Long time solutions for quasi-linear Hamiltonian perturbations of Schr\"odinger and Klein-Gordon equations on tori

We consider quasi-linear, Hamiltonian perturbations of the cubic Schr\"odinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time $\varepsilon^{-2}$. More precisely, concerning the Schr\"odinger equation we show that the lifespan is at least of order $O(\varepsilon^{-4})$, in the Klein-Gordon case, we prove that the solutions exist at least for a time of order $O(\varepsilon^{-{8/3}^{-}})$ as soon as $d\geq3$. Regarding the Klein-Gordon equation, our result presents novelties also in the case of semi-linear perturbations: we show that the lifespan is at least of order $O(\varepsilon^{-{10/3}^-})$, improving, for cubic non-linearities and $d\geq4$, the general results in [17,24].


INTRODUCTION
This paper is concerned with the study of the lifespan of solutions of two classes of quasi-linear, Hamiltonian equations on the d -dimensional torus T d := (R/2πZ) d , d ≥ 1.We study quasi-linear perturbations of the Schrödinger and Klein-Gordon equations.
where C u := u(t , x), x ∈ T d , d ≥ 1, V (x) is a real valued potential even with respect to x, h(x) is a function in C ∞ (R; R) such that h(x) = O(x 2 ) as x → 0. The initial datum u 0 has small size and belongs to the Sobolev space H s (T d ) (see (2.2)) with s 1.
We examine also the Klein-Gordon equation where ψ = ψ(t , x), x ∈ T d , d ≥ 1 and m > 0. The initial data (ψ 0 , ψ 1 ) have small size and belong to the Sobolev space H s (T d ) × H s−1 (T d ), for some s 1.The nonlinearity f (ψ) has the form where F (y 0 , y 1 , . . ., y d ) ∈ C ∞ (R d +1 , R), has a zero of order at least 6 at the origin.The non linear term g (ψ) has the form by other authors in the context of water-waves equations, firstly by Berti-Delort in [6] in a non resonant regime, secondly by Berti-Feola-Pusateri in [8,9] and Berti-Feola-Franzoi [7] in the resonant case.We also mention that this feature has been used in other contexts for the same equations, for instance Feola-Procesi [30] prove the existence of a large set of quasi-periodic (and hence globally defined) solutions when the problem is posed on the circle.This "reduction to constant coefficients" is a peculiarity of one dimensional problems, in higher dimensions new ideas have to be introduced.For quasi-linear equations on tori of dimension two we quote the paper about long-time solutions for water-waves problem by Ionescu-Pusateri [34], where a different normal form analysis has been presented.Historical introduction for (KG).The local existence for (KG) is classical and we refer to Kato [37].Many analysis have been done for global/long time existence.
When the equation is posed on the Euclidean space we have global existence for small and localized data Delort [16] and Stingo [44], here the authors use dispersive estimates on the linear flow combined with quasi-linear normal forms.
For (KG) on compact manifolds we quote Delort [18,19] on S d and Delort-Szeftel [20] on T d .The results obtained, in terms of length of the lifespan of solutions, are stronger in the case of the spheres.More precisely in the case of spheres the authors show the following.If m in (KG) is chosen outside of a set of zero Lebesgue measure, then for any natural number N , any initial condition of size ε (small depending on N ) produces a solution whose lifespan is at least of magnitude ε −N .In the case of tori in [20] they consider a quasi-linear equation, vanishing quadratically at the origin and they prove that the lifespan of solutions is of order ε −2 if the initial condition has size ε small enough.The differences between the two results are due to the different behaviors of the eigenvalues of the square root of the Laplace-Beltrami operator on S d and T d .The difficulty on the tori is a consequence of the fact that the set of differences of eigenvalues of −∆ T d is dense in R if d ≥ 2, this does not happen in the case of spheres.A more general set of manifolds where this does not happen is the Zoll manifolds, in this case we quote the paper by Delort-Szeftel [21] and Bambusi-Delort-Grébert-Szeftel [3] for semi-linear Klein-Gordon equations.For semi-linear Klein-Gordon equations on tori we have the result by Delort [17].In this paper the author proves that if the non-linearity is vanishing at order k at zero then any initial datum of small size ε produces a solution whose lifespan is at least of magnitude ε −k(1+ 2 d ) , up to a logarithmic loss.We improve this result, see Theorem 4, when k = 2 and d ≥ 4.
Statement of the main results.The aim of this paper is to prove, in the spirit of [34], that we may go beyond the trivial time of existence, given by the local well-posedness theorem which is ε −2 since we are considering equations vanishing cubically at the origin and initial conditions of size ε.
In order to state our main theorem for (NLS) we need to make some hypotheses on the potential V .We consider potentials having the following form (1.5) We endow the set O := [−1/2, 1/2] Z d with the standard probability measure on product spaces.Our main theorem is the following.

