Cécile Huneau

Stability in exponential time of Minkowski Space-time with a translation space-like Killing eld

Stability in exponential time of Minkowski Space-time

with a translation space-like Killing field

Introduction

In this paper, we address the quasi stability of the Minkowski solution to the Einstein vacuum equations with a translation space-like Killing eld. In the presence of a translation space-like Killing eld, the 3 + 1 Einstein vacuum equations reduces to the following system in the polarized case (see Appendix A) g φ = 0, R µν = ∂ µ φ∂ ν φ. (1.1) This system has been studied by Choquet-Bruhat and Moncrief in [START_REF] Choquet | Nonlinear stability of an expanding universe with the S 1 isometry group[END_REF] (see also [START_REF] Choquet-Bruhat | General relativity and the Einstein equations[END_REF]) in the case of a space-time of the form Σ × S 1 × R, where Σ is a compact two dimensional manifold of genus G ≥ 2, and R is the time axis, with a space-time metric independent of the coordinate on S 1 . They prove the existence of global solutions corresponding to the perturbations of a particular expanding universe. This symmetry has also been studied in [START_REF] Ashtekar | Asymptotic structure of symmetry-reduced general relativity[END_REF], with an additional rotation symmetry.

In this paper, we consider a space-time of the form R 2 ×R x 3 ×R t , for which ∂ 3 is a Killing vector eld. Minkowski space-time can be seen as a trivial solution of Einstein vacuum equations with this symmetry. The question we address in this paper is the stability of the Minkowski solution in this framework.

In the 3+1 vacuum case, the stability of Minkowski space-time has been proven in the celebrated work of Christodoulou and Klainerman in [START_REF] Christodoulou | The global nonlinear stability of the Minkowski space[END_REF] in the maximal foliation. It has then been proven by Lindblad and Rodnianski using the wave coordinates in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF]. Their proof extends also to the Einstein equations coupled to a scalar eld. In this work we will use wave coordinates.

Einstein equations in wave coordinates

Wave coordinates (x α ) are required to satisfy g x α = 0. In these coordinates (1.1) reduces to the following system of quasilinear wave equations g φ = 0, g g µν = -∂ µ φ∂ ν φ + P µν (∂g, ∂g), (1.2) where P µν is a quadratic form. To understand the diculty, let us rst recall known results in 3 + 1 dimensions. In 3 + 1 dimensions, a semi linear system of wave equations of the form u i = P i (∂u j , ∂u k ) is critical in the sense that if there isn't enough structure, the solutions might blow up in nite time (see the counter examples by John [START_REF] John | Blow-up for quasilinear wave equations in three space dimensions[END_REF]). However, if the right-hand side satises the null condition, introduced by Klainerman in [START_REF] Klainerman | The null condition and global existence to nonlinear wave equations[END_REF], the system admits global solutions. This condition requires that P i be linear combinations of the following forms

Q 0 (u, v) = ∂ t u∂ t v -∇u.∇v, Q αβ (u, v) = ∂ α u∂ β v -∂ α v∂ β u.
In three dimensions, the Einstein equations written in wave coordinates do not satisfy the null condition. However, this is not a necessary condition to obtain global existence. An example is provided by the system φ 1 = 0, φ 2 = (∂ t φ 1 ) 2 .

(1.3)

The non-linearity does not have the null structure, but thanks to the decoupling there is nevertheless global existence. In [START_REF] Lindblad | The weak null condition for Einstein's equations[END_REF], Lindblad and Rodnianski showed that the non linear terms in the Einstein equations in wave coordinates consist of a linear combination of null forms with an underlying structure of the form (1.3). They used the wave condition to obtain better decay for some coecients of the metric. However the decay is slower than for the solution of the wave equation. An example of a quasilinear scalar wave equation admitting global existence without the null condition, but with a slower decay is also studied by Lindblad in [START_REF] Lindblad | Global solutions of nonlinear wave equations[END_REF] in the radial case, and by Alinhac in [START_REF] Alinhac | An example of blowup at innity for a quasilinear wave equation[END_REF] and Lindblad in [START_REF] Lindblad | Global solutions of quasilinear wave equations[END_REF] in the general case. In [START_REF] Lindblad | The weak null condition for Einstein's equations[END_REF], Lindblad and Rodnianski introduced the notion of weak-null structure, which gathers all these examples.

In 2 + 1 dimensions, to show global existence, one has to be careful with both quadratic and cubic terms. Quasilinear scalar wave equations in 3 + 1 dimensions have been studied by Alinhac in [START_REF] Alinhac | The null condition for quasilinear wave equations in two space dimensions I[END_REF]. He shows global existence for a quasilinear equation of the form u = g αβ (∂u)∂ α ∂ β u, if the quadratic and cubic terms in the right-hand side satisfy the null condition. Global existence for a semi-linear wave equation with the quadratic and cubic terms satisfying the null condition has been shown by Godin in [START_REF] Godin | Lifespan of solutions of semilinear wave equations in two space dimensions[END_REF] using an algebraic trick to remove the quadratic terms, which does however not extend to systems. The global existence in the case of systems of semi-linear wave equations with the null structure has been shown by Hoshiga in [START_REF] Hoshiga | The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space[END_REF]. It requires the use of L ∞ -L ∞ estimates for the inhomogeneous wave equations, introduced in [START_REF] Kubo | Scattering for systems of semilinear wave equations with dierent speeds of propagation[END_REF].

To show the quasi global existence for our system in wave coordinates, it will therefore be necessary to exhibit structure in the quadratic and cubic terms. However, as for the vacuum Einstein equations in 3 + 1 dimension in wave coordinates, our system does not satisfy the null structure. It will in particular be important to understand what happens for a system of the form (1.3) in 2 + 1 dimensions. For such a system, standard estimates only give an L ∞ bound for φ 2 , without decay. Moreover, the growth of the energy of φ 2 is like √ t.

One can easily imagine that with more intricate a coupling than for (1.3), it will be very dicult to prove stability without decay for φ 2 . To obtain a more useful estimate, the idea will be to exploit more precisely the fact that φ 1 also satises a wave equation. To understand how this might help, we will look at special solutions of the vacuum Einstein equations with a translation space-like Killing eld : Einstein-Rosen waves. These solutions have been discovered by Beck (see [START_REF] Beck | Zur Theorie binärer Gravitationsfelder[END_REF], and also [START_REF] Ashtekar | Asymptotic structure of symmetry-reduced general relativity[END_REF] and [START_REF] Berger | On asymptotically at space-times with G 2 -invariant Cauchy surfaces[END_REF] for a mathematical description).

Einstein-Rosen waves

Einstein-Rosen waves are solutions of the vacuum Einstein equations with two space-like orthogonal Killing elds : ∂ 3 and ∂ θ . The 3 + 1 metric can be written g = e 2φ (dx 3 ) 2 + e 2(a-φ) (-dt 2 + dr 2 ) + r 2 e -2φ r 2 dθ 2 .

The reduced equations R µν = ∂ µ φ∂ ν φ, g φ = 0, can be written in this setting

R tt = ∂ 2 r a -∂ 2 t a + 1 r ∂ r a = 2(∂ t φ) 2 , (1.4)
R rr = -∂ 2 r a + ∂ 2 t a + 1 r ∂ r a = 2(∂ r φ) 2 , R tr = 1 r ∂ t a = 2∂ t φ∂ r φ.
The equation for φ can be written, since φ is radial

e 2a g φ = -∂ 2 t φ + ∂ 2 r φ + 1 r ∂ r φ = 0,
where g is the metric g = e 2a (-dt 2 + dr 2 ) + r 2 dθ 2 .

