj Λ X, X, Λ, for a.e. X in R n. Then, we first notice that the continuity of the functions in the previous estimate proves that in fact this estimate is fulfilled for all X R n. We obtain from 4..57, and 4..6 that < Q X t 0, X 0, Λ Λ sup α j L R n Q t t 0, X 0, Λ. j=,..., To end the proof of this lemma, it is sufficient to check that Q tt 0, X 0, Λ 0. This is the case because if Q t t 0, X 0, Λ < 0, we would deduce from 4.. and that the function t qt, Λ X0, would change sign from positive values to negative ones at the first order in t 0, which is not possible in view of the assumption... Since from 4.., 4..5, and 4..40, H, 0 w 0 and supp w 0 ], ] [, + [, we deduce from Lemma 4..5 that < δq Q X w 0 Q X H, Φ > S,S Λ 4ε 0 sup j=,..., sup j=,..., α j L < δqq t w 0 Q X H, Φ > S,S α j L < δqq tw 0 Q X H, Φ > S,S, because according to 4..7, Q t H ε 0 ΛH. Let t 0, X 0 R n+ and Λ such that Qt 0, X 0, Λ = 0 and Q Xt 0, X 0, Λ. The result of the lemma 4..5 implies that Q t t 0, X 0, Λ > 0.

26 5 Since from 4..3, Q C [n/]+4 R n+, we can apply the implicit function theorem to get θ Λ a C [n/]+4 function defined on an open neighbourhood V 0 of X 0 such that for all X V 0, Q θλ X, X, Λ = 0 with t 0 = θ Λ X 0. Using a Taylor formula at the first order, we deduce from that for all t, X R V 0, Qt, X, Λ = Qt, X, Λ Q θλ X, X, Λ = Q t s θλ X + st, X, Λ t ds θλ X. Setting gt, X, Λ = 0 0 Q t s θλ X + st, X, Λ ds, we deduce from 4..64, and that there exists an open neighbourhood Ω 0 of the set {Q = 0 and Q X /} such that Ω 0 { Q X > 0}, g C [n/]+3 Ω 0, g > 0 on Ω 0 and Qt, X, Λ = gt, X, Λ t θ Λ X, on Ω 0. It follows from 4.. and that for all t, X Ω 0 and Λ, θ Λ X = θx, Λ and Qt, X, Λ = gt, X, Λ t θx, Λ. This induces that the function t, X t θx, Λ is C [n/]+4 on Ω 0 and that 4..7 H Qt, X, Λ = H t θx, Λ, on Ω 0, where H stands for the Heaviside function. By differentiating 4..7 with respect to t, we obtain according to that on Ω 0, 4..7 Q t δq = δ t θx, Λ. Since from and 4..40, supp w 0 Q X { Q X }, we deduce from 4..6, and 4..7 that < δqq tw 0 Q X H, Φ > S,S = < δ t θx, Λ w 0 Q X H, Φ > S,S = w 0 Q X θx, Λ, X, Λ H θx, Λ, X, Λ Φ θx, Λ, X dx R n H θx, Λ, X, Λ Φ θx, Λ, X dx. R n We obtain from 4..56, and that [ < w 0 Q Q X H. S ] Wicku, u >S,S 4ε 0 sup j=,..., α j L R n H θx, Λ, X, Λ Φ θx, Λ, X dx.

27 6 Using the triangular inequality, it follows from 4..4, 4..4, 4..45, 4..47, and that [ <. S Q ] Wicku, u >S,S 4ε 0 sup j=,..., α j L R n H θx, Λ, X, Λ Φ θx, Λ, X dx + c 9 u L H θx, Λ, X, Λ Φ θx, Λ, X dx + c 9 u L, R n where c 9 = c 7 + c 8. Using again the triangular inequality, we obtain from 4..8 and that ReD t u, is Wick u L + SQ Wick u, u ReD L t u + iq Wick u, is Wick u L + 4π < Q X.S X Wick. u, u > S,S + Rtut,, ut, L R n dt Using always the triangular inequality, we deduce from 4..9, and that π < Q X.S X Wick u, u > S,S nγ, Q + c 9 u 4π L + H θx, Λ, X, Λ Φ θx, Λ, X dx. 4π R n Finally, we obtain with 4..5, 4..30, 4..34, 4..35, and that H θx, Λ, X, Λ Φ θx, Λ, X dx c 6 u L π R n + H t, X, Λ Qt, X, Λ Φt, X dtdx R n+ n D t u + iq Wick u L u L + π + d n γ, Q u L + 4π c 9 u L + H θx, Λ, X, Λ Φ θx, Λ, X dx, 4π R n which induces, since from..4,, that there exists a positive constant c 0 such that for all u C0 Rn+ and Λ, H t, X, Λ Qt, X, Λ Φt, X dtdx D t u+iq Wick u L u L +c 0 u L. R n+ This ends the proof of Lemma 4... Step 3. In this third step, we estimate the first term of the right-hand-side of.3.. Lemma There exist some positive constants c and Λ 0 such that for all u C0 R n+ and Λ Λ 0, Λ + H t, X, Λ Φt, X dtdx c D t u + iq Wick u L u L { Q <Λ /+ } + c u L + c H t, X, Λ Qt, X, Λ Φt, X dtdx + R Λ + u L, n+ if Φt, X = W ut, X. R

