A Stingo

Global existence of small amplitude solutions for a model quadratic quasi-linear coupled wave-Klein-Gordon system in two

Global existence of small amplitude solutions for a model quadratic quasi-linear coupled wave-Klein-Gordon system in two space dimension, with mildly decaying Cauchy data

Annalaura Stingo

To cite this version:

Annalaura Stingo. Global existence of small amplitude solutions for a model quadratic quasi-linear coupled wave-Klein-Gordon system in two space dimension, with mildly decaying Cauchy data. 2018.

Contents

Introduction

The result we present in this paper concerns the global existence of solutions to a quadratic quasi-linear coupled system of a wave equation and a Klein-Gordon equation in space dimension two, when initial data are small smooth and mildly decaying at infinity. We prove this result for a model non-linearity with the aim of extend it, in the future, to the most general case. Keeping this long term objective in mind, we shall try to develop a fairly general approach in spite of the fact that we are treating here a simple model. The Cauchy problem we consider is the following

(1) (∂ 2 t -∆ x )u(t, x) = Q 0 (v, ∂ 1 v) , (∂ 2 t -∆ x + 1)v(t, x) = Q 0 (v, ∂ 1 u) , (t, x) ∈]1, +∞[×R 2
with initial conditions

(2) (u, v)(1, x) = ε(u 0 (x), v 0 (x)) , (∂ t u, ∂ t v)(1, x) = ε(u 1 (x), v 1 (x)) ,
where ε > 0 is a small parameter, and Q 0 is the null form:

Q 0 (v, w) = (∂ t v)(∂ t w) -(∇ x v) • (∇ x w) .
We also suppose that, for some n ∈ N sufficiently large, (∇ x u 0 , u 1 ) is in the unit ball of H n (R 2 , R) × H n (R 2 , R), (v 0 , v 1 ) in the unit ball of H n+1 (R 2 , R) × H n (R 2 , R), and that

(3)

1≤|α|≤3 x α ∇ x u 0 H |α| + x α v 0 H |α|+1 + x α u 1 H |α| + x α v 1 H |α| ≤ 1.
Some physical models, especially related to general relativity, have shown the importance of studying such systems to which several recent works have been dedicated. Most of the results known at present concern wave-Klein-Gordon systems in space dimension 3. One of the first ones goes back to Georgiev [START_REF] Georgiev | Global solution of the system of wave and Klein-Gordon equations[END_REF]. He observed that the vector fields' method developed by Klainerman was not well adapted to handle at the same time massless and massive wave equations because of the fact that the scaling vector field S = t∂ t + x • ∇ x is not a Killing vector field for the Klein-Gordon equation. To overcome this difficulty he adapted Klainerman's techniques, introducing a strong null condition to be satisfied by semi-linear nonlinearities that ensures global existence. In 2012 Katayama [START_REF] Katayama | Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions[END_REF] showed the global existence of small amplitude solutions to coupled systems of wave and Klein-Gordon equations under certain suitable conditions on the non-linearity that include the null condition of Klainerman ([19]) on self-interactions between wave components, and are weaker than the strong null condition of Georgiev. Consequently, the result he obtains applies also to certain other physical systems such as Dirac-Klein-Gordon equations, Dirac-Proca equations and Klein-Gordon-Zakharov equations. Later, this problem was also studied by LeFloch, Ma [START_REF] Lefloch | of Series in Applied and Computational Mathematics[END_REF] and Wang [START_REF] Wang | Global existence for the Einstein equations with massive scalar fields[END_REF] as a model for the full Einstein-Klein-Gordon system (E-KG)

Ric αβ = D α ψD β ψ + 1 2 ψ 2 g αβ g ψ = ψ
The authors prove global existence of solutions to wave-Klein-Gordon systems with quasi-linear quadratic non-linearities satisfying suitable conditions, when initial data are small, smooth and compactly supported, using the so-called hyperboloidal foliation method introduced by Le Foch, Ma in [START_REF] Lefloch | of Series in Applied and Computational Mathematics[END_REF]. Global stability for the full (E-KG) has been then proved by LeFloch-Ma [START_REF] Lefloch | The global nonlinear stability of Minkowski space for selfgravitating massive fields. The Wave-Klein-Gordon model[END_REF][START_REF] Lefloch | The global nonlinear stability of Minkowski space for selfgravitating massive fields[END_REF] in the case of small smooth perturbations that agree with a Scharzschild solution outside a compact set (see also Wang [30]). In a recent paper [START_REF] Ionescu | On the global regularity for a wave-Klein-Gordon coupled system[END_REF] Ionescu and Pausader prove global regularity and modified scattering in the case of small smooth initial data that decay at suitable rates at infinity, but not necessarily compactly supported. The quadratic quasi-linear problem they deal with is the following

-u = A αβ ∂ α v∂ β v + Dv 2 -( + 1)v = uB αβ ∂ α ∂ b v
where A αβ , B αβ , D are real constants. The system keeps the same linear structure as (E-KG) in harmonic gauge, but only keeps quadratic non-linearities that involve the massive scalar field v (semilinear in the wave equation, quasi-linear in the Klein-Gordon equation). Moreover, the non-linearity they consider does not present a null structure but shows a particular resonant pattern. Their result relies, on the one hand, on a combination of energy estimates to control high Sobolev norms and weighted norms involving the admissible vector fields; on the other hand, on a Fourier analysis, in connection with normal forms and analysis of resonant sets, to prove dispersive estimates and decay in suitable lower regularity norms. The only results we know about global existence of small amplitude solutions in lower space dimension are due to Ma. In space dimension 2 he considers the case of compactly supported Cauchy data and adapts the hyperboloidal foliation method mentioned above to 2 + 1 spacetime wave-Klein-Gordon systems (see [START_REF] Ma | Global solutions of quasilinear wave-Klein-Gordon system in two-space dimension: technical tools[END_REF]). In particular, in [START_REF] Ma | Global solutions of quasilinear wave-Klein-Gordon system in two-space dimension: completion of the proof[END_REF] he combines this method with a normal form argument to treat some quasi-linear quadratic non-linearities, while in [START_REF] Ma | Global solutions of non-linear wave-Klein-Gordon system in two space dimension: semi-linear interactions[END_REF] he studies the case of some semi-linear quadratic interactions. In a very recent paper [START_REF] Ma | Global solutions of non-linear wave-Klein-Gordon system in one space dimension[END_REF] he also tackles the one-dimensional problem, studying a model semi-linear cubic wave-Klein-Gordon system. In this work he finally overcomes the restriction on the support of initial data and generalizes the hyperboloidal foliation method, combining the hyperboloidal foliation of the translated light cone with the standard time-constant foliation outside of it. The analysis of the problem and the deduction of the estimates of interest is then made separately inside and outside the mentioned light cone.

The result we prove in this paper is the following:

Theorem 1. There exists ε 0 > 0 such that for any ε ∈]0, ε 0 [, system [START_REF] Alazard | Sobolev estimates for two dimensional gravity water waves[END_REF] with initial data satisfying (2), (3) admits a unique global solution defined on [1, +∞[, with ∂ t,x u ∈ C 0 ([1, +∞[;

H n (R 2 )) and (v, ∂ t v) ∈ C 0 ([1, +∞[; H n+1 (R 2 ) × H n (R 2 )).
We describe below the strategy of the theorem's proof. First of all, we rewrite system (1) in terms of unkowns (4)

u ± = (D t ± |D x |) u, v ± = (D t ± D x ) v,
where D t,x = -i∂ t,x , and introduce the admissible Klainerman vector fields for this problem, i.e.

Ω = x 1 ∂ 2 -x 2 ∂ 1 , Z j = x j ∂ t + t∂ j , j = 1, 2.
We also denote by Z = {Γ 1 , . . . , Γ 5 } the family made by above vector fields together with the two spatial derivatives, and if I = (i 1 , . . . , i p ) is an element of {1, . . . , 5} p , Γ I w is the function obtained letting Γ i 1 , . . . , Γ ip act successively on w. We then set (5)

u I ± = (D t ± |D x |) Γ I u, v I ± = (D t ± D x ) Γ I v,
and introduce the following energies:

E 0 (t; u ± , v ± ) = R 2 |u + (t, x)| 2 + |u -(t, x)| 2 + |v + (t, x)| 2 + |v -(t, x)| 2 dx, then for n ≥ 3, E n (t; u ± , v ± ) = |α|≤n E 0 (t; D α x u ± , D α x v ± ),
which controls the H n regularity of u ± , v ± , and finally, for any integer k between 0 and 2,

E k 3 (t; u ± , v ± ) = |α|+|I|≤3 |I|≤3-k E 0 (t; D α x u I ± , D α x v I ± )
that takes into account the decay in space of u ± , v ± and of at most three of their spatial derivatives. By a local existence argument, an a-priori estimate on E n on a certain time interval will be enough to ensure the extension of the solution to that interval. For this reason, we are led to prove a result as the following one, in which R = (R 1 , R 2 ) denotes the Riesz transform:

Theorem 2. Let K 1 , K 2 two constants strictly bigger than 1. There exist two integers n ρ 1, ε 0 ∈]0, 1[ small enough, some small real 0 < δ δ 2 δ 1 δ 0 1 and two constants A, B sufficiently large such that, if functions u ± , v ± , defined by (4) from a solution to (1), satisfy

D x ρ+1 u ± (t, •) L ∞ + D x ρ+1 Ru ± (t, •) L ∞ ≤ Aεt -1 2 D x ρ v ± L ∞ ≤ Aεt -1 E n (t; u ± , v ± ) ≤ B 2 ε 2 t 2δ E k 3 (t; u ± , v ± ) ≤ B 2 ε 2 t 2δ k , 0 ≤ k ≤ 2, (6)
for every t ∈ [1, T ], then on the same interval [1, T ] we have

D x ρ+1 u ± (t, •) L ∞ + D x ρ+1 Ru ± (t, •) L ∞ ≤ A K 1 εt -1 2 D x ρ v ± L ∞ ≤ A K 1 εt -1 E n (t; u ± , v ± ) ≤ B 2 K 2 2 ε 2 t 2δ E k 3 (t; u ± , v ± ) ≤ B 2 K 2 2 ε 2 t 2δ k , 0 ≤ k ≤ 2. (7)
The proof of the theorem consists, on the one hand, to prove that [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF] implies the latter two estimates in [START_REF] Delort | Semiclassical microlocal normal forms and global solutions of modified onedimensional KG equations[END_REF] by means of an energy inequality. On the other hand, by reduction of the starting problem to a coupled system of an ordinary differential equation and a transport equation, we prove that [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF] implies the first two estimates in [START_REF] Delort | Semiclassical microlocal normal forms and global solutions of modified onedimensional KG equations[END_REF].

In order to recover the mentioned energy inequality that allows us to propagate the a-priori energy estimates, we let family Γ I of vector fields act on [START_REF] Alazard | Sobolev estimates for two dimensional gravity water waves[END_REF] and then pass to unknowns [START_REF] Delort | Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle[END_REF]. We obtain a new system of the form

(D t ∓ |D x |)u I ± = NL w (v I ± , v I ± ) (D t ∓ |D x |)v I ± = NL kg (v I ± , u I ± )
where the non-linearities (whose explicit expression may be found in the right hand side of (2.1.2)) are bilinear quantities of their arguments. Because of the quasi-linear nature of our problem, the first step towards the derivation of the mentioned inequality is to highlight the very quasi-linear contribution to above non-linearities and make sure that it does not lead to a loss of derivatives. For this reason, we write the above system in a vectorial fashion by introducing vectors

U I =     u I + 0 u I - 0     , V I =     0 v I + 0 v I -     , W I = U I + V I ,
and successively para-linearize the vectorial equation satisfied by W I (using the tools introduced in subsection 1.2.1) to stress out the quasi-linear contribution to the non-linearity. Finally, we symmetrize it (in the sense of subsection 2.1.3) by introducing some new unknown W I s comparable to W I . What we would need to show in order to prove the last two inequalities in [START_REF] Delort | Semiclassical microlocal normal forms and global solutions of modified onedimensional KG equations[END_REF] is that, using the estimates in [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF], the derivative in time of the L 2 norm to the square of W I s is bounded by Cε t W I L 2 . By analysing the semi-linear contributions in the symmetrized equation satisfied by W I s , we find out that the L 2 norm of some of those ones can only be estimated making appear the L ∞ norm on the wave factor and the L 2 norm on the Klein-Gordon one. Because of the very slow decay in time of the wave solution (the decay rate being t -1/2 , as assumed in the first inequality of ( 6)), we are hence very far away from the wished estimate. Consequently, the second step for the derivation of the right energy inequality consists in performing a normal form argument to get rid of those quadratic terms and replace them with cubic ones. For that, we first use a Shatah' normal form adapted to quasi-linear equations (see subsection 2.2.1) as already used by several authors (we cite [START_REF] Ozawa | Remarks on the Klein-Gordon equation with quadratic nonlinearity in two space dimensions[END_REF][START_REF] Delort | Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle[END_REF][START_REF] Delort | A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on S 1[END_REF][START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF] for quasi-linear Klein-Gordon equations, and [START_REF] Hunter | Long time solutions for a Burgers-Hilbert equation via a modified energy method[END_REF][START_REF] Hunter | Two dimensional water waves in holomorphic coordinates[END_REF][START_REF] Ionescu | Global solutions for the gravity water waves system in 2d[END_REF][START_REF] Alazard | Sobolev estimates for two dimensional gravity water waves[END_REF][START_REF] Ifrim | Two dimensional water waves in holomorphic coordinates II: Global solutions[END_REF] for quasi-linear equations arising in fluids mechanics), but also a semilinear normal form argument to treat some other terms on which we are allowed to lose some derivatives (see subsection 2.2.2). These two normal forms' steps lead us to define some new energies E n (t; u ± , v ± ), E k 3 (t; u ± , v ± ), equivalent to the starting ones E n (t; u ± , v ± ), E k 3 (t; u ± , v ± ), that we are able to propagate. That concludes the first part of the proof.

The last thing that remains to prove is that [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF] implies the first two estimates in [START_REF] Delort | Semiclassical microlocal normal forms and global solutions of modified onedimensional KG equations[END_REF]. The strategy we employ is very similar to the one developed in [START_REF] Stingo | Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data:46[END_REF]: we deduce from the starting system (1) a new coupled one of an ordinary differential equation, coming from the Klein-Gordon equation, and of a transport equation, derived from the wave one. The study of this system will provide us with the wished L ∞ estimates. We start our analysis by another normal form in order to replace almost all quadratic non-linear terms in the equations satisfied by u ± , v ± with cubic ones. The only contributions that cannot be eliminated are those depending on (v + , v -) which are resonant and should be suitably treated. We do not use directly the normal forms obtained in the previous step. In fact, our aim is basically to obtain an L ∞ estimate for at most ρ derivatives of the solution, having a control on their H s norm for s ρ. This permits us to lose some derivatives in the normal form reduction, so the fact that the system is quasi-linear is no longer important.

We define two new unknowns u N F , v N F by adding some quadratic perturbations to u -, v -, in such a way that they are solution to [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]

(D t + |D x |)u N F = q w + c w + r N F w , (D t + |D x |)v N F = r N F kg ,
where r N F w , c w , r N F kg are cubic terms, whereas q w is the mentioned bilinear expression in v + , v - that cannot be eliminated by normal forms but whose structure will successively provide us with remainder terms. Then, if we define [START_REF] Georgiev | Global solution of the system of wave and Klein-Gordon equations[END_REF] u(t, x) = tu N F (t, tx), v(t, x) = tv N F (t, tx), and introduce h := t -1 the semi-classical parameter, we obtain that u, v verify

(D t -Op w h (x • ξ -|ξ|)) u = h -1 q w (t, tx) + c w (t, tx) + r N F w (t, tx) (D t -Op w h (x • ξ -ξ )) v = h -1 r N F kg (t, tx) (10)
where Op w h is the Weyl quantization introduced, along with the semi-classical pseudo-differential calculus, in subsection 1.2.1. We also consider the following operators

M j = 1 h x j |ξ| -ξ j , L j = 1 h x j - ξ j ξ ,
whose symbols are given respectively (up to the multiplication by |ξ| for the former case) by the derivative with respect to ξ of symbols x • ξ -|ξ| and x • ξ -ξ in [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF]. Using the equation satisfied by u N F (resp. v N F ), we can express M j u (resp. L j v) in terms of Z j u N F (resp. Z j v N F ) and of q w , c w , r N F w (resp. r N F kg ). As done in [START_REF] Stingo | Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data:46[END_REF], we first introduce the lagrangian

Λ kg = (x, ξ) : x - ξ ξ = 0
which is the graph of ξ = -dφ(x), with φ(x) = 1 -|x| 2 , and decompose v into the sum of a contribution micro-localised on a neighbourhood of size √ h of Λ kg , and another one microlocalised out of that neighbourhood (in the spirit of [START_REF] Ifrim | Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension[END_REF]). The second contribution can be basically estimated in L ∞ by h 1 2 -0 times the L 2 norm of some iterates of operator L acting on v (which are controlled by the L 2 hypothesis in theorem 2). The main contribution to v is then represented by v Λ kg , which appears to be solution to

[D t -Op w h (x • ξ -ξ )] v Λ kg = controlled terms.
Developing the symbol in the above left hand side on Λ kg we finally obtain the wished ODE, which combined with the a-priori estimate of the "controlled terms" allows us to deduce from [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF] the second estimate in [START_REF] Delort | Semiclassical microlocal normal forms and global solutions of modified onedimensional KG equations[END_REF] (with ρ = 0, the general case being treated in the same way up to few more technicalities).

The same strategy is employed to obtain some uniform estimates on u. We introduce the lagrangian

Λ w = (x, ξ) : x - ξ |ξ| = 0
which, differently from Λ kg , is not a graph but projects on the basis as an hypersurface. For this reason, the classical problem associated to the first equation in [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF] is rather a transport equation than an ordinary differential equation. It is obtained in a similar way by decomposing u into two contributions: one denoted by u Λw and micro-localised in a neighbourhood of size h 1 2 -σ (for some small σ > 0) of Λ w ; another one micro-localised away from this neighbourhood. As for the Klein-Gordon component, this latter contribution can be easily controlled thanks to the L 2 estimates that the last two inequalities in [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF] infer on the iterates of M j acting on u. By micro-localisation we derive that u Λw satisfies

[D t -Op w h (x • ξ -|ξ|)] u Λw = controlled terms,
and by developing symbol x • ξ -|ξ| on Λ w we obtain the wished transport equation. Integrating this equation by the method of characteristics, we finally recover the first estimate in [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF] and conclude the proof of theorem 2.

Chapter 1

Main Theorem and Preliminary Results

Statement of the main theorem

Notations: We warn the reader that, throughout the paper, we will often denote ∂ t (resp. ∂ x j , j = 1, 2) by ∂ 0 (resp. ∂ j , j = 1, 2), while symbol ∂ without any subscript will stand for one of the three derivatives ∂ a , a = 0, 1, 2. ∇ x f is the classical spatial gradient of f , D := -i∂ and R j denotes the Riesz operator D j |D x | -1 , for j = 1, 2. We will also employ notation ∂ t,x w with the meaning ∂ t w + ∂ x w and Rw = j R j w .

We consider the following quadratic, quasi-linear, coupled wave-Klein-Gordon system (1.1.1)

(∂ 2 t -∆ x )u(t, x) = Q 0 (v, ∂ 1 v) , (∂ 2 t -∆ x + 1)v(t, x) = Q 0 (v, ∂ 1 u) , (t, x) ∈]1, +∞[×R 2
with initial conditions (1.1.2) (u, v)(1, x) = ε(u 0 (x), v 0 (x)) ,

(∂ t u, ∂ t v)(1, x) = ε(u 1 (x), v 1 (x)) ,
where ε > 0 is a small parameter, and Q 0 is the null form:

(1.1.3)

Q 0 (v, w) = (∂ t v)(∂ t w) -(∇ x v) • (∇ x w) .
Our aim is to prove that there is a unique solution to Cauchy problem (1.1.1)-(1.1.2) provided that ε is sufficiently small and u 0 , v 0 , u 1 , v 1 decay rapidly enough at infinity. The theorem we are going to demonstrate is the following:

Theorem 1.1.1 (Main Theorem). There exist an integer n sufficiently large and ε 0 ∈]0, 1[ sufficiently small such that, for any ε ∈]0, ε 0 [, any real valued u 0 , v 0 , u 1 , v 1 satisfying:

(1.1.4)

∇ x u 0 H n + v 0 H n+1 + u 1 H n + v 1 H n ≤ 1, 2 |α|=1 ( x α ∇ x u 0 H |α| + x α v 0 H |α|+1 + x α u 1 H |α| + x α v 1 H |α| ) ≤ 1,
system (1.1.1)-(1.1.2) admits a unique global solution (u, v) with

∂ t,x u ∈ C 0 [1, ∞[; H n (R 2 ) , v ∈ C 0 [1, ∞[; H n+1 (R 2 ) ∩ C 1 [1, ∞[; H n (R 2 ) .
The proof of the main theorem is based on the introduction of four new functions u + , u -, v + , v -, defined in terms of u, v as follows:

(1.1.5)

u + := (D t + |D x |)u , u -:= (D t -|D x |)u , v + := (D t + D x )v , v -:= (D t -D x )v ,
and on the propagation of some a-priori estimates made on them in some interval [1, T ], for a fixed T > 1. In order to state this result we consider the admissible Klainerman vector fields for the wave-Klein-Gordon system:

(1. 1.6)

Ω := x 1 ∂ 2 -x 2 ∂ 1 , Z j := x j ∂ t + t∂ j , j = 1, 2
and denote by Γ a generic vector field in Z = {Ω, Z j , ∂ j , j = 1, 2}. If Z is assumed ordered, i.e.

(1.1.7) Z = {Γ 1 , . . . , Γ 5 }

with Γ 1 = Ω, Γ j = Z j-1 for j = 2, 3, Γ j = ∂ j-3 for j = 4, 5, then for a multi-index I = (i 1 , . . . , i n ), i j ∈ {1, . . . , 5} for j = 1, . . . , n, we define the length of I as |I| := n, and Γ I := Γ i 1 • • • Γ in the product of vector fields Γ i j ∈ Z, j = 1, . . . , n.

Vector fields Γ have two relevant properties: they act like derivations on non-linear terms; they exactly commute with the linear part of both wave and Klein-Gordon equation. This is the reason why we exclude of our consideration the scaling vector field S = t∂ t + j x j ∂ j , which is always considered in the so-called Klainerman vector fields' method for the wave equation, as it does not commute with the Klein-Gordon operator.

We also introduce the energy of (u + , u -, v + , v -) at time t ≥ 1 as (

) := |α|+|I|≤3 |I|≤3-k E 0 (t; D α x u I ± ; D α x v I ± ), 0 ≤ k ≤ 2,
where, for any multi-index I,

(1.1.10)

u I ± := (D t ± |D x |)Γ I u, v I ± := (D t ± D x )Γ I v.
Energy E n (t; u ± , v ± ), for n ≥ 3, is introduced with the aim of controlling the Sobolev norm H n of u ± , v ± for large values of n. The reason of dealing with E k 3 (t; u ± , v ± ) is, instead, to control the L 2 norm of Γ I u ± , Γ I v ± , for any general Γ ∈ Z and |I| ≤ 3. In particular, superscript k indicates that we are considering only products Γ I containing at most 3 -k vector fields in {Ω, Z m , m = 1, 2}. For instance, the L 2 norms of Ω 3 u ± , ΩZ 2 1 v ± are bounded by E 0 3 (t; u ± , v ± ) but not by E 1 3 (t; u ± , v ± ), while the L 2 norms of Z 2 1 u ± , ∂ 2 ΩZ 2 v ± are controlled by both E 1 3 (t; u ± , v ± ), E 0 3 (t; u ± , v ± ), etc. The interest of distinguishing between k = 0, 1, 2, is to take into account the different growth in time of the L 2 norm of such terms depending on the number of vector fields Ω, Z m acting on u ± , v ± , as emerges from a-priori estimate (1.1.11d).

Theorem 1.1.2 (Bootstrap Argument). Let K 1 , K 2 > 1 and H ρ,∞ be the space defined in 1.2.1 (iii). There exist two integers n ρ sufficiently large, some 0 < δ δ 2 δ 1 δ 0 1 small, two constants A, B > 1 sufficiently large and ε 0 ∈]0, (2A + B) -1 [ such that, for any 0 < ε < ε 0 , if (u, v) is solution to (1.1.1)-(1.1.2) on some interval [1, T ], for a fixed T > 1, and u ± , v ± defined in (1.1.5) satisfy:

u ± (t, •) H ρ+1,∞ + Ru ± (t, •) H ρ+1,∞ ≤ Aεt -1 2 , (1.1.11a) v ± (t, •) H ρ,∞ ≤ Aεt -1 , (1.1.11b) E n (t; u ± , v ± ) 1 2 ≤ Bεt δ 2 , (1.1.11c) E k 3 (t; u ± , v ± ) 1 2 ≤ Bεt δ k 2 , ∀ 0 ≤ k ≤ 2, (1.1.11d)
for every t ∈ [1, T ], then in the same interval they verify also

u ± (t, •) H ρ+1,∞ + Ru ± (t, •) H ρ+1,∞ ≤ A K 1 εt -1 2 , (1.1.12a) v ± (t, •) H ρ,∞ ≤ A K 1 εt -1 , (1.1.12b) E n (t; u ± , v ± ) 1 2 ≤ B K 2 εt δ 2 , (1.1.12c) E k 3 (t; u ± , v ± ) 1 2 ≤ B K 2 εt δ k 2 , ∀ 0 ≤ k ≤ 2. (1.1.12d)
The a-priori estimates on the uniform norm of u ± , Ru ± , v ± made in the above theorem translate in terms of u ± , v ± the sharp decay in time we expect for the solution (u, v) to starting problem (1.1.1). Indeed, from definitions (1.1.5) it appears that

D t u = u + + u - 2 , D x u = R u + -u - 2 , D t v = v + + v - 2 , v = D x -1 v + -v - 2 ,
so (1.1.11a), (1.1.11b) imply

∂ t,x u(t, •) H ρ,∞ ≤ Aεt -1 2 , ∂ t v(t, •) H ρ,∞ + v(t, •) H ρ+1,∞ ≤ Aεt -1 .
Furthermore, the following quantity

∂ t u(t, •) H n + ∇ x u(t, •) H n + ∂ t v(t, •) H n + ∇ x v(t, •) H n + v(t, •) H n
is equivalent to the square root of E n (t; u ± , v ± ), which implies that the propagation of a-priori energy estimate (1.1.11c) is equivalent to the propagation of a certain estimate on the above Sobolev norms. For this reason, the propagation of the a-priori estimate on E n (t; u ± , v ± ) and a local existence argument will imply theorem 1.1.1.

Before ending this section and going into the core of the subject, we briefly remind the general definition of null condition for a multilinear form on R 1+n and a result by Hörmander (see [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]).

Definition 1.1.3. A k-linear form G on R 1+n is said to satisfy the null condition if and only if, for all ξ ∈ R n , ξ = (ξ 0 , . . . , ξ n ) such that ξ 2 0 -n j=1 ξ 2 j = 0, Example: The trilinear form ξ 2 0 ξ a -j=1,2 ξ 2 j ξ a associated to Q 0 (v, ∂ a w) satisfies the null condition (1.1.13), for any a = 0, 1, 2. This is the most common example of null form.

Lemma 1.1.4 (Hörmander [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF], Lemma 6.6.5.). Let G be a k-linear form on R 1+n , k = k 1 + • • • + k r , with k j positive integers, and Γ ∈ Z. For all u j ∈ C k+1 (R 1+n ), all α j ∈ N 1+n , |α j | = k j , and u (k j ) j := ∂ α j u j , ΓG(u

(k 1 )
1 , . . . , u (kr) r ) = G((Γu 1 ) (k 1 ) , . . . , u (kr) r ) + . . . + G(u

(k 1 )
1 , . . . , (Γu r ) (kr) ) + G 1 (u

(k 1 )
1 , . . . , u (kr) r ) ,

(1. 1.14) where G 1 satisfies the null condition.

Remark 1.1.5. Previous lemma simplifies when the multi-linear form G satisfying the null condition is Q 0 (v, ∂ a w), for any a = 0, 1, 2. Indeed, the structure of the null form is not modified by the action of vector field Γ in the sense that (1.1.15)

ΓQ 0 (v, ∂ a w) = Q 0 (Γv, ∂ a w) + Q 0 (v, ∂ a Γw) + G 1 (v, ∂w) ,
where G 1 (v, ∂w) = 0 if Γ = ∂ m , m = 1, 2, and

(1.1. [START_REF] Ionescu | Global solutions for the gravity water waves system in 2d[END_REF])

G 1 (v, ∂w) =                -Q 0 (v, ∂ m w), if a = 0, Γ = Z m , m ∈ {1, 2}, 0, if a = 0, Γ = Ω, -Q 0 (v, ∂ t w), if a = 0, Γ = Z a , 0,
if a = 0, Γ = Z m , m ∈ {1, 2} \ {a}, (-1) a Q 0 (v, ∂ m w), with m ∈ {1, 2} \ {a}, if a = 0, Γ = Ω.

If Γ I contains at least k ≤ |I| space derivatives then (1.1.17)

Γ I Q 0 (v, ∂ 1 w) = |I 1 |+|I 2 |=|I| Q 0 (Γ I 1 v, ∂ 1 Γ I 2 w) + k≤|I 1 |+|I 2 |<|I| c I 1 ,I 2 Q 0 (Γ I 1 v, ∂Γ I 2 w),
with c I 1 ,I 2 ∈ {-1, 0, 1}. In the above equality we should think of multi-index I 1 (resp. I 2 ) as obtained by extraction of a |I 1 |-tuple (resp. |I 2 |-tuple) from I = (i 1 , . . . i n ), in such a way that each i j appearing in I and corresponding to a spatial derivative (e.g. Γ i j = D m , for m ∈ {1, 2}), appears either in I 1 or in I 2 , but not in both. For further references, we define (1.1.18) I(I) := {(I 1 , I 2 )|I 1 , I 2 multi-indices obtained as described above} .

Preliminary Results

The aim of this section is to introduce most of the technical tools that will be used throughout the paper. In particular, subsections 1.2.1 and 1.2.2 are devoted to recall some definitions and results about paradifferential and pseudo-differential calculus respectively; subsection 1.2.3 and 1.2.4 are dedicated to the introduction of some special operators that we will frequently use when dealing with the wave and the Klein-Gordon component. Subsections 1.2.1, 1.2.2 barely contain proofs (we refer for that to [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF], [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems, volume 5 of Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series[END_REF], [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF], [START_REF] Zworski | Semiclassical analysis[END_REF]), whereas subsections 1.2.3, 1.2.4 are much longer and richer in proofs and technicalities.

Paradifferential calculus

In the current subsection we recall some definitions and properties that will be useful in chapter 2.

We first recall the definition of some spaces (Sobolev, Lipschitz and Hölder spaces) in dimension d ≥ 1 and afterwards some results concerning symbolic calculus and the action of paradifferential operators on Sobolev spaces (see for instance [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems, volume 5 of Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series[END_REF]). We warn the reader that we will use both notations ŵ(ξ) and F x →ξ w for the Fourier transform of a function w = w(x).

Definition 1.2.1 (Spaces). (i) Let s ∈ R. H s (R d ) denotes the space of tempered distributions w ∈ S (R d ) such that ŵ ∈ L 2 loc (R d ) and

w 2 H s (R d ) := 1 (2π) d (1 + |ξ| 2 ) s | ŵ(ξ)| 2 dξ < +∞;
(ii) For ρ ∈ N, W ρ,∞ (R d ) denotes the space of distributions w ∈ D (R d ) such that ∂ α x w ∈ L ∞ (R d ), for any α ∈ N d with |α| ≤ ρ, endowed with the norm

w W ρ,∞ := |α|≤ρ ∂ α x w L ∞ ;
(iii) For ρ ∈ N, we also introduce H ρ,∞ (R d ) as the space of tempered distributions w ∈ S (R d ) such that w H ρ,∞ := D x ρ w L ∞ < +∞.

Definition 1.2.2. An operator T is said of order ≤ m ∈ R if it is a bounded operator from H s+m (R d ) to H s (R d ) for all s ∈ R.

Definition 1.2.3 (Smooth symbols). Let m ∈ R.

(i) S m 0 (R d ) denotes the space of functions a(x, η) on R d × R d which are C ∞ with respect to η and such that, for all α ∈ N d , there exists a constant C α > 0 and

∂ α η a(•, η) L ∞ ≤ C α (1 + |η|) m-|α| , ∀η ∈ R d .
Σ m 0 (R d ) denotes the subclass of symbols a ∈ S m 0 (R d ) satisfying

(1.2.1) ∃ε < 1 : F x→ξ a(ξ, η) = 0 for |ξ| > ε(1 + |η|).

S m 0 is equipped with seminorm M m 0 (a; n) given by (1.2.2)

M m 0 (a; n) = sup |β|≤n sup η∈R 2 (1 + |η|) |β|-m ∂ β η a(•, η) L ∞ .
(ii) For r ∈ N, S m r (R d ) denotes more generally the space of symbols a ∈ S m 0 (R d ) such that, for all α ∈ N d and all η ∈ R d , function x → ∂ α η a(x, η) belongs to W r,∞ (R d ) and there exists a constant C α > 0 such that

∂ α η a(•, η) W r,∞ ≤ C α (1 + |η|) m-|α| , ∀η ∈ R d .
Σ m r (R d ) denotes the subclass of symbols a ∈ S m r (R d ) satisfying the spectral condition (1.2.1). S m r is equipped with seminorm M m r (a; n), given by (ii) for all (α, β) ∈ N d × N d there is a constant C α,β > 0 such that (1.2.5)

|∂ α ξ ∂ β η ψ(ξ, η)| ≤ C α,β (1 + |η|) -|α|-|β| , ∀(ξ, η).
Example: If χ is a smooth cut-off function such that χ(z) = 1 for |z| ≤ ε 1 and is supported in the open ball B ε 2 (0), with 0 < ε 1 < ε 2 < 1, function ψ(ξ, η) := χ ξ η is an admissible cut-off function in the sense of definition 1.2.4. We will only consider this type of admissible cut-off functions for the rest of the paper and refer (abusively) to χ itself as an admissible cut-off. Definition 1.2.5. Let χ be an admissible cut-off function and a(x, η) ∈ S m r , m ∈ R, r ∈ N. The Bony quantization (or paradifferential quantization) Op B (a(x, η)) associated to symbol a and acting on a test function w is defined as

Op B (a(x, η))w(x) := 1 (2π) d R d e ix•η σ χ a (x, η) ŵ(η)dη , with σ χ a (x, η) := 1 (2π) d R d e i(x-y)•ζ χ ζ η a(y, η)dydζ .
The operator defined above depends on the choice of the admissible cut-off function χ. However, if a ∈ S m r for some m ∈ R, r ∈ N, a change of χ modifies Op B (a) only by the addition of a r-smoothing operator (i.e. an operator which is bounded from H s to H s+r , see [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]), so the choice of χ will be substantially irrelevant as long as we can neglect r-smoothing operators. For this reason, we will not indicate explicitly the dependence of Op B (resp. of σ χ a ) on χ to keep notations as light as possible. Let us also observe that, with such a definition, the Fourier transform of Op B (a)w has the following simple expression (1.2.6)

F x→ξ Op B (a(x, η))w(x) (ξ) = 1 (2π) d χ ξ -η η ây (ξ -η, η) ŵ(η)dη ,
where ây (ξ, η) := F y→ξ a(y, η) , and the product of two functions u, v can be developed as

(1.2.7) uv = Op B (u)v + Op B (v)u + R(u, v),
where remainder R(u, v) writes on the Fourier side as

(1.2.8) R(u, v) (ξ) = 1 (2π) d 1 -χ ξ -η η -χ η ξ -η u(ξ -η) v(η)dη.
We remark that frequencies η and ξ -η in the above integral are either bounded or equivalent, and R(u, v) = R(v, u). With the aim of having uniform notations, we introduce the operator (1.2.9)

For future references, we recall the definition of the Littlewood-Paley decomposition of a function w.

Definition 1.2.6 (Littlewood-Paley decomposition). Let χ : R 2 → [0, 1] be a smooth decaying radial function, supported for |x| ≤ 2 -1 10 and identically equal to 1 for |x| ≤ 1 + 1 10 . Let also ϕ(ξ) := χ(ξ) -χ(2ξ) ∈ C ∞ 0 (R 2 \ {0}), supported for 1 2 < |ξ| < 2, and ϕ k (ξ) := ϕ(2 -k ξ) for all k ∈ N * , with the convention that ϕ 0 := χ. Then k∈N ϕ(2 -k ξ) = 1, and for any w ∈ S (R d )

(1.2.10) w = k∈N ϕ k (D x )w
is the Littlewood-Paley decomposition of w.

The following proposition is a classical result about the action of para-differential operators on Sobolev spaces (see [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] for further details). Proposition 1.2.8 shows, instead, that some results of continuity over L 2 hold also for operators whose symbol a(x, η) is not a smooth function of η, and that map

(u, v) → R(u, v) is continuous from H 4,∞ × L 2 to L 2 .
Proposition 1.2.7 (Action). Let m ∈ R. For all s ∈ R and a ∈ S m 0 , Op B (a) is a bounded operator from H s+m (R d ) to H s (R d ). In particular,

Op B (a(x, η))w L 2 a 1 L ∞ w L 2 .
The same result is true for Op B R (a(x, η)); (ii) Map (u, v) ∈ H 4,∞ × L 2 → R(u, v) ∈ L 2 is well defined and continuous. Proof. As concerns (i) we have that Op B (a(x, η))w(x) = K(x -z, x -y)a 1 (y)w(z)dydz

∂ β ζ χ ζ η b(η) 1 {|η| 1} |g β (ζ)|, ∂ α η ∂ β ζ χ ζ η b(η) 1 {|η| 1} |η| 1-|α| |g β (ζ)|, |α| ≥ 1,
for some bounded and compactly supported functions g β . Lemma A.1 (i) and corollary A.2 (i) of appendix A imply that |K(x, y)| |x| -1 x -2 y -3 for any (x, y), and statement (i) follows by an inequality such as (A.8) with L = L 2 .

In order to prove assertion (ii) we consider a cut-off function ψ ∈ C ∞ 0 (R 2 ) equal to 1 in some closed ball B C (0), for a C 1, and decompose R(u, v) as follows, using (1.2.8):

R(u, v) = K 0 (x -y, y -z)u(y)v(z)dydz + K 1 (x -y, y -z)[ D x 4 u](y)v(z)dydz, with

K 0 (x, y) = 1 (2π) 2 e ix•ξ+iy•η 1 -χ ξ -η η -χ η ξ -η ψ(η)dξdη, K 1 (x, y) = 1 (2π) 2 e ix•ξ+iy•η 1 -χ ξ -η η -χ η ξ -η (1 -ψ)(η) ξ -η -4 dξdη.
Since frequencies ξ, η are both bounded on the support of 1 -χ ξ-η η -χ η ξ-η ψ(η), one can show through some integration by parts that |K 0 (x, y)|

x -3 y -3 for any (x, y), to then deduce that K 0 (x -y, y -z)u(y)v(z)dydz

L 2 (dx) u L ∞ v L 2 .
Kernel K 1 (x, y) can be split using a Littlewood-Paley decomposition as follows

K 1 (x, y) = k≥1 1 (2π) 2 e ix•ξ+iy•η 1 -χ ξ -η η -χ η ξ -η (1 -ψ)(η)ϕ(2 -k η) ξ -η -4 dξdη K 1,k (x,y)
, for a suitable ϕ ∈ C ∞ 0 (R 2 \{0}). On the support of

1 -χ ξ-η η -χ η ξ-η (1-ψ)(η)ϕ(2 -k η),
frequencies η, ξ -η are either bounded or equivalent and of size 2 k (which implies in particular that ξ -η -4 ξ -3 η -1 ). After a change of coordinates and some integration by parts one can show that |K 1,k (x, y)| 2 k x -3 2 k y -3 , for any k ≥ 1, and therefore that e i(x-y)•ξ+i(y-z)•η K 1 (x -y, y -z)[ D x 4 u](y)v(z)dydz

L 2 (dx) k≥1 2 k x -y -3 2 k (y -z) -3 | D x 4 u(y)||w(z)|dydz L 2 (dx) k≥1 2 k y -3 2 k z -3 [ D x 4 u](• -y)w(• -y -z) L 2 (dx) dydz u H 4,∞ w L 2 ,
which concludes the proof of statement (ii).

The last results of this subsection are stated without proofs. All the details can be found in chapter 6 of [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems, volume 5 of Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series[END_REF] (see theorems 6.1.1, 6.1.4, 6.2.1, 6.2.4).

Proposition 1.2.9 (Composition). Consider a ∈ S m r , b ∈ S m r , r ∈ N * , m, m ∈ R.

Op B (a)Op B (b)w -Op(ab)w H s ≤ C M m 1 (a; 3)M m 0 (b; 2) + M m 0 (a; 3)M m 1 (b; 2) w H s+m+m -1 .
Proposition 1.2.12 (Adjoint). Consider a ∈ S m r (R d ), denote by Op B (a) * the adjoint operator of Op B (a) and by a * (x, ξ) = a(x, ξ) the complex conjugate of a(x, ξ).

(i) Symbol b(x, ξ) := |α|<r 1 α! D α x ∂ α ξ a * (x, ξ) is well defined in j<r S m-j r-j ; (ii) Operator Op B (a) * -Op B (b) is of order ≤ m -r. Precisely, for all s ∈ R there is a constant C > 0 such that, for all a ∈ S m r (R d ) and w ∈ H s+m-r (R d ), Op B (a) * w -Op B (b)w H s ≤ CM m r (a; n) w H s+m-r ,
with n 0 = d 2 + 1, n = n 0 + r. These results extend to matrix valued symbols a, with a * (x, ξ) denoting the adjoint of matrix a(x, ξ).

Corollary 1.2.13. For d = 2 and all s ∈ R, there exists a constant C > 0 such that, for a ∈ S m r , r ∈ N * and w ∈ H s+m-1 ,

Op B (a) * w -Op(a * )w H s ≤ CM m 1 (a; 3) w H s+m-1 .

Semi-classical pseudodifferential calculus

In this subsection we recall some definitions and results about semi-classical symbolic calculus in general space dimension d ≥ 1 which will be used in section 3.2. We refer the reader to [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] and [START_REF] Zworski | Semiclassical analysis[END_REF] for more details.

Definition 1.2.14. An order function on

R d × R d is a smooth map from R d × R d to R + : (x, ξ) → M (x, ξ) such that there exist N 0 ∈ N, C > 0 and for any (x, ξ), (y, η) ∈ R d × R d (1.2.13) M (y, η) ≤ C x -y N 0 ξ -η N 0 M (x, ξ) , where x = 1 + |x| 2 .
Definition 1.2.15. Let M be an order function on R d × R d , δ, σ ≥ 0. One denotes by S δ,σ (M ) the space of smooth functions

(x, ξ, h) → a(x, ξ, h) R d × R d ×]0, 1] → C satisfying for any α 1 , α 2 ∈ N d , k, N ∈ N (1.2.14) |∂ α 1 x ∂ α 2 ξ (h∂ h ) k a(x, ξ, h)| M (x, ξ) h -δ(|α 1 |+|α 2 |) (1 + σh σ |ξ|) -N .
A key role in this paper will be played by symbols a verifying (1.2.14

) with M (x, ξ) = x+f (ξ) √ h -N ,
for N ∈ N and a certain smooth function f (ξ). This function M is no longer an order function because of the term h -1 2 , but nevertheless we keep writing a ∈ S δ,σ ( x+f (ξ)

It is clear from the definition that the two quantizations coincide when the symbol does not depend on x. We also introduce a semi-classical version of Sobolev spaces on which the above operators act naturally.

Definition 1.2.18. (i) Let ρ ∈ N. We define the semi-classical Sobolev space H ρ,∞ h (R d ) as the space of tempered distributions w such that hD ρ w :

= Op h ( ξ ρ )w ∈ L ∞ , endowed with norm w H ρ,∞ h = hD ρ w L ∞ ;
(ii) Let s ∈ R. We define the semi-classical Sobolev space H s h (R d ) as the space of tempered distributions w such that hD s w := Op h ( ξ s )w ∈ L 2 , endowed with norm

w H s = hD s w L 2 .
For future references, we write down the semi-classical Sobolev injection in space dimension 2:

(1.2.15) v h H ρ,∞ h (R 2 ) σ h -1 v h H ρ+1+σ h (R 2 ) , ∀σ > 0 .
The following two propositions are stated without proof. They concern the adjoint and the composition of pseudo-differential operators. All related details are provided in chapter 7 of [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] or in chapter 4 of [START_REF] Zworski | Semiclassical analysis[END_REF].

(a) • Op w h (b) = Op w h (a b) ,
where

(1.2.16) a b (x, ξ) := 1 (πh) 2d R d R d R d R d e 2i h σ(y,η; z,ζ) a(x + z, ξ + ζ)b(x + y, ξ + η) dydηdzdζ, and σ(y, η; z, ζ) = η • z -y • ζ .
It is often useful to derive an asymptotic expansion for a b, as it allows easier computations than integral formula (1.2.16). This expansion is usually obtained by applying the stationary phase argument when a, b ∈ S δ,σ (M ), δ ∈ [0, 1 2 [ (as shown in [START_REF] Zworski | Semiclassical analysis[END_REF]). Here we provide an expansion at any order even when one of two symbols belongs to S 1 2 ,σ 1 (M ) (still having the other in S δ,σ 2 (M ) for δ < 1 2 , and σ 1 , σ 2 either equal or, if not, one of them equal to zero), whose proof is based on the Taylor development of symbols a, b, and can be found in the appendix of [START_REF] Stingo | Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data:46[END_REF] (for d = 1).

Proposition 1.2.21. Let M 1 , M 2 be two order functions and a ∈ S δ 1 ,σ 1 (M 1 ), b ∈ S δ 2 ,σ 2 (M 2 ), δ 1 , δ 2 ∈ [0, 1 2 ], δ 1 + δ 2 < 1, σ 1 , σ 2 ≥ 0 such that (1.2.17) σ 1 = σ 2 ≥ 0 or σ 1 = σ 2 and σ i = 0 , σ j > 0 , i = j ∈ {1, 2} . Then a b ∈ S δ,σ (M 1 M 2 ), where δ = max{δ 1 , δ 2 }, σ = max{σ 1 , σ 2 }. Moreover, (1.2.18) a b = α=(α 1 ,α 2 ) |α|=0,...,N -1 (-1) |α 1 | α! h 2i |α| ∂ α 1 x ∂ α 2 ξ a(x, ξ) ∂ α 2 x ∂ α 1 ξ b(x, ξ) + r N , where r N ∈ h N (1-(δ 1 +δ 2 )) S δ,σ (M 1 M 2 ) and (1.2.19) r N (x, ξ) = h 2i N N (πh) 2d α=(α 1 ,α 2 ) |α|=N (-1) |α 1 | α! R 4 e 2i h (η•z-y•ζ) × 1 0 ∂ α 1 x ∂ α 2 ξ a(x + tz, ξ + tζ)(1 -t) N -1 dt ∂ α 2 x ∂ α 1 ξ b(x + y, ξ + η) dydηdzdζ , or (1.2.20) r N (x, ξ) = h 2i N N (πh) 2d α=(α 1 ,α 2 ) |α|=N (-1) |α 1 | α! R 4 e 2i h (η•z-y•ζ) ∂ α 1 x ∂ α 2 ξ a(x + z, ξ + ζ) × 1 0 ∂ α 2 x ∂ α 1 ξ b(x + ty, ξ + tη)(1 -t) N -1 dt dydηdzdζ . More generally, if h N δ 1 ∂ α a ∈ S δ 1 ,σ 1 (M N 1 ), h N δ 2 ∂ α b ∈ S δ 2 ,σ 2 (M N 2 ), for |α| = N and some order functions M N 1 , M N 2 , then r N ∈ h N (1-(δ 1 +δ 2 )) S δ,σ (M N 1 M N 2 ). Remark 1.2.22. From the previous proposition it follows that, if symbols a ∈ S δ 1 ,σ 1 (M 1 ), b ∈ S δ 2 ,σ 2 (M 2 ) are such that suppa ∩ suppb = ∅, then a b = O(h ∞ ), meaning that, for every N ∈ N, a b = r N with r N ∈ h N (1-(δ 1 +δ 2 )) S δ,σ (M 1 M 2 ).
Remark 1.2.23. We draw the reader's attention to the fact that symbol is used simultaneously in Bony calculus (see proposition 1.2.9) and in Weyl semi-classical calculus (as in (1.2.18)) with two different meaning. However, we avoid to introduce different notations as it will be clear by the context if we are dealing with the former or the latter one.

The result of proposition 1.2.21 and remark 1.2.22 are still true even when one of the two order functions, or both, has the form x+f (ξ) √ h

-1 , for a smooth function f (ξ), ∇f (ξ) bounded, as stated below (see the appendix of [START_REF] Stingo | Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data:46[END_REF]).

Lemma 1.2.24. Let f (ξ) : R d → R be a smooth function, with |∇f (ξ)| bounded. Consider a ∈ S δ 1 ,σ 1 ( x+f (ξ) √ h -m ), m ∈ N, and b ∈ S δ 2 ,σ 2 (M ), for M order function or M (x, ξ) = x+f (ξ) √ h -n , n ∈ N, some δ 1 ∈ [0, 1 2 ], δ 2 ∈ [0, 1 2 [, σ 1 , σ 2 ≥ 0 as in (1.2.17). Then a b ∈ S δ,σ ( x+f (ξ) √ h -m M ),
where δ = max{δ 1 , δ 2 }, σ = max{σ 1 , σ 2 }, and the asymptotic expansion (1.2.18) holds, with

r N ∈ h N (1-(δ 1 +δ 2 )) S δ,σ ( x+f (ξ) √ h -m M ) given by (1.2.19) (or equivalently (1.2.20)).

More generally, if h

N δ 1 ∂ α a ∈ S δ 1 ,σ 1 ( x+f (ξ) √ h -m ) and h N δ 2 ∂ α b ∈ S δ 2 ,σ 2 (M N ), |α| = N , M N order function or M N (x, ξ) = x+f (ξ) √ h
-n , for some m , n ∈ N, then remainder r N belongs to

h N (1-(δ 1 +δ 2 )) S δ,σ ( x+f (ξ) √ h -m M N ).

Semi-classical Operators for the Wave Solution: Some Estimates

From now on we place ourselves in space dimension d = 2. This technical subsection focuses on the introduction and the analysis of some particular operators that we will use when dealing with the wave component in the semi-classical framework (subsection 3.2.2). More precisely, lemma 1.2.25 will be often recalled to prove that some operator belongs to L(L

(x, ξ) with ∂ α x ∂ β ξ A ∈ L 2 (R 2 × R 2 ) for |α|, |β| ≤ 3, and any function w ∈ L 2 (R 2 ), (1.2.21) Op w h A(x, ξ) w(x) ≤ C w L 2 R 2 x -y -3 |α|,|β|≤3 ∂ α y ∂ β ξ A x + y 2 , hξ L 2 (dξ)
dy.

Moreover, if A(x, ξ) is compactly supported in x there exists a smooth function, supported in a neighbourhood of suppA, such that

(1.2.22) Op w h A(x, ξ) w(x) ≤ C w L 2 R 2 θ x + y 2 |α|≤3 ∂ α y A x + y 2 , hξ L 2 (dξ)
dy.

Proof. Let us prove the statement for

A ∈ S(R 2 × R 2 ) and w ∈ S(R 2 ). The density of S(R 2 × R 2 ) into {A ∈ L 2 (R 2 × R 2 )|∂ α x ∂ β ξ A ∈ L 2 (R 2 × R 2 ), |α|, |β| ≤ 3} and of S(R 2 ) into L 2 (R 2 ) will then
justify the definition of Op w h (A(x, ξ))w for A and w as in the statement, together with inequalities (1.2.21), (1.2.22).

Using integration by parts, Cauchy-Schwarz inequality, and Young's inequality for convolutions, we can write the following:

|Op w h (A(x, ξ))w(x)| = 1 (2π) 2 R 4 e i(x-y)•ξ A x + y 2 , hξ w(y) dydξ = 1 (2π) 4 R 2 w(η) R 2 R 2 e i(x-y)•ξ+iy•η A x + y 2 , hξ dydξ dη = 1 (2π) 4 R 2 w(η) R 2 R 2 1 -i(x -y) • ∂ ξ 1 + |x -y| 2 3 1 + i(ξ -η) • ∂ y 1 + |ξ -η| 2 3 e i(x-y)•ξ+iy•η ×A x + y 2 , hξ dydξ dη R 2 | ŵ(η)| R 2 R 2 x -y -3 ξ -η -3 |α|,|β|≤3 ∂ α y ∂ β ξ A x + y 2 , hξ dydξ dη ŵ L 2 (dη) η -3 L 1 (dη) R 2 x -y -3 |α|,|β|≤3 ∂ α y ∂ β ξ A x + y 2 , hξ L 2 (dξ) dy w L 2 R 2 x -y -3 |α|,|β|≤3 ∂ α y ∂ β ξ A x + y 2 , hξ L 2 (dξ)
dy .

If symbol A(x, ξ) is compactly supported in x we can consider a smooth function θ ∈ C ∞ 0 (R), identically equal to 1 on the support of A(x, ξ), and write

|Op w h (A(x, ξ))w(x)| = 1 (2π) 2 R 2 w(η)dη R 2 R 2 1 + i(ξ -η) • ∂ y 1 + |ξ -η| 2 3 e i(x-y)•ξ+iy•η ×A x + y 2 , hξ dydξ R 2 | ŵ(η)| dη R 2 R 2 θ x + y 2 ξ -η -3 |α|≤3 ∂ α y A x + y 2 , hξ dydξ w L 2 R 2 θ x + y 2 |α|≤3 ∂ α y A x + y 2 , hξ L 2 (dξ)
dy .

A very important role in this subsection and in subsection 3.2.2 will be played by functions of the form

γ x|ξ|-ξ h 1/2-σ ψ(2 -k ξ), where γ ∈ C ∞ (R 2 ) is such that |∂ α γ(z)| z -|α| , ψ ∈ C ∞ 0 (R 2 -{0}), σ > 0 is a small fixed constant and k is an integer belonging to set K, with (1.2.23) K := {k ∈ Z : h 2 k h -σ }.
In various results, such as proposition 1.2.30, we will need a more decaying smooth function γ 1 verifying that 1+|α|) . We introduce here some notations we will keep throughout the whole paper: n+|α|) , for any α ∈ N 2 . We use the simplest notation γ for γ 0 ; The following lemma is a useful reference when we need to deal with some derivatives of γ x|ξ|-ξ h 1/2-σ . Lemma 1.2.26. Let σ ∈ R and n ∈ N. For any multi-indices α, β ∈ N 2 we have that

|∂ α γ 1 (z)| z -(
Notation 1. For any n ∈ N, γ n denotes a smooth function in R 2 such that |∂ α γ n (z)| α z -(
(1.2.24) ∂ α x ∂ β ξ γ n x|ξ| -ξ h 1/2-σ = |β| k=0 h -(|α|+k)( 1 2 -σ) γ n+|α|+k x|ξ| -ξ h 1/2-σ b |α|+k-|β| (ξ). Furthermore, if θ = θ(x) ∈ C ∞ 0 (R 2
), there exists a set {θ k (x)} 1≤k≤|β| of smooth compactly supported functions such that

(1.2.25) θ(x)∂ α x ∂ β ξ γ n x|ξ| -ξ h 1/2-σ = |β| k=1 h -(|α|+k)( 1 2 -σ) γ n+|α|+k x|ξ| -ξ h 1/2-σ θ k (x)b |α|+k-|β| (ξ).
Proof. Let δ ij be the Kronecker delta and be a concise notation to indicate a linear combination. For i = 1, 2,

∂ ξ i γ n x|ξ| -ξ h 1/2-σ = h -( 1 2 -σ) 2 j=1 (∂ j γ n ) x|ξ| -ξ h 1/2-σ (x j ξ i |ξ| -1 -δ ij ) = 2 j=1 (∂ j γ n ) x|ξ| -ξ h 1/2-σ x j |ξ| -ξ j h 1/2-σ ξ i |ξ| -2 + 2 j=1 h -( 1 2 -σ) (∂ j γ n ) x|ξ| -ξ h 1/2-σ [ξ i ξ j |ξ| -2 -δ ij ], (1.2.26)
which can be summarized saying that

∂ ξ i γ n x|ξ| -ξ h 1/2-σ = γ n x|ξ| -ξ h 1/2-σ b -1 (ξ) + h -( 1 2 -σ) γ n+1 x|ξ| -ξ h 1/2-σ b 0 (ξ),
for some new functions γ n , γ n+1 , b 0 , b -1 . Iterating this argument one finds that, for all β ∈ N 2 ,

∂ β ξ γ n x|ξ| -ξ h 1/2-σ = k=0,...,|β| h -k( 1 2 -σ) γ n+k x|ξ| -ξ h 1/2-σ b k-|β| (ξ),
and obtains (1.2.24) using that, for any

m ∈ N, α ∈ N 2 , (1.2.27) ∂ α x γ m x|ξ| -ξ h 1/2-σ = h -|α|( 1 2 -σ) (∂ α γ m ) x|ξ| -ξ h 1/2-σ |ξ| |α| . Equality (1.2.25) is obtained replacing (1.2.26) with θ(x)∂ ξ i γ n x|ξ| -ξ h 1/2-σ = h -( 1 2 -σ) 2 j=1 (∂ j γ n ) x|ξ| -ξ h 1/2-σ (θ(x)x j ξ i |ξ| -1 -θ(x)δ ij ) = h -( 1 2 -σ) γ n+1 x|ξ| -ξ h 1/2-σ θ 1 (x)b 0 (ξ),
where θ 1 (x) is a new compactly supported function. By iteration one finds that, for any β ∈ N 2 , there is a set of |β| compactly supported functions

θ k (x), 1 ≤ k ≤ |β|, such that θ(x)∂ β ξ γ n x|ξ| -ξ h 1/2-σ = |β| k=1 h -k( 1 2 -σ) γ n+k x|ξ| -ξ h 1/2-σ θ k (x)b k-|β| (ξ),
which combined with (1.2.27) gives (1.2.25).

In some of the following results we denote by Θ h the operator of change of coordinates

Θ h w(x) = w( √ hx),
for any h ∈]0, 1], and use that for any symbol a(x, ξ),

(1.2.28) Op w h a(x, ξ) = Θ h Op w h a(x, ξ) Θ -1 h , with a(x, ξ) = a x √ h , √ hξ .
Proposition 1.2.27 (Continuity on L 2 ). Let σ > 0 be sufficiently small, K be the set defined in

(1.2.23), k ∈ K and p ∈ Z. Let also ψ ∈ C ∞ 0 (R 2 \ {0}
) and a(x) be a smooth function, bounded together with all its derivatives. Then

Op w h γ x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) : L 2 → L 2 is bounded and (1.2.29) Op w h γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) L(L 2 ) 2 kp . Proof. Let A(x, ξ) = γ x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ). For indices k ∈ K such that h 1/2-σ 2 k h -σ the statement follows from the fact that A(x, ξ) ∈ 2 kp S 1 2 ,0 (1
) and by theorem 7.11 of [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF].

For k ∈ K such that h ≤ 2 k ≤ h 1/2-σ , A(x, ξ) := A( x √ h , √ hξ) ∈ 2 kp S 1 2 ,0 (1
) and the result follows by theorem 7.11 of [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] and equality (1.2.28).

Proposition 1.2.28. Let σ, k, p be as in the previous proposition. Let also q ∈ Z, ψ ∈ C ∞ 0 (R 2 \ {0}), a (x) be a smooth function, bounded together with all its derivatives, and f ∈ C(R). Define (1.2.30)

I k p,q (x, ξ) := 1 (πh) 4 e 2i h (η•z-y•ζ) 1 0 γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) | (x+tz,ξ+tζ) f (t)dt × ψ(2 -k (ξ + η))a (x + y)b q (ξ + η) dydzdηdζ and (1.2.31) J k p,q (x, ξ) := 1 (πh) 4 e 2i h (η•z-y•ζ) 1 0 ψ(2 -k (ξ + tζ))a (x + tz)b q (ξ + tζ)f (t)dt × γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) | (x+y,ξ+η) dydzdηdζ.
Then Op w h (I k p,q (x, ξ)) and Op w h (J k p,q (x, ξ)) are bounded operators on L 2 and Op w h (I k p,q (x, ξ))

L(L 2 ) + Op w h (J k p,q (x, ξ)) L(L 2 ) 2 k(p+q) .
The same results holds also if q = 0 and ψ(2 -k ξ)b q (ξ) ≡ 1.

Proof. We show the result for Op w h (I k p,q ), leaving the reader to check that a similar argument can be used for Op w h (J k p,q ). We distinguish between two ranges of frequencies. For indices

k ∈ K such that h 1/2-σ ≤ 2 k h -σ we observe that I k p,q (x, ξ) ∈ 2 k(p+q) S 1 2 ,0 (1). Indeed, γ x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) ∈ 2 kp S 1 2 ,σ (1)
by lemma 1.2.26 while ψ(2 -k ξ)a (x)b q (ξ) ∈ 2 kq S 1 2 -σ,σ [START_REF] Alazard | Sobolev estimates for two dimensional gravity water waves[END_REF]. Hence performing a change of variables

y → √ hy, z → √ hz, η → √ hη, ζ → √ hζ, writing (1.2.32) e 2i(η•z-y•ζ) = 1 + 2iy • ∂ ζ 1 + 4|y| 2 3 1 -2iz • ∂ η 1 + 4|z| 2 3 1 -2iη • ∂ z 1 + 4|η| 2 3 1 + 2iζ • ∂ y 1 + 4|ζ| 2 3 e 2i(η•z-y•ζ) ,
and integrating by parts in all variables, we get that

I k p,q (x, ξ) 2 k(p+q) y -3 z -3 η -3 ζ -3 dydzdηdζ 2 k(p+q) ,
without any loss in h -δ due to the fact that we are considering symbols A(x, ξ) ∈ S δ,σ (1) with δ ∈ {0, 1/2-σ, 1/2}, and the derivatives of

A(x+ √ hy, ξ + √ hη) (resp. of A(x+t √ hz, ξ +t √ hζ)
) with respect to y and η (resp. with respect to z and ζ). In a similar way one can also prove that p+q) , for any α, β ∈ N 2 . Theorem 7.11 of [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] implies then the statement in this case.

|∂ α x ∂ β ξ I k p,q (x, ξ)| α,β h -1 2 (|α|+|β|) 2 k(
For indices k ∈ K such that h 2 k ≤ h 1/2-σ we observe that γ x|ξ| h 1/2-σ -h σ ξ ψ(2 -k √ hξ)a x √ h b p ( √ hξ) ∈ 2 kp S 1 2 ,σ (1)
,

ψ(2 -k √ hξ)a x √ h b q ( √ hξ) ∈ 2 kq S 1 2 ,σ (1)
.

Then I k p,q (x, ξ) = I k p,q x √ h , √ hξ ∈ 2 k(p+q) S 1 2 ,0 (1)
and theorem 7.11 of [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] along with equality (1.2.28) imply that Op w h (I k p,q ) : L 2 → L 2 is bounded, uniformly in h. The last part of the statement can be proved following an analogous scheme, after having previously made an integration in dzdη (or in dydζ if dealing with J k p,0 ).

Proposition 1.2.29 (Continuity on L p ). Let 1 ≤ p ≤ +∞, γ ∈ C ∞ 0 (R 2 ) be radial, ψ ∈ C ∞ 0 (R 2 \ {0}
), a(x) be a smooth function, bounded together with all its derivatives. Let also σ > 0 be small, k ∈ K with K given by (1.2.23) and q ∈ Z.

Then Op w h γ x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b q (ξ) : L p → L p is a bounded operator with Op w h γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b q (ξ) L(L p ) 2 kq .
Proof. In order to prove the result of the statement we need to show that kernel

K k (x, y) asso- ciated to Op w h γ x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b q (ξ) , i.e.
(1.2.33)

K k (x, y) := 1 (2πh) 2 e i h (x-y)•ξ γ x+y 2 |ξ| -ξ h 1/2-σ ψ(2 -k ξ)a x + y 2 b q (ξ)dξ, is such that sup x |K k (x, y)|dy 2 kq , sup y |K k (x, y)|dx 2 kq .
From the symmetry between variables x, y, it will be enough to show that one of the two above inequalities is satisfied. To do that we study K k separately in different spatial regions, distinguishing also between indices k ∈ K such that 2 k ≤ h 1/2-σ and 2 k > h 1/2-σ . We hence introduce three smooth cut-off functions θ s , θ b , θ, supported respectively for |x| ≤ m 1, |x| ≥ M 1, 0 < m ≤ |x| ≤ M < +∞, for some constants m, m , M, M , and such that θ s + θ b + θ ≡ 1.

Denoting concisely by A k (x, ξ) the multiplier in (1.2.33), we split it as follows

A k (x, ξ) = A k s (x, ξ) + A k b (x, ξ) + A k 1 (x, ξ), with A k s (x, ξ) := A k (x, ξ)θ s (x), A k b (x, ξ) := A k (x, ξ)θ b (x) and A k 1 (x, ξ) := A k (x, ξ)θ(x). Case I: Let us consider k ∈ K such that h 2 k ≤ h 1/2-σ . According to the above decomposition we have that K k (x, y) = K k s (x, y) + K k b (x, y) + K k 1 (x, y), with clear meaning of kernels K k s , K k b , K k 1 . Let us first prove that (1.2.34) sup x |K k s (x, y)|dy + sup x |K k b (x, y)|dy 2 kq .
For |x|

1 (resp. |x| 1), x|ξ|-ξ h 1/2-σ h -1/2+σ |ξ| (resp. x|ξ|-ξ h 1/2-σ h -1/2+σ |ξ||x| h -1/2+σ |ξ|) so by lemma 1.2.26 (1.2.35) ∂ β ξ γ x|ξ| -ξ h 1/2-σ |β| j=0 h -j( 1 2 -σ) x|ξ| -ξ h 1/2-σ -j |b j-|β| (ξ)| |ξ| -|β| . Therefore (1.2.36) ∂ β ξ A k s (x, 2 k ξ) |β 1 |≤|β| 2 k|β| |2 k ξ| -|β 1 | 2 -k(|β|-|β 1 |)+kq 1 |ξ|∼1 2 kq 1 |ξ|∼1 ,
so making a change of coordinates ξ → 2 k ξ and some integration by parts we derive that

|K k s (x, y)| 2 kq (2 k h -1 ) 2 2 k h -1 (x -y) -3
, for every (x, y) ∈ R 2 × R 2 . The same argument applies to K k b (x, y), hence taking the L 1 norm we obtain (1.2.34).

As concerns kernel K k 1 (x, y), we deduce from lemma 1.2.26, the fact that θ 1 (x) is supported for |x| ∼ 1, and that 2 k h 1/2-σ , the following inequality:

∂ β ξ A k 1 x + y 2 , 2 k ξ 2 k|β| 2 k(q-|β|) + |β| j=1 h -j( 1 2 -σ) |b j-|β|+q (2 k ξ)| 2 kq .
Performing a change of coordinates ξ → 2 k ξ and making some integration by parts one finds that

|K k 1 (x, y)| 2 kq (2 k h -1 ) 2 2 k h -1 (x -y) -3 , ∀(x, y),
and consequently that

sup x |K k (x, y)|dy 2 kq .
Summing up with (1.2.34), this gives us that

Op w h (A k (x, ξ)) = Op w h (A k s (x, ξ)) + Op w h (A k b (x, ξ)) + Op w h (A k 1 (x, ξ))
is a bounded operator on L p , for every 1 ≤ p ≤ +∞, with norm O(2 kq ).

Case II:

Let us now suppose that k ∈ K is such that h 1/2-σ < 2 k ≤ h -σ . From (1.2.35) we have that A k s (x, ξ) = A k s x √ h , √ hξ satisfies ∂ β ξ A k s (x, ξ) |β 1 |≤|β| h |β| 2 | √ hξ| -|β 1 | 2 -k(|β|-|β 1 |)+kq 1 |ξ|∼2 k h -1/2 ,
for every (x, ξ) ∈ R 2 × R 2 , and hence

∂ β ξ A k s (x, 2 k h -1/2 ξ) |β 1 |≤|β| 2 k|β| |2 k ξ| -|β 1 | 2 -k(|β|-|β 1 |)+kq 1 |ξ|∼1 2 kq 1 |ξ|∼1 .
By making a change of coordinates ξ → 2 k h -1/2 ξ, some integrations by parts and using the above inequality, one can show that kernel

K k s (x, y) associated to Op w h ( A k s (x, ξ)), i.e. K k s (x, y) = 1 (2πh) 2 e i h (x-y)•ξ A k s x + y 2 , ξ dξ, is such that | K k s (x, y)| 2 kq (2 k h -3 2 ) 2 2 k h -3 2 (x -y) -3 ∀(x, y), which implies that sup x | K k s (x,
w h (A 1 (x, ξ)) ∈ L(L p ) for every 1 ≤ p ≤ +∞. Let K k 1 (x, y) be its associated kernel, i.e. (1.2.37) K k 1 (x, y) = 1 (2πh) 2 e i h (x-y)•ξ γ x+y 2 |ξ| -ξ h 1/2-σ ψ(2 -k ξ)a x + y 2 b q (ξ)dξ,
and assume, without loss of generality, that γ(x) = γ(|x| 2 ). Set

x + y 2 = r[cos θ, sin θ],
with m ≤ r ≤ M on the support of θ 1 x+y 2

, and for fixed r, θ let

(1.2.38) ξ = ρ[cos θ, sin θ] + rΩ[-sin θ, cos θ].
We immediately notice that

[ ∂(ξ 1 ,ξ 2 ) ∂(ρ,Ω) ] = r ∼ 1 and that |ξ| 2 = ρ 2 + r 2 Ω 2 . Moreover, x + y 2 |ξ| -ξ 2 = r ρ 2 + r 2 Ω 2 -ρ 2 + r 2 Ω 2 .
If the support of γ is of size 0 < α 1 sufficiently small, from the above equality and the fact that |ξ| ∼ 2 k on the support of ψ(2

-k ξ), with h 1/2-σ < 2 k h -σ , we deduce that rΩ ≤ √ αh 1/2-σ and |ρ| ∼ |ξ| ∼ 2 k and rΩ |ρ| ≤ √ α.
Consequently

αh 1-2σ ≥ r ρ 2 + r 2 Ω 2 -ρ 2 ρ 2 |r -1| 2 .
The above left inequality implies that ρ > 0, inferring so the right one. Moreover

αh 1-2σ ≥ r ρ 2 + r 2 Ω 2 -ρ 2 + r 2 Ω 2 = ρ 2 (r -1) + r 1 + r 2 Ω 2 ρ 2 -1 2 + r 2 Ω 2 = ρ 2 |r -1| 2 + r 2 Ω 2 [1 + a(r, Ω, ρ)] ,
with a(r, Ω, ρ) bounded such that, for any l, m, n ∈ N,

|∂ l r ∂ m Ω ∂ n ρ a(r, Ω, ρ)| = O(ρ -(m+n) ).
If

Γ h := γ ρ 2 |r -1| 2 h 1-2σ + r 2 Ω 2 h 1-2σ [1 + a(r, Ω, ρ)] ψ(2 -k ρ 2 + r 2 Ω 2 )a(r, θ)b q (ρ),
from all the observations made above along with the fact that h -1/2+σ ρ -1 we deduce that, for any m, n ∈ N,

(1.2.39) ∂ m ρ Γ h = O(2 kq ρ -m ) and |∂ n Ω Γ h | = O(2 kq ρ -n ).
With the change of coordinates considered in (1.2.38), and setting

w := x -y, e θ := [cos θ, sin θ], kernel K k 1 (x, y) transforms into 1 (2πh) 2 e i h ρw•e θ + i h rΩw•e ⊥ θ Γ h rdρdΩ and is restricted to |ρ| ∼ 2 k , |Ω| h 1/2-σ
, so by making some integrations by parts, using (1.2.39), and reminding that |r -1| 2 -k h 1/2-σ 1 on the support of Γ h , we find that, for any N ∈ N,

|K k 1 (x, y)| h -3 2 -σ 2 k 2 k h w • e θ -N 2 k h w • e ⊥ θ -N 1 || x+y 2 |-1| 1 .
Now, as w = (x -y), e θ = x+y |x+y| , and |x

+ y| = 2r ∼ 1 on the support of Γ h , we have that |w • e θ | ∼ ||x| 2 -|y| 2 |, |w • e ⊥ θ | ∼ |(x + y)(x + y) ⊥ | ∼ 2|x • y ⊥ | = 2|x 1 y 2 -x 2 y 1 |, and consequently |K k 1 (x, y)| h -3 2 -σ 2 k(1+q) 2 k h |x| 2 -|y| 2 -N 2 k h (x 1 y 2 -x 2 y 1 ) -N 1 || x+y 2 |-1| 1 .
Successively, taking the L 1 (dy) norm of K k 1 (x, y) and using the above estimate we find that:

• if |x| |y| or |x| |x|, 2 k h |x| 2 -|y| 2 -N 1 || x+y 2 |-1| 1 h N ( 1 2 +σ) ,
as follows from the fact that h2 -k < h 1/2+σ . We obtain that

sup x |K k 1 (x, y)|dy h -3 2 2 k(1+q) h N ( 1 2 +σ)
1 by taking N ∈ N sufficiently large (e.g. N > 3) and σ > 0 small.

• if |x| ∼ |y|, we deduce that |x| ≥ c > 0 from the fact that x+y 2 -1 ≤ √ αh 1/2-σ 2 -k on the support of Γ h .
Without loss of generality we can assume that x = λe 1 (this always being possible by making a rotation) and |λ| ≥ c > 0. If w := x + y,

|x| 2 -|y| 2 = w • (x -y) = w • (2x -w) = w • (2λe 1 -w) = 2λw 1 -w 2 1 -w 2 2 ,
and then

|x| 2 -|y| 2 h = - (w 1 -λ) 2 -λ 2 h + w 2 √ h 2 while x 1 y 2 -x 2 y 1 = λw 2 .
This implies that

|K k 1 (x, y)| h -3 2 -σ 2 k(1+q) 2 k h (w 1 -λ) 2 -λ 2 -N 2 k h w 2 -N . Since |K k 1 (x, y)|dy = |K k 1 (
x, y)|dw, from the above estimate (with a fixed N ∈ N sufficiently large) this integral is bounded by 2 kq when restricted to |x| ∼ |y|. Indeed, when the integral is taken in a neighbourhood of w 1 = 0 or w 1 = 2λ, (w 1 -λ 2 ) -λ 2 can be considered as the variable of integration, and by a change of coordinates along with the fact that 2 -k < h -1/2+σ one deduces that

U 0 ∪U 2λ |K k 1 (x, y)|dw h -3 2 -σ 2 k(1+q) h 2 2 -2k 2 kq , where U 0 (resp. U 2λ ) is a neighbourhood of w 1 = 0 (resp. of w 1 = 2λ). Outside of U 0 ∪ U 2λ , 2 k h (w 1 -λ) 2 -λ 2 -N (h2 -k ) N w 1 -N h N ( 1 2 +σ) w 1 -N , so (U 0 ∪U 2λ ) |K k 1 (x, y)|dw h -3 2 -σ 2 k(1+q) h2 -k h N ( 1 2 +σ) 2 kq .
This finally proves that also Op w

h (A k 1 (x, ξ)) is a bounded operator on L p with norm O(2 kq ).
Let us introduce the Euclidean rotation in the semi-classical setting (1.2.40)

Ω h := x 1 hD 2 -x 2 hD 1 = Op w h (x 1 ξ 2 -x 2 ξ 1
∈ L 2 (R 2 ) such that Ω h w ∈ L 2 loc (R 2 ) (1.2.41) Op w h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) w L ∞ 2 kp h -1 2 -σ ( w L 2 + θ 0 Ω h w L 2 ) ,
where θ 0 is a smooth function supported in some annulus centred in the origin.

Proof. We prove the statement distinguishing between three spatial regions. For that, we introduce three cut-off functions:

θ s (x) supported for |x| ≤ m 1; θ b (x) supported for |x| ≥ M 1; θ(x) supported for m ≤ |x| ≤ M , for some 0 < m 1, M 1, such that θ s + θ b + θ ≡ 1. We define respectively A k s (x, ξ) := γ 1 x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)θ s (x), A k b (x, ξ) := γ 1 x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)θ b (x), and A k (x, ξ) := γ 1 x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)θ(x), so that γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) = A k s (x, ξ) + A k b (x, ξ) + A k (x, ξ).

The fact that Op

w h (A k s ), Op w h (A k b ) ∈ L(L 2 ; L ∞ ) and their norm is a O(2 kp h -1/
x|ξ|-ξ h 1/2-σ h -1/2+σ |ξ| (resp. x|ξ|-ξ h 1/2-σ h -1/2+σ |ξ||x| h -1/2+σ
|ξ|), so from lemma 1.2.26 we derive that

∂ α x ∂ β ξ γ 1 x|ξ| -ξ h 1/2-σ |β| j=0 h -(|α|+j)( 1 2 -σ) x|ξ| -ξ h 1/2-σ -1-|α|-j |b |α|+j-|β| (ξ)| h 1 2 -σ |ξ| -1-|β| . Consequently, as 2 -k h ≤ 1, we deduce that |∂ α x ∂ β ξ A k s ( x+y 2 , hξ) | 2 kp h -1/2-σ |ξ| -1 for any α, β ∈ N 2 . Therefore ∂ α y ∂ β ξ A k s x + y 2 , hξ L 2 (dξ) 2 kp h -1 2 -σ |ξ|∼2 k h -1 |ξ| -2 dξ 1 2 2 kp h -1 2 -σ .
The same holds for A k b (x, ξ) so, injecting these estimates in inequality (1.2.21), we derive that

Op w h (A k s (x, ξ))w L ∞ + Op w h (A k b (x, ξ))w L ∞ ≤ C2 kp h -1 2 -σ w L 2 . A different analysis is needed for Op w h (A k (x, ξ))
w, since it is no longer true that there exists a positive constant C such that |x|ξ| -ξ| ≥ C|ξ| on the support of A k (x, ξ). In this case we exploit the fact that A k (x, ξ) is supported in an annulus to perform a change of variables. If

θ 0 ∈ C ∞ 0 (R 2 \ {0}
) is a cut-off function equal to 1 on the support of θ we have that, for any

N ∈ N, A k (x, ξ) = θ 0 (x) A k (x, ξ) + r k N (x, ξ
) by means of proposition 1.2.21, where

r k N (x, ξ) = h 2i N N (πh) 4 |α|=N (-1) |α| α! e 2i h (η•z-y•ζ) 1 0 ∂ α x θ 0 (x + tz)(1 -t) N -1 dt × (∂ α ξ A k )(x, ξ + η) dydzdηdζ.
If we take N sufficiently large it turns out that the quantization of r k N satisfies a better estimate than (1.2.41). Indeed, using lemma 1.2.26 and integrating in dydζ, it can be rewritten as

(1.2.42) r k N (x, ξ) = j≤N h N -j( 1 2 -σ) (πh) 2 e 2i h η•z 1 0 θ 0 (x + tz)(1 -t) N -1 dt × γ 1+j x|ξ + η| -(ξ + η) h 1/2-σ ψ(2 -k (ξ + η))θ j (x)a(x)b p+j-N (ξ + η) dzdη,
for some new functions θ 0 , γ 1+j , ψ, θ j , a, b p+j-N . As it is compactly supported in x, by lemma 1.2.25 there is a new cut-off function (that we still call θ) such that

|Op w h (r k N (x, ξ))w| w L 2 θ x + y 2 |α |≤3 ∂ α y r k N x + y 2 , hξ L 2 (dξ)
dy.