The equation for φ decouples from the equations for the metric. Therefore we can solve the at wave equation φ = 0, with initial data (φ, ∂ t φ)| t=0 = (φ 0 , φ 1 ) and then solve the Einstein equations, which reduce to

∂ r a = r (∂ r φ) 2 + (∂ t φ) 2 , (1.5)
with the boundary condition φ| r=0 = 0 in order to have a smooth solution. Since φ = 0, if (φ 0 , φ 1 ) have enough decay, we have the following decay estimate for φ |∂φ(r, t)|

1 √ 1 + t + r(1 + |t -r|) 3 2
.

Therefore since

a = R 0 r (∂ r φ) 2 + (∂ t φ) 2 dr we have |a| 1 (1 + |r -t|) 2 , f or r < t, |a -E(φ)| 1 (1 + |r -t|) 2 , f or r > t,
where the energy

E(φ) = ∞ 0 r (∂ r φ) 2 + (∂ t φ) 2 dr
does not depend on t. For r > t, we have a ∼ E(γ) and hence is only bounded. In particular, the metric e 2a dr 2 + r 2 dθ 2 exhibits an angle at space-like innity, that is to say the circles of radius r have a perimeter growth of e -E(φ) 2πr instead of 2πr. However, in the interior, the decay we get is far better than the one we could have found with standard estimates, if we had used (1.4) instead of (1.5).

The background metric

We would like to adapt the analysis of Section 1.2 in the case where we only assume one Killing eld (i.e. in the case where ∂ 3 is Killing but not ∂ θ ). Assume that a = R 0 r (∂ r φ) 2 + (∂ t φ) 2 dr is still an approximate solution of (1.3), which will appear to be true in Section 7. As in this case φ also depends on θ, we will have

lim R→∞ a(t, R, θ) = ∞ 0 r (∂ r φ) 2 + (∂ t φ) 2 dr = b(t, θ).
Note that we have to be careful with the dependence on θ. The metric e 2b(θ) (-dt 2 + dr 2 ) + r 2 dθ 2 is no longer a Ricci at metric when b depends on θ. Consequently it is not a good guess for the behavior at innity of our metric solution g. A good candidate should be Ricci at in the region r > t. Indeed if we considered compactly supported initial data for φ, by nite speed propagation, φ should intuitively be supported in the region r < t. Consequently, the equation

R µν = ∂ µ φ∂ ν φ
implies that g should be Ricci at for r > t. Consequently, we are yield to consider the following family of space-time metrics g b = -dt 2 + dr 2 + (r + χ(q)b(θ)q) 2 dθ 2 + J(θ)χ(q)dqdθ, (1.6) where (r, θ) are polar coordinates, q = r -t and χ is a cut-o function such that χ(q) = 0 for q < 1 and χ(q) = 1 for q > 2. In the coordinates s = r + t, q, θ, a tedious calculation yields that all the Ricci coecients are zero except (R b ) qq = -b(θ)∂ 2 q (qχ(q)) r + b(θ)qχ(q) + qχ(q)χ (q)J(θ)∂ θ b (r + b(θ)qχ(q)) 3 + J(θ) 2 χ(q)χ (q) 4(r + b(θ)qχ(q)) 3 -

χ (q)∂ θ J(θ) (r + b(θ)qχ(q)) 2 , = - b(θ)∂ 2 q (qχ(q)) r + O C(b, b , J, J )1 1<q<2 r 2 , (1.7)
(R b ) qθ = -J(θ)χ (q) 2(r + b(θ)qχ(q)) = O C(b, J)1 1<q<2 r .

(1.8)

Therefore, the metrics g b are Ricci at in the region r > t + 2. We will see in the next section that they are compatible with the initial data for g given by the constraint equations.

This choice of background metric will force us to work in generalized wave coordinates, instead of usual wave coordinates. Indeed, for the metric g b dened by (1.6), the coordinates (t, x 1 , x 2 ) are not wave coordinates, not even asymptotically. The generalized wave coordinate condition reads, for g of the form g = g b + g g and F α is dened by the sum of the crossed terms of the form g ∂ θ r g b in g λβ Γ α λβ -Hα b . The reason of this choice for F α will be explained in next section, in the proof of Theorem 1.3.

The form of (1.1) in generalized wave coordinates is given by (2.1).

The initial data

In this section, we will explain how to choose the initial data for φ and g. We will note i, j the space-like indices and α, β the space-time indices.

We will work in weighted Sobolev spaces. Denition 1.1. Let m ∈ N and δ ∈ R. The weighted Sobolev space H m δ (R n ) is the completion of

C ∞ 0 for the norm u H m δ = |β|≤m (1 + |x| 2 ) δ+|β| 2 D β u L 2 .
The weighted Hölder space C m δ is the complete space of m-times continuously dierentiable functions with norm

u C m δ = |β|≤m (1 + |x| 2 ) δ+|β| 2 D β u L ∞ .
Let 0 < α < 1. The Hölder space C m+α δ is the the complete space of m-times continuously dierentiable functions with norm

u C m+α δ = u C m δ + sup x =y, |x-y|≤1 |∂ m u(x) -∂ m u(y)|(1 + |x| 2 ) δ 2
|x -y| α .

We recall the Sobolev embedding with weights (see for example [START_REF] Choquet-Bruhat | General relativity and the Einstein equations[END_REF], Appendix I).

Proposition 1.2. Let s, m ∈ N. We assume s > 1. Let β ≤ δ + 1 and 0 < α < min(1, s -1).

Then, we have the continuous embedding

H s+m δ (R 2 ) ⊂ C m+α β (R 2 ).
Let 0 < δ < 1. The initial data (φ 0 , φ 1 ) for (φ, ∂ t φ)| t=0 are freely given in H N +1 δ × H N δ+1 with 0 < δ < 1. However the initial data for (g µν , ∂ t g µν ) cannot be chosen arbitrarily.

The induced metric and second fundamental form (ḡ, K) must satisfy the constraint equations.

The generalized wave coordinates condition must be satised at t = 0.

Moreover, we want to prescribe the asymptotic behaviour for g : we want it to be asymptotic to g b , where b(θ) is arbitrarily prescribed, except for its components in 1, cos(θ) and sin(θ).

We recall the constraint equations. First we write the metric g in the form g = -N 2 (dt) 2 + ḡij (dx i + β i dt)(dx j + β j dt),

where the scalar function N is called the lapse, the vector eld β is called the shift and ḡ is a Riemannian metric on R 2 . We consider the initial space-like surface R 2 = {t = 0}. We will use the notation

∂ 0 = ∂ t -L β ,
where L β is the Lie derivative associated to the vector eld β. With this notation, we have the following expression for the second fundamental form of R 2

K ij = - 1 2N ∂ 0 g ij .
We will use the notation

τ = g ij K ij
for the mean curvature. We also introduce the Einstein tensor

G αβ = R αβ - 1 2 Rg αβ ,
where R is the scalar curvature R = g αβ R αβ . The constraint equations are given by

G 0j ≡ N (∂ j τ -D i K ij ) = ∂ 0 φ∂ j φ, j = 1, 2, (1.11)
G 00 ≡ N 2 2 ( R -|K| 2 + τ 2 ) = (∂ 0 φ) 2 - 1 2 g 00 g αβ ∂ α φ∂ β φ, (1.12)
where D and R are respectively the covariant derivative and the scalar curvature associated to ḡ.

The following result, proven in Appendix B, gives us the initial data we need.

Theorem 1.3. Let 0 < δ < 1.

Let (φ 0 , φ 1 ) ∈ H N +1 δ (R 2 ) × H N δ+1 (R 2
) and b(θ) ∈ W N,2 (S 1 ) such that bdθ = b cos(θ)dθ = b sin(θ)dθ = 0.