Theorem 1. (Long time existence for NLS).
Consider the (NLS) with d ≥ 2. There exists N ⊂ O having zero Lebesgue measure such that if x ξ in (1.5) is in O \ N , we have the following.There exists s 0 = s 0 (d , m) 1 such that for any s ≥ s 0 there are constants c 0 > 0 and ε 0 > 0 such that for any 0 < ε ≤ ε 0 we have the following.If u 0 H s < c 0 ε, there exists a unique solution of the Cauchy problem (NLS) such that u(t , x) ∈ C 0 [0, T ); H s (T d ) , sup In the one dimensional case the potential V may be disregarded and we obtain the following.
Theorem 2. Consider (NLS) with V ≡ 0 and d = 1.There exists s 0 1 such that for any s ≥ s 0 there are constants c 0 > 0 and ε 0 > 0 such that for any 0 < ε ≤ ε 0 we have the following.If u 0 H s < c 0 ε, there exists a unique solution of the Cauchy problem (NLS) such that u(t , x) ∈ C 0 [0, T ); H s (T d ) , sup This is, to the best of our knowledge, the first result of this kind for quasi-linear Schrödinger equations posed on compact manifolds of dimension greater than one.
Our main theorem regarding the problem (KG) is the following.Theorem 3. (Long time existence for KG).Consider the (KG) with d ≥ 2. There exists N ⊂ [1,2] having zero Lebesgue measure such that if m ∈ [1, 2]\N we have the following.There exists s 0 = s 0 (d , m) 1 such that for any s ≥ s 0 there are constants c 0 > 0 and ε 0 > 0 such that for any 0 < ε ≤ ε 0 we have the following.
We remark that long time existence for quasi-linear Klein-Gordon equations in dimension one are nowadays well known, see for instance [18].The theorem 2 improves the general result in [17] in the particular case of cubic non-linearities in the following sense.First of all we can consider more general equations containing derivatives in the non-linearity (with "small" quasi-linear term), moreover our time of existence does not depend on the dimension.Furthermore, adapting our proof to the semi-linear case (i.e. when f = 0 in (KG) and (1.1) and G in (1.2) does not depend on y 1 ), we obtain the better time of existence ε −10/3 + for any d ≥ 4. In the cases d = 2, 3 we recover the time of existence in [17].This is the content of the next Theorem.Comments on the results.We begin by discussing the (NLS) case.We remark that, beside the mathematical interest, it would be very interesting, from a physical point of view, to be able to deal with the case h(τ) ∼ τ .Indeed, for instance, if we chose h(τ) = 1 + τ − 1; the respective equation (NLS) models the self-channeling of a high power, ultra-short laser pulse in matter, see [11].Unfortunately we need in our estimates h(τ) ∼ τ 1+δ with δ > 0, and since h has to be smooth this leads to h(τ) ∼ τ 2 .
Our method covers also more general cubic terms.For instance we could replace the term |u| 2 u with g (|u| 2 )u, where g (•) is any analytic function vanishing at least linearly at the origin and having a primitive G = g .We preferred not to write the paper in the most general case since the non-linearity |u| 2 u is a good representative for the aforementioned class and allows us to avoid to complicate the notation furtherly.We also remark that we consider a class of potentials V more general than the one we used in [29,26] and more similar to the one used in [4] in a semi-linear context.
We now make some comments on the result concerning (KG).In this case we use normal forms (the same strategy is used for (NLS) as well) and therefore small divisors' problems arise.The small divisors, coming from the four waves interaction, are of the form with Λ KG defined in (1.4).In this case we prove the lower bound (see (2.26)) for almost any value of the mass m in the interval [1,2] and where β is any real number in the open interval (3,4).The second factor in the r.h.s. of the above inequality represents a loss of derivatives when dividing by the quantity (1.9) which may be transformed in a loss of length of the lifespan through partition of frequencies.This is an extra difficulty, compared with the (NLS) case, which makes the problem challenging already in a semi-linear setting.The novelty in (1.10) is that β does not depend on the dimension d .This is why we can improve the result of [17].We also quote [5] where Bernier-Faou-Grébert use a control of the small divisors involving only the largest index (and not max 2 as in (1.10)).They obtained, in the semi-linear case, the control of the Sobolev norm as in (1.8), with a arbitrary large, but assuming that the initial datum satisfies ψ 0 H s +1/2 + ψ 1 H s −1/2 < c 0 ε for some s ≡ s (a) > s, i.e. allowing a loss of regularity.
We notice that also in the (KG) case we are not able to deal with the interesting case of cubic quasilinear term.
Ideas of the proof.In our proof we shall use a quasi-linear normal forms/modified energies approach, this seems to be the only successful one in order to improve the time of existence implied by the local theory.We recall, indeed, that on T d the dispersive character of the solutions is absent.Moreover, the lack of conservation laws and the quasi-linear nature of the equation prevent the use of semi-linear techniques as done by Bambusi-Grébert [4] and Bambusi-Delort-Grébert-Szeftel [3].
The most important feature of equation (NLS) and (KG), for our purposes, is their Hamiltonian structure.This property guarantees some key cancellations in the energy-estimates that will be explained later on in this introduction.
The equation (NLS) may be indeed rewritten as follows: where ∂ x j ∂ u x j P (u, ∇u) . (1.11) The equation (KG) is Hamiltonian as well.Thanks to the (1.1), (1.2) we have that also the nonlinear Klein-Gordon in (KG) can be written as where H KG (ψ, φ) is the Hamiltonian We describe below our strategy in the case of the (NLS) equation.The strategy for (KG) is similar.
In [28] we prove an energy estimate, without any assumption of smallness on the initial condition, for a more general class of equations.This energy estimate, on the equation (NLS) with small initial datum, would read where An estimate of this kind implies, by a standard bootstrap argument, that the lifespan of the solutions is of order at least O(ε −2 ), where ε is the size of the initial condition.To increase the time to O(ε −4 ) one would like to show the improved inequality Our main goal is to obtain such an estimate.PARA-LINEARIZATION OF THE EQUATION (NLS).The first step is the para-linearization, à la Bony [10], of the equation as a system of the variables (u, u), see Prop.3.1.We rewrite (NLS) as a system of the form (compare with (3.4)) -adjoint matrix of para-differential operators of order two (see (3.3), (3.2)),A 1 (U ) is a self-adjoint, diagonal matrix of para-differential operators of order one (see (3.4), (3.2)).These algebraic configuration of the matrices (in particular the fact that A 1 (U ) is diagonal) is a consequence of the Hamiltonian structure of the equation.The summand X H 4 is the cubic term (coming from the para-linearization of |u| 2 u, see (3.5)) and R(U ) H s is bounded from above by U 7 H s for s large enough.Both the matrices A 2 (U ) and A 1 (U ) vanish when U goes to 0. Since we assume that the function h, appearing in (NLS), vanishes quadratically at zero, as a consequence of (3.2), we have that U 6 H s , where by L (X ; Y ) we denoted the space of linear operators from X to Y .We also remark that the summand X H 4 is an Hamiltonian vector field with Hamiltonian function DIAGONALIZATION OF THE SECOND ORDER OPERATOR.The matrix of para-differential operators A 2 (U ) is not diagonal, therefore the first step, in order to be able to get at least the weak estimate (1.14), is to diagonalize the system at the maximum order.This is possible since, because of the smallness assumption, the operator E (−∆ + A 2 (U )) is locally elliptic.In section 4.1.1we introduce a new unknown W = Φ NLS (U )U , where Φ NLS (U ) is a parametrix built from the matrix of the eigenvectors of E (−∆+A 2 (U )), see (4.4), (4.2).The system in the new coordinates reads (1) (U ), where both A (1)  2 (U ), A (1)  1 (U ) are diagonal, see (4.11) and where R (1) (U ) H s U 7 H s for s large enough.We note also that the cubic vector field X H 4 remains the same because the map Φ NLS (U ) is equal to the identity plus a term vanishing at order six at zero, see (4.5).
DIAGONALIZATION OF THE CUBIC VECTOR-FIELD.In the second step, in section 4.1.2,we diagonalize the cubic vector-field X H 4 .It is fundamental for our purposes to preserve the Hamiltonian structure of this cubic vector-field in this diagonalization procedure.In view of this we perform a (approximatively) symplectic change of coordinates generated from the Hamiltonian in (4.22) and (4.21) (note that this is not the case for the diagonalization at order two).Actually the simplecticity of this change of coordinates is one of the most delicate points in our paper.The entire Appendix A is devoted to this.This diagonalization is implemented in order to simplify a low-high frequencies analysis.More precisely we prove that the cubic vector field may be conjugated to a diagonal one modulo a smoothing remainder.The diagonal part shall cancel out in the energy estimate due to a symmetrization argument based on its Hamiltonian character.As a consequence the time of existence shall be completely determined by the smoothing reminder.Being this remainder smoothing the contribution coming from high frequencies is already "small", therefore the normal form analysis involves only the low modes.This will be explained later on in this introduction.
We explain the result of this diagonalization.We define a new variable Z = Φ B NLS (W ), see (4.23), and we obtain the new diagonal system (compare with (4.26)) 5 (U ), where the new vector-field X H 4 (Z ) is still Hamiltonian, with Hamiltonian function defined in (4.29), and it is equal to a skew-selfadjoint and diagonal matrix of bounded para-differential operators modulo smoothing reminders, see (4.27).Here R (2)  5 (U ) satisfies the quintic estimates (4.28).INTRODUCTION OF THE ENERGY-NORM.Once achieved the diagonalization of the system we introduce an energy norm which is equivalent to the Sobolev one.Assume for simplicity s = 2n with n a natural number.Thanks to the smallness condition on the initial datum we prove in Section 5.1.
we are reduced to study the L 2 norm of the function Z n .This has been done in Lemma 5.2.Since the system is now diagonalized, we write the scalar equation, see Lemma 5.3, solved by z n , where we have denoted by T L the element on the diagonal of the self-adjoint operator −∆1 + A 2 (U ) + A 1 (U ), see (5.1) does not give any contribution to the energy estimates.This key cancellation may be interpreted as a consequence of the fact that the super actions where z is defined in (2.1), are prime integrals of the resonant Hamiltonian vector field X +,res H 4 (Z ) in the same spirit 2 of [24].This is the content of Lemma 5.4, more specifically equation (5.16).We are left with the study of the term −∆ n X +,⊥ H 4 .In Lemma 5.3 we prove that −∆ n X +,⊥ H 4 = B (1)  n (Z )+B (2)  n (Z ), where B (1)  n (Z ) does not contribute to energy estimates and B (2)  n (Z ) is smoothing, gaining one space derivative, see (5.11) and Lemma 2.5.The cancellation for B (1)  n (Z ) is again a consequence of the Hamiltonian structure and it is proven in Lemma 5.4, more specifically equation (5.17).Summarizing we obtain the following energy estimate (see (2.3)) 1 2  (5.17).Setting E (t ) = z n (t ) 2  L 2 , the only term which is still not good in order to obtain an estimate of the form (1.15) is the (1.20).
In order to improve the time of existence we need to reduce the size of this new term B (2)  n (Z ) by means of normal forms/integration by parts.We note immediately that, thanks to all the reductions we have performed, the term B (2)  n presents two advantages: it is non-resonant and smoothing.Thanks to the fact that it is smoothing we shall need to perform a normal form only for the low frequencies of B (2)  n (Z ).More precisely, thanks to (5.9) and (5.11), we prove in Lemma 5.8, see (5.34), that the high frequency part of this vector-field is already small, if N therein is chosen large enough inversely proportional to a power 1 To be precise the definition of Z n = (z n , z n ) in 5.1.1 is slightly different than the one presented here, but they coincide modulo smoothing corrections.For simplicity of notation, and in order to avoid technicalities, in this introduction we presented it in this way. 2 More generally, this cancellation can be viewed as a consequence of the commutation of the linear flow with the resonant part of the nonlinear perturbation which is a key of the Birkhoff normal form theory (see for instance [31]).
of the size of the initial condition.The normal form on the non-resonant term, restricted to the low frequencies, is performed in Proposition 5.7.Here we use the lower bound on the small divisor in (5.26) given by Proposition 5.6.
As said before the strategy for (KG) is similar except for the control of the small divisor (1.10) which implies some extra difficulties that we already talk about.Let us just describe how the paper is organized concerning (KG): In Section 3.2 we paralinearize the equation obtaining, passing to the complex variables (3.11), the system of equations of order one (3.31).In Section 4.2 we diagonalize the system: the operator of order one is treated in Prop.4.11 and the order zero in Prop.4.13.As done for (NLS) in the diagonalization of the operator of order zero we preserve its Hamiltonian structure.The energy estimates are given in Section 5.2.The non degeneracy of the linear frequencies is studied in Appendix B.
ACKNOWLEDGEMENTS.We would like to warmly thank prof.Fabio Pusateri for the inspiring discussions.