One can check that the action of ∂ α y on r k N ( x+y 2 , hξ) makes appear factors (h -1/2+σ h|ξ + η|) i , for i ≤ |α |, without changing the underlining structure of r k N , and these are bounded by (h -1/2+σ 2 k ) i on the support of ψ(2 -k h(ξ + η)). After a change of variables η → hη in (1.2.42), we use that

e 2iη•z = 1-2iη•∂z 1+4|η| 2 3 1-2iz•∂η 1+4|z| 2 3 e 2iη•z
, integrate by parts, apply Young's inequality for convolutions, and fix N > 7, in order to deduce the following chain of inequalities:

∂ α y r k N x + y 2 , hξ 2
L 2 (dξ) i≤|α |,j≤N h 2N -2j( 1 2 -σ) h -1 2 +σ 2 k 2i 2 2k(p+j-N ) dξ z -3 η -3 |ψ(2 -k h(ξ + η))|dzdη 2 i≤|α |,j≤N h 2N -2j( 1 2 -σ) h -1 2 +σ 2 k 2i 2 2k(p+j-N ) |ψ(2 -k hξ)| 2 dξ i≤|α |,j≤N h 2N -2j( 1 2 -σ) h -1 2 +σ 2 k 2i 2 2k(p+j-N ) h -1 2 k 2 2 2kp ,
and that Op w h (r k N ) L(L 2 ;L ∞ ) 2 kp . We can then focus on the analysis of the L ∞ norm of θ 0 (x)Op w h (A k (x, ξ))w. In polar coordinates x = ρe iα operator Ω h reads as D α , so using the classical one-dimensional Sobolev injection with respect to variable α, the one-dimensional semiclassical Sobolev injection with respect to variable ρ, and successively returning back to coordinates x, we deduce that

θ 0 (x)Op w h (A k (x, ξ))w h -1 2 Op w h (A k )w L 2 (dx) + Op w h (ξ)Op w h (A k )w L 2 (dx) + Ω h θ 0 Op w h (A k )w L 2 (dx) + Op w h (ξ)Ω h θ 0 Op w h (A k )w L 2 (dx) 2 kp h -1 2 -σ [ w L 2 + θ 0 Ω h w L 2 ] .
The latter of above inequalities is derived observing that the commutator between Ω h and Op w h (A k ) is a semi-classical pseudo-differential operator whose symbol is linear combination of terms of the form

γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)θ(x)b p (ξ),
for some new γ

(I k p,q (x, ξ)) and Op w h (J k p,q (x, ξ)) are bounded operators from L ∞ to L 2 , with (1.2.43) Op w h (I k p,q (x, ξ)) L(L 2 ;L ∞ ) + Op w h (J k p,q (x, ξ)) L(L 2 ;L ∞ ) i≤6 2 k(p+q) (h -1 2 +σ 2 k ) i (h -1 2 k ).
The same result holds if q = 0 and ψ(2 -k ξ)b q (ξ) ≡ 1.

Proof. As in proposition 1.2.28, we prove the statement only for Op w h (I k p,q ), leaving to the reader to check that the result is true also for Op w h (J k p,q ).

Let w ∈ L 2 . After lemma 1.2.25 we should prove that

∂ α y ∂ β ξ I k p,q ( x+y 2 , hξ) L 2 (dξ)
is estimated by the right hand side of (1.2.43), for any |α|, |β| ≤ 3. A change of variables η → hη, ζ → hζ allows us to write I k p,q ( x+y 2 , hξ) as

1 π 4 e 2i(η•z-y •ζ) 1 0 γ h 1 2 +σ (x|ξ| -ξ) ψ(2 -k hξ)a(x)b p (hξ) | ( x+y 2 +tz,ξ+tζ) f (t)dt × ψ(2 -k h(ξ + η))a x + y 2 + y b q (h(ξ + η)) dy dzdηdζ.
We observe that, while on the one hand the action of ∂ α y on the above integral makes appear a factor (h -1 2 +σ |h(ξ + tζ)|) i , with i ≤ |α|, on the other hand that of ∂ β ξ has basically no effect on the L 2 norm that we want to estimate as one can check using lemma 1.2.26 and the fact that 2 -k h ≤ 1. With this in mind, we can reduce to the study of the L 2 (dξ) norm of an integral function as

i≤3 (h -1 2 +σ 2 k ) i e 2i(η•z-y •ζ) 1 0 γ h 1 2 +σ (x|ξ| -ξ) ψ(2 -k hξ)a(x)b p (hξ) | ( x+y 2 +tz,ξ+tζ) f (t)dt × ψ(2 -k h(ξ + η))a x + y 2 + y b q (h(ξ + η)) dy dzdηdζ,
for some new functions γ, ψ, a, b p , ψ, a , b q , with the same properties as their previous homonyms. We use that

e 2i(ηz-y •ζ) = 1 + 2iy • ∂ ζ 1 + 4|y | 2 3 1 -2iη • ∂ z 1 + 4|η| 2 3 1 -2iz • ∂ η 1 + 4|z| 2 3 1 + 2iζ • ∂ y 1 + 4|ζ| 2 3 e 2i(η•z-y •ζ)
and make some integration by parts to obtain the integrability in dy dzdηdζ, up to new factors (h -1 2 +σ |h(ξ + tζ)|) j , with j ≤ 3, coming out from the derivation of the integrand with respect to z. Then, using that functions h j b p-j (h(ξ + tζ)) (resp. h j b q-j (h(ξ + η))), j ≤ 3, appearing from the derivation of b p (h(ξ + tζ)) with respect to ζ (resp. the derivation of b q (h(ξ + η)) with respect to η), are such that

|h j b p-j (h(ξ + tζ))| ≤ h j 2 k(p-j) 2 kp on the support of ψ(2 -k h(ξ + tζ)) (resp. |h j b q-j (h(ξ + η))| ≤ 2 kq on the support of ψ(2 -k h(ξ + η)))
, and the fact that

η -3 | ψ(2 -k h(ξ + η))|dη L 2 (dξ) ≤ ψ(2 -k h•) L 2 h -1 2 k ,
we obtain the result of the statement.

The last part of the statement can be proved following an analogous scheme, after having previously made an integration in dzdη (or in dydζ if dealing with J k p,0 ).

Lemma 1.2.32. Let σ > 0 be sufficiently small, k ∈ K with K given by (1.2.23) and p, q ∈ N.

Let also ψ, ψ ∈ C ∞ 0 (R 2 \ {0}), a(x) be either a smooth compactly supported function or a ≡ 1, and f ∈ C(R). For a fixed integer N > 2(p + q) + 9 we define

(1.2.44) r k N,p (x, ξ) := h N (πh) 4 |α|=N e 2i h (η•z-y•ζ) 1 0 ∂ α x γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) | (x+tz,ξ+tζ) × f (t)dt ∂ α ξ b q (ξ) ψ(2 -k ξ) | (x+y,ξ+η) dydzdηdζ,

and

(1.2.45)

r k N,p (x, ξ) := h N (πh) 4 |α1|+|α2|=N e 2i h (η•z-y•ζ) 1 0 ∂ α1 x ∂ α2 ξ γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) | (x+tz,ξ+tζ) × f (t)dt ∂ α2 x ∂ α1 ξ x n b q (ξ) ψ(2 -k ξ) | (x+y,ξ+η) dydzdηdζ . Then (1.2.46) Op w h (r k N,p ) L(L 2 ) + Op w h ( r k N,p ) L(L 2 ) + Op w h (r k N,p ) L(L 2 ;L ∞ ) + Op w h ( r k N,p ) L(L 2 ;L ∞ ) h p+q .
Proof. We remind definition (1.2.30) of integral I k p,q (x, ξ) for general k ∈ K, p, q ∈ Z. After an explicit development of the derivatives appearing in (1.2.44) we find that r k N,p (x, ξ) may be written as

j≤N h N -j( 1 2 -σ) I k p+j,q-N (x, ξ)
where γ is replaced with γ 1 and a ≡ 1 in I k p+j,q-N . Propositions 1.2.28 and 1.2.31, combined with the fact that h ≤ 2 k ≤ h -σ , imply respectively that

Op w h (r k N,p ) L(L 2 ) j≤N h N -j( 1 2 -σ) 2 k(p+j+q-N ) j≤N p+j+q≤N h N -j( 1 2 -σ)+p+j+q-N + j≤N p+j+q>N h N -j( 1 2 -σ)-σ(p+j+q-N ) h p+q and Op w h (r k N,p ) L(L 2 ;L ∞ ) i≤6,j≤N h N -j( 1 2 -σ) 2 k(p+j+q-N ) (h -1 2 +σ 2 k ) i (h -1 2 k ) i≤6,j≤N p+i+j+q≤N -1 h N -1-(i+j)( 1 2 -σ)+p+i+j+q-N +1 + i≤6,j≤N p+i+j+q>N -1 h N -1-(i+j)( 1 2 -σ)-σ(p+i+j+q-N +1) h p+q , as N > 2(p + q) + 9.
As regards (1.2.45), we first observe that index α 2 is such that

|α 2 | ≤ 1 since x n b q (ξ) ψ(2 -k ξ) is linear in x n .
An explicit development of derivatives in (1.2.45), combined with lemma 1.2.26, shows that r k N,p (x, ξ) splits into two contributions:

J 0 (x, ξ) = h N (πh) 4 i≤N h -i( 1 2 -σ) e 2i h (η•z-y•ζ) 1 0 γ 1+i x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+i (ξ) | (x+tz,ξ+tζ) f (t)dt ×(x n + y n )b q-N (ξ + η) ψ(2 -k (ξ + η)) dydzdηdζ,
for some new functions a, ψ, ψ and clear meaning for γ i , b p+i , b q-N , coming out when |α 2 | = 0;

J 1 (x, ξ) = h N (πh) 4 i≤N -1,j≤1 h -(i+j)( 1 2 -σ) e 2i h (η•z-y•ζ) × 1 0 γ 1+i+j x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+i+j-1 (ξ) | (x+tz,ξ+tζ) f (t)dt × b q-N +1 (ξ + η) ψ(2 -k (ξ + η)) dydzdηdζ,
for some new other a, ψ, ψ, corresponding instead to |α 2 | = 1. One has that

J 1 (x, ξ) = i≤N -1,j≤1 h N -(i+j)( 1 2 -σ) I k p+i+j-1,q-N +1 (x, ξ),
with γ replaced with γ 1 and a ≡ 1, so propositions 1.2.28 and 1.2.31, along with the fact that N > 2(p + q) + 9, imply

Op w h (J 1 (x, ξ)) L(L 2 ) i≤N -1,j≤1 h N -(i+j)( 1 2 -σ) 2 k(p+i+j+q-N ) h p+q , Op w h (J 1 (x, ξ)) L(L 2 ;L ∞ ) i≤N -1,j≤1 l≤6 h N -(i+j)( 1 2 -σ) 2 k(p+i+j+q-N ) (h -1 2 +σ 2 k ) l (h -1 2 k ) h p+q .
In order to derive the same estimates for J 0 (x, ξ) we split the sum x n + y n and analyse separately the two out-coming integrals, that we denote J 0,x (x, ξ), J 0,y (x, ξ). In the latter one, we use that

y n e -2i h y•ζ = -h 2i ∂ ζn e -
(x, ξ) = i≤N,j≤1 h N +1-(i+j)( 1 2 -σ) e 2i h (η•z-y•ζ) × 1 0 γ 1+i+j x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+i+j-1 (ξ) | (x+tz,ξ+tζ) f (t)dt × b q-N (ξ -η) ψ(2 -k (ξ + η)) dydzdηdζ
for some new functions a, ψ, ψ, f . Again by propositions 1.2.28, 1.2.31 and the fact that h ≤ 2 k ≤ h -σ , N > 2(p + q) + 9, we deduce that:

(1.2.48a) Op w h (J 0,y (x, ξ)) L(L 2 ) i≤N,j≤1 h N +1-(i+j)( 1 2 -σ) 2 k(p+i+j+q-N -1) h p+q , (1.2.48b) Op w h (J 0,y (x, ξ)) L(L 2 ;L ∞ ) i≤N,j≤1 l≤6 h N +1-(i+j)( 1 2 -σ) 2 k(p+i+j+q-N -1) (h -1 2 -σ 2 k ) l (h -1 2 k ) h p+q .
In J 0,x (x, ξ) we first integrate in dydζ and then we split the occurring integral into two other contributions, called J 0,x+tz (x, ξ), J 0,tz (x, ξ), by writing

x n = (x n + tz n ) -tz n .
Similarly to what done above, we use that

z n e 2i h η•z = h 2i ∂ ηn e 2i h η
•z in J 0,tz , and successively integrate by parts in dη n : as 2 -k h ≤ 1, we obtain that J 0,tz has the same form as (1.2.47) for some new b q-N , ψ, and verifies (1.2.48). Finally, using that

x n + tz n = h 1 2 -σ (xn+tzn)|ξ|-ξn h 1/2-σ |ξ| -1 + ξ n |ξ| -1 , we derive that J 0,x+tz (x, ξ) = i≤N h N -(i-1)( 1 2 -σ) e 2i h η•z γ i x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+i-1 (ξ) | (x+tz,ξ) f (t)dt × b q-N (ξ + η) ψ(2 -k (ξ + η))dzdη, + i≤N h N -i( 1 2 -σ) e 2i h η•z γ 1+i x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+i (ξ) | (x+tz,ξ) f (t)dt × b q-N (ξ + η) ψ(2 -k (ξ + η))dzdη,
so by propositions 1.2.28 and 1.2.31

Op w h (J 0,x+tz (x, ξ)) L(L 2 ) i≤N h N -i( 1 2 -σ) 2 k(p+i+q-N ) h p+q , Op w h (J 0,x+tz (x, ξ)) L(L 2 ;L ∞ ) i≤N,l≤3 h N -i( 1 2 -σ) 2 k(p+i+q-N ) (h -1 2 +σ 2 k ) l (h -1 2 k ) h p+q .
That concludes the proof as r k N,p = J 0,x+tz + J 0,tz + J 0,y + J 1 .

We introduce the following operator:

(1.2.49)

M j := 1 h Op w h (x j |ξ| -ξ j ), j = 1, 2
and use the notation

M γ w = M γ 1 1 M γ 2 2 w for any γ = (γ 1 , γ 2 ) ∈ N 2 .
We have now all the ingredients to state and prove the following two results. Lemma 1.2.33. Let σ, k, p, ψ, a be as in lemma 1.2.32 and a(x) such that (a ≡ 1) ⇒ ( a ≡ 1), (a compactly supported ) ⇒ [( a ≡ 1) or ( a compactly supported and aa ≡ a)].

We have that

(1.2.50) Op w h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x n |ξ| -ξ n ) = Op w h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) a(x)hM n + Op w h (r k p (x, ξ)),
where

(1.2.51a) Op w h (r k p (x, ξ))w L 2 h 1-β w L 2 , (1.2.51b) Op w h (r k p (x, ξ))w L ∞ h 1 2 -β ( w L 2 + θ 0 Ω h w L 2 ),
for some θ ∈ C ∞ 0 (R 2 \ {0}) and a small β > 0, β → 0 as σ → 0. Moreover

(1.2.52a) Op w h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x n |ξ| -ξ n ) w L 2 h 1-β w L 2 + M n w L 2 , (1.2.52b) Op w h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x n |ξ| -ξ n ) w L ∞ h 1 2 -β 1 µ=0 (θ 0 Ω h ) µ w L 2 + (θ 0 Ω h ) µ M n w L 2 .
Proof. The proof of the statement is basically made of tedious calculations and the application of propositions 1.2.27, 1.2.30 along with lemma 1.2.32.

Let ψ ∈ C ∞ 0 (R 2 \ {0}) such that ψ ≡ 1 on the support of ψ. From formulas (1.2.18), (1.2.19) and the hypothesis of the statement we derive that for a fixed N ∈ N, and up to negligible multiplicative constants,

γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) (x n |ξ| -ξ n ) a(x) ψ(2 -k ξ) = γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x n |ξ| -ξ n ) + h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ), (x n |ξ| -ξ n ) + 2≤|α|<N |α 1 |+|α 2 |=|α| h |α| ∂ α 1 x ∂ α 2 ξ γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) ∂ α 2 x ∂ α 1 ξ (x n |ξ| -ξ n ) + r k N,p (x, ξ), (1.2.53) with
(1.2.54)

r k N,p (x, ξ) = h N (πh) 4 |α1|+|α2|=N e 2i h (η•z-y•ζ) 1 0 ∂ α1 x ∂ α2 ξ γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) | (x+tz,ξ+tζ) × (1 -t) N -1 dt ∂ α2 x ∂ α1 ξ (x n |ξ| -ξ n ) a(x) ψ(2 -k ξ) | (x+y,ξ+η) dydzdηdζ .
If a ≡ 1 above r k N,p can be decomposed into the sum of integrals of the form (1.2.44) and (1.2.45) with q = 1, so (1.2.55)

Op w h (r k N,p ) L(L 2 ) + Op w h (r k N,p ) L(L 2 ;L ∞ ) h 1+p if N is
I k p,q for general k ∈ K, p, q ∈ Z r k N,p (x, ξ) = |α 1 |+|α 2 |=N i≤|α 1 |,1≤j≤|α 2 | h N -(i+j)( 1 2 -σ) I k p+i+j-|α 2 |,1-|α 1 | (x, ξ).
An explicit computation of the Poisson bracket in (1.2.53) shows that it is equal to

(1.2.56) h(∂γ 1 ) x|ξ| -ξ h 1/2-σ x 1 ξ 2 -x 2 ξ 1 h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) + hγ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ),
where is a concise notation to indicate a linear combination, and ψ, a, b p are some new functions with the same features of their homonyms. After writing (1.2.57)

(x 1 ξ 2 -x 2 ξ 1 ) = (x 1 |ξ| -ξ 1 )ξ 2 |ξ| -1 -(x 2 |ξ| -ξ 2 )ξ 1 |ξ| -1 ,
we recognize that the quantization of (

h |α|-i( 1 2 -σ) γ 1+i x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+i+1-|α| (ξ) x µ n ,
for some new functions ψ, a. Observe that µ = 0 if a ≡ 1 because the derivation of γ 1

x|ξ|-ξ h 1/2-σ
|α 1 |-times with respect to x makes appear, inter alia, a factor |ξ| |α 1 | that allows us to rewrite

∂ α 1 ξ (x n |ξ| -ξ n ) from (x n |ξ| -ξ n ) + b 0 (ξ),
for some new b 0 , and

∂ α 1 z γ 1 (z)z n is of the form γ |α 1 | (z)). The second term, corresponding instead to |α 2 | = 1, is given by i≤|α|-1,j≤1 h |α|-(i+j)( 1 2 -σ) γ 1+i+j x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+i+j+1-|α| (ξ),
for some new other functions ψ, a. From propositions 1.2.27, 1.2.30 we then deduce that

Op w h (t k α )w L 2 (h |α| 2 -β + h 1+p ) w L 2 , (1.2.58a) Op w h (t k α )w L ∞ (h |α|-1 2 -β + h 1 2 +p )( w L 2 + θΩ h w L 2 ), (1.2.58b) which concludes that γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) (x n |ξ| -ξ n ) a(x) ψ(2 -k ξ) = γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x n |ξ| -ξ n ) + r k p (x, ξ),
with r k p satisfying (1.2.51). Finally, by symbolic calculus we have that, up to some multiplicative constants,

Op w h (x n |ξ| -ξ n ) a(x) ψ(2 -k ξ) = a(x)Op w h (x n |ξ| -ξ n ) ψ(2 -k ξ) + Op w h (r k (x, ξ)) = Op w h ( ψ(2 -k ξ)) a(x)hM n + h a(x)Op w h ((∂ ψ)(2 -k ξ)(2 -k |ξ|)) + Op w h ( r k (x, ξ))hM n + Op w h (r k (x, ξ)),
where

r k (x, ξ) = h (2π) 2 e 2i h η•z ∂ x a(x + tz)dt ∂ ξ (x n |ξ| -ξ n ) ψ(2 -k ξ) | (x,ξ+η) dzdη, r k (x, ξ) = h2 -k (2π) 2 e 2i h η•z ∂ x a(x + tz)dt (∂ ξ ψ)(2 -k (ξ + η))dzdη, are such that Op w h (r k 1 ) L(L 2 ) = O(h), Op w h ( r k 1 ) L(L 2 ) = O(1). An explicit computation shows also that [Ω h , Op w h (r k )] L(L 2 ) = O(h) and [Ω h , Op w h ( r k )] L(L 2 ) = O(1)
. Therefore, since ψ ≡ 1 on the support of ψ, a ≡ 1 on the support of a, one can use remark 1.2.22 together with propositions 1.2.28, 1.2.31, and also propositions 1.2.27, 1.2.30, to show that

Op w h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) Op w h (x n |ξ| -ξ n ) a(x) ψ(2 -k ξ) = Op w h γ 1 x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) a(x)hM n + Op w h (r k p (x, ξ)),
for a new Op w h (r k p (x, ξ)) satisfying (1.2.51a). This concludes the proof of (1.2.50) and of the entire statement by applying propositions 1.2.27, 1.2.30 to the first operator in the above right hand side. Lemma 1.2.34. Let σ > 0 be small, k ∈ K with K given by (1.2.23) and p ∈ N. Let also

γ ∈ C ∞ 0 (R 2 ) be equal to 1 in a neighbourhood of the origin, ψ ∈ C ∞ 0 (R 2 \ {0}), and a ∈ C ∞ 0 (R 2 ). For any function w ∈ L 2 (R 2 ) such that Mw ∈ L 2 (R 2 ), any m, n = 1, 2, we have that Op w h γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x m |ξ| -ξ m )(x n |ξ| -ξ n ) w = Op w h γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x m |ξ| -ξ m ) [hM n w] + O L 2 h 2-β ( w L 2 + Mw L 2 ) ,
with β > 0 small, β → 0 as σ → 0.

Proof. Let γ(z) := γ(z)z m and ψ ∈ C ∞ 0 (R 2 \ {0}) be identically equal to 1 on the support of ψ. We saw in the proof of the previous lemma that the symbolic product

γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ) [(x n |ξ| -ξ n ) ψ(2 -k ξ)]
develops as in (1.2.53), (1.2.54), with γ 1 replaced with γ and a ≡ 1. From (1.2.56), the fact that

{x m |ξ| -ξ m , x n |ξ| -ξ n } = 0 if m = n, (-1) m+1 (x 1 ξ 2 -ξ 2 x 1 ) if m = n, and that (x 1 ξ 2 -ξ 2 x 1 ) = (x 1 |ξ| -ξ 1 )ξ 2 |ξ| -1 -(x 2 |ξ| -ξ 2 )ξ 1 |ξ| -1
, we derive that the first order term of the mentioned symbolic development is a linear combination of products of the form

h 3 2 γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x j |ξ| -ξ j ),
for some new functions γ, ψ, a, and its quantization acting on w is a remainder as in the statement after estimate (1.2.52a).

The second order term is given, up to some negligible multiplicative constants, by

h 1+2σ |α|=2 (∂ α γ) x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a 1 (x)b p+1 (ξ)(x m |ξ| -ξ m ) + h 3 2 +σ |α|=1 (∂ α γ) x|ξ| -ξ h 1/2-σ ψ 2 (2 -k ξ)a 2 (x)b p+1 (ξ) + h 2 γ x|ξ| -ξ h 1/2-σ ψ 3 (2 -k ξ)a 3 (x)b p+1 (ξ),
for some new smooth compactly supported ψ 2 , ψ 3 , a 1 , a 2 , a 3 , and as the derivatives of γ vanish in a neighbourhood of the origin we can replace (∂ α γ)(z) with j γ j 1 (z)z j , γ 1 j (z) := (∂ α γ)(z)z j |z| -2 , when |α| = 1. The third order one is given by

h 3 2 +3σ |α|=3 (∂ α γ) x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a 1 (x)b p+1 (ξ)(x m |ξ| -ξ m ) + h 2 γ 1 x|ξ| -ξ h 1/2-σ ψ 1 (2 -k ξ)a 2 (x)b p+1 (ξ),
for some other ψ 1 , a 1 , a 2 and a new γ 1 ∈ C ∞ 0 (R 2 ). Using estimate (1.2.52a) for the summations in α and proposition 1.2.27 for the remaining terms in the above expressions we obtain that the quantizations of the second and third order term are also a O L 2 h 2-β ( w L 2 + Mw L 2 ) when acting on w, for a small β > 0, β → 0 as σ → 0.

In all the other |α|-order terms, with 4 ≤ |α| ≤ N -1, and in integral remainder r k N,p , we look at

γ x|ξ|-ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x m |ξ| -ξ m ) as a symbol of the form γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p+1 (ξ)
for a new a 1 ∈ C ∞ 0 (R 2 ). From (1.2.58a) and (1.2.55) when N > 11, we derive that the quantizations of these terms are also a O L 2 h 2-β ( w L 2 + Mw L 2 ) when acting on w.

We finally obtain that

Op w h γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x m |ξ| -ξ m )(x n |ξ| -ξ n ) w = Op w h γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x m |ξ| -ξ m ) Op w h (x n |ξ| -ξ n ) ψ(2 -k ξ) + O L 2 h 2-β ( w L 2 + Mw L 2 ) ,
and the conclusion of the proof comes then from the fact that, by symbolic calculus,

Op w h (x n |ξ| -ξ n ) ψ 1 (2 -k ξ) = hOp w h ( ψ 1 (2 -k ξ))M n - h 2i Op w h (∂ ψ 1 )(2 -k ξ) • (2 -k ξ) ,
and by remark 1.2.22 as all derivatives of ψ vanish on the support of ψ.

The following lemma is introduced especially for the proof of lemma 3.2.13. Even if quite similar to lemma 1.2.33, we are going to see that the particular structure of symbolic product in the left hand side of (1.2.59) allows for a remainder r k p satisfying enhanced estimate (1.2.60b) rather than (1.2.51b). Lemma 1.2.35. Let us take σ > 0 sufficiently small, k ∈ K and p, q ∈ N. Let also γ ∈ C ∞ 0 (R 2 ) such that γ ≡ 1 in a neighbourhood of the origin, ψ, ψ ∈ C ∞ 0 (R 2 \ {0}) such that ψ ≡ 1 on the support of ψ, a(x) be a smooth compactly supported function. Then

(1.2.59) (x n |ξ| -ξ n ) ψ(2 -k ξ)a(x)b p (ξ) γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ) = γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(x n |ξ| -ξ n ) + r k p (x, ξ),
where

(1.2.60a) Op w h (r k p (x, ξ))w L 2 h 3 2 -β ( w L 2 + Mw L 2 ) + h 1+p w L 2 , (1.2.60b) Op w h (r k p (x, ξ))w L ∞ h 1-β 1 µ=0 (θ 0 Ω h ) µ w L 2 + (θ 0 Ω h ) µ Mw L 2 ,
for some θ ∈ C ∞ 0 (R 2 \ {0}), and a small β > 0, β → 0 as σ → 0.

Proof. Using proposition 1.2.21, for a fixed N ∈ N and up to multiplicative constants independent of h, k, we have the following symbolic development:

(x n |ξ| -ξ n ) ψ(2 -k ξ)a(x)b p (ξ) γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ) = γ x|ξ| -ξ h 1/2-σ ψ(2 -k )a(x)b p (ξ)(x n |ξ| -ξ n ) + h (x n |ξ| -ξ n ) ψ(2 -k ξ)a(x)b p (ξ), γ x|ξ| -ξ h 1/2-σ + α=(α 1 ,α 2 ) 2≤|α|<N h |α| ∂ α 1 x ∂ α 2 ξ (x n |ξ| -ξ n ) ψ(2 -k ξ)a(x)b p (ξ) ∂ α 2 x ∂ α 1 ξ γ x|ξ| -ξ h 1/2-σ + r k N,p (x, ξ), (1.2.61)
with

r k N,p (x, ξ) = h N (πh) 4 |α1|+|α2|=N e 2i h (η•z-y•ζ) 1 0 ∂ α1 x ∂ α2 ξ (x n |ξ| -ξ n )a(x)b p (ξ) ψ(2 -k ξ) | (x+tz,ξ+tζ) × (1 -t) N -1 dt ∂ α2 x ∂ α1 ξ γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ) | (x+y,ξ+η) dydzdηdζ.
For sake of simplicity, we denote by t k 1 (resp. t k α , |α| = 2, . . . , N -1) the Poisson brackets (resp. the |α|-th contribution) in (1.2.61). An explicit computation of t k 1 , combined with the fact that

x 1 ξ 2 -x 2 ξ 1 = (x 1 |ξ| -ξ 1 )ξ 2 |ξ| -1 -(x 2 |ξ| -ξ 2 )ξ 1 |ξ| -1 , shows that it is linear combination of terms of the form h(∂γ) x|ξ| -ξ h 1/2-σ x j |ξ| -ξ j h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ),
for j ∈ {1, 2} and some new functions ψ, a, b p , so by inequalities (1.2.52) we derive that (1.2.62a) Op w h (t k 1 )w

L 2 h 3 2 -β ( w L 2 + Mw L 2 ) , (1.2.62b) Op w h (t k 1 )w L ∞ h 1-β 1 µ=0 ( (θ 0 Ω h ) µ w L 2 + (θ 0 Ω h ) µ Mw L 2 ).
The improvement of these estimates with respect to (1.2.51) is attributable to the choice of ψ identically equal to 1 on the support of ψ. All derivatives of ψ vanish against ψ, so in the development of t k 1 we avoid terms like

γ x|ξ|-ξ| h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)(∂ψ)(2 -k ξ)(2 -k |ξ|), coming out from {x n |ξ| -ξ n , ψ(2 -k ξ)}γ x|ξ|-ξ| h 1/2-σ ψ(2 -k ξ)a(x)b p (ξ)
, that do not enjoy estimates like (1.2.62). Using formula (1.2.25) and looking at (x n |ξ| -ξ n ) ψ(2 -k ξ)a(x)b p (ξ) as a linear combination of terms ψ(2 -k ξ)a(x)b p+1 (ξ), for some new ψ, a, b p+1 , we realize that, for any 2 ≤ |α| < N ,

t k α = |α 1 |+|α 2 |=|α| 1≤j≤|α 1 | h |α|-(j+|α 2 |)( 1 2 -σ) γ j+|α 2 | x|ξ| -ξ h 1/2-σ ψ(2 -k ξ)a j (x)b p+j+1-|α 1 | (ξ),
for some new other ψ, a j , with a j compactly supported, and then that

Op w h (t k α )w L 2 |α 1 |+|α 2 |=|α| 1≤j≤|α 1 | h |α|-(j+|α 2 |)( 1 2 -σ) 2 k(p+j+1-|α 1 |) w L 2 , Op w h (t k α )w L ∞ |α 1 |+|α 2 |=|α| 1≤j≤|α 1 | h |α|-(j+|α 2 |)( 1 2 -σ) 2 k(p+j+1-|α 1 |) h -1 2 -σ 1 µ=0 ( (θ 0 Ω h ) µ w L 2 + (θ 0 Ω h ) µ Mw L 2 ),
(t k α )w L ∞ h 1-β 1 µ=0 ( (θ 0 Ω h ) µ w L 2 + (θ 0 Ω h ) µ Mw L 2 )
. For |α| = 2, we exploit the fact that functions γ j+|α 2 | vanish in a neighbourhood of the origin, as they come from γ's derivatives, and define γ i j+|α 2 | (z) := γ j+|α 2 | (z)z i |z| -2 , i = 1, 2, so that

t k α = |α1|+|α2|=|α| 1≤j≤|α1|,i=1,2 h |α|-(j+|α2|)( 1 2 -σ) γ i j+|α2| x|ξ| -ξ h 1/2-σ x i |ξ| -ξ i h 1/2-σ ψ(2 -k ξ)a j (x)b p+j+1-|α1| (ξ),
to which we can then apply lemma 1.2.33. After inequalities (1.2.52), Op w h (t k α ) with |α| = 2 also satisfies (1.2.62).

Finally, reminding definition (1.2.31) of J k p,q (x, ξ) for general k ∈ K, p, q ∈ Z, and developing derivatives in r k N,p using lemma 1.2.26, we observe that

r k N,p = |α 1 |+|α 2 |=N 0≤j≤|α 1 | h N -(|α 2 |+j)( 1 2 -σ) J k p+1-|α 2 |,|α 2 |+j-|α 1 | (x, ξ),
hence propositions 1.2.28 and 1.2.31 give that

Op w h (r k N,p ) L(L 2 ) |α 1 |+|α 2 |=N 0≤j≤|α 1 | h N -(|α 2 |+j)( 1 2 -σ) 2 k(p+1+j-|α 1 |) h 1+p , Op w h (r k N,p ) L(L 2 ;L ∞ ) |α 1 |+|α 2 |=N 0≤j≤|α 1 |,i≤6 h N -(|α 2 |+j)( 1 2 -σ) 2 k(p+1+j-|α 1 |) (h -1 2 +σ 2 k ) i (h -1 2 k ) h 1+p ,
if N is chosen sufficiently large (e.g. N > 10 + 2p). We should also highlight the fact that, at the difference of (1.2.60b), (1.2.60a) does not improve (1.2.51a): if we get a h 3 2 -β factor in front of the first term in the right hand side, the second term h 1+p w L 2 is just a O(h 1-β ) in the case p = 0, coming from |α 1 | = N , j = |α 2 | = 0, p = 0 above.

Operators for the Klein-Gordon solution: some estimates

This subsection is mostly devoted to the introduction of some symbols and operators, along with their properties, that we will often use in the paper when dealing with the Klein-Gordon component of the solution to starting system (1.1.1). From now on we will use the notation p(ξ) := 1 + |ξ| 2 (thus, p (ξ) denotes the gradient of p(ξ), p (ξ) = (∂ 2 ij p(ξ)) ij the 2 × 2 Hessian matrix of p(ξ)).

Proposition 1.2.36 is a general result about continuity on spaces H s h (R 2 ) of operators with symbols of order r ∈ R and generalises theorem 7.11 in [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]. Proposition 1.2.37 is a result of continuity from L 2 to H ρ,∞ h of a particular class of operators that will act on the Klein-Gordon component. In the spirit of [START_REF] Ifrim | Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension[END_REF] for the Schrödinger equation, it allows to pass from uniform norms to the L 2 norm losing only a power h -1 2 -β for a small β > 0 instead of a h -1 as for the semi-classical Sobolev injection. Proposition 1.2.39 is, instead, a result of uniform L p -L p continuity of such operators, for every 1 ≤ p ≤ +∞.

Proposition 1.2.36 (Continuity on H

s h ). Let s ∈ R. Let a ∈ S δ,σ ( ξ r ), r ∈ R, δ ∈ [0, 1 2 ], σ ≥ 0. Then Op w h (a) is uniformly bounded : H s h (R 2 ) → H s-r h (R 2
) and there exists a positive constant C independent of h such that

Op w h (a) L(H s h ;H s-r h ) ≤ C , ∀h ∈]0, 1] . Proposition 1.2.37 (Continuity from L 2 to H ρ,∞ h ). Let ρ ∈ N. Let a ∈ S δ,σ ( x-p (ξ) √ h -1 ), δ ∈ [0, 1 2 ], σ > 0. Then Op w h (a) is bounded : L 2 (R 2 ) → H ρ,∞ h (R 2
) and there exists a positive constant C independent of h such that

Op w h (a) L(L 2 ;H ρ,∞ h ) ≤ Ch -1 2 -β , ∀h ∈]0, 1] ,
where β > 0 depends linearly on σ.

Proof. We first remark that, after definition 1.2.18

(i) of the H ρ,∞ h norm, Op w h (a)w H ρ,∞ h = hD x ρ Op w h (a)w L ∞ ,
and that, by symbolic calculus of lemma 1.2.24, ξ ρ a(x, ξ) belongs to S δ,σ ( ξ ρ x-p (ξ)

√ h -1 ) ⊂ h -ρσ S δ,σ ( x-p (ξ) √ h -1
). This means that estimating the H ρ,∞ h norm of an operator whose symbol is rapidly decaying in |h σ ξ| corresponds actually to estimate the L ∞ norm of an operator associated to another symbol (namely, ã(x, ξ) = ξ ρ a(x, ξ)) which is still in the same class as a, up to a small loss h -ρσ . From definition 1.2.17 (i) of Op w h (a)w, and using a change of coordinates y → √ hy, ξ → √ hξ, integration by part, Cauchy-Schwarz inequality, and Young's inequality for convolutions, we derive what follows:

|Op w h (a)w| = = 1 (2π) 2 e i( x √ h -y)•ξ a x + √ hy 2 , √ hξ w( √ hy) dydξ = 1 (2π) 4 h ŵ η √ h dη e i( x √ h -y)•ξ+iη•y a x + √ hy 2 , √ hξ dydξ = 1 (2π) 4 h ŵ η √ h 1 -i x √ h -y • ∂ ξ 1 + | x √ h -y| 2 3 1 + i(ξ -η) • ∂ y 1 + |ξ -η| 2 3 e i( x √ h -y)•ξ+iη•y × a x + √ hy 2 , √ hξ dydξdη 1 h ŵ η √ h x √ h -y -3 ξ -η -3 h σ √ hξ -N x+ √ hy 2 -p ( √ hξ) √ h -1 dydξdη 1 h ŵ • √ h L 2 η -3 L 1 (η) x √ h -y -3 h σ √ hξ -N x+ √ hy 2 -p ( √ hξ) √ h -1 dy L 2 (dξ) h -1 2 w L 2 x √ h -y -3 h σ √ hξ -N x+ √ hy 2 -p ( √ hξ) √ h -1 L 2 (ξ)
dy ,

(1.2.63)

where N > 0 will be properly chosen later. We draw attention to two facts: in the third equality in (1.2.63) we use that

1 -i( x √ h -y) • ∂ ξ 1 + ( x √ h -y) 2 3 1 + i(ξ -η) • ∂ y 1 + (ξ -η) 2 3 e i( x √ h -y)•ξ+iη•y = e i( x √ h -y)•ξ+iη•y so, integrating by part, derivatives ∂ y , ∂ ξ fall on x √ h -y -1 , ξ -η -1 (

In order to estimate h

σ √ hξ -N x+ √ hy 2 -p ( √ hξ) √ h -1 L 2 ξ we first introduce a smooth cut-off func- tion χ( x+ √ hy 2
), with χ supported in some ball B C (0), to distinguish between the case when x+ | → +∞. In the latter situation, say for

| x+ √ hy 2 | > 2, we have x+ √ hy 2 -p ( √ hξ) √ h -1 √ h and (1 -χ) x + √ hy 2 h σ √ hξ -N x+ √ hy 2 -p ( √ hξ) √ h -1 L 2 (dξ) h -σ .
On the other hand, when x+ √ hy 2 is bounded we consider a Littlewood-Paley decomposition and write

h σ √ hξ -N x+ √ hy 2 -p ( √ hξ) √ h -1 2 L 2 (ξ) = h -1 k≥0 h σ ξ -2N x+ √ hy 2 -p (ξ) √ h -2 ϕ k (ξ)dξ = h -1 k≥0 I k (1.2.64)
where

I 0 = h σ ξ -2N x+ √ hy 2 -p (ξ) √ h -2
ϕ 0 (ξ)dξ and

I k = h σ ξ -2N x+ √ hy 2 -p (ξ) √ h -2 ϕ(2 -k ξ)dξ = 2 2k h σ 2 k ξ -2N x+ √ hy 2 -p (2 k ξ) √ h -2 ϕ(ξ)dξ 2 (-2N +2)k h -2σN x+ √ hy 2 -p (2 k ξ) √ h -2 ϕ(ξ)dξ . k ≥ 1 (1.2.65) For a fixed k 0 and any k ≤ k 0 , | det(p (2 k ξ))| ≥ C > 0 on the support of ϕ. For k ≥ k 0 , function ξ → g k (ξ) = 2 3k ( x+ √ hy 2 ) -2 3k p (2 k ξ) is such that det(g k (ξ)) = 2 4k (1+|2 k ξ| 2 ) 2 and |det(g k (ξ))| ∼ 1
on the support of ϕ. We may thus split the dξ integral in a finite number (independent of k) of integrals, computed on compact domains, on which ξ → g k (ξ) is a change of variables with Jacobian of size 1. We are then reduced to estimate 2

(-2N +2)k h -2σN |z|≤C z+g k (ξ 0 ) 2 3k √ h -2 dz,
where C is a positive constant and ξ 0 is in suppϕ. Since we assumed that x+

√ hy 2 is bounded, |g k (ξ 0 )| = O(2 3k
) and we get

I k 2 (-2N +2)k h -2σN |z| 2 3k z 2 3k √ h -2 dz 2 (-2N +8)k h -2σN h |z| h -1/2 z -2 dz 2 (-2N +8)k h -2σN +1 log(h -1 ) .
Taking the sum of all I k for k ≥ 0 we then deduce that

h σ √ hξ -N x+ √ hy 2 -p ( √ hξ) √ h -1 L 2 (ξ) h -σN -δ k≥0 2 (-2N +8)k 1 2 h -σN -δ ,
for δ > 0 as small as we want, if we choose N > 0 such that -2N + 8 < 0 (e.g. N = 5). Finally

Op w h (a) L(L 2 ;H ρ,∞ h ) = O(h -1 2 -β ) ,
where β(σ) = (N + ρ)σ + δ.

The following lemma is as simple as useful and will be largely recalled from subsection 3.2.1 on. It is also useful to introduce now the following manifold (1.2.66)

Λ kg := {(x, ξ) ∈ R 2 × R 2 : x -p (ξ) = 0}
which appears to be the graph of function ξ = -dφ(x), with φ(x) = 1 -|x| 2 (see picture 3.1).

Lemma 1.2.38. Let γ, χ ∈ C ∞ 0 (R 2 ) be equal to 1 in a neighbourhood of the origin and with sufficiently small support, and σ > 0 be small. There exists a family of smooth functions θ h (x), real valued, equal to 1 for |x| ≤ 1 -ch 2σ and supported for |x| ≤ 1 -c 1 h 2σ , for some 0

< c 1 < c, with ∂ α θ h L ∞ = O(h -2|α|σ ) and (h∂ h ) k θ h bounded for every k ∈ N, such that (1.2.67) γ x -p (ξ) √ h χ(h σ ξ) = θ h (x)γ x -p (ξ) √ h χ(h σ ξ).
Proof. Straightforward after observing that function γ x-p (ξ)

√ h χ(h σ ξ) is localized around man- ifold Λ kg , meaning that its support is included in {(x, ξ)||ξ| h -σ , |x| ≤ 1 -ch 2σ }, for a small c > 0. Proposition 1.2.39 (Continuity from L p to L p ). Let γ, χ ∈ C ∞ 0 (R 2
) be equal to 1 in a neighbourhood of the origin and with sufficiently small support, Σ(ξ) = ξ ρ with ρ ∈ N, and σ > 0.

Then Op w h γ x-p (ξ) √ h χ(h σ ξ)Σ(ξ) : L p → L p is bounded and its L(L p ) norm is estimated by h -σρ-β
, for a small β > 0, β → 0 as σ → 0, for every 1 ≤ p ≤ +∞.

Proof. From lemma 1.2.38 and the fact that γ x-p (ξ) √ h χ(h σ ξ) is supported in a neighbourhood of Λ kg introduced above, we can find a new smooth cut-off function γ 1 , suitably supported, so that

Op w h γ x -p (ξ) √ h χ(h σ ξ)Σ(ξ) = Op w h γ x -p (ξ) √ h χ(h σ ξ)Σ(ξ)γ 1 ξ + dφ(x) h 1/2-β θ h (x)
where β > 0 is a small constant, β → 0 as σ → 0, that takes into account the degeneracy of the equivalence between the two equations of Λ kg when approaching the boundary of suppθ h .

Denoting γ x-p (ξ) √ h χ(h σ ξ)Σ(ξ) concisely by A(x, ξ) and looking at the kernel associated to the above operator

K(x, y) := 1 (2πh) 2 e i h (x-y)•ξ A x + y 2 , ξ γ 1 ξ + dφ( x+y 2 ) h 1/2-β θ h x + y 2 dξ = e -i h (x-y)•dφ( x+y 2 ) (2πh) 2 θ h x + y 2 e i h (x-y)•ξ A x + y 2 , ξ -dφ x + y 2 γ 1 ξ h 1/2-β dξ, we observe that, since x √ h α e i h (x-y)•ξ = √ h i |α| ∂ α ξ e i h (x-y)•ξ and h |α|/2 ∂ α ξ A( x+y 2 , ξ) is bounded by h -σρ for any α ∈ N 2
, by making some integration by parts

x √ h α K(x, y) h -2-σρ |ξ| h 1/2-β dξ h -1-σρ-2β , ∀(x, y) ∈ R 2 × R 2 .
This means in particular that

|K(x, y)| h -1-σρ-2β x √ h -3 , |K(x, y)| h -1-σρ-2β y √ h -3
, ∀(x, y)

implying that sup x |K(x, y)|dy h -σρ-2β , sup y |K(x, y)|dx h -σρ-2β .
The operator associated to K(x, y) is hence bounded on L p with norm O(h -σρ-2β ), for every 1 ≤ p ≤ +∞.

The following lemma shows that we have nice upper bounds for operators whose symbol is supported for large frequencies |ξ| ≥ h -σ , σ > 0, when acting on functions w that belong to H s h , for some large s. We state it in space dimension 2 but it is clear that it holds true in general space dimension d ≥ 1. This result is useful when we want to reduce to symbols rapidly decaying in |h σ ξ|, for example in the intention of using proposition 1.2.37 or when we want to pass from a symbol of a certain positive order to another one of order zero, up to small losses of order O(h -β ), β > 0 depending linearly on σ. We can always split a symbol using that 1 = χ(h σ ξ)+(1-χ)(h σ ξ), for a smooth χ equal to 1 close to the origin, and consider as remainders all contributions coming from the latter. Lemma 1.2.40. Let s ≥ 0 and χ ∈ C ∞ 0 (R 2 ), χ ≡ 1 in a neighbourhood of zero. Then

Op w h ((1 -χ)(h σ ξ))w H s h ≤ Ch σ(s-s ) w H s h , ∀s > s .
Proof. The result is a simple consequence of the fact that

(1-χ)(h σ ξ) is supported for |ξ| h -σ , because Op w h ((1 -χ)(h σ ξ))w 2 H s h = (1 + |hξ| 2 ) s |(1 -χ)(h σ hξ)| 2 | ŵ(ξ)| 2 dξ = (1 + |hξ| 2 ) s (1 + |hξ| 2 ) s -s |(1 -χ)(h σ hξ)| 2 | ŵ(ξ)| 2 dξ ≤ Ch 2σ(s-s ) w 2 H s h ,
where the last inequality follows from an integration on |hξ| h -σ and from the fact that s -s < 0, (1 + |hξ| 2 ) s -s ≤ Ch -2σ(s -s) .

We introduce the following operator:

(1.2.68)

L j := 1 h Op w h (x -p j (ξ)), j = 1, 2,
and use the notation

L γ w = L γ 1 1 L γ 2 2 w for any γ = (γ 1 , γ 2 ) ∈ N 2 . Lemma 1.2.41. Let γ ∈ C ∞ 0 (R 2
) be equal to 1 in a neighbourhood of the origin, c(x, ξ) ∈ S δ,σ (1) with δ ∈ [0, 1 2 [ and σ > 0. Then γ( x-p (ξ)

√ h )c(x, ξ) belongs to S 1 2 ,σ (1) x-p (ξ) √ h -N , for all N ≥ 0.
Proof. Straightforward.

Lemma 1.2.42. Let n ∈ N and γ n (z) be a smooth function such that |∂ α γ n (z)| z -|α|-n for all α ∈ N 2 . Let also c(x, ξ) ∈ S δ,σ (1), with δ ∈ [0, 1 2 [, σ > 0, be supported for |ξ| h -σ . Up to some multiplicative constants independent of h, we have the following equality:

(1.2.69) c(x, ξ)γ n x -p (ξ) √ h x j -p j (ξ) = c(x, ξ)γ n x -p (ξ) √ h x j -p j (ξ) + hγ n x -p (ξ) √ h (∂ ξ j c) + (∂ x c) • (∂ ξ p j ) + h |α|=2 (∂ α γ n ) x -p (ξ) √ h c(x, ξ)(∂ α ξ p j )(ξ) + r(x, ξ), with r ∈ h 3/2-δ S 1 2 ,σ x-p (ξ) √ h -n , and if χ ∈ C ∞ 0 (R 2 ) is such that χ(h σ ξ) ≡ 1 on the support of c(x, ξ), (1.2.70a) Op w h c(x, ξ)γ n x -p (ξ) √ h (x j -p j (ξ)) v L 2 1 |γ|=0 h 1-β Op w h (χ(h σ ξ))L γ v L 2 , (1.2.70b) Op w h c(x, ξ)γ n x -p (ξ) √ h (x j -p j (ξ)) v L ∞ 1 |γ|=0 h 1 2 δn-β Op w h (χ(h σ ξ))L γ v L 2 ,
where δ n = 1 if n > 0, 0 otherwise, and β > 0 is small, β → 0 as δ, σ → 0.

Moreover, if n ∈ N * and ∂ α γ n vanishes in a neighbourhood of the origin whenever |α| ≥ 1, we also have that

(1.2.71a) Op w h c(x, ξ)γ n x -p (ξ) √ h (x i -p i (ξ))(x j -p j (ξ)) v L 2 0≤|γ|≤2 h 2-β Op w h (χ(h σ ξ))L γ v L 2 , (1.2.71b) Op w h c(x, ξ)γ n x -p (ξ) √ h (x i -p i (ξ))(x j -p j (ξ)) v L ∞ 0≤|γ|≤2 h 3 2 -β Op w h (χ(h σ ξ))L γ v L 2 . Proof. As c(x, ξ)γ n x-p (ξ) √ h ∈ S 1 2 ,σ x-p (ξ) √ h -n and ∂ α x,ξ (x j -p j (ξ)) ∈ S 0,0 (
c(x, ξ)γ n x -p (ξ) √ h , x j -p j (ξ) = γ n x -p (ξ) √ h (∂ ξ j c) + (∂ x c) • (∂ ξ p j ) ,
and that, up to some multiplicative negligible,

h 2 |α|=2 ∂ α x c(x, ξ)γ n x -p (ξ) √ h (∂ α ξ p j )(ξ) = h |α|=2 (∂ α γ n ) x -p (ξ) √ h c(x, ξ)(∂ α ξ p j )(ξ) + h 3 2 |α|=2 |α1|,|α2|=1 (∂ α1 γ n ) x -p (ξ) √ h (∂ α2 x c)(x, ξ)(∂ α ξ p j )(ξ) + h 2 |α|=2 γ n x -p (ξ) √ h (∂ α x c)(x, ξ)(∂ α ξ p j )(ξ) ∈h 3 2 -δ S 1 2 ,σ x-p (ξ) √ h -n .
If χ is a cut-off function as in the statement, its derivatives vanish on the support of c(x, ξ), and from remark 1.2.22 1) , which gives that 1) . As ∂ α γ n-1 vanishes in a neighbourhood of the origin for |α| = 2 by the hypothesis made on γ n , we can rewrite it as 2 l=1 γ l n+2 (z)z l , where γ l n+2 (z) := n+2) . Then, using again equality (1.2.69) for all products different from r(x, ξ) in the above right hand side (with c replaced with h δ [(∂ ξ j c)-(∂ x c) • (∂ ξ p j )] in the second addend, and γ n and c replaced with γ l n+2 and c(∂ α ξ p j ) respectively in the third one, l = 1, 2) we find that We will also need the following result, which is detailed in lemma 1.2.6 in [START_REF] Delort | Semiclassical microlocal normal forms and global solutions of modified onedimensional KG equations[END_REF] for the onedimensional case.

(1.2.73) c(x, ξ)γ n x -p (ξ) √ h = c(x, ξ)γ n x -p (ξ) √ h χ(h σ ξ) + r ∞ (x, ξ) with r ∞ ∈ h N S 1 2 ,σ ( x-p (ξ) √ h -n ), N ∈ N
(z) = γ n (z)z i , where |∂ α γ n-1 (z)| z -|α|-(n-
c(x, ξ)γ n x -p (ξ) √ h (x i -p i (ξ))(x j -p j (ξ)) = c(x, ξ)γ n x -p (ξ) √ h (x i -p i (ξ)) (x j -p j (ξ)) -hγ n x -p (ξ) √ h (x i -p i (ξ)) (∂ ξ j c) + (∂ x c) • (∂ ξ p j ) -h 3 2 |α|=2 (∂ α γ n-1 ) x -p (ξ) √ h c(x, ξ)(∂ α ξ p j )(ξ) - √ hr(x, ξ), with r ∈ h 3 2 -δ S 1 2 ,σ x-p (ξ) √ h -(n-
(∂ α γ n-1 )(z)z l |z| -2 is such that |∂ β γ l n+2 (z)| z -|β|-(
c(x, ξ)γ n x -p (ξ) √ h (x i -p i (ξ))(x j -p j (ξ)) = c(x, ξ)γ n x -p (ξ) √ h (x i -p i (ξ)) (x j -p j (ξ)) + hr 1 (x, ξ) (x j -p j (ξ)) - √ hr(x, ξ), for a new r 1 ∈ h -δ S 1 2 ,σ x-p (ξ) √ h -n .
Lemma 1.2.43. Let γ ∈ C ∞ 0 (R 2 ), and φ(x) = 1 -|x| 2 . If the support of γ is sufficiently small, (x k -p k (ξ))γ ξ 2 (x -p (ξ)) = 2 l=1 e k l (x, ξ)(ξ l + d l φ(ξ)), (1.2.74a) (ξ k + d k φ(x))γ ξ 2 (x -p (ξ)) = 2 l=1 e k l (x, ξ)(x l -p l (ξ)), (1.2

.74b)

for any k = 1, 2, where functions e k l (x, ξ), e k l (x, ξ) are such that, for any α, β ∈ N 2 ,

|∂ α x ∂ β ξ e k l (x, ξ)| αβ ξ -3+2|α|-|β| , (1.2.75a) |∂ α x ∂ β ξ e k l (x, ξ)| αβ ξ 3+2|α|-|β| , (1.2.75b) for any k, l = 1, 2.
Chapter 2

Energy Estimates

The aim of this chapter is to write an energy inequality for E n (t; u ± , v ± ) and E k 3 (t; u ± , v ± ) respectively, which allows us to propagate the a-priori energy estimates made in theorem 1.1.2, i.e. to pass from (1.1.11) to (1.1.12c), (1.1.12d). Such an inequality is in general derived by computing and estimating the derivative in time of the energy, i.e. of the L 2 norm to the square of u I ± , v I ± . As this computation makes use of the system of equations satisfied by (u I ± , v I ± ) (see (2.1.2)), two main difficulties arise due to the quasi-linear nature of the starting problem and the very slow decay in time (1.1.11a) of the wave solution.

On the one hand, among all quadratic terms appearing in the right hand side of (2.1.2) we find the quasi-linear ones

Q w 0 (v ± , D 1 v I ± ) and Q kg 0 (v ± , D 1 u I ± ), whose L 2 norm is bounded by v ± (t, •) H 1,∞ ( u I ± (t, •) H 1 + v I ± (t, •) H 1 )
, as usual for this kind of terms. This means that they are at the wrong energy level, in the sense that they cannot be controlled in L 2 by E n (t; u ± , v ± ) or E k 3 (t; u ± , v ± ). This causes a "loss of derivatives" in the energy inequality if we roughly estimate

1 2 ∂ t u I ± (t, •) 2 L 2 + v I ± (t, •) 2 L 2 = - Q w 0 (v ± , D 1 v I ± ), u I ± + Q kg 0 (v ± , D 1 u I ± ), v I ± + . . .
using the Cauchy-Schwarz inequality. This issue is however only technical. In fact, by writing system (2.1.2) in a vectorial fashion and para-linearising it in order to stress out the very troublesome terms (see subsection 2.1.1) we are able to symmetrize it, i.e. to derive an equivalent system in which the quasi-linear contribution is represented by a self-adjoint operator of order 1 (see subsection 2.1.3, proposition 2.1.5). As this operator is self-adjoint it essentially disappears in the energy inequality, replaced with an operator of order 0 whose action on u

I ± , v I ± is bounded in L 2 by E n (t; u ± , v ± ) or E k 3 (t; u ± , v ± )
, depending on the multi-index I we are dealing with. On the other hand, the L 2 norm of some semi-linear contributions to the right hand side of (2.1.2) decays very slowly in time. It is the case, for instance, of

Q kg 0 (v I ± , D 1 u ± ), whose L 2 norm is bounded by u ± (t, •) H 2,∞ v I ± (t,
•) L 2 and only has the slow decay (1.1.11a) of the wave component u ± . Since we want to prove that

∂ t E n (t; u ± , v ± ) = O εt -1+ δ 2 E n (t; u ± , v ± ) 1 2 , ∂ t E k 3 (t; u ± , v ± ) = O εt -1+ δ k 2 E k 3 (t; u ± , v ± ) 1 2
we need to get rid of such terms by means of normal forms (see section 3.1). Because of the quasi-linear nature of our problem, some of them will be eliminated by an adapted quasi-linear normal form argument (see subsection 2.2.1), while the remaining ones can be treated with an usual semi-linear one (see subsection 2.2.2). At that point we will be able to prove proposition 2.2.13 and to derive estimates (1.1.12c), (1.1.12d).

Paralinearization and Symmetrization

As anticipated above, the first step towards the derivation of the right energy inequality is to handle the quasi-linear terms appearing in the right hand side of (2.1.2) in order to avoid any loss of derivatives. We realize that the very quasi-linear contribution to our system appears in equation (2.1.20) through a para-differential operator whose symbol is a real non symmetric matrix. As we need this operator to be self-adjoint (up to an operator of order 0), we symmetrize equation (2.1.20) by defining a new function W I s in terms of W I , that will be solution to a new equation in which the symbol of the quasi-linear contribution is a real symmetric matrix (see subsection 2.1.3). Also, we set aside subsection 2.1.2 to the estimate of the L 2 norms of the non-linear terms in the right hand side of (2.1.20).

Paralinearization

Let us remind definitions (1.1.10) and (1.1.18). Since admissible vector fields considered in Z = {Ω, Z j , ∂ j , j = 1, 2} exactly commute with the linear part of system (1.1.1), we deduce from remark 1.1.5 and (1.1.17) that, for any multi-index I,

(Γ I u, Γ I v) is solution to              ∂ 2 t -∆ x Γ I u = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q 0 (Γ I 1 v, ∂ 1 Γ I 2 v) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q 0 (Γ I 1 v, ∂Γ I 2 v), ∂ 2 t -∆ x + 1 Γ I v = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q 0 (Γ I 1 v, ∂ 1 Γ I 2 u) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q 0 (Γ I 1 v, ∂Γ I 2 u), with coefficients c I 1 ,I 2 ∈ {-1, 0, 1} such that c I 1 ,I 2 = 1 for |I 1 | + |I 2 | = |I|,
Q w 0 (v ± , D a v ± ) := i 4 (v + + v -)D a (v + + v -) - D x D x (v + -v -) • D x D a D x (v + -v -) , Q kg 0 (v ± , D a u ± ) := i 4 (v + + v -)D a (u + + u -) - D x D x (v + -v -) • D x D a |D x | (u + -u -) , (2.1.1)
for any a = 0, 1, 2, we deduce that

(u I + , v I + , u I -, v I -) is solution to (2.1.2)                                      (D t -|D x |)u I + (t, x) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q w 0 (v I 1 ± , D 1 v I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ) (D t -D x )v I + (t, x) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q kg 0 (v I 1 ± , D 1 u I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± ) (D t + |D x |)u I -(t, x) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q w 0 (v I 1 ± , D 1 v I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ) (D t + D x )v I -(t, x) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q kg 0 (v I 1 ± , D 1 u I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± )
The quasi-linear structure of the above system can be emphasized by using (1.2.7) and decomposing

Q w 0 (v ± , D 1 v I ± ), Q kg 0 (v ± , D 1 u I ± ) as follows: (2.1.3) Q w 0 (v ± , D 1 v I ± ) = (QL) 1 + (SL) 1 , Q kg 0 (v ± , D 1 u I ± ) = (QL) 2 + (SL) 2 , with (QL) 1 := i 4 Op B (v + + v -)η 1 (v I + + v I -) -Op B D x D x (v + -v -) • ηη 1 η (v I + -v I -) , (SL) 1 := i 4 Op B D 1 (v I + + v I -) (v + + v -) -Op B D x D 1 D x (v I + -v I -) • η η (v + -v -) +Op B R (v + + v -)η 1 (v I + + v I -) -Op B R D x D x (v + -v -) • ηη 1 η (v I + -v I -) , (QL) 2 := i 4 Op B (v + + v -)η 1 (u I + + u I -) -Op B D x D x (v + -v -) • ηη 1 |η| (u I + -u I -) , (SL) 2 := i 4 Op B D 1 (u I + + u I -) (v + + v -) -Op B D x D 1 |D x | (u I + -u I -) • η η (v + -v -) +Op B R (v + + v -)η 1 (u I + + u I -) -Op B R D x D x (v + -v -) • ηη 1 |η| (u I + -u I -) ,
where the Bony quantization Op B (resp. Op B R ) has been defined in 1.2.5 (resp. in (1.2.9)). We do a similar decomposition also for the semi-linear contribution Q kg 0 (v I ± , D 1 u ± ), for this term will thereafter be the object of the two normal forms mentioned at the beginning of this section:

Q kg 0 (v I ± , D 1 u ± ) = i 4 Op B (v I + + v I -)η 1 (u + + u -) -Op B D x D x (v I + -v I -) • ηη 1 |η| (u + -u -) + i 4 Op B D 1 (u + + u -) (v I + + v I -) -Op B D x D 1 |D x | (u + -u -) • η η (v I + -v I -) + i 4 Op B R (v I + + v I -)η 1 (u + + u -) -Op B R D x D x (v I + -v I -) • ηη 1 |η| (u + -u -) .
(2.1.4)

In order to handle system (2.1.2) in the most efficient way we proceed to write it in a vectorial fashion. To this purpose, we introduce the following matrices:

(2.1.5)

A(η) =     |η| 0 0 0 0 η 0 0 0 0 -|η| 0 0 0 0 -η     , A (V ; η) :=     0 a k η 1 0 b k η 1 a 0 η 1 0 b 0 η 1 0 0 a k η 1 0 b k η 1 a 0 η 1 0 b 0 η 1 0     , (2.1.6) A (V I ; η) :=     0 0 0 0 a I 0 η 1 0 b I 0 η 1 0 0 0 0 0 a I 0 η 1 0 b I 0 η 1 0     , (2.1.7) C (W I ; η) :=     0 c I 0 0 d I 0 0 e I 0 0 f I 0 0 c I 0 0 d I 0 0 e I 0 0 f I 0     , C (U ; η) :=     0 0 0 0 0 e 0 0 f 0 0 0 0 0 0 e 0 0 f 0     where (2.1.8)            a k = a k (v ± ; η) := i 4 (v + + v -) -Dx Dx (v + -v -) • η η b k = b k (v ± ; η) := i 4 (v + + v -) + Dx Dx (v + -v -) • η η a 0 = a 0 (v ± ; η) := i 4 (v + + v -) -Dx Dx (v + -v -) • η |η| b 0 = b 0 (v ± ; η) := i 4 (v + + v -) + Dx Dx (v + -v -) • η |η| (2.1.9)            c 0 = c 0 (v ± ; η) := i 4 D 1 (v + + v -) -DxD 1 Dx (v + -v -) • η η d 0 = d 0 (v ± ; η) := i 4 D 1 (v + + v -) + DxD 1 Dx (v + -v -) • η η e 0 = e 0 (u ± ; η) := i 4 D 1 (u + + u -) -DxD 1 |Dx| (u + -u -) • η η f 0 = f 0 (u ± ; η) := i 4 D 1 (u + + u -) + DxD 1 |Dx| (u + -u -) • η η (2.1.10) a I 0 = a 0 (v I ± ; η), b I 0 = b 0 (v I ± ; η), c I 0 = c 0 (v I ± ; η), d I 0 = d 0 (v I ± ; η), e I 0 = e 0 (u I ± ; η), f I 0 (u I ± ; η), vectors W, U, V : (2.1.11) W :=     u + v + u - v -     , V :=     0 v + 0 v -     , U :=     u + 0 u - 0     ,
along with W I (resp. V I , U I ) defined from W (resp. V, U ) by replacing u ± , v ± with u I ± , v I ± ; and finally (2.1.12)

Q I 0 (V, W ) =           (I 1 ,I 2 )∈I(I) |I 2 |<|I| c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ) (I 1 ,I 2 )∈I(I) |I 1 |,|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± ) (I 1 ,I 2 )∈I(I) |I 2 |<|I| c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ) (I 1 ,I 2 )∈I(I) |I 1 |,|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± )          
The quantization Op B (resp. Op B R ) of a matrix A = (a ij ) 1≤i,j≤n is meant as a matrix of operators

Op B (A) = (Op B (a ij )) 1≤i,j≤n (resp. Op B R (A) = (Op B R (a ij )) 1≤i,j≤n
), and for a vector Y = [y 1 , . . . , y n ],

Op B (A)Y † =          n j=1 Op B (a 1j )y j . . . n j=1 Op B (a nj )y j          , Y † being the transpose of Y . We also remind that A L 2 = i,j |a ij | 2 1 2 , A L ∞ = sup ij |a ij |.
With notations introduced above, system (2.1.2) writes in the following compact fashion which has the merit to well highlight, among all non-linear terms, the very quasi-linear contributions (QL) 1 , (QL) 2 , represented below by Op B (A (V ; η))W I :

D t W I = A(D)W I + Op B (A (V ; η))W I + Op B (C (W I ; η))V + Op B R (A (V ; η))W I + Op B (A (V I ; η))U + Op B (C (U ; η))V I + Op B R (A (V I ; η))U + Q I 0 (V, W ).
(2.1.13)

The energies defined in (1.1.9) take the form

E n (t; u ± , v ± ) = |α|≤n D α x W (t, •) L 2 , ∀ n ∈ N, n ≥ 3, (2.1.14a) E k 3 (t; u ± , v ± ) = |α|+|I|≤3 |I|≤3-k D α x W I (t, •) 2 L 2 , ∀ 0 ≤ k ≤ 2, (2.1.14b)
and we can refer to them, respectively, as E n (t; W ), E k 3 (t; W ). We also notice that, since

(2.1.15a) [Γ, D t ± |D x |] = 0 if Γ ∈ {Ω, ∂ j , j = 1, 2}, ∓ Dm |Dx| (D t ± |D x |) if Γ = Z m , m = 1, 2, (2.1.15b) [Γ, D t ± D x ] = 0 if Γ ∈ {Ω, ∂ j , j = 1, 2}, ∓ Dm Dx (D t ± D x ) if Γ = Z m , m = 1, 2,
and operators

D m |D x | -1 , D m D x -1 are continuous on L 2 for m = 1, 2, there exists a constant C > 0 such that (2.1.16) C -1 I∈I k 3 Γ I W (t, •) 2 L 2 ≤ E k 3 (t; W ) ≤ C I∈I k 3 Γ I W (t, •) 2 L 2 ,
where, for any integer 0 ≤ k ≤ 2,

(2.1.17)

I k 3 := |I| ≤ 3 : Γ I = D α x Γ J with |α| + |J| = |I|, |J| ≤ 3 -k .
For convenience, we also introduce the following set:

(2.1.18)

I n := |I| ≤ n : Γ I = D α x with |α| = |I| , n ∈ N, n ≥ 3.
Matrices A(η), A (V ; η), A (V I ; η) are of order 1 and A (V ; η), A (V I ; η) are singular at η = 0 (i.e. some of their elements are singular at η = 0), while C (W I ; η), C (U ; η) are of order 0. Since we will need to do some symbolic calculus on A (V ; η), we need to isolate the mentioned singularity. We hence define (2.1.19)

A 1 (V ; η) :=     0 a 0 η 1 0 b 0 η 1 a 0 η 1 0 b 0 η 1 0 0 a 0 η 1 0 b 0 η 1 a 0 η 1 0 b 0 η 1 0     , A -1 (V ; η) :=     0 (a k -a 0 )η 1 0 (b k -b 0 )η 1 0 0 0 0 0 0 0 0 0 (a k -a 0 )η 1 0 (b k -b 0 )η 1     , A 1 (V ; η) being a matrix of order 1, A -1 (V ; η) of order -1, both singular at η = 0, and write A 1 (V ; η) = A 1 (V ; η)(1 -χ)(η) + A 1 (V ; η)χ(η), where χ ∈ C ∞ 0 (R 2
) is equal to 1 in the unit ball. Equation (2.1.13) can be the rewritten as follows

D t W I = A(D)W I + Op B (A 1 (V ; η)(1 -χ)(η))W I + Op B (A 1 (V ; η)χ(η))W I + Op B (A -1 (V ; η))W I + Op B (C (W I ; η))V + Op B R (A (V ; η))W I + Op B (A (V I ; η))U + Op B (C (U ; η))V I + Op B R (A (V I ; η))U + Q I 0 (V, W ), (2.1.20)
and the symbol A 1 (V ; η)(1-χ)(η) associated to the quasi-linear contribution is no longer singular at η = 0. We observe that this matrix is real since i

(v + + v -) = 2∂ t v, i Dx Dx (v + -v -) = 2∂
x v and v is a real solution, but it is not symmetric and such a lack of symmetry could lead to a loss of derivatives when writing an energy inequality for W I . The issue is however only technical, in the sense that A 1 (V ; η)(1 -χ)(η) can be replaced with a real, symmetric matrix, as explained in subsection 2.1.3 (see proposition 2.1.5). Before proving such result, we need to derive some L 2 estimates for the semi-linear terms in the right hand side of (2.1.20).

Estimates of quadratic terms

In this subsection we recover some estimates for the L 2 norm of the non-linear terms in the right hand side of equation (2.1.20).

Lemma 2.1.1. Let I be a fixed multi-index and χ ∈ C ∞ 0 (R 2 ) equal to 1 in a neighbourhood of the origin. The following estimates hold:

(2.1.21a) Op B A 1 (V ; η)χ(η) + Op B A -1 (V ; η) W I (t, •) L 2 V (t, •) H 1,∞ W I (t, •) L 2 ; (2.1.21b) Op B (C (W I ; η))V (t, •) L 2 V (t, •) H 6,∞ W I (t, •) L 2 ; (2.1.21c) Op B R (A (V ; η))W I (t, •) L 2 V (t, •) H 7,∞ W I (t, •) L 2 ; (2.1.21d) Op B (A (V I ; η))U (t, •) L 2 + Op B R (A (V I ; η))U (t, •) L 2 R 1 U (t, •) H 6,∞ + U (t, •) H 6,∞ V I (t, •) L 2 ; (2.1.21e) Op B (C (U ; η))V I (t, •) L 2 R 1 U (t, •) H 2,∞ + U (t, •) H 2,∞ W I (t, •) L 2 ;
Proof. • Inequality (2.1.21a) follows applying proposition 1.2.7 to

Op B A -1 (V ; η)(1 -χ)(η) W I whose symbol A -1 (V ; η)(1 -χ)(η) is of order -1 and has M -1 0 seminorm bounded from above by V (t, •) H 1,∞ , after definitions (1.2.2), (2.1.8) and (2.1.19). • Since from definition (2.1.7) of matrix C (W I ; η) Op B (C (W I ; η))V L 2 Op B (D 1 (v I + + v I -))v ± L 2 + Op B D x D 1 D x (v I + -v I -) • η η v ± L 2 + Op B D 1 (u I + + u I -) v ± L 2 + Op B D x D 1 |D x | (u I + -u I -) • η η v ± L 2
, we reduce to prove inequality (2.1.21b) for Op B DxD 1 Dx (v I + -v I -)• η η v + , the same argument being applicable to all other L 2 norms appearing in the above right hand side. Using equality (1.2.6), and considering a new admissible cut-off function χ 1 identically equal to 1 on the support of χ, we first derive that

Op B DxD 1 Dx (v I + + v I -) • η η v + (ξ) = 1 (2π) 2 χ ξ -η η DxD 1 Dx (v I + + v I -) (ξ -η) • Dx Dx v + (η)dη = 1 (2π) 2 χ ξ -η η ξ 1 -η 1 η Dx Dx (v I + + v I -) (ξ -η) • D x v + (η)dη = 1 (2π) 2 χ 1 ξ -η η χ Dx η D 1 η Dx Dx (v I + + v I -) (ξ -η) • D x v + (η)dη = Op B χ Dx η D 1 η Dx Dx (v I + + v I -) D x v + (ξ).
Successively, by decomposition (1.2.7) and the fact that R(u, v) is symmetric in (u, v), we have that

Op B χ D x η D 1 η D x D x (v I + + v I -) D x v + = χ D x η D 1 η D x D x (v I + + v I -) • D x v + -Op B (D x v + ) + Op B R (D x v + ) χ D x η D 1 η D x D x (v I + + v I -) ,
so propositions 1.2.7, 1.2.8 (ii), and the fact that χ Dx η D 1 η

Dx

Dx is an operator uniformly bounded on L 2 , imply that

Op B D x D 1 D x (v I + + v I -) • η η v + L 2 V (t, •) H 6,∞ V I (t, •) L 2 .
• By definition (2.1.5) of A (V ; η),

Op B R A (V ; η) W I (t, •) L 2 Op B R (v + + v -)v I ± L 2 + Op B R D x D x (v + -v -) • η η v I ± L 2 + Op B R (v + + v -)u I ± L 2 + Op B R D x D x (v + -v -) • η |η| u I ± L 2 .
Let us only show that inequality (2.1.21c) holds for

Op B Dx Dx (v + -v -) • ηη 1 |η| u I + .
For a smooth cut-off function φ equal to 1 in the unit ball we write

Op B R D x D x (v + -v -) • ηη 1 |η| u I + = Op B R D x D x (v + -v -) • ηη 1 |η| φ(η) u I + + Op B R D x D x (v + -v -) • ηη 1 |η| (1 -φ)(η) u I + ,
where by proposition 1.2.8 (i)

Op B R D x D x (v + -v -) • ηη 1 |η| φ(η) u I + L 2 D x D x (v + -v -)(t, •) L ∞ u I + (t, •) L 2 V (t, •) H 1,∞ W I (t, •) L 2 .
On the other hand

Op B R D x D x (v + -v -) • ηη 1 |η| (1 -φ)(η) u I + = e ix•ξ m(ξ, η) D x 7 (v + -v-)(ξ -η) ûI + (η)dξdη,
where

m(ξ, η) := 1 (2π) 2 1 -χ ξ -η η -χ η ξ -η (1 -φ)(η) ξ -η ξ -η 8 • ηη 1 |η|
and frequencies ξ -η and η are either bounded or equivalent on the support of m(ξ, η). Therefore m(ξ, η) satisfies the hypothesis of lemma A.

1 (i) |∂ α ξ ∂ β η m(ξ, η)| ξ -3 η -3
for any α, β ∈ N 2 , and by inequality (A.4a)

Op B R D x D x (v + -v -) • ηη 1 |η| (1 -φ)(η) u I + L 2 V (t, •) H 7,∞ W I (t, •) L 2 . • From definition (2.1.6) of A (V ; η), Op B A (V ; η) U (t, •) L 2 Op B (v I + + v I -)η 1 u ± L 2 + Op B D x D x (v I + -v I -) • ηη 1 |η| u ± L 2
, (the same inequality holds evidently when Op B is replaced by Op B R ). As done for previous cases, we reduce to show (2.1.21d) for Op B Dx Dx

(v I + -v I -) • ηη 1 |η| u + (resp. for Op B replaced with Op B R ). Using decomposition (1.2.7) and the fact that R(u, v) is symmetric in (u, v) we find that Op B D x D x (v I + -v I -) • ηη 1 |η| u + = D x D x (v I + -v I -) • D x D 1 |D x | u + -Op B D x D 1 |D x | u + • η η (v I + -v I -) -Op B R D x D 1 |D x | u + • η η (v I + -v I -),
and

Op B R D x D x (v I + -v I -) • ηη 1 |η| u + = Op B R D x D 1 |D x | u + • η η (v I + -v I -),
so a direct application of propositions 1.2.7 and 1.2.8 (ii) gives that the L 2 norm of the above right hand sides is bounded by

DxD 1 |Dx| u + H 4,∞ V I (t, •) L 2 , and hence by R 1 U (t, •) H 6,∞ V I (t, •) L 2 , which gives inequality (2.1.21d). • From definition (2.1.7) of matrix C (U ; η), Op B (C (U ; η))V I L 2 Op B (D 1 (u + + u -))(v I + + v I -) L 2 + Op B D x D 1 |D x | (u + -u -) • η η (v I + -v I -) L 2
, so estimate (2.1.21e) follows immediately from proposition 1.2.7.

Lemmas 2.1.2 and 2.1.3 below are introduced with the aim of deriving an estimate of the L 2 norm of vector Q I 0 (V, W ) defined in (2.1.12) (see corollary 2.1.4). We remind that the summations defining Q I 0 (V, W ) come from the action of family Γ I of admissible vector fields on the quadratic non-linearities

Q 0 (v, ∂ 1 v) and Q 0 (v, ∂ 1 u) in (1.1.1) (or, in terms of u ± , v ± , on Q w 0 (v ± , D 1 v ± ) and Q kg 0 (v ± , D 1 u ± )). According to remark 1.1.5, if I ∈ I n and Γ I is a product of spatial derivatives only the action of Γ I on Q w 0 (v ± , D 1 v ± ) (resp. on Q kg 0 (v ± , D 1 u ± )
) "distributes" entirely on its factors, meaning that

Γ I Q w 0 (v ± , D 1 v ± ) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q w 0 (v I 1 ± , D 1 v I 2 ± ),
(the same for

Γ I Q kg 0 (v ± , D 1 u ± ))
, and all coefficients c I 1 ,I 2 in the right hand side of (2.1.2) are equal to 0. On the contrary, if I ∈ I k 3 for 0 ≤ k ≤ 2 and Γ I contains some Klainerman vector fields Ω, Z m , m = 1, 2, the commutation between Γ I and the null structure gives rise to new quadratic contributions in which the derivative D 1 is eventually replaced with D 2 , D t . As already seen in (1.1.17), in this case we have

(2.1.22) Γ I Q w 0 (v ± , D 1 v ± ) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |=|I| Q w 0 (v I 1 ± , D 1 v I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ),
with some of the coefficients c I 1 ,I 2 being equal to 1 or -1, and D ∈ {D 1 , D 2 , D t } depending on the addend we are considering (similarly for Γ I Q kg 0 (v ± , D 1 u ± )). For our scopes, there will be no difference between the case D = D 1 and D = D 2 , the two associated quadratic contributions enjoying the same L 2 and L ∞ estimates. When D = D t , we should make use of the equation satisfied by v I 2 ± (resp. by u

I 2 ± ) in system (2.1.2) to replace Q w 0 (v I 1 ± , D t v I 2 ± ) (resp. Q kg 0 (v I 1 ± , D t u I 2 ± )) with (2.1.23) Q w 0 (v I 1 ± , D x v I 2 ± ) + Q w 0 v I 1 ± , Γ I 2 Q kg 0 (v ± , D 1 u ± ) , resp. with Q kg 0 (v I 1 ± , |D x |u I 2 ± ) + Q kg 0 v I 1 ± , Γ I 2 Q w 0 (v ± , D 1 v ± ) ,
where the left hand side quadratic terms are given by (2.1.24)

Q w 0 (v I 1 ± , D x v I 2 ± ) = (v I 1 + + v I 1 -) D x (v I 2 + -v I 2 -) - D x D x (v I 1 + -v I 1 -) • D x (v I 2 + + v I 2 -), resp. Q kg 0 (v I 1 ± , |D x |u I 2 ± ) = (v I 1 + + v I 1 -)|D x |(u I 2 + -u I 2 -) - D x D x (v I 1 + -v I 1 -) • D x (u I 2 + + u I 2 -) ,
while the right hand side ones in (2.1.23) are cubic. On the Fourier side, these new quadratic contributions write as

j 1 ,j 2 ∈{+,-} j 2 1 -j 1 j 2 ξ -η ξ -η • η η η vI 1 j 1 (ξ -η)v I 2 j 2 (η)dξdη,   resp. j 1 ,j 2 ∈{+,-} j 2 1 -j 1 j 2 ξ -η ξ -η • η |η| |η|v I 1 j 1 (ξ -η)û I 2 j 2 (η)dξdη   ,
and have basically the same nature of the starting ones, as

Q w 0 (v I 1 ± , D 1 v I 2 ± ) (ξ) = j 1 ,j 2 ∈{+,-} 1 -j 1 j 2 ξ -η ξ -η • η η η 1 vI 1 j 1 (ξ -η)v I 2 j 2 (η)dξdη,   resp. Q kg 0 (v I 1 ± , D 1 u I 2 ± ) (ξ) = j 1 ,j 2 ∈{+,-} 1 -j 1 j 2 ξ -η ξ -η • η |η| η 1 vI 1 j 1 (ξ -η)û I 2 j 2 (η)dξdη   .
For this reason, as long as we can neglect the cubic terms in (2.1.23), we will not pay attention to the value of D ∈ {D

K := {I = (i 1 , i 2 ) : i 1 , i 2 = 1, 2, 3}
as the set of indices I such that Γ I is the product of two Klainerman vector fields only, together with (2.1.26)

V k := {I ∈ I k 3 : ∃(I 1 , I 2 ) ∈ I(I) with I 1 ∈ K}, which is evidently empty when k = 2. We also warn the reader that, in inequality (2.1.30) with k = 2, E 3 3 (t; W ) stands for E 3 (t; W ), this double notation allowing us to combine in one line all cases k = 0, 1, 2.