28 7 Proof of Lemma For Λ, let us consider the set E Λ = {t, X R n+ : Qt, X, Λ < Λ + }. If t, X E Λ, it follows from 4.. that qt, Λ X < Λ +. Since from..4,, we deduce from..3 and that there exists a constant Λ 0 such that for all Λ Λ 0, 4..8 E Λ [ A, A] R n. Then, using..4,..5, and 4..8, we obtain that if t, X E Λ and Λ Λ 0, M 0 t α j Λ X et, Λ X j= j= which implies that there exists j 0 {,..., } such that 4..8 M 0 t α j0 Λ X < Λ +. Let us consider χ 3 a C0 R, [0, ] function such that supp χ 3 [, ] and χ 3 = on [, ]. t α j Λ X < Λ +, We deduce from 4..8 and that for all t, X E Λ and Λ Λ 0, χ 3 M 0 Λ + t αj Λ X, j= which implies according to that Λ + H t, X, Λ Φt, X dtdx j= { Q <Λ /+ } R n+ Λ + H t, X, Λχ 3 M 0 Λ Let j 0 {,..., }, we define the following multiplier mt, X, Λ = H t, X, Λ t Λ + t αj Λ X Φt, X dtdx. + χ3 M 0 Λ + s αj0 Λ X ds. Using a change of variables, we first notice from 4..6, and that for all t, X R n+ and Λ, i.e. mt, X, Λ M 0 M 0 M 0 Λ + t α j0 Λ X R χ 3 udu χ 3 udu < +, m L R n+ M 0 χ 3 L R < +. One has from that t mt, X, Λ = H t, X, ΛΛ + χ3 M 0 Λ + t αj0 Λ X + At, X, Λ,

29 8 where At, X, Λ = t H t, X, Λ t Λ + χ3 M 0 Λ + s αj0 Λ X ds. Since from 4.. and 4..5, a direct computation gives t H t, X, Λ = χ Q t t, X, Λε 0 Λ Q ttt, X, Λε 0 Λ = χ Q t t, X, Λε 0 Λ q ttt, Λ Xε 0 ε 0 χ L R q tt L R n+, we deduce using the same change of variables as in from.. and 4..3 that there exists a positive constant c such that for all Λ, A L R n+ ε 0 M 0 χ 3 L R χ L R q tt L R n+ c. We also need to make some estimates on the gradient of the multiplier mt, X, Λ with respect to X. We get from that 4..9 m X t, X, Λ = Bt, X, Λ + Ct, X, Λ, where 4..9 Bt, X, Λ = X H t, X, Λ and t Λ Ct, X, Λ = H t, X, Λα j 0 Λ X t Λ + M 0 χ 3 + χ3 M M 0 Λ 0 Λ + s αj0 Λ X ds + s αj0 Λ X ds. According to 4..83, we can compute the integral in to obtain that Ct, X, Λ = H t, X, Λα j 0 Λ XΛ + χ 3 M 0 Λ + t αj0 Λ X, because d [χ 3 M 0 ds Λ + s αj0 Λ ] X = M 0 Λ + χ 3 M 0 Λ + s αj0 Λ X. Using 4..3, 4..6, 4..9 and the same change of variables as in 4..86, we get that for all Λ, B.Q X L R n+ γ 0, H Λ M 0 χ 3 L RΛ γ, Q γ 0, H γ, QM 0 χ 3 L R < +. To estimate the term C.Q X L R n+, we start by using the Taylor formula at the first order to write that Q X t, X, Λ = Q X αj0 Λ X, X, Λ + Q X,t sαj0 Λ t X + st, X, Λ ds αj0 Λ X. Since from.. and 4.., Q t S [n/]+3λ, Λ dx, uniformly with respect to the parameter t in R, we deduce using the triangular inequality and that for all t, X R n+ and Λ, Q X t, X, Λ Q X αj0 Λ X, X, Λ + γ, Q t Λ t αj0 Λ X.