PRELIMINARIES
We denote by H s (T d ; C) (respectively H s (T d ; C 2 )) the usual Sobolev space of functions T d x → u(x) ∈ C (resp.C 2 ).We expand a function u(x), x ∈ T d , in Fourier series as We set , where 〈D〉e i j •x = 〈 j 〉e i j •x , for any j ∈ Z d , and (•, •) L 2 denotes the standard complex L 2 -scalar product Notation.We shall use the notation A B to denote A ≤ C B where C is a positive constant depending on parameters fixed once for all, for instance d and s.We will emphasize by writing q when the constant C depends on some other parameter q.Basic Paradifferential calculus.We follow the notation of [28].We introduce the symbols we shall use in this paper.We shall consider symbols The constant m ∈ R indicates the order of the symbols, while s denotes its differentiability.Let 0 < < 1/2 and consider a smooth function χ : R → [0, 1] For a symbol a(x, ξ) in N m s we define its (Weyl) quantization as where a(η, ξ) denotes the Fourier transform of a(x, ξ) in the variable x ∈ T d .Thanks to the choice of χ in (2.5) we have that, if j = 0 then χ (| j − k|/〈 j + k〉) ≡ 0 for any k ∈ Z d \ {0}.Moreover, the function T a h − (T a h)(0) depends only on the values of a(x, ξ) for |ξ| ≥ 1.In view of this fact, if a(x, ξ) = b(x, ξ) for |ξ| > 1/2 then T a − T b is a finite rank operator.Therefore, without loss of generality, we write a = b if a(x, ξ) = b(x, ξ) for |ξ| > 1/2.Moreover the definition of the operator T a is independent of the choice of the cut-off function χ up to smoothing terms, see, for instance, Lemma 2.1 in [28].
Notation.Given a symbol a(x, ξ) we shall also write to denote the associated para-differential operator.
We now collects some fundamental properties of para-differential operators.For details we refer the reader to section 2 in [28].
where {a, b} := (2.9) s 0 +2 we have (recall (2.9)) where R j (a, b) are remainders satisfying, for any s ∈ R, where (2.15) Proof.We start by proving the following claim: the term is a remainder of the form (2.15).By (2.6) this is actually true with coefficients a(ξ, η, ζ) of the form In order to prove this, we consider the following partition of the unity:  (2.15).A similar property holds also for T g h f and T f h g .At this point we write One concludes by using the claim at the beginning of the proof.