(I 1 ,I 2 )∈I(I) |I 2 |<n Q w 0 (v I 1 ± , D x v I 2 ± ) L 2 + (I 1 ,I 2 )∈I(I) |I 1 |≤[ n 2 ],|I 2 |<n Q kg 0 (v I 1 ± , D x u I 2 ± ) L 2 V (t, •) H [ n 2 ]+2,∞ E n (t; W ) 1 2 ,
(2.1.28)

(I 1 ,I 2 )∈I(I) |I 1 |>[ n 2 ] Q kg 0 (v I 1 ± , D x u I 2 ± ) L 2 U (t, •) H [ n 2 ]+2,∞ + R 1 U (t, •) H [ n 2 ]+2,∞ E n (t; W ) 1 2 . (ii) Let 0 ≤ k ≤ 2 and I ∈ I k 3 .
There exists a constant C > 0 such that, if we assume a-priori estimates (1.1.11a), (1.1.11b) satisfied and 0 < ε 0 < (2A + B) -1 small, for any χ ∈ C ∞ 0 (R 2 ) equal to 1 in a neighbourhood of the origin and σ > 0 small we have

(I 1 ,I 2 )∈I(I) |I 2 |<3 Q w 0 (v I 1 ± , D x v I 2 ± ) = R k 3 (t, x), (2.1.29a) (I 1 ,I 2 )∈I(I) |I 1 |,|I 2 |<3 Q kg 0 (v I 1 ± , D x u I 2 ± ) = δ V k (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 Q kg 0 v I 1 ± , χ(t -σ D x )D x u I 2 ± + R k 3 (t, x), (2.1.29b) where δ V k = 1 if I ∈ V k , 0 otherwise, and (2.1.30) R k 3 (t, •) L 2 ≤ C(A + B)εt -1 E k 3 (t, W ) 1 2 + CBεt -5 4 , with β > 0 small, β → 0 as σ → 0, for all t ∈ [1, T ]. The same result holds with D x v I 2 ± (resp. D x u I 2 ± ) replaced with D x v I 2 ± (resp. |D x |u I 2 ± ).
Proof. (i) The proof of follows straightly from (2.1.1) with a = 1, 2, by bounding the L 2 norm of each product with the L ∞ norm of the factor indexed in J ∈ {I 1 , I 2 } such that |J| ≤ |I| 2 , times the L 2 norm of the remaining one.

(ii) Let I ∈ I k 3 . One immediately sees that:

(2.1.31)

(J,0)∈I(I) Q w 0 (v J ± , D x v ± ) L 2 + (J,0)∈I(I) |J|<3 Q w 0 (v ± , D x v J ± ) L 2 + Q kg 0 (v ± , D x u J ± ) L 2 V (t, •) H 2,∞ E k 3 (t; W ) 1 2
;

if (I 1 , I 2 ) ∈ I(I) is such that |I 2 | < 3 and either Γ I 1 or Γ I 2 is a product of spatial derivatives only (2.1.32) Q w 0 (v I 1 ± , D x v I 2 ± ) L 2 V (t, •) H 4,∞ E k 3 (t; W ) 1 2
;

if (I 1 , I 2 ) ∈ I(I) is such that |I 2 | < 3 and Γ I 1 is a product of spatial derivatives only (2.1.33) Q kg 0 (v I 1 ± , D x u I 2 ± ) L 2 V (t, •) H 3,∞ E k 3 (t; W ) 1 2 .
Hence, the remaining quadratic contributions to be estimated are those corresponding to indices (I 1 , I 2 ) ∈ I(I), with |I 2 | < 3, such that: both Γ I 1 and Γ I 2 contain at least one Klainerman vector field, in the left hand side of (2.1.29a); Γ I 1 contains one or two Klainerman vector fields, in the left hand side of (2.1.29b).

The idea to estimate the L 2 norm of the Q w 0 (v I 1 ± , Dv I 2 ± ), for indices I 1 , I 2 just mentioned above, is to decompose the Klein-Gordon component carrying exactly one Klainerman vector field in frequencies, by means of a truncation χ(t -σ D x ) for some smooth cut-off function χ and σ > 0 small. Basically, the L ∞ norm of the contribution truncated for large frequencies |ξ| t σ can be bounded by making appear a power of t as negative as we want, while that of the remaining one, localized for |ξ| t σ , enjoys the sharp Klein-Gordon decay t -1 as proved in lemma B.4.14 in appendix B. The same argument can be applied to

Q kg 0 (v I 1 ± , D x u I 2 ± )
with I 1 such that Γ I 1 contains exactly one Klainerman vector field. Then, by lemma B.2.4 in appendix B with L = L 2 we find that, for some χ ∈ C ∞ 0 (R 2 ), the following: if Γ I 1 contains exactly one Klainerman vector field,

Q w 0 (v I 1 ± , D x v I 2 ± )(t, •) L 2 χ(t -σ D x )v I 1 ± (t, •) H 1,∞ v I 2 ± (t, •) H 1 + t -N (s) ( v ± (t, •) H s + D t v ± (t, •) H s ) 1 |µ|=0 x µ v I 2 ± (t, •) H 1 + t v I 2 ± (t, •) H 1 and Q kg 0 (v I 1 ± , D x u I 2 ± )(t, •) L 2 χ(t -σ D x )v I 1 ± (t, •) H 1,∞ u I 2 ± (t, •) H 1 + t -N (s) ( v ± (t, •) H s + D t v ± (t, •) H s ) 1 |µ|=0 x µ D x u I 2 ± (t, •) L 2 + t u I 2 ± (t, •) H 1 ;
if Γ I 2 contains exactly one Klainerman vector field,

Q w 0 (v I 1 ± , D x v I 2 ± )(t, •) L 2 χ(t -σ D x )v I 2 ± (t, •) H 2,∞ v I 1 ± (t, •) L 2 + t -N (s) ( v ± (t, •) H s + D t v ± (t, •) H s ) 1 |µ|=0 x µ v I 1 ± (t, •) L 2 + t v I 1 ± (t, •) L 2 ,
where, in all above inequalities, N (s) ≥ 3 if s > 0 is large enough. From inequalities (B.1.5a), (B.1.6a), estimates (B.1.17), lemma B.4.14 and the boostrap assumptions (1.1.11), together with the fact δ, δ j 1 are small, for j = 0, 1, 2, we derive that there is a positive constant C such that, for multi-indices I 1 , I 2 considered in above inequalities,

Q w 0 (v I 1 ± , Dv I 2 ± )(t, •) L 2 + Q kg 0 (v I 1 ± , Du I 2 ± )(t, •) L 2 ≤ CBεt -1 E k 3 (t; W ) 1 2 + CBεt -5 4 .
The remaining quadratic terms are

Q kg 0 (v I 1 ± , D x u I 2 ± ) with I 1 ∈ K (and hence |I 2 | ≤ 1) if V k is non empty. Applying lemma B.2.4 with L = L 2 ,
w = u and the same s as before, and making use of estimates (1.1.11), (B.1.17), together with inequality (B.1.5a), we see that

Q kg 0 (v I 1 ± , D x u I 2 ± )(t, •) L 2 Q kg 0 v I 1 ± , χ(t -σ D x )D x u I 2 ± (t, •) L 2 + t -3 1 |µ|=0 x µ v I 1 ± (t, •) L 2 + t v I 1 ± (t, •) L 2 ( u ± (t, •) H s + D t u ± (t, •) H s ) Q kg 0 v I 1 ± , χ(t -σ D x )D x u I 2 ± (t, •) L 2 + CBεt -5 4 ,
which hence concludes the proof of (ii). We should highlight the fact that the quadratic contribution in the above left hand side is treated differently from the previous ones, because we do not have a sharp decay O(t -1 ) for v I 1 ± when I 1 ∈ K (neither when truncated for moderate frequencies), but only a control in O(t -1+β ), for some small β > 0 (see lemma B.4.2). Moreover, the decay enjoyed by the uniform norm of χ(t -σ D x )D x u I 2 ± , appearing in the quadratic term in the above right hand side, is very weak (only t -1/2+β , see lemma B.2.10). Such terms, that contribute to the energy and decay slowly in time, will be successively eliminated by a normal form argument (see subsection 2.2.2).

Lemma 2.1.3. Let 0 ≤ k ≤ 2 and I ∈ I k 3 . For any χ ∈ C ∞ 0 (R 2 )
equal to 1 in a neighbourhood of the origin and σ > 0 small

(I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 Q w 0 (v I 1 ± , D t v I 2 ± ) = R k 3 (t, x), (2.1.34a) (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 Q kg 0 (v I 1 ± , D t u I 2 ± ) = δ V k (J,0)∈I(I) J∈K Q kg 0 (v J ± , χ(t -σ D x )|D x |u ± ) + R k 3 (t, x), (2.1.34b) with δ V k = 1 if I ∈ V k , 0 otherwise, and R k 3 (t, x) satisfying (2.1.30).
Proof. Using the equation satisfied by v I 2 ± and u I 2 ± respectively in system (2.1.2) with I = I 2 we see that

(2.1.35a)

(I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 Q w 0 (v I 1 ± , D t v I 2 ± ) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 Q w 0 (v I 1 ± , D x v I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 (J 1 ,J 2 )∈I(I 2 ) c J 1 ,J 2 Q w 0 v I 1 ± , Q kg 0 (v J 1 ± , Du J 2 ± ) , (2.1.35b) (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 Q kg 0 (v I 1 ± , D t u I 2 ± ) = (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 Q kg 0 (v I 1 ± , |D x |u I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 (J 1 ,J 2 )∈I(I 2 ) c J 1 ,J 2 Q kg 0 v I 1 ± , Q w 0 (v J 1 ± , Dv J 2 ± ) , with coefficients c J 1 ,J 2 ∈ {-1, 0, -1} such that c J 1 ,J 2 = 1 whenever |J 1 | + |J 2 | = |I 2 |, and
Q w 0 (v I 1 ± , D x v I 2 ± ) (in which case D = D 1 ), Q kg 0 (v I 1 ± , |D x |u I 2 ± )
given explicitly by (2.1.24). After lemma 2.1.2 (ii) we know that [START_REF] Wang | An intrinsic hyperboloid approach for Einstein Klein-Gordon equations[END_REF]. The only thing to prove is that the cubic terms in the right hand side of (2.1.35) are remainders R k 3 . We focus on those in the right hand side of (2.1.35a) as the same argument applies to the ones in (2.1.35b).

(I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2 Q w 0 (v I 1 ± , D x v I 2 ± ) + Q kg 0 (v I 1 ± , |D x |u I 2 ± ) = (J,0)∈I(I) J∈K Q kg 0 (v J ± , |D x |u ± ) + R k 3 (t, x), with R k 3 verifying (2.1.
Q w 0 v I 1 ± , Q kg 0 (v ± , D 1 u ± ) L 2 v I 1 ± (t, •) L 2 Q kg 0 (v ± , D 1 u ± ) H 1,∞ ≤ CBεt -3 2 +β ,
for some β > 0 small as long as σ, δ 0 are small.

Let us now consider indices

I 1 , I 2 such that Γ I 1 ∈ {Ω, Z m , m = 1, 2}.
As we also require that

(I 1 , I 2 ) ∈ I(I) with |I 2 | ≤ 2, we have in this case that |I 2 | ≤ 1 and consequently, for each (J 1 , J 2 ) ∈ I(I 2 ), either |J 1 | = 0 or |J 2 | = 0. Using lemma B.2.4 in appendix B with L = L 2 and w = v, we derive that for any χ ∈ C ∞ 0 (R 2 )
as in the statement and σ > 0 small

(J 1 ,J 2 )∈I(I 2 ) Q w 0 v I 1 ± , Q kg 0 (v J 1 ± , Du J 2 ± ) (t, •) L 2 (J 1 ,J 2 )∈I(I 2 ) χ(t -σ D x )v I 1 ± (t, •) L ∞ Q kg 0 (v J 1 ± , Du J 2 ± )(t, •) L 2 + (J 1 ,J 2 )∈I(I 2 ) t -N (s) ( v ± (t, •) H s + D t v ± (t, •) H s ) × 1 |µ|=0 x µ Q kg 0 (v J 1 ± , Du J 2 ± )(t, •) L 2 + t Q kg 0 (v J 1 ± , Du J 2 ± )(t, •) L 2 , with N (s) ≥ 3 is s > 0 is sufficiently large. Here (J 1 ,J 2 )∈I(I 2 ) xQ kg 0 (v J 1 ± , Du J 2 ± )(t, •) L 2 |µ|=0,1 |J|≤1 x D x D x µ v ± (t, •) L ∞ u J ± (t, •) H 1 + D t u J ± (t, •) L 2 + xv J ± (t, •) L 2 ( R µ u ± (t, •) H 2,∞ + D t R µ u ± (t, •) H 1,∞ ) ≤ C(A + B)Bε 2 t 1 2 + δ 2 2

and

(2.1.36)

(J 1 ,J 2 )∈I(I 2 ) Q kg 0 (v J 1 ± , Du J 2 ± )(t, •) L 2 |J|≤1 v J ± (t, •) L 2 1 |µ|=0 R µ u ± (t, •) H 2,∞ + D t R µ u ± (t, •) H 1,∞ + v ± (t, •) H 1,∞ u J ± (t, •) H 1 + D t u J ± (t, •) L 2 ≤ C(A + B)Bε 2 t -1 2 + δ 2
(J 1 ,J 2 )∈I(I 2 ) Q w 0 v I 1 ± , Q kg 0 (v J 1 ± , Du J 2 ± ) (t, •) L 2 ≤ CBεt -3 2 +β ,
for some new β > 0 small, β → 0 as σ, δ 0 → 0.

Finally, for indices

I 1 , I 2 such that Γ I 1 ∈ {D α x , |α| ≤ 1} (2.1.37) (J 1 ,J 2 )∈I(I 2 ) Q w 0 v I 1 ± , Q kg 0 (v J 1 ± , Du J 2 ± ) L 2 (J 1 ,J 2 )∈I(I 2 ) v ± (t, •) H 2,∞ Q kg 0 (v J 1 ± , Du J 2 ± ) L 2 . For (J 1 , J 2 ) ∈ I(I 2 ) such that |J 1 | + |J 2 | = |I 2 |
we have by lemma 2.1.2 (ii) and a-priori estimates (1.1.11) that

Q kg 0 (v J 1 ± , D 1 u J 2 ± ) L 2 R k 3 (t, •) L 2 + J∈K Q kg 0 (v J ± , D 1 χ(t -σ D x )u ± ) L 2 R k 3 (t, •) L 2 + t β 1 |µ|=0 R µ 1 u ± (t, •) L ∞ E 1 3 (t; W ) 1 2 ≤ CBεt -1 2 +β+ δ 1 2 , with β > 0 small, β → 0 as σ → 0, while for (J 1 , J 2 ) ∈ I(I 2 ) such that |J 1 | + |J 2 | < |I 2 | (hence < 2)
an estimate such as (2.1.36) holds. These estimates, together with (1.1.11b), imply that the right hand side of (2.1.37) is bounded by CABε 2 t -3 2 +β , for a new small β > 0, β → 0 as σ, δ 0 → 0, and that concludes the proof of the statement.

Corollary 2.1.4. Let Q I 0 (V, W ) be the vector defined in (2.1.12). There exists a constant C > 0 such that, if we assume that a-priori estimates (1.1.11) are satisfied in interval [1, T ], for some fixed T > 1, with ε 0 < (2A + B) -1 small:

(i) if I ∈ I n with n ≥ 3: (2.1.38) Q I 0 (V, W ) L 2 ≤ CAεt -1 2 + δ 2 ; (ii) if I ∈ I k 3 , with 0 ≤ k ≤ 2, (2.1.39) Q I 0 (V, W ) L 2 ≤ C(A + B)εt -1 2 + δ k 2 .
Proof. (i) Inequality (2.1.38) is straightforward after definition (2.1.12) (all coefficients c I 1 ,I 2 are equal to 0 when I ∈ I n ), lemma 2.1.2 (i), and a-priori estimates (1.1.11a), (1.1.11b).

(ii) If I ∈ I k 3 for a fixed 0 ≤ k ≤ 2 we have by definition (2.1.12) and lemmas 2.1.2, 2.1.3 that (2.1.40)

(I 1 ,I 2 )∈I(I) |I 2 |<|I| Q w 0 (v I 1 ± , D x v I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2,|I 2 |<|I| Q w 0 (v I 1 ± , D t v I 2 ± ) = R k 3 (t, x), with R k 3 (t, x) satisfying (2.1.30). Moreover, for some smooth χ ∈ C ∞ 0 (R 2 )
, equal to 1 in a neighbourhood of the origin and σ > 0 small, (2.1.41)

(I 1 ,I 2 )∈I(I) |I 2 |<|I| Q kg 0 (v I 1 ± , D x u I 2 ± ) = δ V k (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 Q kg 0 v I 1 ± , χ(t -σ D x )D x u I 2 ± + R k 3 (t, x), (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2,|I 2 |<|I| Q kg 0 (v I 1 ± , D t u I 2 ± ) = δ V k (J,0)∈I(I) J∈K Q kg 0 v J ± , χ(t -σ D x )|D x |u ± + R k 3 (t, x), with sets K, V k given, respectively, by (2.1.25), (2.1.26), δ V k = 1 if I ∈ V k , 0 otherwise (remind that V 2 is empty). Observe that, if k = 0, 1, I ∈ I k 3 and (I 1 , I 2 ) ∈ I(I) with I 1 ∈ K, two situations may occur: if Γ I 2 ∈ {D α
x , |α| ≤ 1} then product Γ I 1 contains exactly the same number of Klainerman vector fields as in Γ I and V I 1 would be at the same energy level as V I (i.e. its L 2 norm being controlled by E k 3 (t; W ) 1/2 ). In this case, from a-priori estimates (1.1.11a)

v I 1 ± (t, •) L 2 χ(t -σ D x )u I 2 ± (t, •) H ρ,∞ + χ(t -σ D x )Ru I 2 ± (t, •) H ρ,∞ ≤ Aεt -1 2 E k 3 (t; W ) 1 2 .
(2.1.42)

If instead I 2 is such that Γ I 2 ∈ {Ω, Z m , m = 1,
2} is a Klainerman vector field, we automatically have that Γ I is a product of three Klainerman vector fields and that V I 1 is at an energy level strictly lower than V I (i.e. its L 2 norm is controlled by

E 1 3 (t; W ) 1/2 whereas that of V I is bounded by E 0 3 (t; W ) 1/2 ). From lemma B.2.10 we deduce that (2.1.43) v I 1 ± (t, •) L 2 χ(t -σ D x )u I 2 ± (t, •) H ρ,∞ + χ(t -σ D x )Ru I 2 ± (t, •) H ρ,∞ ≤ C(A + B)εt -1 2 +β+ δ 1 2 E 1 3 (t; W ) 1 2 ,
for a small β > 0, β → 0 as σ → 0. Summing up (2.1.40) to (2.1.43) and using (2.1.30) we obtain that there is a positive constant C such that (2.1.44)

Q I 0 (V, W ) L 2 ≤ δ k C(A + B)εt -1 2 E k 3 (t; W ) 1 2 + δ 0 t β+ δ 1 2 E 1 3 (t; W ) 1 2 + CBεt -5 4 ,
with δ k = 1 for k = 0, 1, equal to 0 when k = 2, and δ 0 = 1 only when k = 0, 0 otherwise. Finally, taking σ > 0 small so that β + δ 1 /2 δ 0 /2 and using a-priori estimates (1.1.11d) we deduce estimate (2.1.39) from (2.1.44).

Symmetrization

Proposition 2.1.5. As long as H 1,∞ norm of V (t, •) is sufficiently small, there exists a real matrix P (V ; η) of order 0 and a real symmetric matrix A 1 (V ; η) of order 1, vanishing at order 1 at V = 0, such that

(2.1.45) W I s := Op B P (V ; η) W I is solution to D t W I s = A(D)W I s + Op B ( A 1 (V ; η))W I s + Op B (A (V I ; η))U + Op B (C (U ; η))V I + Op B R (A (V I ; η))U + Q I 0 (V, W ) + R(U, V ), (2.1.46) where R(U, V ) satisfies, for any θ ∈]0, 1[, R(U, V )(t, •) L 2 V (t, •) H 7,∞ + V (t, •) 1-θ H 1,∞ V (t, •) θ H 3 ( U (t, •) H 2,∞ + R 1 U (t, •) H 2,∞ ) + V (t, •) L ∞ U (t, •) 1-θ H 2,∞ + R 1 U (t, •) 1-θ H 2,∞ U (t, •) θ H 4 W I (t, •) L 2 + V (t, •) H 1,∞ W (t, •) H 7,∞ + RU (t, •) H 6,∞ W I (t, •) L 2 + V (t, •) H 1,∞ Q I 0 (V, W ) L 2 .
(2.1.47)

Moreover, for any n, r ∈ N,

M 0 r (P (V ; η) -I 4 ; n) V (t, •) H 1+r,∞ , (2.1.48) M 1 r A 1 (V ; η); n V (t, •) H 1+r,∞ , (2.1.49) and as long as the H 2,∞ norm of V (t, •) is small there is a constant C > 0 such that (2.1.50) C -1 W I (t, •) L 2 ≤ W I s (t, •) L 2 ≤ C W I (t, •) L 2 .
In order to prove proposition 2.1.5, we first need to introduce the following lemma.

Lemma 2.1.6. Let α, β ∈ R, L ∈ M 2 (R) and M 0 , N (α, β) ∈ M 4 (R) given by L = 0 1 1 0 , M 0 = I 2 0 0 -I 2 , N (α, β) = αL βL αL βL =     0 α 0 β α 0 β 0 0 α 0 β α 0 β 0     .
There exist a small δ > 0 and a smooth function defined on open ball B δ (0) of radius δ,

(α, β) ∈ B δ (0) → P (α, β) ∈ Sym 4 (R),
with values in the space of real, symmetric, 4 × 4 matrices Sym 4 (R), such that P (0, 0

) = I 4 , P (α, β) = I 4 + O(|α| + |β|) and P (α, β) -1 M 0 + N (α, β) P (α, β) is symmetric for any (α, β) ∈ B δ (0). Furthermore P -1 (α, β) = I 4 + O(|α| + |β|).
Proof. Let E be the vector space of 2 × 2 matrices B(α, β) = αI 2 + βL and F be the set of 4 × 4 matrices of the form

F 11 F 12 F 21 F 22
with F ij ∈ E. We look for a matrix P of the form (2.1.51)

P (B) = (I 2 -B 2 ) -1 2 I 2 -B -B I 2
with B ∈ E close to zero (so that in particular (I 2 -B 2 ) 1/2 is well defined). We remark that matrix P (B) -1 has the form

P (B) -1 = (I 2 -B 2 ) -1 2 I 2 B B I 2
and that P (0

) = P -1 (0) = I 4 . We consider Φ : R 2 × E → F defined by Φ(α, β, B) := P (B) -1 M 0 + N (α, β) P (B) = Φ ij (α, β, B) 1≤i,j≤2
, where

Φ ij ∈ E as E is a commutative sub-algebra of M 2 (R). We also define Ψ(α, β, B) := Φ 12 (α, β, B) -Φ † 21 (α, β, B) with Φ † 21 denot- ing the transpose of Φ 21 .
We have that Ψ(0, 0, 0) = 0 and

D B Φ(0, 0, 0) • B = 0 B B 0 M 0 -M 0 0 B B 0 = 2 0 -B B 0 from which follows that D B Ψ(0, 0, 0) • B = -4B, i.e. D B Ψ(0, 0, 0) = -4I.
Therefore, there exist a small δ > 0 and a smooth function

(α, β) ∈ B δ (0) → B(α, β) ∈ E such that B(0, 0) = 0 (which implies P (B(0, 0)) = I 4 ), and Ψ(α, β, B(α, β)) = 0 ∀(α, β) ∈ B δ (0). This is equivalent to say that Φ(α, β, B(α, β)) is symmetric and moreover P (B(α, β)), P (B(α, β)) -1 = I 4 + O(|α| + |β|). Defining P (α, β) := P (B(α, β)) concludes the proof of the statement.
Proof of proposition 2.1.5. With notations introduced in lemma 2.1.6 and in (2.1.5), (2.1.19),

A(η) = η M 0 + S(η) and A 1 (V ; η)(1 -χ)(η) = η N (α, β), with S(η) =     |η| -η 0 0 0 0 0 0 0 0 0 -(|η| -η ) 0 0 0 0 0     whose elements are O(|η| -1 ), |η| → +∞,
and α = a 0 (v ± ; η) η 1 η (1 -χ)(η), β = b 0 (v ± ; η) η 1 η (1 -χ)(η), a 0 , b 0 defined in (2.1.8). Since sup η |α| + |β| V (t, •) H 1,∞
, by lemma 2.1.6 we have that, as long as V (t, •) H 1,∞ is sufficiently small, there exists a real symmetric matrix P = P (V ; η) of the form (2.1.51) such that P (V ; η) -1 M 0 + N (α, β) P (V ; η) is real and symmetric. Moreover P = I 4 + Q(V ; η) and P -1 = I 4 + Q (V ; η), where Q(V ; η), Q (V ; η) are matrices depending smoothly on α, β (which are symbols of order 0), null at order 1 at V = 0, verifying for any n, r ∈ N

M 0 r (Q(V ; η); n) + M 0 r Q (V ; η); n V (t, •) H 1+r,∞ .
We define the following matrix of order 1

A 1 (V ; η) := P (V ; η) -1 η M 0 + N (α, β) P (V ; η) -η M 0 and W I s := Op B (P -1 (V ; η))W I . From the fact that A 1 (V ; η) also writes as η Q (V ; η)M 0 + P -1 (V ; η)M 0 Q(V ; η) + P -1 (V ; η)N (α, β)P (V ; η)
we see that it vanishes at order 1 at V = 0 and is such that

M 1 r ( A 1 (V ; η); n) V (t, •) H 1+r,∞ . Moreover, from proposition 1.2.9 (ii) with r = 1 it follows that (2.1.52) I = Op B (P (V ; η))Op B (P -1 (V ; η)) + T -1 (V )
,

where operator T -1 (V ) is of order less or equal than -1 whose L(L 2 ) norm is a O( V (t, •) H 2,∞ ). Therefore W I = Op B (P (V ; η))W I s + T -1 (V )W I and from proposition 1.2.7 the L 2 norms of W I , W I
s are equivalent as long as the H 2,∞ norm of V is small. Using equation (2.1.20) we find that:

D t W I s = Op B (P -1 (V ; η))Op B A(η) + A 1 (V ; η)(1 -χ)(η) W I + Op B (P -1 (V ; η)) Op B A 1 (V ; η)χ(η) + Op B A -1 (V ; η) W I + Op B (P -1 (V ; η)) Op B (C (W I ; η))V + Op B R (A (V ; η))W I + Op B (P -1 (V ; η)) Op B (A (V I ; η))U + Op B (C (U ; η))V I + Op B R (A (V I ; η))U + Op B (P -1 (V ; η))Q I 0 (V, W ) + Op B (D t P -1 (V ; η))W I (2.1.53)
where

Op B (P -1 (V ; η))Op B A(η) + A 1 (V ; η)(1 -χ)(η) W I = Op B (P -1 (V ; η))Op B η M 0 + N (α, β) W I + Op B (S(η))W I + Op B (Q (V ; η))Op B (S(η))W I = Op B (P -1 (V ; η))Op B η M 0 + N (α, β) Op B (P (V ; η))W I s + Op B (P -1 (V ; η))Op B η M 0 + N (α, β) T -1 (V )W I + Op B (S(η))W I s + Op B (S(η))Op B (Q(V ; η))W I s + Op B (S(η))T -1 (V )W I + Op B (Q (V ; η))Op B (S(η))W I =Op B (A(η) + A 1 (V ; η))W I s + T 0 (V )W I s + T 0 (V )W I (2.1.54) with T 0 (V ), T 0 (V ) operators of order 0 and L(L 2 ) norm O( V (t, •) H 2,∞
). Last equality follows indeed from the fact that, by proposition 1.2.9 (ii) with r = 1 and proposition 1.2.7,

Op B (P -1 (V ; η))Op B η M 0 + N (α, β) Op B (P (V ; η)) = Op B P (V ; η) -1 η M 0 + N (α, β) P (V ; η) + T 0 (V ) and Op B (S(η))Op B (Q(V ; η)), Op B (Q (V ; η))Op B (S(η)
) are operator of order 0, too (the former of the form T 0 (V ), the latter of the form T 0 (V )), while Op B (S(η))T -1 (V ) is of order -1 (and can be included in T 0 (V )). The equivalence between the L 2 norms of W I s and W I implies that T 0 (V )W I s + T 0 (V )W I in (2.1.54) is a remainder R(U, V ). All operators appearing in the second and third line of (2.1.53) are also remainders R(U, V ) because, from proposition 1.2.7, the fact that

M 0 0 (P -1 (V ; η); 2) = O(1) and lemma 2.1.1, their L 2 norm is bounded by V (t, •) H 7,∞ W I (t, •) L 2 . Last term in (2.1.53) also contributes to R(U, V ) for matrix D t P -1 (V ; η) is of order 0, its M 0 0 (•, 2) seminorm is bounded by D t V (t, •) H 1,∞ and for any θ ∈ [0, 1] D t V (t, •) H 1,∞ V (t, •) H 2,∞ + V (t, •) 1-θ H 1,∞ V (t, •) θ H 3 ( U (t, •) H 2,∞ + R 1 U (t, •) H 2,∞ ) + V (t, •) L ∞ U (t, •) 1-θ H 2,∞ + R 1 U (t, •) 1-θ H 2,∞ U (t, •) θ H 4 ,
as follows from (B.1.6b) with s = 1. Finally, in remaining contributions in (2.1.53) we replace Op B (P -1 (V ; η)) with I +Op B (Q (V ; η)) and observe that the terms on which Op B (Q (V ; η)) acts are remainders R(U, V ) after proposition 1.2.7, the fact that

M 0 0 (Q (V ; η); 2) = O( V (t, •) H 1,∞
) and lemma 2.1.1. Interchanging the notation of P (V ; η) and P -1 (V ; η), we obtain the result of the statement.

Normal forms and energy estimates

Before going further in writing an energy inequality for W I s we should make few remarks. As we previously anticipated, the L 2 norm of some of the semi-linear terms appearing in equation (2.1.46) have a very slow decay in time. On the one hand, it is the case of Op B (A (V I ; η))U , Op B (C (U ; η))V I and Op B R (A (V ; η))U , whose L 2 norms are estimated in (2.1.21d), (2.1.21e) in terms of the uniform norms of U, R 1 U . On the other hand, also some of the contributions to

Q I 0 (V, W ) are only a O L 2 (t -1/2+β
), for some small β > 0, after corollary 2.1.4. Nevertheless, we are going to see that Op B (A (V I ; η))U , Op B R (A (V ; η))U and the mentioned contributions to Q I 0 (V, W ) can be easily eliminated by performing a semi-linear normal form argument in the energy inequality (see subsection 2.2.2). Such an argument is however not well adapted to handle Op B (C (U ; η))V I , for it leads to a loss of derivatives linked to the quasi-linear nature of the problem, i.e. to the fact that matrix A 1 (V ; η) in the right hand side of (2.1.46) is of order 1. This latter contribution should instead be eliminated through a suitable normal form applied directly on equation (2.1.46), which is the object of the subsection 2.2.1.

A first normal forms transformation and the energy inequality

First of all, we replace Op B (C (U ; η))V I in equation (2.1.46) with Op B (C (U ; η))V I s , having defined V I s := Op B (P -1 (V ; η))V I , and remind that from (2.1.48) with r = 0 and (2.1.52) the L 2 norm of V I and V I s are equivalent as long as the H 2,∞ norm of V (t, •) is small (assumption compatible with (1.1.11b) if ρ ≥ 2). We will rather deal with

(D t -A(D))W I s = Op B ( A 1 (V ; η))W I s + Op B (A (V I ; η))U + Op B (C (U ; η))V I s + Op B R (A (V I ; η))U + Q I 0 (V, W ) + R(U, V ), (2.2.1)
for a new R(U, V ) satisfying (2.1.47) and show how to get rid of Op B (C (U ; η))V I s in the above right hand side. More precisely, we are going to prove the following result: Proposition 2.2.1. Let N ∈ N * . There exist three matrices E 0 d (U ; η), E -1 d (U ; η), E nd (U ; η) linear in (u + , u -), with E 0 d (U ; η) real diagonal of order 0 and E -1 d (U ; η), E nd (U ; η) of order -1, and, as long as

R 1 U (t, •) H 2,∞ is small, a real diagonal matrix F 0 d (U ; η) of order 0 such that, if (2.2.2) W I s := Op B (I 4 + E(U ; η))W I s , with E(U ; η) := E 0 d (U ; η) + E -1 d (U ; η) + E nd (U ; η), then (D t -A(D)) W I s = Op B (I 4 + E 0 d (U ; η)) A 1 (V ; η)(I 4 + F 0 d (U ; η)) W I s + Op B (A (V I ; η))U + Op B R (A (V I ; η))U + Q I 0 (V, W ) + T -N (U )W I s + R (U, V ).
(2.2.3)

In the above right hand side T -N (U ) = (σ ij (U, D x )
) ij is a pseudo-differential operator of order less or equal than -N , with

(2.2.4) T -N (U ) L(H s-N ;H s ) R 1 U (t, •) H N +2,∞ + U (t, •) H N +6,∞ ,
for any s ∈ R and such that (2.2.5a)

F x →ξ (σ ij (U, η))(ξ) = σ + ij (ξ, η)û + (ξ) + σ - ij (ξ, η)û -(ξ), i, j ∈ {2, 4}, 0,
otherwise, with σ ± ij (ξ, η) supported for |ξ| ≤ ε η for a small ε > 0, and for any α, β ∈ N 2

(2.2.5b)

|∂ α ξ ∂ β η σ ± ij (ξ, η)| α,β |ξ| N +1-|α| η -N -|β| , i, j ∈ {2, 4}. Also, R (U, V ) is a remainder satisfying, for any θ ∈]0, 1[ R (U, V )(t, •) L 2 (1 + U (t, •) H 5,∞ ) R(U, V ) L 2 + ( R 1 U (t, •) H 1,∞ + U (t, •) H 5,∞ ) Q I 0 (V, W ) L 2 + ( RU (t, •) H 6,∞ + U (t, •) H 6,∞ ) W I (t, •) L 2 + V (t, •) 2-θ H 5,∞ V (t, •) θ H 7 W I (t, •) L 2 , (2.2.6) with R(U, V ) verifying (2.1.47).
For any n, r ∈ N, any χ ∈ C ∞ 0 (R 2 ) equal to 1 close to the origin and supported in open ball B ε (0), with ε > 0 sufficiently small, we have that

(2.2.7a) M 0 r E 0 d χ D x η U ; η ; n R 1 U (t, •) H 1+r,∞ , (2.2.7b) M -1 r E -1 d χ D x η U ; η ; n U (t, •) H 5+r,∞ , (2.2
.7c) M -1 r E nd χ D x η U ; η ; n U (t, •) H 5+r,∞ ; and (2.2.8) M 0 r F 0 d χ D x η U ; η ; n R 1 U (t, •) H 1+r,∞ .
Finally, as long as

R 1 U (t, •) H 2,∞ + U (t, •) H 5,∞ is small, there is a constant C > 0 such that (2.2.9) C -1 W I s (t, •) L 2 ≤ W I s (t, •) L 2 ≤ C W I s (t, •) L 2 .
Remark 2.2.2. From propositions 2.1.5 and 2.2.1 it follows that, as long as

R 1 U (t, •) H 2,∞ , U (t, •) H 5,∞ and V (t, •) H 2,∞ are small, there is a constant C > 0 such that (2.2.10) C -1 W I (t, •) L 2 ≤ W I s (t, •) L 2 ≤ C W I (t, •) L 2 .
This implies that, if

E n (t; W ) := |α|≤n Op B (I 4 + E(U ; η))Op B (P (V ; η))D α x W (t, •) L 2 , ∀ n ∈ N, n ≥ 3, (2.2.11a) E k 3 (t; W ) := |α|+|I|≤3 |I|≤3-k Op B (I 4 + E(U ; η))Op B (P (V ; η))D α x W I (t, •) L 2 , ∀ 0 ≤ k ≤ 2, (2.2.11b) there exists a constant C 1 > 0 such that (2.2.12) C -1 1 E n (t; W ) ≤ E n (t; W ) ≤ C 1 E n (t; W ), ∀ n ≥ 3, C -1 1 E k 3 (t; W ) ≤ E k 3 (t; W ) ≤ C 1 E k 3 (t; W ), ∀ 0 ≤ k ≤ 2.
Thanks to the above equivalence, the propagation of some suitable estimates on E n (t; W ) and E k 3 (t; W ) will provide us with (1.1.12c) and (1.1.12d) respectively, so we can rather focus on the derivation of an energy inequality for E n (t; W ), E k 3 (t; W ).

In order to get rid of Op

B (C d (U ; η))V I s in (2.2.1)we introduce matrices (2.2.13) C d (U ; η) =     0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 f 0     , C nd (U ; η) =     0 0 0 0 0 0 0 f 0 0 0 0 0 0 e 0 0 0     so that C (U ; η) = C d (U ; η) + C nd (U ; η),
and proceed to eliminate Op B (C d (U ; η))V I s and Op B (C nd (U ; η))V I s separately. Lemma 2.2.3. Let N ∈ N * . There exists a diagonal matrix E d (U ; η) of order 0, linear in

(u + , u -), such that (2.2.14) Op B (C d (U ; η))V I s + Op B (D t E d (U ; η))W I s -[A(D), Op B (E d (U ; η))]W I s = T -N (U )W I s + R (V, V ),
where R (V, V ) satisfies, for any θ ∈]0, 1[,

(2.2.15) R (V, V )(t, •) L 2 V (t, •) 2-θ H 5,∞ V (t, •) θ H 7 V I (t, •) L 2 ,
and T -N (U ) is a pseudo-differential operator of order less or equal than -N such that, for any

s ∈ R, (2.2.16
)

T -N (U ) L(H s-N ;H s ) R 1 U (t, •) H N +2,∞ + U (t, •) H N +6,∞ , whose symbol σ(U, η) = (σ ij (U, η)) 1≤i,j≤4 is such that (2.2.17a) F x →ξ (σ ij (U, η))(ξ) = σ + ii (ξ, η)û + (ξ) + σ - ii (ξ, η)û -(ξ), i = j ∈ {2, 4}, 0,
otherwise, with σ ± ii (ξ, η) supported for |ξ| ≤ ε η for a small ε > 0, and verifying, for any α, β ∈ N 2 ,

(2.2.17b) |∂ α ξ ∂ β η σ ± ii (ξ, η)| α,β |ξ| N +1-|α| η -N -|β| , for i = 2, 4. Moreover, if χ ∈ C ∞ 0 (R 2
) is equal to 1 close to the origin and has a sufficiently small support,

(2.2.18) E d χ D x η U ; η = E 0 d χ D x η U ; η + E -1 d χ D x η U ; η ,
the former matrix in the above right hand side being real of order 0 and satisfying (2.2.7a), the latter being of order -1 and verifying (2.2.7b).

Proof. Because of the diagonal structure of A(η) and C d (U ; η) we look for a matrix E d = (e ij ) 1≤i,j≤4 satisfying (2.2.14) such that e ij = 0 for all i, j but i = j ∈ {2, 4}, and we also require symbols e 22 , e 44 to be of order 0 and linear in (u + , u -). If we remind that matrix A(η) in (2.1.5) is of order 1 and make the ansatz that E d is of order 0, then by symbolic calculus of proposition 1.2.9 we have that

(2.2.19) -[A(D), Op B (E d (U ; η))] = - N |α|=1 1 α! Op B ∂ α η A(η)D α x E d (U ; η) + T -N (U )
with T -N (U ) pseudo-differential operator of order less or equal than -N such that, for any s ∈ R,

(2.2.20) T -N (U ) L(H s-N ;H s ) M 1 N +1 (A(η); N + 3)M 0 0 (E d (U ; η); 2) + M 1 0 (A(η); N + 3)M 0 N +1 (E d (U ; η); 2)
and whose symbol σ(U, η) = (σ ij (U, η)) ij is such that σ ij (U, η) = 0 for all i, j but i = j ∈ {2, 4}.

Therefore, for any fixed

χ ∈ C ∞ 0 (R 2 ) equal to 1 in B ε 1 (0) and supported in B ε 2 (0), for some 0 < ε 1 < ε 2 1, we look for E d (U ; η) such that χ D x η   C d (U ; η) + D t E d (U ; η) - N |α|=1 1 α! ∂ α η A(η)D α x E d (U ; η)   = 0.
Since E d (U ; η) is required to be linear in (u + , u -), we should write it rather as E d (u + , u -; η) to then realize that, as u + (resp. u -) is solution to the first (resp. to the third) equation in (2.1.2) with |I| = 0,

D t E d (u + , u -; η) = E d (|D x |u + , -|D x |u -; η) + E d Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ); η , D α x E d (u + , u -; η) = E d (D α x u + , D α x u -; η), ∀α ∈ N 2 .
If we temporarily neglecting contribution

E d Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± )
; η , we are lead to solve the following equation

χ D x η   C d (U ; η) + E d (|D x |u + , -|D x |u -; η) - N |α|=1 1 α! ∂ α η A(η)E d (D α x u + , D α x u -; η)   = 0,
which is equivalent to system

                           e 22   χ Dx η |D x | - N |α|=1 1 α! ∂ α η ( η )D α x u + , -χ D x η |D x | + N |α|=1 1 α! ∂ α η ( η )D α x u -; η   = -χ Dx η e 0 e 44   χ Dx η |D x | + N |α|=1 1 α! ∂ α η ( η )D α x u + , -χ D x η |D x | - N |α|=1 1 α! ∂ α η ( η )D α x u -; η   = -χ Dx η f 0 ,
with e 0 , f 0 defined in (2.1.9). Then, if we look for e ii of the form (2.2.21)

e ii (u + , u -; η) = e ix•ξ α ii (ξ, η)û + (ξ)dξ + e ix•ξ β ii (ξ, η)û -(ξ)dξ,
this system implies, inter alia, that

e ix•ξ χ ξ η |ξ| - N |α|=1 1 α! ∂ α η ( η )ξ α α 22 (ξ, η)û + (ξ)dξ = - i 4 e ix•ξ χ ξ η 1 - η η • ξ |ξ| ξ 1 û+ (ξ)dξ. As   1 ∓ N |α|=1 1 α! ∂ α η ( η ) ξ α |ξ|   = 1 ∓ N k=1 1 k! (ξ • ∇ η ) k ( η )
and

(∂ η 1 ξ 1 + ∂ η 2 ξ 2 ) k η = |ξ| k η k-1 1 - η η • ξ |ξ| 2 b k (ξ, η), 2 ≤ k ≤ N, with b k (ξ, η) polynomial of degree k -2 in η η • ξ |ξ| , we derive that (2.2.22)   1 ∓ N |α|=1 1 α! ∂ α η ( η ) ξ α |ξ|   = 1 ∓ η η • ξ |ξ| (1 ∓ b ± (ξ, η)) with (2.2.23) b ± (ξ, η) := N k=2 1 k! |ξ| k-1 η -(k-1) 1 ± η η • ξ |ξ| b k (ξ, η), |∂ µ ξ ∂ ν η b ± (ξ, η)| µ,ν |ξ| 1-|µ| η -1-|ν| , ∀µ, ν ∈ N 2 ,
and we can then choose

α 22 (ξ, η) in (2.2.21) such that, when |ξ| ≤ ε 2 η , (2.2.24) α 22 (ξ, η) = - i 4 (1 -b + (ξ, η)) -1 ξ 1 |ξ| .
Similarly, we choose multipliers β 22 , α 44 , β 44 such that, as long as |ξ| ≤ ε 2 η ,

β 22 (ξ, η) = i 4 (1 + b -(ξ, η)) -1 ξ 1 |ξ| , α 44 (ξ, η) = - i 4 (1 + b -(ξ, η)) -1 ξ 1 |ξ| , β 44 (ξ, η) = i 4 (1 -b + (ξ, η)) -1 ξ 1 |ξ| .
These multipliers are all well defined for |ξ| ≤ ε 2 η as b ± (ξ, η) = O(|ξ| η -1 ). Moreover, using that

(1 ± b ∓ (ξ, η)) -1 = 1 ∓ b ∓ (ξ, η) + O(|ξ| 2 η -2
) as long as |ξ| ≤ ε 2 η , we have that

α 22 (ξ, η) = - i 4 ξ 1 |ξ| + α -1 22 (ξ, η), β 22 (ξ, η) = i 4 ξ 1 |ξ| + β -1 22 (ξ, η), α 44 (ξ, η) = - i 4 ξ 1 |ξ| + α -1 44 (ξ, η), β 44 (ξ, η) = i 4 ξ 1 |ξ| + β -1 44 (ξ, η), with |∂ µ ξ ∂ ν η α -1 ii | + |∂ µ ξ ∂ ν η β -1 ii | µ,ν |ξ| 1-|µ| η -1-|ν| for any µ, ν ∈ N 2 . Injecting the above α ii , β ii , i ∈ {2, 4}, in (2.2.21) we find that e 22 χ D x η u + , χ D x η u -; η = - i 4 R 1 (u + -u -) + e -1 22 χ D x η u + , χ D x η u -; η , e 44 χ D x η u + , χ D x η u -; η = - i 4 R 1 (u + -u -) + e -1 44 χ D x η u + , χ D x η u -; η ,
where, for i ∈ {2, 4},

e -1 ii χ D x η u + , χ D x η u -; η = e ix•ξ χ ξ η α -1 ii (ξ, η)û + (ξ)dξ + e ix•ξ χ ξ η β -1 ii (ξ, η)û -(ξ)dξ.
After lemma A.1 (i) and above estimates for α -1 ii , β -1 ii , kernels

K i + (x, η) := e ix•ξ χ ξ η α -1 ii (ξ, η) ξ -4 dξ, K i -(x, η) := e ix•ξ χ ξ η β -1 ii (ξ, η) ξ -4 dξ are such that, for any β ∈ N 2 , |∂ β η K i ± (x, η)| |x| -1
x -2 η -1-|β| for every (x, η). This implies that

∂ β η e -1 ii χ D x η u + , χ D x η u -; η ≤ ∂ β η K i + (x -y, η)[ D x 4 u + ](y)dy + ∂ β η K i -(x -y, η)[ D x 4 u -](y)dy U (t, •) H 4,∞ η -1-|β| and e -1
ii is a symbol of order -1, for i = 2, 4. Moreover, using definition (1.2.3) and the fact that space W r,∞ injects in H r+1,∞ , one can check that for any r, n ∈ N,

M -1 r e -1 ii χ D x η u + , χ D x η u -; η ; n U (t, •) H 5+r,∞
and therefore that

M 0 r e ii χ D x η u + , χ D x η u -; η ; n R 1 U (t, •) H 1+r,∞ + U (t, •) H 5+r,∞ . Defining E 0 d (U ; η) =     0 0 0 0 0 -i 4 R 1 (u + -u -) 0 0 0 0 0 0 0 0 0 -i 4 R 1 (u + -u -)     , E -1 d (U ; η) =     0 0 0 0 0 e -1 22
0 0 0 0 0 0 0 0 0 e -1 44     ,
E d Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ); η = E -1 d Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ); η
for any n ∈ N and θ ∈]0, 1[, we derive from (B.1.3d) with s = 4 that

M 0 0 E d Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ); η ; n Q w 0 (v ± , D 1 v ± ) H 4,∞ V (t, •) 2-θ H 5,∞ V (t, •) θ H 7 ,
and hence that the quantization of

E d Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ); η
(2.2.25) Op B (C nd (U ; η))V I s + Op B (D t E nd (U ; η))W I s -[A(D), Op B (E nd (U ; η))]W I s = T -N (U )W I s + R (V, V ),
where R (V, V ) is a remainder satisfying (2.2.15) and T -N (U ) is a pseudo-differential operator of order less or equal than -N such that, for any s ∈ R,

(2.2.26) T -N (U ) L(H s-N ;H s ) U (t, •) H N +6,∞ . Moreover, its symbol σ(U, η) = (σ ij (U, η)) 1≤i,j≤4 is such that (2.2.27a) F x →ξ (σ ij (U, η))(ξ) = σ + ij (ξ, η)û + (ξ) + σ - ij (ξ, η)û -(ξ), (i, j) ∈ {(2, 4), (4, 2)}, 0,
otherwise, with σ ± ij supported for |ξ| ≤ ε η for a small ε > 0, and verifying, for any α, β ∈ N 2 , (2.2.27b)

|∂ α ξ ∂ β η σ ± ij (ξ, η)| α,β |ξ| N +2-|α| η -N -1-|β| , for (i, j) ∈ {(2, 4), (4, 2)}.
Proof. Because of the structure of C nd (U ; η), we seek for a matrix E nd (U ; η) satisfying (2.2.25), of the form E nd (U ; η) = (e ij ) 1≤i,j≤4 with e ij = 0 for all i, j, except (i, j) ∈ {(2, 4), (4, 2)}. If we make the ansatz that E nd (U ; η) is linear in (u + , u -), of order -1, and remind that A(η) in (2.1.5) is of order 1, from symbolic calculus of proposition 1.2.9 we have that

-[A(D), Op B (E nd (U ; η))] = -Op B (A(η)E nd (U ; η) -E nd (U ; η)A(η)) - N |α|=1 1 α! Op B (∂ α η A(η) • D α x E nd (U ; η)) + T -N (U ),
where T -N (U ) is a pseudo-differential operator of order less or equal than -N , such that, for any s ∈ R,

(2.2.28) T -N (U ) L(H s-N ;H s ) M 1 N +1 (A(η); N + 3)M -1 0 (E nd (U ; η); 2) + M 1 0 (A(η); N + 3)M -1 N +1 (E nd (U ; η); 2),
and whose symbol σ(U, η)

= (σ ij (U, η)) ij is such that σ ij = 0 for all i, j but (i, j) ∈ {(2, 4), (4, 2)}. Hence, for any fixed χ ∈ R 2 equal to 1 in B ε 1 (0) and supported in B ε 2 (0), for some 0 < ε 1 < ε 2 1, we look for E nd (U ; η) such that (2.2.29) χ D x η C nd (U ; η) + D t E nd (U ; η) -A(η)E nd (U ; η) + E nd (U ; η)A(η) - N |α|=1 1 α! ∂ α η A(η) • D α x E nd (U ; η) = 0.
Furthermore, as E nd (U ; η) = E nd (u + , u -; η) is linear in (u + , u -) and u + (resp. u -) is solution to the first (resp. the third) equation in (2.1.2) with |I| = 0, we have that

D t E nd (u + , u -; η) = E nd (|D x |u + , -|D x |u -; η) + E nd Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ); η , D α x E nd (u + , u -; η) = E nd (D α x u + , D α x u -; η), ∀α ∈ N 2 while -A(η)E nd (U ; η) + E nd (U ; η)A(η) =     0 0 0 0 0 0 0 -2 η e 24 0 0 0 0 0 2 η e 42 0 0     .
Then we rather search for symbols e 24 and e 42 such that

                           χ Dx η e 2,4   |D x | - N |α|=1 1 α! ∂ α ( η )D α x -2 η u + , -|D x | + N |α|=1 1 α! ∂ α ( η )D α x + 2 η u -; η   = -χ Dx η f 0 , χ Dx η e 4,2   |D x | + N |α|=1 1 α! ∂ α ( η )D α x + 2 η u + , -|D x | - N |α|=1 1 α! ∂ α ( η )D α x -2 η u -; η   = -χ Dx η e 0 ,
with e 0 , f 0 given by (2.1.9), neglecting contribution

E nd Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± )
; η whose quantization acting on W I s gives rise to a remainder R (V, V ), as we will see at the end of the proof. We look for e ij of the form

e ij (u + , u -; η) = e ix•ξ α ij (ξ, η)û + (ξ)dξ + e ix•ξ β ij (ξ, η)û -(ξ)dξ,
for (i, j) ∈ {(2, 4), (4, 2)}, and reminding (2.2.22), (2.2.23) we choose the above multipliers such that, as long as |ξ| ≤ ε 2 η ,

α 24 (ξ, η) = - i 4 1 + η η • ξ |ξ| 1 - η η • ξ |ξ| (1 -b + (ξ, η)) -2 η |ξ| -1 ξ 1 |ξ| , β 24 (ξ, η) = - i 4 1 - η η • ξ |ξ| 1 + η η • ξ |ξ| (1 + b -(ξ, η)) + 2 η |ξ| -1 ξ 1 |ξ| , α 42 (ξ, η) = β 24 , β 42 (ξ, η) = α 24 (ξ, η).
One can check that, on the support of χ ξ η and for any µ, ν

∈ N 2 , |∂ µ ξ ∂ ν η α ij | + |∂ µ ξ ∂ ν η β ij | µ,ν |ξ| 1-|µ| η -1-|ν| ,
and then that, if

K ij + (x, η) := e ix•η χ ξ η α ij (ξ, η) ξ -4 dξ, K ij -(x, η) := e ix•η χ ξ η β ij (ξ, η) ξ -4 dξ, for (i, j) ∈ {(2, 4), (4, 2)}, |∂ β η K ij ± (x, η)| |x| -1 x -2 η -1-|β| , for any β ∈ N 2 and (x, η) ∈ R 2 × R 2 , as a consequence of lemma A.1. Therefore ∂ β η e ij χ D x η u + , χ D x η u -; η ≤ ∂ β η K ij + (x -y, η)[ D x 4 u + ](y)dy + ∂ β η K ij -(x -y, η)[ D x 4 u -](y)dy U (t, •) H 4,∞ η -1-|β| ,
which implies that e 24 , e 42 are symbols of order -1. Also, for (i, j) ∈ {(2, 4), (4, 2)} and any n, r ∈ N, one can prove that

M -1 r e ij χ D x η u + , χ D x η u -; η ; n U (t, •) H 5+r,∞
M -1 0 E nd Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ); η ; n V (t, •) 2-θ H 5,∞ V (t, •) θ H 7
, and the quantization of

E nd Q w 0 (v ± , D 1 v ± ), Q w 0 (v ± , D 1 v ± ) acting
(D t -A(D)) W I s = Op B ( A 1 (V ; η))W I s + Op B (A (V I ; η))U + Op B R (A (V I ; η))U + Q I 0 (V, W ) + R(U, V ) + Op B (E(U ; η)) Op B ( A 1 (V ; η))W I s + Op B (A (V I ; η))U + Op B (C (U ; η))V I s + Op B R (A (V I ; η))U + Q I 0 (V, W ) + R(U, V ) + T -N (U )W I s + R (V, V )
where R(U, V ) satisfies (2.1.47), R (V, V ) satisfies (2.2.15), and T -N (U ) is a pseudo-differential operator of order less or equal than -N verifying (2.2.5), (2.2.4). Contribution

Op B (E(U ; η)) Op B (A (V I ; η))U + Op B (C (U ; η))V I s + Op B R (A (V I ; η))U + Q I 0 (V, W ) + R(U, V )
is a remainder of the form R (U, V ) satisfying estimate (2.2.6) as a consequence of proposition 1.2.7, estimates (2.2.7) with r = 0, lemma 2.1.1, and the fact that the L 2 norms of V I s and V I are equivalent as long as

V (t, •) H 2,∞ is small.
According to the definition of E(U ; η) and decomposition (2.2.18)

Op B (E(U ; η))Op B ( A 1 (V ; η)) = Op B (E 0 d (U ; η))Op B ( A 1 (V ; η)) + Op B E -1 d (U ; η) + E nd (U ; η) Op B ( A 1 (V ; η)).
Proposition 1.2.7 and estimates (2.1.49), (2.2.7b), (2.2.7c) with r = 0, imply that the latter addend in the above right hand side is a bounded operator on

L 2 whose L(L 2 ) norm is esti- mated by U (t, •) H 5,∞ V (t, •) H 1,∞ .
The former one writes instead as Op B (E 0 d (U ; η) A 1 (V ; η))+ T 0 (U, V ), for an operator T 0 (U, V ) of order less or equal than 0 and L(L 2 ) norm controlled by

R 1 U (t, •) H 2,∞ V (t, •) H 2,∞ ,
Op B (E(U ; η))Op B ( A 1 (V ; η))W I s = Op B (E 0 d (U ; η) A 1 (V ; η))W I s + R (U, V ), for a new R (U, V ) satisfying (2.2.6).
After (2.2.7a) matrix

I 4 + E 0 d χ( Dx η )U ; η is invertible as long as R 1 U (t, •) H 1,∞ is small and F 0 d (U ; η) := I 4 + E 0 d χ( Dx η )U ; η -1 -I 4 is such that, for any n, r ∈ N, M 0 r F 0 d χ D x η U ; η ; n R 1 U (t, •) H 1+r,∞ .
Moreover, F 0 d (U ; η) is a real diagonal matrix of order 0, and by corollary 1.2.11 with r = 1

Op B (I 4 + F 0 d (U ; η))Op B (I 4 + E 0 d (U ; η)) = Id + T -1 (U ),
with T -1 (U ) of order less or equal than 0 and L(H

s-1 ; H s ) norm bounded by R 1 U (t, •) H 2,∞ , for any s ∈ R. This implies that Op B (I 4 + F 0 d (U ; η)) W I s = W I s + T -1 (U )W I s , T -1 (U ) = T -1 (U ) + Op B (E -1 d (U ; η) + E nd (U ; η))
with T -1 (U ) of order less or equal than -1 and

(2.2.30)

T -1 (U ) L(H s-1 ;H s ) R 1 U (t, •) H 2,∞ + U (t, •) H 5,∞
for any s ∈ R. Hence, as long as this quantity is small, there exists a positive constant C such that (2.2.9) holds. Also,

Op B (I 4 + E 0 d (U ; η))Op B ( A 1 (V ; η))W I s = Op B (I 4 + E 0 d (U ; η))Op B ( A 1 (V ; η))Op B (I 4 + F 0 d (U ; η)) W I s -Op B (I 4 + E 0 d (U ; η))Op B ( A 1 (V ; η)) T -1 (U )W I s ,
where from proposition 1.2.

V (t, •) H 1,∞ ( R 1 U (t, •) H 2,∞ + U (t, •) H 5,∞ ) W I (t, •) L 2 .
On the other hand, by corollary 1.2.11 with r = 1 we get that

Op B (I 4 + E 0 d (U ; η))Op B ( A 1 (V ; η))Op B (I 4 + F 0 d (U ; η)) W I s = Op B (I 4 + E 0 d (U ; η)) A 1 (V ; η)(I 4 + F 0 d (U ; η)) W I s + Op B (I 4 + E 0 d (U ; η))T 0 (U, V ) W I s + T 0 (U, V ) W I s ,
with T 0 (U, V ), T 0 (U, V ) operators of order less or equal than 0 and L(L 2 ) norm controlled, respectively, by

R 1 U (t, •) H 2,∞ V (t, •) H 2,
∂ t E n (t; W ) = O εt -1+ δ 2 E n (t; W ) 1 2 , ∂ t E k 3 (t; W ) = O εt -1+ δ 2 E k 3 (t; W ) 1 2
. This is do to the fact that we still have to deal with semi-linear slow-decaying contributions

Op B (A (V I ; η))U , Op B R (A (V I ; η))U , Q I 0 (V, W ) to the right hand side of (2.2.3), together with the new T -N (U )W I s whose L 2 norm is also a O(t -1 2 W I (t, •) L 2 ) after (2.2.4
) and (1.1.11a). The aim of the current subsection is hence to perform a new normal form argument to replace the mentioned terms with more decaying ones. This is actually done at the energy level, meaning that we are going to add some suitable cubic perturbations to E n (t; W ) and E k 3 (t; W ) so that the new energies so defined satisfy estimates as in (2.2.32).

Let us first focus on the slow decaying terms that appear when computing

∂ t E n (t; W ) = I∈In ∂ t W I s , W I s
for any integer n ≥ 3. Using equation (2.2.3) and rewriting W I s in terms of W I we find, on the one hand, the contribution (2.2.33) -

I∈In Op B (A (V I ; η))U + Op B R (A (V I ; η))U, W I + T -N (U )W I , W I ,
which is a O(εt -1/2 E n (t; W )) after Cauchy-Schwarz inequality, lemma 2.1.1 and a-priori estimates (1.1.11). But we also have (2.2.34) -

I∈In (I 1 ,I 2 )∈I(I) [ |I| 2 ]<|I 1 |<|I| Q kg 0 (v I 1 ± , D 1 u I 2 ± ), v I + + v I -,
which enjoys the same decay as the previous one, as can be immediately seen using again Cauchy-Schwarz inequality along with (2.1.28) and (1.1.11a). From definition (2.1.6) of matrix A (V I , η), Plancherel's formula, (1.2.6) and the fact that

v I + = -v I - Op B (A (V I ; η))U, W I = Op B (a 0 (v I ± ; η)η 1 )u + + Op B (b 0 (v I ± ; η)η 1 )u -, v I + + v I - = - i 4(2π) 2 χ ξ -η η (v I + + v I -)(ξ -η)(u + + u -)(η) - ξ -η ξ -η • η |η| (v I + -v I -)(ξ -η) ×(u + -u -)(η) η 1 (v I -+ v I -)(-ξ)dξdη,
with χ denoting a smooth function equal to 1 in B ε 1 (0) and supported in B ε 2 (0), for some

0 < ε 1 < ε 2 1. Hence (2.2.35) - Op B (A (V I ; η))U, W I = j k ∈{+,-} C I (j 1 ,j 2 ,j 3 )
with (2.2.36)

C I (j 1 ,j 2 ,j 3 ) = 1 4(2π) 2 χ ξ -η η 1 -j 1 j 2 ξ -η ξ -η • η |η| η 1 vI j 1 (ξ -η)û j 2 (η)v I j 3 (-ξ)dξdη, for any j 1 , j 2 , j 3 ∈ {+, -}. Analogously, from equality (1.2.8) (2.2.37) - Op B R (A (V I ; η))U, W I = j k ∈{+,-} C I,R (j 1 ,j 2 ,j 3 ) with (2.2.38) C I,R (j 1 ,j 2 ,j 3 ) = 1 4(2π) 2 1 -χ ξ -η η -χ η ξ -η 1 -j 1 j 2 ξ -η ξ -η • η |η| η 1 × vI j 1 (ξ -η)û j 2 (η)v I j 3 (-ξ)dξdη. After proposition 2.2.1, T -N (U ) = (σ ij (U, D x )) ij with symbols σ ij (U, η) satisfying (2.2.5
). Introducing ρ : {+, -} → {2, 4} such that ρ(+) = 2, ρ(-) = 4 and using the convention that -j k ∈ {+, -} \ {j k }, we have that

T -N (U )W I , W I = i,j∈{+,-} σ ρ(i)ρ(j) (U, D x )v I j , v I i = - 1 (2π) 2 j k ∈{+,-} σ j 2 ρ(j 3 ),ρ(j 1 ) (η, ξ -η)v I j 1 (ξ -η)û j 2 (η)v I -j 3 (-ξ)dξdη, (2.2.39)
where multipliers σ j 2 ρ(j 3 ),ρ(j 1 ) (η, ξ -η) are supported for |η| ≤ ε|ξ -η| and such that, for any

α, β ∈ N 2 , ∂ α ξ ∂ β η σ j 2 ρ(j 3 ),ρ(j 1 ) (η, ξ -η) α,β |η| N +1-|β| ξ -η -N -|α| ,
for any (ξ, η) ∈ R 2 × R 2 , any j 1 , j 2 , j 3 ∈ {+, -}. Moreover, by (2.1.1) we have that

- Q kg 0 (v I 1 ± , D 1 u I 2 ± ), v I + + v I - = j k ∈{+,-} C I 1 ,I 2 (j 1 ,j 2 ,j 3 ) (2.2.40) with (2.2.41) C I 1 ,I 2 (j 1 ,j 2 ,j 3 ) := 1 4(2π) 2 1 -j 1 j 2 ξ -η ξ -η • η |η| η 1 vI 1 j 1 (ξ -η)û I 2 j 2 (η)v I j 3 (-ξ)dξdη.
The above equalities lead us to introduce the following multipliers

(2.2.42) B k (j 1 ,j 2 ,j 3 ) (ξ, η) := 1 j 1 ξ -η + j 2 |η| + j 3 ξ 1 -j 1 j 2 ξ -η ξ -η • η |η| η k , k = 1, 2 and (2.2.43) σ N (j 1 ,j 2 ,j 3 ) (ξ, η) := σ j 2 ρ(j 3 ),ρ(j 1 ) (η, ξ -η) j 1 ξ -η + j 2 |η| -j 3 ξ ,
together with the following integrals (2.2.44a)

D I (j 1 ,j 2 ,j 3 ) := i 4(2π) 2 χ ξ -η η B 1 (j 1 ,j 2 ,j 3 ) (ξ, η)v I j 1 (ξ -η)û j 2 (η)v I j 3 (-ξ) dξdη, (2.2.44b) D I,R (j 1 ,j 2 ,j 3 ) := i 4(2π) 2 1 -χ ξ -η η -χ η ξ -η B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) × vI j 1 (ξ -η)û j 2 (η)v I j 3 (-ξ)dξdη, (2.2.44c) D I,T -N (j 1 ,j 2 ,j 3 ) := Re 1 (2π) 2 σ N (j 1 ,j 2 ,j 3 ) (ξ, η)v I j 1 (ξ -η)û j 2 (η)v I -j 3 (-ξ)dξdη and (2.2.45) D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) := i 4(2π) 2 B 1 (j 1 ,j 2 ,j 3 ) (ξ, η)v I 1 j 1 (ξ -η)û I 2 j 2 (η)v I j 3 (-ξ) dξdη
for any triplet (j 1 , j 2 , j 3 ) ∈ {+, -}

I∈In j i ∈{+,-} D I (j 1 ,j 2 ,j 3 ) + D I,R (j 1 ,j 2 ,j 3 ) + D I,T -N (j 1 ,j 2 ,j 3 ) + I∈In j i ∈{+,-} (I 1 ,I 2 )∈I(I) [ |I| 2 ]<|I 1 |<|I| D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) .
Let us now analyse the time derivative of E k 3 (t; W ) for integers 0 ≤ k ≤ 2. As in the previous case, from equation (2.2.3) we see appear the same contribution as in (2.2.33), but with the sum over I n replaced with that on I k 3 . We also find (2.2.47) -

I∈I k 3 [ Q I 0 (V, W ), W I ] which is a O(εt -(1+δ k )/2 E k 3 (t; W ) 1/2
) from Cauchy-Schwarz inequality and estimate (2.1.39). To be more precise, the slow decay in time of the above scalar product is due to some particular quadratic term appearing in Q I 0 (V, W ). In fact, according to definition (2.1.12) and to (2.1.29a), (2.1.30), (2.1.34a), (1.1.11d), for any

I ∈ I k 3 (2.2.48) (I 1 ,I 2 )∈I(I) |I 2 |<|I| Q w 0 (v I 1 ± , D x v I 2 ± ), u I + + u I -+ (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2,|I 2 |<|I| Q w 0 (v I 1 ± , D t v I 2 ± ), u I + + u I - R k 3 (t, •) L 2 U I (t, •) L 2 ≤ C(A + B)εt -1+ δ k 2 E k 3 (t; W ) 1 2 .
Also, after (2.1.29b) and (2.1.34b) we have that for all I / ∈ V k , with V k defined in (2.1.26),

(2.2.49)

(I 1 ,I 2 )∈I(I) |I 1 |,|I 2 |<|I| Q kg 0 (v I 1 ± , D x u I 2 ± ), v I + + v I - + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2,|I 1 |,|I 2 |<|I| Q kg 0 (v I 1 ± , D t u I 2 ± ), v I + + v I - R k 3 (t, •) L 2 V I (t, •) L 2 ≤ C(A + B)εt -1+ δ k 2 E k 3 (t; W ) 1 2 .
Observe that the decay rate O(t -1+δ k /2 ) in the right hand side of the two above inequalities is the slowest one that allows us to propagate a-priori estimate (1.1.11d) and it gives us back exactly the slow growth in time

t δ k /2 enjoyed by E k 3 (t; W ) 1/2 , for 0 ≤ k ≤ 2.
On the other hand, for I ∈ V k with k = 0, 1, we have that, for some smooth cut-off function χ and some σ > 0 small,

(I 1 ,I 2 )∈I(I) |I 1 |,|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , D x u I 2 ± ) = (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 c I 1 ,I 2 Q kg 0 v I 1 ± , χ(t -σ D x )D x u I 2 ± + R k 3 (t, x), (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤2,|I 1 |,|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , D t u I 2 ± ) = (J,0)∈I(I) J∈K c J,0 Q kg 0 (v J ± , χ(t -σ D x )|D x |u ± ) + R k 3 (t, x).
The L 2 norms of the summations in the above right hand sides are bounded by

J|≤1 χ(t -σ D x )u J ± (t, •) H 2,∞ + χ(t -σ D x )Ru J ± (t, •) H 2,∞ E k 3 (t; W ) 1 2
and hence by εt -1/2 E k 3 (t; W ) 1/2 as follows by sharp estimate (B.2.57) derived in appendix B. Therefore, the very contribution to (2.2.47) that has to be eliminated from ∂ t E k 3 (t; W ) appears only for k = 0, 1 and is (2.2.50) -

I∈V k (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 c I 1 ,I 2 Q kg 0 v I 1 ± , χ(t -σ D x )D x u I 2 ± , v I + + v I - - I∈V k (J,0)∈I(I) J∈K c J,0 Q kg 0 v J ± , χ(t -σ D x )|D x |u ± , v I + + v I - . As (2.2.51) - Q kg 0 (v I 1 ± , χ(t -σ D x )D l u I 2 ± ), v I + + v I - = j i ∈{+,-} F I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) , l = 1, 2 - Q kg 0 (v I 1 ± , χ(t -σ D x )|D x |u I 2 ± ), v I + + v I - = j i ∈{+,-} F I 1 ,I 2 ,3 (j 1 ,j 2 ,j 3 ) , with (2.2.52) F I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) = 1 4(2π) 2 1 -j 1 j 2 ξ -η ξ -η • η |η| η l vI 1 j 1 (ξ -η)χ(t -σ D x )u I 2 j 2 (η)v I j 3 (-ξ)dξdη,
for any j i ∈ {+, -}, l = 1, 2, 3, and η 3 := j 2 |η|, we introduce a new multiplier (2.2.53)

B 3 (j 1 ,j 2 ,j 3 ) (ξ, η) := j 2 j 1 ξ -η + j 2 |η| + j 3 ξ 1 -j 1 j 2 ξ -η ξ -η • η |η| |η| together with integrals (2.2.54) G I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) = i 4(2π) 2 B l (j 1 ,j 2 ,j 3 ) (ξ, η) vI 1 j 1 (ξ -η)χ(t -σ D x )u I 2 j 2 (η)v I j 3 (-ξ)dξdη
for any l = 1, 2, 3, (j 1 , j 2 , j 3 ) ∈ {+, -} 3 , with multipliers B l (j 1 ,j 2 ,j 3 ) given by (2.2.42) when l = 1, 2, and by (2.2.53) when l = 3. We warn the reader that in what follows we will sometimes refer to multipliers B l (j 1 ,j 2 ,j 3 ) (resp. integrals F I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) and G I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) ) simply as B (j 1 ,j 2 ,j 3 ) (resp.

F I 1 ,I 2 (j 1 ,j 2 ,j 3 )
and G I 1 ,I 2 (j 1 ,j 2 ,j 3 ) ) forgetting about superscript l. This choice reveals to be convenient when we do not need to distinguish between l = 1, 2, 3. Definition 2.2.6. For every integer 0 ≤ k ≤ 2 we define the second modified energy

E k, † 3 (t; W ) as (2.2.55) E k, † 3 (t; W ) := E k 3 (t; W ) + I∈I k 3 j i ∈{+,-} D I (j 1 ,j 2 ,j 3 ) + D I,R (j 1 ,j 2 ,j 3 ) + D I,T -N (j 1 ,j 2 ,j 3 ) + δ k<2 I∈V k j i ∈{+,-} (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 c I 1 ,I 2 G I 1 ,I 2 (j 1 ,j 2 ,j 3 ) ,
with δ k<2 = 1 if k = 0, 1, 0 otherwise, and coefficients c I 1 ,I 2 ∈ {-1, 0, 1}.

In view of the lemmas to follow it is useful to remind that, after system (2.1.2), for any multiindex I vector ). We are also going to see that, if N ∈ N * is chosen sufficiently large (e.g. N = 18), all these quartic terms suitably decay in time, and that modified energies

(û I + , vI + , ûI -, vI -) is solution to (2.2.56)              (D t -|ξ|)û I + (t, ξ) = |I 1 |+|I 2 |=|I| Q w 0 (v I 1 ± , D 1 v I 2 ± ) + |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ) (D t -ξ )v I + (t, ξ) = |I 1 |+|I 2 |=|I| Q kg 0 (v I 1 ± , D 1 u I 2 ± ) + |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± ) (D t + |ξ|)û I -(t, x) = |I 1 |+|I 2 |=|I| Q w 0 (v I 1 ± , D 1 v I 2 ± ) + |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ) (D t + ξ )v I -(t, x) = |I 1 |+|I 2 |=|I| Q kg 0 (v I 1 ± , D 1 u I 2 ± ) + |I 1 |+|I 2 |<|I| c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± ) with coefficients c I 1 ,I 2 ∈ {-1, 0, 1}
E † n (t; W ), E k, † 3 (t; W ) are equivalent, respectively, to E n (t; W ), E k 3 (t, W ).
We point out the fact that the normal form's step performed in previous section was necessary to avoid here some problematic quartic contributions coming from quasi-linear terms in (2.2.56) and that could lead to some loss of derivatives. Before proving the mentioned lemmas, we need to introduce two preliminary results, that will be useful in the proof of lemmas 2.2.9, 2.2.11. Lemma 2.2.7. For any j i ∈ {+, -}, i = 1, 2, 3, let B k (j 1 ,j 2 ,j 3 ) (ξ, η) be the multiplier defined in (2.2.42) when k = 1, 2 and in (2.2.53) when k = 3, and ψ 1 , ψ 2 , ψ 3 be three smooth cut-off functions such that ψ

1 (x) is supported for |x| ≤ c, ψ 2 (x) is supported for c ≤ |x| ≤ C , ψ 3 (x) is supported for |x| ≥ C, for some 0 < c, c 1, C, C 1, and ψ 1 + ψ 2 + ψ 3 ≡ 1.
Let also δ k be equal to 1 for k = 1, 2, 0 for k = 3.

(i) For any j 1 , . . . , j 5 ∈ {+, -}, i = 1, 2, and any

u 1 , u 2 , u 3 , u 4 such that u 1 ∈ H 4,∞ (R 2 ), u 2 , u 4 ∈ L 2 (R 2 ), u 3 ∈ H 11,∞ (R 2 ) and δ k R k u 3 ∈ H 7,∞ (R 2 ),
(2.2.57)

ψ i ξ -η η B k (j1,j2,j3) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 û1 (ξ -η -ζ)û 2 (ζ)û 3 (η)û 4 (-ξ)dξdηdζ u 1 H 4,∞ u 2 L 2 ( u 3 H 11,∞ + δ k R k u 3 H 7,∞ ) u 4 L 2 ;
(ii) For any j 1 , . . . , j 5 ∈ {+, -}, and any

u 1 , u 2 , u 3 , u 4 such that u 1 ∈ H 7,∞ (R 2 ), u 2 ∈ H 1 (R 2 ), u 4 ∈ L 2 (R 2 ), u 3 ∈ H 4,∞ (R 2 ) and δ k R k u 3 ∈ L ∞ (R 2 ),
(2.2.58)

ψ 3 ξ -η η B k (j1,j2,j3) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 û1 (ξ -η -ζ)û 2 (ζ)û 3 (η)û 4 (-ξ)dξdηdζ u 1 H 7,∞ u 2 H 1 ( u 3 H 4,∞ + δ k R k u 3 L ∞ ) u 4 L 2 .
Proof. Let k = 1, 2. We are going to refer to B k (j 1 ,j 2 ,j 3 ) (resp. η k and R k ) simply as B (j 1 ,j 2 ,j 3 ) (resp. η and R) and rather use a superscript to define a decomposition of this multiplier (see (2.2.60))

Let us observe that, as

B (j 1 ,j 2 ,j 3 ) (ξ, η) = j 1 ξ -η + j 2 |η| -j 3 ξ 2j 1 j 2 ξ -η |η| η,
we can write (2.2.59)

B (j 1 ,j 2 ,j 3 ) (ξ, η) = B 0 (j 1 ,j 2 ,j 3 ) (ξ, η) η |η| + B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) η 4 ,
where for any smooth cut-off function φ, equal to 1 in a neighbourhood of the origin, (2.2.60)

B 0 (j 1 ,j 2 ,j 3 ) (ξ, η) := j 1 ξ -η + j 2 |η| -j 3 ξ 2j 1 j 2 ξ -η φ(η), B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) := j 1 ξ -η + j 2 |η| -j 3 ξ 2j 1 j 2 ξ -η |η| η η -4 (1 -φ)(η).
According to decomposition (2.2.59) we have that, for any i = 1, 2, 3,

ψ i ξ -η η B (j1,j2,j3) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 û1 (ξ -η -ζ)û 2 (ζ)û 3 (η)û 4 (-ξ)dξdηdζ = ψ i ξ -η η B 0 (j1,j2,j3) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 û1 (ξ -η -ζ)û 2 (ζ) Ru 3 (η)û 4 (-ξ)dξdηdζ + ψ i ξ -η η B 1 (j1,j2,j3) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 û1 (ξ -η -ζ)û 2 (ζ) D x 4 u 3 (η)û 4 (-ξ) dξdηdζ =: I 0 i + I 1 i .
(2.2.61)

(i) The first thing we observe concerning integral

I k i for k = 0, 1, i = 1, 2, is that |ξ -η|, |ξ| η on the support of ψ i ξ-η η
and that |ζ| ≤ ξ -η -ζ η . Therefore, introducing the following multipliers

B i,k (j 1 ,...,j 5 ) (ξ, η, ζ) := ψ i ξ -η η B k (j 1 ,j 2 ,j 3 ) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 η -7 ξ -η -ζ -4 ,
for any j 1 , . . . j 5 ∈ {+, -}, k = 0, 1, i = 1, 2, a straight computation shows that, for any α, β, γ ∈ N 2 , (2.2.62)

∂ α ξ ∂ β η B i,k (j 1 ,...,j 5 ) (ξ, η, ζ) ζ -3 |g α,β (ξ, η)|, ∂ α ξ ∂ β η ∂ γ ζ B i,k (j 1 ,...,j 5 ) (ξ, η, ζ) (|ζ| ζ -1 ) 1-|γ| ζ -3 |g α,β (ξ, η)|, |γ| ≥ 1, with (2.2.63) |g α,0 (ξ, η)| α η -3 ξ -3 , |g α,β (ξ, η)| α,β (|η| η -1 ) 1-|β| η -3 ξ -3 , |β| ≥ 1. If K i,k (j 1 ,...,j 5 ) (x, y, z) := e ix•ξ+iy•η+iz•ζ B i,k (j 1 ,...,j 5 ) (ξ, η, ζ)dξdηdζ, by lemma A.1 (i) we first find that, for any α, β ∈ N 2 , ∂ α ξ ∂ β η e iz•ζ B i,k (j 1 ,...,j 5 ) (ξ, η, ζ)dζ |z| -1 z -2 |g α,β (ξ, η)|
and successively that

∂ α ξ e iy•η+iz•ζ B i,k (j 1 ,...,j 5 ) (ξ, η, ζ)dηdζ |y| -1 y -2 |z| -1 z -2 ξ -3 , for every ξ ∈ R 2 , (y, z) ∈ R 2 × R 2 . Corollary A.2 (i) hence implies that |K i,k (j 1 ,...,j 5 ) (x, y, z)| x -3 |y| -1 y -2 |z| -1 z -2 , ∀(x, y, z) ∈ (R 2 ) 3 .
As for i = 1, 2

I 0 i = B i,0 (j 1 ,...,j 5 ) (ξ, η, ζ) D x 4 u 1 (ξ -η -ζ)û 2 (ζ) D x 7 Ru 3 (η)û 4 (-ξ) dξdηdζ, = K i,0 (j 1 ,...,j 5 ) (t -x, x -z, x -y)[ D x 4 u 1 ](x)u 2 (y)[ D x 7 Ru 3 ](z)u 4 (t)dxdydzdt, I 1 i = B i,1 (j 1 ,...,j 5 ) (ξ, η, ζ) D x 4 u 1 (ξ -η -ζ)û 2 (ζ) D x 11 u 3 (η)û 4 (-ξ) dξdηdζ = K i,1 (j 1 ,...,j 5 ) (t -x, x -z, x -y)[ D x 4 u 1 ](x)u 2 (y)[ D x 11 u 3 ](z)u 4 (t)
f (t -x)g(x -z)h(x -y)| u 1 (x)|| u 2 (y)|| u 3 (z)|| u 4 (t)|dxdydzdt
can be bounded from above by the product of the L 2 norm of any two functions u k times the L ∞ norm of the remaining ones.