We assume

φ 0 H N +1 δ + φ 1 H N δ+1 ε, b W N,2 ε 2 .
If ε > 0 is small enough, there exists b

0 , b 1 , b 2 ∈ R × R × S 1 , J ∈ W N,2 (S 1
) and

(g αβ ) 0 , (g αβ ) 1 ∈ H N +1 δ × H N δ+1
such that the initial data for g given by

g = g b + g 0 , ∂ t g = ∂ t g b + g 1 ,
where g b is dened by (1.6) with

b(θ) = b 0 + b 1 cos(θ) + b 2 sin(θ) + b(θ),
are such that g ij , K ij = L β g ij satisfy the constraint equations (1.11) and (1.12).

the following generalized wave coordinates condition is satised at t = 0

g λβ Γ α λβ = g λβ b (Γ b ) α λβ + F α ,
where F α is the sum of all the crossed term of the form g 0 ∂ θ r g b in g λβ Γ α λβ -g λβ b (Γ b ) α λβ . Moreover, we have the estimates

J W N,2 (S 1 ) + g 0 H N +1 δ + g 1 H N δ+1 ε 2 , b 0 = 1 4π φ2 + |∇φ| 2 dx + O(ε 4 ), b 1 = 1 π φ∂ 1 φdx + O(ε 4 ), b 2 = 1 π φ∂ 2 φdx + O(ε 4 ).
Let us make a remark on the choice of F .

Remark 1.4. The initial data ∂ t g 00 and ∂ t g 0i are constructed so that the generalized wave coordinate condition is satised at t = 0. The choice of F is here to prevent terms of the form g∂ U (g b ) in this gauge condition, and therefore allows us to have

∂ t g 00 , ∂ t g 0i ∈ H N δ+1 .
Before stating our main result, we will recall some notations and basic tools in the study of wave equations.

Some basic tools Coordinates and frames

We note x α the standard space-time coordinates, with t = x 0 . We note (r, θ) the polar spacelike coordinates, and s = t + r, q = r -t the null coordinates. The associated one-forms are ds = dt + dr, dq = dr -dt, and the associated vector elds are

∂ s = 1 2 (∂ t + ∂ r ), ∂ q = 1 2 (∂ r -∂ t ).
We note dx the Lebesgue measure with respect to the space coordinates x i .

We note ∂ the space-time derivatives, ∇ the space-like derivatives, and by ∂ the derivatives tangent to the future directed light-cone in Minkowski, that is to say ∂ t + ∂ r and ∂ θ r .

We introduce the null frame

L = ∂ t + ∂ r , L = ∂ t -∂ r , U = ∂ θ r .
In this frame, the Minkowski metric takes the form

m LL = -2, m U U = 1, m LL = m LL = m LU = m LU = 0.
The collection T = {U, L} denotes the vector elds of the frame tangent to the light-cone, and the collection V = {U, L, L} denotes the full null frame.

The at wave equation Let φ be a solution of φ = 0, (φ, ∂ t φ)| t=0 = (φ 0 , φ 1 ).

(1. [START_REF] John | Blow-up for quasilinear wave equations in three space dimensions[END_REF] The following proposition establishes decay for the solutions of the 2 + 1 dimensional at wave equation.

Proposition 1.5 (Proposition 2.1 in [START_REF] Kubota | Existence of a global solution to a semi-linear wave equation with initial data of noncompact support in low space dimensions[END_REF]). Let µ > 1 2 . We have the estimate |φ(x, t)| M µ (φ 0 , φ 1 ) (1 + |t -r|) [1-µ] + √ 1 + t + r 1 + |t -r| where M µ (φ 0 , φ 1 ) = sup and where we used the notation

A [α] + = A max(α,0) if α = 0 and A [0] + = ln(A).
Minkowski vector elds We will rely in a crucial way on the Klainerman vector eld method.

We introduce the following family of vector elds

Z = {∂ α , Ω αβ = -x α ∂ β + x β ∂ α , S = t∂ t + r∂ r } ,
where x α = m αβ x β . These vector elds satisfy the commutation property

[ , Z] = C(Z) ,
where

C(Z) = 0, Z = S, C(S) = 2.
Moreover some easy calculations give

∂ t + ∂ r = S + cos(θ)Ω 0,1 + sin(θ)Ω 0,2 t + r , 1 r ∂ θ = Ω 1,2 r = cos(θ)Ω 0,2 -sin(θ)Ω 0,1 t , ∂ t -∂ r = S -cos(θ)Ω 0,1 -sin(θ)Ω 0,2 t -r .
With this calculations, and the commutations properties in the region

-t 2 ≤ r ≤ 2t [Z, ∂] ∼ ∂, [Z, ∂] ∼ ∂, we obtain |∂ k ∂l u| ≤ 1 (1 + |q|) k (1 + s) l |Z k+l u|, (1.14)
where here and in the rest of the paper, Z I u denotes any product of I of the vector elds of Z.

Estimates (1.14) and Proposition 1.5 yield Corollary 1.6. Let φ be a solution of (1.13). We have the estimate

|∂ k ∂l φ(x, t)| M k+l µ (φ 0 , φ 1 ) (1 + |t -r|) [1-µ] + (1 + t + r) l+ 1 2 (1 + |t -r|) k+ 1 2
where

M j µ (φ 0 , φ 1 ) = sup y∈R 2 (1 + |y|) µ+j |∇ s φ 0 (y)| + (1 + |y|) µ+1+j (|∇ s φ 1 (y)| + |∇ 1+j φ 0 (y)|).
Weighted energy estimate We consider a weight function w(q), where q = r -t, such that w (q) > 0 and w(q)

(1 + |q|) 1+µ w (q) w(q) 1 + |q| , for some 0 < µ < 1 2 .
Proposition 1.7. We assume that φ = f . Then we have

1 2 ∂ t w(q) (∂ t φ) 2 + |∇φ| 2 dx + 1 2 w (q) (∂ s φ) 2 + ∂ θ u r 2 dx w(q)|f ∂ t φ|dx.
For the proof of Proposition 1.7, we refer to the proof of Proposition 9.1 which is the quasilinear equivalent of Proposition 1.7.

Weighted Klainerman-Sobolev inequality

The following proposition allows us to obtain L ∞ estimates from the energy estimates. It is proved in Appendix F. The proof is inspired from the corresponding 3 + 1 dimensional proposition (Proposition 14.1 in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF]).

Proposition 1.8. We denote by v any of our weight functions. We have the inequality

|f (t, x)v 1 2 (|x| -t)| 1 1 + t + |x| 1 + ||x| -t| |I|≤2 v 1 2 (. -t)Z I f L 2 .
Weighted Hardy inequality If u is solution of u = f , the energy estimate allows us to estimate the L 2 norm of ∂u. To estimate the L 2 norm of u, we will use a weighted Hardy inequality.

Proposition 1.9. Let α < 1 and β > 1. We have, with q = r -t v(q)

1 2

(1 + |q|) f L 2 v(q) 1 2 ∂ r f L 2 ,
where

v(q) = (1 + |q|) α , f or q < 0, v(q) = (1 + |q|) β , f or q > 0.
This is proven in Appendix E. The proof is inspired from the 3 + 1 dimensional analogue (Lemma 13.1 in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF]).