Matrices of symbols and operators.
Let us consider the subspace U defined as Along the paper we shall deal with matrices of linear operators acting on H s (T d ; C 2 ) preserving the subspace U .Consider two operators R 1 , R 2 acting on C ∞ (T d ; C).We define the operator F acting on where the linear operators . We say that an operator of the form (2.18) is real-to-real.It is easy to note that real-to-real operators preserves U in (2.17).
Consider now a symbol a(x, ξ) of order m and set A := T a .Using (2.6) one can check that By (2.20) we deduce that the operator A is self-adjoint with respect to the scalar product (2.3) if and only if the symbol a(x, ξ) is real valued.We need the following definition.Consider two symbols a, b ∈ N m s and the matrix Define the operator (recall (2.7))

.21)
The matrix of paradifferential operators defined above have the following properties: • Reality: by (2. 19) we have that the operator M in (2.21) has the form (2.18), hence it is real-to-real; • Self-adjointeness: using (2.20) the operator M in (2.21) is self-adjoint with respect to the scalar product on (2.17)

.23)
Non-homogeneous symbols.In this paper we deal with symbols satisfying (2.4) which depends nonlinearly on an extra function u(t , x) (which in the application will be a solution either of (NLS) or a solution of (KG)).We are interested in providing estimates of the semi-norms (2.4) in terms of the Sobolev norms of the function u.Consider a function F (y 0 , y 1 , . . ., for some j , k = 1, . . ., d , α, β ∈ {0, 1} and σ, σ ∈ {±} where we used the notation u + = u and u − = u.The following lemma is proved in section 2 of [28]. where a is the symbol in (2.24).Moreover, for any h ∈ H s+s 0 +1 , the map h → (∂ u a)(u; x, ξ)h extends as a linear form on H s+s 0 +1 and satisfies The same holds for ∂ u a. Moreover if the symbol a does not depend on ∇u, then the same results are true with s 0 + 1 s 0 .
Trilinear operators.Along the paper we shall deal with trilinear operators on the Sobolev spaces.We shall adopt a combination of notation introduced in [6] and [34].In particular we are interested in studying properties of operators of the form where the coefficients q(ξ, η, ζ) ∈ C for any ξ, η, ζ ∈ Z d .We introduce the following notation: given j 1 , . . .,

.26)
We now prove that, under certain conditions on the coefficients, the operators of the form (2.25) extend as continuous maps on the Sobolev spaces.
Proof.By (2.2) we have which is the (2.28).The bounds of I and I I are similar.
In the following lemma we shall prove that a class of "para-differential" trilinear operators, having some decay on the coefficients, satisfies the hypothesis of the previous lemma.
Proof.First of all we write q(ξ, )

Hamiltonian formalism in complex variables. Given a Hamiltonian function
Hamiltonian vector field has the form where Ω is the non-degenerate symplectic form The Poisson brackets between two Hamiltonians H ,G are defined as The nonlinear commutator between two Hamiltonian vector fields is given by Hamiltonian formalism in real variables.Given a Hamiltonian function H R : H 1 (T d ; R 2 ) → R, its hamiltonian vector field has the form where J is in (2.33).Indeed one has where Ω is the non-degenerate symplectic form Ω( We introduce the complex symplectic variables where where Ω is in (2.35).In these coordinates the vector field X H R in (2.38) assumes the form X H as in (2.33) with We now study some algebraic properties enjoyed by the Hamiltonian functions previously defined.Let us consider a homogeneous Hamiltonian H : for some coefficients By (2.43) one can check that the Hamiltonian H is real valued and symmetric in its entries.Recalling (2.33) we have that its Hamiltonian vector field can be written as where the coefficients f (ξ, η, ζ) have the form We need the following definition.
Definition 2.7.(Resonant set).We define the following set of resonant indexes: (2.47) Consider the vector field in (2.45).We define the field X +,res where 1 R is the characteristic function of the set R.
In the next lemma we prove a fundamental cancellation.
Lemma 2.8.For n ≥ 0 one has (recall (2.2)) Re(〈D〉 n X +,res Proof.Using (2.47)-(2.49)one can check that By an explicit computation we have Re(D s X +,res By (2.51), (2.46) and using the symmetries (2.43) we have Remark 2.9.We remark that along the paper we shall deal with general Hamiltonian functions of the form where we used the notation However, by the definition of the resonant set (2.47), we can note that the resonant vector field has still the form (2.48) and it depends only on the monomials in the Hamiltonian H (U ) which are gauge invariant, i.e. of the form (2.42).

PARA-DIFFERENTIAL FORMULATION OF THE PROBLEMS
In this section we rewrite the equations in a para-differential form by means of the para-linearization formula (à la Bony see [10]).In subsection 3.1 we deal with the problem (NLS) and in the 3.2 we deal with (KG).
3.1.Para-linearization of the NLS.In the following proposition we para-linearize (NLS), with respect to the variables (u, u).We shall use the following notation throughout the rest of the paper Define the following real symbols We define also the matrix of functions with a 2 (x) and b 2 (x) defined in (3.2).We have the following.
where V is the convolution potential in (1.5), the matrix A 2 (x) is the one in (3.3), the symbol a 1 (x) • ξ is in (3.2) and the vector field X H (4) NLS (U ) is defined as follows The semi-norms of the symbols satisfy the following estimates where we have chosen s ) The remainder R(U ) has the form (R + (U ), R + (U )) T .Moreover, for any s > 2d + 2, we have the estimates Proof.By Proposition 3.3 in [28] we obtain that at the positive orders the symbols are given by Im (∂ uu x j P )ξ j , then one obtains formulae (3.2) by direct inspection by using the second line in (1.11).The estimates (3.6) are obtained as consequence of the fact that h (s) ∼ s when s goes to 0 and using Lemma 2.4.The estimate on R(U ) in (3.9) may be deduced from (2.10), (2.8), (2.12) and (3.6), for more details one can follow Proposition 3.3 in [28].Formula (3.5) is obtained by using Lemma 2.3 applied to |u| 2 u.

5) is the Hamiltonian vector field of the Hamiltonian function
are self-adjoint thanks to (2.23) and (3.2).