(ii) For a cut-off function φ as the one introduced at the beginning of the proof we decompose integral I k 3 , k = 0, 1, distinguishing between |ζ| 1 and |ζ| 1. On the one hand, for any j 1 , . . . , j 5 , k = 0, 1, we consider

B 3,k (j 1 ,...,j 5 ) (ξ, η, ζ) := ψ 3 ξ -η η φ(ζ)B k (j 1 ,j 2 ,j 3 ) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 ξ -η -ζ -3
and observe that, since |ξ| ≤ ξ -η -ζ on the support of ψ 3 ξ-η η φ(ζ), the above multiplier satisfies estimates (2.2.62), (2.2.63). From the same argument as before this implies that (2.2.65)

J 0 3 := B 3,0 (j 1 ,...,j 5 ) (ξ, η, ζ) D x 3 u 1 (ξ -η -ζ)û 2 (ζ) Ru 3 (η)û 4 (-ξ)dξdηdζ u 1 H 3,∞ u 2 L 2 Ru 3 L ∞ u 4 L 2 together with (2.2.66) J 1 3 := B 3,1 (j 1 ,...,j 5 ) (ξ, η, ζ) D x 3 u 1 (ξ -η -ζ)û 2 (ζ) D x 4 u 3 (η)û 4 (-ξ)dξdηdζ u 1 H 3,∞ u 2 L 2 u 3 H 4,∞ u 4 L 2 .
On the other hand, we make a further decomposition on the integral restricted to |ζ| 1 by means of functions ψ i , i = 1, 2, 3, distinguishing between three regions: for |ζ| ≤ c ξ -η , for c ξ -η ≤ |ζ| ≤ C ξ -η and |ζ| > C ξ -η . For any j 1 , . . . , j 5 ∈ {+, -}, k = 0, 1, we hence introduce the following multipliers

B 3,i,k (j 1 ,...,j 5 ) (ξ, η, ζ) := ψ 3 ξ -η η (1 -φ)(ζ)ψ i ζ ξ -η × B k (j 1 ,j 2 ,j 3 ) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 ξ -η -ζ -7 ;
for i = 1m, 3, and

(2.2.67) B 3,2,k (j 1 ,...,j 5 ) (ξ, η, ζ)

:= ψ 3 ξ -η η (1 -φ)(ζ)ψ 2 ζ ξ -η × B k (j 1 ,j 2 ,j 3 ) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 ζ -1 . Since |ξ| ∼ |ξ -η| ∼ |ξ -η -ζ| on the support of ψ 3 ξ-η η (1 -φ)(ζ)ψ 1 ζ ξ-η (resp. |ξ| ∼ |ξ -η| |ζ| ∼ |ξ -η -ζ| on the support of ψ 3 ξ-η η (1 -φ)(ζ)ψ 3 ζ ξ-η ),
:= B 3,i,0 (j 1 ,...,j 5 ) (ξ, η, ζ) D x 7 u 1 (ξ -η -ζ)û 2 (ζ) Ru 3 (η)û 4 (-ξ)dξdηdζ u 1 H 7,∞ u 2 L 2 Ru 3 L ∞ u 4 L 2 along with (2.2.69) J i,1 3 := B 3,i,1 (j 1 ,...,j 5 ) (ξ, η, ζ) D x 7 u 1 (ξ -η -ζ)û 2 (ζ) D x 4 u 3 (η)û 4 (-ξ)dξdηdζ u 1 H 7,∞ u 2 L 2 u 3 H 4,∞ u 4 L 2 ,
∂ α ξ ∂ γ ζ B 3,2,k (j 1 ,...,j 5 ) (ξ, η, ξ -ζ) α,γ η -3 ξ -|α| , ∂ α ξ ∂ β η ∂ γ ζ B 3,2,k (j 1 ,...,j 5 ) (ξ, η, ξ -ζ) (|η| η -1 ) 1-|β| η -3 ξ -|α| , |β| ≥ 1.
If we introduce a Littlewood-Paley decomposition such that

B 3,2,k (j 1 ,...,j 5 ) (ξ, η, ξ -ζ) = l≥1 B 3,2,k (j 1 ,...,j 5 ) (ξ, η, ξ -ζ)ϕ(2 -l ξ),
one can check, using lemma A.1 (i) to obtain the decay in y, making a change of coordinates ξ → 2 l ξ, some integration by parts, and using inequalities (2.2.70), that

K k,l (j 1 ,...,j 5 ) (x, y, z) := e ix•ξ+iy•η+iz•ζ B 3,2,k (j 1 ,...,j 5 ) (ξ, η, ξ -ζ)ϕ(2 -l ξ)dξdηdζ is such that (2.2.71) |K k,l (j 1 ,...,j 5 ) (x, y, z)| 2 2l 2 l x -3 |y| -1 y -2 z -3 , ∀(x, y, z) ∈ (R 2 ) 3 .
Moreover, since |ξ| ∼ |ξ -ζ| on the support of B 3,2,k (j 1 ,...,j 5 ) (ξ, η, ζ) there are two other suitably supported cut-off functions ϕ 1 , ϕ 2 such that ϕ(2

-l ξ) = ϕ(2 -l ξ)ϕ 1 (2 -l ξ)ϕ 2 (2 -l (ξ -ζ)), for any l ≥ 1. If ∆ l j w := ϕ j (2 -l D x )w, we finally obtain that J 2,0 3 := B 3,2,0 (j 1 ,...,j 5 ) (ξ, η, ζ)û 1 (ξ -η -ζ) D x u 2 (ζ) Ru 3 (η)û 4 (-ξ)dξdηdζ = B 3,2,0 (j 1 ,...,j 5 ) (ξ, η, ξ -ζ)û 1 (ζ -η) D x u 2 (ξ -ζ) Ru 3 (η)û 4 (-ξ)dξdηdζ = l≥1 K 0,l (j 1 ,...,j 5 ) (t -y, x -z, y -x)u 1 (x)[∆ l 1 D x u 2 ](y)[Ru 3 ](z)[∆ l 2 u 4 ](t)dxdydzdt,
and by (2.2.71) together with Cauchy-Schwarz inequality we derive that (2.2.72)

| J 2,0 3 | u 1 L ∞ R 1 u 3 L ∞ l≥1 ∆ l 1 D x u 2 L 2 ∆ l 2 u 4 L 2 u 1 L ∞ u 2 H 1 R 1 u 3 L ∞ u 4 L 2 .
In a similar way we obtain that

J 2,1 3 := B 3,2,1 (j 1 ,...,j 5 ) (ξ, η, ξ -ζ)û 1 (ζ -η) D x u 2 (ξ -ζ) D x 4 u 3 (η)û 4 (-ξ)dξdηdζ satisfies (2.2.73) | J 2,1 3 | u 1 L ∞ u 2 H 1 u 3 H 4,∞ u 4 L 2 .
The result of statement (ii) follows then from inequalities (2.

ψ 3 ξ -η η B (j1,j2,j3) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 û1 (ξ -η -ζ)û 2 (ζ)û 3 (η)û 4 (-ξ)dξdηdζ = 1 k=0 J k 3 + 1 k=0 3 i=1 J i,k 3 .
In conclusion, the same proof of above applies to multiplier B 3 (j 1 ,j 2 ,j 3 ) introduced in (2.2.53), which can be decomposed as

j 2 B 0 (j 1 ,j 2 ,j 3 ) (ξ, η) + B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) η 4
with the same B 0 (j 1 ,j 2 ,j 3 ) as in (2.2.60) and

B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) := j 1 ξ -η + j 2 |η| -j 3 ξ 2j 1 ξ -η η -4 (1 -φ)(η).
The lack of factor η 1 |η| -1 against B 0 (j 1 ,j 2 ,j 3 ) , in comparison to decomposition (2.2.59), is the reason why inequality (2.2.57) (resp. (2.2.58)) holds with

u 3 H 11,∞ + Ru 3 H 7,∞ (resp. u 3 H 4,∞ + Ru 3 L ∞ ) replaced with u 3 H 11,∞ (resp. with u 3 H 4,∞ ).
Lemma 2.2.8. Under the same assumptions as in lemma 2.2.7 we have that:

(i) for any j 1 , . . . , j 5 ∈ {+, -}, i = 1, 2, and any

u 1 , u 2 , u 3 , u 4 such that u 1 ∈ H 4,∞ (R 2 ), u 2 , u 4 ∈ L 2 (R 2 ), u 3 ∈ H 11,∞ (R 2 ) and δ k R k u 3 ∈ H 7,∞ (R 2 ), (2.2.74) ψ i ξ -η η B k (j1,j2,j3) (ξ, η) 1 + j 4 j 5 ξ + ζ ξ + ζ • ζ |ζ| ζ 1 û1 (-ξ -ζ)û 2 (ζ)û 3 (η)û 4 (ξ -η)dξdηdζ u 1 H 4,∞ u 2 L 2 ( u 3 H 11,∞ + δ k R k u 3 H 7,∞ ) u 4 L 2 ;
(ii) for any j 1 , . . . , j 5 ∈ {+, -}, and any

u 1 , u 2 , u 3 , u 4 such that u 1 ∈ H 7,∞ (R 2 ), u 2 ∈ L 2 (R 2 ), u 4 ∈ H 1 (R 2 ), u 3 ∈ H 4,∞ (R 2 ) and δ k R k u 3 ∈ L ∞ (R 2 ), (2.2
.75)

ψ 3 ξ -η η B k (j1,j2,j3) (ξ, η) 1 + j 4 j 5 ξ + ζ ξ + ζ • ζ |ζ| ζ 1 û1 (-ξ -ζ)û 2 (ζ)û 3 (η)û 4 (ξ -η)dξdηdζ u 1 H 7,∞ u 2 L 2 ( u 3 H 4,∞ + δ k R k u 3 L ∞ ) u 4 H 1 .
Proof. The proof of the statement is analogous to that of lemma 2.2.7 after a change of coordinates -ξ → ξ -η. In (2.2.75) we take the H 1 norm on u 4 instead of u 2 , as done in (2.2.58), by replacing multiplier B 3,2,k (j 1 ,j 2 ,j 3 ) in (2.2.67) with

ψ 3 ξ -η η (1 -φ)(ζ)ψ 2 ζ ξ -η B k (j 1 ,j 2 ,j 3 ) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 ξ -1 .
Lemma 2.2.9 (Analysis of quartic terms. I). For any general multi-index I, any j k ∈ {+, -}, k = 1, 2, 3, let C I (j 1 ,j 2 ,j 3 ) , C I,R (j 1 ,j 2 ,j 3 ) be the integrals defined in (2.2.36), (2.2.38) respectively, and

D I (j 1 ,j 2 ,j 3 ) , D I,R (j 1 ,j 2 ,j 3 ) introduced in (2.2.44a), (2.2.44b). Then (2.2.76) ∂ t D I (j 1 ,j 2 ,j 3 ) + D I,R (j 1 ,j 2 ,j 3 ) = -C I (j 1 ,j 2 ,j 3 ) -C I,R (j 1 ,j 2 ,j 3 ) + D I quart ,
where D I quart satisfies

D I quart (t) V (t, •) 7 4
H 10,∞ V (t, •)

1 4 H 12 + V (t, •) H 4,∞ ( U (t, •) H 11,∞ + R 1 U (t, •) H 7,∞ ) W I (t, •) 2 L 2 + (I1,I2)∈I(I) |I2|<|I| Q kg 0 (v I1 ± , Du I2 ± )(t, •) L 2 ( U (t, •) H 11,∞ + R 1 U (t, •) H 7,∞ ) V I (t, •) L 2 .
(

-4(2π) 2 ∂ t D I (j 1 ,j 2 ,j 3 ) + C I (j 1 ,j 2 ,j 3 ) = χ ξ -η η B 1 (j 1 ,j 2 ,j 3 ) (ξ, η)   (I 1 ,I 2 )∈I(I) c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± )   (ξ -η)û j 2 (η)v I j 3 (-ξ)dξdη + χ ξ -η η B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) vI j 1 (ξ -η)Q w 0 (v ± , D 1 v ± ) (η)v I j 3 (-ξ)dξdη + χ ξ -η η B 1 (j 1 ,j 2 ,j 3 ) (ξ, η)v I j 1 (ξ -η)û j 2 (η)   (I 1 ,I 2 )∈I(I) c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± )   (-ξ)dξdη =: S 1 + S 2 + S 3 , (2.2.78)
where coefficients c

I 1 ,I 2 ∈ {-1, 0, 1} are such that c I 1 ,I 2 = 1 when |I 1 | + |I 2 | = |I| (in which case D = D 1 ) and χ ∈ C ∞ 0 (R 2
) is equal to 1 close to the origin and has a sufficiently small support. All integrals in the above right hand side are quartic terms for they involve the quadratic nonlinearities of (2.2.56).

The fact that S 2 is a remainder D I quart satisfying (2.2.77) follows by inequalities (A.17), (B.1.3d) with s = 7, and the fact that

(2.2.79) R 1 Q w 0 (v ± , D 1 v ± ) H 7,∞ V (t, •) 2-(2-θ)θ H 10,∞ V (t, •) (2-θ)θ H 12 ,
for any θ ∈]0, 1[. The above inequality is justified by the fact that, for any function

w ∈ W 1,∞ ∩ H 1 , ρ ∈ N and any θ ∈]0, 1[, setting p = 2 θ ∈]2, ∞[, (2.2.80) D x ρ R 1 w L ∞ D x ρ R 1 w W 1,p D x ρ w W 1,p D x ρ w 1-θ W 1,∞ D x ρ w θ H 1 D x ρ w 1-θ H 2,∞ D x ρ w θ H 1 ,
as a consequence of Morrey's inequality, continuity of R 1 : L p → L p for p < +∞, interpolation inequality, and the injection of

W 1,∞ into H 2,∞ . This implies that (2.2.81) R 1 Q w 0 (v ± , D 1 v ± ) H ρ,∞ Q w 0 (v ± , D 1 v ± ) 1-θ H ρ+2,∞ Q w 0 (v ± , D 1 v ± ) θ H ρ+1 ,
for any ρ ∈ N, and gives (2.2.79) when ρ = 7 after inequalities (B.1.3c) with s = 8, (B.1.3d) with s = 9. Therefore, for any θ ∈]0, 1[,

|S 2 | V (t, •) 2-θ H 8,∞ V (t, •) θ H 10 + V (t, •) 2-(2-θ)θ H 10,∞ V (t, •) (2-θ)θ H 12 V I (t, •) 2 L 2 ,
so choosing θ 1 small (e.g. θ ≤ 1/8) and keeping in mind estimates (1.1.11b), (1.1.11c) we deduce that S 2 is controlled by the first term in the right hand side of (2.2.77). Inequality (A.17) allows also to estimate all integrals in summations S 1 , S 3 corresponding to indices (I 1 , I 2 ) ∈ I(I) with |I 2 | < |I|, and to bound them with the latter term in the right hand side of (2.2.77). This is not the case for integrals with I 2 = I involving quasi-linear term

Q kg 0 (v ± , D 1 u I ± )
, because a straight application of that inequality would give a bound at the wrong energy level n + 1, as

Q kg 0 (v ± , D 1 u I ± ) L 2 V (t, •) H 1,∞ D 1 U I (t, •) L 2 . Instead, since (2.2.82) Q kg 0 (v ± , D 1 u I ± ) (ξ) = i 4 j 4 ,j 5 ∈{+,-} 1 -j 4 j 5 ξ -ζ ξ -ζ • ζ |ζ| ζ 1 vj 4 (ξ -ζ)û I j 5 (ζ)dζ,
we can rather write those integrals as the sum over j k ∈ {+, -}, k = 1, . . . 4, of the following:

(2.2.83a)

χ ξ -η η B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 vj 4 (ξ -η -ζ)û I j 5 (ζ)û j 2 (η)v I j 3 (-ξ) dξdηdζ, (2.2.83b) χ ξ -η η B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) 1 + j 4 j 5 ξ + ζ ξ + ζ • ζ |ζ| ζ 1 vI j 1 (ξ -η)û j 2 (η)v j 4 (-ξ -ζ)û I j 5 (ζ) dξdηdζ,
and estimate them by using inequalities (2.2.57) and (2.2.74) respectively. We hence obtain that

|S 1 | + |S 3 | V (t, •) H 4,∞ ( U (t, •) H 11,∞ + R 1 U (t, •) H 7,∞ ) W I (t, •) 2 L 2 + (I 1 ,I 2 )∈I(I) |I 2 |<|I| Q kg 0 (v I 1 ± , Du I 2 ± )(t, •) L 2 ( U (t, •) H 11,∞ + R 1 U (t, •) H 7,∞ ) V I (t, •) L 2 ,
and, since the same argument applies to ∂ t D I,R (j 1 ,j 2 ,j 3 ) , this also concludes the proof of the statement.

Lemma 2.2.10 (Analysis of quartic terms. II). For any general multi-index I, any

j k ∈ {+, -}, k = 1, 2, 3, let D I,T -N (j 1 ,j 2 ,j 3 ) be defined as in (2.2.44c). Then (2.2.84) ∂ t D I,T -N (j 1 ,j 2 ,j 3 ) = T -N (U )W I , W I + D I,N quart and if N ≥ 18 D I,N quart satisfies D I,N quart V (t, •) 7 4 H N +4,∞ V (t, •) 1 4 H N +6 W I (t, •) 2 L 2 + (I 1 ,I 2 )∈I(I) |I 2 |<|I| Q kg 0 (v I 1 ± , Du I 2 ± ) L 2 U (t, •) H N +3,∞ V I (t, •) L 2 .
(2.2.85)

Proof. For any triplet (j 1 , j 2 , j 3 ), we compute the time derivative of D I,T -N by making use of system (2.2.56). Recalling (2.2.39) and (2.2.43), we find that

∂ t   j k ∈{+,-} D I,T -N (j 1 ,j 2 ,j 3 )   -[ T -N (U )W I , W I ] = = Re   1 (2π) 2 σ N (j 1 ,j 2 ,j 3 ) (ξ, η)   (I 1 ,I 2 )∈I(I) c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± ) (ξ -η)   ûj 2 (η)v I -j 3 (-ξ)dξdη + 1 (2π) 2 σ N (j 1 ,j 2 ,j 3 ) (ξ, η)v I j 1 (ξ -η)Q w 0 (v ± , D 1 v ± ) (η)v I -j 3 (-ξ)dξdη + 1 (2π) 2 σ N (j 1 ,j 2 ,j 3 ) (ξ, η)v I j 1 (ξ -η)û j 2 (η)   (I 1 ,I 2 )∈I(I) c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± ) (-ξ)   dξdη   =: S T -N 1 + S T -N 2 + S T -N 3 , (2.2
T -N 2 | V (t, •) 2-θ H N +4,∞ V (t, •) θ H N +6 V I (t, •) 2 L 2 .
Choosing θ 1 small (e.g. θ ≤ 1/8) we then obtain that S T -N 2 is a remainder D I,N quart satisfying (2.2.85). Also, the same lemma implies that each contribution in S

T -N 1 , S T -N 3 corresponding to (I 1 , I 2 ) ∈ I(I) with |I 2 | < |I| is bounded by Q kg 0 (v I 1 ± , Du I 2 ± ) L 2 U (t, •) H N +3,∞ V I (t, •) L 2 .
Reminding instead (2.2.82), we find that the remaining contribution to S

T -N 1
, corresponding to I 2 = I, is equal to the sum over j 1 , . . . , j 5 ∈ {+, -} of the (imaginary part) of the following integrals:

(2.2.87)

σ N (j 1 ,j 2 ,j 3 ) (ξ, η) 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 vj 4 (ξ -η -ζ)û I j 5 (ζ)û j 2 (η)v I j 3 (-ξ) dξdηdζ.
Analogously, the contribution corresponding to

I 2 = I in S T -N 3
is the sum over

j k ∈ {+, -}, k = 1, . . . , 5 of (2.2.88) σ N (j 1 ,j 2 ,j 3 ) (ξ, η) 1 + j 4 j 5 ξ + ζ ξ + ζ • ζ |ζ| ζ 1 vI j 1 (ξ -η)û j 2 (η)v j 4 (-ξ -ζ)û I j 5 (ζ) dξdηdζ.
Since σ N (j 1 ,j 2 ,j 3 ) (ξ, η) satisfies (A.23) and is supported for |η| ≤ ε|ξ -η|, for a small 0 < ε 1, we rewrite above integrals, respectively, as

(2.2.89) σ N (j 1 ,j 2 ,j 3 ) (ξ, η) η -N -3 1 -j 4 j 5 ξ -η -ζ ξ -η -ζ • ζ |ζ| ζ 1 ξ -η -ζ -4 × D x 4 v j 4 (ξ -η -ζ)û I j 5 (ζ) D x N +3 u j 2 (η)v I j 3 (-ξ) dξdηdζ,

and

(2.2.90) σ N (j 1 ,j 2 ,j 3 ) (ξ, η) η -N -7 1 + j 4 j 5

ξ + ζ ξ + ζ • ζ |ζ| ζ 1 ξ + ζ -4 × vI j 1 (ξ -η) D x N +7 u j 2 (η) D x 4 v j 4 (-ξ -ζ)û I j 5 (ζ) dξdηdζ.
With such a choice, the new multipliers, that we denote concisely by σ N,k (j 1 ,...,j 5 ) (ξ, η, ζ), k = 0, 1, verify, for any α, β, γ ∈ N 2 ,

∂ α ξ ∂ β η σ N,k (j 1 ,...,j 5 ) (ξ, η, ζ) ζ -3 |g N α,β (ξ)|, ∂ α ξ ∂ β η ∂ γ ζ σ N,k (j 1 ,...,j 5 ) (ξ, η, ζ) (|ζ| ζ -1 ) 1-|γ| ζ -3 |g N α,β (ξ)|, |γ| ≥ 1, with g N α,β (ξ, η) supported for |η| ≤ ε|ξ -η| and such that |g N α,β (ξ, η)| ξ -η 6-N +|α|+2|β| |η| N -|β| η -N -3 , ∀(ξ, η) ∈ R 2 × R 2 .
If N ∈ N * is sufficiently large (e.g. N ≥ 18), the above estimate implies that, for any α, β ∈ N 2 of length less or equal than 3,

|g N α,β (ξ, η)| η -3 ξ -3 ,
so by lemma A.1 (i) together with corollary A.2 (i) we obtain that, for any k = 0, 1,

K N,k (j 1 ,...,j 5 ) (x, y, z) := e ix•ξ+iy•η+iz•ζ σ N,k (j 1 ,...,j 5 ) (ξ, η, ζ)dξdηdζ is such that (2.2.91) |K N,k (j 1 ,...,j 5 ) (x, y, z)| x -3 |y| -1 y -2 |z| -1 z -2 , ∀(x, y, z) ∈ (R 2 ) 3 .
K N,0 (j 1 ,...,j 5 ) (t -x, x -z, x -y)[ D x 4 v j 4 ](x)u I j 5 (y)[ D x N +3 u j 2 ](z)v I j 3 (t)dxdydzdt and K N,1 (j 1 ,...,j 5 ) (z -x, x -y, z -t)v I j 1 (x)[ D x N +7 u j 2 ](y)[ D x 4 v j 4 ](z)u I j 5 (t)dxdydzdt.
Using (2.2.91) and the fact that integrals such as (2.2.64) can be bounded from above by the product of the L 2 norm of any two functions u k times the L ∞ norm of the remaining ones, they are estimated by (j 1 ,j 2 ,j 3 ) , D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) be the integrals defined, respectively, in (2.2.41), (2.2.45), for any

V (t, •) H 4,∞ U (t, •) H N +7,∞ W I (t, •) 2 L 2 ,
j k ∈ {+, -}, k = 1, 2, 3. Then (2.2.92) ∂ t D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) = -C I 1 ,I 2 (j 1 ,j 2 ,j 3 ) + D I 1 ,I 2 quart ,
where

D I 1 ,I 2 quart satisfies (2.2.93) D I 1 ,I 2 quart (t) W (t, •) H [ n 2 ]+12,∞ + R 1 U (t, •) H [ n 2 ]+8,∞ 2 + V (t, •) 7 4 H [ n 2 ]+11,∞ V (t, •) 1 4 H [ n 2 ]+12 E n (t; W ).
Proof. We compute the time derivative of D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) by making use of system (2.2.56). We remind that, after remark 1.1.5 and definition (1.1.18), if Γ I is a product of spatial derivatives then all couples of indices (I 1 , I 2 ) belonging to I(I) are such that |I 1 | + |I 2 | = |I| and Γ I 1 , Γ I 2 are also products of spatial derivatives. Therefore, all coefficients c I 1 ,I 2 appearing in the right hand side of (2.2.56) are equal to 0. By definitions (2.2.42) with k = 1, (2.2.41), (2.2.45), we find that

-4(2π) 2 ∂ t D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) + C I 1 ,I 2 (j 1 ,j 2 ,j 3 ) = B 1 (j 1 ,j 2 ,j 3 ) (ξ, η)   (J 1 ,J 2 )∈I(I 1 ) Q kg 0 (v J 1 ± , D 1 u J 2 ± ) (ξ -η)   ûI 2 j 2 (η)v I j 3 (-ξ)dξdη + B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) vI 1 j 1 (ξ -η)   (J 1 ,J 2 )∈I(I 2 ) Q w 0 (v J 1 ± , D 1 v J 2 ± ) (η)   vI j 3 (-ξ)dξdη + B 1 (j 1 ,j 2 ,j 3 ) (ξ, η)v I 1 j 1 (ξ -η)û I 2 j 2 (η)   (J 1 ,J 2 )∈I(I) Q kg 0 (v J 1 ± , D 1 u J 2 ± )   (-ξ)dξdη =:S I 1 ,I 2 1 + S I 1 ,I 2 2 + S I 1 ,I 2 3 .
(2.2.94)

Since |J 1 | + |J 2 | = |I 1 | < |I| ≤ n in S I 1 ,I 2 1
, we can estimate all its contributions using inequality (A.17). Using lemma 2.1.2 (i), the fact that |I 2 | ≤ [ n 2 ] by the hypothesis and, hence, that

u I 2 ± (t, •) H 7,∞ + R 1 u I 2 ± (t, •) H 7,∞ U (t, •) H [ n 2 ]+8,∞ ,
we deduce that

S I 1 ,I 2 1 W (t, •) H [ n 2 ]+2 + R 1 U (t, •) H [ n 2 ]+2,∞ U (t, •) H [ n 2 ]+8,∞ E n (t; W ),
and above estimate holds also for all integrals in S I 1 ,I 2 3 corresponding to |J 2 | < |I|. The same inequality (A.17), combined with (2.2.81) applied to

Q w 0 (v J 1 ± , D 1 v J 2 ±
) and with corollary A.4 in appendix A, gives that, for any θ ∈]0, 1[,

|S I 1 ,I 2 2 | |J 1 |+|J 2 |=|I 2 | Q w 0 (v J 1 ± , D 1 v J 2 ± ) H 7,∞ + Q w 0 (v J 1 ± , D 1 v J 2 ± ) 1-θ H 9,∞ Q w 0 (v J 1 ± , D 1 v J 2 ± ) θ H 8 E n (t; W ) V (t, •) 2-(2-θ)θ H [ n 2 ]+11,∞ V (t, •) (2-θ)θ H [ n 2 ]+12 E n (t; W ).
Finally, the last remaining integral in S I 1 ,I 2

3

, corresponding to indices J 1 = 0, J 2 = I, can be written using (2.2.82) as

j 4 ,j 4 ∈{+,-} B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) 1 + j 4 j 5 ξ + ζ ξ + ζ • ζ |ζ| ζ 1 vI 1 j 1 (ξ -η)û I 2 j 2 (η)v j 4 (-ξ -ζ)û I j 5 (ζ)dξdηdζ,
and is estimated, after lemma 2.2.8 and the fact that

|I 1 | < |I|, by V (t, •) H 7,∞ U (t, •) H [ n 2 ]+12,∞ + R 1 U (t, •) H [ n 2 ]+8,∞ E n (t; W ).
This gives that

S I 1 ,I 2 3 W (t, •) H [ n 2 ]+12,∞ + R 1 U (t, •) H [ n 2 ]+8,∞ 2 E n (t; W )
and concludes the proof of the statement.

Lemma 2.2.12 (Analysis of quartic terms. IV). Let k = 0, 1, K, V k be the sets introduced in (2.1.25), (2.1.26) respectively, I ∈ V k and (I 1 , I 2 ) ∈ I(I) be such that

I 1 ∈ K, |I 2 | ≤ 1.
Let also F I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) , G I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) be the integrals defined in (2.2.52), (2.2.54), for any l = 1, 2, 3, j i ∈ {+, -}, i = 1, 2, 3. For any l = 1, 2, 3, any triplet (j 1 , j 2 , j 3 ), we have that

(2.2.95) ∂ t G I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) = -F I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) + G I 1 ,I 2 quart ,
and there is a constant C > 0 such that, if a-priori estimates (1.1.11) are satisfied in interval

[1, T ] for a fixed T > 1, with ε 0 < (2A + B) -1 small, (2.2.96) |G I 1 ,I 2 quart (t)| ≤ C(A + B) 2 ε 2 t -1+ δ k 2 E k 3 (t; W ) 1 2 + δ V 0 t β+ δ 1 2 E 1 3 (t; W ) 1 2 + t -1 4 - δ k 2 ,
for every t ∈ [1, T ], with δ V 0 = 1 if I ∈ V 0 , 0 otherwise, and β > 0 as small as we want.

Proof. First of all, it is useful to remind that from (2.1.42), (2.1.43) and a-priori estimate (1.1.11d), for any k = 0, 1, I ∈ I k 3 , (I 1 , I 2 ) ∈ I(I) such that I 1 ∈ K, |I 2 | ≤ 1, and σ > 0 sufficiently small (2.2.97)

V I 1 (t, •) L 2 χ(t -σ D x )U I 2 (t, •) H ρ,∞ + χ(t -σ D x )RU I 2 (t, •) H ρ,∞ ≤ C(A + B)Bε 2 t -1 2 + δ k 2 , for every t ∈ [1, T ].
For any fixed (j 1 , j 2 , j 3 ), any l = 1, 2, 3, we compute ∂ t G I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) recurring to system (2.2.56) along with its compact form

(D t ∓ D x )v I ± = Γ I Q w 0 (v ± , D 1 v ± ), (D t ∓ |D x |)u I ± = Γ I Q kg 0 (v ± , D 1 u ± ),
and using that

[D t , χ(t -σ D x )] = t -1 χ 1 (t -σ D x ) with χ 1 (ξ) := iσ(∂χ)(ξ) • ξ. We find that -4(2π) 2 ∂ t G I1,I2,l (j1,j2,j3) + F I1,I2,l (j1,j2,j3) = B l (j1,j2,j3) (ξ, η) Γ I1 Q kg 0 (v ± , D 1 u ± ) (ξ -η) χ(t -σ D x )u I2 j2 (η)v I j3 (-ξ)dξdη + B l (j1,j2,j3) (ξ, η)v I1 j1 (ξ -η) χ(t -σ D x )Γ I2 Q w 0 (v ± , D 1 v ± ) (η) + t -1 χ 1 (t -σ D x )u I2 j2 (η) vI j3 (-ξ)dξdη + B l (j1,j2,j3) (ξ, η)v I1 j1 (ξ -η)χ(t -σ D x )u I2 j2 (η) (J1,J2)∈I(I) c J1,J2 Q kg 0 (v J1 ± , Du J2 ± ) (-ξ) dξdη =: S I1,I2,l 1 + S I1,I2,l 2 + S I1,I2,l 3 ,
with B l (j 1 ,j 2 ,j 3 ) given by (2.2.42) when l = 1, 2 or (2.2.53) when l = 3.

Applying (A.17) to S I 1 ,I 2 ,l 2 , using (2.2.80) with w = Γ I 2 Q w 0 (v ± , D 1 v ± ) and ρ = 7, together with the fact that operators χ(t -σ D x ), χ 1 (t -σ D x ) are bounded from L ∞ to H ρ,∞ for any ρ ≥ 0 with norm O(t σρ ), and from L 2 to H s for any s ≥ 0 with norm O(t σs ), we deduce that, for any

θ ∈]0, 1[, (2.2.98) |S I 1 ,I 2 ,l 2 | t β V I 1 (t, •) L 2 V I (t, •) L 2 × Γ I 2 Q w 0 (v ± , D 1 v ± ) L ∞ + δ l Γ I 2 Q w 0 (v ± , D 1 v ± ) 1-θ L ∞ Γ I 2 Q w 0 (v ± , D 1 v ± ) θ L 2 +t -1 χ 1 (t -σ D x )u I 2 ± (t, •) L ∞ + χ 1 (t -σ D x )Ru I 2 ± (t, •) L ∞ ,
for some β > 0 small, β → 0 as σ → 0, and with

δ l = 1 if l = 1,
Γ I 2 Q w 0 (v ± , D 1 v ± ) = Q w 0 (v I 2 ± , D 1 v ± ) + Q w 0 (v ± , D 1 v I 2 ± ) + G w 1 (v ± , Dv ± ) with G w 1 (v ± , Dv ± ) = G 1 (v, ∂v)
given by (1.1.16). Using lemma B.2.4 in appendix B with L = L ∞ to estimate the L ∞ norm of the first two quadratic terms in the above right hand side, we find that, for some new χ ∈ C ∞ 0 (R 2 ) and σ > 0 small, there is a constant C > 0 such that

Γ I 2 Q w 0 (v ± , D 1 v ± ) L ∞ χ(t -σ D x )v I 2 ± (t, •) H 2,∞ v ± (t, •) H 2,∞ + t -N (s) ( v ± (t, •) H s + D t v ± (t, •) H s ) 1 |µ|=0 x µ v ± (t, •) H 1 + t v ± (t, •) H 1 + v ± (t, •) H 1,∞ ( v ± (t, •) H 2,∞ + D t v ± (t, •) H 1,∞ ) ≤ CABε 2 t -2 ,
where last inequality is obtained by picking s > 0 sufficiently large so that N (s) ≥ 4 and using (B.1.6a), (B.1.6b), (B.1.10a), lemma B.4.14, together with a-priori estimates. Also, by (B.1.6a) with s = 0 and a-priori estimates

Γ I 2 Q w 0 (v ± , D 1 v ± ) L 2 V (t, •) H 2,∞ V I 2 (t, •) H 1 + D t V (t, •) L 2 ≤ CABε 2 t -1+ δ 2 2 .
Therefore, using lemma B.2.10 and taking θ, σ > 0 sufficiently small we deduce from (2.2.98) and the above estimates that, for any l = 1, 2, 3 and a new C > 0,

(2.2.99)

|S I 1 ,I 2 ,l 2 | ≤ CABε 2 t -5 4 E k 3 (t; W ) 1 2 .
We make use of inequality (A.17) to estimate S I 1 ,I 2 ,l 1 , too. From (1.1.17) we have that

Γ I 1 Q kg 0 (v ± , D 1 u ± ) = Q kg 0 (v I 1 ± , D 1 u ± ) + (J 1 ,J 2 )∈I(I 1 ) |J 1 |<|I 1 | c J 1 ,J 2 Q kg 0 (v J 1 ± , Du J 2 ± )
with c J 1 ,J 2 ∈ {-1, 0, 1}, and then from (2.1.29b), (2.1.34b) and the fact that I 1 ∈ K,

Γ I 1 Q kg 0 (v ± , D 1 u ± ) = Q kg 0 (v I 1 ± , χ(t -σ D x )D 1 u ± ) + R k 3 (t, x),
with R k 3 satisfying (2.1.30) and

Q kg 0 (v I 1 ± , χ(t -σ D x )D 1 u ± ) L 2 ≤ ( U (t, •) H 2,∞ + RU (t, •) H 2,∞ ) V I 1 (t, •) L 2 .
So from (2.1.30), (2.2.97), lemma B.2.10 and priori estimates (1.1.11)

|S I 1 ,I 2 ,l 1 | ( U (t, •) H 2,∞ + RU (t, •) H 2,∞ ) V I 1 (t, •) L 2 + R k 3 (t, •) L 2 × χ(t -σ D x )U I 2 (t, •) H 7,∞ + χ(t -σ D x )RU I 2 (t, •) H 7,∞ V I (t, •) L 2 ≤ CABε 2 t -1+ δ k 2 E k 3 (t; W ) 1 2 .
(2.2.100)

Let us now consider all the addends in S I 1 ,I 2 ,l 3 with |J 2 | < |I|, which by inequality (A.17) are bounded by

V I 1 (t, •) L 2 1 |µ|=0 χ(t -σ D x )R µ U I 2 (t, •) H 7,∞ (J 1 ,J 2 )∈I(I) |J 2 |<|I| c J 1 ,J 2 Q kg 0 (v J 1 ± , Du J 2 ± ) L 2 .
As the latter above factor is bounded by the L 2 norm of Q I 0 (V, W ) (see definition (2.1.12)), inequalities (2.1.44) and (2.2.97) imply that those integrals are remainders G I 1 ,I 2 quart satisfying (2.2.96). Finally, the last contribution to S I 1 ,I 2 ,l 3 , corresponding to |J 1 | = 0, J 2 = I, for which D = D 1 , can be rewritten using (2.2.82) as the sum over j 4 , j 5 ∈ {+, -} of

B 1 (j 1 ,j 2 ,j 3 ) (ξ, η) 1 + j 4 j 5 ξ + ζ ξ + ζ • ζ |ζ| ζ 1 v4 (-ξ -ζ)û I j 5 (ζ)χ(t -σ D x )u I 2 j 2 (η)v I 1 j 1 (ξ -η)dξdη.
By means of lemma 2.2.8 it is bounded by

V (t, •) H 7,∞ 1 |µ|=0 χ(t -σ D x )D 1 R µ U J 2 (t, •) H 11,∞ V I 1 (t, •) H 1 U I (t, •) L 2
for every t ∈ [1, T ], and hence by CA(A + B)ε 2 t -3 2 +β E k 3 (t; W ), with β > 0 small as long as σ, δ 0 are small, as follows by a-priori estimate (1.1.11b) and lemma B.2.10.

Propagation of the energy estimate

Proposition 2.2.13 (Propagation of the energy estimate). Let us fix K 2 > 0. There exist two integers n ρ 1 sufficiently large, two constants A, B > 1 sufficiently large, ε 0 ∈ ]0, (2A + B) -1 [ sufficiently small, and some 0 < δ δ 2 δ 1 δ 0 1 small such that, for any Proof. For any integer k, n ∈ N, with n ≥ 3 and 0 ≤ k ≤ 2, let E n (t; W ), E k 3 (t; W ) be the first modified energies introduced in (2.2.11) and E † n (t; W ), E k, † 3 (t; W ) be the second modified energies, introduced in (2.2.46) and (2.2.55) respectively. Let also D I (j 1 ,j 2 ,j 3 ) , D I,R (j 1 ,j 2 ,j 3 ) , D

0 < ε < ε 0 , if (u, v) is solution to (1.
I,T -N (j 1 ,j 2 ,j 3 )
be the integrals defined in (2.2.44), D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) in (2.2.45), and G I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) in (2.2.54). Fix N = 18. The first thing we observe is that, as long as estimates (1.1.11a), (1.1.11b) are satisfied and ρ ∈ N is sufficiently large (e.g. ρ ≥ max{[ n 2 ] + 8, 21}), there is a constant C > 0 such that for every t ∈ [1, T ]

C -1 E n (t; W ) ≤ E † n (t; W ) ≤ CE n (t; W ), (2.2.101a) C -1 E k 3 (t; W ) ≤ E k, † 3 (t; W ) ≤ CE k 3 (t; W ). (2.2.101b)
Above equivalences follow from (2.2.12), a-priori estimates (1.1.11a), (1.1.11b), the fact that for a general multi-index

I (I ∈ I n or I ∈ I k 3 for 0 ≤ k ≤ 2) j i ∈{+,-} D I (j 1 ,j 2 ,j 3 ) + D I,R (j 1 ,j 2 ,j 3 ) ( U (t, •) H 7,∞ + R 1 U (t, •) H 7,∞ ) V I (t, •) 2 L 2
by inequality (A.17),

j k ∈{+,-} D I,T -18 (j 1 ,j 2 ,j 3 ) U (t, •) H 21,∞ W I (t, •) 2 L 2
by inequality (A.24), and:

• as concerns especially (2.2.101a), from the fact that for any I ∈ I n , any (I 1 , I 2 ) ∈ I(I) with

[ |I| 2 ] < |I 1 | < |I|, by (A.17) j i ∈{+,-} D I 1 ,I 2 (j 1 ,j 2 ,j 3 ) U I 2 (t, •) H 7,∞ + R 1 U I 2 (t, •) H 7,∞ V I 1 (t, •) L 2 V I (t, •) L 2 U (t, •) H [ n 2 ]+8,∞ + R 1 U (t, •) H [ n 2 ]+8,∞ E n (t;
j i ∈{+,-} G I 1 ,I 2 ,l (j 1 ,j 2 ,j 3 ) 1 |µ|=0 χ(t -σ D x )R µ U I 2 (t, •) H 7,∞ V I 1 (t, •) L 2 V I (t, •) L 2 ≤ C(A + B)Bε 2 t -1 2 + δ k 2 E k 3 (t; W ) 1 2 .
(2.2.102)

Let us now consider a general multi-index I. From equation (2.2.3) we deduce the following equality:

1 2 ∂ t W I s (t, •) 2 L 2 = - D t W I s , W I s = - A(D) W I s , W I s + Op B (I 4 + E 0 d (U ; η)) A 1 (V ; η)(I 4 + F 0 d (U ; η)) W I s , W I s + Op B (A (V I ; η))U + Op B R (A (V I ; η))U, W I s + Q I 0 (V, W ), W I s + T -18 (U )W I s , W I s + R (U, V ), W I s (2.2.103)
and immediately notice that [ A(D) W I s , W I s ] = 0 because of the fact that A(η), introduced in (2.1.5), is real diagonal matrix and its quantization is a self-adjoint operator.

Matrix (I 4 + E 0 d (U ; η)) A 1 (V ; η)(I 4 + F 0 d (U ; η)
) is real, symmetric, of order 1, with semi-norm

M 1 1 I 4 + E 0 d (U ; η)) A 1 (V ; η)(I 4 + F 0 d (U ; η) , 3 (1 + R 1 U (t, •) H 2,∞ ) 2 V (t, •) H 2,∞
as follows by estimate (2.2.7a) on E 0 d , (2.2.8) of F 0 d , and (2.1.49) on A 1 (V ; η). Corollary 1.2.13 and a-priori estimates (1.1.11a), (1.1.11b) imply then that the second term in the right hand side of (2.2.103) reduces to T 0 (U, V ) W I s , W I s , with T 0 (U, V ) operator of order less or equal than 0 such that

T 0 (U, V ) L(L 2 ) M 1 1 I 4 + E 0 d (U ; η)) A 1 (V ; η)(I 4 + F 0 d (U ; η) , 3 ≤ CAεt -1 ,
so after Cauchy-Schwarz inequality and equivalence (2.2.10) it is a remainder R(t) satisfying, for

every t ∈ [1, T ] (2.2.104) |R(t)| ≤ CAεt -1 W I (t, •) 2 L 2 .
Observe that, by the definition of W I s in (2.2.2) and of W I s in (2.1.45), we have that

( W I s -W I )(t, •) L 2 ≤ Op B (P (V ; η) -I 4 )W I L 2 + Op B (E(U ; η))W I s L 2 ( V (t, •) H 1,∞ + U (t, •) H 5,∞ + R 1 U (t, •) H 1,∞ ) W I (t, •) L 2 , ( 2
M 0 0 E χ D x η U ; η ; n U (t, •) H 5,∞ + R 1 U (t, •) H 1,∞ ,
and equivalence (2.1.50). Therefore, third and fifth brackets in the right hand side of (2.2.103) can be replaced with

Op B (A (V I ; η))U + Op B R (A (V I ; η))U, W I + T -18 (U )W I ,
1 2 ∂ t W I s (t, •) L 2 = - Op B (A (V I ; η))U + Op B R (A (V I ; η))U, W I + Q I 0 (V, W ), W I s + T -18 (U )W I s , W I s + R (U, V ), W I s + R(t).
(

Q I 0 (V, W ), W I s = Q I 0 (V, W ), W I + R n (t)
where, for a new constant C > 0 and every t ∈ [1, T ],

(2.2.108)

|R n (t)| ≤ CAεt -1+ δ 2 E n (t; W ) 1 2 .
Reminding definition (2.1.12) of Q I 0 (V, W ) and the fact that coefficients c I 1 ,I 2 are all equal to 0 when I ∈ I n , we notice that some of the contributions to the scalar product in the right hand side of (2.2.107) are also remainders R n (t). These are precisely the following ones:

(I 1 ,I 2 )∈I(I) Q w 0 (v I 1 ± , D 1 v I 2 ± ), u I + + u I -+ (I 1 ,I 2 )∈I(I) |I 1 |≤[ |I| 2 ] Q kg 0 (v I 1 ± , D 1 u I 2 ± ), v I + + v I - in consequence
1 2 ∂ t W I s (t, •) 2 L 2 = - Op B (A (V I ; η))U + Op B R (A (V I ; η))U, W I + (I 1 ,I 2 )∈I(I) [ |I| 2 ]<|I 1 |<|I| Q kg 0 (v I 1 ± , D 1 u I 2 ± ), v I + + v I -+ T -18 (U )W I , W I + R n (t).
From definition (2.

1 2 ∂ t E † n (t; W ) |R n (t)| + I∈In D I quart (t) + D I,18 quart (t) + I∈In (I 1 ,I 2 )∈I(I) [ |I| 2 ]<|I 1 |<|I| D I 1 ,I 2 quart (t) ,
where quartic terms D I quart , D I,18 quart , D I 1 ,I 2 quart satisfy, respectively, (2.2.77), (2.2.85) with N = 18, (2.2.93). These latter ones can also be considered as remainders R n (t) thanks to lemma 2.1.2 (i) and a-priori estimates (1.1.11), which implies that, for some new C > 0 and every t ∈ [1, T ],

∂ t E † n (t; W ) ≤ CAεt -1+ δ 2 E n (t; W ) 1 2
.

Then

E † n (t; W ) 1 2 ≤ E † n (1; W ) 1 2 + t 1
CAετ -1+ δ 2 dτ, so after equivalence (2.2.101a) and a-priori estimate (1.1.11c)

E n (t; W ) 1 2 ≤ CE n (1; W ) 1 2 + t 1 CAετ -1+ δ 2 dτ ≤ CE n (1; W ) 1 2 + 2CAε δ t δ 2 ,
again for a new C > 0. Taking B > 1 sufficiently large so that E n (1; W )

1 2 ≤ Bε 2CK 2 and 2CA δ < B 2K 2
we finally obtain (1.1.12c).

Propagation of a-priori estimate (1.1.11d): Let us now suppose that I ∈ I k 3 for 0 ≤ k ≤ 2. After (2.1.39) and (2.2.105) we have that

Q I 0 (V, W ), W I s = Q I 0 (V, W ), W I + R k 3 (t) with (2.2.109) |R k 3 (t)| ≤ CA(A + B)ε 2 t -1+ δ k 2 E k 3 (t; W ) 1 2 ,
and moreover

- Q I 0 (V, W ), W I = -δ V k (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 c I 1 ,I 2 Q kg 0 v I 1 ± , χ(t -σ D x )D x u I 2 ± , v I + + v I - -δ V k (J,0)∈I(I) J∈K c J,0 Q kg 0 v J ± , χ(t -σ D x )|D x |u ± , v I + + v I - + R k 3 (t), (2.2.110)
with δ V k = 1 if I ∈ V k , 0 otherwise, as already seen in (2.2.50). Also, R (U, V ), W I s in the right hand side of (2.2.106) and R(t) are remainders R k 3 (t) in consequence of the same argument used in the previous case, but with estimate (2.1.38) replaced with (2.1.39). Therefore, we can further reduce (2.2.106) to the following equality:

1 2 ∂ t W I s (t, •) 2 L 2 = - Op B (A (V I ; η))U + Op B R (A (V I ; η))U, W I + T -18 (U )W I , W I -δ V k (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 c I 1 ,I 2 Q kg 0 v I 1 ± , χ(t -σ D x )Du I 2 ± , v I + + v I - -δ V k (J,0)∈I(I) J∈K c J,0 Q kg 0 v J ± , χ(t -σ D x )|D x |u ± , v I + + v I - + R k 3 (t)
∂ t E k, † 3 (t; W ) |R k 3 (t)| + I∈I k 3 D I quart (t) + D I,18 quart (t) + δ k<2 I∈V k j i ∈{+,-} (I 1 ,I 2 )∈I(I) I 1 ∈K,|I 2 |≤1 G I 1 ,I 2 (j 1 ,j 2 ,j 3 )
with δ k<2 = 1 for k < 2, 0 otherwise. On the one hand, quartic terms D I quart , D I,18 quart satisfy, respectively, (2.2.77) and (2.2.85) with N = 18, and are remainders R k 3 (t) after (2.1.39) and a-priori estimates. On the other hand, G I 1 ,I 2 (j 1 ,j 2 ,j 3 ) verifies estimate (2.2.96). Consequently, there is a constant C > 0 such that

E k, † 3 (t; W ) ≤ E k, † 3 (1; W ) + C(A + B) 2 ε 2 t 1 τ -1+ δ k 2 E k 3 (tτ ; W ) 1 2 dτ + δ k<2 C(A + B) 2 ε 2 δ k=0 t 1 τ -1+ δ 0 2 +β+ δ 1 2 E 1 3 (τ ; W ) 1 2 dτ + t 1 τ -5 4 dτ
with δ k=0 = 1 if k = 0, 0 otherwise, β > 0 as small as we want, and after equivalence (2.2.101b)

E k 3 (t; W ) ≤ CE k 3 (1; W ) + C(A + B) 2 ε 2 t 1 τ -1+ δ k 2 E k 3 (τ ; W ) 1 2 dτ + δ k<2 C(A + B) 2 ε 2 δ k=0 t 1 τ -1+ δ 0 2 +β+ δ 1 2 E 1 3 (τ ; W ) 1 2 dτ + t 1 τ -5 4 dτ ,
for a new C > 0. Injecting (1.1.11d) in the above inequality and integrating in dτ , we obtain that

E k 3 (t; W ) ≤ CE k 3 (1; W ) + C(A + B) 2 Bε 3 1 δ k t δ k + δ k=0 1 δ 0 2 + β + δ 1 t δ 0 2 +β+δ 1 ,
and taking β sufficiently small so that β + δ 1 ≤ δ 0 /2, B > 1 sufficiently large so that

E k 3 (1; W ) ≤ B 2 ε 2 2CK 2 2
and B ≥ A, and ε 0 > 0 sufficiently small so that

ε 0 ≤ 1 8BCK 2 2 1 δ k + δ k=0 1 δ 0 2 + β + δ 1 -1
, we finally derive enhanced estimate (1.1.12d) and the conclusion of the proof.

Chapter 3

Uniform Estimates

Semilinear normal forms

In proposition 2.2.13 of the previous chapter we proved the propagation of the a-priori the energy estimates, i.e. that there exist some constants A, B > 1 large and ε 0 > 0 small, such that (1.1.11) implies (1.1.12c), (1.1.12d). To conclude the proof of theorem 1.1.2 it only remains to show that (1.1.11) also implies (1.1.12a), (1.1.12b). In particular, as u + = -u -and v + = -u -, it will be enough to prove this result for (u -, v -), which is solution to

(3.1.1) (D t + |D x |) u -= Q w 0 (v ± , D 1 v ± ), (D t + D x ) v -= Q kg 0 (v ± , D 1 u ± ), with Q w 0 (v ± , D 1 v ± ), Q kg 0 (v ± , D 1 u ± )
given by (2.1.1). As for the simpler case of the one-dimensional Klein-Gordon equation (see [START_REF] Stingo | Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data:46[END_REF]), the main idea is to reformulate system (3.1.1) in terms of two new functions u, v, defined from u -, v -and living in a new framework (the semi-classical framework ), and to deduce a new simpler system, made of a transport equation and an ODE. Through this new system we will be able to recover the required enhanced estimates (1.1.12a), (1.1.12b).

Before introducing the semi-classical framework in which we will work for the rest of the paper, we need to replace almost all quadratic non-linearities in (3.1.1) with cubic ones by a normal forms. This is the object of the following two subsections. We highlight the fact that we do not make use directly of the normal forms obtained in the proof of the energy inequality, because our goals and constraints are henceforth different. In fact, we want to obtain a L ∞ estimate for essentially ρ derivatives of our solution, having a control on its H s norm for s ρ. Therefore, we are allowed to lose some derivatives in the normal form reduction, which means that we do not care any more about the quasi-linear nature of our problem.

We warn the reader that, for seek of compactness, we will often use the notation NL w (resp. NL kg ) when referring to

Q w 0 (v ± , D 1 v ± ) (resp. to Q kg 0 (v ± , D 1 u ± )).

Normal forms for the Klein-Gordon equation

The aim of this subsection is to introduce a new unknown v N F , defined in terms of v -, in such a way it is solution to a cubic half Klein-Gordon equation instead of the quadratic one satisfied by v -in (3.1.1). This normal form is motivated by the fact that the L 2 norm of Q kg 0 (v ± , D 1 u ± ) decays too slowly in time (only t -1+δ/2 ), as follows from (B.1.4a) and a-priori estimates (1.1.11b), (1.1.11c), and this decay is not enough in view of proposition 3.2.7 (the required one being strictly faster than t -3/2 ).

It is fundamental to observe that, after (1.1.11) and inequality (3.1.7b) below with θ 1 small enough (e.g. θ < (2 + δ) -1 ), v N F and v -are comparable, in the sense that there is a positive constant C such that

(3.1.2) v -(t, •) H ρ,∞ -v N F (t, •) H ρ,∞ ≤ Cε 2 t -1 .
Then bootstrap assumption (1.1.11b) implies that the new unknown v N F disperses in time at the same rate t -1 as v -, and the propagation of a suitable estimate of the H ρ,∞ norm of v N F will provide us with enhanced (1.1.12b).

Proposition

v N F := v -- i 4(2π) 2 j 1 ,j 2 ∈{+,-} e ix•ξ B 1 (j 1 ,j 2 ,+) (ξ, η)v j 1 (ξ -η)û j 2 (η)dξdη,
with B 1 (j 1 ,j 2 ,+) (ξ, η) given by (2.2.42) with k = 1 and j 3 = +. Then for every t ∈

[1, T ] v N F is solution to (3.1.4) (D t + D x ) v N F (t, x) = r N F kg (t, x),
where

(3.1.5) r N F kg (t, x) = - i 4(2π) 2 j 1 ,j 2 ∈{+,-} e ix•ξ B 1 (j 1 ,j 2 ,+) (ξ, η) × NL kg (ξ -η)û j 2 (η) + vj 1 (ξ -η) NL w (η) dξdη satisfies r N F kg (t, •) L 2 1 µ=0 V (t, •) H 1,∞ R µ 1 U (t, •) L ∞ U (t, •) H 1 + V (t, •) 2 H 2,∞ V (t, •) H 2 , (3.1.6a) χ(t -σ D x )r N F kg (t, •) L ∞ V (t, •) H 1,∞ 1 µ=0 R µ 1 U (t, •) H 2,∞ 2 + t σ V (t, •) 3 H 2,∞ , (3.1.6b) for any χ ∈ C ∞ 0 (R 2 ), σ > 0. Moreover, for every s, ρ ≥ 0, any θ ∈]0, 1[, (3.1.7a) (v N F -v -)(t, •) H s 1 µ=0 V (t, •) H s R µ 1 U (t, •) L ∞ + V (t, •) L ∞ U (t, •) H s+1 , (v N F -v -)(t, •) H s,∞ 1 µ=0 V (t, •) 1-θ H s,∞ V (t, •) θ H s+2 R µ 1 U (t, •) L ∞ + 1 µ=0 V (t, •) L ∞ R µ 1 U (t, •) 1-θ H s+1,∞ U (t, •) θ H s+3 , (3.1.7b
)

Ω(v N F -v -)(t, •) L 2 1 µ,ν=0 [ Ω µ V (t, •) L 2 R ν 1 U (t, •) L ∞ + V (t, •) L ∞ Ω µ U (t, •) H 1 ] + ΩV (t, •) H 2 U (t, •) H 1 + V (t, •) L 2 ΩU (t, •) H 2 , (3.1.7c) and (3.1.8a) χ(t -σ D x )(v N F -v -)(t, •) L 2 t σ V (t, •) H 1,∞ U (t, •) L 2 , (3.1.8b) χ(t -σ D x )Ω(v N F -v -)(t, •) L 2 t σ   1 µ=0 ΩV (t, •) L 2 R µ 1 U (t, •) L ∞ + V (t, •) H 1,∞ Ω µ U (t, •) L 2   .
Proof. From definition (3.1.3) of v N F , system (2.1.2) with |I| = 0, and the fact that

(3.1.9) Q kg 0 (v ± , D 1 u ± ) = i 4(2π) 2 j 1 ,j 2 ∈{+,-} e ix•ξ 1 -j 1 j 2 ξ -η ξ -η • η |η| η 1 vj 1 (ξ -η)û j 2 (η)dξdη,
it immediately follows that v N F is solution to (3.1.4) with r N F kg given by (3.1.5). We observe that, after formula (A.15), we have the following explicit expressions:

v N F -v -= - i 8 (v + + v -)R 1 (u + -u -) - D 1 D x (v + -v -)(u + + u -) + D 1 [ D x -1 (v + -v -)](u + + u -) -D x [ D x -1 (v + -v -)]R 1 (u + -u -) (3.1.10) and
(3.1.11)

r N F kg = - i 4 NL kg R 1 (u + -u -) - D x D x (v + -v -) NL w + D 1 D x -1 (v + -v -) NL w . Inequalities (
(t -σ D x ), with χ ∈ C ∞ 0 (R 2 ) and σ > 0, is L 2 -H 1 continuous with norm O(t σ ).
As concerns r N F kg , from (3.1.11) and corollary A.4 we find that

r N F kg (t, •) L 2 1 µ=0 NL kg (t, •) L 2 R µ 1 U (t, •) L ∞ + V (t, •) L 2 NL w (t, •) L ∞ + V (t, •) L ∞ NL w (t, •) H 1 and χ(t -σ D x )r N F kg (t, •) L ∞ 1 µ=0 NL kg (t, •) L ∞ R µ 1 U (t, •) L ∞ + t σ V (t, •) H 1,∞ NL w (t, •) L ∞ .

Inequalities

Normal forms for the wave equation

We now focus on the wave equation satisfied by u -:

(D t + |D x |)u -(t, x) = Q w 0 (v ± , D 1 v ± ),
and perform a normal form argument in order to replace (a part of) the quadratic non-linearity in the above right hand side with a cubic non-local one. The fundamental reason for that is to be found in lemma 3.2.14, where we have to prove that the L 2 norm of some operator, acting on the non-linearity of the equation satisfied by u -, decays like t -1/2+β , for a small β > 0. That decay is obtained if the L 2 norm of the mentioned non-linearity is a O(t -3/2+β ), for some new small β > 0, which is not the case for Q w 0 (v ± , D 1 v ± ), as follows from (B.1.3a), (1.1.11b), (1.1.11c). This normal form can be actually performed only on contributions depending on (v + , v + ) and (v -, v -) but not on (v + , v -), which are resonant. Nevertheless, the structure of these latter contributions allows us to recover the right mentioned time decay for their L 2 norm (see lemmas 3.2.15 and 3.2.16).

Thanks to inequalities (3.1.20b), (3.1.20c) and a-priori estimates (1.1.11), the new unknown u N F we define in (3.1.15) below is equivalent to the former u -, meaning that there exists a positive constant C such that (3.1.12)

1 κ=0 R κ 1 u -(t, •) H ρ+1,∞ -R κ 1 u N F (t, •) H ρ,∞ ≤ Cε 2 t -1+ δ 2 .
After (1.1.11a) this means that u N F and R 1 u N F decay in the H ρ+1,∞ norm at the same rate t -1/2 as u -, R 1 u -, and the propagation of a suitable estimate of this norm will provide us with enhanced (1.1.12a).

Let us rewrite Q w 0 (v ± , D 1 v ± ) as follows

Q w 0 (v ± , D 1 v ± ) = - 1 2 v + D 1 v -+ D x D x v + • D x D 1 D x v - + i 4(2π) 2 j∈{+,-} e ix•ξ 1 - ξ -η ξ -η • η η η 1 vj (ξ -η)v j (η)dξdη, (3.1.13)
and introduce, for any j ∈ {+, -},

(3.1.14) D j (ξ, η) := 1 -ξ-η ξ-η • η η η 1 j ξ -η + j η + |ξ| . Proposition 3.1.2. Assume that (u, v) is solution to (1.1.1) in [1, T ] for a fixed T > 1, consider (u + , v + , u -, v -) defined in (1.1.
u N F := u -- i 4(2π) 2 j∈{+,-} e ix•ξ D j (ξ, η)v j (ξ -η)v j (η)dξdη, with multiplier D j defined in (3.1.14). For every t ∈ [1, T ] u N F is solution to (3.1.16) (D t + |D x |)u N F (t, x) = q w (t, x) + c w (t, x) + r N F w (t, x),
where quadratic term q w is given by

(3.1.17) q w (t, x) = 1 2 v N F D 1 v N F - D x D x v N F • D x D 1 D x v N F ,
while cubic terms c w , r N F w are equal, respectively, to

c w (t, x) = 1 2 (v --v N F ) D 1 v -+ v N F D 1 (v --v N F ) - D x D x (v --v N F ) • D x D 1 D x v -- D x D x v N F • D x D 1 D x (v --v N F ) , (3
r N F w (t, x) = - i 4(2π) 2 j∈{+,-} e ix•ξ D j (ξ, η) NL kg (ξ -η)v j (η) + vj (ξ -η) NL kg (η) dξdη.
For any s, ρ ≥ 0, any t ∈ [1, T ],

(3.1.20a) u N F (t, •) -u -(t, •) H s V (t, •) L ∞ V (t, •) H s+15 , (3.1.20b) u N F (t, •) -u -(t, •) H ρ+1,∞ V (t, •) L ∞ V (t, •) H ρ+18 , (3.1.20c) R j u N F (t, •) -R j u -(t, •) H ρ+1,∞ V (t, •) L ∞ V (t, •) H ρ+8 , j = 1, 2.
Moreover, for any cut-off function χ ∈ C ∞ 0 (R 2 ) and σ > 0 there exists some χ 1 ∈ C ∞ 0 (R 2 ) and s > 0 such that

χ(t -σ D x )c w (t, •) L 2 t σ χ 1 (t -σ D x )(v N F -v -)(t, •) L 2 V (t, •) H 2,∞ + v N F (t, •) H 1,∞ + t -N (s) (v N F -v -)(t, •) H 1 V (t, •) H s + v N F (t, •) H s , (3.1.21a) χ(t -σ D x )c w (t, •) L ∞ t σ χ 1 (t -σ D x ) v N F -v -(t, •) L ∞ V (t, •) H 2,∞ + v N F (t, •) H 1,∞ + t -N (s) (v N F -v -)(t, •) H 1 V (t, •) H s + v N F (t, •) H s (3.1.21b) χ(t -σ D x )Ωc w (t, •) L 2 t σ χ 1 (t -σ D x )Ω(v N F -v -)(t, •) L 2 V (t, •) H 2,∞ + v N F (t, •) H 1,∞ + t -N (s) Ω(v N F -v -)(t, •) L 2 V (t, •) H s + v N F (t, •) H s + t σ (v N F -v -)(t, •) H 1,∞ × 1 µ=0 Ω µ V (t, •) H 1 + Ω µ v N F (t, •) L 2 (3.1.21c)
with N (s) > 0 as large as we want as long as s > 0 is large, and

(3.1.22a) χ(t -σ D x )r N F w (t, •) L 2 V (t, •) 2 H 13,∞ U (t, •) H 1 , (3.1.22b) χ(t -σ D x )r N F w (t, •) L ∞ V (t, •) 2 H 13,∞ ( U (t, •) H 2,∞ + R 1 U (t, •) H 2,∞
) , and for any θ ∈]0, 1[,

χ(t -σ D x )Ωr N F w (t, •) L 2 t β V (t, •) 1-θ H 15,∞ V (t, •) θ H 17 ( U (t, •) H 1,∞ + R 1 U (t, •) H 1,∞ ) + V (t, •) L ∞ U (t, •) 1-θ H 16,∞ + R 1 U (t, •) 1-θ H 16,∞ U (t, •) θ H 18 ΩV (t, •) L 2 + t β V (t, •) H 1,∞ ( U (t, •) H 1 + ΩU (t, •) H 1 ) + ( U (t, •) H 2,∞ + R 1 U (t, •) H 2,∞ ) ( V (t, •) L 2 + ΩV (t, •) L 2 ) V (t, •) H 17,∞ . (3
(D t + |D x |)u N F (t, x) = - 1 2 v + D 1 v -+ D x D x v + • D x D 1 D x v -+ r N F w (t, x),
with r N F w given by (3.1.19). Reminding that v + = -v -and replacing each occurrence of v -in the quadratic contribution to the above right hand side, we find that u N F is solution to (3.1.16).

The first part of lemma A.8 and the fact that any H ρ+1,∞ injects into H ρ+3 by Sobolev inequality immediately imply estimates (3.1.20) and

χ(t -σ D x )r N F w (t, •) L 2 NL kg (t, •) L 2 V (t, •) H 13,∞ , χ(t -σ D x )r N F w (t, •) L ∞ NL kg (t, •) L ∞ V (t, •) H 13,∞ ,
for any s, ρ ≥ 0. Moreover, from (A.37a) we derive that

χ(t -σ D x )Ωr N F w (t, •) L 2 t β ( NL kg (t, •) L 2 + ΩNL kg (t, •) L 2 ) V (t,
L = L 2 (resp. L = L ∞ ), w = v --v N F , and the fact that χ 1 (t -σ D x ) is continuous from L 2 to H 1 (resp. from L ∞ to H 1,∞ ) with norm O(t σ ). Inequality (3.1.21c
) is deduced applying Ω to (3.1.18) and using the Leibniz rule. The L 2 norm of products in which Ω is acting on v

--v N F is estimated by means of lemma B.2.2 with L = L 2 , w = v --v N F
, whereas the L 2 norm of the remaining products is simply estimated by taking the L ∞ norm on v --v N F times the L 2 norm of the remaining factor.

From PDEs to ODEs

In the previous section we showed that, if (u -, v -) is solution to system (3.1.1) in some interval [1, T ], for a fixed T > 1, one can define two new functions, v N F as in (3.1.3) and u N F as in (3.1.15), respectively comparable to v -and u -in the sense of (3.1.2) and (3.1.12), such that (u N F , v N F ) is solution to a new wave-Klein-Gordon system:

(3.2.1)

(D t + |D x |) u N F (t, x) = q w (t, x) + c w (t, x) + r N F w (t, x), (D t + D x ) v N F (t, x) = r N F kg (t, x), for every (t, x) ∈ [1, T ] × R 2
, where quadratic inhomogeneous term q w is given by (3.1.17 ). As anticipated before, our aim is to deduce from (3.2.1) a system made of a transport equation and an ODE, from which it will be possible to deduce suitable estimates on (u N F , v N F ) (and consequently on (u -, v -)). Thanks to (3.1.2) and (3.1.12) these estimates will allow us to close the bootstrap argument and prove theorem 1.1.2.

In subsection 3.2.1 we focus on the deduction of the mentioned ODE starting from the Klein-Gordon equation satisfied by v N F , while in subsection 3.2.2 we show how to derive a transport equation from the wave equation satisfied by u N F . The framework in which this plan takes place is the semi-classical framework, introduced below.

Let us introduce the semi-classical parameter

u(t, •) H ρ+1,∞ h + Op w h (ξ|ξ| -1 ) u(t, •) H ρ+1,∞ h ≤ Cεh -1 2 , (3.2.3a) v(t, •) H ρ,∞ h ≤ Cε, (3.2.3b)
for some positive constant C. A suitable propagation of the above estimates will therefore provide us with (1.1.12a) and (1.1.12b).

A straight computation shows that ( u, v) satisfies the following coupled system of semi-classical pseudo-differential equations:

(3.2.4) D t -Op w h (x • ξ -|ξ|) u(t, x) = h -1 q w (t, tx) + c w (t, tx) + r N F w (t, tx) D t -Op w h (x • ξ -ξ ) v(t, x) = h -1 r N F kg (t, tx),
where Op w h denotes the semi-classical Weyl quantization introduced in 1.2.17 (i). Moreover, if M j (resp. L j ), j = 1, 2, is the operator introduced in (1.2.49) (resp. (1.2.68)), M j u (resp. L j v) can be expressed in term of Z j u N F (resp. Z j v N F ). We have the following general result: Lemma 3.2.1. (i) Let w(t, x) be a solution to the inhomogeneous half wave equation

(3.2.5) [D t + |D x |] w(t, x) = f (t,
D t -Op w h (x • ξ -|ξ|) w(t, x) = h -1 f (t, tx),
so, for any i = 1, 2,

Z j w(t, y) = ih -1 x j D t + Op w h (ξ j -x j x • ξ) + 3h 2i x j 1 t w(t, x) x= y t = i x j D t + Op w h (ξ j -x j x • ξ) + h 2i x j w(t, x) x= y t = i x j Op w h (x • ξ -|ξ|) w(t, x) + Op w h (ξ j -x j x • ξ) w(t, x) + h 2i x j u(t, x) + h -1 x j f (t, tx) x= y t = ih -M j w(t, x) + 1 2i Op w h ξ j |ξ| w(t, x) | x= y t + iy j f (t, y).
We should specify that last equality is obtained by a trivial version of symbolic calculus (1.2.18), that applies also to symbols b(ξ) singular at ξ = 0. Indeed, if symbol a = a(x, ξ) is linear in x, and b(ξ) is lipschitz, the development a b is actually finite:

a b(x, ξ) = a(x, ξ)b(ξ) - h 2i ∂ x a(x, ξ) • ∂ ξ b(ξ).
(ii) The result is analogous to the previous one, after observing that w satisfies

D t -Op w h (x • ξ -ξ ) w(t, x) = h -1 f (t, tx).
As a straight consequence of lemma 3.2.1 and system (3.2.4) we have that (3.2.9a)

Z j u N F (t, y) = ih -M j u(t, x) + 1 2i Op w h ξ j |ξ| u(t, x) | x= y t + iy j q w + c w + r N F w (t, y), (3.2.9b) Z j v N F (t, y) = ih -Op w h ( ξ )L j v(t, x) + 1 i Op w h ξ j ξ v(t, x) x= y t + iy j r N F kg (t, y).
In view of lemma 3.2.14, it is also useful to write down the analogous relation between

(Z m Z n u) - and M[t(Z n u) -(t, tx)]. As (Z n u) -is solution to D t + |D x | (Z n u) -= Z n NL w (t,
(Z m Z n u) -(t, y) = ih -M m u J (t, x) + 1 2i Op w h ξ m |ξ| u J (t, x) x= y t + iy m Z n NL w (t, y) - D m |D y | (Z n u) -(t, y),
where J is the index such that Γ J = Z n and u J (t, x) := t(Z n u) -(t, tx). Also, observe that from (1.1.15), (1.1.16), (1.1.5) and (1.1.10)

Z n NL w = Q w 0 (Z n v) ± , D 1 v ± + Q w 0 v ± , D 1 (Z n v) ± -δ 1 n Q w 0 (v ± , D t v ± ) with δ 1 n = 1 for n = 1
, and that from inequality (B.1.6a) with s = 0,

(3.2.11) Z n NL w (t, •) L 2 Z n V (t, •) H 1 V (t, •) H 2,∞ + V (t, •) H 1 + V (t, •) L 2 ( U (t, •) H 1,∞ + R 1 U (t, •) H 1,∞ ) + V (t, •) L ∞ U (t, •) H 1 V (t, •) H 1,∞ .
Moreover, from the definition of M j and L j we see that

hM j w(t, x) = y j |D y | -tD j + 1 2i D j |D y | w(t, y)| y=tx , hOp w h ( ξ )L j w(t, x) = y j D y -tD j -i D j D y w(t,

Derivation of the ODE and propagation of the uniform estimate on the Klein-Gordon component

Let us firstly deal with the semi-classical Klein-Gordon equation satisfied by v:

(3.2.13) D t -Op w h (x • ξ -p(ξ)) v(t, x) = h -1 r N F kg (t, tx)
, where p(ξ) = ξ and r N F kg is given by (3.1.5) and satisfies (3.1.6). We remind that p (ξ) denotes the gradient of p(ξ) while p (ξ) is its 2×2 Hessian matrix, and that L j is the operator introduced in (1.2.68) for j = 1, 2. We also remind definition (1.2.66) of manifold Λ kg , represented in dimension 1 by picture 3.1 below, and decompose v into the sum of two contributions: one localized in a neighbourhood of Λ kg of size √ h (in the spirit of [START_REF] Ifrim | Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension[END_REF]), the other localized out of this neighbourhood. The contribution localized away from Λ kg appears to be a O(h 1/2-0 ) if we ). Thus the main contribution to v is the one localized around Λ kg . We are going to show that this latter one is solution to an ODE (see proposition 3.2.6) and that its H ρ,∞ h norm is uniformly bounded in time, which will finally enable us to propagate (3.2.3b) and obtain (1.1.11b) (see proposition 3.2.7).

For any fixed ρ ∈ Z let Σ(ξ) := ξ ρ , and for some γ, χ ∈ C ∞ 0 (R 2 ) equal to 1 close to the origin, σ > 0 small (e.g. σ < 1 4 ) let (3.2.14)

Γ kg := Op w h γ x -p (ξ) √ h χ(h σ ξ) .
We also introduce the following notations:

(3.2.15) v Σ := Op w h (Σ(ξ)) v, together with (3.2.16a) v Σ Λ kg := Γ kg v Σ , (3.2.16b) v Σ Λ c kg := Op w h 1 -γ x -p (ξ) √ h χ(h σ ξ) v Σ , so that v Σ = v Σ Λ kg + v Σ Λ c kg
, and remind that

L γ w = L γ 1 1 L γ 2 2 w , for any γ = (γ 1 , γ 2 ) ∈ N 2 . Lemma 3.2.2. Let γ ∈ C ∞ (R 2
) vanish in a neighbourhood of the origin and be such that

|∂ α z γ(z)| z -|α| . Let c(x, ξ) ∈ S δ,σ (1
) with δ ∈ [0, 1 2 ], σ > 0, be supported for |ξ| h -σ . For any χ ∈ C ∞ 0 (R 2 ) such that χ(h σ ξ) ≡ 1 on the support of c(x, ξ),

Op w h γ x -p (ξ) √ h c(x, ξ) w L 2 1 |µ|=0 h 1 2 -β Op w h (χ(h σ ξ))L µ w L 2 , (3.2.17a) Op w h γ x -p (ξ) √ h c(x, ξ) w L ∞ 1 |µ|=0 h -β Op w h (χ(h σ ξ))L µ w L 2 , (3.2.17b) and Op w h γ x -p (ξ) √ h c(x, ξ) w L 2 2 |µ|=0 h 1-β Op w h (χ(h σ ξ))L µ w L 2 , (3.2.18a) Op w h γ x -p (ξ) √ h c(x, ξ) w L ∞ 2 |µ|=0 h 1 2 -β Op w h (χ(h σ ξ))L µ w L 2 , (3.2.18b)
for a small β > 0, β → 0 as σ → 0.

Proof. The proof of (3.2.17) (resp. of (3.2.18)) follows straightly by inequalities (1.2.70) (resp. (1.2.71)), after observing that, as γ vanishes in a neighbourhood of the origin,

γ x -p (ξ) √ h c(x, ξ) = 2 j=1 γ j 1 x -p (ξ) √ h x j -p j (ξ) √ h c(x, ξ), where γ j 1 (z) := γ(z)z j |z| -2 is such that |∂ α z γ j 1 (z)| z -1-|α| (resp. γ x -p (ξ) √ h c(x, ξ) = 2 j=1 γ 2 x -p (ξ) √ h x -p (ξ) √ h 2 c(x, ξ), where γ 2 (z) := γ(z)|z| -2 is such that |∂ α z γ(z)| z -2-|α| ).
Corollary 3.2.3. There exists s > 0 sufficiently large such that

(3.2.19a) v Σ Λ c kg (t, •) L 2 h 1-β   v(t, •) H s h + 1≤|µ|≤2 Op w h (χ(h σ ξ))L µ v(t, •) L 2   , (3.2.19b) v Σ Λ c kg (t, •) L ∞ h 1 2 -β   v(t, •) H s h + 1≤|µ|≤2 Op w h (χ(h σ ξ))L µ v(t, •) L 2   .
for a small β > 0, β → 0 as σ → 0.

Proof. Since symbol 1 -γ x-p (ξ) √ h χ(h σ ξ) is supported for | x-p (ξ) √ h | ≥ d 1 > 0 or |h σ ξ| ≥ d 2 > 0,
for some small d 1 , d 2 > 0, we may consider a smooth cut-off function χ equal to 1 close to the origin and such that χχ ≡ χ, so that 1 -γ x-p (ξ) √ h χ(h σ ξ) writes as

1 -γ x -p (ξ) √ h χ(h σ ξ) + 1 -γ x -p (ξ) √ h χ(h σ ξ) (1 -χ)(h σ ξ),
the first symbol being supported in {(x, ξ) :

| x-p (ξ) √ h | ≥ d 1 , |ξ| h -σ }, the second one for large frequencies |ξ| h -σ .
Using lemma 1.2.24 and the fact that γ x-p (ξ)

√ h χ(h σ ξ) ∈ S 1 2 ,σ x-p (ξ) √ h -M , for any M ∈ N,
we have that, for a fixed N ∈ N * ,

1 -γ x -p (ξ) √ h χ(h σ ξ) 1 -χ(h σ ξ) = 1 -χ(h σ ξ) 1 -γ x -p (ξ) √ h χ(h σ ξ) + 1≤j<N χ j (h σ ξ) a j (x, ξ) + r N (x, ξ),
where function χ j (h σ ξ) is still supported for large frequencies |ξ| h -σ , for every 1 ≤ j < N , up to negligible multiplicative constants,

a j (x, ξ) = h j( 1 2 +σ) |α|=j (∂ α γ) x -p (ξ) √ h χ(h σ ξ) ∈ h j( 1 2 +σ) S 1 2 ,σ x -p (ξ) √ h -M , and r N ∈ h N ( 1 2 +σ) S 1 2 ,σ x-p (ξ) √ h -M .
Op w h 1 -γ x -p (ξ) √ h χ(h σ ξ) (1 -χ)(h σ ξ) v Σ (t, •) L 2 h N (s) v(t, •) H s h , Op w h 1 -γ x -p (ξ) √ h χ(h σ ξ) (1 -χ)(h σ ξ) v Σ (t, •) L ∞ h N (s) v(t, •) H s h ,
where N (s), N (s) ≥ 1 if s > 2 is sufficiently large.