L ∞ -L ∞ estimate With the condition w (q) > 0 for the energy inequality, we are not allowed to take weights of the form (1 + |q|) α , with α > 0 in the region q < 0. Therefore, the Klainerman-Sobolev inequality cannot give us more than the estimate

|∂u| 1 1 + |q| √ 1 + s ,
in the region q < 0, for a solution of u = f . However, we know that for suitable initial data, the solution of the wave equation

u = 0 satises |u| 1 1 + |q| √ 1 + s , |∂u| 1
(1 + |q|)

3 2 √ 1 + s .
To recover some of this decay we will use the following proposition Proposition 1.10. Let u be a solution of

u = F, (u, ∂ t u)| t=0 = (0, 0). For µ > 3 2 , ν > 1 we have the following L ∞ -L ∞ estimate |u(t, x)|(1 + t + |x|) 1 2 ≤ C(µ, ν)M µ,ν (F )(1 + |t -|x|||) -1 2 +[2-µ] + ,
where

M µ,ν (F ) = sup(1 + |y| + s) µ (1 + |s -|y||) ν F (y, s),
and where we used the convention

A [α] + = A max(α,0) if α = 0 and A [0] + = ln(A).
This is proven in Appendix D. This inequality has been introduced by Kubo and Kubota in [START_REF] Kubo | Scattering for systems of semilinear wave equations with dierent speeds of propagation[END_REF].

An integration lemma The following lemma will be used many times in the proof of Theorem 1.12, to obtain estimates for u when we only have estimates for ∂u.

Lemma 1.11. Let α, β, γ ∈ R with β < -1. We assume that the function u : R 2+1 → R satises

|∂u| (1 + s) γ (1 + |q|) α , f or q < 0, |∂u| (1 + s) γ (1 + |q|) β f or q > 0,
and for t = 0

|u| (1 + r) γ+β .
Then we have the following estimates

|u| (1 + s) γ max(1, (1 + |q|) α+1 ), f or q < 0, |u| (1 + s) γ (1 + |q|) β+1 f or q > 0.
Proof. We assume rst q > 0. We integrate the estimate

|∂ q u| (1 + s) γ (1 + |q|) β , from t = 0. We obtain, since β < -1, for q > 0 |u| (1 + s) γ (1 + |q|) β+1 .
Consequently, we have, for q = 0, |u| (1 + s) γ . We now assume q < 0. We integrate

|∂ q u| (1 + s) γ (1 + |q|) α , from q = 0. We obtain |u| (1 + s) γ max(1, (1 + |q|) α+1 ).
This concludes the proof of Lemma 1.11.

Main Result

We introduce an other cut-o function Υ : R + → R + such that Υ(ρ) = 0 for ρ ≤ 1 2 and ρ ≥ 2 and Υ = 1 for 3 4 ≤ ρ ≤ 3 2 . Theorem 1.12 is our main result, in which we prove stability of Minkowski space-time with a translational symmetry in exponential time T exp 1

√ ε

where ε > 0 is the size of the small initial data.

Theorem 1.12. Let 0

< ε < 1. Let 1 2 < δ < 1 and N ≥ 40. Let (φ 0 , φ 1 ) ∈ H N +1 δ (R 2 ) × H N δ+1 (R 2 ).
We assume

φ 0 H N +1 δ + φ 1 H N δ+1 ≤ ε. Let T exp( 1 √ ε ). Let ε ρ σ µ δ. If ε is small enough, there exists b(θ), J(θ) ∈ W N,2 (S 1
) and there exists a global coordinate chart (t, x 1 , x 2 ) such that, for t ≤ T , there exists a solution (φ, g) of (1.1) that we can write

g = g b + Υ r t g LL 4 dq 2 + g U L 2
rdqdθ + g such that we have the estimates

|I|≤N α 2 w 1 2 0 (q)∂Z I φ L 2 + 1 √ 1 + t α 2 w 1 2 3 (q)∂Z I g LL L 2 + 1 √ 1 + t α 2 w 1 2 3 (q)∂Z I g LU L 2 + α 2 w 1 2 2 (q)∂Z I g L 2 ε(1 + t) C √ ε .
with

w 0 (q) = (1 + |q|) 2+2δ , q > 0 w 0 (q) = 1 + 1 (1+|q|) 2µ , q < 0, w 2 (q) = (1 + |q|) 2+2δ , q > 0 w 2 (q) = 1 (1+|q|) 1+2µ , q < 0, w 3 (q) = (1 + |q|) 3+2δ , q > 0 w 3 (q) = 1 + 1 (1+|q|) 2µ , q < 0, α 2 (q) = (1 + |q|) -2σ , q > 0 α 2 (q) = 1, q < 0,
Moreover, for all ρ > 0, we have the L ∞ estimate, for |I| ≤ N 2 + 2 and r < t

|Z I φ(x, t)| εC(ρ) (1 + t + r) 1 2 (1 + |t -r|) 1 2 -4ρ
,

|Z I g LL | εC(ρ) (1 + |t -r|) 1 2 -ρ , |Z I g LU | + |Z I g| εC(ρ) (1 + t + r) 1 2 -ρ
.

and we have the estimate for b b(θ)

+ Σ T,θ (∂ q φ) 2 rdr ε 2 √ T ,
where we have used the notation

Σ T,θ (∂ q φ) 2 rdr = ∞ 0 (∂ q φ(T, r, θ)) 2 rdr.
(1.15)

Comments on Theorem 1.12

We consider perturbations of 3 + 1 dimensional Minkowski space-time with a translational space-like Killing eld. These perturbations are not asymptotically at in 3 + 1 dimensions, therefore the result of Theorem 1.12 does not follow from the stability of Minkowski spacetime by Christodoulou and Klainerman [START_REF] Christodoulou | The global nonlinear stability of the Minkowski space[END_REF].

As our gauge, we choose the generalized wave coordinates, which are picked such that the generalized wave coordinates condition is satised by g b . Therefore, the method we use has a lot in common with the method of Lindblad and Rodnianski in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF] where they proved the stability of Minkowski space-time in harmonic gauge. It is an interesting problem to investigate the stability of Minkowski with a translation symmetry using a strategy in the spirit of [START_REF] Christodoulou | The global nonlinear stability of the Minkowski space[END_REF] or [START_REF] Klainerman | The evolution problem in general relativity[END_REF].

The function J(θ), We can make a rapprochement of these quantities with the ADM mass and linear momentum. The remaining Fourier coecients of b are chosen to ensure the convergence to Minkowski in the direction of time-like innity, and is an essential element in the proof of the quasi stability. In the subsequent paper [START_REF] Huneau | Stability of Minkowski Space-time with a translation space-like Killing eld[END_REF], it is shown that these remaining Fourier coecient correspond actually to a gauge choice. This remark is the key ingredient in [START_REF] Huneau | Stability of Minkowski Space-time with a translation space-like Killing eld[END_REF] to

show the full stability.

The logarithmic growth of w

1 2 (q)∂Z N φ L 2 , and the condition b(θ) ∼ Σ T,θ (∂ q φ) 2 rdr, (1.16)
give the estimate |∂ N b| ε 2 (1 + T ) Cε . To avoid factors of the form (1 + T ) Cε in all our estimate, we are forced to assume (1 + T ) Cε 1. This is the only place where we need (1 + T ) Cε 1, and this is what prevents us to prove the stability.

The condition (1.16) is not necessary to control the metric in the exterior region r > t. For this reason we believe that the stability holds in the exterir region, without the condition

T exp 1 √ ε .
As we said in the second comment, we use a method similar than Lindblad and Rodnianski method in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF]. Let us list some of the similarities and dierences with their method.

Similarities with [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF] We use the vector eld method. The vector elds we use are the Klainerman vector elds of Minkowski space-time.

We use the wave coordinate condition to obtain more decay on the coecients g T T of the metric.

We exhibit the structure corresponding to the model problem (1.3).

Dierences with [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF] The asymptotic behaviour given by the solutions of the constraint equations prevent us to work in wave coordinates. Instead we work in generalised wave coordinates.

In the exterior region, our solution does not converge to Minkowski, but to a family of Ricci at metrics g b .

The decay of the free wave is weaker in 2 + 1 dimension. Consequently, the coecient g LL of the metric does not have any decay near the light cone. We have to rely on the null decomposition at all steps in our proof to isolate this behaviour, even in the L 2 estimates.