Para-linearization of the KG.
In this section we rewrite the equation (KG) as a paradifferential system.This is the content of Proposition 3.6.Before stating this result we need some preliminaries.In particular in Lemma 3.3 below we analyze some properties of the cubic terms in the equation (KG).Define the following real symbols We define also the matrices of symbols ) and the Hamiltonian function with G the function appearing in (1.13).First of all we study some properties of the vector field of the Hamiltonian H (4)  KG .Lemma 3.3.We have that ) T and (recall (2.52)) ) for any σ 1 , σ 2 , σ 3 ∈ {±}.Finally, for s > 2d + 1, we have ) Proof.By and explicit computation and using (1.2) we get The function g is a homogeneous polynomial of degree three.Hence, by using Lemma 2.3, we obtain iX + where and Q −ρ is a cubic smoothing remainder of the form (2.15) whose coefficients satisfy the bound (3.17).
The symbols of the the paradifferential operators have the form (using that G is a polynomial) where k, j ∈ {y 0 , y 1 } and where the coefficients We claim that the term in (3.24) is a cubic remainder of the form (3.16) with coefficients satisfying (3.17).By (2.6) we have which implies that A −1 has the form (3.16) with coefficients where We note that Then we deduce Again by Lemma 2.6 one can conclude that r σ 1 ,σ (3.29) Moreover one has |a The main result of this section is the following.Proposition 3.6.(Paralinearization of KG).The system (1.12) is equivalent to where U : KG (U ) is the Hamiltonian vector field of (3.14).
The operator R(u) has the form (R + (u), R + (u)) T .Moreover we have that where we have chosen s 0 > d .Finally there is µ > 0 such that, for any s > 2d + µ, the remainder R ≥5 (u) satisfy R(u) H s u 5 H s . (3.33) Proof.First of all we note that system (1.12) in the complex coordinates (2.41) reads with is the first component of the vector field X H (4) KG (U ) which has been studied in Lemma 3.3.By using the Bony para-linearization formula (see [10]), passing to the Weyl quantization and (1.1) we get where R −ρ (ψ) satisfies R −ρ (ψ) H s+ρ ψ 5 H s for any s ≥ s 0 > d + ρ.By Lemma 2.12 in [28], and recalling that F (ψ, ∇ψ) ∼ O(ψ 6 ), we have that where s 0 > d .Recall that ∂ x j = Op BW (ξ j ).Then, by Proposition 2.2, we have By using again Lemma 2.1 and Proposition 2.2 we get that where a 2 is in (3.11) and R(ψ) is a remainder satisfying (3.33).The symbol a 2 (x, ξ) satisfies (3.32) by (3.37).Moreover up to remainders satisfying (3.33).Here we used Proposition 2.2 to study the composition operator . By the discussion above and formula (3.34) we deduce the (3.31).Remark 3.7.In the semi-linear case, i.e. when f = 0 and g does not depend on y 1 (see (1.1), (1.2)) , the equation (3.31) reads and where the vector field X H (4) KG has the particular structure described in Remark 3.5.

DIAGONALIZATION
4.1.Diagonalization of the NLS.In this section we diagonalize the system (3.4).We first diagonalize the matrix E (1 + A 2 (x)) in (3.4) by means of a change of coordinates as the ones made in the papers [28,29].After that we diagonalize the matrix of symbols of order 0 at homogeneity 3, by means of an approximatively symplectic change of coordinates.Throughout the rest of the section we shall assume the following.Hypothesis 4.1.We restrict the solution of (NLS) on the interval of times [0, T ), with T such that for some 0 < c 0 (s) < 1.
Note that such a time T > 0 exists thanks to the local existence theorem in [28].
4.1.1.Diagonalization at order 2. We consider the matrix E (1 + A 2 (x)) in (3.4).We define and we note that ±λ NLS (x) are the eigenvalues of the matrix E (1 + A 2 (x)).We denote by S matrix of the eigenvectors of E (1 + A 2 (x)), more explicitly Since ±λ NLS (x) are the eigenvalues and S(x) is the matrix eigenvectors of E (1 + A 2 (x)) we have that where we have used the notation (3.1).In the following lemma we estimate the semi-norms of the symbols defined above.
We now study how the system (3.4) transforms under the maps where the constant C depends on s; The map as a consequence the map Φ NLS is invertible and where the constant C depends on s; . Since u solves (3.4) and satisfies Hypothesis 4.1, then using Lemma 2.1 and (3.9) we deduce that u H s u H s+2 .Hence the estimates (4.9) follow by direct inspection by using the explicit structure of the symbols s 1 , s 2 in (4.2), Lemma 2.4 and (2.10).
We are now in position to state the following proposition.with Φ NLS defined in (4.4).Then W solves the equation where the vector field X H (4) NLS is defined in (3.5).The symbols a (1)  2 and a (1)  1 • ξ are real valued and satisfy the following estimates where we have chosen s 0 > d .The remainder R (1) has the form (R (1,+) , R (1,+) ) T .Moreover, for any s > 2d +2, it satisfies the estimate R (1) (U ) Proof.The function W defined in (4.10) satisfies here we have used items (i i ) and (i i i ) of Lemma 4.3.
We are going to analyze each term in the r.h.s. of the equation above.Because of estimates (4.7), (4.5) (applied for the map Φ NLS ), Lemma 4.2 (applied for the symbols a 2 , b 2 and a 1 • ξ) and finally item (i i ) of Lemma 2.1 we may absorb term (4.18) in the remainder R (1) (U ) verifying (4.13).The term in (4.17) may be absorbed in R (1) (U ) as well because of (3.9) and (4.5) for the first addendum, because of (4.9) and item (i i ) of Lemma 2.1 for the second one.
Reasoning analogously one can prove that the term in (4.15) equals to −iOp BW diag( a 1 (U )•ξ) W , modulo contributions to R (1) (U ).We are left with studying (4.16).First of all we note that X H (4)
4.1.2.Diagonalization of cubic terms at order 0. The aim of this section is to diagonalize the cubic vector field X H (4) NLS in (4.11) (see also (3.5)) up to smoothing remainder.In order to do this we will consider a change of coordinates which is symplectic up to high degree of homogeneity.We reason as follows.Define the following frequency localization: for some 0 < < 1, where χ is defined in (2.5).Consider the matrix of symbols and the Hamiltonian function where S ξ W := (S ξ w, S ξ w) T .The presence of truncation on the high modes (S ξ ) will be decisive in obtaining Lemma A.1 (see comments in the proof of this Lemma). Let where X B NLS is the Hamiltonian vector field of (4.22).We note that Φ B NLS is not symplectic, nevertheless it is close to the flow of B NLS (W ) which is symplectic.This is a consequence of the estimates (4.5), (4.8), (A.7), (A.4), (A.9).
We introduce the following notation.We define the operator Λ NLS as the Fourier multiplier acting on periodic functions as follows: where V (ξ) are the real Fourier coefficients of the convolution potential V (x) given in (1.5).We prove the following.
where a (1)  2 (x), a ( Then we have that the equation (4.11) reads where we set Hence by (4.23) we get We study each summand separately.First of all we have that Let us now analyze the first summand in the r.h.s. of (4.32).We write where R 5 is a remainder satisfying the quintic estimate (A.11).By Lemma A.4 we also have that In this section we diagonalize the system (3.31) up to a smoothing remainder.This will be done into two steps.We first diagonalize the matrix E (1 + A 1 (x, ξ)) in (3.31) by means of a change of coordinates similar to the one made in the previous section for the (NLS) case.
After that we diagonalize the matrix of symbols of order 0 at homogeneity 3, by means of an approximatively symplectic change of coordinates.Consider the Cauchy problem associated to (KG).Throughout the rest of the section we shall assume the following.
Hypothesis 4.8.We restrict the solution of (KG) on the interval of times [0, T ), with T such that for some 0 < c 0 (s) < 1 with ψ(0, x) = ψ 0 (x) and Note that such a T exists thanks to the local well-posedness proved in [37].
Remark 4.9.Recall the (2.41).Then one can note that ψ 1. Diagonalization at order 1.Consider the matrix of symbols (see (3.11), (3.12)) Notice that the symbol λ KG (x, ξ) is well-defined by taking u H s 1 small enough.The matrix of eigenvectors associated to the eigenvalues of E (1 , By a direct computation one can check that We shall study how the system (3.31)transforms under the maps We prove the following result.
Lemma 4.10.Assume Hypothesis 4.8.We have the following: for some remainder satisfying (2.12) with a s 1 and b s 2 .Therefore the (4.45) follows by using (2.8), (2.10) and (4.43).(i v) It is similar to the proof of item (i i i ) of Lemma 4.3.
where the vector field X H (4) KG is defined in (3.15).The symbol ã+ 2 is defined in (4.39).The remainder R (1) has the form (R (1,+) , R (1,+) ) T .Moreover, for any s > 2d + µ, for some µ > 0, it satisfies the estimate R (1) where we used items (i i ), (i i i ) in Lemma 4.10.We study the first summand in the r.h.s of (4.50).By direct inspection, using Lemma 2.1 and Proposition 2.2 we get where R(u) is a remainder satisfying (4.49).Thanks to the discussion above and (4.39) we obtain the highest order term in (4.48).All the other summands in the r.h.s. of (4.50) may be analyzed as done in the proof of Prop.4.4 by using Lemma 4.10.