On the other hand, as function

(1 -γ) x-p (ξ) √ h
vanishes in a neighbourhood of the origin and is such that |∂ α z (1 -γ)(z)| z -|α| , by inequalities (3.2.18) and the fact that, using symbolic calculus to commute L with Σ(ξ),

(3.2.20) Op w h (χ(h σ ξ))L µ v Σ (t, •) L 2 h -ν |µ 1 |≤|µ| Op w h (χ(h σ ξ))L µ 1 v(t, •) L 2
with ν = ρσ if ρ ≥ 0, 0 otherwise, we have that

Op w h (1 -γ) x -p (ξ) √ h χ(h σ ξ) v Σ (t, •) L 2 |µ|≤2 h 1-β Op w h (χ(h σ ξ))L µ v(t, •) L 2 , Op w h (1 -γ) x -p (ξ) √ h χ(h σ ξ) v Σ (t, •) L ∞ |µ|≤2 h 1 2 -β Op w h (χ(h σ ξ))L µ v(t, •) L 2 ,
for a small β > 0, β → 0 as σ → 0.

In the following lemma we show how to develop the symbol a(x, ξ) associated to an operator acting on Γ kg w, for some suitable function w, at ξ = -dφ(x), where φ(x) = 1 -|x| 2 .

Lemma 3.2.4. Let a(x, ξ) be a real symbol in S δ,0 ( ξ q ), q ∈ R, for some δ > 0 small, Σ(ξ) = ξ ρ for some fixed ρ ∈ Z, Γ kg the operator introduced in (3.2.14) and w = w(t, x) such that L µ w(t, •) ∈ L 2 (R 2 ) for any |µ| ≤ 2. Let us also introduce w Σ Λ kg := Γ kg Op w h (Σ)w. There exists a family (θ h (x)) h of C ∞ 0 functions real valued, equal to 1 on the closed ball B 1-ch 2σ (0) and supported in B 1-c 1 h 2σ (0), for some small 0 < c 1 < c, σ > 0, with

∂ α x θ h L ∞ = O(h -2|α|σ
) and (h∂ h ) k θ h bounded for every k, such that

(3.2.21) Op w h (a)w Σ Λ kg = θ h (x)a(x, -dφ(x))w Σ Λ kg + R 1 (w) ,
where R 1 (w) satisfies

(3.2.22a) R 1 (w)(t, •) L 2 h 1-β   w(t, •) H s h + |γ|=1 Op w h (χ(h σ ξ))L γ w(t, •) L 2   , (3.2.22b) R 1 (w)(t, •) L ∞ h 1 2 -β   w(t, •) H s h + |γ|=1 Op w h (χ(h σ ξ))L γ w(t, •) L 2   , with β = β(σ, δ) > 0, β → 0 as σ, δ → 0. Moreover, if ∂ ξ a(x, ξ) vanishes at ξ = -dφ(x)
, the above estimates can be improved and R 1 (w) is rather a remainder R 2 (w) such that

(3.2.23a) R 2 (w)(t, •) L 2 h 2-β   w(t, •) H s h + 1≤|γ|≤2 Op w h (χ(h σ ξ))L γ w(t, •) L 2   , (3.2.23b) R 2 (w)(t, •) L ∞ h 3 2 -β   w(t, •) H s h + 1≤|γ|≤2 Op w h (χ(h σ ξ))L γ w(t, •) L 2   .
Proof. After lemma 1.2.38 we know that there exists a family of functions θ h (x) as in the statement such that equality (1.2.67) holds. We highlight the fact that any derivative of θ h vanishes on the support of γ x-p (ξ) √ h χ(h σ ξ) and its derivatives. After remark 1.2.22, this implies that

w Σ Λ kg = θ h (x)w Σ Λ kg + r ∞ , r ∞ ∈ h N S 1 2 ,σ ( x -∞ ) and hence that Op w h (a)w Σ Λ kg = Op w h (a)θ h (x)w Σ Λ kg + Op w h (r a ∞ )w Σ Λ kg , with r a ∞ = a r ∞ ∈ h N -γ S 1 2 ,σ ( x -∞
) and γ = qσ if q ≥ 0, 0 otherwise. From proposition 1.2.36 and the semi-classical Sobolev injection it follows at once that Op w h (r a ∞ )w Σ Λ kg satisfies enhanced estimates (3.2.23) if N is taken sufficiently large. Up to negligible multiplicative constants, a further application of symbolic calculus gives also that

Op w h (a(x, ξ))θ h (x)w Σ Λ kg = Op w h (a(x, ξ)θ h (x))w Σ Λ kg + N -1 |α|=1 h |α| Op w h ∂ α ξ a(x, ξ)∂ α x θ h (x) w Σ Λ kg + Op w h (r N (x, ξ))w Σ Λ kg ,
where r N ∈ h N -β S δ ,0 ( ξ q-N x -∞ ) for a new small β = β(δ, σ) and δ = max{δ, σ}. From the same argument as above Op w h (r N )w Σ Λ kg verifies enhanced estimates (3.2.23) if N is suitably chosen. Also, since the support of ∂ α ξ a(x, ξ) • ∂ α x θ h (x) has empty intersection with that of γ x-p (ξ) √ h χ(h σ ξ) for any |α| ≥ 1, all the |α|-order terms in the above equality are remainders R 1 (w). Now, as symbol a(x, ξ)θ h (x) is supported for |x| ≤ 1 -c 1 h 2σ < 1, we are allowed to develop it at ξ = -dφ(x):

a(x, ξ)θ h (x) = a(x, -dφ(x))θ h (x) + |α|=1 1 0 (∂ α ξ a)(x, tξ + (1 -t)dφ(x))dt θ h (x)(ξ + dφ(x)) α = a(x, -dφ(x))θ h (x) + 2 j=1 b j (x, ξ)(x j -p j (ξ)), (3.2.24) with (3.2.25) b j (x, ξ) = |α|=1 1 0 (∂ α ξ a)(x, tξ + (1 -t)dφ(x))dt θ h (x) (ξ + dφ(x)) α (x j -p j (ξ)) |x -p (ξ)| 2 , j = 1, 2. If χ 1 ∈ C ∞ 0 (R 2 )
is a new cut-off function equal to 1 close to the origin, we can reduce ourselves to the analysis of symbol b j (x, ξ)(x j -p j (ξ))χ 1 (h σ ξ). In fact, as b j (x, ξ)(x j -p j (ξ))(1χ 1 )(h σ ξ) is supported for large frequencies, one can prove that its operator acting on w

Σ Λ kg is a O L 2 ∩L ∞ (h N w(t, •) H s h )
with N > 0 large as long as s > 0 is large, by using the semi-classical Sobolev injection, symbolic calculus of proposition 1.2.21, lemma 1.2.40 and proposition 1.2.36. Furthermore, if we consider a smooth cut-off function γ ∈ C ∞ 0 (R 2 ), equal to 1 close to the origin and such that γ ξ 2 (x -p (ξ)) ≡ 1 on the support of γ x-p (ξ) [START_REF] Alazard | Sobolev estimates for two dimensional gravity water waves[END_REF], for some new small β, δ > 0, and its support has empty intersection with that of γ x-p (ξ) √ h

√ h χ(h σ ξ) (which is possible if σ < 1/4), we have that b j (x, ξ)(x j -p j (ξ))χ 1 (h σ ξ) = b j (x, ξ)(x j -p j (ξ))χ 1 (h σ ξ) γ ξ 2 (x -p (ξ)) + b j (x, ξ)(x j -p j (ξ))χ 1 (h σ ξ)(1 -γ) ξ 2 (x -p (ξ)) . Since b j (x, ξ)(x j -p j (ξ))χ 1 (h σ ξ)(1-γ) ξ 2 (x-p (ξ)) ∈ h -β S δ,σ
(which instead belongs to class

S 1 2 ,0 ( x-p (ξ) √ h -M
), for M ∈ N as large as we want), its quantization acting on w Σ Λ kg is also an enhanced remainder R 2 (w).

The very contribution that only enjoys estimates (3.2.22) is Op

w h c(x, ξ)(x j -p j (ξ)) w Σ Λ kg , with c(x, ξ) := b j (x, ξ)χ 1 (h σ ξ) γ ξ 2 (x -p (ξ)) ∈ h -β S 2σ,σ (1
) and β depending linearly on σ. In fact, if we assume that the support of χ 1 is sufficiently small so that χ 1 χ ≡ χ 1 and all derivatives of χ vanish on that support, by using symbolic development (1.2.18) until a sufficiently large order N and observing that

c(x, ξ)(x j -p j (ξ)), γ x -p (ξ) √ h = c(x, ξ), γ x -p (ξ) √ h (x j -p j (ξ)) = (∂ ξ c) • (∂γ) x -p (ξ) √ h + (∂ x c) • (∂γ) x -p (ξ) √ h p (ξ) x j -p j (ξ) √ h
does not lose any power h -1/2 , we derive that, up to negligible constants,

c(x, ξ)(x j -p j (ξ)) γ x -p (ξ) √ h χ(h σ ξ) = γ x -p (ξ) √ h χ(h σ ξ)c(x, ξ)(x j -p j (ξ)) + h γ x -p (ξ) √ h c(x, ξ) + r N (x, ξ).
In the above equality is a concise notation to indicate a linear combination, γ ∈ C ∞ 0 (R 2 \{0}), c ∈ h -β S δ,σ (1) for some new small β, δ > 0, and

r N ∈ h (N +1)/2-β S 1 2 ,σ x-p (ξ) √ h -(M -1) as c(x, ξ)(x j -p j (ξ)) ∈ h 1/2-β S 2σ,σ x-p (ξ) √ h
. From inequalities (1.2.70) and (3.2.20) we deduce that Op w h γ x-p (ξ)

√ h χ(h σ ξ)c(x, ξ)(x j -p j (ξ)) Op w h (Σ)w is a remainder R 1 (w) satisfying (3.2.22
). The quantization of all the addends in acting on Op w h (Σ)w is estimated by using that γ(z) vanishes in a neighbourhood of the origin and can be rewritten as j=1,2 γ 2 (z)z 2 j , with γ 2 (z) := γ(z)|z| -2 such that |∂ α z γ 2 (z)| z -2-|α| . Inequalities (1.2.71) and the successive commutation of L γ with Σ, for |γ| = 1, 2, give then that hOp w h γ x-p (ξ)

√ h c(x, ξ) Op w h (Σ)w is a remainder R 2 (w). Finally, as r N (x, ξ) Σ(ξ) ∈ h N 2 -β-µ S 1 2 ,σ ( x -p (ξ) √ h -(M -1) )
with µ = σρ if ρ ≥ 0, 0 otherwise, Op w h (r N )Op w h (Σ)w is also a remainder R 2 (w) just from 1.2.36, 1.2.37, fixing N ∈ N sufficiently large (e.g. N = 3).

If symbol a(x, ξ) is such that ∂ ξ a| ξ=-dφ = 0, instead of equality (3.2.24) with b j given by (3.2.25), we have

a(x, ξ)θ h (x) = a(x, -dφ(x))θ h (x) + j=1,2 b(x, ξ)(x j -p j (ξ)) 2 , with b(x, ξ) = |α|=2 2 α! 1 0 (∂ α ξ a)(tξ -(1 -t)dφ(x))(1 -t)dt θ h (x) (ξ + dφ(x)) α |x -p (ξ)| 2 .
The same argument as before can be applied to Op w h b(x, ξ)θ h (x)(x j -p j (ξ)) 2 w Σ Λ kg to show that it reduces to

Op w h b(x, ξ)θ h (x)(x j -p j (ξ)) 2 χ 1 (h σ ξ) γ ξ 2 (x -p (ξ)) w Σ Λ kg + R 2 (w), with R 2 (w) satisfying (3.2.23). If B(x, ξ) := b(x, ξ)θ h (x)χ 1 (h σ ξ) γ ξ 2 (x -p (ξ)) then B(x, ξ)(x j -p j (ξ)) 2 ∈ h -β S δ ,σ (1
) by lemma 1.2.43, for some new small β, δ > 0 depending on σ, δ. Using lemma 1.2.24, symbolic development (1.2.18) until order 4, and assuming that the support of χ 1 is sufficiently small so that χχ 1 ≡ χ, we derive that

B(x, ξ)(x j -p j (ξ)) 2 γ x -p (ξ) √ h χ(h σ ξ) = B(x, ξ)γ x -p (ξ) √ h (x j -p j (ξ)) 2 + h i 2 i=1 (∂ i γ) x -p (ξ) √ h x j -p j (ξ) √ h   (∂ ξ i B) + j (∂ x j B)p ij (ξ)   (x j -p j (ξ)) + 2≤|α|≤3 h |α| 2 -2δ -β γ α x -p (ξ) √ h B α (x, ξ) + r 4 (x, ξ), where γ α ∈ C ∞ 0 (R 2 \ {0}), B α (x, ξ) ∈ S δ ,σ (1)
, and

r 4 (x, ξ) ∈ h 2-4δ -β S 1 2 ,σ x-p (ξ) √ h -M . As r 4 (x, ξ) Σ(ξ) ∈ h 2-β S 1 2 ,σ x-p (ξ) √ h -M , for β = 2 -4δ -β -ρσ if ρ ≥ 0, β = 2 -4δ - β otherwise,
(z) = ∂ i γ(z)z j and c = h δ [(∂ ξ i B) + (∂ x B) • (∂ ξ p 1 + ∂ ξ p 2 )] ∈ S δ ,σ (1)
, for i, j = 1, 2), and (3.2.20), we deduce that the quantization of the first (resp. the second) contribution in above symbolic development is a remainder R 2 (w), when acting on Op w h (Σ)w. Finally, as γ α vanishes in a neighbourhood of the origin, we write

γ α x -p (ξ) √ h = 2 k=1 h -1 γ α x -p (ξ) √ h x -p (ξ) √ h -2 γα x-p (ξ) √ h ×(x k -p k (ξ)) 2 , |α| = 2, γ α x -p (ξ) √ h = 2 k=1 h -1 2 γ α x -p (ξ) √ h x k -p k (ξ) √ h x -p (ξ) √ h -2 γ k α x-p (ξ) √ h ×(x k -p k (ξ)), |α| = 3
and obtain that the quantization of α-th order term with |α| = 2 (resp. |α| = 3) is a remainder R 2 (w) when acting on Op w h (Σ)w, after inequalities (1.2.71) (resp. (1.2.70)) with γ n = γ α (resp.

γ n = γ k α , k = 1, 2) and c = B α .
The following two results allow us to finally derive the ODE satisfied by v Σ Λ kg .

Lemma 3.2.5. We have that

(3.2.26) D t -Op w h (x • ξ -p(ξ)), Γ kg = Op w h (b), where b(x, ξ) = - h 2i (∂γ) x -p (ξ) √ h • x -p (ξ) √ h χ(h σ ξ) - σh i γ x -p (ξ) √ h (∂χ)(h σ ξ) • (h σ ξ) + i 24 h 3 2 |α|=3 (∂ α γ) x -p (ξ) √ h (∂ α ξ p (ξ))χ(h σ ξ) + r(x, ξ) (3.2.27) and r ∈ h 5/2 S 1 2 ,σ ( x-p (ξ) √ h -N ) for any N ≥ 0. Therefore, function v Σ Λ kg is solution to (3.2.28) D t -Op w h (x • ξ -p(ξ)) v Σ Λ kg = Γ kg Op w h (Σ(ξ)) h -1 r N F kg (t, tx) + R 2 ( v) with R 2 ( v) satisfying estimates (3.2.

23).

Proof. Recalling the definition (3.2.14) of Γ kg , one can prove by a straight computation that

D t , Γ kg = h i Op w h (∂γ) x -p (ξ) √ h • p (ξ)ξ √ h χ(h σ ξ) + h 2i Op w h (∂γ) x -p (ξ) √ h • x -p (ξ) √ h χ(h σ ξ) - (1 + σ)h i Op w h γ x -p (ξ) √ h (∂χ)(h σ ξ) • (h σ ξ) .
Since the development of a commutator's symbol only contains odd-order terms, lemma 1.2.24 gives that the symbol associated to Γ kg , Op w h (x • ξ -p(ξ)) writes as

h i γ x -p (ξ) √ h χ(h σ ξ), x • ξ -p(ξ) + i 24 h 3 2 |α|=3 (∂ α γ) x -p (ξ) √ h χ(h σ ξ)(∂ α ξ p(ξ)) + r 5 (x, ξ) with r 5 ∈ h 5/2 S 1 2 ,σ ( x-p (ξ) √ h -N
) for any N ≥ 0. Developing the above Poisson bracket one finds that

Γ kg , Op w h (x • ξ -p(ξ)) = - h i Op w h (∂γ) x -p (ξ) √ h • p (ξ)ξ √ h χ(h σ ξ) - h i Op w h (∂γ) x -p (ξ) √ h • x -p (ξ) √ h χ(h σ ξ) + h i Op w h γ x -p (ξ) √ h (∂χ)(h σ ξ) • (h σ ξ) + i 24 h 3 2 |α|=3 Op w h (∂ α γ) x -p (ξ) √ h (∂ α ξ p (ξ))χ(h σ ξ) + Op w h (r 5 (x, ξ)),
which summed to the previous commutator gives (3.2.27).

The last part of the statement follows applying to equation (3.2.13) operators Op w h (Σ(ξ)) (which commutes exactly with the linear part of the equation, evident in non semi-classical coordinates) and Γ kg . Since

hOp w h (∂γ) x -p (ξ) √ h • x -p (ξ) √ h χ(h σ ξ) v Σ = 2 k=1 Op w h γ k x -p (ξ) √ h • (x -p (ξ))(x k -p k (ξ)) v Σ with γ k (z) := (∂γ)(z)z k |z| -2
, and

h 3 2 Op w h (∂ α γ) x -p (ξ) √ h (∂ α ξ p (ξ)) = hOp w h γ k α ξ -p (ξ) √ h (∂ α ξ p (ξ))(x k -p k (ξ)) v Σ with γ k α (z) := (∂ α γ)(z)z k |z| -2 ,
(∂γ) x-p (ξ) √ h • x-p (ξ) √ h χ(h σ ξ) v Σ (resp. h 3/2 Op w h (∂ α γ) x-p (ξ) √ h (∂ α ξ p (ξ)) , |α| = 3) is a remainder R 2 ( v).
The same holds true for Op w h γ x-p (ξ) √ h

(∂χ)(h σ ξ) • (h σ ξ) v Σ , as follows combining symbolic calculus and lemma 1.2.40, because its symbol is supported for large frequencies |ξ| h -σ . From propositions 1.2.36 and 1.2.37 it immediately follows that Op w h (r 5 ) v Σ satisfies (3.2.23a) and (3.2.23b).

Proposition 3.2.6 (Deduction of the ODE).

There exists a family (θ h (x)) h of C ∞ 0 functions, real valued, equal to 1 on the closed ball B 1-ch 2σ (0) and supported in B 1-c 1 h 2σ (0), for some small

0 < c 1 < c, σ > 0, with ∂ α x θ h L ∞ = O(h -2|α|σ
) and (h∂ h ) k θ h bounded for every k, such that

(3.2.29) Op w h (x • ξ -p(ξ)) v Σ Λ kg = -φ(x)θ h (x) v Σ Λ kg + R 2 ( v),
where φ(x) = 1 -|x| 2 and R 2 ( v) satisfies estimates (3.2.23). Therefore, v Σ Λ kg is solution of the following non-homogeneous ODE:

(3.2.30) D t v Σ Λ kg = -φ(x)θ h (x) v Σ Λ kg + Γ kg Op w h (Σ(ξ)) h -1 r N F kg (t, tx) + R 2 ( v),
with r N F kg given by (3.1.5).

Proof. The proof of the statement follows directly from lemma 3.2.4 if we observe that ∂ ξ (x • ξp(ξ)) = 0 at ξ = -dφ(x) and x • (-dφ(x)) -p(-dφ(x)) = -φ(x). Therefore, (3.2.29) holds and, injecting it in (3.2.28), we obtain (3.2.30).

Proposition 3.2.7 (Propagation of the uniform estimate on V ). Let us fix K 1 > 0. There exist two integers n ρ 1 sufficiently large, two constants A, B > 1 sufficiently large, ε 0 ∈]0, (2A + B) -1 [ sufficiently small, and 0 δ δ 2 δ 1 δ 0 1 small, such that, for any 0 < ε < ε 0 , if (u, v) is solution to (1.1.1)-(1.1.2) in some interval [1, T ] for a fixed T > 1, and u ± , v ± defined in (1.1.5) satisfy a-priori estimates (1.1.11) for every t ∈ [1, T ], then it also verify (1.1.12b) in the same interval [1, T ].

Proof. We warn the reader that, throughout the proof, we will denote by C, β (resp. β ) two positive constants such that β → 0 as σ → 0 (resp. β → 0 as δ 0 , σ → 0). These constants may change line after line. We also remind that h = 1/t.

In proposition 3.1.1 we introduced function v N F , defined from v -through (3.1.3), and proved that its H ρ,∞ norm differs from that of v -by a quantity satisfying (3.1.7b). Hence, from a-priori estimates (1.1.11a), (1.1.11b), (1.1.11c) and for θ ∈]0, 1[ sufficiently small (e.g. θ < 1/4)

(3.2.31) v -(t, •) H ρ,∞ ≤ v N F (t, •) H ρ,∞ + CA 2-θ B θ ε 2 t -5 4 .
We successively introduced v in (3.2.2) and decomposed it into the sum of functions v Σ Λ kg and v Σ Λ c kg (see (3.2.16)). We will show in lemma B.2.14 of appendix B that, for any s ≤ n,

(3.2.32) v(t, •) H s h + 2 |γ|=1 Op w h (χ(h σ ξ))L γ v(t, •) L 2 ≤ CBεh -β for all t ∈ [1, T ], so inequality (3.2.19b) gives that (3.2.33) v Σ Λ c kg (t, •) L ∞ ≤ CBεh 1 2 -β .
As concerns v Σ Λ kg , we proved in proposition 3.

( v)(t, •) L ∞ ≤ CBεt -3 2 +β .
We also have that

(3.2.34) Γ kg Op w h (Σ(ξ))[tr N F kg (t, tx)] L ∞ (dx) ≤ C(A + B)ABε 3 t -3 2 +β .
In fact, by symbolic calculus of lemma 1.2.24 we derive that, for a fixed N ∈ N (e.g. N > ρ) and up to negligible multiplicative constants,

Γ kg Op w h (Σ(ξ)) = N -1 |α|=0 h |α| 2 Op w h (∂ α γ) x -p (ξ) √ h χ(h σ ξ)(∂ α Σ)(ξ) + Op w h (r N (x, ξ)), where r N ∈ h N 2 S 1 2 ,σ ( x-p (ξ) √ h -1
). Choosing N sufficiently large, we deduce from proposition 1.2.37, the fact that tw(t, t•) L 2 = w(t, •) L 2 , inequality (3.1.6a) and a-priori estimates, that for every

t ∈ [1, T ] Op w h (r N (x, ξ))[tr N F kg (t, tx)] L ∞ (dx) ≤ CA 2 Bε 3 t -2 .
Using, instead, proposition 1.2.39 with p = +∞, inequality (B.3.25) in appendix B, and that h = t -1 , we deduce that

N -1 |α|=0 h |α| 2 Op w h (∂ α γ) x -p (ξ) √ h χ(h σ ξ)(∂ α Σ)(ξ) Op w h (χ 1 (h σ ξ))[tr N F kg (t, tx)] L ∞ t 1+β χ(t -σ D x )r N F kg (t, •) L ∞ ≤ C(A + B)ABε 3 t -3 2 +β . Summing up, Γ kg Op w h (Σ(ξ))[t -1 r N F kg (t, tx)] + R 2 ( v) = F kg (t, x) with F kg (t, •) L ∞ ≤ [C(A + B)ABε 3 + CBε]t -3 2 +β ,
Using equation (3.2.30) we deduce that

(3.2.35) 1 2 ∂ t | v Σ Λ kg (t, x)| 2 = v Σ Λ kg D t v Σ Λ kg ≤ | v Σ Λ kg (t, x)||F kg (t, x)|
and hence that

v Σ Λ kg (t, •) L ∞ ≤ v Σ Λ kg (1, •) L ∞ + t 1 F kg (τ, •) L ∞ dτ ≤ v Σ Λ kg (1, •) L ∞ + C(A + B)ABε 3 + CBε. As v Σ Λ kg (1, •) L ∞ v(1, •) L 2 ≤ CBε
v N F (t, •) L ∞ ≤ (C(A + B)ABε 3 + CBε)t -1 ,
which injected in (3.2.31) leads finally to (1.1.12b) if we take A > 1 sufficiently large such that CB < A 3K 1 , and ε 0 > 0 sufficiently small to verify

C(A + B)Bε 2 0 + CA 1-θ B θ ε 0 ≤ 1 3K 1 .

The derivation of the transport equation

We now focus on the semi-classical wave equation satisfied by u:

(3.2.36) D t -Op w h (x • ξ -|ξ|) u(t, x) = h -1
q w (t, tx) + c w (t, tx) + r N F w (t, tx) , with q w , c w , r N F w given by (3.1.17), (3.1.18), (3.1.19) respectively, and on the derivation of the mentioned transport equation. As we will make use several times of proposition 1.2.30 and inequalities (1.2.52), we remind the reader about definition (1.2.40) of Ω h and (1.2.49) of M j . Also, θ 0 (x) denotes a smooth radial cut-off function (often coming with operator Ω h ) while χ ∈ C ∞ 0 (R 2 ) is equal to 1 in a neighbourhood of the origin and suitably supported. In order to recover a sharp estimate for u such as (3.2.3a), we study the behaviour of this function separately in different regions of the phase space (x, ξ) ∈ R 2 × R 2 . We start by fixing ρ ∈ Z, and by introducing (3.2.37)

Σ j (ξ) := ξ ρ , for j = 0, ξ ρ ξ j |ξ| -1 , for j = 1, 2.
Taking a smooth cut-off function χ 0 equal to 1 in a neighbourhood of the origin, a Littlewood-Paley decomposition, and a small σ > 0, we write the following for any j ∈ {0, 1, 2}:

(3.2.38)

Op w h (Σ j (ξ)) u = Op w h (Σ j (ξ)χ 0 (h -1 ξ)) u + k Op w h Σ j (ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ) u + Op w h (Σ j (ξ)(1 -χ 0 )(h σ ξ)) u,
observing that the sum over k is actually finite and restricted to set of indices K := {k ∈ Z : h 2 k h -σ }. From the classical Sobolev injection and the continuity on L 2 of the Riesz operator

(3.2.39) Op w h (Σ j (ξ)χ 0 (h -1 ξ)) u(t, •) L ∞ = Σ j (hD)χ 0 (D) u(t, •) L ∞ u(t, •) L 2 ,
while from the semi-classical Sobolev injection along with lemma 1.2.40

(3.2.40) Op w h (Σ j (ξ)(1 -χ 0 )(h σ ξ)) L ∞ h N u(t, •) H s h
, where N = N (s) ≥ 0 if s > 0 is sufficiently large. The remaining terms in the right hand side of (3.2.38), localised for frequencies |ξ| ∼ 2 k , need a sharper analysis because a direct application of semi-classical Sobolev injection only gives that

Op w h Σ j (ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ) u L ∞ ≤ 2 k h -1-µ u L 2 ,
with µ = σρ if ρ ≥ 0, 0 otherwise, and factor 2 k h -1-µ may grow too much when h → 0.

For any fixed k ∈ K, ρ ∈ Z and j ∈ {0, 1, 2}, let us introduce

(3.2.41) u Σ j ,k (t, x) := Op w h Σ j (ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ) u(t, x)
and observe that, from the commutation of the above operator with the linear part of equation (3.2.36), we get that u Σ j ,k is solution to

[D t -Op w h (x • ξ -|ξ|)] u Σ j ,k (t, x) = h -1 Op w h Σ j (ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ) q w (t, tx) + c w (t, tx) + r N F w (t, tx) -ih Op w h Σ j (ξ)(∂χ 0 )(h -1 ξ) • (h -1 ξ)ϕ(2 -k ξ) u -iσh Op w h Σ j (ξ)ϕ(2 -k ξ)(∂χ 0 )(h σ ξ)) • (h σ ξ) u. ( 3
Γ w,k := Op w h γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ) ,
for some γ ∈ C ∞ 0 (R 2 ) equal to 1 close to the origin and ψ ∈ C ∞ 0 (R 2 \ {0}) equal to 1 on suppϕ, whose symbol is localized in a neighbourhood of Λ w ∩ {|ξ| ∼ 2 k } of size h 1/2-σ . We also define

(3.2.45a) u Σ j ,k Λw := Γ w,k u Σ j ,k , (3.2.45b) u Σ j ,k Λ c w := Op w h 1 -γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ) u Σ j ,k , so that u Σ j ,k = u Σ j ,k Λw + u Σ j ,k Λ c
w . We are going to prove that, if we suitably control the L 2 norm of

(θ 0 Ω h ) µ M ν u Σ j ,k , for any µ, |ν| ≤ 1, then u Σ j ,k Λw c is a O L ∞ (h -0 ) (see proposition 3.2.8). As h = t -1 , this means that u Σ j ,k Λ c w
grows in time at a rate slower than the one expected for u Σ j ,k (that is t 1/2 after (3.2.3a)). Analogously to the Klein-Gordon case discussed in the previous subsection, the main contribution to u Σ j ,k is hence the one localized around Λ w and represented by u

Σ j ,k
Λw . We will show that this function is solution to a transport equation (see proposition 3.2.17), from which we will be able to derive a suitable estimate of its uniform norm and to finally propagate (3.1.12) (see proposition 3.3.7).

∈]0, 1], k ∈ K, (3.2.46a) u Σ j ,k Λ c w (t, •) L 2 ≤ Ch 1 2 -β u Σ j ,k (t, •) L 2 + M u Σ j ,k (t, •) L 2 , (3.2.46b) u Σ j ,k Λ c w (t, •) L ∞ ≤ Ch -β 1 µ=0 (θ 0 Ω h ) µ u Σ j ,k (t, •) L 2 + (θ 0 Ω h ) µ M u Σ j ,k (t, •) L 2 ,
for a small β > 0, β → 0 as σ → 0.

Proof. The proof is straightforward if one writes |α|+1) , and uses inequalities (1.2.52) with a

u Σ j ,k Λ c w = 2 j=1 Op w h γ j 1 x|ξ| -ξ h 1/2-σ x j |ξ| -ξ j h 1/2-σ ψ(2 -k ξ) u Σ j ,k , where γ j 1 (z) := (1-γ)(z)z j |z| 2 is such that |∂ α z γ j 1 (z)| z -(
(x) = b p (ξ) ≡ 1. Lemma 3.2.9. Let ϕ ∈ C ∞ 0 (R 2 \ {0}
) be such that ϕ ≡ 1 on suppϕ and have a sufficiently small support so that ψ ϕ ≡ ψ. Then for any k ∈ K

(3.2.47) Γ w,k , D t -Op w h (x • ξ -|ξ|) ϕ(2 -k ξ) Op w h (ϕ(2 -k ξ)) = Op w h (b(x, ξ)),
where, for any w ∈ L 2 such that θ 0 Ω h w,

(θ 0 Ω h ) µ Mw ∈ L 2 (R 2 ), for µ = 0, 1, (3.2.48a) Op w h (b(x, ξ))w L 2 h 1-β ( w L 2 + Mw L 2 ) , (3.2.48b) Op w h (b(x, ξ))w L ∞ h 1-β 1 µ=0 (θ 0 Ω h ) µ w L 2 + (θ 0 Ω h ) µ Mw L 2 ,
with β > 0 small, β → 0 as σ → 0.

Proof. We warn the reader that most of the terms arising from the development of the commutator in the left hand side of (3.2.47) satisfy a better L 2 estimate than (3.2.48a), namely (3.2.49)

• L 2 h 3 2 -β w L 2 + Mw L 2 .
The only contribution whose L 2 norm is only a O(h w L 2 ) is the integral remainder called r k N , appearing in symbolic development (3.2.51).

Since ∂ t = -h 2 ∂ h , an easy computation shows that [Γ w,k , D t ] = 1 2 + σ h i Op w h (∂γ) x|ξ| -ξ h 1/2-σ • x|ξ| -ξ h 1/2-σ ψ(2 -k ξ) + h i Op w h γ x|ξ| -ξ h 1/2-σ (∂ψ)(2 -k ξ) • (2 -k ξ) .
(3.2.50)

The first term in the above right hand side satisfies (3.2.49) and (3.2.48b) after inequalities (1.2.52). The same estimates hold also for the latter one when it acts on Op w h (ϕ(2 -k ξ))w, for the derivatives of ψ vanish on the support of ϕ (and then of ϕ) as a consequence of our assumptions. In fact, if we introduce a smooth function ψ ∈ C ∞ 0 (R 2 \ {0}), equal to 1 on the support of ∂ψ and such that supp ψ ∩ suppϕ = ∅, and use symbolic calculus we find that, for any fixed N ∈ N,

Op w h γ x|ξ| -ξ h 1/2-σ (∂ψ)(2 -k ξ) • (2 -k ξ) Op w h (ϕ(2 -k ξ)) = Op w h γ x|ξ| -ξ h 1/2-σ (∂ψ)(2 -k ξ) • (2 -k ξ) Op w h ψ(2 -k ξ)ϕ(2 -k ξ) -Op w h (r k N ),
where the first term in the above right hand side is 0, and integral remainder r k N is given by

r k N = h 2i N |α|=N N (-1) |α| α!(πh) 4 e 2i h (η•z-y•ζ) 1 0 ∂ α x γ x|ξ| -ξ h 1/2-σ (∂ψ)(2 -k ξ) • (2 -k ξ) | (x+tz,ξ+tζ) dt × ∂ α ξ ψ(2 -k ξ) | (ξ+η) dydzdηdζ.
Developing explicitly the above derivatives and reminding definition (1.2.30) of integrals I k p,q , for general k ∈ K, p, q ∈ Z, one recognizes that, up to some multiplicative constants, r k N has the form

h N -N ( 1 2 -σ) 2 -kN I k N,0 (x, ξ), with a, a , b q ≡ 1, p = N and ψ(2 -k ξ) replaced with (∂ψ)(2 -k ξ) • (2 -k ξ). Propositions 1.2.

and 1.2.31 imply then that

Op w h (r k N ) L(L 2 ) + Op w h (r k N ) L(L 2 ;L ∞ ) h
if N ∈ N is chosen sufficiently large (e.g. N > 9), which implies that the L(L 2 ) and L(L 2 ; L ∞ ) norms of the latter operator in the right hand side of (3.2.50) is bounded by h 2 .

As regards

[Γ w,k , Op w h ((x • ξ -|ξ|) ϕ(2 -k ξ))],
we first remind that the symbolic development of a commutator's symbol only contains odd order terms. Consequently, for a new fixed N ∈ N and up to multiplicative constants independent of h, k, the symbol of the considered commutator writes as

(3.2.51) h γ x|ξ| -ξ h 1/2-σ , (x • ξ -|ξ|) ϕ(2 -k ξ) + 3≤|α|<N |α|=|α 1 |+|α 2 | h |α| ∂ α 1 x ∂ α 2 ξ γ x|ξ| -ξ h 1/2-σ ∂ α 2 x ∂ α 1 ξ (x • ξ -|ξ|) ϕ(2 -k ξ) + r k N (x, ξ), with r k N (x, ξ) = h 2i N |α1|+|α2|=N N (-1) |α1| α!(πh) 4 e 2i h (η•z-y•ζ) 1 0 ∂ α1 x ∂ α2 ξ γ x|ξ| -ξ h 1/2-σ ψ(2 -k ξ) (x+tz,ξ+tζ) dt × ∂ α2 x ∂ α1 ξ (x • ξ -|ξ|) ϕ(2 -k ξ) | (x+y,ξ+η) dydzdηdζ .
Since γ x|ξ|-ξ h 1/2-σ , x • ξ -|ξ| = 0 the Poisson braket in the above sum reduces to

h j,l (∂ j γ) x|ξ| -ξ h 1/2-σ (∂ j ϕ)(2 -k ξ) x l |ξ| -ξ l h 1/2-σ (2 -k ξ l )
and its quantization acting on Op w h (ϕ(2 -k ξ))w satisfies (3.2.49), (3.2.48b) because ∂ ϕ vanishes on the support of ϕ.

An explicit calculation of terms of order 3 ≤ |α| < N , with the help of lemma 1.2.26 and the observation that

|α 2 | ≤ 1 because (x • ξ -|ξ|) ϕ(2 -k ξ) is affine in x,
shows that they are linear combination of products

h |α|-|α|( 1 2 -σ) γ |α| x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)x ν b 1 (ξ)
and

h |α|-(|α|-1)( 1 2 -σ) γ x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)b 0 (ξ) for two new cut-off functions γ, ϕ, |∂ β b 0 (ξ)| β |ξ| -|β| ,
and ν ∈ N 2 of length at most 1. Furthermore, for j = 1, 2,

h |α|-|α|( 1 2 -σ) γ |α| x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)x j b 1 (ξ) = h |α|-(|α|-1)( 1 2 -σ) γ j |α| x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)b 0 (ξ) + h |α|-|α|( 1 2 -σ) γ |α| x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)ξ j b 0 (ξ),
with γ j |α| (z) := γ |α| (z)z j . From propositions 1.2.27, 1.2.30, the fact that |α| ≥ 3 and 2 k ≤ h -σ we deduce that the quantization of these |α|-order terms acting on Op w h (ϕ(2 -k ξ))w satisfies (3.2.49), (3.2.48b).

Finally, we notice that integral remainder r k N can be actually seen as the sum of two contributions, one of the form (1.2.44), the other like (1.2.45), with a ≡ 1 and p = 1. Lemma 1.2.32 implies then that the L(L 2 ) and L(L 2 ; L ∞ ) norms of Op w h ( r k N ) are bounded by h as foretold, which concludes the proof of the statement.

Lemma 3.2.10. Function u Σ j ,k
Λw is solution to the following equation:

D t -Op w h (x • ξ -|ξ|) ϕ(2 -k ξ) u Σ j ,k Λw (t, x) = f w k (t, x) + h -1 Γ w,k Op w h Σ j (ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ) q w (t, tx) + c w (t, tx) + r N F w (t, tx) -ih Γ w,k Op w h Σ j (ξ)(∂χ 0 )(h -1 ξ) • (h -1 ξ)ϕ(2 -k ξ) u -iσh Γ w,k Op w h Σ j (ξ)ϕ(2 -k ξ)(∂χ 0 )(h σ ξ)) • (h σ ξ) u, (3.2.52) where ϕ ∈ C ∞ 0 (R 2 \ {0}
) is equal to 1 on suppϕ, and there exist two constants C, C > 0 such that, for any h ∈]0, 1], k ∈ K,

(3.2.53a) f w k (t, •) L 2 ≤ Ch 1-β u Σ j ,k (t, •) L 2 + M u Σ j ,k (t, •) L 2 , (3.2.53b) f w k (t, •) L ∞ ≤ C h 1-β 1 µ=0 (θ 0 Ω h ) µ u Σ j ,k (t, •) L 2 + (θ 0 Ω h ) µ M u Σ j ,k (t, •) L 2 ,
with β > 0 small, β → 0 as σ → 0.

Proof. If we consider a cut-off function ϕ ∈ C ∞ 0 (R 2 \ {0}) such that ϕ ≡ 1 on the support of ϕ (ϕ being the truncation on u Σ j ,k 's frequencies), we have the exact equality

Op w h (x • ξ -|ξ|) u Σ j ,k = Op w h ((x • ξ -|ξ|) ϕ(2 -k ξ)) u Σ j ,k
. Moreover, if we assume that its support is sufficiently small so that ψ ϕ ≡ ϕ, and apply operator Γ w,k to equation (3.2.42), lemma 3.2.9 gives us the result of the statement.

The transport equation we talked about at the beginning of this section will be deduced from equation (3.2.52) by suitably developing symbol (x • ξ -|ξ|) ϕ(2 -k ξ). To do that, we first need to restrict the support of that symbol to bounded values of x through the introduction of a new cut-off function θ(x). We remind that Σ is a concise notation that we use to indicate a linear combination of a finite number of terms of the same form.

(3.2.54) Op w h (x • ξ -|ξ|) ϕ(2 -k ξ) = Op w h θ(x)(x • ξ -|ξ|) ϕ(2 -k ξ) + (1 -θ)(x)Op w h ((x • ξ -|ξ|) ϕ(2 -k ξ)) + θ(x)Op w h ( ϕ 1 (2 -k ξ)) + Op w h (r(x, ξ)),
where θ is a smooth function supported for

D 1 < |x| < D 2 , ϕ 1 ∈ C ∞ 0 (R 2 \ {0}) and Op w h (r) L(L 2 ) + Op w h (r) L(L 2 ;L ∞ ) h.
Therefore, u

Σ j ,k Λ kg verifies D t -Op w h θ(x)(x • ξ -|ξ|) ϕ(2 -k ξ) u Σ j ,k Λw (t, x) = f w k (t, x) + (1 -θ)(x)Op w h ((x • ξ -|ξ|) ϕ(2 -k ξ)) u Σ j ,k Λw + θ(x)Op w h ( ϕ 1 (2 -k ξ)) u Σ j ,k Λw + h -1 Γ w,k Op w h Σ(ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ) q w (t, tx) + c w (t, tx) + r N F w (t, tx) -ih Γ w,k Op w h Σ(ξ)(∂χ 0 )(h -1 ξ) • (h -1 ξ)ϕ(2 -k ξ) u -iσh Γ w,k Op w h Σ(ξ)ϕ(2 -k ξ)(∂χ 0 )(h σ ξ)) • (h σ ξ) u, (3.2

.55)

where f w k satisfies estimates (3.2.53).

Proof. Let θ(x) be the cut-off function of the statement. By proposition 1.2.21 we have that

(1 -θ)(x)(x • ξ -|ξ|) ϕ(2 -k ξ) = (1 -θ)(x) (x • ξ -|ξ|) ϕ(2 -k ξ) - h 2i ∂θ(x) • x - ξ |ξ| ϕ(2 -k ξ) - 2 -k h 2i (x • ξ -|ξ|)∂θ(x) • (∂ ϕ)(2 -k ξ) + r k 2 (x, ξ) = (1 -θ)(x) (x • ξ -|ξ|) ϕ(2 -k ξ) - h 2i ∂θ(x) • x ϕ(2 -k ξ) + h 2i 2 l=1 ∂ l θ(x) ξ l |ξ| ϕ(2 -k ξ) - h 2i 2 j,l=1 ∂ j θ(x)x l (∂ j ϕ)(2 -k ξ)(2 -k ξ l ) + h 2i 2 l=1 ∂ l θ(x) (2 -k |ξ|)(∂ l ϕ)(2 -k ξ) + r k 2 (x, ξ) + r k 2 (x, ξ), (3.2.56)
where ∂θ is supported for D 1 < |x| < D 2 , and r k 2 (t, x) (resp. r k 2 (t, x)) is a linear combination of integrals of the form

h 2 2 -k (πh) 2 e 2i h η•z 1 0 θ(x + tz)(1 -t) 2 dt x ν ϕ(2 -k (ξ + η))dzdη,
with |ν| = 0, 1 (resp. |ν| = 0), for some new θ, ϕ ∈ C ∞ 0 (R 2 \ {0}). By writing x as (x + tz) -tz, using that ze

2i h η•z = h 2i ∂ ξ e 2i h η•z
, and making an integration by parts, one can expressr k 2 (t, x) as the sum over |ν| = 0, 1 of integrals such as

h 2 2 -k (h2 -k ) ν (πh) 2 e 2i h η•z 1 0 θ(x + tz)f (t)dt ϕ(2 -k (ξ + η))dzdη,
for some new smooth θ, f, ϕ, and show that for any α, β ∈ N 2

∂ α x ∂ β ξ (r k 2 + r k 2 )(x, hξ) α,β h 2 2 -k α,β h.
Thus (r k 2 + r k 2 )(x, hξ) ∈ hS 0 (1)
, which means, by classical results on pseudo-differential operators (see for instance [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]), that

Op w h ((r k 2 + r k 2 )(x, ξ)) = Op w ((r k 2 + r k 2 )(x, hξ)) ∈ L(L 2 )
with norm O(h). Furthermore, one can also show that Op w h (r k 2 + r k 2 ) L(L 2 ;L ∞ ) h using lemma 1.2.25 and the fact that, by making some integrations by parts, for any multi

-indices α, β ∈ N 2 and a new ϕ ∈ C ∞ 0 (R 2 \ {0}) ∂ α y ∂ β ξ (r k 2 + r k 2 )
x + y 2 , hξ

L 2 (dξ) h 2 2 -k η -3 | ϕ(2 -k h(ξ + η))|dη L 2 (dξ)
h.

These considerations, along with the continuity of Γ w,k on L 2 , uniformly in h and k (see proposition 1.2.27), imply that Op w h (r

k 2 + r k 2 ) u Σ j ,k
Λw is a remainder f w k .

Lemma 3.2.12. We have that

|ξ| -x • ξ = 1 2 (1 -|x| 2 )x • ξ + e(x, ξ) with (3.2.57) e(x, ξ) = 1 2 |ξ| x - ξ |ξ| 2 + 1 2 x - ξ |ξ| • ξ x - ξ |ξ| • x + ξ |ξ| .
Proof.

|ξ| -xξ = 1 2 |ξ| x - ξ |ξ| 2 + 1 2 |ξ|(1 -|x| 2 ) = 1 2 |ξ| x - ξ |ξ| 2 + 1 2 (|ξ| -x • ξ)(1 -|x| 2 ) + 1 2 (1 -|x| 2 )x • ξ = 1 2 |ξ| x - ξ |ξ| 2 + 1 2 ξ |ξ| -x • ξ ξ |ξ| -x • ξ |ξ| + x e(x,ξ) + 1 2 (1 -|x| 2 )x • ξ . Lemma 3.2.13. Let γ, θ ∈ C ∞ 0 (R 2 ) and ϕ ∈ C ∞ 0 (R 2 \ {0}
) be such that ϕ ≡ 1 on the support of ϕ and have a sufficiently small support so that ψ ϕ ≡ ϕ. Let also

(3.2.58) B(x, ξ) := γ x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)θ(x) x m - ξ m |ξ| , m ∈ {1, 2}. For any function w ∈ L 2 (R 2 ) such that Mw ∈ L 2 (R 2 ), any m, n ∈ {1, 2}, (3.2.59a) Op w h θ(x) ϕ(2 -k ξ) x m - ξ m |ξ| (x n |ξ| -ξ n ) Γ w,k w L 2 h 1-β w L 2 + Mw L 2 , (3.2.59b) Op w h θ(x) ϕ(2 -k ξ) x m - ξ m |ξ| (x n |ξ| -ξ n ) Γ w,k w L ∞ h 1-β w L 2 + Mw L 2 + h -β Op w h B(x, ξ)ξ Mw L 2 ,
with β > 0 small, β → 0 as σ → 0.

Proof. After lemma 1.2.35 with p = 0 we have that

Op w h θ(x) ϕ(2 -k ξ) x m - ξ m |ξ| (x n |ξ| -ξ n ) Γ w,k w = Op w h (B(x, ξ)(x n |ξ| -ξ n )) w + Op w h (r k 0 (x, ξ))w,
and the L 2 (resp. L ∞ ) norm of the latter term in the above right hand side is bounded by the right hand side of (3.2.59a) (resp. of (3.2.59b)) after inequality (1.2.60a) (resp. (1.2.60b)). Moreover, the L 2 norm of Op w h B(x, ξ)(x n |ξ| -ξ n ) w is also bounded by the right hand side of (3.2.59a) as straightly follows from emma 1.2.33. It only remains to prove that the L ∞ norm of this term is bounded by the right hand side of (3.2.59b).

We first consider a new cut-off function

ϕ 1 ∈ C ∞ 0 (R 2 \ {0}
), equal to 1 on supp ϕ so that its derivatives vanish against ϕ, and use symbolic calculus to write

Op w h B(x, ξ)(x n |ξ| -ξ n ) = Op w h ( ϕ 1 (2 -k ξ))Op w h B(x, ξ)(x n |ξ| -ξ n ) + Op w h (r k N,1 (x, ξ)),
where r k N,1 (x, ξ) is obtained using (1.2.20). Up to interchange the role of variables y and z (resp. η and ζ) and to consider e r k N,1 is analogous to integral (1.2.45) with p = 1. Therefore, if N ∈ N is chosen sufficiently large (e.g. N > 11), lemma 1.2.32 implies that Op w h (r k N,1 ) L(L 2 ;L ∞ ) = O(h). Since ϕ 1 localises frequencies ξ in an annulus, the classical Sobolev injection gives that

Op w h ( ϕ 1 (2 -k ξ))Op w h B(x, ξ)(x n |ξ| -ξ n ) w L ∞ log h Op w h B(x, ξ)(x n |ξ| -ξ n ) w L 2 + D x Op w h B(x, ξ)(x n |ξ| -ξ n ) w L 2 .
As previously said, the former norm in the above right hand side satisfies inequality (3.2.59a).

As concerns the latter one, we remark that thanks to the specific structure of symbol B(x, ξ) its first derivative with respect to x does not lose any factor h -1/2+σ , as (3.2.60)

∂ x γ x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)θ(x) x m - ξ m |ξ| = (∂γ) x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)θ(x) x m |ξ| -ξ m h 1/2-σ .
Consequently, by using symbolic calculus we derive that

D x Op w h B(x, ξ)(x n |ξ| -ξ n ) w = h -1 Op w h B(x, ξ)(x n |ξ| -ξ n )ξ w + Op w h γ x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)a(x)b 0 (ξ)(x j |ξ| -ξ j ) w,
where is a concise notation to indicate linear combinations, j ∈ {m, n} and γ, ϕ, a are some new smooth functions with a(x) compactly supported. Again by lemma 1.2.33 the L 2 norms of latter contributions in the above right hand side are bounded by h 1-β

( w L 2 + Mw L 2 ).
Finally, we observe that symbol B(x, ξ)ξ can be seen as

(3.2.61) γ x|ξ| -ξ h 1/2-σ (x m |ξ| -ξ m ) ϕ(2 -k ξ)θ(x)b 0 (x),
which implies, after lemma 1.2.34, that

h -1 Op w h B(x, ξ)(x n |ξ| -ξ n )ξ w = Op w h B(x, ξ)ξ M n w + O L 2 (h 1-β ( w L 2 + Mw L 2 )).
Lemma 3.2.14. Let e(x, ξ) be the symbol defined in (3.2.57), θ ∈ C ∞ 0 (R 2 ), and ϕ ∈ C ∞ 0 (R 2 \{0}) with sufficiently small support so that ψ ϕ ≡ ϕ. If a-priori estimates (1.1.11) are satisfied for every t ∈ [1, T ], for some fixed T > 1, there exists a constant C > 0 such that (3.2.62)

Op w h θ(x) ϕ(2 -k ξ)e(x, ξ) u Σj ,k Λw (t, •) L 2 + Op w h θ(x) ϕ(2 -k ξ)e(x, ξ) u Σj ,k Λw (t, •) L ∞ ≤ CBεh 1-β for every t ∈ [1, T ], with β > 0 small, β → 0 as σ → 0.
Proof. We warn the reader that, throughout this proof, C, β and β will denote three positive constants that may change line after line, with β → 0 as σ → 0 (resp. β → 0 as σ, δ 1 → 0).

Since symbol e(x, ξ) writes as

e(x, ξ) = 1 2 2 m=1 x m - ξ m |ξ| (x m |ξ| -ξ m ) + 1 2 2 m,n=1 x m - ξ m |ξ| (x n |ξ| -ξ n ) ξ m |ξ| ξ n |ξ| + x n ξ m |ξ| , it follows that the L 2 norm of Op w h θ(x) ϕ(2 -k ξ)e(x, ξ) u Σ j ,k
Λw satisfies inequality (3.2.62) after lemmas 3.2.13 and B.2.1 in appendix B. Moreover, from lemma 3.2.13

Op w h θ(x) ϕ(2 -k ξ)e(x, ξ) u Σ j ,k Λw L ∞ h 1-β u Σ j ,k (t, •) L 2 + M u Σ j ,k (t, •) L 2 + h -β Op w h B(x, ξ)ξ M u Σ j ,k (t, •) L 2 ,
with B(x, ξ) defined in (3.2.58). The aim of the proof is then to show that the L 2 norm of Op w h B(x, ξ)ξ M u Σ j ,k is estimated by the right hand side of (3.2.62). First of all, we remind that B(x, ξ)ξ can be seen as a symbol of the form (3.2.61). From proposition 1.2.27 we hence have that

(3.2.63a) Op w h B(x, ξ)ξ L(L 2 ) = O(h 1 2 -β ), while from inequality (1.2.52a) (3.2.63b) Op w h B(x, ξ)ξ w L 2 h 1-β ( w L 2 + Mw L 2 ).
We also recall definition (3.2.41) of u Σ j ,k , use the concise notation φ j k (ξ) for its symbol

Σ j (ξ)(1 - χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ), and observe that (3.2.64) M n , Op w h (φ j k (ξ)) = - 1 2i Op w h |ξ|∂ n φ j k (ξ) , M n , Op w h (φ j k (ξ)) L(L 2 ) = O(h -σ ),
Op w h B(x, ξ)ξ M n u Σ j ,k (t, •) L 2 Op w h B(x, ξ)ξ Op w h (φ j k (ξ))[t(Z n u N F )(t, tx)] L 2 (dx) + Op w h B(x, ξ)ξ Op w h ξ n |ξ| -1 φ j k (ξ)) u(t, •) L 2 + Op w h B(x, ξ)ξ Op w h |ξ|∂ n φ j k (ξ) u(t, •) L 2 + Op w h B(x, ξ)ξ Op w h (φ j k (ξ)) t(tx n ) q w (t, tx) + c w (t, tx) + r N F w (t, tx) L 2 (dx)
, with u N F defined in (3.1.15), q w , c w and r N F w given by (3.1.17), (3.1.18) and (3.1.19) respectively. Evidently, after (3.2.63b) and a further commutation of M with Op w h ξ n |ξ| -1 φ j k (ξ) and Op w h |ξ|∂ n φ j k (ξ) respectively, the second and third L 2 norm in the above right hand side are estimated by

h 1-β ( u(t, •) L 2 + Op w h (χ(h σ ξ))M u(t, •) L 2 ),
for some χ ∈ C ∞ 0 (R 2 ). They are hence bounded by CBεh 1-β by lemma B.2.1.

• Estimate of Op w h B(x, ξ)ξ)Op w h (φ j k (ξ))[t(Z n u N F )(t, tx)] L 2 :
This L 2 norm is basically estimated in terms of the L 2 norm of (Z µ u) -, for |µ| ≤ 2. In fact, after definition (3.1.15) and equality (2.1.15a)

(3.2.65) (Z n u N F )(t, tx) = (Z n u) -(t, tx) + D n |D x | u -(t, tx) - i 4(2π) 2 l∈{+,-} Z n e iy•ξ D l (ξ, η)v l (ξ -η)v l (η)dξdη y=tx ,
with D l given by (3.1.14). On the one hand, taking a new smooth cut-off function θ 1 equal to 1 on the support of θ, using (1.2.50) with a = θ 1 , together with (1.2.51a), proposition 1.2.27, and (3.2.64), we deduce that

Op w h B(x, ξ)ξ)Op w h (φ j k (ξ))[t(Z n u) -(t, tx)] L 2 (dx) 2 m=1 h θ 1 (x)Op w h (φ j k (ξ))M m [t(Z n u) -(t, tx)] L 2 (dx) + h 1-β (Z n u) -(t, •) L 2 .
After relation (3.2.10),

θ 1 (x)Op w h (φ j k (ξ))M m [t(Z n u) -(t, tx)] L 2 (Z m Z n u) -(t, •) L 2 + (Z n u) -(t, •) L 2 + θ 1 x t φ j k (D x ) [x m Z n NL w ] (t, •) L 2 .
Moreover,

θ 1 x t φ j k (D x )x m = tθ 1,m x t φ j k (D x ) + θ 1 x t [φ j k (D x ), x m ],
where θ 1,m (z) = θ 1 (z)z m , and [φ j k (D x ), x m ] is a bounded operator on L 2 with norm O(t), as one can check computing its associated symbol and using that 2 -k h -1 = t. Therefore, using also inequality (3.2.11) together with a-priori estimates (1.1.11) we deduce that

Op w h B(x, ξ)ξ)Op w h (φ j k (ξ)) t(Z n u) -(t, tx) L 2 (dx) 2 |µ|=1 h (Z µ u) -(t, •) L 2 + Z n V (t, •) H 1 V (t, •) H 2,∞ + V (t, •) H 1 + V (t, •) L 2 ( U (t, •) H 1,∞ + R 1 U (t, •) H 1,∞ ) + V (t, •) L ∞ U (t, •) H 1 V (t, •) H 1,∞ ≤ CBεh 1-δ 1 2 . ( 3
Op w h B(x, ξ)ξ)Op w h (φ j k (ξ))[t(D n |D x | -1 u) -(t, tx)] L 2 h 1-β ( u(t, •) L 2 + Op w h (χ(h σ ξ))M u(t, •) L 2 ) ≤ CBεh 1-δ 2 2 .
Finally, by symbolic calculus and (3.2.60) we have that

(3.2.68) Op w h (B(x, ξ)ξ) = Op w h (B(x, ξ))(hD x ) + h 2i Op w h ∂ x B(x, ξ) ,
where

∂ x B is of the form (3.2.69) γ x|ξ| -ξ h 1/2-σ ϕ(2 -k ξ)θ(x)b 0 (ξ) for some new γ, θ ∈ C ∞ 0 (R 2 ). Consequently, by proposition 1.2.27 (3.2.70) Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ)) tZ n e iy•ξ D l (ξ, η)v l (ξ -η)v l (η)dξdη y=tx L 2 (dx) χ(t -σ D x )D x Z n e ix•ξ D l (ξ, η)v l (ξ -η)v l (η)dξdη L 2 (dx) + h χ(t -σ D x )Z n e ix•ξ D l (ξ, η)v l (ξ -η)v l (η)dξdη L 2 (dx)
and the above right hand side is bounded by

h -β ( V (t, •) H 1 + V (t, •) L 2 ( U (t, •) H 1,∞ + R 1 U (t, •) H 1,∞ ) + V (t, •) L ∞ U (t, •) H 1 ) × ( V (t, •) H 14,∞ + h V (t, •) H 13 ) + h -β Z n V (t, •) L 2 V (t, •) H 17,
tq w (t, tx) = h 2 v Op w h (ξ 1 ) v -Op w h ξ 1 ξ v • Op w h ξξ 1 ξ v (t, x) =: q w (t, x), where (3.2.72) q w (t, •) L 2 h v(t, •) H 1,∞ v(t, •) H 1 .
Then

Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ)) [t(tx n )q w (t, tx)] L 2 (dx) = h -1 Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ)) [x n q w (t, x)] L 2 (dx) .
Since B(x, ξ) is compactly supported in x and

Op w h B(x, ξ)ξ Op w h (φ j k (ξ)), x n L(L 2 ) = O(h 1 2 -β ),
as follows from symbolic calculus, (3.2.63a), equality (1.2.25) and proposition 1.2.27, we can morally reduce ourselves to the study of the L2 norm of

h -1 Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ)) q w (t, x) up to a O L 2 (h -1/2-β q w L 2 ).
h -1 Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ)) q w (t, •) L 2 h -1 Op w h (φ j k (ξ))(hD x ) q w (t, •) L 2 + q w (t, •) L 2 ,
h -1 Op w h (φ j k (ξ))(hD x ) q w (t, •) L 2 h 1-β v(t, •) H s + 2 |µ|=1 Op w h (χ(h σ ξ))L µ v(t, •) L 2 v(t, •) H 1,∞ ≤ CBεh 1-β ; • Estimate of Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ))(t(tx n )c w (t, tx) L 2 (dx) :
As for the previous estimate, we can reduce to the study of the L 2 norm of

Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ))[t 2 c w (t, tx)], up to a O L 2 h -1/2-β Op w h (χ(h σ ξ))[tc w (t, tx)] L 2 (dx) for some χ ∈ C ∞ 0 (R 2
). So using (3.2.63a), the fact that tw(t, t•) L 2 = w(t, •) L 2 , and (3.1.21a) with s > 0 sufficiently large so that N (s) > 2, we obtain that for a new

χ 1 ∈ C ∞ 0 (R 2 ) Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ))[t 2 c w (t, t•)] L 2 h -1 2 -β χ(t -σ D x )c w (t, •) L 2 h -1 2 -β χ 1 (t -σ D x )(v N F -v -)(t, •) L 2 V (t, •) H 2,∞ + v N F (t, •) H 1,∞ + h 3 2 (v N F -v -)(t, •) H 1 V (t, •) H s + v N F (t, •) H s .
(3.2.74)

Then inequalities (3.1.7a) with s = 1 and (3.1.8a), together with a-priori estimates, give that

Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ))[t 2 c w (t, t•)] L 2 ≤ CBεh 1-β . • Estimate of Op w h (B(x, ξ)ξ)Op w h (φ j k (ξ))(t(tx n )r N F w (t,
(B(x, ξ)ξ)Op w h (φ j k (ξ))[t 2 r N F w (t, t•)] L 2 h -1 2 -β χ(t -σ D x )r N F w (t, •) L 2 h -1 2 -β V (t, •) 2 H 13,∞ U (t, •) H 1 ≤ CBεh Lemma 3.2.15. Let ϕ ∈ C ∞ 0 (R 2 \ {0}
), k ∈ K and a j (ξ) be two smooth real symbols of order j = 0, 1.

Then (3.2.76) Op w h (ϕ(2 -k ξ))(hD x ) a 0 (hD x ) v a 1 (hD x ) v (t, •) L 2 h 1-β v(t, •) H s h + 2 |µ|=1 Op w h (χ(h σ ξ))L µ v(t, •) L 2 v(t, •) H 1,∞ h . Proof.
a 0 (hD x ) v(t, •) L ∞ + a 0 (hD x ) v Λ kg (t, •) L ∞ h -β v(t, •) H 1,∞ h ,
for a small β > 0, β → 0 as σ → 0, as follows from lemma 1.2.39 with p = +∞ and the uniform continuity of a 0 (hD x ) from H 1,∞ to L ∞ . Therefore, using the continuity on

L 2 of Op w h (ϕ(2 -k ξ))(hD x ) with norm O(2 k
) and the fact that 2 k h -σ we deduce that, for any

w 1 , w 2 ∈ { v, v Λ kg , v Λ c
kg } with at least one w j equal to v Λ c kg ,

Op w h (ϕ(2 -k ξ))(hD x ) a 0 (hD x )w 1 a 1 (hD x )w 2 L 2 h 1-β v(t, •) H s h + 2 |µ|=1 Op w h (χ(h σ ξ))L µ v(t, •) L 2 v(t, •) H 1,∞ h .
We are thus reduced to proving inequality (3.2.76) for

Op w h (ϕ(2 -k ξ))(hD x ) a 0 (hD x ) v Λ kg a 1 (hD x ) v Λ kg (t, •) L 2 .
Furthermore, by means of lemma 3.2.4 we can replace the action of a j (hD x ) in the above L 2 norm, for j = 0, 1, with the multiplication operator by a real function, up to new remainders bounded in L 2 by the right hand side of (3.2.76). In fact,

a j (hD x ) v Λ kg = θ h (x)a j (-dφ(x)) v Λ kg + R 1 ( v), j = 0, 1,
where θ h is a smooth cut-off function as in the statement of lemma 3.2.4 and R 1 ( v) satisfies (3.2.22a). Now

hD x | v Λ kg | 2 = Op w h (ξ + dφ(x)θ h (x)) v Λ kg v Λ kg -v Λ kg Op w h (ξ + dφ(x)θ h (x)) v Λ kg ,
and from lemma 3.2.16 below

Op w h (ξ + dφ(x)θ h (x)) v Λ kg (t, •) L 2 h 1-β 1 |µ|=0 Op w h (χ(h σ ξ))L µ v(t, •) L 2 .
This implies, after having applied the Leibniz rule and proposition 1.2.39, that

hD x a 0 (-dφ(x))a 1 (-dφ(x))θ 2 h (x)| v Λ kg | 2 (t, •) L 2 h 1-β v(t, •) H s h + 2 |µ|=1 Op w h (χ(h σ ξ))L µ v(t, •) L 2 v(t, •) L ∞
and the conclusion of the statement.

Lemma 3.2.16. Let γ, χ ∈ C ∞ 0 (R 2 ) be equal to 1 in a neighbourhood of the origin, σ > 0 small, (θ h (x)) h be a family of C ∞ 0 (B 1 (0)) functions, equal to 1 on the support of γ x-p (ξ)

√ h χ(h σ ξ), with ∂ α x θ h L ∞ = O(h -2|α|σ
) and (h∂ h ) k θ h bounded for every k. Let also φ(x) = 1 -|x| 2 . Then for every j = 1, 2

Op w h (ξ j + d j φ(x)θ h (x))Op w h γ x -p (ξ) √ h χ(h σ ξ) v(t, •) L 2 h 1-β 2 |µ|=0 Op w h (χ(h σ ξ))L µ v(t, •) L 2 ,
with β > 0 small, β → 0 as σ → 0.

Proof. By symbolic calculus of lemma 1.2.24 and the fact that θ h ≡ 1 on the support of γ x-p (ξ) √ h χ(h σ ξ), we have that, for any j = 1, 2,

Op w h (ξ j +d j φ(x)θ h (x))Op w h γ x -p (ξ) √ h χ(h σ ξ) v = Op w h γ x -p (ξ) √ h χ(h σ ξ)(ξ j + d j φ(x)) v + √ h 2i Op w h (∂ j γ) x -p (ξ) √ h χ(h σ ξ) v - √ h 2i 2 k,l=1 Op w h (∂ l γ) x -p (ξ) √ h p k,l (ξ)∂ k (d j φ(x)θ h (x))χ(h σ ξ) v + h 1+σ 2i 2 k=1 Op w h γ x -p (ξ) √ h ∂ k (d j φ(x)θ h (x))(∂ k χ)(h σ ξ) v + Op w h (r 2 (x, ξ)) v, (3.2.77) with r 2 ∈ h 1-4σ S 1 2 ,σ ( x-p (ξ) √ h -1
). On the one hand, as

Op w h γ x -p (ξ) √ h χ(h σ ξ)(ξ j -d j φ(x)) v = 2 k=1 Op w h γ x -p (ξ) √ h χ(h σ ξ) e j k (x, ξ)(x k -p k (ξ)) v,
with e j k satisfying (1.2.75b) on the support of γ x-p (ξ) √ h χ(h σ ξ), the L 2 norm of the first term in the right hand side of (3.2.77) can be estimated using (1.2.71a).

On the other hand, as ∂γ vanishes in a neighbourhood of the origin, the L 2 norm of the second and third term in the right hand side of (3.2.77) can be estimated using (3.2.17a).

The two remaining contributions to the right hand side of (3.2.77), that already carry the right power of h, can be estimated with h 1-β v(t, •) L 2 simply by proposition 1.2.36.

We can finally state the following result: Proposition 3.2.17 (Deduction of the transport equation). For any fixed T > 1, D > 0, let C T D := {(t, x) : 1 ≤ t ≤ T, |x| ≤ D} be the truncated cylinder, and assume that estimates (1.1.11) are satisfied in time interval [1, T ]. Then function

(3.2.78) u Σ j Λw (t, x) := k u Σ j ,k Λw (t, x)
is solution to the following transport equation:

(3.2.79) D t + 1 2 (1 -|x| 2 )x • (hD x ) + h 2i (1 -2|x| 2 ) u Σ Λw (t, x) = F w (t, x), ∀(t, x) ∈ C T D ,
and there exists some constant C > 0 such that

(3.2.80) F w (t, •) L ∞ ≤ CBεh 1-β
for some β > 0 small, β → 0 as σ, δ 1 → 0.

Proof. By the assumption in the statement, all that we are going to say is to be meant in time interval [1, T ]. We remind the reader that, by the definition of u

Σ j ,k
Λw in (3.2.45a) and of u Σ j ,k in (3.2.41), the sum defining u

Σ j
Λw is finite and restricted to indices k ∈ K := {k ∈ Z : h 2 k h -σ }. Also, we warn the reader that, throughout the proof, C and β will denote two positive constants that may change line after line, with β → 0 as σ → 0.

In lemma 3.2.11 we proved that function u

Σ j ,k
Λw is solution to (3.2.55) with f w k verifying (3.2.53). Hence, by lemma B.2.1 we derive that f w k is a remainder of the form F w satisfying (3.2.80). For seek of compactness, we denote symbol

Σ j (ξ)(1-χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ) in
the right hand side of (3.2.55) by φ j k (ξ). On the one hand, reminding (3.2.71) and using the L ∞ -L ∞ continuity of operator Γ w,k (see proposition 1.2.29), together with the classical Sobolev injection, the fact that (3.2.81)

Op w h (φ j k (ξ)) L(L 2 ) = O(h -µ ),
with µ = σρ if ρ ≥ 0, 0 otherwise, estimates (3.2.72),(3.2.73) and (3.2.3b), we find that

Γ w,k Op w h (Σ j (ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ)) [tq w (t, tx)] L ∞ h -β q w (t, •) L 2 + h -1-β Op w h (ϕ(2 -k ξ))(hD x ) q w (t, •) L 2 ≤ CBεh 1-β . (3.2.82)
On the other hand, using proposition 1.2.30, estimate (3.2.81), the fact that the commutator between Op w h (φ j k (ξ)) and Ω h is also continuous on

L 2 with norm O(h -µ ), equality tw(t, t•) L 2 = w(t, •) L 2 , and (3.1.21a), (3.1.21c) (in which we choose s > 0 large enough to have, say, N (s) ≥ 2), we deduce that there is a χ ∈ C ∞ 0 (R 2 ) such that Γ w,k Op w h φ j k (ξ) (h -1 c w (t, tx)) L ∞ (dx) t 1 2 +β χ(t -σ D x )(v N F -v -)(t, •) L 2 V (t, •) H 2,∞ + v N F (t, •) H 1,∞ + t -3 2 +β (v N F -v -)(t, •) H 1 V (t, •) H s + v N F (t, •) H s + t 1 2 +β χ(t -σ D x )Ω(v N F -v -)(t, •) L 2 V (t, •) H 2,∞ + v N F (t, •) H 1,∞ + t -3 2 +β Ω(v N F -v -)(t, •) L 2 V (t, •) H s + v N F (t, •) H s + t 1 2 +β (v N F -v -)(t, •) H 1,∞ × 1 µ=0 Ω µ V (t, •) H 1 + Ω µ v N F (t, •) L 2 .
(3.2.83) Also, from (3.1.22a), (3.1.22c) we get that for every θ ∈]0, 1[

Γ w,k Op w h φ j k (ξ) (h -1 r N F w (t, tx)) L ∞ t 1 2 +β V (t, •) 2 H 13,∞ U (t, •) H 1 + t 1 2 +β V (t, •) 1-θ H 15,∞ V (t, •) θ H 17 ( U (t, •) H 1,∞ + R 1 U (t, •) H 1,∞ ) + V (t, •) L ∞ U (t, •) 1-θ H 16,∞ + R 1 U (t, •) 1-θ H 16,∞ U (t, •) θ H 18 ΩV (t, •) L 2 + t 1 2 +β V (t, •) H 1,∞ ( U (t, •) H 1 + ΩU (t, •) H 1 ) + ( U (t, •) H 2,∞ + R 1 U (t, •) H 2,∞ ) ΩV (t, •) L 2 V (t, •) H 17,∞ . (3.2.84)
Therefore, using (3.1.7) with s = 1, (3.1.8a), (1.1.11), and choosing θ 1 sufficiently small, we derive that h

-1 Γ w,k Op w h φ j k (ξ) (c w (t, tx)+r N F w (t, tx)) is a remainder F w (t, x) satisfying (3.2.80). Since function (∂χ 0 )(h -1 ξ) is localized for frequencies of size h, its product with ψ(2 -k ξ) is non-zero only for values of k ∈ Z such that 2 k ∼ h. In that case, by commutating Γ w,k with Op w h (∂χ 0 )(h -1 ξ)•(h -1 ξ)ψ(2 -k ξ)
and using the classical Sobolev injection, together with proposition 1.2.27, we find that

(3.2.85) ih Γ w,k Op w h (∂χ 0 )(h -1 ξ) • (h -1 ξ)ψ(2 -k ξ) u(t, •) L ∞ h u(t, •) L 2 .
Since (∂χ 0 )(h σ ξ) is, instead, localized for frequencies larger than h -σ , by applying the semiclassical Sobolev injection and lemma 1.2.40 we find that

(3.2.86) iσh Γ w,k Op w h ψ(2 -k ξ)(∂χ 0 )(h σ ξ)) • (h σ ξ) u(t, •) L ∞ h N u(t, •) H s h ,
with N = N (s) > 1 as long as s > 0 is sufficiently large. By lemma B.2.1 we obtain that also the fifth and sixth addend in the right hand side of (3.2.55) are remainders F w (t, x).

Finally, after lemma 3.2.12

-Op w h θ(x)(x • ξ -|ξ|) ϕ(2 -k ξ) u Σ j ,k Λw = 1 2 Op w h θ(x)(1 -|x| 2 )x • ξ ϕ(2 -k ξ) u Σ j ,k Λw + Op w h (θ(x)e(x, ξ) ϕ(2 -k ξ)) u Σ j ,k Λw
with e(x, ξ) given by (3.2.57), and latter term in the above right hand side satisfies (3.2.62). Using symbolic calculus of proposition 1.2.21 until order N ∈ N we find that

1 2 Op w h θ(x)(1 -|x| 2 )x • ξ ϕ(2 -k ξ) u Σj ,k Λw = θ(x) 1 2 (1 -|x| 2 )x • (hD x ) + h 2i (1 -2|x| 2 ) Op w h ( ϕ(2 -k ξ)) u Σj ,k Λw + h 4i (∂θ)(x) • x(1 -|x| 2 )Op w h ( ϕ(2 -k ξ)) u Σj ,k Λw + hθ 1 (x)Op w h ( ϕ 1 (2 -k ξ)) u Σj ,k Λw + Op w h (r( N x, ξ)) u Σj ,k Λw ,
with being a concise notation to indicate a linear combination, ∂θ(x)

supported for |x| > D 1 , θ 1 ∈ C ∞ 0 (R 2 ), ϕ 1 ∈ C ∞ 0 (R 2 \ {0}
) coming out from the derivatives of ϕ, and r N (x, ξ) integral remainder of the form

h N (πh) 2 e 2i h η•z 1 0 θ N (x + tz)(1 -t) N -1 dt ϕ N (2 -k (ξ + η))dzdη, for some other θ N ∈ C ∞ 0 (R 2 ), ϕ N ∈ C ∞ 0 (R 2 \ {0}), verifying that (3.2.87) Op w h (r N (x, ξ)) L(L 2 ;L ∞ ) = O(h)
if N is taken sufficiently large. Therefore, from proposition 1.2.27, (3.2.81) and (B.2.1a)

Op w h (r(x, ξ)) u Σ j ,k Λw (t, •) L ∞ h 1-β u(t, •) L 2 ≤ CBεh 1-β .
Moreover, since ϕ ≡ 1 on the support of ϕ (which defines u Σ j ,k ), by commutating Op w h ( ϕ(2 -k ξ)) with Γ w,k and using remark 1.2.22 we find that, for any N ∈ N as large as we want,

Op w h ( ϕ(2 -k ξ)) u Σ j ,k Λw = u Σ j ,k Λw + O L ∞ (h N u L 2 )
. Also, since ϕ 1 is obtained from the derivatives of ϕ and vanishes on the support of ϕ,

θ 1 (x)Op w h ( ϕ 1 (2 -k ξ)) u Σ j ,k Λw = O L ∞ (h N u L 2 )
. Therefore, again from (B.2.1a) we deduce that

-Op w h θ(x)(x • ξ -|ξ|) ϕ(2 -k ξ) u Σ j ,k Λw = θ(x) 1 2 (1 -|x| 2 )x • (hD x ) + h 2i (1 -2|x| 2 ) u Σ j ,k Λw + h 4i (∂θ)(x) • x(1 -|x| 2 ) u Σ j ,k Λw + Op w h θ(x) ϕ(2 -k ξ)e(x, ξ) u Σ j ,k Λw + O L ∞ (h 1-β ),
which implies, summed up with estimates from (3.2.82) to (3.2.86), that u

Σ j ,k
Λw is solution to

D t + θ(x) 1 2 (1 -|x| 2 )x • (hD x ) + θ(x) h 2i (1 -2|x| 2 ) u Σ j ,k Λw (t, x) = F k w (t, x) + (1-θ)(x)Op w h ((x•ξ -|ξ|) ϕ(2 -k ξ))+ θ(x)Op w h ( ϕ 1 (2 -k ξ))- h 4i (∂θ)(x)•x(1-|x| 2 ) u Σ j ,k Λw (t, x) ,
where F k w (t, x) satisfies (3.2.80). Choosing D 1 = D, we obtain that u

Σ j Λw is solution to (3.2.79) in cylinder C T D , with F w (t, x) := k F k w (t,
x) (this sum being finite and restricted to indices k ∈ K) satisfying the same L ∞ estimate as F k w , up to an additional factor h -σ .

Analysis of the transport equation and end of the proof

In previous section (see proposition 3.2.7) we firstly showed how to propagate a-priori uniform estimate (1.1.11b) on the Klein-Gordon component v -, in the sense of deducing (1.1.12b) from estimates (1.1.11). We then passed to the study of the wave equation and proved that, if

(u -, v -) is solution to (3.1.1) in some interval [1, T ], function u Σ j
Λw defined in (3.2.78) is solution to transport equation (3.2.79) in truncated cylinder C T D := {(t, x) : 1 ≤ t ≤ T, |x| ≤ D}, for any D > 0. The aim of this section is to study such a transport equation in order to deduce some information on the uniform norm of its solutions. This will allow us to finally propagate a-priori estimate (1.1.11a) on the wave component u -and to close the bootstrap argument. A short proof of main theorem 1.1.1 is given at the end of this section.

The inhomogeneous transport equation

The aim of this subsection is to study the behaviour of a solution w to the following transport equation

(3.3.1) D t + 1 2 (1 -|x| 2 )x • (hD x ) - i 2t (1 -2|x| 2 ) w = f , in a cylinder C = {(t, x) : t ≥ 1, |x| ≤ D} for a large constant D 1, where the inhomogeneous term f is a O L ∞ (εt -1+β
), for some ε > 0 small and 0 ≤ β < 1/2. We distinguish in C two subregions: and denote by I 1,t , I 2,t their sections at a fixed time t ≥ 1,

I 1 := (t, x) : t ≥ 1, |x| < t t -1 1 2 , |x| ≤ D , I 2 := (t, x) : t > 1, t t -1 1 2 ≤ |x| ≤ D ,
I 1,t := x : |x| < t t -1 1 2 , |x| ≤ D , I 2,t := x : t t -1 1 2 ≤ |x| ≤ D .
The result we prove is the following. Proposition 3.3.1. Let ε > 0 be small and w be the solution to the following Cauchy problem

(3.3.2) D t + 1 2 (1 -|x| 2 )x • (hD x ) -i 2t (1 -2|x| 2 ) w = f , w(1, x) = εw 0 (x) , with f = O L ∞ (εt -1+β ), for some fixed 0 ≤ β < 1/2. Let us suppose that |w 0 (x)|
x -2 and that |w(t, x)| εt β for some β > 0 whenever |x| > D 1. Therefore,

(3.3.3) |w(t, x)| ε w 0 L ∞ t β (1 + |x|) -1 2 (t -1 + |1 -|x||) -1 2 +β , for every (t, x) ∈ C D = {(t, x)|t ≥ 1, |x| ≤ D}, with β = max{β, β }.
We observe that, if W (t, x) = t -1 w(t, t -1 x), the above inequality implies that

|W (t, x)| ε w 0 L ∞ (t + |x|) -1 2 (1 + |t -|x||) -1 2 +β ,
showing that the uniform norm of W (t, •) decays in time at a rate t -1/2 , enhanced to t -1+β out of the light cone t = |x|.