We have to t b(θ) so that the condition (1.16) is satised. This leads to regularity issues for b, which prevent us from proving the global existence.

The structure of the paper is as followed. In Section 2 we describe the structure of the equations (1.1) in generalized wave coordinates. We exhibit the structure of our system in Section 2. We also describe the interactions between g b and g. In Section 1.3 we outline the main issues of the proof by discussing some model problems. In section 4 we give our bootstrap assumptions. In section 5

we derive preliminaries estimates thanks to the wave coordinate condition. In section 6 we derive preliminaries estimate for the angle and the linear momentum. In section 7, we will exploit the analysis begun in section 1.2. In section 8.4 we will improve the L ∞ estimate. In section 9 we will derive the weighted energy estimate. In section 10 we will improve the L 2 estimates and in section 11 we will adjust the parameter b(θ).

Structure of the equations

In this section, we provide a discussion of the specic features of the structure of the equations, which will be relevant for the proof of Theorem 1.12. For sake of simplicity, we assume the result of Proposition 6.1, and we will go backward in Section ??.

The generalized wave coordinates

Wave coordinates allow to recast Einstein equations as a system of non-linear wave equations. The wave coordinates condition, which consists in choosing coordinates such that g x α = 0 can be rewritten as

g λβ Γ α λβ = 0.
However, for the metric g b dened by (1.6), the coordinates (t, x 1 , x 2 ) are not wave coordinates, not even asymptotically. We will therefore work with generalized wave coordinates. We will impose that our metric satises

g λβ Γ α λβ = H α b
where H α b is dened by (1.9)

H α b = (g b ) λβ (Γ b ) α λβ + F α , with F α of the form g qχ(q)∂ θ b r 2 .
The role of F α was explained in section 1.4. In generalized wave coordinates, the expression (C.4)

of Appendix C allows us to write the system (1.1) under the form

g φ = 0 g g µν = -2∂ µ φ∂ ν φ + P µν (∂g, ∂g) + g µρ ∂ ν H ρ + g νρ ∂ µ H ρ , (2.1)
where

P µν (g)(∂g, ∂g) = 1 2 g αρ g βσ ∂ µ g ρσ ∂ α g βν + ∂ ν g ρσ ∂ α g βµ -∂ β g µρ ∂ α g νσ - 1 2 ∂ µ g αβ ∂ ν g ρσ + 1 2 g αβ g λρ ∂ α g νρ ∂ β g µρ . (2.2)
Remark 2.1. In generalized wave coordinates, the wave operator can be expressed as

g = g αρ ∂ α ∂ ρ -H ρ b ∂ ρ .
The expression (C.4) yields also

(R b ) µν = - 1 2 g b (g b ) µν + 1 2 P µν (g b )(∂g b , ∂g b ) + 1 2 (g b ) µρ ∂ ν Hρ b + (g b ) µρ ∂ µ Hρ b .
(2.3) Therefore, subtracting twice the equation (2.3) to the second equation of (2.1) we obtain

g φ = 0, g g µν = -2∂ µ φ∂ ν φ + 2(R b ) µν + P µν (g)(∂ g, ∂ g) + P µν ( g, g b ), (2.4)
where P µν (g)(∂ g, ∂ g) is dened by (2.2) and

P µν ( g, g b ) = g αβ b -g αβ ∂ α ∂ β (g b ) µν + F ρ ∂ ρ (g b ) µν + P µν (g)(∂g, ∂ g ) -P µν (g)(∂ g, ∂ g) -P µν (g b )(∂g b , ∂g b ) + (g b ) µρ ∂ ν F ρ + (g b ) νρ ∂ µ F ρ + g µρ ∂ ν H ρ b + g νρ ∂ µ H ρ b .
(2.5)

Let us note that P µν ( g, g b ) contains only crossed terms between g b and g.

The weak null structure

To exhibit the main terms in the structure of (2.4), let us neglect for a moment P µν , P µν , H b . We will see in the next section that this approximation is relevant. Let us also neglect the nonlinear terms involving ∂ derivatives. Then we obtain the following approximate system

φ + g LL ∂ 2 q φ = 0, g T V + g LL ∂ 2 q g T V = 0, g LL + g LL ∂ 2 q g LL = 4 -2(∂ q φ) 2 -2b(θ) ∂ 2 q (χ(q)q) r ,
where we also have used the approximation

(R b ) qq ∼ - b(θ)∂ 2 q (qχ(q)) r + O C(b, b , J, J )1 1<q<2 r 2 ,
as shown in (1.7). In 2+1 dimensions, a term of the form g LL ∂ 2 q φ is impossible to handle if one only relies on the decay for g LL provided by the fact of being a solution of a wave equation. However, as in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF], we can exploit the wave condition to obtain better decay for some coecients of the metric.

More precisely, we have roughly ∂g T T ∼ ∂g. This is done properly in Proposition 5.1 for the coecient g LL and in Proposition 5.2 for the coecients g LU and g U U . Therefore, the g T T coecients have a better decay in t than the solutions of the wave equation (the challenges of the quasilinear terms of the form g LL ∂ 2 q φ, g LL ∂ 2 q g T V are presented in Section 3.4).

Remark 2.2. The other quasilinear terms are of the form

g T V ∂ T ∂ V φ, g T V ∂ T ∂ V g.
Consequently, they involved at least one "good derivative" of φ, g. Thus, they are easier to estimate, and we can always focus on the terms

g LL ∂ 2 q φ, g LL ∂ 2 q g.
Assuming that we can also neglect the terms involving g LL , we are reduced to the following system φ = 0,

g LL = 4 -2(∂ q φ) 2 -2b(θ) ∂ 2 q (χ(q)q) r , (2.6)
which is a system of the form (1.3) and displays the weak null structure.

The second component of the solution of (1.3) does not have any decay near the light cone in 2 + 1 dimensions (see Section 1.2 for the radial case). Therefore, the coecient g LL will not display any decay at all near the light cone (see the estimates of Theorem 1.12). To obtain decay for g LL in the q variable, we will approximate g LL 4 by the solution h 0 of the following transport equation

∂ q h 0 = -2r(∂ q φ) 2 -2b(θ)∂ 2 q (qχ(q)).
The ideas of this approximation are presented in Section 3.2, and are exploited in Section 7.

Non-commutation of the wave operator with the null frame

The structure of Einstein equations can only be seen in the null frame. However it is well known that the wave operator does not commute with the null frame. In Theorem 1.12 we have decomposed our metric in the following way

g = g b + g + Υ r t g LL 4 dq 2 + g U L 2 rdqdθ .
The problems of non-commutation induced by g LL and g U L are totally similar. Consequently, we can neglect the second one. We expressed the 2-forms dq 2 in the coordinate (t, x 1 , x 2 )

dq 2 = (dr -dt) 2 = (cos(θ)dx 1 + sin(θ)dx 2 -dt) 2
Therefore, we will have, in the coordinates

x 1 , x 2 Υ r t g LL dq 2 µν - Υ r t g LL (dq 2 ) µν = Υ r t 1 r 2 u 1 µν (θ)g LL + u 2 µν (θ)∂ θ g LL (2.7)
where u 1 µν and u 2 µν are some trigonometric functions. The challenges of the terms involving u 1 µν and u 2 µν are explained in Section 3.3.

2.4

The semi linear term P µν (g)(∂ g, ∂ g).

Recall the form of the term P µν (g)(∂ g, ∂ g).