Diagonalization of cubic terms at order 0.
In the previous section we showed that if the function U solves (3.31) then W in (4.47) solves (4.48).The cubic terms in the system (4.48) are the same appearing in (3.31) and have the form (3.15).The aim of this section is to diagonalize the matrix of symbols of order zero A 0 (x, ξ).We must preserve the Hamiltonian structure of the cubic terms in performing this step.In order to do this, in analogy with the (NLS) case, we reason as follows.Consider the matrix of symbols with a 0 (x, ξ) in (3.11), and define the Hamiltonian function where S ξ W := (S ξ w, S ξ w) T where S ξ is in (4.20).Let us define where X B KG is the Hamiltonian vector field of ( where H (4)  KG is in (3.14), and B KG is in (4.52), (4.51).Proof.Recalling (3.14) and (the second equation in) (4.58) we define and we rewrite the equation (4.48) as Then, using (4.53), we get

.62)
By estimates (A.5) and (4.49) we have that the term in (4.62) can be absorbed in a remainder satisfying the (4.57).Consider the term in (4.61).We write modulo remainders that can be absorbed in R (2)  5 satisfying (4.57).The (4.64), (4.60)-(4.62)and the discussion above imply the (4.55)where the cubic vector field has the form where X H (4)

ENERGY ESTIMATES
5.1.Estimates for the NLS.In this section we prove a priori energy estimates on the Sobolev norms of the variable Z in (4.23).In subsection 5.1.1 we introduce a convenient energy norm on H s (T d ; C) which is equivalent to the classic H s -norm.This is the content of Lemma 5.2.In subsection 5.1.2,using the non-resonance conditions of Proposition 5.6, we provide bounds on the non-resonant terms appearing in the energy estimates.We deal with resonant interactions in Lemma 5.4.

Energy norm. Let us define the symbol
where the symbols a (1)  2 (x), a (1)  1 (x) are given in Proposition 4.4.We have the following.
Lemma 5.1.Assume the Hypothesis 4.1 and let γ > 0. Then for ε > 0 small enough we have the following.
for some C > 0 depending on s 0 .(i i ) For any s ∈ R and any h ∈ H s (T d ; C), one has for some C > 0 depending on s.
The operators T L , T L γ are self-adjoint with respect to the L 2 -scalar product (2.3).
In the following we shall construct the energy norm.By using this norm we are able to achieve the energy estimates on the previously diagonalized system.For s ∈ R we define for some constant C depending on s.The discussion above implies the (5.6) by taking ε > 0 in Hyp.4.1 small enough.
(5.13)By differentiating (5.5) and using the (5.1) and (5.7) we get By using Lemmata 2.1, 5.1 and Proposition 2.2, and the (5.6), (4.24) one proves that the last summand gives a contribution to R 5,n (U ) satisfying (5.12).By using (5.4), (4.24), (4.28) we deduce that Secondly we write By (5.3), (4.24), and recalling (1.5) we conclude H s .We now study the third summand in (5.14).We have (see (5.13)) By (5.3), (4.27), (2.10), Lemma 2.5 and using the estimate (A.18), one obtains Recalling (4.37) and (5.13) we write We now consider the operator C 1 with coefficients c 1 (ξ, η, ζ).First of all we remark that it can be written as C 1 = M (z, z, z) where M is a trilinear operator of the form (2.25).Moreover, setting we can write C 1 = B (1)  n (Z ) − M (z, z, h n ) , where B (1)  n has the form (5.9) with coefficients as in (5.10).Using that |c 1 (ξ, η, ζ)| 1, Lemma 2.5 (with m = 0) and (5.3) we deduce that M (z, z, h n ) L 2 u 9 H s .Therefore this is a contribution to R 5,n (U ) satisfying (5.12).The discussion above implies formula (5.8) by setting B (2)  n as the operator of the form (5.9) with coefficients b (2)    In the following lemma we prove a key cancellation due to the fact that the super actions are prime integrals of the resonant Hamiltonian vector field X +,res H 4 (Z ) in the same spirit of [24].We also prove an important algebraic property of the operator B (1)  n in (5.8).
Proof.The (5.16) follows by Lemma 2.8.Let us check the (5.17).By an explicit computation using (2.3), (5.9) we get Re(B (1) By (5.10) we have where we used the form of the resonant set R in (2.47).This proves the lemma.
We conclude the section with the following proposition.