In order to prove the result of proposition 3.3.1 we fix T ≥ 1, x ∈ B D (0), and look for the characteristic curve of (3.3.2) with initial point (T, x), i.e. map t → X(t; T, x) solution of

(3.3.4) d dt X(t; T, x) = 1 2t 1 -|X(t; T, x)| 2 X(t; T, x) X(T ; T, x) = x t ≥ T.
Lemma 3.3.2. Solution X(t; T, x) to (3.3.4) writes explicitly as

(3.3.5) X(t; T, x) = √ tx (T -(T -t)|x| 2 ) 1 2
and it is well defined for all t > T (1 -|x| -2 ). Moreover, for any fixed

t > T , map x ∈ R 2 → X(t; T, x) ∈ |x| < t t-T 1 2 is a diffeomorphism of inverse Y (t, y) = √ T y (t+(T -t)|y| 2 ) 1 2
.

Proof. Multiplying equation (3.3.4) by 2X(t; T, x) we deduce that |X(t; T, x)| 2 satisfies the equation

d dt |X(t; T, x)| 2 = 1 t 1 -|X(t; T, x)| 2 |X(t; T, x)| 2 , from which follows that 1 -|X(t; T, x)| 2 = T (1-|x| 2 ) T -(T -t)|x| 2 .
Injecting this result in (3.3.4) and integrating in time, we obtain expression (3.3.5) and observe that the obtained map is well defined for all t > T (1 -|x| -2 ).

In order to prove the second part of the statement, we fix t > T , y ∈ |x| ≤ t t-T

1 2
and look for Y (t, y) such that X(t; T, Y (t, y)) = y. In other words,

y = √ tY (t, y) (T -(T -t)|Y (t, y)| 2 ) 1 2
,

which implies that Y (t, y) = √ T y (t+(T -t)|y| 2 ) 1 2
. This map is well defined as long as |y|

< t t-T 1 2 .
Along the characteristic curve X(t; T, x) function w satisfies

d dt w t, X(t; T, x) = - 1 2t 1 -2|X(t; T, x)| 2 w t, X(t; T, x) + if t, X(t; T, x) = - 1 2t T -T |x| 2 -t|x| 2 T -(T -t)|x| 2 w t, X(t; T, x) + if t, X (t; T, x) and hence (3.3
.6) d dt exp t T 1 2τ T -T |x| 2 -τ |x| 2 T -(T -τ )|x| 2 dτ w t, X(t; T, x) = i exp t T 1 2τ T -T |x| 2 -τ |x| 2 T -(T -τ )|x| 2 dτ f t, X(t; T, x) . Lemma 3.3.3. (3.3.7) exp t T 1 2τ T -T |x| 2 -τ |x| 2 T -(T -τ )|x| 2 dτ = t T 1 2 T -T |x| 2 + t|x| 2 T -1
.

Proof. The result follows writing

1 2τ T -T |x| 2 -τ |x| 2 T -(T -τ )|x| 2 = 1 2τ - |x| 2 T -T |x| 2 + τ |x| 2 ,
taking the integral over τ ∈ [T, t] and then passing to its exponential.

Let us first study the behaviour of w, solution to (3.3.2), in region I 1 . We fix T = 1 and, integrating equality (3.3.6) over [1, t], we find that (3.3.8) exp

t 1 1 2τ 1 -|x| 2 -τ |x| 2 1 -(1 -τ )|x| 2 dτ w t, X(t; 1, x) = w(1, x) + i t 1 exp s 1 1 2τ 1 -|x| 2 -s|x| 2 1 -(1 -s)|x| 2 ds f s, X(s; 1, x) ds.
Using (3.3.7) and the fact that f = O L ∞ (εt -1+β ), we then obtain that

(3.3.9) w(t, X(t; 1, x)) ≤ t -1 2 (1 -|x| 2 + t|x| 2 )|w(1, x)| + Cεt -1 2 (1 -|x| 2 + t|x| 2 ) t 1 ds (1 -|x| 2 + s|x| 2 )s 1 2 -β
, for some positive constant C.

Lemma 3.3.4. For any fixed 0 ≤ β < 1/2

(3.3.10) t 1 ds (1 -|x| 2 + s|x| 2 )s 1 2 -β t 1 2 +β (1 + √ t|x|) 1+2β (1 + |x|) -1+2β+β ,
for all t ≥ 1 and β > 0 as small as we want.

Proof. For √ t|x| ≤ 1, we have that

t 1 ds (1 -|x| 2 + s|x| 2 )s 1 2 -β t 1 2 +β t 1 2 +β (1 + √ t|x|) 1+2β (1 + |x|) -1+2β+β , for any β ≥ 0. Suppose then that √ t|x| > 1. For t ≤ 2 t 1 ds (1 -|x| 2 + s|x| 2 )s 1 2 -β (1 + |x|) -2 log(1 + |x| 2 ) and |x| -2 log(1 + |x| 2 ) (1 + √ t|x|) 1+2β t 1 2 +β (1 + |x|) -1+2β log(1 + |x| 2 ),
which immediately implies inequality (3.3.10). For t ≥ 2

t 1 ds (1 -|x| 2 + s|x| 2 )s 1 2 -β = 2 1 ds (1 -|x| 2 + s|x| 2 )s 1 2 -β + t 2 ds (1 -|x| 2 + s|x| 2 )s 1 2 -β ,
where the first integral is bounded from the right hand side of (3.3.10). The second one is less or equal than

t-1 1 ds (1+s|x| 2 )s 1 2 -β , so for |x| ≥ 1 it follows that t-1 1 ds (1 + s|x| 2 )s 1 2 -β ≤ |x| -2 t-1 1 ds s 3 2 -β (1 + |x|) -2 . Since (1+ √ t|x|) 1+2β t 1 2 +β ≤ (1 + |x|) 1+2β
, from the above inequality we deduce the right bound of the statement. For |x| < 1, a change of variables gives that t-1

1 ds (1 + s|x| 2 )s 1 2 -β = |x| -1-2β (t-1)|x| 2 |x| 2 ds (1 + s)s 1 2 -β |x| -1-2β (t|x| 2 ) 1 2 +β (1 + t|x| 2 ) 1 2 +β ≤ t 1 2 +β (1 + t|x| 2 ) 1 2 +β .
If initial condition w 0 (x) is sufficiently decaying in space, e.g. |w 0 (x)|

x -2 , we deduce from inequalities (3.3.9) and (3.3.10) the following bound for w along the characteristic curve X(t; 1, x):

(3.3.11) w(t, X(t; 1, x)) ε w 0 L ∞ t β (1 + √ t|x|) 1-2β (1 + |x|) -1+2β+β ,
for any β > 0 as small as we want.

x t 1

• (T 1 , x 1 )
•(T2, x 2 )

1 x = ( t t-1 ) 1 2 Figure 3.4: Characteristic curves of initial point (T i , x i ) ∈ I 1 , i = 1, 2, in space dimension 1
Proposition 3.3.5. Let w be the solution to transport equation

(3.3.2), with f (t, •) L ∞ εt -1+β for some fixed 0 ≤ β < 1/2, and initial condition |w 0 (x)| x -2 , ∀x ∈ R 2 . Then (3.3.12) |w(t, x)| εt β t -1 + |1 -|x|| -1 2 +β for every (t, x) ∈ I 1 = {(t, x) : t ≥ 1, |x| < t t-1 1 2 , |x| ≤ D}.
Proof. In lemma 3.3.2 we proved that, for any fixed

t > T = 1, map x ∈ R 2 → X(t; 1, x) ∈ x : |x| < ( t t-1 ) 1 2 is a diffeomorphism with inverse Y (t, y) = y(t + (1 -t)|y| 2 ) -1/2 .
From inequality (3.3.11) we hence deduce that, for any y such that |y|

< t t-1 1 2 , |w(t, y)| εt β 1 + √ t|Y (t, y)| 1-2β 1 + |Y (t, y)| -1+2β+β .
In particular, as t(1 1-2β

-|y| 2 ) + |y| 2 ) ∼ t|1 -|y| 2 | + |y| 2 when |y| < t t-1
εt β t -1 + |1 -|y|| -1 2 +β , simply using that (1 + |Y (t, y)|) -1+2β+β ≤ 1. Moreover, for t → 1 and |y| ≤ D, |w(t, y)| ε εt β t -1 + |1 -|y|| -1 2 +β .
Proposition 3.3.6. Let ε > 0 be small and w be the solution to transport equation

(3.3.2), with f (t, •) L ∞ εt -1+β
for some fixed 0 ≤ β < 1/2, and suppose that |w(t, x)| εt β for some β > 0 whenever |x| ≥ D. Then

|w(t, x)| εt β (|x| 2 -1) β -1 2 , for every (t, x) ∈ I 2 = {(t, x) : t > 1, t t-1 1 2 ≤ |x| ≤ D}, where β = max{β, β }. 1 • (T, x) (T * , x * ) 1 D x t Figure 3.5: Characteristic curve of initial point (T, x) ∈ I 2
Proof. For a fixed (T, x) ∈ I 2 we look at X(t; T, x), solution to (3.3.4) and given by the explicit expression (3.3.5). We observe that there exists a time T * , 1 < T * < T , such that X(t; T, x) hits the boundary |y| = D at t = T * . In other words, t = T * is the first time when X(t; T, x) enters in the region {(t, x) : t ≥ 1, |x| ≤ D}, to never leave it again for function t → |X(t; T, x)| is strictly decreasing. A simple computation shows that

(3.3.13) T * = D 2 D 2 -1 (1 -|x| -2 )T < T .
Integrating expression (3.3.6) over [T * , T ] and using (3.3.7), we find that

(3.3.14) w(T, x) = T * T 1 2 T -T (1 -|x| -2 ) T * -T (1 -|x| -2
) w(T * , X(T * ; T, x))

+ i T T * t T 1 2 T -T (1 -|x| -2 ) t -T (1 -|x| -2 ) f t, X(t; T, x) dt .
From (3.3.13)

T * -T (1 -|x| -2 ) = 1 D 2 -1 (1 -|x| -2 )T and T * T = D 2 D 2 -1 (1 -|x| -2 )
so since |w(t, x)| εt β whenever |x| ≥ D, for some β > 0 by the hypothesis, we find that the first term in right hand side of (3.3.14) is bounded by

Cε(|x| 2 -1) -1 2 (T * ) β , for some constant C > 0. Setting c = 1 D 2 -1
, by the hypothesis on f we derive that

T T * t T 1 2 T -T (1 -|x| -2 ) t -T (1 -|x| -2 ) f t, X(t; T, x) dt εT 1 2 T T * t -T (1 -|x| -2 ) -1 t -1 2 +β dt = εT 1 2 T T * t -T * + c(1 -|x| -2 )T -1 t -1 2 +β dt ≤ εT 1 2 T -T * 0 dt t + c(1 -|x| -2 )T t 1 2 -β εT 1 2 (1 -|x| -2 )T β-1 2 = εT β (1 -|x| -2 ) β-1 2 .

Propagation of the uniform estimate on the wave component

Proposition 3.3.7 (Propagation of the a-priori estimate on U, RU ). Let us fix K 1 > 0. There exist two integers n ρ 1 sufficiently large" two constants A, B > 1 sufficiently large, some small 0 < δ δ 2 δ 1 δ 0 , and ε 0 ∈]0, 1[ sufficiently small, such that, for any

0 < ε < ε 0 , if (u, v) is solution to (1.1.1)-(1.1.2) in some interval [1, T ],
for a fixed T > 1, and u ± , v ± defined in (1.1.5) satisfy a-priori estimates (1.1.11), for every t ∈ [1, T ], then it also verify (1.1.12a) in the same interval [1, T ].

Proof. We warn the reader that, throughout the proof, C, β, β will denote some positive constants that may change line after line, such that β → 0 as σ → 0 (resp. β → 0 as δ 1 , σ → 0). We also remind that h = 1/t.

In proposition 3.1.2 we introduced function u N F , defined from u -through (3.1.15), and showed that its H ρ+1,∞ norm (resp. the H ρ+1,∞ norm of Ru N F ) differs from that of u -(resp. of Ru -) by a quantity satisfying (3.1.20b) (resp. (3.1.20c)). If n is sufficiently large with respect to ρ (at least n ≥ ρ + 18), a-priori estimates (1.1.11b), (1.1.11c) give that, for every t ∈ [1, T ],

(3.3.15) u -(t, •) H ρ+1,∞ + 2 j=1 R j u -(t, •) H ρ+1,∞ ≤ u N F (t, •) H ρ+1,∞ + 2 j=1 R j u N F (t, •) H ρ,∞ + 2ABε 2 t -1+ δ 2 .
We successively considered u(t, x) := t u N F (t, tx) and decomposed it as in (3.2.38), with Σ j given by (3.2.37), showing that it satisfies (3.2.39) (resp. (3.2.40)) when restricted to small frequencies |ξ| t -1 (resp. large frequencies |ξ| t σ ). We then focused on u Σ j ,k defined in (3.2.41), which is localized for frequencies supported in an annulus of size

2 k with k ∈ K = {k ∈ Z : h 2 k h -σ },
and further split it into the sum of functions u

Σ j ,k Λw , u Σ j ,k Λ c w (see (3.2.45)
). On the one hand, from inequality (3.2.46b) and lemma B.2.1 we have that there is a positive constant C such that, for every

t ∈ [1, T ], u Σ j ,k Λ c w (t, •) L ∞ ≤ Cεt β .
On the other hand, we proved in proposition 3.2.17 that, for any D > 0 and any (t, x) in truncated cylinder

C T D = {(t, x) : 1 ≤ t ≤ T, |x| ≤ D}, u Σ j Λw (t, x) defined in (3.2.78
) is solution to inhomogeneous transport equation (3.2.79), with inhomogeneous term F w (t, x) satisfying (3.2.80). Observe that, by definition (1.2.49) of M, symbolic calculus, and proposition 1.2.36, we have that

u Σ j Λw (1, •) L 2 + x u Σ j Λw (1, •) L 2 u(1, •) L 2 + Op w h (χ(h σ ξ))M u(1, •) L 2 ≤ Cε, which means that ε -1 x u Σ j Λw (1, x) ∈ L 2 . That hence implies that | u Σ j Λw (1, x)| ε x -2 for every x ∈ R 2 (if not, we would have • -1 L 2 ≤ ε -1 • u Σ j Λw (1, •) L 2 )
. Moreover, if D 1 is sufficiently large, from lemma 3.3.9 below and B.2.1 in appendix B we deduce that

(3.3.16) |1 |x|≥D u Σ j Λw (t, x)| ≤ C log |x| |x| h -β Op w h (χ(h σ ξ)) u(t, •) L 2 + Op w h (χ(h σ ξ))M u(t, •) L 2 ≤ Cε log |x| |x| t β .
Therefore, from proposition 3.3.1 we obtain that

| u Σ j Λw (t, x)| Cεt β (1 + |x|) -1 2 t -1 + |1 -|x|| -1 2 +β , ∀(t, x) ∈ C T D .

Summing up, denoting by

1 C T D the characteristic function of cylinder C T D , | u Σ (t, x)| ≤ Cε1 C T D t β (1 + |x|) -1 2 t -1 + |1 -|x|| -1 2 +β + Cεt β , ∀(t, x) ∈ [1, T ] × R 2 .
Returning back to function u N F via (3.2.2), this means that, for every

(t, x) ∈ [1, T ] × R 2 , (3.3.17) D x ρ u N F (t, x) + 2 j=1 D x ρ R j u N F (t, x) ≤ Cε1 {|x|≤Dt} (t + |x|) -1 2 (1 + |t -|x||) -1 2 +β + Cεt -1+β .
Finally, reminding definition 1.2.1 (iii) of space H ρ,∞ , injecting the above inequality in (3.3.15), and choosing A > 1 sufficiently large such that C < A 3K 1 , ε 0 > 0 sufficiently small so that CBε 0 < (3K 1 ) -1 , we deduce enhanced estimate (1.1.12a).

Remark 3.3.8. Beside the propagation of estimate (1.1.11a), by combining inequalities (3.3.15), (3.3.17), and (1.1.5), we also deduce the following inequality

|∂ t u(t, x)| + |∇ x u(t, x)| ≤ Cε1 {|x|≤Dt} (t + |x|) -1 2 (1 + |t -|x||) -1 2 +β + Cεt -1+β ,
with β > 0 small as long as σ, δ 1 are small, which almost corresponds to the optimal decay in time and space enjoyed by the linear wave in space dimension two.

Lemma 3.3.9. Let χ ∈ C ∞ 0 (R 2 ) be equal to 1 in a neighbourhood of the origin and σ > 0 be small. Let also ϕ ∈ C ∞ 0 (R 2 \ {0}). There exists a constant C > 0 such that for every h ∈]0, 1[, R 1, and any function w(t, x) with w(t, •),

Op w h (χ(h σ ξ))Mw(t, •) ∈ L 2 (R 2 ), (3.3.18) ϕ • R Op w h (χ(h σ ξ))w(t, •) L ∞ ≤ CR -1 (log R + | log h|) 1 |γ|=0 Op w h (χ(h σ ξ))M γ w(t, •) L 2 .
Proof. Let us fix R 1 and, for seek of compactness, denote Op w h (χ(h σ ξ))w by w χ . For a new smooth cut-off function χ 1 equal to 1 on the support of χ, we have that

ϕ x R Op w h (χ(h σ ξ))w = Op w h (χ 1 (h σ ξ)) ϕ x R w χ + ϕ x R , Op w h (χ 1 (h σ ξ)) w χ ,
where the symbol associated to above commutator is given by

r R (x, ξ) = - h 1+σ R -1 i(πh) 2 e 2i h η•z 1 0 (∂ϕ) x + tz R dt (∂χ 1 )(h σ (ξ + η))dzdη,
as follows from (1.2.19) and integration in dy, dζ. Since (∂χ 1 )(h σ ξ) is supported for frequencies |ξ| ≤ h -σ , and R -1 , h 1+σ ≤ 1, by making a change of coordinates η/h → η and using that

e 2iη•z = 1-2iη•∂z 1+4|η| 2 1-2iz•∂η
1+4|z| 2 e 2iη•z , together with some integration by parts, one can check that

∂ α y ∂ β ξ r R ( x + y 2 , hξ) L 2 (dξ) R -1
for any α, β ∈ N 2 , and hence obtain from lemma 1.2.25 that

Op w h (r k R (x, ξ))w χ (t, •) L ∞ R -1 w χ (t, •) L 2 .
Successively, taking a Littlewood-Paley decomposition such that

χ 1 (h σ ξ) ≡   φ R h ξ + hR -1 ≤2 j ≤h -σ (1 -φ) R h ξ ψ(2 -j ξ)   χ 1 (h σ ξ), with φ ∈ C ∞ 0 (R 2 )
, equal to 1 close to the origin and ψ ∈ C ∞ 0 (R 2 \ {0}), we derive that

(3.3.19) Op w h (χ 1 (h σ ξ)) ϕ x R w χ (t, •) L ∞ Op w h φ R h ξ χ 1 (h σ ξ) ϕ x R w χ (t, •) L ∞ + hR -1 ≤2 j ≤h -σ Op w h (1 -φ) R h ξ ψ(2 -j ξ)χ 1 (h σ ξ) ϕ x R w χ (t, •) L ∞
, and immediately notice that

(3.3.20) Op w h φ R h ξ χ 1 (h σ ξ) ϕ x R w χ (t, •) L ∞ = φ(RD x )Op w h (χ 1 (h σ ξ)) ϕ x R w χ (t, •) L ∞ R -1 w χ (t, •) L 2 ,
just by the classical Sobolev injection and the uniform continuity of Op w h (χ

1 (h σ ξ))ϕ x R on L 2 . Introducing operators Θ R , Θ -1 R , where Θ R u(x) := u(Rx), Θ -1 R u(x) := u x R , we have the following equality (3.3.21) Op w h (1 -φ) R h ξ ψ(2 -j ξ)χ 1 (h σ ξ) ϕ x R w χ = Θ -1 R Op w h Rj (1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ) ϕ(x)Θ R w χ
with h Rj := h R2 j ≤ 1, and by h Rj -symbolic calculus (that is proposition 1.2.21 with h replaced by h Rj ),

Op w h Rj (1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ) ϕ(x) = Op w h Rj (1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ)ϕ(x) + Op w h Rj (r(x, ξ)) with r(x, ξ) = h Rj 2i(πh Rj ) 2 e -2i h Rj y•ζ 1 0 ∂ ξ (1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ) (ξ+tζ) dt (∂ϕ)(x + y)dydζ.
Similarly as before, one can prove that

∂ α x ∂ β ξ r( x + y 2 , hξ) L 2 (dξ) 1
for any α, β ∈ N 2 , observing that no negative power of h Rj appears in the right hand side of this inequality for the product of ψ(ξ) with any derivative of (1 -φ)( ξ h Rj ) is supported for

h Rj ∼ |ξ| ∼ 1. Hence lemma 1.2.25 gives that operator Op w h Rj (r(x, ξ)) is uniformly bounded from L 2 to L ∞ and Op w h Rj (r(x, ξ))Θ R w χ (t, •) L ∞ Θ R w χ (t, •) L 2 R -1 w χ (t, •) L 2 . Since symbol (1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ)ϕ(x) is supported for |x| ∼ |ξ| ∼ 1, (1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ)ϕ(x) = 2 l=1 (1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ)ϕ(x)(Rx l |2 j ξ| -2 j ξ l ) |Rx|2 j ξ| -2 j ξ| 2 a l (x,ξ) Rx l |2 j ξ| -2 j ξ l , with a l (x, ξ) ∈ R -1 2 -j S 0,0 (1
) as long as R 1, and by h Rj -symbolic calculus

(1 -φ) ξ h Rj ψ(ξ)χ 1 (h σ 2 j ξ)ϕ(x) = 2 l=1 a l (x, ξ) (Rx l |2 j ξ| -2 j ξ l ) ψ(ξ) + r Rj (x, ξ), with ψ ∈ C ∞ 0 (R 2 \ {0}) such that ψψ ≡ ψ, and r Rj ∈ h Rj S 0,0 (1). From semi-classical Sobolev injection Op w h Rj (r Rj (x, ξ))Θ R w χ (t, •) L ∞ Θ R w χ (t, •) L 2 ≤ R -1 w χ (t, •) L 2 while Op w h Rj (a l (x, ξ))Op w h Rj (Rx l |2 j ξ| -2 j ξ) ψ(ξ) Θ R w χ = Op w h Rj (a l (x, ξ))Θ R Op w h (x l |ξ| -ξ) ψ(2 -j ξ) w χ = Op w h Rj (a l (x, ξ))Θ R Op w h ( ψ(2 -j ξ))Op w h (x l |ξ| -ξ)w χ - h 2i Op w h ((2 -j ξ) • (∂ ψ)(2 -j ξ))w χ . (3.3.22)
The last thing to do to conclude the proof of the statement is to study continuity of operator Op w h Rj (a l (x, ξ)).

Lemma 3.3.10. We have that Op w h Rj (a l (x, ξ))

: L 2 → L ∞ is bounded with norm Op w h Rj (a l (x, ξ)) L(L 2 ;L ∞ ) h -1 .
Proof. The result comes straightly from lemma 1.2.25. Indeed, since symbol a l (x, ξ) is compactly supported in x there is a smooth cut-off function

ϕ 1 ∈ C ∞ 0 (R 2 \ {0}), with ϕ 1 ϕ ≡ ϕ, such that Op w h Rj (a l (x, ξ))w w L 2 (dx) ϕ 1 x + y 2 |α|≤3 ∂ α y a l x + y 2 , h R j ξ L 2 (dξ)
dy, and for |α| ≤ 3

∂ α y a l x + y 2 , h R j ξ L 2 (dξ) R h ∂ α y (1 -φ)(ξ)ψ(h Rj ξ)χ 1 (h Rj h σ 2 j ξ)ϕ 1 ( x+y 2 ) |R( x+y 2 )|ξ| -ξ| 2 R x l + y l 2 |ξ| -ξ l L 2 (dξ) | ϕ( x+y 2 )| h |ψ(h Rj ξ)| 2 |ξ| 2 dξ 1 2 | ϕ( x+y 2 )| h , where ϕ ∈ C ∞ 0 (R 2 \ {0}).
Finally, summing up all formulas from (3.3.21) to (3.3.22) and using lemma 3.3.10, we obtain that

Op w h (1 -φ) R h ξ ψ(2 -j ξ)χ 1 (h σ ξ) ϕ x R w χ (t, •) L ∞ R -1 ( w χ (t, •) L 2 + Mw χ (t, •) L 2 ),
for any index j ∈ Z such that hR -1 ≤ 2 j ≤ h -σ . Injecting (3.3.20) and the above inequality in (3.3.19), and using that [M, Op w h (χ(h σ ξ))] = iOp w h ((∂χ)(h σ ξ)(h σ |ξ|)) is uniformly continuous on L 2 , we deduce (3.3.18) (the loss in log R + | log h| arising from the fact that we are considering a sum over indices j, with log h -log R j log(h -1 )).

Proof of the main theorems

Proof of theorem 1.1.2. Straightforward after propositions 2.2.13, 3.2.7, 3.3.7.

Proof of theorem 1.1.1. Let us prove that, for small enough data satisfying (1.1.4), Cauchy problem (1.1.1)-(1.1.2) has a unique global solution. This result follows by a local existence argument, after having proved that there exist two integers n ρ 1, two constants A , B > 1 sufficiently large, ε 0 > 0 sufficiently small, and 0 < δ δ 2 δ 1 δ 0 small, such that, for any

0 < ε < ε 0 , if (u, v) is solution to (1.1.1)-(1.1.2) in [1, T ] × R 2 , for some T > 1, with ∂ t,x u ∈ C 0 ([1, T ]; H n (R 2 )), v ∈ C 0 ([1, T ]; H n+1 (R 2 )) ∩ C 1 ([1, T ]; H n (R 2
)), and satisfies

∂ t u(t, •) H ρ+1,∞ + ∇ x u(t, •) H ρ+1,∞ + |D x |u(t, •) H ρ+1,∞ + 2 j=1 R j ∂ t u(t, •) H ρ+1,∞ ≤ A εt -1 2 , (3.3.23a)
∂ t v(t, •) H ρ,∞ + v(t, •) H ρ+1,∞ ≤ A εt -1 , (3.3.23b) ∂ t u(t, •) H n + ∇ x u(t, •) H n + ∂ t v(t, •) H n + ∇ x v(t, •) H n + v(t, •) H n ≤ B εt δ 2 , (3.3.23c) (3.3.23d) |I|=k ∂ t Γ I u(t, •) L 2 + ∇ x Γ I u(t, •) L 2 + ∂ t Γ I v(t, •) L 2 + ∇ x Γ I v(t, •) L 2 + Γ I v(t, •) L 2 ≤ B εt δ 3-k 2 , 1 ≤ k ≤ 3, for every t ∈ [1, T ], then in the same interval it satisfies ∂ t u(t, •) H ρ+1,∞ + ∇ x u(t, •) H ρ+1,∞ + |D x |u(t, •) H ρ+1,∞ + 2 j=1 R j ∂ t u(t, •) H ρ+1,∞ ≤ A 2 εt -1 2 , (3.3.24a)
∂ t v(t, •) H ρ,∞ + v(t, •) H ρ+1,∞ ≤ A 2 εt -1 , (3.3.24b) ∂ t u(t, •) H n + ∇ x u(t, •) H n + ∂ t v(t, •) H n + ∇ x v(t, •) H n + v(t, •) H n ≤ B 2 εt δ 2 , (3.3.24c) (3.3.24d) (3.3.24e) |I|=k ∂ t Γ I u(t, •) L 2 + ∇ x Γ I u(t, •) L 2 + ∂ t Γ I v(t, •) L 2 + ∇ x Γ I v(t, •) L 2 + Γ I v(t, •) L 2 ≤ B 2 εt δ 3-k 2 , 1 ≤ k ≤ 3.
is obvious in the unit ball. Out of the unit ball we consider a Littlewood-Paley decomposition in frequencies so that

φ(ξ) = φ(ξ)   ϕ 0 (2 -L 0 ξ) + 0 k=L 0 +1 ϕ(2 -k ξ)   , with suppϕ 0 ⊂ B 1 (0), ϕ ∈ C ∞ 0 (R 2 \ {0}) and L 0 < 0 such that 2 L 0 ∼ |x| -1
, and write

xK 0 (x) = K 0 0 (x) + 0 k=L 0 +1 K k 0 (x) with K 0 0 (x) := e ix•ξ a 1 (ξ)ϕ 0 (2 -L 0 ξ)dξ, K k 0 (x) := e ix•ξ a 1 (ξ)ϕ k (2 -k ξ)dξ.
Performing a change of coordinates and making some integrations by parts we deduce that

|K 0 0 (x)| 2 2L 0 and |K k 0 (x)| 2 2k 2 k x -3 , L 0 + 1 ≤ k ≤ 0 for any x ∈ R 2 , which finally implies |xK 0 (x)| 2 2L 0 ∼ |x| -2 .
(ii) The result follows splitting ǎ as in (A.1) and applying to K 0 (x) the same argument previously used for xK 0 (x).

(iii) The result follows straightly from integration by parts and the fact that

f α ∈ L 1 (R 2 ) for any |α| ≤ N . Corollary A.2. Let d ∈ N * , N ∈ N and g β ∈ L 1 (R d ) for every |β| ≤ N . (i) If a(ξ, η) : R 2 × R d → C is such that, for any β ∈ N d with |β| ≤ N , (A.2) |∂ β η a(ξ, η)| β ξ -3 |g β (η)|, |∂ α ξ ∂ β η a(ξ, η)| α,β (|ξ| ξ -1 ) 1-|α| ξ -3 |g β (η)|, 1 ≤ |α| ≤ 4.
for any (ξ, η) ∈ R 2 × R d , then

(A.3) e ix•ξ+iy•η a(ξ, η)dξdη |x| -1 x -2 y -N , ∀(x, y) ∈ R 2 × R d . Moreover, if d = 2 and N = 3, for any u, v ∈ L 2 (R 2 ) ∩ L ∞ (R 2 ) (A.4a) e ix•ξ a(ξ, η)û(ξ -η)v(η)dξdη L 2 (dx) u L 2 v L ∞ (or u L ∞ v L 2 )
and

(A.4b) e ix•ξ a(ξ, η)û(ξ -η)v(η)dξdη L ∞ (dx) u L ∞ v L ∞ . (ii) If a(ξ, η) is such that, for any α ∈ N 2 with |α| ≤ 3, β ∈ N d with |β| ≤ N , (A.5) |∂ α ξ ∂ β η a(ξ, η)| α,β (|ξ| ξ -1 ) -|α| ξ -3 |g β (η)|, for any (ξ, η) ∈ R 2 × R d , then (A.6) e ix•ξ+iy•η a(ξ, η)dξdη x -2 y -N , ∀(x, y) ∈ R 2 × R d . Moreover, if d = 2, N = 3, for any u, v ∈ L 2 (R 2 ) (A.7a) e ix•ξ a(ξ, η)û(ξ -η)v(η)dξdη L 2 (dx) u L 2 v L 2 while if u ∈ L 2 (R 2 ), v ∈ L ∞ (R 2 ), (A.7b) e ix•ξ a(ξ, η)û(ξ -η)v(η)dξdη L ∞ (dx) u L 2 v L ∞ .
Proof. Let K(x, η) := e ix•ξ a(ξ, η)dξ and K(x, y) := e ix•ξ K(x, η)dη.

By the hypothesis on a(ξ, η) and lemma A.1 (i) (resp. (ii)) we derive that, for any β ∈ N d with |β| ≤ N ,

|∂ β η K(x, η)| |x| -1 x -2 |g β (η)| resp. |∂ β η K(x, η)| x -2 |g β (η)| ∀(x, η) ∈ R 2 × R d .
Hence (A.3) (resp. (A.6)) follows applying lemma A.1 (iii) to K(x, y).

(i) If d = 2, N = 3, inequality (A.4a) from the fact that e ix•ξ a(ξ, η)û(ξ -η)v(η)dη = K(x -y, y -z)u(y)v(z)dydx,
and by (A.3), for

L = L 2 or L = L ∞ , K(x -y, y -z) u(y) v(z)dydz L(dx) |x -y| -1 x -y -2 y -z -3 u(y) v(z)dydz L(dx) |y| -1 y -2 z -3 u(• -y) v(• -y -z) L(dx) dydz u L ∞ v L (or u L v L ∞ ). (A.8)
(ii) By inequality (A.6)

K(x -y, y -z)u(y)v(z)dydz L 2 (dx) x -y -2 y -z -3 |u(y)||v(z)|dydz L 2 (dx) y -z -3 |u(y)||v(z)|dydz |v(z)| y -z -3 dy 1 2 y -z -3 |u(y)| 2 dy 1 2 dz v L 2 y -z -3 |u(y)| 2 dydz 1 2 u L 2 v L 2 and K(x -y, y -z)u(y)v(z)dydz L ∞ (dx) x -y -2 y -z -3 |u(y)||v(z)|dydz L ∞ (dx) v L ∞ x -y -2 |u(y)|dy L ∞ (dx) u L 2 v L ∞ . Lemma A.3 (Sobolev norm of a product). Let s ∈ N * . For any u, v ∈ H s (R 2 ) ∩ L ∞ (R 2 ), (A.9) uv H s u H s v L ∞ + u L ∞ v H s ; for any u, v ∈ H s,∞ (R 2 ) ∩ H s+2 (R 2 ), any θ ∈]0, 1[, (A.10) uv H s,∞ u 1-θ H s,∞ u θ H s+2 v L ∞ + u L ∞ v 1-θ H s,∞ v θ H s+2 .
Proof. Inequality (A.9) is a classical result (see, for instance, [START_REF] Alinhac | Pseudo-differential operators and the Nash-Moser theorem[END_REF]).

In order to deduce (A.10) we decompose product uv as follows:

(A.11) uv = T u v + T v u + R(u, v),
where T u v is the para-product of u times v defined by

T u v := S -3 uS 0 v + k≥1 S k-3 u∆ k v, with S k = χ(2 -k D x ), χ ∈ C ∞ 0 (R 2 ) such that χ(ξ) = 1 for |ξ| ≤ 1/2, χ(ξ) = 0 for |ξ| ≥ 1, ∆ 0 = S 0 and ∆ k = S k -S k-1 for k ≥ 1, and R(u, v) = k ∆ k u ∆ k v, with ∆ k = ∆ k-1 + ∆ k + ∆ k+1 . Since T u v = j≥0 ∆ j (T u v) = j,k |j-k|≤N 0 ∆ j [S k-3 u∆ k v] for a certain N 0 ∈ N, by definition 1.2.1 (iii) of the H s,∞ norm and the fact that ∆ k v L ∞ 2 k ∆ k v L 2 we deduce that, for any fixed θ ∈]0, 1[, T u v H s,∞ = D x s T u v L ∞ ≤ j,k |j-k|≤N 0 2 js ∆ j [S k-3 u∆ k v] L ∞ ≤ j,k |j-k|≤N 0 2 js S k-3 u L ∞ ∆ k v L ∞ ≤ j,k |j-k|≤N 0 2 js u L ∞ (2 -ks ∆ k D x s v L ∞ ) 1-θ (2 k ∆ k v L 2 ) θ j,k |j-k|≤N 0 2 (j-k)s u L ∞ ∆ k D x s v 1-θ L ∞ 2 -k ∆ k D x s+2 v L 2 θ u L ∞ v 1-θ H s,∞ v θ H s+2 . (A.12) Similarly, T v u H s,∞ + R(u, v) H s,∞ u 1-θ H s,∞ u θ H s+2 v L ∞ . Corollary A.4. Let s ∈ N * , a 1 (ξ) ∈ S m 1 0 (R 2 ), a 2 (ξ) ∈ S m 2 0 (R 2 ), for some m 1 , m 2 ≥ 0. For any u ∈ H s+m 1 (R 2 ) ∩ H m 1 ,∞ (R 2 ), v ∈ H s+m 2 (R 2 ) ∩ H m 2 ,∞ (R 2 ), (A.13) [a 1 (D x )u] [a 2 (D x )v] H s u H s+m 1 v H m 2 ,∞ + u H m 1 ,∞ v H s+m 2 ; for any u ∈ H s+m 1 ,∞ (R 2 ) ∩ H s+m 1 +2 (R 2 ), v ∈ H s+m 2 ,∞ (R 2 ) ∩ H s+m 2 +2 (R 2 ), any θ ∈]0, 1[, (A.14) [a 1 (D x )u] [a 2 (D x )v] H s,∞ u 1-θ H s+m 1 ,∞ u θ H s+m 1 +2 v H m 2 ,∞ + u H m 1 ,∞ v 1-θ H s+m 2 ,∞ v θ H s+m 2 +2 .
Proof. The result of the statement follows writing [a 1 (D x )u] [a 2 (D x )v] in terms of para-products as in (A.11), and using that

T a 1 (D)u (a 2 (D)v), T a 2 (D)v (a 1 (D)u) and remainder R a 1 (D)u, a 2 (D)v can be written from u = D x m 1 u, v = D x m 2 v
, as done below for the former of these terms:

T a 1 (D)u (a 2 (D)v) = [S -3 a 1 (D) D x -m 1 u][S 0 a 2 (D) D x -m 2 v] + k [S k-3 a 1 (D) D x -m 1 u][∆ k a 2 (D) D x -m 2 v]. Since a j (ξ) ξ -m j ∈ S 0 0 (R 2 ), j = 1, 2, operators S k a j (D) D x -m j , ∆ k a j (D) D
x -m j have the same spectrum (i.e. the support of the Fourier transform) up to a negligible constant of S k and ∆ k respectively.

In the following lemma we prove a result of continuity for a trilinear integral operator defined from multiplier B k (j 1 ,j 2 ,j 3 ) (ξ, η) given by (2.2.42) (resp. by (2.2.53)) for k = 1, 2 (resp. k = 3), any j 1 , j 2 , j 3 ∈ {+, -}. It is useful to observe that, since

B k (j 1 ,j 2 ,j 3 ) (ξ, η) = j 1 ξ -η + j 2 |η| -j 3 ξ 2j 1 j 2 ξ -η |η| η k , k = 1, 2 from (2.2.42), while B 3 (j 1 ,j 2 ,j 3 ) (ξ, η) = j 1 j 1 ξ -η + j 2 |η| -j 3 ξ 2 ξ -η (2.2
.53), we have that

(A.15) 1 (2π) 2 e ix•ξ B k (j 1 ,j 2 ,j 3 ) (ξ, η)û(ξ -η)v(η)dξdη = j 2 2 (uR k v)(x) - j 1 2 D 1 D x u v (x) + j 1 2 D 1 ( D x -1 u)v (x) - j 3 2j 1 j 2 D x [( D x -1 u)(R k v)](x)
for k = 1, 2, while for k = 3

(A.16) 1 (2π) 2 e ix•ξ B 3 (j 1 ,j 2 ,j 3 ) (ξ, η)û(ξ -η)v(η)dξdη = 1 2 (uv)(x) + j 1 j 2 2 [( D x -1 u)|D x |v](x) - j 1 j 3 2 D x ( D x -1 u)v (x).
Lemma A.5. Let B k (j 1 ,j 2 ,j 3 ) (ξ, η) be given by (2.2.42) when k = 1, 2, and by (2.2.53) when k = 3, for any j 1 , j 2 , j 3 ∈ {+, -}. Let also

δ k = 1 if k ∈ {1, 2}, δ k = 0 if k = 3. For any u, w ∈ L 2 (R 2 ), v ∈ H 2,∞ (R 2 ) such that δ k R k v ∈ H 2,∞ (R 2 ), (A.17) B k (j 1 ,j 2 ,j 3 ) (ξ, η)û(ξ -η)v(η) ŵ(-ξ)dξdη u L 2 ( v H 7,∞ + δ k R k v H 7,∞ ) w L 2 .
Proof. First of all we observe that for k ∈ {1, 2}

(A.18a) B k (j1,j2,j3) (ξ, η)û(ξ -η)v(η) ŵ(-ξ)dξdη = j 2 2 u(R k v) (ξ) ŵ(-ξ)dξ - j 1 2 Dx Dx u v (ξ) ŵ(-ξ)dξ + j 1 2 ξ 1 ξ -η û(ξ -η)v(η) ŵ(-ξ)dξdη - j 3 2j 1 j 2 ξ ξ -η û(ξ -η) R k v(η) ŵ(-ξ)dξdη, while for k = 3, (A.18b) B 3 (j 1 ,j 2 ,j 3 ) (ξ, η)û(ξ -η)v(η) ŵ(-ξ)dξdη = 1 2 uv (ξ) ŵ(-ξ)dξ + j 1 j 2 2 ( D x -1 u)|D x |v (ξ) ŵ(-ξ)dξ - j 1 j 3 2 ξ ξ -η û(ξ -η)v(η) ŵ(-ξ)dξ.
Hölder's inequality shows immediately that the first two addends in both above right hand sides are bounded by the right hand side of (A.17). Then the result of the statement follows by proving that inequality (A.17) is satisfied by integrals such as

a(ξ, η)û 1 (ξ -η)û 2 (η)û 3 (-ξ)dξdη with a(ξ, η) = ξ 1 ξ -η -1 or a(ξ, η) = ξ ξ -η -1
, and some general functions

u 1 , u 3 ∈ L 2 (R 2 ), u 2 ∈ L ∞ (R 2 )
. By taking a Littlewood-Paley decomposition we can split the above integral as

(A.19) k,l≥0 a(ξ, η)ϕ k (ξ)ϕ l (η)û 1 (ξ -η)û 2 (η)û 3 (-ξ)dξdη, with ϕ 0 ∈ C ∞ 0 (R 2 ), ϕ ∈ C ∞ 0 (R 2 \ {0}) and ϕ k (ζ) = ϕ(2 -k ζ)
for any k ∈ N * . Since frequencies ξ, η are bounded on the support of ϕ 0 (ξ)ϕ 0 (η), kernel

K 0 (x, y) := e ix•ξ+iy•η a(ξ, η)ϕ 0 (ξ)ϕ 0 (η)dξdη is such that |K 0 (x, y)|
x -3 y -3 for any (x, y), after the first part of corollary A.2 (i). Therefore

a(ξ, η)ϕ 0 (ξ)ϕ 0 (η)û 1 (ξ -η)û 2 (η)û 3 (-ξ)dξdη = K 0 (z -x, x -y)u 1 (x)u 2 (y)u 3 (z)dxdydz z -x -3 x -y -3 |u 1 (x)||u 2 (y)||u 3 (z)|dxdydz u 2 L ∞ x -3 |u 1 (z -x)||u 3 (z)|dxdz u 1 L 2 u 2 L ∞ u 3 L 2 ,
where last inequality obtained by Hölder inequality.

For positive indices l, k such that l > k + N 0 ≥ 0 (resp. |l -k| ≤ N 0 ), for a suitably large integer N 0 > 1, we have that |ξ| < |η| ∼ |ξ -η| (resp. |ξ| ∼ |η|) on the support of ϕ k (ξ)ϕ l (η). If we define a l>k+N0 (ξ, η) := a(ξ, η) η -1 and a |l-k|≤N0 (ξ, η) := a(ξ, η) η -7 , it is a computation to check that, for any α, β ∈ N 2 with |α|, |β| ≤ 3,

|∂ α ξ ∂ β η [a l>k+N0 (2 k ξ, 2 l η)]| + |∂ α ξ ∂ β η a |l-k|≤N0 (ξ, η)| 2 -l .
Hence, their associated kernels K l>k+N0 (x, y) and K |l-k|≤N0 (x, y) are such that

|K l>k+N0 (x, y)| + |K |l-k|≤N0 (x, y)| 2 2k 2 l 2 k x -3 2 l y -3 , ∀(x, y) ∈ R 2 × R 2
as follows after a change of coordinates and some integrations by parts, and for any l > k

+ N 0 a(ξ, η)ϕ k (ξ)ϕ l (η)û 1 (ξ -η)û 2 (η)û 3 (-ξ)dξdη = K l>k+N0 (z -x, x -y)u 1 (x)[ D x u 2 ](y)u 3 (z)dxdydz 2 2k 2 l 2 k (z -x) -3 2 l (x -y) -3 |u 1 (x)|| D x u 2 (y)||u 3 (z)|dxdydz 2 -k 2 2 -l 2 u 1 L 2 u 2 H 1,∞ u 3 L 2 , (A.20)
We observe that

[j 1 j 2 ξ -η |η| -(ξ -η) • η] -1 ξ -η |η| -1 , ∀(ξ, η) ∈ R 2 × R 2
and that for any multi-indices α, β ∈ N 2 of positive length

∂ α ξ (j 1 j 2 ξ -η |η| -(ξ -η) • η) -1 1≤|α 1 |≤|α| |j 1 j 2 ξ -η |η| -(ξ -η) • η| -1-|α 1 | |η| |α 1 | ξ -η -(|α|-|α 1 |) , ∂ β η (j 1 j 2 ξ -η |η| -(ξ -η) • η) -1 0≤|β 1 |<|β| |j 1 j 2 ξ -η |η| -(ξ -η) • η| -1-(|β|-|β 1 |) i+j=|β|-2|β 1 | i,j≤|β|-|β 1 | ξ -η i |η| j .
From above inequalities we hence deduce that on the support of σ N (j 1 ,j 2 ,j 3 ) (η, ξ -η) (i.e. for |η| ≤ ε|ξ -η|), for any α, β ∈ N 2 ,

∂ α ξ ∂ β η (j 1 j 2 ξ -η |η| -(ξ -η) • η) -1 α,β ξ -η 1+|α|+2|β| |η| -1-|β| ,
and therefore that

∂ α ξ ∂ β η j 1 ξ -η + j 2 |η| + j 3 ξ 2j 1 j 2 ξ -η |η| -2(ξ -η) • η α,β ξ -η 2+|α|+2|β| |η| -1-|β| + ξ ξ -η 1+|α|+2|β| |η| -1-|β| .
The above estimates, summed up with the fact

|∂ α ξ ∂ β η [σ N (j 1 ,j 2 ,j 3 ) (η, ξ -η)]| α,β ξ -η -N -|α| |η| N +1-|β| ,
gives the first part of the statement.

Let us now suppose that N ≥ 15 and take χ ∈ C ∞ 0 (R 2 ) equal to 1 in a neighbourhood of the origin. We have that

σ N (j 1 ,j 2 ,j 3 ) (ξ, η)û(ξ -η)v(η) ŵ(-ξ)dξdη = K N 0 (z -x, x -y)u(x)v(y)w(z)dxdydz + K N 1 (z -x, x -y)u(x)[ D x N +3 v](y)w(z)dxdydz, with K N k (x, y) := e ix•ξ+iy•η σ N,k (j 1 ,j 2 ,j 3 ) (ξ, η)dξdη, σ N,0 (j 1 ,j 2 ,j 3 ) (ξ, η) = σ N (j 1 ,j 2 ,j 3 ) (ξ, η)χ(η) and σ N,1 (j 1 ,j 2 ,j 3 ) (ξ, η) = σ N (j 1 ,j 2 ,j 3 ) (ξ, η) η -N -3 (1 -χ)(η).
Then inequality (A.23) is obtained using the fact that, for any u, w

∈ L 2 , v ∈ L ∞ , z -x -3 x -y -3 | u(x)|| v(y)|| w(z)|dxdydz v L ∞ z -3 | u(x)|| w(z -x)|dxdz u L 2 v L ∞ w L 2 .
In the following lemma we derive some results on the Sobolev continuity of the bilinear integral operator

(u, v) → e ix•ξ D (j 1 ,j 2 ) (ξ, η)û(ξ -η)v(η)dξdη,
with D (j 1 ,j 2 ) defined in (3.1.14). We warn the reader that we are not going to take advantage of factor (1 -ξ-η ξ-η • η η ) in D (j 1 ,j 2 ) (ξ, η) when deriving the estimates mentioned below, since the Sobolev continuity of the above integral operator does not depend on the null structure Q 0 (v, ∂ 1 v) we chose for the Klein-Gordon self-interaction in the wave equation in system (1.1.1).

Lemma A.7. Let ρ ∈ N and D(ξ, η) a function satisfying, for any multi-indices α, β ∈ N 2 , the following:

(i) if |ξ| 1, |∂ β η D(ξ, η)| β η ρ+|β| , |∂ α ξ ∂ β η D(ξ, η)| α,β η ρ+|α|+|β| + |α 1 |+|α 2 |=|α| |ξ| 1-|α 1 | η ρ+|α 2 |+|β| , |α| ≥ 1; (ii) for |ξ| 1, |η| ξ -η , |∂ α ξ ∂ β η D(ξ, η)| α,β ξ -η ρ+|α|+|β| ; (iii) for |ξ| 1, |η| ξ -η : |∂ α ξ ∂ β η D(ξ, η)| α,β η ρ+|α|+|β| . Then for any s ≥ 0, any u, v ∈ H s+ρ+13 (R 2 ) ∩ L ∞ (R 2 ) (resp. u, v ∈ H s+ρ+13,∞ (R 2 ) ∩ L 2 (R 2 )) e ix•ξ D(ξ, η)û(ξ -η)v(η)dξdη H s (dx) u H s+ρ+13 v L ∞ + u L ∞ v H s+ρ+13 (or u H s+ρ+13,∞ v L 2 + u L 2 v H s+ρ+13,∞ ), (A.25a)
and for any u, v ∈ H s+ρ+13,∞ (R 2 ) (A.25b)

e ix•ξ D(ξ, η)û(ξ -η)v(η)dξdη H s,∞ (dx) u H s+ρ+13,∞ v L ∞ + u L ∞ v H s+ρ+13,∞ .
Furthermore, if φ ∈ C ∞ 0 (R 2 ), t ≥ 1, σ > 0 small, there exists δ > 0 depending linearly on σ, such that

φ(t -σ D x ) e ix•ξ D(ξ, η)û(ξ -η)v(η)dξdη H s (dx) t δ u H ρ+13 v L ∞ (or t δ u H ρ+13,∞ v L 2 ) (or t δ u L ∞ v H ρ+13 ), (or t δ u L 2 v H ρ+13,∞ ), (A.26a) φ(t -σ D x ) e ix•ξ D(ξ, η)û(ξ -η)v(η)dξdη H s,∞ t δ u H ρ+13,∞ v L ∞ (or t δ u L ∞ v H ρ+13,∞ ). (A.26b)
Finally, if for any α, β ∈ N 2 D(ξ, η) satisfies (ii), (iii) when |ξ| 1, together with:

( i) if |ξ| 1 |∂ α ξ ∂ β η D(ξ, η)| α,β η ρ+|α|+|β| + |α 1 |+|α 2 |=|α| |ξ| -|α 1 |+1 η ρ+|α 2 |+|β| , then, for any u, v ∈ H s+ρ+13 (R 2 ) ∩ L ∞ (R 2 ), (A.27a) e ix•ξ D(ξ, η)û(ξ -η)v(η)dξdη H s (dx) u H ρ+10 v L 2 + u H s+ρ+13 v L ∞ + u L ∞ v H s+ρ+13 (or u L 2 v H ρ+10 + u H s+ρ+13,∞ v L 2 + u L 2 v H s+ρ+13,∞ ),
and for any

u, v ∈ H s+ρ+13,∞ (R 2 ), with u ∈ H ρ+10 (R 2 ) (or u ∈ L 2 (R 2 )), (A.27b) e ix•ξ D(ξ, η)û(ξ -η)v(η)dξdη H s,∞ (dx) u H ρ+10 v L ∞ + u H s+ρ+13,∞ v L ∞ + u L ∞ v H s+ρ+13,∞ (or u L 2 v H ρ+10,∞ + u H s+ρ+13,∞ v L ∞ + u L ∞ v H s+ρ+13,∞ ).
Proof. Let L(R 2 ) denote either the L 2 (R 2 ) space or the L ∞ (R 2 ) one. After definition 1.2.1 (i) of space H s (resp. (iii) of H s,∞ ), we should prove that the L 2 norm (resp. the L ∞ ) norm of (A.28)

e ix•ξ D s (ξ, η)û(ξ -η)v(η)dξdη,
with D s (ξ, η) := D(ξ, η) ξ s , is bounded by the right hand side of (A.25a) and (A.26a) (resp. (A.25b) and (A.26b)). Let us first take χ ∈ C ∞ 0 (R 2 ) equal to 1 in a neighbourhood of the origin and split the above integral, distinguishing between bounded and unbounded frequencies ξ, as (A.29)

e ix•ξ D s (ξ, η)χ(ξ)û(ξ -η)v(η)dξdη + e ix•ξ D s (ξ, η)(1 -χ)(ξ)û(ξ -η)v(η)dξdη.
On the support of χ(ξ) frequencies ξ -η, η are either bounded or equivalent, thus if

a s 0 (ξ, η) :=      D s (ξ, η)χ(ξ) ξ -η -ρ-10 or D s (ξ, η)χ(ξ) η -ρ-10
a s 0 (ξ, η) satisfies (A.2) with g β (η) = η -3 for any |β| ≤ 3, after hypothesis (i) on D(ξ, η). Then by (A.4) and depending on the choice of a s 0 (ξ, η), we have that

(A.30a) e ix•ξ D s (ξ, η)χ(ξ)û(ξ -η)v(η)dξdη L(dx) = e ix•ξ a s 0 (ξ, η) D x ρ+10 u (ξ -η)v(η)dξdη L(dx) D x ρ+10 u L v L ∞ (or D x ρ+10 u L ∞ v L ), or (A.30b) e ix•ξ D s (ξ, η)χ(ξ)û(ξ -η)v(η)dξdη L(dx) = e ix•ξ a s 0 (ξ, η)û(ξ -η) D x ρ+10 v (η)dξdη L(dx) u L ∞ D x ρ+10 v L (or u L D x ρ+10 v L ∞ ).
Successively, we consider a Littlewood-Paley decomposition in order to write

(A.31) e ix•ξ D s (ξ, η)(1 -χ)(ξ)û(ξ -η)v(η)dξdη = k≥1,l≥0 e ix•ξ D s (ξ, η)(1 -χ)(ξ)ϕ k (ξ)ϕ l (η)û(ξ -η)v(η)dξdη, where ϕ 0 ∈ C ∞ 0 (R 2 ), ϕ k (ζ) = ϕ(2 -k ζ) with ϕ ∈ C ∞ 0 (R 2 \ {0}
) for any k ∈ N * . When positive indices l, k are such that k > l + N 0 for a certain large N 0 ∈ N * , we have that |η| < |ξ -η| and |ξ -η| ∼ |ξ| ∼ 2 k on the support of ϕ k (ξ)ϕ l (η). If

a s k>l+N 0 (ξ, η) := D s (ξ, η)ϕ k (ξ)ϕ l (η) ξ -η -s-ρ-13 ,
by hypothesis (ii) we deduce that, for any α, β ∈ N 2 of length less or equal than 3,

|∂ α ξ ∂ β η [a s k>l+N 0 (2 k ξ, 2 l η)]| 2 -k , ∀(ξ, η) ∈ R 2 × R 2
and its associated kernel

K s k>l+N 0 (x, y) := e ix•ξ+iy•η a s k>l+N 0 (ξ, η)dξdη verifies that |K s k>l+N 0 (x, y)| 2 k 2 2l 2 k x -3 2 l y -3 , ∀(x, y) ∈ R 2 × R 2
as one can check doing some integration by parts. Therefore

e ix•ξ D s (ξ, η)ϕ k (ξ)ϕ l (η)û(ξ -η)v(η)dξdη L(dx) = K s k>l+N 0 (x -y, y -z)[ D x s+ρ+13 u](y)v(z)dydz L(dx) 2 k 2 2l 2 k (x -y) -3 2 l (y -z) -3 | D x s+ρ+13 u(y)||v(z)|dydz L(dx) 2 k 2 2l 2 k y -3 2 l z -3 [ D x s+ρ+13 u](• -y)v(• -y -z) L(dx) dydz 2 -k 2 2 -l 2 D x s+ρ+13 u L v L ∞ ( or 2 -k 2 2 -l 2 D x s+ρ+13 u L ∞ v L ).
(A.32)

For indices l, k such that 1 ≤ k ≤ l + N 0 we have that |ξ -η| |η| on the support of ϕ k (ξ)ϕ l (η). If a s k≤l+N 0 (ξ, η) := D s (ξ, η)ϕ k (ξ)ϕ l (η) η -s-ρ-13 by hypothesis (iii) for any multi-indices α, β of length less or equal than 3,

|∂ α ξ ∂ β η [a s k≤l+N 0 (2 k ξ, 2 l η)]| α,β 2 -l ,
and its associated kernel K s k≤l+N 0 (x, y) is such that

K s k≤l+N 0 (x, y)| 2 2k 2 l 2 k x -3 2 l y -3 , ∀(x, y) ∈ R 2 × R 2 .
Consequently

e ix•ξ D s (ξ, η)ϕ k (ξ)ϕ l (η)û(ξ -η)v(η)dξdη L(dx) 2 -k 2 2 -l 2 u L ∞ D x s+ρ+13 v L (or 2 -k 2 2 -l 2 u L D x s+ρ+13 v L ∞ ), (A.33)
and inequality (A.25a) (resp. (A.25b)) is hence obtained by combining inequalities (A.30), (A.32), (A.33) with L = L 2 (resp. L = L ∞ ), and taking the sum over k ≥ 1, l ≥ 0.

In order to derive inequalities (A.26), we first observe that we can reduce to study the L 2 and L ∞ norm of (A.28) with s = 0 and D(ξ, η) multiplied by φ(t -σ ξ), up to a factor t sσ . Here we use again decompositions (A.29), (A.31), and only need to modify some of the multipliers defined above, depending on if we want derivatives falling entirely on u or rather on v. In fact, in order to prove the first two inequalities in (A.26a) and the first one in (A.26b) we introduce a φ l≤k+N 0 (ξ, η) := D(ξ, η)χ(t -σ ξ)ϕ k (ξ)ϕ l (η) a φ l>k+N 0 (ξ, η) := D(ξ, η)χ(t -σ ξ)ϕ k (ξ)ϕ l (η) ξ -η -ρ-13 and deduce from hypothesis (ii) -(iii) on D(ξ, η) and the fact that |ξ| t σ on the support of φ(t -σ ξ) that, for any α, β ∈ N 2 of length less or equal than 3,

|∂ α ξ ∂ β η [a φ l≤k+N 0 (2 k ξ, 2 l η)]| t δ 2 -k and |∂ α ξ ∂ β η [a φ l>k+N 0 (2 k ξ, 2 l η)]| 2 -l
with δ > 0, δ → 0 as σ → 0. On the one hand, kernel K φ l≤k+N 0 (x, y) associated to a φ l≤k+N 0 (ξ, η)

verifies |K φ l≤k+N 0 (x, y)| t δ 2 k 2 2l 2 k x -3 2 l y -3 , ∀(x, y) ∈ R 2 × R 2 and then for any l, k such that l ≤ k + N 0 e ix•ξ D(ξ, η)φ(t σ ξ)ϕ k (ξ)ϕ l (η)û(ξ -η)v(η)dξdη L(dx) = K φ l≤k+N 0 (x -y, y -z)u(y)v(z)dydz L(dx) t δ 2 -k 2 2 -l 2 u L v L ∞ (or t δ 2 -k 2 2 -l 2 u L ∞ v L ).
(A.34)

On the other hand, kernel K φ l>k+N 0 (x, y) associated to a φ l>k+N 0 (ξ, η) satisfies

|K φ l>k+N 0 (x, y)| 2 2k 2 l 2 k x -3 2 l y -3 , ∀(x, y) ∈ R 2 × R 2 so for indices l, k such that l > k + N 0 e ix•ξ D(ξ, η)φ(t σ ξ)ϕ l (ξ)ϕ l (η)û(ξ -η)v(η)dξdη L(dx) = K φ l>k+N 0 (x -y, y -z)[ D x ρ+13 u](y)v(z)dydz L(dx) 2 -k 2 2 -l 2 D x ρ+13 u L v L ∞ (or 2 -k 2 2 -l 2 D x ρ+13 u L ∞ v L ) .
Combining these two inequalities with (A.30a) and taking the sum over k ≥ 1, l ≥ 0 we obtain the wished estimates.

Last two inequalities in (A.26a) and last one in (A.26b) are instead obtained combining (A.30b) with (A.33) (that evidently holds for D s (ξ, η) replaced with D(ξ, η)φ(t σ ξ)) and (A.34).

Finally, last part of the statement follows from the same argument of above, with the only difference that, after hypothesis ( i), multiplier a s 0 (ξ, η) := D(ξ, η)χ(ξ) η -ρ-10 satisfies (A.5) with |g β (η)| η -3 for any |β| ≤ 3, then by (A.7) we have that

e ix•ξ D s (ξ, η)χ(ξ)û(ξ -η)v(η)dξdη L(dx) = e ix•ξ a s 0 (ξ, η) D x ρ+10 u (ξ -η)v(η)dξdη L(dx) D x ρ+10 u L 2 v L ,
Proof. The statement follows essentially from the observation that, for j ∈ {+, -}, functions D j (ξ, η) and [(ξ i ∂ ξ j ) k 1 (η i ∂ η j ) k 2 D j ](ξ, η) satisfy hypothesis (i) -(iii) of lemma A.7 with ρ = 2 and ρ = 2 + 2(k 1 + k 2 ) respectively, while ∂ ξ D j (ξ, η) satisfies ( i), (ii), (iii) with ρ = 3. In fact, we first remark that, for every ξ, η, denominator 1 + ξ -η η -(ξ -η) • η is bounded from below by a positive constant; secondly, the derivation of that denominator gives rise to losses in ξ -η , η , as

∂ ξ k (1 + ξ -η η -(ξ -η) • η) = ξ k -η k ξ -η η + η k , ∂ η k (1 + ξ -η η -(ξ -η) • η) = ξ k -η k ξ -η η + ξ -η η k η + η k -(ξ k -η k ).
For |ξ| 1 we have that ξ -η η , so for any α, β ∈ N 2

∂ α ξ ∂ β η j ξ -η + j η 1 + ξ -η η -(ξ -η) • η η 1 α,β η 2+|α|+|β| , while ∂ β η |ξ| 1 + ξ -η η -(ξ -η) • η η 1 β η 1+|β| , ∂ α ξ ∂ β η |ξ| 1 + ξ -η η -(ξ -η) • η η 1 α,β |α 1 |+|α 2 |=|α| |ξ| 1-|α 1 | η 1+|α 2 |+|β| , |α| ≥ 1.
For |ξ| 1 and |η| ξ -η (resp. |η| ξ -η ) we have that |ξ| |ξ -η| (resp.|ξ| |η|), so each time a derivative hits the denominator of D j (ξ, η) we lose a factor ξ -η (resp. η ). Hence lemma A.7 immediately implies inequalities (A.25), (A.26) with D = D j and ρ = 2, together with (A.35), (A.36), while inequalities (A.37) follow from (A.26) and the fact that, after some integration by parts,

Ω e ix•ξ D j (ξ, η)û(ξ -η)v(η)dξdη = k 1 +k 2 +k 3 +k 4 =1 e ix•ξ [(ξ 1 ∂ ξ 2 -ξ 2 ∂ ξ 1 ) k 1 (η 1 ∂ η 2 -η 2 ∂ η 1 ) k 2 D j ](ξ, η) Ω k 3 u(ξ -η)Ω k 4 v (η)dξdη, Z n e ix•ξ D j (ξ, η)û(ξ -η)v(η)dξdη = e ix•ξ [∂ ξn D j ](ξ, η)D t û(ξ -η)v(η) dξdη + e ix•ξ [∂ ηn D j ](ξ, η)û(ξ -η) D t v(η)dξdη + e ix•ξ D j (ξ, η) Z n u(ξ -η)v(η)dξdη + e ix•ξ D j (ξ, η)û(ξ -η) Z n v(η)dξdη,
and, if δ jn denotes the Kronecker delta,

D j Z n e ix•ξ D j (ξ, η)û(ξ -η)v(η)dξdη = δ jn e ix•ξ D j (ξ, η)D t [û(ξ -η)v(η)] dξdη + e ix•ξ ∂ ξn [ξ j D j ](ξ, η)D t û(ξ -η)v(η) dξdη + e ix•ξ ∂ ηn [ξ j D j ](ξ, η)û(ξ -η) D t v(η)dξdη + e ix•ξ ξ j D j (ξ, η) Z n u(ξ -η)v(η)dξdη + e ix•ξ ξ j D j (ξ, η)û(ξ -η) Z n v(η)dξdη.
. Proof. Straight consequence of the previous lemma and the fact that (u + , v + , u -, v -) is solution to system (3.1.1). Observe that inequality (B.1.5c) is derived using that

R 1 NL w (t, •) H s,∞ NL w (t, •) H s+2
after classical Sobolev injection and continuity of R 1 : H s → H s , for any s ≥ 0.

Lemma B.1.3. Let |I| = 1 be such that Γ I ∈ {Ω, Z m , m = 1, 2}. Then (B.1.7) D t U I (t, •) L 2 U I (t, •) H 1 + V (t, •) H 2,∞ V I (t, •) H 1 + V (t, •) H 1 1 + 1 µ=0 R µ 1 U (t, •)) H 1,∞ + V (t, •) L ∞ U (t, •) H 1 , (B.1.8) D t V I (t, •) L 2 V I (t, •) H 1 + 1 µ=0 R µ 1 U (t, •) H 2,∞ V I (t, •) L 2 + V (t, •) H 1,∞ U I (t, •) H 1 + U (t, •) H 1 + V (t, •) H 1,∞ V (t, •) H 1 .
Proof. The result of the statement follows using the equation satisfied, respectively, by u I ± and v I ± , together with (B.1.5a), (B.1.6a) with s = 0. In fact, by (1.1.15)

with |I| = 1, D t u I ± = ±|D x |u I ± + Q w 0 (v I ± , D 1 v ± ) + Q w 0 (v ± , D 1 v I ± ) + G w 1 (v ± , Dv ± ), D t v I ± = ± D x v I ± + Q kg 0 (v I ± , D 1 u ± ) + Q kg 0 (v ± , D 1 u I ± ) + G kg 1 (v ± , Du ± ), with G w 1 (v ± , ∂v ± ) = G 1 (v, ∂v), G kg 1 (v ± , Du ± ) = G 1 (v, ∂u)
and G 1 given by (1.1.16). Hence one can estimate the L 2 norm of the first two quadratic terms in above equalities with the L 2 norm of factors indexed in I times the L ∞ norm of the remaining one, while the L 2 norm of the latter quadratic terms can be instead bounded by taking the L 2 norm of one of the two factors times the L ∞ norm of the remaining one, indifferently. We choose here to consider the L 2 norm of factors Du ± , Dv ± , and use (B.1.5a), (B.1.6a) if the derivative D is a time derivative.

It is useful to remind that, if w(t, x) is solution to inhomogeneous half wave equation (3.2.5) from (3.2.12a) we have that for any j, k ∈ {1, 2} and |µ| ≤ 1

x j D k D x |D x | µ w = D k |D x | D x |D x | µ x j |D x | -tD j + 1 2i D j |D x | w + t D j D k |D x | D x |D x | µ w - 1 2i D j D k |D x | 2 D x |D x | µ w + iOp ∂ j ξ k |ξ| ξ |ξ| µ |ξ| w = i D k |D x | D x |D x | µ Z j w + D k |D x | D x |D x | µ [x j f (t, x)] + t D j D k |D x | D x |D x | µ w + iOp ∂ j ξ k |ξ| ξ |ξ| µ |ξ| w. (B.1.9a)
Analogously, if w(t, x) is solution to inhomogeneous half Klein-Gordon (3.2.7), from (3.2.12b) we have that

x j D x D x µ w = 1 D x D x D x µ [ D x x j -tD j ] w + t D j D x D x D x µ w + iOp w h ∂ j ξ ξ µ w = i 1 D x D x D x µ Z j w -i D j D x 2 D x D x µ w + 1 D x D x D x µ [x j f (t, x)] + t D j D x D x D x µ w + iOp w h ∂ j ξ ξ µ w. (B.1.9b)
We also remind the reader about equivalence (2.1.16), so we won't particularly care if we are dealing with Γ I u ± , Γ I v ± instead of (Γ I u) ± , (Γ I v) ± , when we bound the L 2 norm of those terms with the energy defined in (1.1.9).

Lemma B.1.4. There exists a positive constant C > 0 such that, for every j = 1, 2, t ∈ [1, T ],

1 |µ|=0 x j D x D x µ v ± (t, •) H 1 ≤ CBεt 1+ δ 2 , (B.1.10a) 1 |µ|=0 x j D x D x µ v ± (t, •) H 1,∞ ≤ C(A + B)εt
1 |µ|=0 x j D x D x |D x | µ u ± (t, •) L 2 ≤ CBεt 1+ δ 2 .
Proof. We warn the reader that, throughout the proof, C will denote a positive constant that may change line after line. As v + = -v -(resp. u + = -u -), it is enough to prove the statement for v -(resp. for u -).

Since v -is solution to equation (3.2.7) with f = NL kg , from (B.1.9b) it immediately follows that, for any |µ| ≤ 1,

(B.1.12a) x j D x D x µ v -(t, •) H 1 Z j v -(t, •) L 2 + t v -(t, •) H 1 + x j NL kg (t, •) L 2 (dx)
along with (B.1.12b)

x j D x D x µ v -(t, •) H 1,∞ ≤ Z j v -(t, •) H 2 + t v -(t, •) H 2,∞ + x j NL kg (t, •) L ∞ (dx) ,
derived by using the classical Sobolev injection. Observe that (B.1.13a)

x j NL kg (t, •) L ∞ x j v -(t, •) L ∞ + x j D x D x v -(t, •) L ∞ 1 µ=0 R µ 1 U (t, •) H 2,∞ , but also (B.1.13b) x j NL kg (t, •) L 2 1 |µ|=0 x j D x D x µ v ± (t, •) L 2 ( U (t, •) H 2,∞ + R 1 U (t, •) H 2,∞ ) .
Thus, if ε 0 > 0 is assumed sufficiently small to verify ε 0 < (2A) -1 , by injecting (B.1.13b) (resp. (B.1.13a)) into (B.1.12a) (resp. in (B.1.12b)), and using a-priori estimates (1.1.11), we obtain that

1 |µ|=0 x j D x D x µ v ± (t, •) H 1 ≤ C E 2 3 (t; W ) 1 2 + tE 3 (t; W ) 1 2 + R 1 U (t, •) H 2,∞ E 0 (t; W ) 1 2 ≤ CBεt 1+ δ 2 resp. 1 |µ|=0 x j D x D x µ v ± (t, •) H 1,∞ ≤ CE 2 3 (t; W ) 1 2 + t V (t, •) H 2,∞ ≤ C(A + B)εt δ 2 2
, and the conclusion of the proof of (B.1.10).

Analogously, from (B.1.9a) with w = u -and f = NL w ,

1 |µ|=0 x j D k D x |D x | µ u -(t, •) L 2 Z j u ± (t, •) L 2 + t u ± (t, •) L 2 + x j NL w (t, •) L 2 (dx) ≤ CBεt 1+ δ 2 ,
as follows (1.1.11c), (1.1.11d), (B.1.10b) and the fact that (B.1.14)

x j NL w (t, •) L 2 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ v ± (t, •) H 1 .
Corollary B.1.5. There exists a constant C > 0 such that, for every j = 1, 2, t ∈ [1, T ],

x j NL kg (t, •) L 2 ≤ C(A + B)Bε 2 t δ+δ 2
2 , (B.1.15a)

x j NL kg (t, •) L ∞ ≤ C(A + B)Bε 2 t -1 2 + δ 2 2 , (B.1.15b) and x j NL w (t, •) L 2 ≤ C(A + B)Bε 2 t δ+δ 2 2 , (B.1.16a) x j NL w (t, •) L ∞ ≤ C(A + B)Bε 2 t -1+ δ 2 2 . (B.1.16b) Proof. From x j NL kg (t, •) L 2 1 µ=0 x j (D x D x -1 ) µ v ± (t, •) L ∞ u ± (t, •) H 1 ,
and (B.1.13a), together with (B.1.14) and

x j NL w (t, •) L ∞ 1 µ=0 x j (D x D x -1 ) µ v ± (t, •) L ∞ v ± (t, •) H 2,∞ ,
we immediately derive the estimates of the statement using (B.1.10b) and a-priori estimates.

Lemma B.1.6. There exists a positive constant C > 0 such that, for any multi-index

I of length k, with 1 ≤ k ≤ 2, any j = 1, 2, t ∈ [1, T ],
(B.1.17)

1 |µ|=0 x j D x D x µ (Γ I v) ± (t, •) H 1 + x j D x D x |D x | µ (Γ I u) ± (t, •) L 2 ≤ CBεt 1+ δ 3-k 2 .
Proof. We warn the reader that, throughout the proof, C will denote a positive constant that may change line after line. As Γ I w + = -Γ I w -, for any I and w ∈ {v, u}, it is enough to prove the statement for Γ I v -, Γ I u -.

From equalities (B.1.9) together with the fact that, for any multi-index I, (Γ I v) -, (Γ I u) -are solution to (B.1.18a)

[D t + D x ](Γ I v) -(t, x) = Γ I NL kg and (B.1.18b) [D t + D x ](Γ I u) -(t, x) = Γ I NL w
respectively, we derive that, for any j, k ∈ {1, 2}, (B.1.19a)

1 |µ|=0 x j D x D x µ (Γ I v) ± (t, •) H 1 ≤ Z j (Γ I v) -(t, •) L 2 + t (Γ I v) -(t, •) L 2 + x j Γ I NL kg (t, •) L 2 together with (B.1.19b) 1 |µ|=0 x j D x D x |D x | µ (Γ I u) ± (t, •) L 2 ≤ Z j (Γ I u) -(t, •) L 2 +t (Γ I u) -(t, •) L 2 + x j Γ I NL w (t, •) L 2 .
The first two quantities in above right hand sides are bounded by CBεt 1+δ 3-k /2 after (1.1.11d), so the quantities that need to be estimated in order to prove the statement are the L 2 norms of

x j Γ I NL kg , x j Γ I NL w , for 1 ≤ |I| ≤ 2.
We first prove (B.1.17) for |I| = 1 and Γ I = Γ, reminding that from (1.1.15), (B.1.20a)

ΓNL kg = Q kg 0 (Γv) ± , D 1 u ± + Q kg 0 v ± , D 1 (Γu) ± + G kg 1 v ± , Du ± and (B.1.20b) ΓNL w = Q w 0 (Γv) ± , D 1 v ± + Q w 0 v ± , D 1 (Γv) ± + G w 1 v ± , Dv ± , with G kg 1 v ± , Du ± = G 1 (v, ∂u), G w 1 v ± , Dv ± = G 1 (v, ∂v)
, and G 1 given by (1.1.16). By multiplying x j against the Klein-Gordon component in each product of ΓNL kg we find that (B.1.21)

x j ΓNL kg (t, •) L 2 1 |µ|=0 x j D x D x µ (Γv) -(t, •) L 2 ( U (t, •) H 2,∞ + R 1 U (t, •) H 2,∞ ) + 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ ( (Γu) ± (t, •) H 1 + u ± (t, •) H 1 + D t u ± (t, •) L 2 ) ,
which injected into (B. 1.19a) with Γ I = Γ, together with (B.1.5a) with s = 0, (B.1.10b), and a-priori estimates (1.1.11), gives that

1 |µ|=0 x j D x D x µ (Γ I v) ± (t, •) H 1 ≤ CBεt 1+ δ 2 2 .
Similarly, using the above estimate together with (B.1.6a) with s = 0, (B.1.10b) and a-priori estimates, we derive that

x j ΓNL w (t, •) L 2 1 |µ|=0 x j D x D x µ (Γv) -(t, •) L 2 v ± (t, •) H 2,∞ + 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ ( (Γv) ± (t, •) H 1 + v ± (t, •) H 1 + D t v ± (t, •) L 2 ) ≤ C(A + B)Bε 2 t δ 2 .
(B. 1.22) Plugging the above inequality in (B. 1.19b) for Γ I = Γ and using again a-priori estimates we deduce that

1 |µ|=0 x j D k D x |D x | µ (Γu) -(t, •) L 2 ≤ CBεt 1+ δ 2 2 ,
and conclude the proof of (B.1.17) when |I| = 1.

When |I| = 2 we observe that, from (1.1.17), (B.1.23)

Γ I NL kg = Q kg 0 (v I ± , D 1 u ± ) + Q kg 0 (v ± , D 1 u I ± ) + (I 1 ,I 2 )∈I(I) |I 1 |=|I 2 |=1 Q kg 0 (v I 1 ± , D 1 u I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |≤1 c I 1 ,I 2 Q kg 0 (v I 1 ± , Du I 2 ± ),
with c I 1 ,I 2 ∈ {-1, 0, 1}. Since the L 2 norm of terms indexed in I 1 , I 2 with |I 1 | = |I 2 | = 1 can be estimated using the Sobolev injection as follows:

(B.1.24)

x j Q kg 0 (v I 1 ± , D 1 u I 2 ± ) L 2 1 |µ|=0 v I 1 ± (t, •) H 2 x j D 1 D x |D x | µ u I 2 ± (t, •) L 2
, from (B.1.23) we derive that

x j Γ I NL kg L 2 1 |J|≤2 |µ|,ν=0 x j D x D x µ (Γ J v) -(t, •) L 2 R ν 1 u ± (t, •) H 2,∞ + D t R ν 1 u ± (t, •) H 1,∞ + 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ u I ± (t, •) H 1 + |J|≤1 u J ± (t, •) H 1 + D t u J ± (t, •) L 2 + |I 1 |=|I 2 |=1 |µ|=0,1 v I 1 ± (t, •) H 2 x j D 1 D x |D x | µ u I 2 ± (t, •) L 2 .
As before, injecting the above inequality into (B.1.19a), using a-priori estimates (1.1.11) and the fact that ε 0 < (2A)

1 |µ|=0 x j D x D x µ (Γ I v) -(t, •) H 1 ≤ CBεt 1+ δ 1 2 .
Analogously, since

Γ I NL w = Q w 0 (v I ± , D 1 v ± ) + Q w 0 (v ± , D 1 v I ± ) + (I 1 ,I 2 )∈I(I) |I 1 |=|I 2 |=1 Q w 0 (v I 1 ± , D 1 v I 2 ± ) + (I 1 ,I 2 )∈I(I) |I 1 |+|I 2 |<2 c I 1 ,I 2 Q w 0 (v I 1 ± , Dv I 2 ± ),
we have that

x j Γ I NL w L 2 |J|≤2 |µ|=0,1 x j D x D x µ (Γ J v) ± (t, •) L 2 v ± (t, •) H 2,∞ + D t v ± (t, •) H 1,∞ + 1 |µ|=0 x j D x D x µ v ± (t, •) H 1,∞   |J|≤2 (Γ J v) ± (t, •) H 1 + |J|≤1 D t v J ± (t, •) L 2   + |I 1 |=|I 2 |=1 |µ|=0,1 (Γ I 1 v) ± (t, •) H 2 x j D x D x µ (Γ I 2 v) ± (t, •)
x j D k D x |D x | µ (Γ I u) -(t, •) L 2 CBεt 1+ δ 1 2 ,
and hence conclude the proof of inequality (B.1.17) also for the case |I| = 2.

Corollary B.1.7. There exists a positive constant C > 0 such that, for any Γ ∈ Z, j = 1, 2, and every t ∈ [1, T ], Lemma B.1.8. There exists a constant C > 0 such that, for every i, j = 1, 2, every t ∈ [1, T ],

x j ΓNL kg (t, •) L 2 ≤ C(A + B)Bε 2 t 1 2 + δ 2 2 , (B.1.26a) x j ΓNL w (t, •) L 2 ≤ C(A + B)Bε 2 t δ
1 |µ|=0 x j x k D x D x µ v ± (t, •) L 2 ≤ CBεt 2+ δ 2 2 , (B.1.27a) 1 |µ|=0 x j x k D x D x µ v ± (t, •) L ∞ ≤ C(A + B)εt 1+ δ 2 2 . (B.1.27b)
Moreover, for any Γ ∈ Z, (B.1.28)

1 |µ|=0 x i x j D x D x µ (Γv) ± (t, •) L 2 ≤ CBεt 2+ δ 2 2 .
Proof. The proof of the statement follows from the fact that, by multiplying (B.1.9b) by x i and using that

x i x j NL kg (t, •) L 2 1 |µ|=0 x i x j D x D x µ v -(t, •) L 2 ( u ± (t, •) H 2,∞ + R 1 u ± (t, •) H 2,∞ )
together with

x i x j NL kg (t, •) L ∞ 1 |µ|=0 x j x k D x D x µ v -(t, •) L ∞ ( u ± (t, •) H 2,∞ + R 1 u ± (t, •) H 2,∞ ) ,
we derive that

1 |µ|=0 x j x k D x D x µ v ± (t, •) L 2 1 |µ|=0 ( x µ i (Z j v) -(t, •) L 2 + t x µ i v -(t, •) L 2 ) + 1 µ=0 x i x j D x D x µ v -(t, •) L 2 ( u ± (t, •) H 2,∞ + R 1 u ± (t, •) H 2,∞ ) ,
and using that operator

D x -1 is bounded from H 1 to L ∞ 1 |µ|=0 x i x j D x D x µ v -(t, •) L ∞ 1 k=0 x k i (Z j v) -(t, •) H 1 + t x k i v -(t, •) H 1,∞ + 1 k,|µ|=0 x k i x j D x D x µ v -(t, •) L ∞ ( u ± (t, •) H 2,∞ + R 1 u ± (t, •) H 2,∞ ) .
As ε 0 > 0 verifies that ε 0 < (2A) -1 , inequality (B.1.10a), (B.1.17) with k = 1, and a-priori estimates (1.1.11) imply that

1 |µ|=0 x i x j D x D x µ v -(t, •) L 2 CBεt 2+ δ 2 2 ,
while from (B.1.10b), (B.1.17) with k = 1 and a-priori estimates,

1 |µ|=0 x i x j D x D x µ v -(t, •) L ∞ ≤ C(A + B)εt 1+ δ 2 2 .
As v + = -v -, that implies the first part of the statement.