P µν (g)(∂ g, ∂ g) = 1 2 g αρ g βσ ∂ µ g ρσ ∂ α g βν + ∂ ν g ρσ ∂ α g βµ -∂ β g µρ ∂ α g νσ - 1 2 ∂ µ g αβ ∂ ν g ρσ + 1 2 g αβ g λρ ∂ α g νρ ∂ β g µρ .
The quadratic terms In the null frame (L, L, U ) the only non zero coecients of the Minkowski metric are m LL = -1 2 and m U U = 1. Thanks to this remark, we can describe the terms appearing in the dierent components of P µν .

In P T T (g)(∂ g, ∂ g), there can not be strictly more than 2 occurrences of the vector eld L.

Therefore, the quadratic terms are of one of these form

∂ V g VT ∂ T g T T , ∂ T g VV ∂ T g T T , (2.8)
where we have used the fact, proved in Section 5 that

∂ V g T T ∼ ∂ T g VV .
These terms all have the classical null structure. How this structure can be used to show global existence is explained in Section 3.1. Since they are by far easier to handle than the one we will describe in the following, they will be neglected in the proof of Theorem 1.12.

In P T V (g)(∂ g, ∂ g), there can not be strictly more than 3 occurrences of the vector eld L.

Therefore, the quadratic terms are of one of these form

∂ V g T V ∂ T g T V , ∂ V g VV ∂ T g T T , ∂ T g VV ∂ V g T T , ∂ T g T V ∂ T g VV
where we have used the fact, proved in Section 5 that

∂ V g T T ∼ ∂ T g VV .
These terms all have the null structure. However, since g LL does not decay at all in t (see the estimates of Theorem 1.12), one has to be more careful with the terms of the form ∂ T g T T ∂ L g LL These terms have a good structure since ∂ T g T T is a "good derivative" of a "good component". However, one needs two steps to exploit this structure, which can be dicult to achieve if there is no regularity left. Thankfully, these terms have three occurrences of L, therefore they can only intervene in P T L .

In P LL we will have to be careful with

∂ L g LL ∂ L g LL .
This term can be converted in ∂ L g LL ∂ L g LL with the help of the algebraic trick

(uv) = u v + v u + ∂ L u∂ L v + ∂ L v∂ L u + ∂ U u∂ U v.
This fact will be used only in the proof of Lemma 10.6.

In P LU we will have to be careful with

∂ U g LL ∂ L g LL .
This term can not be removed with the previous trick. We will have to single out its inuence thanks to the decomposition

g = g b + χ r t hdq 2 + χ r t krdqdθ + g 4 ,
where k satises

g k = ∂ U g LL ∂ L g LL .
This will also be used only in the proof of Lemma 10.6.

The terms in P LL which are not of the previous form can be written

∂ L g LL ∂ L g LL , ∂ L g LL ∂ L g LL .
(2.9)

We note the crucial cancellation of terms of the form (∂ L g LL ) 2 in P LL . The contributions (2.9) will be singled out in (2.12).

The cubic terms In two dimensions, cubic terms could be troublesome. However, in the form P VT , if there are 4 occurrences of the vector eld L, or in P LL if there are 5 occurrences of the vector eld L, then we have a factor g LL , which has a decay equivalent to g LL . Therefore we can neglect the cubic terms in this nonlinearity.

The crossed terms

In this section, we discuss the structure of the crossed terms between b and ( g, φ).

The crossed terms involving two derivatives of b are absent In the expression

g g µν -g µρ ∂ ν H ρ b + g νρ ∂ µ H ρ b ,
there could be terms involving two derivatives of b(θ), which would be troublesome since they would lead to a loss of a derivative (recall that we only have the regularity b ∈ W N,2 ). However, the terms involving two derivatives of b in this expression, are the same than the terms involving two derivatives of b in R µν (g). Thus, these terms cancel in the expression

g αβ b -g αβ ∂ α ∂ β (g b ) µν + (g b ) µρ ∂ ν F ρ + (g b ) νρ ∂ µ F ρ + g µρ ∂ ν H ρ b + g νρ ∂ µ H ρ b ,
which appears in P µν ( g, g b ) dened by (2.5). These cancellations can be checked for example with Mathematica.

The crossed terms in P µν We recall from (1.6) that g b = dsdq + (r + χ(q)qb(θ)) 2 dθ 2 + J(θ)χ(q)dqdθ.

Therefore in P µν we can nd terms involving

(g b ) U U = 1 + χ(q)qb(θ) r 2 and (g b ) U L = -2 J(θ)χ(q) r .
Since (g b ) U L decays faster than (g b ) U U let us focus on the crossed terms between (g b ) U U and g. The problem with the term (g b ) U U is that far from the light cone, it does not decay at all. This is one of the causes of the logarithmic growth of the energy in the statement of Theorem 1.12. However, these terms are present only in the exterior region. Moreover they display also a special structure.

Since the terms involving two derivatives of b are absent, and the terms involving two derivatives of g are only present in g g, the terms in P µν are of the form

g --∂ -(g b ) U U ∂ -g --.
In P T V the crossed terms involving ∂ L (g b ) U U can not contain more than two occurrences of L. They must be of the following form

∂ L (g b ) U U ∂ T g T V , ∂ T (g b ) U U ∂ V g T V , ∂ T (g b ) U U ∂ T g VV ,
where we have used the wave coordinates condition ∂ V g T T ∼ ∂ T g T V . We have the following inequalities, thanks to (1.14)

|∂ L (g b ) U U ∂ T g T V | 1 q>0 (|b| + |∂ θ b|) 1 + r |∂ T g T V | 1 q>0 (|b| + |∂ θ b|) (1 + r) 2 |Z 1 g T V |, |∂ T (g b ) U U ∂ V g T V | 1 q>0 (1 + |q|)(|b| + |∂ θ b|) (1 + r) 2 |∂ V g T V | 1 q>0 (|b| + |∂ θ b|) (1 + r) 2 |Z 1 g T V |.
These two contributions are therefore quite similar. In the following, it will be sucient to study the term

∂ L (g b ) U U ∂ T g T V .
(2.10)

The challenges of this term will be discussed in Section 3.5

In P LL , we may have three occurrences of L. Therefore there are terms of the form

∂ L (g b ) U U ∂ T g LL , ∂ L (g b ) U U ∂ L g LL , ∂ T g U U ∂ L g LL .
We have the following inequalities, thanks to (1.14)

|∂ L (g b ) U U ∂ T g LL | 1 q>0 (|b| + |∂ θ b|) 1 + r |∂ T g LL | 1 q>0 (|b| + |∂ θ b|) (1 + r) 2 |Z 1 g LL | |∂ L (g b ) U U ∂ L g LL | 1 q>0 (|b| + |∂ θ b|) 1 + r |∂ L g L | 1 q>0 (|b| + |∂ θ b|) (1 + r)(1 + |q|) |Z 1 g LL | |∂ T (g b ) U U ∂ L g LL | 1 q>0 (1 + |q|)(|b| + |∂ θ b|) (1 + r) 2 |∂ L g LL | 1 q>0 (|b| + |∂ θ b|) (1 + r) 2 |Z 1 g LL |.
Consequently, the worst term is

∂ L (g b ) U U ∂ L g LL .
(2.11)

We introduce the following notation, to single out the contributions of (2.11) and (2.9)

Q LL (h, g) = ∂ L g LL ∂ L h + ∂ L g LL ∂ L h + ∂ L (g b ) U U ∂ L g LL .
(2.12)

The crossed terms involving two derivatives of g With our choice of coordinates, these terms only appear in g g. They are of the form

1 q>0 b(1 + |q|) 1 + r ∂ 2 U g.
Their contribution is most of the time similar than the one of (2.10), except in the energy estimate, where they require a special treatment because of their lack of decay far from the light cone (see Section 9).