Estimates of non-resonant terms.
In this subsection we provide estimates on the term B(t ) appearing in (5.18).Proposition 5.6.(Non-resonance conditions).Consider the phase ω NLS (ξ, η, ζ) defined as where Λ NLS is in (4.25) and the potential V is in (1.5).We have the following.
There exists N ⊂ O with zero Lebesgue measure such that, for any Proof.Item (i ) follows by Proposition 2.8 in [25].Item (i i ) is classical.
We are now in position to state the main result of this section.

5.34)
There is s 0 (N 0 ) > 0 (N 0 > 0 given by Proposition 5.6) such that for s ≥ s 0 (N 0 ) one has (5.36) Proof.Using (5.32), (5.19), (5.11) we get that Then, by reasoning as in the proof of Lemma 2.5, one obtains the (5.34).Let us prove the bound (5.35) for p = 0, the others are similar.Using (5.33), (5.26), (5.19), (5.11) we have Again, by reasoning as in the proof of Lemma 2.5, one obtains the (5.35).The (5.36)  Consider now the first summand in the r.h.s. of (5.37).We claim that we have the following identity: (5.39) We use the claim, postponing its proof.Consider the first summand in the r.h.s. of (5.39).Using the self-adjointness of T |ξ| 2 and the (5.7) we write We estimate the first summand in the r.h.s. by means of the Cauchy-Schwarz inequality, the (5.35) with p = 2 and the (5.3); analogously we estimate the second summand by means of the Cauchy-Schwarz inequality, (5.36), the (4.27) and the (4.28), obtaining The other terms in (5.39) are estimated in a similar way.We eventually obtain the (5.28).
We now prove the claim (5.39).Recalling (5.7) we have that We define g (ξ) := e it Λ NLS (ξ) z(ξ), ∀ξ ∈ Z d .One can note that g (ξ) satisfies (5.40) According to this notation and using (5.29) and (5.25) we have By integrating by parts in σ and using (5.40) one gets the (5.39) with The remainder above is bounded from above by u 4 L ∞ H s using Cauchy-Schwarz and the (5.36).

5.2.
Estimates for the KG.In this section we provide a priori energy estimates on the variable Z solving (4.55).This implies similar estimates on the solution U of the system (3.31)thanks to the equivalence (4.54).In subsection 5.2.1 we introduce an equivalent energy norm and we provide a first energy inequality.This is the content of Proposition 5.11.Then in subsection 5.2.2 we give improved bounds on the non-resonant terms.
5.2.1.First energy inequality.We recall that the system (4.55) is diagonal up to smoothing terms plus some higher degree of homogeneity remainder.Hence, for simplicity, we pass to the scalar equation where (recall (4.56)) X + For n ∈ R we define (5.42) We have the following.
Lemma 5.9.Fix n := n(d ) 1 large enough and recall (5.41).One has that the function z n defined in (5.42) solves the problem KG (Z ) + B (1)  n (Z ) + B (2)  n (Z ) + R 5,n (U ) , (5.43) where the resonant vector field X +,res KG is defined as in Def.2.7 (see also Rmk. 2.9), the cubic terms B (i ) n , i = 1, 2, have the form B (1)  n (Z which represents the non resonant terms in the cubic vector field of (5.41).By differentiating in t the (5.42) and using the (5.41) we get KG (Z ) (5.51) We analyse each summand above separately.First of all we remark that we have the equivalence between the two norms (see with B (1)  n (Z ) as in (5.44) and coefficients as in (5.46), the term C 1 has the form and the term C 2 has the form (5.45) with coefficients (see (A.23)) (5.55) In order to conclude the proof we need to show that the coefficients in (5.54), (5.55) satisfy the bound (5.47).This is true for the coefficients in (5.55) thanks to the bound (A.24).Moreover notice that Then the coefficients in (5.54) satisfy (5.47) by using Remark 3.4 and Lemma 2. In view of Lemma 5.9 we deduce the following.Proposition 5.14.Let N > 0 and let β be as in Proposition 5. 13.Then there is s 0 = s 0 (N 0 ), where N 0 > 0 is given by Proposition 5.13, such that, if Hypothesis 4.8 holds with s ≥ s 0 , one has where B(t ) is in (5.57).
Proof.The proof is similar to the one of Lemma 5.8.
Remark 5.16.In view of Remark 5.12, if (KG) is semi-linear we may improve (5.75) with (5.76) We are now in position to prove the main Proposition 5.14.
Proof of Proposition 5.14.By (5.68), (5.70), (5.71), and recalling the definition of B in (5.57), we can write Proof.For ε small enough the bound (6.1) holds true, and we fix N := ε −3 .Therefore, there is C = C (s) > 0 such that, for any t ∈ [0, T ), where in the last inequality we have chosen c 0 and ε sufficiently small.This implies the thesis.
Proof of Theorem 2. One has to follow almost word by word the proof of Theorem 1.The only difference relies on the estimates on the small divisors which in this case are given by item (i i ) of Proposition 5.6.
Proof of Theorem 3. Consider (KG) and let (ψ 0 , ψ 1 ) as in the statement of Theorem 3. Let ψ(t , x) be a solution of (KG) satisfying the condition in Hyp.4.8.By Proposition 3.6, recall (2.41), the function . Moreover, by Hyp.4.8 one has sup t ∈[0,T ) u H s ≤ ε.By Remark 4.9, in order to get the (1.8), we have to show that the bound on the function u above holds for a longer time T ε −3 + if d = 2 and T ε −8/3 + if d ≥ 3. Fix β as in Proposition 5.13 and let m ∈ C β .By Propositions 4.11, 4.13 and Lemma 5.9 we can construct a function z n with n = s such that if ψ(t , x) solves the (KG) then z n solves the equation (5.43).By Proposition 5.11 we get Propositions 5.11 and 5.14 apply, therefore, by (5.67) and (5.59), we obtain the following a priori estimate: fix any 0 < N , then for any t ∈ [0, T ), with T as in Hyp.4.8, one has ) Proof.We start with d ≥ 3.For ε small enough the bound (6.6) holds true.Let 0 < σ 1. Define By (6.6), (6.7), (6.8), there is C = C (s) > 0 such that, for any t ∈ [0, T ), where we recall (4.20).By (4.21) we obtain that ∇ w G 1 (W ) = −iOp BW (b NLS (S ξ w))w.We compute the gradient with respect w of the term G 2 (W ).We have Recalling (2.33) and the computations above, after some changes of variables in the summations, we obtain where the remainder R 1 (W ) has the form (R + 1 (W ), R + 1 (W )) T where (recall (2.5) One can check, for 0 < < We write 1 • r 2 (ξ, η, ζ) and we use the partition of the unity in (2.16).Hence using the (2.5) one can check that each summand satisfies the bound in (2.15).Therefore the operator Q G := R 1 +R 2 has the form (A.2) and (A.1) is proved.The estimates (A.3) follow by Lemma 2.5.We note that Then the estimates (A.4) with k = 0, 1, follow by using (A.3), the explicit formula of B (W ; x, ξ) in (4.21) and Lemma 2.1.Reasoning similarly one can prove the (A.4) with k = 2, 3.
In the next proposition we define the changes of coordinates generated by the Hamiltonian vector fields X B NLS and X B KG and we study their properties as maps on Sobolev spaces.Proposition A.2.For any s ≥ s 0 > 2d + 2 there is r 0 > 0 such that for 0 ≤ r ≤ r 0 , the following holds.Define where R + 3 is in (A.28).By the discussion above and by Lemma 3.3 we have that the remainders R 2 , R 4 and Q 3 have the form (A.23) with coefficients satisfying (A.24).To conclude the prove we need to show that F 3 has the same property.This will be a consequence of the choice of the symbol b KG (W ; x, ξ) in (4.51).Indeed, by (4.51), Remark 3.4, (A.31), (A.28), we have where By Taylor expanding the symbol Λ KG (ξ) in (1.4) (see also Remark 3.4) one deduces that Therefore, using Lemma 2.6, we have that the coefficients f σ 1 ,σ 2 ,− 3 (ξ, η, ζ) in (A.32) satisfy the (A.24).This implies the (A.22).