Analogously, using (B.1.9b) with w = (Γv) -and multiplying that relation by x i we find that (B.1.29)

1 |µ|=0 x i x j D x D x µ (Γv) -(t, •) L 2 1 µ=0 x µ i Z j (Γv) -(t, •) L 2 + t x µ i (Γv) -(t, •) L 2 + x µ i x j ΓNL kg (t, •) L 2 ,
and after (B.1.17), (B.1.26a) and a-priori estimates, (B.1.30)

1 µ=0 x µ i Z j (Γv) -(t, •) L 2 + t x µ i (Γv) -(t, •) L 2 + x j ΓNL kg (t, •) L 2 ≤ CBεt 2+ δ 2 2 .
By multiplying both x i , x j against each Klein-Gordon factor in ΓNL kg (see equality (B.1.20a)) we derive that

x i x j ΓNL kg (t, •) L 2 1 |µ|ν=0 x i x j D x D x µ (Γv) -(t, •) L 2 R ν 1 u ± (t, •) H 2,∞ + 1 |µ|=0 x i x j D x D x µ v ± (t, •) L ∞ ( (Γu) ± (t, •) H 1 + u ± (t, •) H 1 + D t u ± (t, •) L 2 ) ,
so by (B.1.5a) with s = 0, (B.1.27b), a-priori estimates and the fact that ε 0 < (2A) -1 ,

x i x j ΓNL kg (t, •) L 2 ≤ 1 2 x i x j (Γv) -(t, •) L 2 + C(A + B)Bε
x i x j NL kg (t, •) L 2 + x i x j NL w (t, •) L 2 ≤ C(A + B)Bε 2 t 1+ δ+δ 2 2 .
Proof. Straightforward after (1.1.12c), (B.1.27b) and the following inequality

x i x j NL kg (t, •) L 2 + x i x j NL w (t, •) L 2 1 |µ|=0 x i x j D x D x µ v ± (t, •) L ∞ ( u ± (t, •) H 1 + v ± (t, •) H 1 ) .
Lemma B.1.10. There exists a constant C > 0 such that, for any i, j, k = 1, 2, every t ∈ [1, T ], B.1.4, (B.1.32)

1 |µ|=0 x i x j x k D x D x µ v ± (t, •) L 2 ≤ CBεt 3+ δ 2 2 .
Proof. Using equality (B.1.9b) we derive that

x i x j x k v -(t, •) L 2 1 µ 1 ,µ 2 =0 x µ 1 i x µ 2 j (Z k v) -(t, •) L 2 + t x µ 1 i x µ 2 j v -(t, •) L 2 + 1 µ 1 ,µ 2 ,|µ|=0 x µ 1 i x µ 2 j x k D x D x µ v -(t, •) L 2 ( u ± (t, •) H 2,∞ + R 1 u ± (t, •) H 2,∞ ) ,
so the result of the statement is a straight consequence of (B.1.10a), (B.1.17), (B.1.27a), (B.1.28), a-priori estimates, and the fact that ε 0 is smaller than (2A) -1 .

B.2 First range of estimates

The aim of this section is to show that, if a-priori estimates (1.1.11) are satisfied for every t ∈ [1, T ], for some fixed T > 1, then in the same interval the semi-classical Sobolev norms of the semi-classical functions u, v introduced in (3.2.2) grow in time at a moderate rate t β , for some small β > 0. More precisely, in lemma B.2.1 we prove that this is the case for the H s h (R 2 ) norm of u, u Σ j ,k (see definition (3.2.41)) for any s ≤ n -15, and for the L 2 (R 2 ) norm of those functions when operators Ω h and M, introduced in (1.2.40) and (1.2.49) respectively, are acting on them and frequencies are less or equal than h -σ , for some small σ > 0. Lemma B.2.14 shows that this moderate growth is also enjoyed by the H s h (R 2 ) norm of v, again for s ≤ n = 15, and by the L 2 (R 2 ) norm of L v (see (1.2.68)) when restricted to frequencies |ξ| h -σ . The proof of this latter lemma will require some intermediate results, among which lemma B.2.8 that provides us with a first non-sharp estimate of the L ∞ (R 2 ) norm of Klein-Gordon functions v ± when one Klainerman vector field is acting on them (and again frequencies are localized for |ξ| t σ ). This estimate will successively improved to the sharpest one (B.4.50) in lemma B.4.14 of section B.4.

As said at the beginning of this chapter, we prove the below results under the hypothesis that a-priori estimates (1.1.11) are satisfied in some fixed [1, T ], with ε 0 < (2A + B) -1 . We remind here that, if χ ∈ C ∞ 0 (R 2 ) and σ > 0, χ(t -σ D x ) is a bounded operator from H s to L 2 with norm O(t σs ), and on L ∞ uniformly in time.

Lemma B.2.1. Let u, u Σ j ,k be defined, respectively, in (3.2.2) and (3.2.41), and s ≤ n -15. There exists a constant C > 0 such that, for any θ 0 , χ ∈ C ∞ 0 (R 2 ) and every t ∈ [1, T ], Proof. We warn the reader that, throughout the proof, C and β will denote positive constants that may change line after line, with β → 0 as σ → 0. We will also use the following concise notation

u(t, •) H s h + u Σ j ,k (t, •) H s h ≤ CBεt δ 2 +κ , (B.2.1a) Ω h u Σ j ,k (t, •) L 2 ≤ CBεt δ 2 2 +κ , (B.2.1b) |µ|=1 Op w h (χ(h σ ξ))M µ u(t, •) L 2 + M µ u Σ j ,k (t, •) L 2 ≤ C(A + B)εt δ 2 2 +κ , (B.2.1c) |µ|=1 θ 0 (x)Ω h M µ u Σ j ,k (t, •) L 2 ≤ CBεt
φ j k (ξ) := Σ(ξ)(1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 0 (h σ ξ), reminding that (B.2.2) Op w h (φ j k (ξ)) L(L 2 ) = O(h -κ ), with κ = σρ if ρ ≥ 0,
some χ ∈ C ∞ 0 (R 2 ) such that Ω h u Σ j ,k (t, •) L 2 h -κ Op w h (χ 0 (h σ ξ))Ω h u(t, •) L 2 , M u Σ j ,k (t, •) L 2 h -κ 1 |ν|=0 Op w h (χ(h σ ξ))M ν u(t, •) L 2 , θ 0 (x)Ω h M u Σj ,k (t, •) L 2 θ 0 (x)Op w h (φ j k (ξ))Ω h M u(t, •) L 2 + h -κ 1 µ=0 Op w h (χ(h σ ξ))Ω µ h u(t, •) L 2 .
Therefore, as h = t -1 , in order to prove (B.2.1b)-(B.2.1d) it is enough to show that, for any

χ ∈ C ∞ 0 (R 2 ), Op w h (χ(h σ ξ))Ω h u(t, •) L 2 ≤ CBεt δ 2 2 , (B.2.3a) Op w h (χ(h σ ξ))M u(t, •) L 2 ≤ C(A + B)εt δ 2 2 , (B.2.3b) θ 0 (x)Op w h (φ j k (ξ))Ω h M u(t, •) L 2 ≤ CBεt
Op w h (χ(h σ ξ))Ω h u(t, •) L 2 (Ωu) -(t, •) L 2 + χ(t -σ D x )Ω(u N F -u -)(t, •) L 2 ΩU (t, •) L 2 + t β ( V (t, •) L 2 + ΩV (t, •) L 2 ) V (t, •) H 17,∞ ≤ C(1 + Aεt -1+β )E 2 3 (t; W ) 1 2 ≤ CBεt δ 2 2 .
From equality (3.2.9a) and definition (3.1.15) of u N F we deduce that

Op w h (χ(h σ ξ))M n u(t, •) L 2 Z n U (t, •) L 2 + χ(t -σ D x )Z n (u N F -u -)(t, •) L 2 + u(t, •) L 2 + Op w h (χ(h σ ξ))[t(tx j )[q w + c w ](t, tx)] L 2 (dx) + χ(t -σ D x )(x n r N F w )(t, •) L 2 , (B.2.4)
with q w , c w and r N F w given by (3.1.17), (3.1.18) and (3.1.19) respectively. We first notice that, after inequality (A.37b) with u = v = v ± , (B.1.6a) with s = 0, a-priori estimates, and the fact that Aε 0 ≤ 1,

(B.2.5) χ(t -σ D x )Z n (u N F -u -)(t, •) L 2 t β ( D t V (t, •) L 2 V (t, •) H 13 + V (t, •) H 15,∞ Z n V (t, •) L 2 ) ≤ CBεt β+δ .
Let us also observe that from (3.1.17), (3.1.18) we have that

q w (t, x) + c w (t, x) = 1 2 v -D 1 v -- D x D x v -• D x D 1 D x v -(t, x) = h 2 2 V Op w h (ξ 1 ) V -Op w h ξ 1 ξ V • Op w h ξξ 1 ξ V t, x t , (B.2.6)
where V (t, x) := tv -(t, tx) is such that, for every s, ρ ≥ 0,

V (t, •) H s h = v -(t, •) H s , V (t, •) H ρ,∞ h = t v -(t, •) H ρ,∞ .
Moreover, by (3.2.8) with w = v -and f = NL kg

L j V (t, •) H 1 h Z j v -(t, •) L 2 + v -(t, •) L 2 + x j v ± (t, •) L ∞ + x j D x D x v ± (t, •) L ∞ U (t, •) H 1 . (B.2.7)
Using (B.2.6) along with the definition of L j in (1.2.68) we derive that

t(tx j )[q w + c w ](t, tx) = 1 2 V Op w h (ξ 1 )(hL j V ) + V Op w h ξ 1 ξ j ξ V + V [x j , Op w h (ξ 1 )] V -Op w h ξ ξ V • Op w h ξξ 1 ξ (hL j V ) -Op w h ξ ξ V • Op w h ξξ 1 ξ j ξ 2 V -Op w h ξ ξ V • x j , Op w h ξξ 1 ξ V (t,
+ c w ](t, t•) L 2 (dx) V (t, •) H 1 h + h L j V (t, •) H 1 h V (t, •) H 1,∞ h ≤ CA(A + B)ε 2 t δ 2 .
(B.2.9) Moreover, from (3.1.19), the fact that x j e ix•ξ = D ξ j e ix•ξ , integration by parts, and inequalities (A.26a) with ρ = 2 (after the first part of lemma A.8), (A.36a), we get that Let us now apply θ 0 x t φ j k (D x )Ω to both sides of (3.2.9a) to deduce that

χ(t -σ D x )(x n r N F w )(t, •) L 2 t β [ x n v -(t, •) L ∞ NL kg (t, •) H 15 + V (t, •) H 15 x n NL kg (t, •) L ∞ + NL kg (t, •) L 2 ( V (t, •) H 13 + V (t, •) H 13,∞ ) + V (t, •) H 13 NL kg (t, •) L ∞ ] ≤ CBεt
θ 0 (x)Op w h (φ j k (ξ))Ω h M n u(t, •) L 2 ΩZ n U (t, •) L 2 + θ 0 x t φ j k (D x )ΩZ n (u N F -u -)(t, •) L 2 + 1 µ=0 Op w h (χ 0 (h σ ξ))Ω µ h u(t, •) L 2 + θ 0 (x)Op w h (φ j k (ξ))Ω h [t(tx j )(q w + c w )(t, tx)] L 2 (dx) + θ 0 (x)Op w h (φ j k (ξ))Ω h [t(tx n )r N F w ](t, tx)] L 2 (dx)
.

(B.2.11)

In order to estimate the second addend in the above right hand side we first commute Z n to Ω, reminding that

[Ω, Z 1 ] = -Z 2 and [Ω, Z 2 ] = Z 1 ,
and use that

θ 0 x t φ j k (D x )Z j = tθ j 0 x t φ j k (D x ) + θ 0 x t [φ j k (D x ), x j ] ∂ t + tθ 0 x t φ j k (D x )∂ j ,
with θ j 0 (z) := θ 0 (z)z j . Observe that commutator [φ j k (D x ), x j ] is bounded on L 2 with norm O(t), and that its symbol is still supported for moderate frequencies |ξ| t -σ . Therefore, for some new χ ∈ C ∞ 0 (R 2 ) we have that

θ 0 x t φ j k (D x )ΩZ n (u N F -u -)(t, •) L 2 t χ(t -σ D x )∂ t,x (u N F -u -)(t, •) L 2 + t χ(t -σ D x )∂ t,x Ω(u N F -u -)(t, •) L 2 ,
so using (A.26a) with ρ = 2 (because of first part of lemma A.8) and (A.37a), both considered with u = ∂ t,x v ± , v = v ± , and u = v ± , v = ∂ t,x v ± , we obtain that the above right hand side is estimated by

t 1+β [( ∂ t,x V (t, •) L 2 + Ω∂ t,x V (t, •) L 2 ) V (t, •) H 17,∞ + ( V (t, •) L 2 + ΩV (t, •) L 2 ) ∂ t,x V (t, •) H 17,∞ ]
From (B.1.6a) and (B.1.6b) with s = 0, (B.1.6c) and a-priori estimates, we hence deduce that (B.2.12)

θ 0 x t φ j k (D x )ΩZ n (u N F -u -)(t, •) L 2 ≤ CBεt β+ δ 2 2 .
As concerns, instead, the estimate of the fourth L 2 norm in the right hand side of (B. h

(ξ))Ω h [t(tx j )[q w + c w ](t, tx)] L 2 1 µ=0 h -κ V (t, •) H 2,∞ h Ω µ h V (t, •) H 1 h + 1 µ=0 h 1-κ V (t, •) H 1,∞ h Ω µ h L j V (t, •) H 1 + h 1-κ Ω h V (t, •) L ∞ L j V (t, •) H 1 h , with κ = σρ if ρ ≥ 0,
Ω h V (t, •) L ∞ L j V (t, •) H 1 h Ω V (t, •) H 2 h L j V (t, •) H 1 h ≤ CBεt 3δ 2 2 .
Also, from (3.2.8) with w = v -and f = NL kg

Ω h L j V (t, •) L 2 ΩZ j v -(t, •) L 2 + 1 µ=0 Ω µ v -(t, •) L 2 + Ω (x j NL kg ) (t, •) L 2 ≤ C(A + B)Bε 2 t 1 2 + δ 2 2 ,
where last inequality is obtained using (1.1.11c), (1.1.11d) and estimates (B.1.15a), (B.1.26a). Therefore

h V (t, •) H 1,∞ h Ω h L j V (t, •) L 2 ≤ CAB(A + B)ε 3 t -1 2 + δ 2
θ 0 (x)Op w h (φ j k (ξ))x n = θ n 0 (x)Op w h (φ j k (ξ)) + θ 0 (x)[Op w h (φ j k (ξ)), x n ],
where

[Op w h (φ j k (ξ)), x n ] = -ihOp w h (∂ n φ j k (ξ))
is uniformly bounded on L 2 . After (3.1.22a), (3.1.22c) with θ 1 small, and a-priori estimates (1.1.11) we derive that, for some χ ∈ C ∞ 0 (R 2 ), In the following lemma we explain how we estimate the L 2 or the L ∞ norm of products supported for moderate frequencies |ξ| t σ , when we have a control on high Sobolev norms of, at least, all factors but one. This type of estimate will be frequently used in most of the results that follow.

θ 0 (x)Op w h (φ j k (ξ))Ω h [t(tx n )r N F w ](t, tx)] L 2 (dx) 1 µ=0 t χ(t -σ D x )Ω µ r N F w (t, •) L 2 ≤ CBε.
Γw) ± = (D t ± |D x |)(Γu), if w = u, (D t ± D x )(Γv), if w = v. (
Let also n ∈ N * , w 1 , . . . , w n be such that w

1 , xw 1 ∈ L 2 (R 2 ) ∩ L ∞ (R 2 ), w j ∈ L ∞ (R 2 ) for j = 2, . . . , n, χ ∈ C ∞ 0 (R 2 ), σ > 0,
a(D x )(Ωw) ± w 1 . . . w n L(dx) χ(t -σ D x )a(D x )(Ωw) ± n j=1 w j L(dx) + t -N (s) w ± (t, •) H s 1 |µ|=0 x µ w 1 L(dx) n j=2 w j L ∞ (dx) (B.2.20a)
and, for m = 1, 2,

(B.2.20b) a(D x )(Z m w) ± w 1 . . . w n L(dx) χ(t -σ D x )a(D x )(Z m w) ± n j=1 w j L(dx) + t -N (s) ( w ± (t, •) H s + D t w ± (t, •) H s ) 1 µ=0 x µ m w 1 L(dx) + t w 1 L(dx) n j=2 w j L ∞ (dx) ,
with N (s) as large as we want as long as s > 0 is large.

Proof. Let us remind definition (1.1.6) of Klainerman vector fields Ω, Z m , for m = 1, 2, and decompose factor a(D x )(Γw) ± in frequencies by means of operator χ(t -σ D x ). When dealing with product

(B.2.21) (1 -χ)(t -σ D x )a(D x )(Γw) ± w 1 • • • w n
the idea is to discharge on w 1 factors x and/or t defining Γ, after a previous commutation between

D t ± |D x | if w = u (resp. D t ± D x if w = v)
and Γ, and between (1 -χ)(t -σ D x )a(D x ) and the mentioned factors x, t. For instance, if w = u and Γ

= Z 1 (B.2.22) (1 -χ)(t -σ D x )a(D x )(Z 1 u) ± w 1 = (1 -χ)(t -σ D x )a(D x )(∂ t u) ± (x 1 w 1 ) + (1 -χ)(t -σ D x )a(D x )(∂ 1 u) ± (tw 1 ) + (1 -χ)(t -σ D x )a(D x ) D 1 |D x | u ± w 1 + (1 -χ)(t -σ D x )a(D x ), x 1 D t u ± w 1 ,
from which we deduce, using the Sobolev injection together with (B.1.2), that

(1 -χ)(t -σ D x )a(D x )(Z 1 u) ± w 1 L t -N (s) ( u ± (t, •) H s + D t u(t, •) H s ) 1 µ=0 x µ 1 w 1 L + t w 1 L ,
with N (s) large as long as s is large. Analogous inequalities can be obtained for Γ = Ω, Z 2 and/or w = v. This concludes the proof of the statement since the L norm of (B.

(D x )(Ωw) ± w 1 • • • w n L χ(t -σ D x )a(D x )(Ωw) ± n j=1 χ(t -σ D x )w j L + t -N (s) w ± (t, •) H s (dx) 1 |µ|=0 x µ w 1 L n j=2 w j L ∞ + t -N (s) (Ωw) ± (t, •) L 2 n j=1 k =j w k L ∞ w j H s (B.2.23a)
and, for m = 1, 2,

a(D x )(Z m w) ± w 1 • • • w n L χ(t -σ D x )a(D x )(Z m w) ± n j=1 χ(t -σ D x )w j L + t -N (s) ( w ± (t, •) H s + D t w ± (t, •) H s ) 1 µ=0 x µ m w 1 L + t w 1 L n j=2 w j L ∞ + t -N (s) (Z m w) ± (t, •) L 2 n j=1 k =j w k L ∞ w j H s , (B.2.23b)
with N (s) as large as we want as long as s > 0 is large. Moreover, there exists χ 1 ∈ C ∞ 0 (R 2 ) such that, for any fixed j 0 ∈ {1, . . . , n},

χ(t -σ D x ) a(D x )(Ωw) ± w 1 • • • w n L χ(t -σ D x )a(D x )(Ωw) ± χ 1 (t -σ D x )w j 0 j=1,...,n j =j 0 χ(t -σ D x )w j L + t -N (s) w ± (t, •) H s 1 |µ|=0 x µ w 1 L n j=2 w j L ∞ + t -N (s) (Ωw) ± (t, •) L 2 j=1,...,n j =j 0 k =j w k L ∞ w j H s (B.2.24a)
and, for m = 1, 2,

χ(t -σ D x ) a(D x )(Z m w) ± w 1 • • • w n L χ(t -σ D x )a(D x )(Z m w) ± χ 1 (t -σ D x )w j 0 j=1,...,n j =j 0 χ(t -σ D x )w j L + t -N (s) ( w ± (t, •) H s + D t w ± (t, •) H s ) 1 µ=0 x µ m w 1 L + t w 1 L n j=2 w j L ∞ + t -N (s) (Z m w) ± (t, •) L 2 j=1,...,n j =j 0 k =j w k L ∞ w j H s . (B.2.24b)
Proof. The inequalities of the statement mainly follows from (B.2.20). In fact, by decomposing each factor w j appearing in the first norm in the right hand sides of (B.2.20) as in (B.2.16), and then using the following inequality, for Γ ∈ {Ω, Z m , m = 1, 2} and w k either equal to w k or to

χ(t -σ D x )w k , [χ(t -σ D x )a(D x )(Γw) ± ] k=1,...,n k =j w k (1 -χ)(t -σ D x )w j L t -N (s) (Γw) ± (t, •) L 2 k=1,...,n k =j w k L ∞ w j H s ,
χ(t -σ D x )w(t, •) L ∞ t -1+β 1 |µ|=0 Op w h (χ(h σ ξ))L µ w(t, •) L 2 , with β > 0 small, β → 0 as σ → 0. Proof. Since χ(t -σ D x )w(t, y) = t -1 Op w h (χ(h σ ξ)) w(t, x)| x= y t , the goal is to prove that (B.2.26) Op w h (χ(h σ ξ)) w(t, •) L ∞ h -β 1 |µ|=0 Op w h (χ(h σ ξ))L µ w(t, •) L 2 ,
for a small β > 0, β → 0 as σ → 0. So let w χ := Op w h (χ(h σ ξ)) w and take χ 1 ∈ C ∞ 0 (R 2 ) equal to 1 on the support of χ, so that

Op w h (χ(h σ ξ)) w = Op w h (χ 1 (h σ ξ)) w χ .
For a γ ∈ C ∞ 0 (R 2 ), equal to 1 in a neighbourhood of the origin and with sufficiently small support, we consider the following decomposition

Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ + Op w h (1 -γ) x -p (ξ) √ h χ 1 (h σ ξ) w χ
and immediately observe that, from inequality (3.2.17b),

Op w h (1 -γ) x -p (ξ) √ h χ 1 (h σ ξ) w χ (t, •) L ∞ h -β 1 |µ|=0 Op w h (χ 1 (h σ ξ))L µ w χ (t, •) L 2 .
After lemma 1.2.38 there exists a family of smooth cut-off functions θ h (x) such that equality (1.2.67) holds. Then, if we also consider a new cut-off function χ 2 equal to 1 on the support of χ 1 and a small σ 1 > σ, by symbolic calculus and remark 1.2.22 we derive that for any N ∈ N

Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ = θ h (x)Op w h (χ 2 (h σ ξ))Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ + Op w h (r ∞ (x, ξ)) w χ + θ h (x)Op w h (r 1 ∞ (x, ξ)) w χ with r ∞ , r 1 ∞ ∈ h N S 1 2 ,σ ( x-p (ξ) √ h -1 ). It is enough to take N = 1 to have, by proposition 1.2.37, that Op w h (r ∞ ) w χ (t, •) L ∞ + θ h (x)Op w h (r 1 ∞ ) w χ (t, •) L ∞ ≤ h -β w χ (t, •) L 2 .
As function φ(x) := 1 -|x| 2 is well defined on the support of θ h we are allowed to to write the following:

θ h (x)Op w h (χ 2 (h σ 1 ξ))Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ (t, •) L ∞ = e i h φ θ h (x)Op w h (χ 2 (h σ 1 ξ))Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ (t, •) L ∞ Op w h (χ 2 (h σ 1 ξ)) e i h φ θ h (x)Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ L ∞ + Op w h (r ∞ ) w χ (t, •) L ∞ , for a new r ∞ ∈ h N S 1 2 ,σ x-p (ξ) √ h -1
. This latter r ∞ comes out from the commutation between e i h φ θ h (x) and Op w h (χ 2 (h σ 1 ξ)), whose symbol is computed using (1.2.18) until a large enough order M . We notice that we gain a factor h |α|(σ 1 -σ) at each order of the mentioned asymptotic development as σ 1 > σ. Moreover, those terms write in terms of the derivatives of χ 2 and hence vanish on the support of χ 1 . By proposition 1.2.21 and remark 1.2.22 we then deduce that the composition of the mentioned commutator with Op w h γ x-p (ξ) √ h χ 1 (h σ ξ) is an operator of symbol r ∞ , with N as large as we want.

Using the classical Sobolev injection, symbolic calculus and lemma 3.2.16 we find that

Op w h (χ 2 (h σ1 ξ)) e i h φ θ h (x)Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ L ∞ | log h|   w χ (t, •) L 2 + 2 j=1 D j e i h φ θ h (x)Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ L 2   | log h|   w χ (t, •) L 2 + 2 j=1 h -1 Op w h (ξ j + d j φ(x))θ h (x) Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) w χ L 2   | log h|   w χ (t, •) L 2 + h -β 1 |µ|=0 Op w h (χ 1 (h σ ξ))L µ w χ (t, •) L 2   .
Finally, commutating L with Op w h (χ(h σ ξ)) defining w χ , and reminding that χ 1 ≡ 1 on the support of χ, we obtain

Op w h (χ(h σ ξ)) w χ (t, •) L ∞ h -β 1 |µ|=0 Op w h (χ(h σ ξ))L µ w(t, •) L 2 ,
with χ(ξ) := (1 -χ)(ξ)|ξ| -1 and using relation (3.2.12a) with w = u ± . The proof for |α| = 2 is analogous. It is based on the commutation of x α with (1 -χ)(t κ D x )χ(t -σ D x ) (the commutator is here a L 2 -L ∞ bounded operator with norm O(t κ )), on the fact that we can rewrite

(1 - χ)(t κ D x )χ(t -σ D x )x α R µ 1 making appear (x|D x | -tD x + 1 2i Dx |Dx| ) α by considering χ 2 (ξ) := (1 - χ)(ξ)|ξ| -2 instead
of previous χ 1 , and use relation (3.2.12a). Doing so we derive the following inequality

(1 -χ)(t κ D x )χ(t -σ D x ) x α R µ 1 u ± (t, •) L ∞ t 2κ   |µ|=2 Z µ u ± (t, •) L 2 + |µ|≤1 t 2-|µ| Z µ u ± (t, •) H 1 + 2 |µ|=1 x µ NL w (t, •) L 2   ≤ CBεt 2+2κ+ δ 2 ,
last estimate following from a-priori estimates, (B.

[(1 -χ)(t -σ D x )χ 1 (t -σ D x )(Γ I v) ± (t, •)](1 -χ)(t κ D x )χ(t -σ D x )R µ 1 u ± (t, •) L ∞ ≤ CBεt -3 2 .
Therefore, from (B.2.31), (B.2.33),(B.2.38), and the uniform continuity on L ∞ of χ(t -σ D x ), we find that

χ(t -σ D x ) v I,NF -v I -(t, •) L ∞ 1 µ=0 t σ χ(t -σ D x )(Γ I v) ± (t, •) L ∞ R µ 1 u ± (t, •) L ∞ + CBεt -1 ,
and as σ is small and ε 0 < (2A) -1 , from (1.1.11a) we obtain (B.2.28).

In order to prove (B.2.29) we apply Z m to equality (B.2.30) and apply the Leibniz rule. As

[Z m , D t ] = -D m , [Z m , D 1 ] = -δ m1 D t , [Z m , D x ] = -D m D x -1 D t , (B.2.39)
with δ m1 the Kronecker delta, we find that

2iχ(t -σ D x )Z m (v I,NF -v I -) = χ(t -σ D x ) (D t Z m Γ I v)(D 1 u) -(D 1 Z m Γ I v)(D t u) + D 1 [(Z m Γ I v)(D t u)] -D x [(Z m Γ I v)(D 1 u)] + (D t Γ I v)(D 1 Z m u) -(D 1 Γ I v)(D t Z m u) + D 1 [(Γ I v)(D t Z m u)] -D x [(Γ I v)(D 1 Z m u)] -(D m Γ I v)(D 1 u) + δ m1 (D t Γ I v)(D t u) -δ m1 D t [(Γ I v)(D t u)] + D m D x D t [(Γ I v)(D 1 u)] -δ m1 (D t Γ I v)(D t u) + (D 1 Γ I v)(D m u) -δ m1 D 1 [(Γ I v)(D t u)] + δ m1 D x [(Γ I v)(D t u)] . (B.2.40)
The L 2 norm of all products in the above second, fourth and fifth line, i.e. those in which Z m is not acting on the wave component u, is estimated by (B.2.41)

1 µ=0 t σ (Z m Γ I v) ± (t, •) L 2 + (Γ I v) ± (t, •) L 2 ( R µ 1 u ± (t, •) L ∞ + D t u ± (t, •) L ∞ ) ≤ CABε 2 t -1 2 + δ 0 2 +σ ,
after inequality (B.1.5b) with s = 0 and a-priori estimates. The L 2 norm of products appearing in the second line are, instead, estimated by using (1.1.10) and (B.2.24b) with L = L 2 , Γw = Z m u, s > 0 sufficiently large so that N (s) ≥ 2. It is hence bounded by

t σ χ(t -σ D x )(Γ I v) ± (t, •) L ∞ (Z m u) ± (t, •) L 2 + t -2 1 |µ|=0 x µ (Γ I v) ± (t, •) L 2 + t (Γ I v) ± (t, •) L 2 ( u ± (t, •) H s + D t u ± (t, •) H s ) ≤ CB 2 ε 2 t 2σ+ δ 3-j +δ 2 2
, where the latter estimate is obtained using the fact that χ(t

χ(t -σ D x )V I (t, •) H ρ,∞ ≤ CBεt -1+β+ δ 1 2 ,
with β > 0 small such that β → 0 as σ → 0.

Proof. Since χ(t -σ D x ) is a bounded operator from L ∞ to H ρ,∞ with norm O(t σρ ), for any ρ ∈ N, it is enough to prove that the L ∞ norm of χ(t -σ D x )V I (t,
•) is bounded by the right hand side of (B.2.42). Moreover, as this latter inequality is automatically satisfied when Γ is a spatial derivative after a-priori estimate (1.1.11b) and the fact that operator χ(t -σ D x ) is uniformly bounded on L ∞ , for the rest of the proof we will assume that Γ ∈ {Ω, Z j , j = 1, 2} is a Klainerman vector field. We also warn the reader that, throughout the proof, C and β will denote some positive constants that may change line after line, with β → 0 as σ → 0.

Instead of proving the result of the statement directly on χ(t -σ D x )v I ± we do it for χ(t -σ D x )v I,NF , where v I,NF has been introduced in (B.2.27) and is considered here for |I| = 1 and Γ I = Γ. In fact, by (B.2.28)

(B.2.43) χ(t -σ D x )v I -(t, •) L ∞ ≤ 2 χ(t -σ D x )v I,NF (t, •) L ∞ + CBεt -1 .
The advantage of dealing with this new function is related to the fact that it is solution to a half Klein-Gordon equation with a more suitable non-linearity (see (B.

= r I,NF kg (t, x) + Q kg 0 (v ± , D 1 u I ± ) + G kg 1 (v ± , Du ± ), G kg 1 (v ± , Du ± ) = G 1 (v,
(t, x) = - i 4(2π) 2 × j 1 ,j 2 ∈{+,-} e ix•ξ B 1 (j 1 ,j 2 ,+) (ξ, η) Γ I NL kg (ξ -η)û j 2 (η) -v I j 1 (ξ -η) NL w (η) dξdη, with B 1 (j 1 ,j
χ(t -σ D x )v I,NF (t, •) L ∞ t -1+β 1 |µ|=0 χ(t -σ D x )Z µ v I,NF (t, •) L 2 + 2 j=1 t -1+β χ(t -σ D x ) x j NL I,NF kg (t, •) L 2 .
From equality (B.2.30), along with (1.1.5), (1.1.10), and a-priori estimates (1.1.11a), (1.1.11d), we immediately see that

χ(t -σ D x )(v I,NF -v I -)(t, •) L 2 t σ v I ± (t, •) L 2 ( u ± (t, •) L ∞ + R 1 u ± (t, •) L ∞ ) ≤ CABε 2 t -1 2 + δ 2 2 +σ , (B.2.49)
and as σ, δ 2 1 are small

χ(t -σ D x )v I,NF (t, •) L 2 ≤ χ(t -σ D x )v I -(t, •) L 2 + χ(t -σ D x )(v I,NF -v I -)(t, •) L 2 ≤ CBεt δ 2
χ(t -σ D x ) x j r I,NF kg (t, •) L 2 x j Γ I NL kg (t, •) L 2 ( u ± (t, •) L ∞ + R 1 u ± (t, •) L ∞ ) + 1 µ=0 t σ x µ j v ± (t, •) L ∞ + x µ j D x D x v ± (t, •) L ∞ v I ± (t, •) L 2 v ± (t, •) H 2,∞ ≤ C(A + B)Bε 2 t δ 2
χ(t -σ D x ) x j Q kg 0 v ± , D 1 u I ± (t, •) L 2 + χ(t -σ D x ) x j G kg 1 (v ± , Du ± ) (t, •) L 2 x j v ± (t, •) L ∞ + x j D x D x v ± (t, •) L ∞ u I ± (t, •) H 1 + D t u ± (t, •) L 2 ≤ C(A + B)Bεt δ 2 .
Therefore, from (B.

χ(t -σ D x )Q w 0 ((Ωv) ± , D 1 (Γv) ± ) L 2 + χ(t -σ D x )Q w 0 ((Γv) ± , D 1 (Ωv) ± ) L 2 t σ χ(t -σ D x )(Ωv) ± (t, •) L ∞ (Γv) ± (t, •) H 1 + 1 |µ|=0 t -3 v ± (t, •) H s x µ (Γv) ± (t, •) H 1 ≤ CB 2 ε 2 t -1+β+ δ 1 +δ 2 2 ,
with β > 0 small such that β → 0 as σ → 0. All remaining quadratic contributions to ΩΓNL w are estimated with

(ΩΓv) ± (t, •) H 1 v ± (t, •) H 2,∞ + (Ωv) ± (t, •) L 2 ( v ± (t, •) H 1,∞ + D t v ± (t, •) L ∞ ) + v ± (t, •) H 1,∞ ( (Ωv) ± (t, •) H 1 + D t (Ωv) ± (t, •) L 2 ) ,
and are hence bounded by C(A+B)Bε 2 t -1+ δ 1 2 after (B.1.6b), (B.1.6c) and the a-priori estimates. This finally implies that

t χ(t -σ D x )ΩΓNL w (t, •) L 2 ≤ C(A + B)Bε 2 t β+ δ 1 +δ 2 2 ,
which, together with (B.2.56) and the fact that β + δ 1 +δ 2 2 ≤ δ 0 2 , as δ 2 δ 1 δ 0 and β > 0 is as small as we want provided that σ is small, gives

θ 0 x t χ(t -σ D x )Ω[x n ΓNL w ](t, •) L 2 ≤ CBεt δ 0 2 .
Lemma B.2.10. There exists a constant C > 0 such that, for any ρ ∈ N, χ ∈ C ∞ 0 (R 2 ) equal to 1 in a neighbourhood of the origin, σ > 0 small, and every t ∈ [1, T ], (B.2.57)

|J|=1 1 |µ|=0 χ(t -σ D x )R µ U J (t, •) H ρ,∞ ≤ C(A + B)εt -1 2 +β+ δ 1 2 ,
for a small β > 0, β → 0 as σ → 0.

Proof. We warn the reader that, throughout the proof, C and β will denote two positive constants that may change line after line, with β → 0 as σ → 0. Moreover, since χ(t -σ D x ) is a bounded operator from L ∞ to H ρ,∞ with norm O(t σρ ), for any ρ ∈ N, we can reduce to prove that the L ∞ norm of χ(t -σ D x )R µ U J (t, •) is bounded by the right hand side of (B.2.57). We observe that this estimate is automatically satisfied when J is such that Γ J is a spatial derivative, as a consequence of a-priori estimate (1.1.11a). We therefore assume that Γ J is one of the Klainerman vector fields Ω, Z m , for m ∈ {1, 2}.

Introducing u J (t, x) := tu J -(t, tx), passing to the semiclassical setting (t → t, x → x t , and h := 1/t), and reminding that u J + = -u J -, inequality (B.2.57) becomes (B.2.58)

1 |µ|=0 Op w h χ(h σ ξ)(ξ|ξ| -1 ) µ u J -(t, •) L ∞ ≤ C(A + B)εh -1 2 -β- δ 1 2 .
We consider a Littlewood-Paley decomposition such that (B.2.59)

χ(h σ ξ) = χ(h -1 ξ) + k (1 -χ)(h -1 ξ)ψ(2 -k ξ)χ(h σ ξ), for some suitably supported χ ∈ C ∞ 0 (R 2 ), ψ ∈ C ∞ 0 (R 2 \ {0}
), and immediately observe that the above sum is restricted to indices k such that h 2 k h -σ . By the classical Sobolev injection, the uniform continuity of Op w h (ξ|ξ| -1 ) on L 2 , and a-priori estimate (1.1.11d), we derive that for any |µ| ≤ 1, every

t ∈ [1, T ], Op w h χ(h -1 ξ)(ξ|ξ| -1 ) µ u J (t, •) L ∞ = χ(D x )Op w h ((ξ|ξ| -1 ) µ ) u J (t, •) L ∞ u J -(t, •) L 2 ≤ CBεt δ 2 2 .
(B.2.60)

If we concisely denote by φ k (ξ) the k-th addend in decomposition (B.2.59) and introduce two smooth cut-off functions χ 0 , γ, with χ 0 radial and equal to 1 on the support of φ k , γ with sufficiently small support, we can write

Op w h φ k (ξ)(ξ|ξ| -1 ) µ u J = Op w h γ x|ξ| -ξ h 1/2-σ φ k (ξ)(ξ|ξ| -1 ) µ Op w h (χ 0 (h σ ξ)) u J + Op w h (1 -γ) x|ξ| -ξ h 1/2-σ φ k (ξ)(ξ|ξ| -1 ) µ Op w h (χ 0 (h σ ξ)) u J .
On the one hand, after proposition 1.2.30, the fact that 2 k h -σ , a-priori estimate (1.1.11d), and the uniform L 2 continuity of Op w h (χ 0 (h σ ξ)), we have that for any |µ| ≤ 1

(B.2.61) Op w h γ x|ξ| -ξ h 1/2-σ φ k (ξ)(ξ|ξ| -1 ) µ Op w h (χ 0 (h σ ξ)) u J (t, •) L ∞ h -1 2 -β Op w h (χ 0 (h σ ξ)) u J (t, •) L 2 + θ 0 (x)Ω h Op w h (χ 0 (h σ ξ)) u J (t, •) L 2 h -1 2 -β u J -(t, •) L 2 + Ωu J -(t, •) L 2 ≤ CBεh -1 2 -β- δ 1
2 .

On the other hand, using that (1 -γ)(z) = γ j 1 (z)z j , where γ j 1 (z) := (1 -γ)(z)z j |z| -2 is such that |∂ α z γ j 1 (z)| ≤ z -1-|α| , we derive from (1.2.52b), the commutation between M with Op w h (χ 0 (h σ ξ)), and lemma B.2.9, that

Op w h (1 -γ) x|ξ| -ξ h 1/2-σ φ k (ξ)(ξ|ξ| -1 ) µ Op w h (χ 0 (h σ ξ)) u J L ∞ h -β 1 γ,|ν|=0 (θ 0 (x)Ω h ) γ M ν Op w h (χ 0 (h σ ξ)) u J (t, •) L 2 ≤ CBεt β+ δ 0 2 .
Combining this estimate with (B.2.61) we deduce that

Op w h φ k (ξ)(ξ|ξ| -1 ) µ u J (t, •) L ∞ ≤ C(A + B)εh -1 2 -β- δ 1
1 |µ|=0 χ(t -σ D x ) x j D x D x µ (Γv) ± (t, •) L ∞ ≤ CBεt β+ δ 1 2 ,
with β > 0 small, β → 0 as σ → 0.

Proof. We warn the reader that, throughout the proof, C and β will denote two positive constants that may change line after line, with β → 0 as σ → 0. As Γv + = -Γv -, it is enough to prove the statement for Γv -.

If Γ is a spatial derivative, estimate (B.2.62) is just consequence of the uniform continuity of χ(t -σ D x ) on L ∞ and of (B.1.10b). We then assume that Γ ∈ {Ω, Z m , m = 1, 2} is a Klainerman vector field. First of all, we observe that by (B.1.9b) with w = (Γv) -and f = ΓNL kg , along with the classical Sobolev injection, (B.2.63)

1 |µ|=0 x j D x D x µ (Γv) -(t, •) L ∞ Z j (Γv) -(t, •) H 1 +t (Γv) -(t, •) H 2 + 1 µ=0 x µ j ΓNL kg (t, •) L ∞ .
From equality (B.

ΓNL kg (t, •) L ∞ 1 µ=0 χ(t -σ D x )(Γv) ± (t, •) H 1,∞ R µ 1 u ± (t, •) H 2,∞ + v ± (t, •) H 1,∞ χ(t -σ D x )(Γu) ± (t, •) H 2,∞ + v ± (t, •) H 1,∞ × 1 |µ|=0 ( R µ u ± (t, •) H 2,∞ + D t R µ u ± (t, •) H 1,∞ ) + t -3 ( v ± (t, •) H s + D t v ± (t, •) H s ) 1 |µ|,|ν|=0 x µ D 1 D x |D x | ν u ± (t, •) L 2 + t u ± (t, •) L 2 + t -3 1 |µ|=0 x µ v ± (t, •) L 2 + t v ± (t, •) L 2 ( u ± (t, •) H s + D t u ± (t, •) H s ) ≤ CABε 2 t -3 2 +β+ δ 1 2 . (B.2.64)
Moreover, as (B.2.65)

x j Q kg 0 (Γv) ± , D 1 u ± L ∞ 1 |µ|,ν=0 x j D x D x µ (Γv) -(t, •) L ∞ R ν 1 u ± (t, •) H 2,∞ , (B.2.66) x j G kg 1 v ± , Du ± L ∞ 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ ( u ± (t, •) H 2,∞ + D t u ± (t, •) H 1,∞ ) ,
and by lemma B.2.4 with L = L ∞ , w = u, and s > 0 large enough so that N (s) ≥ 3,

x j Q kg 0 (v ± , D 1 (Γu) ± ) L ∞ 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ χ(t -σ D x )(Γu) ± (t, •) H 2,∞ + t -3 1 |µ|,ν=0 x µ x ν j v ± (t, •) L 2 + t x ν j v ± (t, •) L 2 ( u ± (t, •) H s + D t u ± (t, •) H s ) , (B.2.67)
we derive that (B.2.68)

x j ΓNL kg (t, •) L ∞ ≤ CAεt -1 2 1 |µ|,ν=0 x j D x D x µ (Γv) -(t, •) L ∞ + C(A + B)Bε
(Γv) -(t, •) L ∞ + x j D x D x (Γv) -(t, •) L ∞ ≤ CBεt 1+ δ 2 2 .
If we take any smooth cut-off function χ and use equality (B.1.9b), instead of (B.2.63) we find that (B.2.70)

1 |µ|=0 χ(t -σ D x ) x j D x D x µ (Γv) -(t, •) L ∞ Z j (Γv) -(t, •) H 1 + t χ(t -σ D x )(Γv) -(t, •) L ∞ + 1 µ=0 χ(t -σ D x ) x µ j ΓNL kg (t, •) L ∞ ,
where now

χ(t -σ D x ) x j ΓNL kg (t, •) L ∞ x j ΓNL kg (t, •) L ∞ ≤ C(A + B)Bε 2 t 1 2 + δ 2
χ(t -σ D x ) x j (Γv) -(t, •) L ∞ + χ(t -σ D x ) x j D x D x (Γv) -(t, •) L ∞ ≤ CBεt 1 2 + δ 2 2 .
Finally, by means of lemma B.2.2 with L = L ∞ , w 1 = x(Γv) ± , and s > 0 such that N (s) ≥ 2, we derive that for any χ

∈ C ∞ 0 (R 2 ) there is some χ 1 ∈ C ∞ 0 (R 2 ) such that χ(t -σ D x )x j Q kg 0 ((Γv) ± , D 1 u ± ) L ∞ 1 |µ|,ν=0 χ 1 (t -σ D x ) x j D x D x µ (Γv) -(t, •) L ∞ χ(t -σ D x )R ν 1 u ± (t, •) H 2,∞ + 1 µ=0 t -2 x µ j (Γv) ± (t, •) L 2 u ± (t, •) H s .
Then, combining such inequality with (B.2.66), (B.2.67), together with (B.1.17), (B.2.71), and all the other inequalities to which we already referred before, from (B.1.20a) we find that . This improvement, that will be useful to derive (B.4.30), is showed in the following lemma. Lemma B.2.12. Let I be a multi-index of length 1 and r I,NF kg be given by (B.2.46). There exists a constant C > 0 such that, for any ρ ∈ N, χ ∈ C ∞ 0 (R 2 ), equal to 1 in a neighbourhood of the origin, σ > 0 small, j = 1, 2, and every t ∈ [1, T ],

χ(t -σ D x ) x j ΓNL kg (t, •) L ∞ ≤ C(A + B)ε 2 t δ 2 ,
(B.2.72) χ(t -σ D x ) x j r I,NF kg (t, •) L 2 ≤ C(A + B)ABε 3 t -1 2 +β+ δ+δ 1 2 ,
with β > 0 small, β → 0 as σ → 0.

Proof. Let us remind the explicit expression (B.2.47) of r I,NF kg and consider the cubic term x j Γ I NL kg (D 1 u). Reminding (1.1.5) and applying lemma B.2.2 with L = L 2 and s > 0 sufficiently large so that N (s) ≥ 2, together with (B.1.26a) and a-priori estimates, we derive that there is some χ

1 ∈ C ∞ 0 (R 2 ) such that χ(t -σ D x ) x j Γ I NL kg (D 1 u) (t, •) L 2 χ 1 (t -σ D x ) x j Γ I NL kg (t, •) L 2 R 1 u ± (t, •) L ∞ + t -2 x j Γ I NL kg (t, •) L 2 u ± (t, •) H s ≤ CAεt -1 2 χ 1 (t -σ D x ) x j NL I kg (t, •) L 2 + C(A + B)Bε 2 t -1 . (B.2.73)
Then, recalling (B.1.20a) and using again lemma B.2.2 with L = L 2 , w 1 = (Γv) ± , and s as before, in order to estimate the contribution coming from the first quadratic term in the right hand side of (B.1.20a), we find that there is a new

χ 2 ∈ C ∞ 0 (R 2 ) such that χ 1 (t -σ D x ) x j NL I kg (t, •) L 2 1 |µ|=0 χ 2 (t -σ D x ) x j D x D x µ (Γv) ± (t, •) L ∞ u ± (t, •) H 1 + t -2 x j (Γv) ± (t, •) L 2 u ± (t, •) H s + 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ ( (Γu) ± (t, •) H 1 + u ± (t, •) H 1 + D t u ± (t, •) L 2 ) ≤ C(A + B)Bε 2 t β+ δ+δ 1 2 ,
where the latter estimate is obtained from (B.1.5a) with s = 0, (B.1.10b), (B.1.17) with k = 1, (B.2.62) and a-priori estimates. This implies, combined with (B.2.73), that

χ(t -σ D x ) x j NL I kg (D 1 u) (t, •) L 2 ≤ C(A + B)ABε 3 t -1 2 +β+ δ+δ 1 2 ,
and from (B.2.47), (B.1.10b) and a-priori estimates,

χ(t -σ D x ) x j r I,NF kg (t, •) L 2 χ(t -σ D x ) x j NL I kg (D 1 u) (t, •) L 2 + 1 µ=0 t σ x µ j v ± (t, •) L ∞ + x µ j D x D x v ± (t, •) L ∞ v I ± (t, •) L 2 v ± (t, •) H 2,∞ ≤ C(A + B)ABε 3 t -1 2 +β+ δ+δ 1 2 ,
which concludes the proof of the statement.

Lemma B.2.13. Let I be a multi-index of length 2. There exists a constant C > 0 such that, for every j = 1, 2, t ∈ [1, T ],

(B.2.74)

x j Γ I NL kg (t, •) L 2 ≤ C(A + B)Bε 2 t 1 2 +β+ δ 1 +δ 2 2
, with β > 0 small, β → 0 as σ → 0.

Proof. We remind the reader about (B.1.23). Instead of using (B.1.24), which was obtained by Sobolev injection, we apply lemma B.2.4 with L = L 2 , Γw = Γ I 2 u, s > 0 sufficiently large so that N (s) ≥ 3, and exploit the fact that we have an estimate of the H ρ,∞ norm of D 1 u I 2 when truncated for frequencies less or equal than t σ (see lemma B.2.10). Therefore, for (I 1 , I 2 ) ∈ I(I)

such |I 1 | = |I 2 | = 1 we obtain that x j Q kg 0 v I 1 ± , D 1 u I 2 ± (t, •) L 2 1 µ=0 x µ j v I 1 ± (t, •) L 2 χ(t -σ D x )u I 2 ± (t, •) H 2,∞ + t -3 ( u ± (t, •) H s + D t u ± (t, •) H s )   2 |µ|=0 x µ v I 1 ± (t, •) L 2 + 1 |µ|=0 t x µ v I 1 ± (t, •) L 2   ≤ C(A + B)Bε 2 t 1 2 +β+ δ 1 +δ 2 2
, last estimate following from lemma B.2.10 together with (B.1.5a), (B.1.17) with k = 1, (B.1.28), a-priori estimates, and the fact that δ 1 , δ 2 1 are small. Consequently, from the following inequality

x j Γ I NL kg (t, •) L 2 1 µ=0 R µ 1 u ± (t, •) H 2,∞ |J|≤2 µ=0,1 x µ j (Γ J v) -(t, •) L 2 + 1 |µ|=0 x j D x D x µ v ± (t, •) L ∞ u I ± (t, •) H 1 + |J|<2 u J ± (t, •) H 1 + D t u J ± (t, •) L 2 + |I 1 |=|I 2 |=1 x j Q kg 0 v I 1 ± , D 1 u I 2 ± (t, •) L 2 ,
(χ(h σ ξ))L µ v(t, •) L 2 ≤ CBεt δ 2 2 , (B.2.75b) for every t ∈ [1, T ].
Proof. We warn the reader that, throughout the proof, C and β will denote two positive constants that may change line after line, with β > 0 is small as long as σ is small.

(χ(h σ ξ))L m v(t, •) L 2 Z m V (t, •) L 2 + χ(t -σ D x )Z m (v N F -v -)(t, •) L 2 + v(t, •) L 2 + χ(t -σ D x )[x m r N F kg ](t, •) L 2 , (
v N F -v -= - i 2 [(D t v)(D 1 u) -(D 1 v)(D t u) + D 1 [vD t u] -D x [vD 1 u]] (B.3.2) χ(t -σ D x ) x m Z n (v N F -v -)(t, •) L 2 + χ(t -σ D x ) x m (Z n v) --v I,NF (t, •) L 2 ≤ C(A + B)Bε 2 t β+ δ 1 +δ 2 2 .
The same estimates hold true when Z n is replaced with Ω.

Proof

t σ 1 µ,ν=0 x µ m D x D x ν v ± (t, •) L ∞ ( (Z m u) ± (t, •) L 2 + u ± (t, •) L 2 + D t u ± (t, •) L 2 ) ,
and then by the right hand side of (B.

µ,ν=0 χ 1 (t -σ D x ) x µ m D x D x ν (Z m v) ± (t, •) L ∞ u ± (t, •) L 2 + 1 µ=0 t -N (s) x µ m (Z m v) ± (t, •) L 2 u ± (t, •) H s ,
for some smooth cut-off χ 1 , and hence by the right hand side of (B.

2iΩ v N F -v -= (D t Ωv)(D 1 u) -(D 1 Ωv)(D t u) + D 1 [(Ωv)(D t u)] -D x [(Ωv)(D 1 u)] + (D t v)(D 1 Ωu) -(D 1 v)(D t Ωu) + D 1 [v(D t Ωu)] -D x [v(D 1 Ωu)] -(D t v)(D 2 u) + (D 2 v)(D t u) -D 2 [v(D t u)] + D x [v(D 2 u)]
and applies the same argument as above to recover the wished estimates.

Lemma B.3.2. Let v NF be defined as in (3.1.3). There exists a constant C > 0 such that, for any

χ ∈ C ∞ 0 (R 2 ), σ > 0 small, m = 1, 2, (B.3.3a) Op w h (χ(h σ ξ))[tZ n v N F (t, tx)] L 2 (dx) ≤ CBεt δ 2 2 , (B.3.3b) Op w h (χ(h σ ξ))L m [tZ n v N F (t, tx)] L 2 (dx) ≤ CBεt δ 1 2 , for every t ∈ [1, T ].
Proof. Let us write Z n v N F as follows:

(B.3.4) Z n v N F = Z n (v N F -v -) + (Z n v) --v I,NF + v I,NF + D n D x v N F + D n D x (v --v N F ),
χ(t -σ D x )D n D x -1 v N F (t, •) L 2 ≤ χ(t -σ D x )D n D x -1 v -(t, •) L 2 + χ(t -σ D x )D n D x -1 (v --v N F )(t, •) L 2 ≤ CBεt δ 2 ,
we immediately obtain (B.3.3a).

From (B.3.4) we also derive that

Op w h (χ(h σ ξ))L m [tZ n v N F (t, tx)] L 2 (dx) Op w h (χ(h σ ξ))L m tZ n (v N F -v -)(t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m t (Z n v) --v I,NF (t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m tv I,NF (t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m tD n D x -1 v N F (t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m tD n D x -1 (v --v N F )(t,
(χ(h σ ξ))L m tD n D x -1 v N F (t, tx) L 2 ≤ CBεt δ 2 2 .
The remaining L 2 norms in the right hand side of (B.3.5) are estimated reminding definition (1.2.68) of L m and using the fact that

(B.3.6) Op w h (χ(h σ ξ))L m [tw(t, tx)] L 2 (dx) χ(t -σ D x )[x m w(t, •)] L 2 + t χ(t -σ D x )w(t, •) L 2 .
Therefore, by (B.2.81) and lemma B.3.1 we derive that

Op w h (χ(h σ ξ))L m tZ n (v N F -v -)(t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m t (Z n v) --v I,NF (t, tx) L 2 (dx) ≤ C(A + B)Bε 2 t β+ δ+δ 1 2 ,
while from (3.1.8a), a-priori estimates, together with the following inequality

χ(t -σ D x )[x m (v --v N F )](t, •) L 2 1 µ,ν=0 t σ x µ m D x D x ν v -(t, •) L ∞ u ± (t, •) L 2 ≤ C(A + B)Bε
(χ(h σ ξ))L m tD n D x -1 (v --v N F )(t, tx) L 2 ≤ C(A + B)Bε 2 t σ+ (δ+δ 2 ) 2 .
In the following lemma we are going to prove that the product of the semiclassical wave function u with the Klein-Gordon one v enjoys a better L 2 (resp. L ∞ ) estimate than the one roughly obtained by taking the L 2 (resp. L ∞ ) norm of the former times the L ∞ norm of the latter. Estimates

v u(t, •) L 2 v(t, •) L ∞ u(t, •) L 2 ≤ CABε 2 h -δ 2 , v u(t, •) L ∞ v(t, •) L ∞ u(t, •) L ∞ ≤ CA 2 ε 2 h -1 2 -δ 2 ,
which follows from (B.2.1a), (B.3.8), (B.3.9), can be in fact improved of a factor h 1/2 (see (B.3.7)). This comes from the fact that the main contribution to u is localized around manifold Λ w introduced in (3.2.43), whereas v concentrates around Λ kg defined in (1.2.66), and these two manifolds are disjoint. Lemma B.3.3. Let h = t -1 , u, v be defined in (3.2.2), a 0 (ξ) ∈ S 0,0 (1), and b 1 (ξ) = ξ j or b 1 (ξ) = ξ j ξ k |ξ| -1 , with j, k ∈ {1, 2}. There exists a constant C > 0 such that, for any χ, χ 1 ∈ C ∞ 0 (R 2 ), σ > 0, and every t ∈ [1, T ], we have that

(B.3.7a) [Op w h (χ(h σ ξ)a 0 (ξ)) v(t, •)][Op w h (χ 1 (h σ ξ)b 1 (ξ)) u(t, •)] L 2 ≤ C(A + B)Bε 2 h 1 2 -β- δ+δ 1 2 , (B.3.7b) [Op w h (χ(h σ ξ)a 0 (ξ)) v(t, •)][Op w h (χ 1 (h σ ξ)b 1 (ξ)) u(t, •)] L ∞ ≤ C(A + B)Bε 2 h -β-δ+δ 1 2 ,
with β > 0 small, β → 0 as σ → 0.

Proof. Before entering in the details of the proof, we warn the reader that C and β denote two positive constants that may change line after line, with β → 0 as σ → 0. Also, we will denote by R(t, x) any contribution, in what follows, that satisfies inequalities (B.3.7), and by χ 2 a smooth cut-off function, identically equal to 1 on the support of χ 1 , so that First of all, we take γ ∈ C ∞ 0 (R 2 ) equal to 1 in a neighbourhood of the origin and with sufficiently small support, and define

Op w h (χ 1 (h σ ξ)) u = Op w h (χ 1 (h σ ξ))Op w h (χ 2 (h σ ξ)) u,
v Λ kg (t, x) := Op w h γ x -p (ξ) √ h χ(h σ ξ)a 0 (ξ) v(t, x), v Λ c kg (t, x) := Op w h (1 -γ) x -p (ξ) √ h χ(h σ ξ)a 0 (ξ) v(t, x), with p(ξ) := ξ , so that (B.3.10) Op w h (χ(h σ ξ)a 0 (ξ)) v = v Λ kg + v Λ c kg .
The following estimates hold:

v Λ kg (t, •) L ∞ ≤ CAεh -β , (B.3.11a) v Λ c kg (t, •) L ∞ ≤ CBεh 1 2 -β- δ 1 2 . (B.3.11b)
The former one is a straight consequence of proposition 1.2.39 with p = +∞ and (B.3.9). On the other hand, if we write

(1 -γ)
x -p (ξ) √ h χ(h σ ξ)a 0 (ξ) = x -p (ξ) √ h χ(h σ ξ)a 0 (ξ) L j v(t, •)

L ∞ + 2 j=1 √ h Op w h γ j 1 x -p (ξ) √ h ∂ j χ(h σ ξ)a 0 (ξ) v(t, •) L ∞ + 2 j=1 |α|=2 √ h Op w h (∂ α γ j 1 )
x -p (ξ) √ h χ(h σ ξ)a 0 (ξ)(∂ α ξ p )(ξ) v(t, •)

L ∞ + Op w h (r(x, ξ)) v(t, •) L ∞ , (B.3.12)
with r ∈ h 1-β S 1 2 ,σ ( x-p (ξ) √ h -1 ). Since γ j 1 vanishes in a neighbourhood of the origin, we derive from inequality (3. For some suitably supported χ 0 ∈ C ∞ 0 (R 2 ), ϕ ∈ C ∞ 0 (R 2 \ {0}), we also consider the following decomposition x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) u(t, x), and observe that Moreover, using lemma 1. Then the end of the proof relies on the fact that θ h (x)(|x| 2 -1) u k Λw can be expressed in terms of hM u. In fact, for a fixed N ∈ N and up to some negligible multiplicative constants, we have from proposition 1. with δ ij = 1 being the Kronecker delta. One the one hand, since the first contribution to the above right hand side is still supported for |x| < 1-ch 2σ , we can multiply and divide it by |x| 2 -1 so that it writes as linear combination of terms of the form h

u k Λ c w (t, •) L 2 h 1 2 -β   u(t, •) L 2 + 1 µ,|ν|=0 (θ 0 (x)Ω h ) µ M ν Op w h (χ 2 (h σ ξ)) u(t, •) L 2   ≤ CBεh
  h |α|-1 2 -β 2 -k(|α|-1) + 1≤j≤|α| h |α|-j( 1 2 -σ) 2 k(j+1-|α|) h -1 2 -β   ×   u(t, •) L 2 + 1 µ,|ν|=0 (θΩ h ) µ Op w h (χ 1 (h σ ξ) u(t, •) L 2   ≤ CBεh 1 2 -β- δ 1 2 .
Finally, by integrating in dydζ and using (1.2.24) in (B.3.16) we find that r N (x, ξ) can be written as for some new smooth compactly supported θ N , γ j , φ j k .From the last part of proposition 1.2.31 then follows that the quantization of the above integral is a bounded operator from L 2 to L ∞ , with norm controlled by as follows by (B.2.77) and (3.1.15), (A.26b) with ρ = 2 (as consequence of lemma A.8), together with a-priori estimates, we can also suppose v -(resp. u -) be replaced with v N F (resp. u N F ), up to some new R(t, x). This reduces us to prove that

j≤N i≤6 h N -j( 1 2 -σ) 2 k(1+j-N ) (h -1 2 +σ 2 k ) i (h -1 2 k ) h if N
[χ(t -σ D x )a 0 (D x )v N F ][χ(t -σ D x )b 1 (D x )u N F ][χ(t -σ D x )b 0 (D x )u -](t, •) L ∞ ≤ C(A + B)ABε 3 t -5 2 +β+ δ+δ 1 2 ,
or rather, reminding (1.1.11a), to show that

[χ(t -σ D x )a 0 (D x )v N F ][χ(t -σ D x )b 1 (D x )u N F ](t, •) L ∞ ≤ C(A + B)Bε 2 t -2+β+ δ+δ 1 2 .
But after writing the above product in the semi-classical setting and reminding definition (3.2.2), one can immediately check that this estimate is satisfied thanks to (B.3.7b), which concludes the proof of (B. 3.24).

The last part of the statement follows from (3.1.11), the fact that

χ(t -σ D x ) - D x D x (v + -v -) NL w + D 1 D x -1 (v + -v -) NL w (t, •) L ∞ ≤ CA 3 ε 3 t -3+σ
for every t ∈ [1, T ], which is consequence of (B.1.3b) and a-priori estimate (1.1.11b), and from the observation that the remaining contributions to r N F kg are products of the form

[a 0 (D x )v -][b 1 (D x )u -]R 1 u -,
with a 0 (ξ) equal to 1 or to ξ j ξ -1 , and b 1 (ξ) equal to ξ 1 or to ξ j ξ 1 |ξ| -1 , for j = 1, 2. Proof. We warn the reader that we will denote by C and β two positive constants that may change line after line, with β → 0 as σ → 0. We also denote by R(t, x) any contribution verifying

χ(t -σ D x ) x n R(t, •) L 2 (dx) ≤ C(A + B) 2 Bε 3 t -1+β+
χ(t -σ D x ) -x n D x D x (v + -v -) NL w + x n D 1 D x -1 (v + -v -) NL w (t, •) L 2 t σ 1 µ=0 x µ n v ± (t, •) L 2 NL w (t, •) L ∞ ≤ CA 2 Bε 3 t -1+σ+ δ and χ(t -σ D x ) -x m x n D x D x (v + -v -) NL w + x m x n D 1 D x -1 (v + -v -) NL w (t, •) L 2 t σ 1 µ 1 ,µ 2 =0 x µ 1 m x µ 2 n v ± (t, •) L 2 NL w (t, •) L ∞ ≤ C(A + B)ABε 3 t σ+ δ 2 .
Therefore, since from (3. This means that we can actually replace u -by u N F up to some new R(t, x). Furthermore, we can also substitute χ 1 (t -σ D x )[x k m x n a 0 (D x )v -] with χ(t -σ D x )[x k m x n a 0 (D x )v N F -], for any k ∈ {0, 1}, up to a new remainder R(t, x) in consequence of a-priori estimate (1.1.11a), the fact that (B.3.30)

χ(t -σ D x ) [χ 1 (t -σ D x )[x n a 0 (D x )v -]][χ(t -σ D x )b 1 (D x )(u N F -u -)][χ(t -σ D x )b 0 (D x )u -] (t, •)(t, •) L 2 1 |µ|=0 x n D x D x µ v ± (t, •) L ∞ χ(t -σ D x )b 1 (D x )(u N F -u -) L ∞ u ± (t, •) L 2 ≤ C(A + B)A 2 Bε 4 t -
u N F (t, •) L 2 ≤ CBεt δ 2 ,
(see (B.2.1a) in semi-classical coordinates), and the following inequalities .

(B.3.31a) χ 1 (t -σ D x ) x n a 0 (D x )(v N F -v -) (t, •) L ∞ 1 µ,ν,κ=0 t σ x µ n D x D x ν v ± (t, •) L ∞ R κ 1 u ± (t, •) L ∞ ≤ C(A + B)Aε 2 t -1 2 +σ+ δ 2 2 and (B.3.31b) χ 1 (t -σ D x ) x m x n a 0 (D x )(v N F -v -) (t, •) L ∞ 1 µ 1 ,µ 2 ,ν,κ=0 x µ 1 m x µ 2 n D x D x ν v ± (t, •) L ∞ R κ 1 u ± (t, •) L ∞ ≤ C(A + B)Aε 2 t
Passing to the semi-classical setting, this corresponds to prove that .

First of all let us notice that, from the commutation of x n with Op w h (a 0 (ξ)) and definition (1.2.68) of L n ,

x n Op w h (a 0 (ξ)) v = hOp w h (a 0 (ξ))L n v + Op w h a 0 (ξ) .

ξ n ξ v - h 2i
The first norm in the above right hand side satisfies an inequality analogous to (B.

B.4 The sharp decay estimate of the Klein-Gordon solution with a Klainerman vector field

This last section is devoted to prove that, for any admissible vector field Γ, the L ∞ (R 2 ) norm of functions (Γv) ± , when restricted to moderate frequencies less or equal than t σ , for some small σ > 0, decays in time at the same sharp rate t -1 of the two-dimensional linear Klein-Gordon solution. This result is proved in lemma B.4.14 under the hypothesis that a-priori estimates (1.1.11) are satisfied in some fixed interval [1, T ], with ε 0 < (2A + B) -1 and 0 < δ δ 2 δ 1 δ 0 1 sufficiently small, and is fundamental when proving lemmas 2.1.2 and 2.1.3. All the other lemmas of this section are to be meant as preparatory intermediate results.

|I 1 |+|I 2 |≤2 |I 1 |<2 χ(t -σ D x ) x n Q kg 0 (v I 1 ± , Du I 2 ± ) (t, •) L 2 (dx) ≤ C(A + B)Bε 2 t β+ δ+δ 2 2 ,
with β > 0 small such that β → 0 as σ → 0.

Proof. We estimate the L 2 norms in the left hand side of (B.4.1) separately. ) µ v I 1 ± with |µ| = 0, 1, and s large enough so that N (s) ≥ 2, to derive that

χ(t -σ D x ) x n Q kg 0 (v I 1 ± , D 1 u I 2 ± ) (t, •) L 2 (dx) 1 |µ|=0 χ 1 (t -σ D x ) x n D x D x µ v I 1 ± (t, •) L ∞ u I 2 ± (t, •) H 1 + |µ|=0,1,2 |ν|=0,1 t -2 x µ v I 1 ± (t, •) L 2 + t x ν v I 1 ± (t, •) L 2 ( u ± (t, •) H s + D t u ± (t, •) H s ) ≤ CB 2 ε 2 t δ 1 +δ 2 2
, where last estimate is deduced using (B.1.5a), (B.1.17 Proof. We warn the reader that, throughout the proof, we denote by C and β two positive constants that may change line after line, with β → 0 as σ → 0.

Let v I,NF be the function defined in (B.2.27) for a generic multi-index I of length 1 or 2, and I 1 , I 2 two multi-indices such that Γ I 1 = Γ, Γ I 2 = Z n Γ. Using (2.1.15b) we rewrite Z n V NF Γ as follows: x µ 1 x δ k1 µ 2 m

Z n V NF Γ = Z n V NF Γ -(Γv) -+ (Z n Γv) --v I 2 ,
x µ 3 n v -(t, •)

L 2 (dx) + t x µ 2 n v -(t, •) L 2 u ± (t, •) H s R 1 u -(t, •) L ∞ + t -3 x k m x n a 0 (D x )v -(t, •) L ∞ (dx) (Γu) -(t, •) L 2 u -(t, •) H s χ 1 (t -σ D x )[x k m x n a 0 (D x )v -] χ(t -σ D x )b 1 (D x )(Γu) -χ(t -σ D x )R 1 u - L 2 (dx) + CAB 2 ε 2 t -(1-k)+ δ+δ 2 2 .
Using instead (B.2.24) with L = L ∞ along with (1.1.11) and (B.1.10b),

χ(t -σ D x ) [[a 0 (D x )v -] [b 1 (D x )(Γu) -] R 1 u -] L ∞ χ 1 (t -σ D x )a 0 (D x )v -χ(t -σ D x )b 1 (D x )(Γu) -χ(t -σ D x )R 1 u -L ∞ + CAB 2 ε 2 t -5 2 + δ+δ 2 2 .
Secondly, we can assume that in (B.

u N F (t, •) H ρ,∞ + Ru N F (t, •) H ρ,∞ ≤ CBεt -1 2 .
Therefore, in order to conclude the proof we must prove that, for some χ, χ 1 ∈ C ∞ 0 (R 2 ) and k ∈ {0, 1},

χ 1 (t -σ D x )[x k m x n a 0 (D x )(Γv) -] χ(t -σ D x )b 1 (D x )u N F χ(t -σ D x )R 1 u - L 2 (dx) + χ 1 (t -σ D x )[x k m x n a 0 (D x )v -] χ(t -σ D x )b 1 (D x )(Γu) -χ(t -σ D x )R 1 u - L 2 (dx) + χ 1 (t -σ D x )[x k m x l n a 0 (D x )v -] χ(t -σ D x )b 1 (D x )u N F χ(t -σ D x )R 1 (Γu) - L 2 (dx) ≤ C(A + B) 2 Bε 3 t -1+k+β
and

χ 1 (t -σ D x )a 0 (D x )(Γv) -χ(t -σ D x )b 1 (D x )u N F χ(t -σ D x )R 1 u -L ∞ (dx) + χ 1 (t -σ D x )a 0 (D x )v -χ(t -σ D x )b 1 (D x )(Γu) -χ(t -σ D x )R 1 u -L ∞ (dx) + χ 1 (t -σ D x )a 0 (D x )v -χ(t -σ D x )b 1 (D x )u N F χ(t -σ D x )R 1 (Γu) -L ∞ (dx)
≤ C(A + B) 2 Bε 3 t -5 2 +β .

Actually, using (1.1.11a), (B.2.57), and passing to the semi-classical framework and unknowns with V Γ introduced in (B.4.20), u, v in (3.2.2), and u I (t, x) = t -1 (Γu) -(t, t -1 x), above inequalities will follow respectively from

Moreover, one can check that

Op w h (χ 1 (h σ ξ))[x n Op w h (a 0 (ξ)) V Γ ] Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •)

L 2 (dx)
≤ Op w h χ 1 (h σ ξ)a 0 (ξ)

ξ n ξ V Γ [Op w h (χ(h σ ξ)b 1 (ξ)) u](t, •) L 2 + CABε 2 h 1 2 -β , Op w h (χ 1 (h σ ξ))[x m x n Op w h (a 0 (ξ)) V Γ ] Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) L 2 (dx)
Op w h χ 1 (h σ ξ)a 0 (ξ)

ξ n ξ V Γ Op w h χ(h σ ξ)b 1 (ξ)
≤ C(A + B) 2 Bε 3 t β , Op w h (χ(h σ ξ))L m t(tx n )Q kg 0 v ± , Q w 0 (v ± , D 1 v ± ) (t, tx) L 2 (dx) ≤ C(A + B)ABε 3 t δ+δ 2 2 ,
with β > 0 such that β → 0 as σ, δ 0 → 0. Proof. As this estimate is evidently satisfied when I is such that Γ I is a spatial derivative after a-priori estimate (1.1.11b), we focus on proving the statement for Γ I ∈ {Ω, Z m , m = 1, 2} being a Klainerman vector field. For simplicity, we refer to Γ I simply by Γ.

Instead of proving the result of the statement directly on (Γv) ± we show that Γ . The most important feature that will provide us with (B.4.51) is that the uniform norm of all involved non-linear terms is integrable in time. Before going into the details, we also remind the reader our choice to denote by C, β and β some positive constants that may change line after line, with β → 0 (resp. β → 0) as σ → 0 (resp. as σ, δ 0 → 0).

Let us consider V Γ (t, x) := tV NF Γ (t, tx), operator Γ kg as follows x -p (ξ) √ h χ 1 (h σ ξ) Op w h (χ(h σ ξ)) V Γ (t, x),

so that Op w h (χ(h σ ξ)) V Γ (t, •) = V Γ Λ kg + V Γ Λ c
L ∞ 2 |µ|=0 h 1 2 -β Op w h (χ(h σ ξ))L µ V Γ (t, •) L 2 ≤ CBεt -1 2 +β .
On the other hand, as V NF Γ is solution to (B.4.23) an explicit computation shows that V Γ satisfies the following semi-classical pseudo-differential equation:

[D t -Op w h (x • ξ -ξ )] V Γ (t, x) = h -1 NL kg,c Γ (t, tx) -δ Z 1 h -1 Q kg 0 v ± , Q w 0 (v ± , D 1 v ± ) (t,
L ∞ ≤ h 1+σ Γ kg θ h (x)Op w h ((∂χ(h σ ξ) • (h σ ξ)) V Γ (t, •) L ∞ + h N V Γ (t, •) L 2 ,
where θ h (x) is a smooth cut-off function supported in closed ball B 1-ch 2σ (0), with c > 0 small. Denoting (∂χ)(ξ)•ξ concisely by χ(ξ), we observe from proposition 1.2.39 with p = +∞, together with the uniform continuity on L ∞ of operator χ(t -σ D x ), the definition of V Γ in terms of V NF Γ , and (B.4.16), that h 1+σ Γ kg θ h (x)Op w h ( χ(h σ ξ)) V Γ (t, •)

L ∞ h 1-β θ h (x)Op w h ( χ(h σ ξ)) V Γ (t, •) L ∞ ≤ t β θ h • t χ(t -σ D x )(Γv) -(t, •) L ∞ + C(A + B)Bε 2 t -5 4 +β .
Using the fact that, for θ j h (z) := θ h (z)z j ,

θ h x t (Ωv) -= t θ 1 h x t ∂ 2 v --θ 2 h x t ∂ 1 v - and θ h x t (Z m v) -= t θ m h x t ∂ t v -+ θ h x t ∂ m v -+ θ h x t D m D x v -, m = 1, 2,
and making some commutations, we can express (Γv) -in terms of v -and its derivatives up to a loss in t. Thus, from the classical Sobolev injection combined inequality (B.1.2), we obtain that

t -β χ(t -σ D x )θ h • t (Γv) -(t, •) L ∞ t -N (s)+1+β ( D t v ± (t, •) H s + v ± (t, •) H s ) ≤ CBεt -3 2 ,
last estimate following by taking s > 0 large enough to have N (s) ≥ 3 and using a-priori estimates along with (B.

Introduction 1 Main

 1 Theorem and Preliminary Results 1.1 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Paradifferential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Semi-classical pseudodifferential calculus . . . . . . . . . . . . . . . . . . . 1.2.3 Semi-classical Operators for the Wave Solution: Some Estimates . . . . . 1.2.4 Operators for the Klein-Gordon solution: some estimates . . . . . . . . . . 2 Energy Estimates 2.1 Paralinearization and Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Paralinearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Estimates of quadratic terms . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Normal forms and energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 A first normal forms transformation and the energy inequality . . . . . . . 2.2.2 A second normal forms transformation. . . . . . . . . . . . . . . . . . . . 2.2.3 Propagation of the energy estimate . . . . . . . . . . . . . . . . . . . . . . 3 Uniform Estimates 3.1 Semilinear normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Normal forms for the Klein-Gordon equation . . . . . . . . . . . . . . . . 3.1.2 Normal forms for the wave equation . . . . . . . . . . . . . . . . . . . . .

3. 2 From

 2 PDEs to ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Derivation of the ODE and propagation of the uniform estimate on the Klein-Gordon component . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The derivation of the transport equation . . . . . . . . . . . . . . . . . . . 3.3 Analysis of the transport equation and end of the proof . . . . . . . . . . . . . . 3.3.1 The inhomogeneous transport equation . . . . . . . . . . . . . . . . . . . .

( 1 .

 1 1.13) G(ξ, . . . , ξ k ) = 0.

( 1 . 2 ( 1 +

 121 2.3) M m r (a; n) = sup |β|≤n sup η∈R |η|) |β|-m ∂ β η a(•, η) W r,∞ .These definitions extend to matrix valued symbolsa ∈ S m r (a ∈ Σ m r ), m ∈ R, r ∈ N. If a ∈ S m r (resp. a ∈ Σ m r ), it is said of order m. Definition 1.2.4. An admissible cut-off function ψ(ξ, η) is a C ∞ function on R d × R d such that (i) there are 0 < ε 1 < ε 2 < 1 and (1.2.4) ψ(ξ, η) = 1, for |ξ| ≤ ε 1 (1 + |η|) ψ(ξ, η) = 0, for |ξ| ≥ ε 2 (1 + |η|);

  Op B R associated to a symbol a(x, η) and acting on a function w asOp B R (a(x, η))w(x) := 1 (2π) d e ix•η δ χ a (x, η) ŵ(η)dη , with δ χ a (x, η) := 1 (2π) d e i(x-y)•ζ 1 -χ ζ η -χ η ζ a(y, η)dydζ .

( 1 .

 1 2.11) Op B (a)w H s M m 0 a; d 2 + 1 w H s+m . Proposition 1.2.8. (i) Let a(x, η) = a 1 (x)b(η), with a 1 ∈ L ∞ (R2 ) and b(η) bounded, supported in some ball centred in the origin and such that |∂ α b(η)| α |η| -|α|+1 for any α ∈ N 2 with |α| ≥ 1. Then Op B (a(x, η)) : L 2 → L 2 is bounded and for any w ∈ L 2 (R 2 )

  with K(x, y) := 1 (2π) 4 e ix•η+iy•ζ χ ζ η b(η)dηdζ and χ is an admissible cut-off function. After the hypothesis on b we have that for every α, β ∈ N 2 ,

Remark 1 . 2 . 10 .Corollary 1 . 2 . 11 .

 12101211 x, ξ)D α x b(x, ξ) is well defined in j<r S m+m -j r-j ; (ii) Op B (a)Op B (b) -Op B (a b) is an operator of order ≤ m + m -r, and for all s ∈ R, there exists a constant C > 0 such that, for all a∈ S m r (R d ), b ∈ S m r (R d ), and w ∈ H s+m+m -r (R d ), Op B (a)Op B (b)w -Op B (a b)w H s ≤ C M m r (a; n)M m 0 (b; n 0 ) + M m 0 (a; n)M m r (b; n 0 ) w H s+m+m -r ,wheren 0 = d 2 + 1, n = n 0 + r. Moreover, Op B (a)Op B (b) -Op B (a b) = σ r (x, D x ) with σ r (x, ξ) = (σ a σ b )(x, ξ) -σ a b (x, ξ) + |α|=r 1 r!(2π) d e i(x-y)•ζ 1 0 ∂ α ξ σ a (x, ξ + tζ)(1 -t) r-1 dt θ(ζ, ξ)D α x σ b (y, ξ)dydζ with θ ≡ 1 in a neighbourhood of the support of F y →η σ b (η, ξ).These results extend to matrix valued symbols and operators. If symbol a(x, ξ) only depends on ξ then σ a σ b -σ a b = 0 and σ r reduces to the only integral term. Moreover, (1.2.12)F x →η σ r (η, ξ) = ξ + tη)(1 -t) [r] + -1 dt χ η ξ η α by (η, ξ),where χ η ξ is the admissible cut-off function defining σ b . For d = 2 and all s ∈ R, there exists a constant C > 0 such that, for a ∈ S m r , b ∈ S m r , r ∈ N * , and w ∈ H s+m+m -1 ,

Notation 2 .