The crossed terms in g φ The crossed terms between g b and ∂φ are of the form

g --∂ -(g b ) U U ∂ -φ.
Consequently, they must be of the following form

∂ V (g b ) U U ∂ T φ, ∂ T (g b ) U U ∂ V φ.
Like for P VT , it will be sucient to study

∂ V (g b ) U U ∂ T φ.
(2.13)

The crossed terms between g b and ∂ 2 φ are of the form

1 q>0 b(1 + |q|) 1 + r ∂ 2 U φ.
As for g, their contribution is most of the time similar than the one of (2.13), except in the energy estimate, where they require a special treatment because of their lack of decay far from the light cone (see Section 9).

Remark 2.3. In the region q > 0 it is generally sucient to study the crossed terms. Indeed, the crossed terms are the one presenting the less decay far from light cone.

Model problems

The proof relies on a bootstrap scheme. Roughly speaking, we will assume some estimates on the coecients Z I φ, Z I g LL and Z I g T V :

L ∞ estimates for I ≤ N 2 ,
L 2 estimates for I ≤ N .

We rewrite the bootstrap assumptions in the condensed form

|φ| X 1 ≤ 2C 0 ε, |g| X 2 ≤ 2C 0 ε,
where C 0 is a constant depending only on the quantities ρ, σ, µ, δ, N introduced in the statement of Theorem 1.12 and such that at t = 0

|φ| X 1 ≤ C 0 ε, |g| X 2 ≤ C 0 ε.
Thanks to the L ∞ -L ∞ estimate and the energy estimate, we will be able to prove

|φ| X 1 ≤ C 0 ε + Cε 2 , |g| X 2 ≤ C 0 ε + Cε 2 .
Therefore, for ε chosen small enough so that Cε ≤ C 0 2 , this improves the bootstrap assumptions.

We will rst consider a toy model, which exhibits some of the mechanisms involved in the proof.

3.1 Global well posedness for a semi linear wave equation with the null structure

We consider the following 2 + 1 dimensional semi-linear wave equation u = ∂u ∂u, (u, ∂ t u)| t=0 = (u 0 , u 1 ).

(3.1) Note that the nonlinearity satises the null condition. Consequently, this model will show us how to treat the terms of the form (2.8). The following result is proved in [START_REF] Hoshiga | The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space[END_REF]. We will give a proof of it for sake of completeness, and because it exhibits some of the mechanisms involved in the proof of Theorem 1.12.

Proposition 3.1. Let 0 < δ < 1 2 . Let N ≥ 8. Let u 0 , u 1 ∈ ×H N +1 -1 2 +δ × H N δ+ 1 2 such that u 0 H N +1 -1 2 +δ + u 1 H N δ+ 1 2 ≤ ε.
If ε > 0 is small enough, the equation (3.1) has a global solution u.

Proof. Let 0 < µ < 1 4 . We introduce the weight function

w(q) = 1 + 1 (1+|q|) 2µ , q < 0, w(q) = (1 + |q|) 1+2δ q > 0. Let 0 < ρ < δ 2 .
To prove global existence for equation (3.1), we consider a time T > 0 such that, on 0 ≤ t ≤ T

|Z I u| ≤ 2C 0 ε √ 1 + s(1 + |q|) δ , I ≤ N 2 , (3.2)
|Z I u| ≤ 2C 0 ε √ 1 + s(1 + |q|) δ 2 , I ≤ N 2 + 1, (3.3)
w 1 2 ∂Z I u L 2 ≤ 2C 0 (1 + t) ρ ε, I ≤ N. (3.4)
Thanks to the Klainerman-Sobolev inequality, the assumption (3.4) yields, for I ≤ N -2

|∂Z I u| ε(1 + t) ρ √ 1 + s 1 + |q| , f or q < 0, |∂Z I u| ε(1 + t) ρ √ 1 + s(1 + |q|) 1+δ
, f or q > 0.

(3.5) and consequently, thanks to Lemma 1.11

|Z I u| ε 1 + |q| (1 + s) 1 2 -ρ
, f or q < 0, |Z I u| ε

(1 + s) 1 2 -ρ (1 + |q|) δ
, f or q > 0.

(3.6)

We use the L ∞ -L ∞ estimate to improve the estimates (3.2) and (3.3). We write

Z I u = I 1 +I 2 ≤I ∂Z I 1 u ∂Z I 2 u. (3.7)
We rst treat the case I ≤ N 2 . We assume I 1 ≤ N 4 (the case I 2 ≤ N 4 can be treated in the same way). Therefore, we can estimate thanks to (1.14)

|∂Z I 1 u| ≤ 1 1 + |q| |Z I 1 +1 u|. Since N 4 + 1 ≤ N
2 we obtain thanks to (3.2)

|∂Z I 1 u| ε (1 + |q|) 1+δ √ 1 + s .
To estimate ∂Z I 2 u we use (1.14) and the bootstrap assumption (3.3) to obtain

| ∂Z I 2 u| 1 1 + s |Z I 2 +1 u| ε (1 + s) 3 2 (1 + |q|) δ 2
.

This yields

| Z I u| ε 2 (1 + s) 2 (1 + |q|) 1+ 3δ 2 .
We can now use the L ∞ -L ∞ estimate of Proposition 1.10, together with the estimate of Proposition 1.5 and the Sobolev injection of Proposition 1.2, which gives

|Z I u| ≤ C 0 ε √ 1 + s(1 + |q|) δ + Cε 2 ln(1 + |q|) √ 1 + s 1 + |q| . This implies, since ln(1 + |q|) (1 + |q|) 1 2 -δ |Z I u| ≤ C 0 ε √ 1 + s(1 + |q|) δ + Cε 2 √ 1 + s(1 + |q|) δ . (3.8)
We now treat the case I = N 2 + 1. We assume I 1 ≤ N +2 4 ≤ N 2 so we have the same estimate as before for ∂Z I 1 u. To estimate ∂Z I 2 u, since N 2 + 2 ≤ N -2 we use (3.6). We obtain

| ∂Z I 2 u| 1 1 + s |Z I 2 +1 u| ε 1 + |q| (1 + s) 3 2 -ρ
.

Therefore we obtain

| Z I u| ε 2 (1 + s) 2-ρ (1 + |q|) 1 2 +δ ε 2 (1 + s) 3 2 + δ 2 (1 + |q|) 1+ δ 2 -ρ
.

Therefore, like for (3.8), the L ∞ -L ∞ estimate yields

|Z I u| ≤ C 0 ε √ 1 + s(1 + |q|) δ + Cε 2 √ 1 + s(1 + |q|) δ 2
.

(3.9)

We now use the weighted energy estimate to imrpove (3.4). Let I ≤ N . In view of (3.7), it implies

d dt w(q) 1 2 ∂Z I u 2 L 2 + w (q) 1 2 ∂Z I u 2 L 2 I 1 +I 2 ≤I w 1 2 ∂Z I 1 u ∂Z I 2 u L 2 w 1 2 ∂Z I u L 2 .
(3.10)

We rst assume I 2 ≤ N 2 . Then we estimate

| ∂Z I 2 u| ε (1 + s) 3 2 (1 + |q|) δ 2
.