APPENDIX B. NON-RESONANCE CONDITIONS FOR (KG)
In this section we prove Proposition 5.13 providing lower bounds on the phase in (5.25).Recall the symbol Λ KG ( j ) in (1.4).Throughout this subsection, in order to lighten the notation, we shall write Λ KG ( j ) Λ j for any j ∈ Z d .The main result of this section is the following.The Proposition B.1 will implies Proposition 5.6.Its proof is done in three steps.
We are now in position to prove the main result of this subsection.
Proof of Proposition B.7.Let g be the rational function introduced in Lemma B.9.We write, with σ =

Theorem 4 .
Consider (KG) with f = 0 and g independent of y 1 .Then the same results of Theorem 3 holds true for T ≥ c 0 ε −a + , with a = 4 if d = 2, and a = 10/3 if d ≥ 3.

Lemma 2 . 6 .
Let µ ≥ 0 and m ∈ R. Consider a trilinear map Q as in (2.25) with coefficients satisfying q

. 4 ) 4 . 3 .
Lemma Let U = u u be a solution of (3.4) and assume Hyp.4.1.Then for any s ≥ 2s 0 + 2, N s 0 > d , we have the following.(i ) One has the upper bound

. 6 )
The thesis of Theorem 3 follows from the following lemma.Lemma 6.2.(Main bootstrap).Let u(t , x) be a solution of (3.31) with t ∈ [0, T ) and initial condition u 0 ∈ H s (T d ; C).Define a = 3 if d = 2 and a = 8/3 if d ≥ 3.Then, for s 1 large enough, there exist ε 0 , c 0 > 0 such that, for any 0
4 ∈ Z d satisfy | j 1 | > | j 2 | > | j 3 | > | j 4 |.First of all, by reasoning as in Lemma 3.2 in [23], one can deduce the following.Lemma B.3.Consider the matrix D whose entry at place (p, q) is given by d p d m p Λ j q , p, q = 1, . . ., 4. The modulus of the determinant of D is bounded from below: one has |det(D)| ≥ C | j 1 | −µ where C > 0 and µ > 0 are universal constants.
, (2.6); X + H 4 (Z ) is the first component of the Hamiltonian vector-field X H 4 (Z ) and R n (U ) is a bounded remainder satisfying the quintic estimate (5.12).CANCELLATIONS AND NORMAL-FORMS.At this point, always in Lemma 5.3, we split the Hamiltonian vector-field X + H 4 = X +,res The first important fact, which is an effect of the Hamiltonian and Gauge preserving structure, is that the resonant term ∆ n X +,res Actually to prove that such a remainder has the form (3.16) with coefficients (3.17) it is more convenient to compute the composition operator explicitly.In particular, recalling (2.6), we get .26) By Lemma 2.6 we have that the coefficients in (3.26) satisfy(3.17).This prove the claim for the operator A −1 .We now study the term in(3.23).We remark that, by Proposition 2.2 (see the composition formula (2.11)), we have that A −1/2 = Op BW (Λ − 1 2 KG (ξ)∂ y 0 y 1 G) up to a smoothing operator of order −3/2.
We remark that the symbol a 0 (x, ξ) in(3.11) is homogenenous of degree two in the variables u, u.In particular, by(3.25),we have U A 0 (x, ξ)h| N 0 p u H p+s 0 h H p+s 0 , p + s 0 ≤ s .Then the second summand in (3.28) verify the bound (3.20) again by Lemma 2.1.The estimate on the third summand in (3.28) follows by (3.16), (3.17) and Lemma 2.5.Remark 3.4.
Then by(2.8),(3.37)and(2.10)(seeLemma 2.1 and Proposition 2.2) we deduce that the terms in (3.36) can be absorbed in a remainder satisfying (3.33) with s 2d large enough.We now consider the first term in the r.h.s.of(3.35).We have We apply Proposition 2.2 to the maps in(4.4), in particular the first part of the item follows by using the expansion (2.13) and recalling that symbols s 1 (x) and s 2 (x) do not depend on ξ.The (4.7) is obtained by Neumann series by using that (see Hyp. 4.1) u H s 1.
.9)Proof.(i ) The bounds (4.5) follow by (2.10) and Lemma 4.2.(i i ) The properties of X B NLS and the estimates of Φ B NLS are discussed in Lemma A.1 and in Proposition A.2.
Remark 4.5.Recall (4.10) and (4.23).One can note that, owing to Hypothesis 4.1, for s > 2d + 2, we have U H s ∼ s W H s ∼ s Z H s .(4.24) .46) Proof.(i ) The (4.43) follows by (3.32) using the explicit formulae (4.40), (4.39).(i i ) It follows by using (4.43) and item (i i ) in Lemma 2.1.(i i i ) By formula (2.11) in Proposition 2.2 one gets 4.52) and W is the function in (4.47).The properties of X B KG and the estimates of Φ B KG are discussed in Lemma A.1 and in Proposition A.2.
Propositions 5.5 and 5.7 apply, therefore, by (5.28) and (5.20), we obtain the (6.1).The thesis of Theorem 1 follows from the following lemma.Let u(t , x) be a solution of (NLS) with t ∈ [0, T ) and initial condition u 0 ∈ H s (T d ; C).Then, for s 1 large enough, there exist ε 0 , c 0 > 0 such that, for any 0