 2 For any integer m ∈ Z, b m (ξ) will denote any function satisfying |∂ β b m (ξ)| β |ξ| m-|β| , for any ξ in its domain, any β ∈ N 2 .

√ hy 2

 2 is bounded from the one where | x+ √ hy 2

Lemma 2 . 1 . 2 .

 212 (i) Let n ∈ N, n ≥ 3 and I ∈ I n . Then(2.1.27)

First, let us

  consider cubic terms corresponding to indices I 1 , I 2 such that |I 1 | = 2 and |I 2 | = 0. In this case we evidently have that |J 1 | = |J 2 | = 0, and by (B.1.4e) with s = 1 and θ 1 small, together with a-priori estimate (1.1.11),

for i = 1 , 3 .

 13 Finally, on the support of ψ 3 we have that |ξ| ∼ |ξ -η| ∼ |ζ| and |ξ -η -ζ| |ζ|. Replacing ζ with ξ -ζ by a change of coordinates we find that, for any α, β, γ ∈ N 2 , (2.2.70)

  .86) with coefficients c I 1 ,I 2 ∈ {-1, 0, 1} such that c I 1 ,I 2 = 1 whenever |I 1 | + |I 2 | = |I| (in which case D = D 1 ). After lemma A.6 and inequality (B.1.3d) with s = N + 3 we deduce that, if N ≥ 15, for any θ ∈]0, 1[ |S

  which concludes the proof of the statement. Lemma 2.2.11 (Analysis of quartic terms. III). Let n ∈ N, n ≥ 3, I ∈ I n and (I 1 , I 2 ) ∈ I(I) be such that [ |I| 2 ] < |I 1 | < |I|. Let also C I 1 ,I 2

  2, 0 otherwise. When |I 2 | = 0 the above right hand side can be estimated using (B.1.3a), (B.1.3b) and a-priori estimates (1.1.11). When |I 2 | = 1 we derive from (1.1.15) that

  1.1)-(1.1.2) in some interval [1, T ] for a fixed T > 1, and u ± , v ± defined in (1.1.5) satisfy a-priori estimates (1.1.11) for every t ∈ [1, T ], then they also verify (1.1.12c), (1.1.12d) on the same interval [1, T ].

  (3.1.6a) and (3.1.6b) follow then by (B.1.3c) with s = 1, (B.1.3b), (B.1.4a) and (B.1.4b).

5 )

 5 and solution to (2.1.2) with |I| = 0, remind definition (2.1.11) of vectors U, V and (3.1.3) of v N F . Let (3.1.15)

  ) and cubic ones c w , r N F w and r N F kg respectively by (3.1.18), (3.1.19) and (3.1.5

Figure 3 . 1 :

 31 Figure 3.1: Lagrangian for the Klein-Gordon equation assume a moderate growth for the L 2 norm of L µ v, with 0 ≤ |µ| ≤ 2, and has hence a better decay in time than the one expected for v (remind h = t -1 ). Thus the main contribution to v is the one localized around Λ kg . We are going to show that this latter one is solution to an ODE (see proposition 3.2.6) and that its H ρ,∞

Figure 3 . 2 :

 32 Figure 3.2: Lagrangian for the wave equation

Lemma 3 . 2 . 11 .

 3211 Let 0 < D 1 < D 2 and θ = θ(x) be a smooth function equal to 1 for |x| ≤ D 1 and supported for |x| ≤ D 2 . Then,

2ih

  (y•ζ-η•ζ) instead of e 2i h (η•z-y•ζ) (which does not affect estimate (1.2.46)),

Figure 3 . 3 :

 33 Figure 3.3: Regions I 1 and I 2 in space dimension 1

1 2

 1 and t ≥ t 0 > 1, and t|1 -|y| 2 | + |y| 2 ∼ t|1 -|y|| + |y| when |y| ≤ D, we find for those values of (t, y) that |w(t, y)| εt β 1 + √ t|y| (t|1 -|y|| + |y|) 1 2

- 1 ,

 1 together with (B.1.5a) with s = 0, (B.1.5b), (B.1.5c) with s = 1, (B.1.7), (B.1.10b), and (B.1.17) with k = 1, we obtain that (B.1.25)

L 2 ,

 2 so from (B.1.6a) with s = 0, (B.1.6b) with s = 1, (B.1.8), (B.1.10b), (B.1.17) with |I| = 1, (B.1.25) and a-priori estimates (1.1.11), we deduce 1 |µ|=0

  2 . (B.1.26b) Proof. Estimate (B.1.26a) follows straightly from (B.1.21), (B.1.5a) with s = 0, and estimates (1.1.11), (B.1.10b), and (B.1.17) with k = 1, while (B.1.26b) has already been proved in (B.1.22).

δ 1 2

 2 +κ , (B.2.1d) with κ = σρ if ρ ≥ 0, 0 otherwise.

  .2.3a) follows from definitions (3.2.2) and (3.1.15), inequality (A.37a) with u = v = v ± , and a-priori estimates (1.1.11), as

  follows from (B.1.10b), (B.1.15a), inequalities (B.1.4a), (B.1.4b), (B.1.4c) with s = 15, and a-priori estimates (1.1.11). Consequently, from (B.2.4), (B.2.5), (B.2.9), (B.2.10), (B.2.1a) and a-priori estimate (1.1.11d) with k = 2, we obtain (B.2.3b).

Lemma B. 2 . 4 .

 24 Let w ∈ {u, v} and for any Γ ∈ {Ω, Z m , m = 1, 2}

γ

  and a(D x ) = D for any α, β, γ ∈ N 2 with |α|, |β|, |γ| ≤ 1. Then for L = L 2 or L = L ∞ we have that

2 ,

 2 (B.2.52) while from (B.1.5a) with s = 0, (B.1.10b) and a-priori estimates

Figure B. 1 :

 1 Figure B.1: Manifolds Λ kg and Λ w .

2 j=1 γ j 1 x 2 j=1√ h Op w h γ j 1

 2121 -p (ξ) √ h χ(h σ ξ)a 0 (ξ) x j -p j (ξ) √ h with γ j 1 (z) := (1 -γ)(z)z j |z| -2 such that |∂ α z γ j 1 (z)| z -1-|α| , and use (1.2.69) with c(x, ξ) = χ(h σ ξ)a 0 (ξ), we obtain that v Λ c kg (t, •) L ∞

  2.17b), equation (3.1.4) and relation (3.2.8) with w = v N F , lemmas B.2.14, B.3.2, and estimate (B.2.79), that the first sum in the above right hand side is bounded by the right hand side of (B.3.7b). The same is true for the above second and third sums after (3.2.17b) and lemma B.2.14, and for the above latter L ∞ norm because of proposition 1.2.37 and estimate (B.2.75a). After decomposition (B.3.10) and estimates (B.2.1a), (B.3.8), and (B.3.11b), we see thatOp w h (χ(h σ ξ)a 0 (ξ)) v Op w h (χ(h σ ξ)b 1 (ξ)) u = v Λ kg Op w h (χ(h σ ξ)b 1 (ξ)) u + R(t, x).

Op w h (χ 1 2 .

 12 (h σ ξ)b 1 (ξ)) u = Op w h (χ 0 (h -1 ξ)b 1 (ξ)) u + k Op w h (1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ 1 (h σ ξ)b 1 (ξ) u,and observe that, from proposition 1.2.36 and the classical Sobolev injection,Op w h (χ 0 (h -1 ξ)b 1 (ξ)) u(t, •) L 2 + Op w h (χ 0 (h -1 ξ)b 1 (ξ)) u(t, •) L ∞ h u(t, •) LCombining the above decomposition and estimate with (B.3.11a) and (B.2.1a) we derive that(B.3.13) v Λ kg Op w h (χ(h σ ξ)b 1 (ξ)) u = k v Λ kg Op w h (φ k (ξ)b 1 (ξ)) u + R(t, x),whereφ k (ξ) := (1 -χ 0 )(h -1 ξ)ϕ(2 -k ξ)χ(h σ ξ). We can further decompose Op w h (φ k (ξ)b 1 (ξ)) u by defining u k Λw (t, x) := Op w h γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) u(t, x), u k Λ c w (t, x) := Op w h (1 -γ)

(θ 0 2 ≤ 2 , 2 j=1 γ j 1 x|ξ| -ξ h 1 / 2 -σ x j |ξ| -ξ j h 1 / 2 -

 022211212 (x)Ω h ) µ M ν Op w h (χ 2 (h σ ξ)) u(t, •) L CBεh -β-δ 1as follows by using the following equality(1 -γ) x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) = σ φ k (ξ)b 1 (ξ), with γ j 1 (z) := (1 -γ)(z)z j |z| -2, together with (1.2.52) with a ≡ 1, p = 1, and lemma B.2.1. Then, as the sum over k in the right hand side of (B.3.13) is actually restricted to indices k such that h 2 k h -σ , the above estimates and (B.3.11a) imply thatk v Λ kg Op w h (φ k (ξ)b 1 (ξ)) u = k v Λ kg u k Λw + R(t, x).

- 1 |α|=2hi h 1 2

 11 2.21 thatθ h (x)(|x| 2 -1) γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) = γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ)θ h (x)(|x| 2 -1) + h θ h (x)(|x| 2 -1), γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) + N |α| ∂ α x θ h (x)(|x| 2 -1) ∂ α ξ γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) + r N (x, ξ), h (x)(|x| 2 -1)]| (x+tz) (1 -t) N -1 dt × ∂ α ξ γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) | (x+y,ξ+η) dydzdηdζ. As |x| 2 -1 = x • x -ξ • ξ |ξ| 2 = (x|ξ| -ξ) • x |ξ| + (x|ξ| -ξ) • ξ |ξ| 2 ,the first term in the right hand side of (B.3.15) appears to be linear combination of products of the form γ x|ξ|-ξ h 1/2-σ φ k (ξ)a(x)b 0 (ξ)(x j |ξ| -ξ j ), for some smooth compactly supported function a(x), and b 0 (ξ) such that |∂ α b 0 (ξ)| |ξ| -|α| . From (1.2.52b) and lemma B.2.1, we hence deduce thatOp w h γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ)θ h (x)(|x| 2 -1) u(t, •) shows that h θ h (x)(|x| 2 -1), γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) = +σ ∂ x i [θ h (x)(|x| 2 -φ k (ξ)b 1 (ξ) + hγ x|ξ| -ξ h 1/2-σ ∂ x [θ h (x)(|x| 2 -1)]∂ ξ [φ k (ξ)b 1 (ξ)],

1 2 -σ γ 1 + |β 1 |+|β 2 |=|α| |β 1 |≥1 |β 1 |

 1111 x|ξ|-ξ h 1/2-σ φ k (ξ)a(x)b 0 (ξ)(x j |ξ| -ξ j ), for a new γ 1 ∈ C ∞ 0 (R 2 ), and some new a(x), b 0 (ξ) with the same properties as the ones we considered before. On the other hand, as ∂ ξ [φ k (ξ)b 1 (ξ)] is uniformly bounded and supported for frequencies of size 2 k , the second term in the above right hand side writes as linear combination of products of the formhγ x|ξ|-ξ h 1/2-σ φ 1 k (ξ)a(x)b 0 (ξ), for some new φ 1 k ∈ C ∞ 0 (R 2 \ {0}). Therefore, inequality (1.2.52b), proposition 1.2.30, and lemma B.2.1, give thathOp w h θ h (x)(|x| 2 -1), γ x|ξ| -ξ h 1/2-σ φ k (ξ)b 1 (ξ) u(t, •)As concerns |α|-order terms, for each fixed 2 ≤ |α| ≤ N -1, we find using (1.2.25) that they are given byh |α| γ x|ξ| -ξ h 1/2-σ ∂ α x [θ h (x)(|x| 2 -1)]∂ α ξ (φ k (ξ)b 1 (ξ)) j=1 h |α|-j( 1 2 -σ) γ j x|ξ| -ξ h 1/2-σ θ j (x)b j-|β 1 | (ξ)∂ β 2 ξ (φ k (ξ)b 1 (ξ)), for some γ j , θ j ∈ C ∞ 0 (R 2 ). Since |α| ≥ 2 and |∂ µ ξ (φ k (ξ)b 1 (ξ))| 2 -k(|µ|-1) , for any µ ∈ N 2 , by proposition 1.2.30 and lemma B.2.1 we obtain that the action of their quantization on u is estimated in the uniform norm by (B.3.19)

θ

  N (x + tz)(1 -t) N -1 dt × γ j x|ξ + η| -(ξ + η) h 1/2-σ φ j k (ξ + η)b j+1-N (ξ + η)dzdη,

Lemma B. 3 . 6 .

 36 Under the same assumptions as in lemma B.3.5,χ(t -σ D x ) x n [a 0 (D x )v -][b 1 (D x )u -]b 0 (D x )u -(t, •) L 2 (dx) ≤ C(A + B) 2 Bε 3 t -1+β+ δ 1 2 , (B.3.27a) χ(t -σ D x ) x m x n [a 0 (D x )v -][b 1 (D x )u -]b 0 (D x )u -(t, •) L 2 (dx) ≤ C(A + B) 2 Bε 3 t β+ δ 1 2 , (B.3.27b)for every t ∈ [1, T ], m, n = 1, 2, with β > 0 small, β → 0 as σ → 0. Moreover,χ(t -σ D x ) x n r N F kg (t, •) L 2 (dx) ≤ C(A + B) 2 Bε 3 t -1+β+ δ+δ 1 2 , (B.3.28a) χ(t -σ D x ) x m x n r N F kg (t, •) L 2 (dx) ≤ C(A + B)2 Bε 3 t β+ δ+δ 1 2 . (B.3.28b)

δ+δ 1 2 ,

 2 (B.3.29a) χ(t -σ D x ) x m x n R(t, •) L 2 (dx) ≤ C(A + B) 2 Bε 3 t β+ δ+δ 1 2 . (B.3.29b)Let us first notice that, after (B.1.3b), (B.1.10a), (B.1.27a) and a-priori estimates, we have that

  1.11) and (B.1.1b) the remaining contributions to r N F kg are of the form[a 0 (D x )v -][b 1 (D x )u -]R 1 u -with a 0 (ξ) equal to 1 or to ξ j ξ -1 , and b 1 (ξ) equal to ξ 1 or to ξ j ξ 1 |ξ| -1 , for j = 1, 2, estimates (B.3.28) will follow from (B.3.27). Our aim is hence to prove that the above product is a remainder R(t, x).Applying lemma B.2.2 with L = L 2 , w 1 = x n a 0 (D x )v -(resp. w 1 = x m x n a 0 (D x )v -), s > 0 sufficiently large so that N (s) > 2,and using estimates (B.1.10a) (resp. (B.1.27a)), (1.1.11a), (1.1.11c), we can suppose all above factors truncated for moderate frequencies less or equal than t σ , up to remainders R(t, x). Let us also observe that, from (B.1.10b), (B.3.26b) and (1.1.11c),

2+β+ δ+δ 2 2 ,

 2 and that, using additionally estimate (B.1.27b),χ(t -σ D x ) [χ 1 (t -σ D x )[x m x n a 0 (D x )v -][χ(t -σ D x )b 1 (D x )(u N F -u -)][χ(t -σ D x )b 0 (D x )u -] (t, •)(t, •) -σ D x )b 1 (D x )(u N F -u -) L 2 R ν u ± (t, •) L ∞ ≤ C(A + B)A 2 Bε 4 t -1+β+ δ+δ 2 2 .

2 , 2 ,

 22 derived from (3.1.10), (B.1.10b), (B.1.27b),(1.1.11a) and(1.1.11b). This reduces us to prove that, for k = 0, 1,χ 1 (t -σ D x )[x k m x n a 0 (D x )v N F -] χ(t -σ D x )b 1 (D x )u N F χ(t -σ D x )b 0 (D x )u -(t, •) L 2 (dx) ≤ C(A + B) 2 Bε 3 t -1+k+β+ δ 1or rather, after (1.1.11a), thatχ 1 (t -σ D x )[x k m x n a 0 (D x )v N F -] [χ(t -σ D x )b 1 (D x )u N F ](t, •) L 2 (dx) ≤ C(A + B)Bε 2 t -

1 2

 1 1 (h σ ξ))[x k m x n Op w h (a 0 (ξ)) v [Op w h (χ(h σ ξ)b 1 (ξ)) u](t, •) L 2 (dx) ≤ C(A + B)Bε 2 h

2 ≤ 2 + CABε 2 h 1 2 -β- δ 2 2 ≤ 1 2

 22121 Op w h ∂ ξn a 0 (ξ) v, (B.3.33) while from the commutation of x m with Op w h (χ(h σ ξ)b 1 (ξ)), definition (1.2.49) of M m , and symbolic calculus,x m Op w h (χ(h σ ξ)b 1 (ξ)) u = hOp w h (χ(h σ ξ)b 1 (ξ)|ξ| -1 )M m u -h 2i Op w h ∂ ξm (χ(h σ ξ)b 1 (ξ)|ξ| -1 )|ξ| u + Op w h (χ(h σ ξ)b 1 (ξ)ξ m |ξ| -1 ) u -h 2i Op w h ∂ ξm (χ(h σ ξ)b 1 (ξ)) u. (B.3.34)On the one hand, using equality (B.3.33), lemma B.2.14, and estimates (B.3.7a), (B.3.8), we deduce that[Op w h (χ 1 (h σ ξ)[x n Op w h (a 0 (ξ)) v]] [Op w h (χ(h σ ξ)b 1 (ξ)) u](t, •) L Op w h χ 1 (h σ ξ)a 0 (ξ) ξ n ξ v(t, •) [Op w h (χ(h σ ξ)b 1 (ξ)) u(t, •)]L C(A + B)Bε 2 h hand, when we deal with the L 2 norm in the left hand side of (B.3.32) corresponding to k = 1 we first commute x m with Op w h (χ 1 (h σ ξ)) and see, using symbolic calculus, that[Op w h (χ 1 (h σ ξ))[x m x n Op w h (a 0 (ξ)) v] [Op w h (χ(h σ ξ)b 1 (ξ)) u](t, •) L 2 (dx) ≤ [h σ Op w h ((∂χ 1 )(h σ ξ))[x n Op w h (a 0 (ξ)) v] [Op w h (χ(h σ ξ)b 1 (ξ)) u](t, •) L 2 (dx) + [Op w h (χ 1 (h σ ξ))[x n Op w h (a 0 (ξ)) v] [x m Op w h (χ(h σ ξ)b 1 (ξ)) u](t, •) L 2 (dx)

Lemma B. 4 . 1 .

 41 With the convention that D = D 1 whenever |I 1 |+|I 2 | = 2, D ∈ {D j , D t , j = 1, 2} otherwise, there exists a positive constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, n = 1, 2, and every t ∈ [1, T ], (B.4.1)

• When |I 1 | 2 ± 1 ≤ 1 +δ 2 2 ;•

 12112 = 0, |I 2 | = 2, we derive from (B.1.10b) and (1.1.11d) thatχ(t -σ D x ) x n Q kg 0 (v ± , D 1 u I 2 ± ) (t, •) (t, •) H C(A + B)Bεt δ When |I 1 | = |I 2 | = 1 and Γ I 2 ∈ {Ω, Z m , m = 1,2} is a Klainerman vector field we use inequalities (B.2.24) with L = L 2 , w j 0 = x n (D x D x -1

  ), (B.1.28), (B.2.62) and (1.1.11d);• When |I 1 | = |I 2 | = 1 and Γ I 2 is a spatial derivative we use lemma B.2.2 with L = L 2 , w 1 = x n (D x D x -1 ) µ v I 1 ± with |µ| = 0,1, s large enough so that N (s) ≥ 1, and again estimates and estimate (B.4.32b) is obtained by choosing s > 0 large so that N (s) > 1 and using (B.1.5a), (B.1.7), (B.1.10b), (B.1.17) with k = 1, (B.1.28), (B.2.62), (B.4.31b) and a-priori estimates. Lemma B.4.8. Let Γ ∈ {Ω, Z m , m = 1, 2} be a Klainerman vector field and V NF Γ be the function defined in (B.4.15). There exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, m, n = 1, 2, and every t ∈ [1, T ], (B.4.33) Op w h (χ(h σ ξ))L m tZ n V NF Γ (t, tx) L 2 (dx) ≤ CBεt δ 0 2 .

δ 1 2 .+ t - 3 1|µ 1 |,µ 2 ,µ 3

 23123 Finally, the remaining norms in the right hand side of (B.4.34) are estimated by the right hand side of (B.4.33) after (B.3.6) and lemma B.4.7. As concerns instead products (B.4.45b) and (B.4.45c), this follows applying inequalities (B.2.24) with w = u, w j 0 = x k m x n a 0 (D x )v -for k = 0, 1, and s > 0 such that N (s) ≥. In fact, for L = L 2 we use estimates (1.1.11), (B.1.10), (B.1.27), together with (B.1.32), to derive that for k ∈ {0, 1}χ(t -σ D x ) x k m x n [a 0 (D x )v -] [b 1 (D x )(Γu) -] R 1 u - L 2 (dx) χ 1 (t -σ D x )[x k m x n a 0 (D x )v -] χ(t -σ D x )b 1 (D x )(Γu) -χ(t -σ D x )R 1 u - L 2 (dx) =0

  4.45a) (resp. in (B.4.45c)) b 1 (D x )u -is replaced with b 1 (D x )u N F (with u N F introduced in (3.1.15)). This is justified up to some R(t, x) terms that satisfy (B.4.40) as consequence of (1.1.11a), (B.1.17), (B.1.28) (resp. (B.1.10a), (B.1.27a)), (B.3.26b), and also (B.4.41) because of (1.1.11a), (B.2.42) (resp. (1.1.11b)) and (B.3.26b). Hence we are led to estimate the L 2 norm ofχ 1 (t -σ D x )[x k m x l n a 0 (D x )(Γv) -] χ(t -σ D x )b 1 (D x )u N F χ(t -σ D x )R 1 u - (B.4.46a) χ 1 (t -σ D x )[x k m x l n a 0 (D x )v -] χ(t -σ D x )b 1 (D x )(Γu) -χ(t -σ D x )R 1 u - (B.4.46b) χ 1 (t -σ D x )[x k m x l n a 0 (D x )v -χ(t -σ D x )b 1 (D x )u N F χ(t -σ D x )R 1 (Γu) - (B.4.46c) for k = 0, 1, l = 1, and the L ∞ norm of above products when k = l = 0. Thirdly, we can think of a 0 (D x )(Γv) -in (B.4.46a) and of a 0 (D x )v -in (B.4.46b), (B.4.46c) as replaced with a 0 (D x )V NF Γ and a 0 (D x )v N F respectively, where V NF Γ has been introduced in (B.4.15) and v N F in (3.1.3). For (B.4.46a) (resp. (B.4.46c)) this substitution is justified up to some R(t, x) terms that satisfy (B.4.40) and (B.4.41), the former because of a-priori estimate (1.1.11a), (B.3.30) and (B.4.37) (resp. (B.2.57), (B.3.30) and (B.3.31)), the latter after (1.1.11a), (B.4.16) (resp. (B.3.26a), (B.2.57)) and the classical translation of the semi-classical (B.3.8)

1 k=0 1 k=0

 11 Op w h (χ 1 (h σ ξ))[x k m x n Op w h (a 0 (ξ)) V Γ ] Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) L 2 (dx) + Op w h (χ 1 (h σ ξ))[x k m x n Op w h (a 0 (ξ)) v] Op w h (χ(h σ ξ)b 1 (ξ)) u I (t, •) L 2 (dx) + Op w h (χ 1 (h σ ξ))[x k m x n Op w h (a 0 (ξ)) v] Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) L 2 (dx) ≤ C(A + B)Bε 3 h -1 2 -β (B.4.47)andOp w h (χ 1 (h σ ξ)a 0 (ξ)) V Γ Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) L ∞ (dx) + Op w h (χ 1 (h σ ξ)a 0 (ξ)) v Op w h (χ(h σ ξ)b 1 (ξ)) u I (t, •) L ∞ (dx) + Op w h (χ 1 (h σ ξ)a 0 (ξ)) v Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) L ∞ (dx) ≤ C(A + B)Bε 3 h -β . (B.4.48) We immediately obtain from inequalities (B.3.35) and (B.3.39) that Op w h (χ 1 (h σ ξ))[x k m x n Op w h (a 0 (ξ)) v] Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) L 2 (dx) ≤ C(A + B)Bε 3 h -1 2 -β .

  (χ 1 (h σ ξ)) x n Op w h (a 0 (ξ)) v Op w h (χ(h σ ξ)b 1 (ξ)) u I (t, •) L 2 (dx

Proof.t δ+δ 2 2 , 2 ≤ 1 h 1 µ=0 2 +

 22112 Straightforward after (B.3.6), lemma B.4.11, estimate (B.4.27) and the following inequalityχ(t -σ D x ) x m x n Q kg 0 v ± , Q w 0 (v ± , D 1 v ± ) (t, •) NL w (t, •) L 2 ≤ C(A + B)ABε 3 deduced from (1.1.11), (B.1.3a) and (B.1.27b).Lemma B.4.13. Let V Γ be the function defined in (B.4.20). There exists some positive constant C such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, and every t ∈ [1, T ],(B.4.49) |µ|=2 Op w h (χ(h σ ξ)L µ V Γ (t, •) L CBεt β , with β > 0 small, β → 0 as σ, δ 0 → 0.Proof. First of all we remind that V NF Γ is solution to (B.4.23). From relation (3.2.9b) and the commutation between L m and Op w h ( ξ ) we deduce that, for any m, n = 1, 2,Op w h (χ(h σ ξ))L m L n V Γ (t, •) H Op w h (χ(h σ ξ))L µ m tZ n V NF Γ (t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L µ m Op w h ξ n ξ V Γ (t, •) L Op w h (χ(h σ ξ))L µ m t(tx n )NL kg,c Γ (t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L µ m t(tx n )Q kg 0 v ± , Q w 0 (v ± , D 1 v ± ) (t, tx) L 2 (dx).The result of the statement follows then from (B.4.26), (B.4.27), (B.4.30), (B.4.33), and lemmas B.4.6, B.4.12. Lemma B.4.14. There exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ) equal to 1 in a neighbourhood of the origin, σ > 0 small, and every t ∈ [1, T ], (B.4.50) |I|=1 χ(t -σ D x )V I (t, •) L ∞ ≤ CBεt -1 .

(B. 4

 4 .51) V NF Γ (t, •) L ∞ ≤ CBεt -1 , where V NF Γ has been introduced in (B.4.15). After (B.4.16), the above inequality evidently implies the statement. The main idea to derive the sharp decay estimate in (B.4.51) is to use the same argument that, in subsection 3.2.1, led us to the propagation of a-priori estimate (1.1.11b), i.e. to move to the semi-classical setting and deduce an ODE from equation (B.4.23) satisfied by V NF

Γ

  kg := Op w h γ x -p (ξ) √ h χ 1 (h σ ξ) , with γ, χ 1 ∈ C ∞ 0 (R 2) such that γ ≡ 1 close to the origin, χ 1 ≡ 1 on the support of χ, p(ξ) := ξ , andV Γ Λ kg (t, x) := Γ kg Op w h (χ(h σ ξ)) V Γ (t, x), x) := Op w h (1 -γ)

  Proposition 1.2.19 (Self-Adjointness). If a(x, ξ) is a real symbol its Weyl quantization is selfadjoint, i.e.

	Op w h (a)

* = Op w h (a) . Proposition 1.2.20 (Composition for Weyl quantization). Let a, b ∈ S(R d ). Then Op w h

  are bounded on L 2 (see proposition 1.2.27), with norm O(2 kp ), O(2 k(p+1) ) respectively, and that 2 k ≤ h -σ . Proposition 1.2.31. Under the same hypothesis as proposition 1.2.28, Op w h

1 , ψ, a, θ, b p , and from the fact that operators Op w h (A k (x, ξ)), Op w h (ξA k (x, ξ))

  ≤ |α| < N . As factor x n |ξ| -ξ n is affine in x n , the length of multi-index α 2 is less or equal than 1 and, using lemma 1.2.26, t k α appears to be the sum of two terms. The first one corresponds to |α 2 | = 0 and has the form

	1.2.56) verifies estimates (1.2.51) thanks to propositions
	1.2.27, 1.2.30 and the fact that 2 kp ≤ h -σp .
	Let us denote concisely by t k α the |α|-order contributions in (1.2.53), for 2 i≤|α|
	µ=0,1

  in which case the derivative ∂ acting on Γ I 2 v (resp. on Γ I 2 u) is equal to ∂ 1 , and ∂ representing one of the partial derivatives ∂ a , a ∈ {0, 1, 2}. Let us remind that, if Γ I contains at least k (≤ |I|) space derivatives, above summations are taken over indices I 1 , I 2 such that k ≤ |I 1 | + |I 2 | ≤ |I|.

	Hence, introducing
	from (1.1.3), (1.1.5),

  acting on V I s verifies (2.2.15) after proposition 1.2.7. Also, (2.2.16) is deduced from (2.2.20) while properties (2.2.17) are obtained using essentially (1.2.12). 2.2.4. Let N ∈ N * . There exists a purely imaginary matrix E nd (U ; η), linear in (u + , u -) and of order -1, satisfying estimate (2.2.7c), such that

	Lemma

  as follows from corollary 1.2.11 and estimates (2.1.49), (2.2.7a) with r = 1. Hence,

  7, (2.1.49), (2.2.7a), (2.2.30) and (2.1.50) the L 2 norm of the latter term in the above right hand side is estimated by (2.2.31)

  ∞ and (2.2.31), so the last two terms in above right hand side are also remainders R (U, V ) by proposition 1.2.7 and estimate (2.2.7a). That concludes the proof of the statement.

2.2.2 A second normal forms transformation. In proposition 2.2.1 in previous subsection we showed that one can get rid of the slow-decayingin-time semi-linear contribution Op B C (U ; η) V I s in (2.1.46) by introducing a new function W I s , defined in (2.2.2) in terms of W I s and solution to equation (2.2.3). That naturally led us to the introduction of new energies E n (t; W ), for n ∈ N, n ≥ 3, and E k 3 (t; W ), for k ∈ N, 0 ≤ k ≤ 2, (see (2.2.11a)) which are respectively equivalent to starting E n (t; W ) and E k 3 (t; W ) whenever some uniform norms of U, V are sufficiently small. However, these new energies do not allow us yet to recover enhanced estimates (1.1.12c) and (1.1.12d) as it is not true that (2.2.32)

  and indices I 1 , I 2 in above right hand side such that (I 1 , I 2 ) ∈ I(I). In lemmas 2.2.9 and 2.2.10 we will check that, with definitions 2.2.5, 2.2.6, the slow decaying contributions highlighted in (2.2.33) are replaced in ∂ t E † n (t; W ), ∂ t E k, † 3 (t; W ) by some new quartic terms. These latter ones are obtained from integrals (2.2.44) by replacing each factor vI j 1 , ûj 2 , vI

	j 3
	at a time with the non-linearity appearing in the equation that factor satisfies in (2.2.56). Lemma
	2.2.11 (resp. lemma 2.2.12) shows that the same is for troublesome contributions (2.2.34) in
	∂ t E † n (t; W ) (resp. for (2.2.50) in ∂ t E k, † 3 (t; W )

  dxdydzdt, inequality (2.2.57) follows by the fact that, for any u 1 , . . . u 4 ∈ L 2 ∩L ∞ , any f, g, h ∈ L 1 , integrals such as

(2.2.64)

  .2.77) Proof. Using definitions (2.2.36), (2.2.44a), (2.2.42) with k = 1, and system (2.2.56), we find that

  By (2.2.89), (2.2.90), integrals (2.2.87), (2.2.88) are respectively equal to

  W ); • as concerns especially (2.2.101b), the fact that for any I ∈ V k (see definition (2.1.26)), any (I 1 , I 2 ) ∈ I(I) with I 1 ∈ K (see (2.1.25)) and |I 2 | ≤ 1, and any l = 1, 2, 3, by (A.17) and (2.2.97)

  .2.106) In order to analyse the behaviour of the second and fourth brakets in above right hand side we need, at this point, to distinguishing between indices I ∈ I n and I ∈ I k 3 . Propagation of a-priori estimate (1.1.11c): Let us suppose that I ∈ I n . Using (2.2.105) and estimate (2.1.38) we find that (2.2.107)

  3.1.1. Assume that (u, v) is solution to (1.1.1) in [1, T ] for a fixed T > 1, consider (u + , v + , u -, v -) defined in (1.1.5) and solution to (2.1.2) with |I| = 0, and remind definition (2.1.11) of vectors U, V . Let

	(3.1.3)

  .1.22c) Proof. By definition (3.1.15) of u N F , system (2.1.2) with |I| = 0, (3.1.13) and (3.1.14), it follows that u N F is solution to

  h := t -1 together with the following two new functions:

	(3.2.2)	u(t, x) := tu N F (t, tx),	v(t, x) := tv N F (t, tx),
	and observe that, from definition (3.2.2) and inequalities (3.1.12), (3.1.2), a-priori estimates
	(1.1.11a), (1.1.11b) are equivalent respectively to

  by proposition 1.2.37 and a-priori estimate (1.1.11c), the above inequality together with(3.2.33) and definition (3.2.2) of v, gives that

  after propositions 1.2.21 and 1.2.27. Using (3.2.64) and recalling relation (3.2.9a), we find that for any n = 1, 2,

  Let us split both v in the left hand side of (3.2.76) into the sum vΛ kg + v Λ c kg , with v Λ kg , v Λ c

	kg
	introduced in (3.2.16) with Σ j ≡ 1. Remind that v Λ c kg satisfies inequality (3.2.19a) and that

  2 t 1+δ 2 , which injected in (B.1.29), together with (B.1.30), implies (B.1.28).

	Corollary B.1.9. There exists a constant C > 0 such that, for every i, j = 1, 2, every t ∈ [1, T ],
	B.1.4,
	(B.1.31)

  0 otherwise.

	Inequality (B.2.1a) is straightforward after (B.2.2), definitions (3.2.2) and (3.1.15), inequality
	(3.1.20a), and a-priori estimate (1.1.11b). By commutating Op w h (φ j k (ξ)) with M (the commutator
	with Ω h being zero if ϕ, χ 0 are supposed to be radial) and using (B.2.2) we observe that there is

  0 otherwise. Using the semi-classical Sobolev injection, (B.2.7) and the fact that Ω h V (t, •) H s h = Ωv -(t, •) H s for any s ≥ 0, together with (B.1.10b) and a-priori estimates, we see that (B.2.14)

  Corollary B.2.5. If the hypothesis of lemma B.2.4 are satisfied and in addition w 1 , . . . , w n ∈ H s (R 2 ), we have that

	a

2.21

) is bounded by the L norm of (1 -χ)(t -σ D x )a(D x )(Z 1 u) ± w 1 times the L ∞ norm of the remaining factors.

  with N (s) as large as we want as long as s > 0, which is obtained from (B.1.2) together with the L 2 -L ∞ and L ∞ -L ∞ continuity of operator χ(t -σ D x ), we obtain (B.2.23).On the other hand, if the product in the left hand side of (B.2.20) is localized in frequencies by means of operator χ(t -σ D x ), so it is for the product in the first norm of the same inequalities. Inequalities (B.2.24) are then derived by bounding these L norms by means lemma B.2.2, where the role of w 1 is here played by w j 0 , for some fixed j 0 ∈ {1, . . . , n}.The following two lemmas are stated and proved in view of lemma B.2.8, in which we recover a first non-sharp estimate on the L ∞ norm of the Klein-Gordon component when one Klainerman vector field is acting on it and its frequencies are less or equal than t σ , for some small σ > 0. This estimate will be successively refined in lemma B.4.14.

Lemma B.2.6. Let χ ∈ C ∞ 0 (R 2 ), σ > 0 small, and w = w(t, x) such that, if w(t, x) := tw(t, tx), Op w h (χ(h σ ξ))L µ w(t, •) ∈ L 2 (R

2 ) for any |µ| ≤ 1. Then (B.2.25)

  -σ D x ) is a bounded operator from L 2 to L ∞ with norm O(t σ ), together with (B.1.5a), (B.1.17) and a-priori estimates. That concludes, together with (B.2.41), the proof of (B.2.29) and of the statement. Lemma B.2.8. There exists a constant C > 0 such that, for any ρ ∈ N, χ ∈ C ∞ 0 (R 2 ), equal to 1 in a neighbourhood of the origin, σ > 0 small, and every t ∈ [1, T ],

	(B.2.42)
	|I|=1

  2 ,+) given by (2.2.42) when j 3 = + and k = 1. After (B.2.27) and (A.15) it appears that r I,NF NL kg )D1 u -(D 1 Γ I v)NL w + D 1 [(Γ I v)NL w ] .

	kg	has the following nice explicit expression
	(B.2.47) (Γ I Using lemma B.2.6 and relation (3.2.8) with w = v I,NF , and reminding that tw(t, t•) L 2 = r I,NF kg = -i 2
	w(t, •) L 2 , we find the following
	(B.2.48)

  B.1.17), (B.2.42) and a-priori estimates, we obtain that

		2.45) we deduce that
	(B.2.53)	χ(t -σ D

x ) x j NL I,NF kg (t, •) L 2 ≤ C(A + B)Bε

2 t δ 2 , so injecting (B.2.50), (B.2.51), (B.2.53) into (B.2.48), and summing it up with (B.2.43), we obtain the result of the statement.

(

2 ,

 2 for any |µ| ≤ 1, and hence (B.2.58) after (B.2.59), (B.2.60), up to a further loss | log h|, as a consequence of the fact that the sum in (B.2.59) is finite and taken over indices k such that log h k log h -1 .

	Lemma B.2.11. There exists a positive constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ) equal
	to 1 in a neighbourhood of the origin, σ > 0, and every t ∈ [1, T ],
	(B.2.62)

  1.20a) and lemma B.2.4 with L = L ∞ and s > 0 large enough so that N (s) ≥

	3, together with estimates (1.1.11), (B.1.5a), (B.1.5b), (B.1.5c), (B.1.6a), (B.1.10a), (B.1.11),
	(B.2.42), and (B.2.57), we get that

  2 t -1

	as follows using (B.1.5a), (B.1.5b) with s = 1, (B.1.10a), (B.1.10b),(B.1.27a), (B.2.57) and a-
	priori estimates. By injecting the above inequality into (B.2.63) and using the fact that ε 0 <
	(2CA) -1 , we initially obtain that
	(B.2.69)	x j
		2 +β+	δ 1 +δ 2 2	,

  It is straightforward to check that the H s h norm of v is bounded by energy E n (t; W )

	1 2 whenever
	n ≥ s + 2, after definitions (3.2.2), (3.1.3), inequality (3.1.7a), and a-priori estimates (1.1.11a),
	(1.1.11b).
	In order to prove (B.2.75b) we first use relation (3.2.9b) and definition (3.1.3) to derive that
	Op w h

  . By comparing equality (B.2.30), with |I| = 1 and Γ I = Z n , with (B.2.80) we see that χ(t -σ D x )(v I,NF -(Z n v) -) corresponds to the first line in the right hand side of (B.2.80). Therefore, inequality (B.3.1) is automatically satisfied after (B.2.81), which was obtained by estimating the right hand side of (B.2.80) term by term. In order to prove (B.3.2), let us consider equality (B.2.80) but with χ(t -σ D x ) replaced with χ(t -σ D x )x m . The L 2 norm of each product in the second to fourth line is bounded by

  3.2) after (1.1.11), (B.1.5a) with s = 0, and (B.1.10b). Using lemma B.2.2 with L = L 2 and s > 0 large enough to have N (s) ≥ 2, we obtain that the L 2 norm of products in the first line of (the modified) (B.2.80) is bounded by

	1

  with v I,NF given by (B.2.27) with |I| = 1 and Γ I = Z n . From the fact that tw(t, t•) L 2 = w(t, •) L 2 and estimates (1.1.11), (3.1.8a), (B.2.50),(B.2.81), (B.3.1), along with the following one

  tx) L 2 (dx) .

	(B.3.5)
	By relation (3.2.8) with w = v I,NF and estimates (B.2.50), (B.2.51), (B.2.53), it follows that
	Op w h (χ(h δ 1 2 ,
	while from (3.2.2) and (B.2.75b) we have that
	Op w h

σ ξ))L m tv I,NF (t, tx) L 2 ≤ CBεt

  assuming that at any time u can be replaced with Op w h (χ 2 (h σ ξ)) u. Finally, it is useful to remind that from (3.2.2), (3.1.15), (3.1.20b), (3.1.20c), and a-priori estimates,

	while by (3.1.3), (3.1.7b) (with θ	1 small enough) and a-priori estimates,
	(B.3.9)			v(t, •) H ρ,∞
	(B.3.8)	u(t, •) H ρ+1,∞ h	+	Op w h (ξ|ξ| -1 ) µ u(t, •) H ρ+1,∞ h	≤ CAεh -1 2 ,
				|µ|=1

h ≤ CAε, for every t ∈ [1, T ].

  2.38, symbolic calculus and remark 1.2.22, each v Λ kg u k Λw in the above right hand side can be replaced withθ h (x) |x| 2 -1 v Λ kg (|x| 2 -1) u k Λw up to a new remainder R(t, x). Since |θ h (x)(|x| 2 -1) -1 | h -2σon the support of θ h (x), from proposition 1.2.36 and estimates (B.2.75a), (B.3.11a), we get thatθ h (x) v Λ kg (t, •) u k Λw (t, •) L 2 ≤ CBεh -δ 2 -β θ h (x)(|x| 2 -1) u k Λw (t, •) L ∞ , (B.3.14a) θ h (x) v Λ kg (t, •) u k Λw (t, •) L ∞ ≤ CAεh -β θ h (x)(|x| 2 -1) u k Λw (t, •) L ∞ . (B.3.14b)

  is sufficiently large (e.g. N ≥ 10), and consequently that(B.3.20) Op w h (r N (x, ξ)) u(t, •) L ∞ h u k (t, •) L 2 ≤ CBεh 1-δ 2 .Finally, summing up the above estimates with formulas from (B.3.15) to (B.3.19) we obtain that θ

h (x)(|x| 2 -1) u Λw (t, •) L ∞ CBεh 1 2 -β-δ 1 2 , which injected in (B.3.14) gives that θ h (x) v Λ kg u k

Λw is a remainder R(t, x). That concludes the proof of the statement.

.

  3.35). In order to derive an estimate for the latter one, we first use equality (B.3.33) and observe the following: from the semi-classical Sobolev injection and estimates (B.2.1c), (B.2.75b), we have that(B.3.36) h 2 Op w h (χ 1 (h σ ξ)a 0 (ξ))L n v Op w h (χ(h σ ξ)b 1 (ξ)ξ m |ξ| -1 )M m u (t, •) L 2 h Op w h (χ 1 (h σ ξ)a 0 (ξ))L n v(t, •) L 2 Op w h (χ(h σ ξ)b 1 (ξ)ξ m |ξ| -1 )M m u(t, •) L 2 ≤ C(A + B)Bε 2 h 1-δ 2 -β ;a similar chain of inequalities as in (2.2.80), together with (3.1.20a), (3.1.20b) and (1.1.11), gives that for any θ ∈]0, 1[(B.3.37) Op w h (b 1 (ξ)ξ m |ξ| -1 ) u(t, •) L ∞ = t b 1 (D x )D m |D x | -1 u N F (t, •) L ∞ t u N F (t, •) 1-θ H 3,∞ u N F (t, •) H 2 ≤ CA 1-θ B θ εt Op w h (χ 1 (h σ ξ)a 0 (ξ))L n v x m Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) L 2 ≤ C(A + B)Bε 2 hMoreover, using again (B.3.34) along with(B.2.1a), (B.2.1c), (B.3.7a) and (B.3.9),Op w h χ 1 (h σ ξ)a 0 (ξ) ξ n ξ v + hOp w h χ 1 (h σ ξ)∂ ξn a 0 (ξ) v x m Op w h (χ(h σ ξ)b 1 (ξ)) u (t, •) Lemma B.3.7.There exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, and everyt ∈ [1, T ], L 2 ≤ CBεt β+ δ+δ 1 2 ,with β > 0 small, β → 0 as σ → 0.Proof. From relation (3.2.9b) and the commutation between L m and Op w h ( ξ ) we deduce that Op w h (χ(h σ ξ))L m L n v(t, •) H 1

						1 2 +	(1+δ) 2 θ .
	Therefore, from equality (B.3.34) and estimates (B.2.75b), (B.3.8), (B.3.37), (B.3.36), we find
	that
	(B.3.38)
	h 1 2 -	δ 2 2 -	(1+δ)θ 2
						L 2 (dx)
	≤ Op w h χ 1 (h σ ξ)a 0 (ξ)	ξ n ξ	v Op w h χ(h σ ξ)b 1 (ξ)	ξ m |ξ|	u (t, •)
		1 2 -β-	δ+δ 1 2 .
	Choosing θ	1 small enough, this concludes that
	(B.3.39)
	[Op w					1 2 -β-	δ+δ 1 2
	and, together with (B.3.35), the proof of (B.3.32).
	We can finally prove the following:
	(B.3.40)
	(B.3.41)
						1
					h	Op w h (χ(h σ ξ))L µ m tZ n v N F (t, tx) L 2 (dx)
						µ=0
	+ Op w h (χ(h σ ξ))L µ m Op w h	ξ n ξ	v(t, •)	L 2 (dx)	+ Op w h (χ(h σ ξ))L µ m t(tx n )r N F kg (t, tx) L 2 (dx) ,

L 2 (dx)

+ C(A + B)Bε 2 h 1-β-δ 2 2 ≤ C(A + B)Bε 2 h h (χ 1 (h σ ξ))[x m x n Op w h (a 0 (ξ)) v] [Op w h (χ(h σ ξ)b 1 (ξ)) u](t, •) L 2 (dx) ≤ C(A + B)Bε 2 h |µ|=2 Op w h (χ(h σ ξ))L µ v(t,

•) so the result of the statement follows from lemmas B.2.14, B.3.2, and inequalities (B.3.6), (B.3.28).

  NF +v I 2 ,NF + D n D x v I 1 ,NF + D n D x (Γv) --v I 1 ,NF so that Op w h (χ(h σ ξ))L m tZ n V NF Γ (t, tx) L 2 (dx) Op w h (χ(h σ ξ))L m tZ n V NFSince v I 2 ,NF satisfies (B.4.4) with I = I 2 , we derive from relation (3.2.8) with w = v I 2 ,NF thatOp w h (χ(h σ ξ))L m tv I 2 ,NF (t, tx) L 2 (dx) χ(t -σ D x )Z m (Γ I 2 v) -(t, •) L 2 + χ(t -σ D x )Z m v I 2 ,NF -(Γ I 2 v) -(t, •) L 2 + χ(t -σ D x )v I 2 ,NF (t, •) L 2 + χ(t -σ D x ) x m NL I 2 ,NF -priori estimate (1.1.11d) with k = 0, (B.2.29) with I = I 2 , (B.4.5), (B.4.7), the fact that δ δ 2 δ 1 δ 0 and that β is small as long as σ is small, implyOp w h (χ(h σ ξ))L m tv I 2 ,NF (t, tx) L 2 (dx) ≤ CBεtAnalogously, commutating L m with Op w h (ξ n ξ -1 ), using (3.2.8) with w = v I 1 ,NF and the fact that v I 1 ,NF is solution to (B.2.44) with non-linear term given by (B.2.45), together with inequalities (B.2.50), (B.2.51), (B.2.53), we derive that

	(B.4.34)
					L 2 (dx)
	+ Op w h (χ(h σ ξ))L m t	D n D x	(Γv) L 2 (dx)	.
					δ 0 2 .
	Op w h (χ(h σ ξ))L m t	D n D x	v I 1 ,NF (t, tx)

Γ -(Γv) -(t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m t (Z n Γv) --v I 2 ,NF (t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m tv I 2 ,NF (t, tx) L 2 (dx) + Op w h (χ(h σ ξ))L m t D n D x v I 1 ,NF (t, tx) --v I 1 ,NF (t, tx) kg (t, •) L 2 . AL 2 (dx)

  Op w h (χ 1 (h σ ξ)) x m x n Op w h (a 0 (ξ)) v Op w h (χ(h σ ξ)b 1 (ξ)) u J (t, •) L 2 Op w h χ 1 (h σ ξ)a 0 (ξ)This can be done using a similar argument to the one that led us to (B.3.35) and (B.3.39), up to replacing v with V Γ in (B.3.33), referring to lemma B.4.6 instead of B.2.14, and to estimate (B.4.36) instead of (B.3.9), in order to derive the former two inequalities; up to replacing u with u I in (B.3.34), using lemma B.2.9 instead of (B.2.1a), (B.2.1c), estimate (B.3.23) instead of (B.3.8), and the fact that for any θ ∈]0, 1[Op w h (χ(h σ ξ)b 1 (ξ)ξ m |ξ| -1 ) u I (t, •) L ∞ = t χ(t -σ D x )b 1 (D x )D m |D x | -1 (Γu) -(t, •) L ∞ t χ(t -σ D x )(Γu) -(t, •) 1-θ H 3,∞ (Γu) -(t, •) θ H 2 ≤ C(A + B)1-θ B θ εt which is the analogous of (B.3.37) (last estimate deduced using (B.2.57) and (1.1.11d) with k = 1), to demonstrate the latter two ones. Therefore, above inequalities and (B.3.22a), (B.4.35a) imply (B.4.47). Finally, (B.4.48) is consequence of (B.3.7b), (B.3.22b) and (B.4.35b). That concludes the proof of the statement. Lemma B.4.12. Let NL kg,c Γ be given by (B.4.28). There exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, m, n = 1, 2, and every t ∈ [1, T ], Op w h (χ(h σ ξ))L m t(tx n )NL kg,c Γ (t, tx)

	ξ n ξ	v Op w h χ(h σ ξ)b 1 (ξ)	ξ m |ξ|	u I (t, •)	L 2 (dx)	+ C(A + B)Bε 2 h	1 2 -β .
						1 2 +β+	δ 1 2 +	(1+δ 1 +δ 2 ) 2	θ ,
						+ CABε 2 h	1 2 -β ,
						L 2

≤ Op w h χ 1 (h σ ξ)a 0 (ξ) ξ n ξ v [Op w h (χ(h σ ξ)b 1 (ξ)) u I ](t, •) L 2 (dx)

  It immediately follows from inequality (3.2.18b) and lemmas B.4.6, B.4.13, that

	(B.4.52)	V Γ Λ c kg	(t, •)

kg

.

  kg to the above equation we find, from symbolic calculus and the first part of lemma 3.2.5, thatV Γ Λ kg satisfies (B.4.53) [D t -Op w h (x • ξ -ξ )] V Γ Λ kg (t, x) = h -1 Γ kg Op w h (χ(h σ ξ)) NL kg,c Γ (t, tx) -δ Z 1 h -1 Γ kg Op w h (χ(h σ ξ)) Q kg 0 v ± , Q w 0 (v ± , D 1 v ± ) (t, tx) -Op w h (b(x, ξ))Op w h (χ(h σ ξ)) V Γ (t, x) + iσh 1+σ Γ kg Op w h (∂χ)(h σ ξ) • (h σ ξ) V Γ ,with symbol b given by (3.2.27). Since γ's derivatives vanish in a neighbourhood of the origin and ∂χ 1 ≡ 0 on the support of χ, from symbolic calculus of lemma 1.2.24 and remark 1.2.22, 1.2.42, together with inequalities (3.2.17b), (3.2.18b), thatOp w h (b(x, ξ))Op w h (χ(h σ ξ)) V Γ (t, •) (χ(h σ ξ))L µ V Γ (t, •) L 2 + h 2 V Γ (t, •) L 2 ≤ CBεt -3 2 +β ,where last estimate is obtained using lemmas B.4.6, B.4.13. Moreover, reminding lemma 1.2.38 and using symbolic calculus we see that, for any N ∈ N as large as we want, (B.4.54) h 1+σ Γ kg Op w h ((∂χ(h σ ξ) • (h σ ξ)) V Γ (t, •)

			L ∞
		2
	h	3 2 -β	Op w h
		|µ|=0

tx), with NL kg,c Γ given explicitly by (B.4.28). Applying successively operators Op w h (χ(h σ ξ)) and Γ

  Index Admissible cut-off function, 16 b m (ξ), function, 23 c w , cubic term in the wave equation after a normal form, 101 E 0 (t; u ± , v ± ), energy, 12 E k 3 (t; u ± , v ± ), generalized energy, 12 E k 3 (t; W ), generalized energy, 53 E n (t; u ± , v ± ), generalized energy, 12 E n (t; W ), generalized energy, 53 E k 3 (t; W ), first modified energy, 67 E n (t; W ), first modified energy, 67 Γ I , product of admissible vector fields, 12 Γ kg , operator, 105 γ n , function, 23 h, semi-classical parameter, 102 H ρ,∞ (R d ), space, 15 H ρ,∞ h (R d ), space, 20 H s (R d ), space, 15 H s h (R d ), space, 20 I(I), set of multi-indices, 14 I k 3 , set of multi-indices, 53 I n , set of multi-indices, 53 K, set of integers, 23 K, set of multi-indices, 57 Klainerman vector fields, 12 Λ kg , manifold associated to the Klein-Gordon equation, 44 Λ w , manifold associated to the wave equation, 115 Littlewood Paley decomposition, 17 L j , operator, 46 M j , operator, 35 M m 0 (a; n), seminorm, 15 NL kg , quadratic non-linearity in the Klein-Gordon equation satisfied by v ± , 157 NL w , quadratic non-linearity in the wave equation satisfied by u ± , 157 Ω, Euclidean rotation, 12 Ω h , semi-classical Euclidean rotation, 30 Op B , para-differential operator, 16 Op B R , remainder para-differential operator, 16 Op h , standard semi-classical quantization, 20 Op w h , semi-classical Weyl quantization, 20 Order function, 19 p(ξ), function, 42Q 0 (v, w), null form, 11 Q kg 0 (v ± , D a u ± ), null form, 50 q w , quadratic term in the wave equation after a normal form, 100Q w 0 (v ± , D a v ± ),null form, 50 r N F kg , cubic term in the Klein-Gordon equation after a normal form, 98 r N F w , cubic term in the wave equation after a normal form, 101 S δ,σ (M ), class of symbols, 19 Σ m 0 (R d ), class of symbols, 15 Σ m r (R d ), class of symbols, 15 S m 0 (R d ), class of symbols, 15 S m r (R d ), class of symbols, 15 Sobolev injection, semi-classical, 20 U , wave vector, 52 U I , wave vector with admissible vector fields, 52 u I ± , wave components with admissible vector fields, 12

1.6a

) with s = 0. From (B.4.21a) and (B.4.54) we hence derive that

h 1+σ Γ kg Op w h ((∂χ(h σ ξ) • (h σ ξ)) V Γ (t, •) L ∞ ≤ CBεt -3

-β .

 [START_REF] Alazard | Sobolev estimates for two dimensional gravity water waves[END_REF]The author is supported by a PhD fellowship funded by the FSMP and the Labex MME-DII, and by For Women in Science fellowship funded by Fondation L'Oréal-UNESCO. Keywords: Global solution of coupled wave-Klein-Gordon systems, Klainerman vector fields, Semiclassical

We remind that, if I = (i 1 , . . . , i n ) is a multi-index of length |I| = n, with i j ∈ {1, . . . , 5}, Γ I = Γ i 1 • • • Γ in is a product of vector fields in family Z = {Ω, Z j , ∂ j |j = 1, 2}.

We can immediately observe that the above bounds are verified at time t = 1 after (1.1.4) and Sobolev injection. By definition (1.1.5) we also notice that (3.3.25a)

and, conversely, (3.3.26a)

Moreover, reminding definition (1.1.9) of generalized energies E n (t; u ± , v ± ), E k 3 (t; u ± , v ± ), for n ≥ 3 and 0 ≤ k ≤ 2, and of set I k 3 in (2.1.17), there is a constant C > 0 such that

and for any 0 ≤ k ≤ 2,

Therefore, after (3.

Appendix A

The aim of this appendix is to prove the continuity of some trilinear integral operators (see lemmas A.5 and A.6) that arise in subsection 2.2.2 when performing a normal form argument at the energy level, and of some bilinear integral operators (see lemma A.8) that instead appear in subsection 3.1.2 when we perform a normal form the wave equation (see proposition 3.1.2). All the other results of this chapter are stated and proved in view of the above mentioned lemmas.

Lemma A.1. Let ǎ(x) denote the inverse transform of a function a(ξ).

(i) If a : R 2 → C is such that, for any α ∈ N 2 with 1 ≤ |α| ≤ 4,

(ii) If a is such that, for any α ∈ N 2 with |α| ≤ 3,

(iii) Let N ∈ N. If for any α ∈ N 2 with |α| ≤ N there exists

Proof. (i) We consider a cut-off function φ ∈ C ∞ 0 (R 2 ) equal to 1 in the unit ball and write (A.1)

with K 0 (x) := 1 (2π) 2 e ix•ξ a(ξ)φ(ξ)dξ, K 1 (x) := 1 (2π) 2 e ix•ξ a(ξ)(1 -φ)(ξ)dξ.

On the one hand, since |∂ α a(ξ)| α ξ -3 on the support of (1 -φ)(ξ) for any |α| ≤ 4, we immediately deduce by integration by parts that |K 1 (x)|

x -4 for any x ∈ R 2 . On the other hand, again an integration by parts gives that xK 0 (x) = e ix•ξ a 1 (ξ)dξ with a 1 (ξ) supported for |ξ| 1 and such that |∂ α a 1 (ξ)| α |ξ| -|α| for any ξ ∈ R 2 , any |α| ≤ 3. This implies that |xK 0 (x)| 1 for any x ∈ R 2 . Moreover, |x α x K 0 (x)| α 1 for any |α| ≤ 3. This while for |l -k| ≤ N 0 a(ξ, η)ϕ l (ξ)ϕ l (η)û 1 (ξ -η)û 2 (η)û 3 (-ξ)dξdη = K |l-k|≤N0 (z -x, x -y)u 1 (x)[ D x 7 u 2 ](y)u 3 (z)dxdydz

(A.21)

Finally, when positive indices l, k are such that k > l -N 0 we observe that frequencies ξ and ξ -η are equivalent and of size 2 k on the support of ϕ k (ξ)ϕ l (η). If we take a k>l-N0 (ξ, η) equal to a l>k+N0 (ξ, η), denote by K k>l-N0 (x, y) its associated kernel (which is hence equal to K l>k+N0 (x, y)), and introduce two new smooth cut-off function ϕ 1 , ϕ 2 ∈ C ∞ 0 (R 2 ) equal to 1 on the support of ϕ, together with operators

Combining decomposition (A. [START_REF] Klainerman | The null condition and global existence to nonlinear wave equations[END_REF]) together with (A.20), (A.21) and Cauchy-Schwarz inequality we finally obtain that

Lemma A. [START_REF] Delort | Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres[END_REF]. Let ε > 0 be small, N ∈ N * , and σ N (ξ, η) : R 2 × R 2 → C be supported for |ξ| ≤ ε η and such that, for any α, β ∈ N 2 ,

For any (j 1 , j 2 , j 3 ) ∈ {+, -} 3 let also (A.22) σ N (j 1 ,j 2 ,j 3 ) (ξ, η) := σ N (η, ξ -η) j 1 ξ -η + j 2 |η| -j 3 ξ .

Then for any α, β ∈ N 2 (A. [START_REF] Ma | Global solutions of non-linear wave-Klein-Gordon system in one space dimension[END_REF])

Proof. From definition (A.22) function σ N (j 1 ,j 2 ,j 3 ) can be written as follows

or

and D j (ξ, η) be the multiplier introduced in (3.1.14). For any s ≥ 0, i = 1, 2, D j (ξ, η) and ξ i |ξ| D j (ξ, η) satisfy inequalities (A.25), (A.26) with ρ = 2, and

together with

with δ > 0, δ → 0 as σ → 0.

Appendix B

The aim of this chapter is to show how, from the bootstrap assumptions (1.1.11), it is possible to derive a moderate growth in time for the L 2 norm of L µ v, with 0 ≤ |µ| ≤ 2, and of Ω µ h M ν u Σ,k , with µ, |ν| = 0, 1. These estimates are fundamentally used in propositions 3.2.7 and 3.3.7. Moreover, we also prove in lemma B.4.14 a sharp decay estimate for the uniform norm of the Klein-Gordon solution when one Klainerman vector field is acting on it (and when considered for frequencies less or equal than t σ , with σ > 0 small). We are hence going to assume for the rest of this chapter that a-priori estimates (1.1.11) are satisfied in interval [1, T ], for some fixed T > 1, and that ε 0 < (2A+B) -1 . We remind that Γ generally denotes one of the admissible vector fields belonging to Z (see (1.1.7)) and that, for a multi-index

) is here considered with respect to spatial variable x. We will often write • X in place of • X(dx) .

B.1 Some preliminary lemmas

In the current section we list, on the one hand, some inequalities concerning the H s and H s,∞ norm of the quadratic non-linearities

2), as they are very frequently recalled in the second part of the paper. On the other hand, we introduce some preliminary small results that will be useful in sections B.2 and B.3.

For seek of compactness, we denote Q w 0 (v ± , D 1 v ± ) and Q kg 0 (v ± , D 1 u ± ) by NL w and NL kg respectively, i.e.

We recall the result of lemma 1.2.40, that can be also stated in the classical setting and says that, for any real positive s > s and w ∈ H s (R 2 ),

It is also useful to remind, in view of upcoming lemmas, that the L 2 norm of (Γ I u) ± and (Γ I v) ± is estimated with:

, whenever |I| ≤ n and Γ I is a product of spatial derivatives;

, whenever |I| ≤ 3 and at most 3 -k vector fields in Γ I , with

As assumed in (1.1.11c), (1.1.11d), such energies have a moderate growth in time and a hierarchy is established among them in the sense that

We warn the reader that this hierarchy is often implicitly used throughout this chapter.

Lemma B.1.1. For any s ≥ 0, any θ ∈]0, 1[, NL w satisfies the following inequalities:

while for NL kg we have that:

), for some large positive s. Let also χ ∈ C ∞ 0 (R 2 ) and σ > 0. There exists some χ 1 ∈ C ∞ 0 (R 2 ), equal to 1 on the support of χ, such that for

with N (s) as large as we want as long as s > 0 is large.

Proof. The idea of the proof is to decompose each factor w j , for j = 2, . . . , n into (B.2.16)

and to estimate the L 2 norm of product (B.2.17)

where w k is either w k or χ(t -σ D x )w k , with the L 2 norm of w 1 times the L ∞ norm of all remaining factors, reminding that χ(t -σ D x ) is uniformly bounded on L ∞ and that by Sobolev injection and (B.1.2), (B.2.18)

with N (s) as large as we want as long as s > 0 is large. The L ∞ norm of (B.2.17) is estimated in the same way, using firstly the L 2 -L ∞ continuity of operator χ(t -σ D x ) acting on the entire product.

The end of the statement follows from the observation that, if suppχ ⊂ B C (0) for some C > 0, then

We have seen at the beginning of section B.1, and already used in the previous lemma's proof, that, if w ∈ H s (R 2 ) for some large s > 0, the L 2 norm (resp. L ∞ norm) of this function when restricted to large frequencies |ξ| t σ decays fast in time as t -σs (resp. t -σ(s-1)-1 after the semi-classical Sobolev injection). The aim of the following lemma is to show that, even if we don't have a control on the H s (R 2 ) norm of (Γu) ± , (Γv) ± , for Γ ∈ {Ω, Z m , m = 1, 2} and s larger than 2, the L 2 norm (resp. L ∞ norm) of products as in (B.

with B 1 (j 1 ,j 2 ,+) given by (2.2.42) with j 3 = + and k = 1. Then there exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, and every t ∈ [1, T ],

(B.2.28)

Moreover, for every

.

Proof. We first notice that, after definition (B.2.27) and inequalities (A.15), (1.1.10), we have the following explicit expression: (B.2.30)

From the above equality together with lemma B.2.2 with L = L ∞ and w 1 = (Γ I v) ± , and (1.1.5), (1.1.10), we deduce that there exists some χ 1 ∈ C ∞ 0 (R 2 ), equal to 1 on the support of χ, and s > 0 sufficiently large such that

where the latter inequality follows from after a-priori energy estimates (1.1.11c), (1.1.11d). Our aim is to truncate factor (Γ I v) ± in the above right hand side rather with the same operator χ(t -σ D x ) appearing on the left hand side. We hence proceed by picking some κ ≥ 1 and decomposing

noticing that, as χ(t κ ξ) is supported for very small frequencies |ξ| t -κ , by Sobolev injection we have that

along with a-priori estimates (1.1.11c), (1.1.11d), we have that for any for µ = 0, 1

we deduce from (B.2.32) and the above inequality that

We then decompose (Γ I v) ± in frequencies using the wished operator χ(t -σ D x ). In order to estimate the L ∞ norm of

and successively look at it as a linear combination of derivations of the form

, multiplying it against the wave factor, and successively combining the classical Sobolev injection with inequality (B.1.2), we find that

Using system (2.1.2) with |I| = 0, (B.1.4c) with s = 1, (B.1.6a) and a-priori estimates, it is straightforward to check that

and for |α| ∈ {1, 2} we have that (B.2.37)

In fact, when |α| = 1 this can be proved by commutating

is bounded from L 2 to L ∞ uniformly in t, and together with estimates (1.1.11d), (B.1.16a), and the following inequality

which is obtained by writing

As done for the Klein-Gordon component in the above lemma, we also derive an estimate for the uniform norm of the wave component when a Klainerman vector field acts on it and its frequencies less or equal than t σ (see lemma B.2.10). We first need the following result.

Lemma B.2.9. Let Γ ∈ Z, index J be such that Γ J = Γ, and u J (t, x) := t(Γu) -(t, tx). There exists a constant C > 0 such that, for any θ 0 , χ ∈ C ∞ 0 (R 2 ), σ > 0, and every t ∈ [1, T ],

Proof. We warn the reader that, throughout the proof, C will denote a positive constant that may change line after line. We also recall that

Estimates (B.2.54a) and (B.2.54b) are straightforward after (1.1.11d) and the fact that

From (3.2.6) with w = (Γu) -and f = ΓNL w , estimates (1.1.11d), (B.1.26b), along with the fact that δ 2 δ 1 (e.g. 2δ 2 ≤ δ 1 ), and (A + B)ε 0 < 1, we obtain (B.2.54c). By (3.2.6) we also derive that, for any n = 1, 2,

The first two norms in the above right hand side are controlled by E 0 3 (t; W ) 1/2 and are hence bounded by CBεt δ 0 2 . By commutating x n with χ(t -σ D x )Ω, and using that θ 0

). On the one hand, using (B.1.20b), (B.1.6a) with s = 0 and a-priori estimates we derive that

On the other hand, when we compute ΩΓNL w we find among the out-coming quadratic terms the following ones

which we estimate in the L 2 norm (when truncated for frequencies less or equal than t σ ) by means of (B.2.24a) with L = L 2 , Γw = Ωv, and s > 0 large enough to have N (s) ≥ 3. From and (B.2.78)

From (B.2.78) and (1.1.5), together with estimates (1.1.11) and (B.1.10b),

We bound the L 2 norm of all products in the first line of the above equality by means of lemma B.2.2, and all the others by the L ∞ norm of the Klein-Gordon factor times the L 2 norm of the wave one. In this way we get that, for some new χ 1 ∈ C ∞ 0 (R 2 ) and s > 0 sufficiently large, we derive that

Consequently, using estimates (1.1.11), (B.1.5a) with s = 0, and (B.2.42), we obtain that (B.2.81)

B.3 Last range of estimates

The aim of this section is to show that a-priori estimates (1.1.11) also infer a moderate growth in time of the L 2 (R 2 ) norm of L µ v, for |µ| = 2, when this function is restricted to frequencies less or equal than h -σ , for σ > 0 small. This is proved in lemma B.

There exists a constant C > 0 such that, for any χ, χ 1 ∈ C ∞ 0 (R 2 ), σ > 0, and every t ∈ [1, T ], we have that

Proof. The proof of this result is analogous to that of lemma B.

which is the semiclassical translation of (B.2.57), and to lemma B.2.9 instead of lemma B.2.1.

There exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, and every t ∈ [1, T ],

where r N F kg is given by (B.2.78).

Proof. We warn the reader that we denote by C and β two positive constants that may change line after line during this proof, with β → 0 as σ → 0. Moreover, we are going to denote generically by R(t, x) each term satisfying

From lemma B.2.2 with L = L ∞ and s > 0 large enough to have N (s) ≥ 3, and a-priori estimates (1.1.11), we can reduce ourselves to estimate the L ∞ norm of the product in the left hand side of (B.3.24) when all its factors are supported for moderate frequencies less or equal than t σ , up to remainders R(t, x). Moreover, since

(B.1.17), (B.2.62) and (1.1.11d). We obtain that

Lemma B.4.2. There exists a positive constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, ρ ∈ N, and every t ∈ [1, T ],

with β > 0 small such that β → 0 as σ → 0.

Proof. Estimate (B.4.3) is evidently satisfied when Γ I contains at least one spatial derivative thanks to lemma B.2.8. We then focus on the case when Γ I is the product of two Klainerman vector fields. As v I + = -v I -, we prove the statement for χ(t

which is consequence of (1.1.11d) with k = 0 and of (B.2.29) with j = 2, we derive that

.

The only thing we need to show in order to prove the statement is hence that

But from (B.4.4) and (B.2.47) with |I| = 2 we have that

Lemma B.4.3. There exists a positive constant C > 0 such that, for any multi-index I of length 2, any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, j = 1, 2, and every t ∈ [1, T ]

with β > 0 small, β → 0 as σ → 0.

Proof

Reminding (B.1.23) and applying lemma B.2.2 with L = L ∞ and w 1 = (D x D x -1 ) µ v I ± , for |µ| = 0, 1, to the contribution coming from the first quadratic term in the right hand side of (B.1.23), we find that there is some

(B.4.10) Therefore, picking s > 0 large so that N (s) > 1 and using the L 2 -L ∞ continuity of χ 1 (t -σ D x ) with norm O(t σ ), together with the estimates (1.1.11), (B.1.17) with k = 2, along with (B.4.1), we find at first that

Injecting the above estimate, together with (1.1.11d) and (B.4.3), into (B.4.9) we derive that

The above inequality holds for any χ ∈ C ∞ 0 (R 2 ), so injecting it into (B.4.10) and using again a-priori estimates, (B.1.17), (B.4.1), together with the fact that β + (δ + δ 2 )/2 ≤ δ 1 /2 as β is as small as we want as long as σ is small and δ, δ 2 δ 1 , we find the following enhanced estimate

Consequently, summing up this latter one with (1.1.11d) and (B.4.3), we end up with (B.4.8).

Lemma B.4.4. Let Γ ∈ Z be an admissible vector field. There exists a positive constant C such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, i, j = 1, 2, and every t ∈ [1, T ],

with β > 0 such that β → 0 as σ → 0.

Proof. Since (Γv) + = -(Γv) + we reduce to prove that inequality (B.4.11) holds true for (Γv) -. Moreover, we only focus on the case where Γ ∈ {Ω, Z m , m = 1, 2} is a Klainerman vector field, as (B.4.11) with Γ being a spatial derivative is simply a consequence of (B.1.27b).

We remind that (Γv) -is solution to non-linear Klein-Gordon equation (B.1.18a) with Γ I = Γ, and that the non-linearity ΓNL kg is given by (B.1.20a). Hence, multiplying x i to relation (B.1.9b) with w = (Γv) -and making some commutations we find that

(B.4.12)

At first, we estimate the latter contribution in the above right hand side using that χ(t

Now, if we change the approach of bounding the L ∞ (dx) norm of x µ i x j Q kg 0 ((Γv) -, D 1 u ± ), which is one of the contributions to x µ i x j ΓNL kg after (B.1.20a), and make use of lemma B.2.2 with L = L ∞ instead of (B.4.13), we see that

Then, choosing s > 0 sufficiently large so that N (s) ≥ 3 and using again (

where δ Ω (resp. δ Z 1 ) is equal to 1 if Γ = Ω (resp. if Γ = Z 1 ), 0 otherwise. There exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0, and every t ∈ [1, T ],

(B.4. 16)

Moreover, for every m = 1, 2 and t ∈ [1, T ] (B.4.17)

Proof. From definition (B.4.15) of V NF Γ and equalities (A.15), (A.16), we find that

) for µ = 0, 1, and s > 0 large enough to have N (s) ≥ 2, in order to estimate the L ∞ norm of products appearing in (B.2.30) (resp. in the second line in the above right hand side). For some new

) µ Z m v for |µ| = 0, 1, and s > 0 large enough to have N (s) > 1. We bound instead the L 2 norm of all other remaining products with the L ∞ norm of factor that does not contain any vector field times the L 2 norm of the remaining one. Hence V Γ (t, x) := tV NF Γ (t, tx).

There exists a positive constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, and every t ∈ [1, T ],

Proof. Let us recall equalities (B.2.30) with Γ I = Γ and (B. 4.18). From a-priori estimates we immediately derive that, for every t ∈

and consequently that

Using definition (B.4.15) one can check that

After (1.1.11d) with k = 1,(B.4.17), and the fact that σ can be chosen sufficiently small so that 3σ + δ 2 ≤ δ 1 /2, as δ 2 δ 1 , it is straightforward to see that

Moreover, from (B.1.3a), (B.1.10b) and a-priori estimates, (B.4.27)

Using instead equalities (A.15) and (A.16) we derive the following explicit expression for NL kg,c Γ :

we find that

with β > 0 small such that β → 0 as σ → 0. Moreover, if V NF Γ is the function defined in (B.4.15), then for every t ∈ [1, T ]

Proof. We warn the reader that throughout the proof we denote by C and β two positive constants that may change line after line, with β → 0 as σ → 0.

We refer to equality (B.2.40) with I = I 1 and bound the L 2 norm of each product in the first, third and fifth line of its right hand side by means of lemma B.2.2 with L = L 2 . The L 2 norm of the remaining products in the second line of the mentioned equality is instead estimated using inequalities (B.2.24) with L = L 2 and w j 0 = (D x D x -1 ) µ (Γ I 1 v) ± . In this way we obtain that there is some

Choosing s > 0 large so that N (s) > 1 and using estimates (1.1.11), (B.1.5a), (B.1.17), together with lemmas B.2.8 and B.4.2, we hence find that

Analogously, In order to derive estimate (B.4.32a) we apply Z m to both sides of equality (B.4.18), use (B.4.31), formulas (B.2.39), and successively proceed as follows: products in which Z m acts on v and Γ on u, that arise from the action of Z m on the second line of (B.4.18), are estimated using inequalities (B.2.24) with L = L 2 and w = u; products in which Z m is acting on v and there are no Klainerman vector fields acting on u are estimated applying lemma B.2.2 with L = L 2 ; the L 2 norm of the remaining ones are controlled by the L ∞ norm of the Klein-Gordon factor times the L 2 norm of the wave one. In this way we get that

Choosing s > 0 large so that N (s) > 2 and using (1. ). An analogous procedure leads us to the following inequality Lemma B.4.9. Let h = t -1 , u, V Γ be respectively defined in (3.2.2) and (B.4.20), a 0 (ξ) ∈ S 0,0 (1), and b 1 (ξ) = ξ j or b 1 (ξ) = ξ j ξ k |ξ| -1 , with j, k ∈ {1, 2}. There exists a constant C > 0 such that, for any χ, χ 1 ∈ C ∞ 0 (R 2 ), σ > 0, and every t ∈ [1, T ], we have that

with β > 0 small, β → 0 as σ, δ 0 → 0.

Proof. The proof of this result has the same structure as that of lemma B.3.3. Only few differences occur due to to the fact that we are replacing v with V Γ . We limit here to indicate them.

Instead of referring to estimate (B.3.9) we use the fact that, after (B.2.42) in classical coordinates, there exists a constant C > 0 such that for any ρ ∈ N (B.4.36)

with β > 0 small such that β → 0 as σ → 0. We successively decompose

On the one hand, from the fact that above operators are supported for frequencies |ξ| h σ , together with proposition 1.2.39 with p = +∞ and (B.4.36), we have that

On the other hand, combining the analogous of (B.

Lemma B.4.10. Let Γ ∈ {Ω, Z m , m = 1, 2} be a Klainerman vector field and V NF Γ be the function defined in (B. 4.15). There exists a constant C > 0 such that, for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, m, n = 1, 2, and every t ∈ [1, T ],

with β > 0 small such that β → 0 as σ → 0.

Proof. We remind the reader about explicit expression (B.4.18) of the difference V NF Γ -(Γv) -, and (B.2.30), here considered with |I| = 1 such that Γ I = Γ.

We first use equalities (1.1.5), (1.1.10), and, after some commutations, multiply x m (together with x n when proving (B.4.37b)) against each Klein-Gordon factor. Successively, we estimate the contribution coming from v I,NF -(Γv) -using lemma B.2.2 with L = L ∞ , and all products coming from the second line of (B.4.18) by means of inequalities (B.2.24) with L = L ∞ , w = u and w j 0 = (D x D x -1 ) µ Z m v for |µ| = 0, 1. On the one hand, we obtain that

so picking the same s as before and using (B. Γ be given by (B.4.28). There exists a constant C > 0 such that for any χ ∈ C ∞ 0 (R 2 ), σ > 0 small, m, n = 1, 2, and every t ∈ [1, T ],

with β > 0 small such that β → 0 as σ, δ 0 → 0. Moreover, in the same time interval

Proof. We warn the reader that we will denote by C, β, β some positive constants that may change line after line, with β → 0 (resp. β → 0) as σ → 0 (resp. as σ, δ 0 → 0). For a seek of compactness we also denote by R(t, x) any contribution verifying

Let us introduce NL cub v as follows

From the mentioned inequalities and the additional (B.1.27b), it also satisfies

Those inequalities make NL cub v a contribution of the form R(t, x), so from (B.4.43) we are left to prove that the same is true for ΓNL kg (D 1 u), NL kg (D 1 Γu), NL kg (D 2 u) and NL kg (D t u).

We immediately observe, from (B.1.1b) and (1.1.5), that the cubic contributions to NL kg (D 2 u) and NL kg (D t u) are of the form

with δ Ω (resp. δ Z 1 ) equal to 1 if Γ = Ω (resp. Γ = Z 1 ), 0 otherwise. Estimates (1.1.11a) and (B.4.27) imply that

Also, for any θ ∈]0, 1[,

as follows from (B.1.3d) with s = 1 and a-priori estimates. Thus Q kg 0 v ± , Q w 0 (v ± , D 1 v ± ) (D 1 u) is a remainder R(t, x). The same holds true for Firstly, we can assume that all factors in (B.4.45) are truncated for moderate frequencies less or equal than t σ , up to R(t, x) contributions. As regards (B.4.45a), this comes out from the application of lemma B.2.2. In fact, taking L = L 2 , w 1 = x k m x n a 0 (D x )(Γv) -for k ∈ {0, 1}, s > 0 large enough to have N (s) > 2, and using a-priori estimates and (B.1.17), (B.1.28), we find that there is some χ 1 ∈ C ∞ 0 (R 2 ) such that, for k = 0, 1,

where δ k1 is the Kronecker delta. Taking instead L = L ∞ , from a-priori estimates we derive that

+ CAB 2 ε 3 t -

≤ CA 3-θ B θ ε 3 t -3+θ(1+ δ 2 ) , as follows by (B.1.3c) with s = 1 and a-priori estimates, we deduce (up to taking θ 1 small in the above inequality) that also the first two non-linear terms in the right hand side of (B.4.53) satisfy (B.4.55) and can be included into R(t, x). Therefore, V Γ Λ kg satisfies