This yields

w 1 2 ∂Z I 1 u ∂Z I 2 u L 2 ε (1 + t) 3 2 w 1 2 ∂Z I 1 u L 2 .
We now assume I 1 ≤ N 2 . Then, we estimate

|∂Z I 1 u| ε √ 1 + s(1 + |q|) 1+ δ 2 .
Therefore we obtain

w 1 2 ∂Z I 1 u ∂Z I 2 u L 2 ε √ 1 + t w 1 2 (1 + |q|) 1+ δ 2 ∂Z I 2 u L 2 . Since w 1 2 (1 + |q|) 1+ δ 2 ≤ w (q) 1 2 ,
we infer

w 1 2 ∂Z I 1 u ∂Z I 2 u L 2 w 1 2 ∂Z I u L 2 ≤ ε 1 + t w 1 2 ∂Z I u 2 L 2 + ε w (q) 1 2 ∂Z I 2 u 2 L 2 .
Therefore (3.10) writes

d dt w(q) 1 2 ∂Z I u 2 L 2 + w (q) 1 2 ∂Z I u 2 L 2 ε 1 + t w 1 2 ∂Z I u 2 L 2 + ε w (q) 1 2 ∂Z I 2 u 2 L 2 , so for ε small enough d dt w(q) 1 2 ∂Z I u 2 L 2 + 1 2 w (q) 1 2 ∂Z I u 2 L 2 ε 1 + t w 1 2 ∂Z I u 2 L 2 .
We obtain w(q)

1 2 ∂Z I u L 2 ≤ C 0 ε(1 + t) Cε . (3.11)
For ε small enough so that

Cε ≤ C 0 2 , (1 + t) Cε ≤ 3 2 (1 + t) ρ ,
we have proved, in view of (3.8), (3.9) and (3.11) that for t ≤ T we have

|Z I u| ≤ 3 2 C 0 ε √ 1 + s(1 + |q|) δ , |I| ≤ N 2 , |Z I u| ≤ 3 2 C 0 ε √ 1 + s(1 + |q|) δ 2 , |I| ≤ N 2 + 1, w 1 2 ∂Z I u L 2 ≤ 3 2 C 0 (1 + t) ρ ε, |I| ≤ N,
which concludes the proof.

Remark 3.2. Actually, only the highest order energy w

1 2 ∂Z N u L 2 grows in t.
To see this, we estimate

w 1 2 ∂Z I 1 u ∂Z I 2 u L 2 for I 1 ≤ N 2 and I 2 ≤ N -1. Since | ∂Z I 2 u| ≤ 1 1 + s |Z I 2 +1 |,
we obtain, together with the weighted Hardy inequality

w 1 2 ∂Z I 1 u ∂Z I 2 u L 2 ε (1 + t) 3 2 w 1 2 (1 + |q|) Z I 2 +1 u L 2 ε (1 + t) 3 2 w 1 2 ∂Z I 2 +1 u L 2 .
Therefore, the weighted energy estimate yields, for |I| ≤ N -1

d dt w 1 2 ∂Z I u 2 L 2 ε 2
(1 + t) Remark 3.3. The use of the term w (q)

1 2 ∂Z I u 2
L 2 to exploit the structure in the energy estimate has been introduced by Alinhac in [START_REF] Alinhac | The null condition for quasilinear wave equations in two space dimensions I[END_REF] and is sometimes called Alinhac ghost weight method. It has also been used in the case of Einstein equations in wave coordinates in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF].

Unfortunately, the Einstein equations in wave coordinates do not have the null structure, but only a weak form of it. In the next sections, we will see what problems this creates and the method we used to tackle them. We will be less precise than in this rst example, since full details will be provided when we proceed with the proof of Theorem 1.12.

The coecient g LL

To understand how to deal with g LL , let us consider the question of global existence for the following system, which is of the form (2.6) .12) with initial data for φ of size ε and zero initial data for h. We recall b L 2 (S 1 ) ε 2 . We have the following estimates for φ w

φ = 0, h = -2(∂ q φ) 2 -2 b(θ)∂ 2 q (qχ(q)) r . ( 3
1 2 ∂φ L 2 ε, |∂φ| ε √ 1 + s(1 + |q|) 1+δ .
Therefore, the energy estimate for h can be written

d dt w 1 2 ∂h 2 L 2 w 1 2 (∂ q φ) 2 L 2 + w 1 2 b(θ)∂ 2 q (qχ(q)) r L 2 w 1 2 ∂h L 2 ,
and thus

d dt w 1 2 ∂h L 2 ε √ 1 + t w 1 2 ∂φ L 2 + ε 2 √ 1 + t ε 2 √ 1 + t .
We infer

w 1 2 ∂h L 2 ≤ ε 2 √ 1 + t. (3.13)
This estimate is not sucient. To obtain more information on h, we will approximate it by the solution h 0 of the following transport equation (this procedure will be made more precise in Section 7)

∂ q h 0 = -2r(∂ q φ) 2 -2b(θ)∂ 2 q (qχ(q)), (3.14)
with initial data h 0 = 0 at t = 0. The L ∞ estimate for φ, and the fact that χ is supported in [START_REF] Alinhac | The null condition for quasilinear wave equations in two space dimensions I[END_REF][START_REF] Alinhac | An example of blowup at innity for a quasilinear wave equation[END_REF] yield

|∂ q h 0 | ε 2 (1 + |q|) 2+2δ .
To estimate h 0 we write

h 0 (Q, s, θ) = Q s -2(∂ q φ) 2 r -2b(θ)∂ 2 q (qχ(q)) dq, so we obtain h 0 (s, Q, θ) = O ε 2 (1 + |Q|) 1+2δ , Q > 0, h 0 (s, Q, θ) = -s s -2(∂ q φ) 2 r -2b(θ)∂ 2 q (qχ(q)) dq + O ε 2 (1 + |Q|) 1+2δ , Q < 0. Therefore, since -s s ∂ 2 q (qχ(q))dq = -1, f or s ≥ 2
to maximize the decay in q for h 0 (and hence for h, provided one has a suitable control over h -h 0 ) we will choose b such that b(θ)

-s s (∂ q φ) 2 rdq. (3.15) Remark 3.4. b(θ) is a free parameter, except from b(θ), b(θ) cos(θ)
and b(θ) sin(θ) which are prescribed by the resolution of the constraint equations, and correspond intuitively to the ADM angle (energy) and linear momentum. Let Π be the projection dened by

Π : W 2,N (S 1 ) → {u ∈ W 2,N (S 1 ), u = cos(θ)u = sin(θ)u = 0}. (3.16) Then Π(b(θ)) Π -s s (∂ q φ) 2 rdq ,
will be forced in the course of the bootstrap procedure. On the other hand, the fact that

b(θ) -s s (∂ q φ) 2 rdqdθ, b(θ) cos(θ) -s s (∂ q φ) 2 cos(θ)rdqdθ, b(θ) sin(θ) -s s (∂ q φ) 2 sin(θ)rdqdθ,
will be obtained by integrating the constraint equations at any time t (see Section 7).

Non commutation of the wave operator with the null frame

In this section, we will discuss the inuence of the terms appearing in (2.7). We have seen in the previous section that h 0 does not decay at all with respect to the s variable. In turn, we will show that this is also the case for h, and nally for the coecient g LL . We do not want this behavior to propagate to the other coecients of the metric. To this end, we will rely on a decomposition of the type

g = g b + Υ r t g LL 4 dq 2 + g i .
However, since the wave operator does not commute with the null decomposition, we have to control the solution g i of an equation of the form

g i = Υ r t ∂h r ,
where h is the solution of (3.12). The term Υ r t ∂h r has the form of the terms appearing in (2.7).

Provided that we can approximate h by the solution h 0 of the transport equation (3.14), we obtain decay with respect to q for h. The decay we will be able to get is

|h| ε 2 1 + |q| .
With this decay we infer

| g i | ε 2 (1 + s) 2 1 + |q| ,
and therefore, with the L ∞ -L ∞ estimate, we deduce

| g i | ε (1 + s) 1 2 -ρ
, for all ρ > 0.

On the other hand, assume we are only allowed to use the energy estimate for h, which is the case when deriving L 2 type estimates for g i at the level of the highest energy. When applying the weighted energy estimate for g i , we obtain

d dt w(q) 1 2 ∂ g i 2 L 2 ≤ w(q) 1 2 Υ r t ∂h r L 2 w(q) 1 2 ∂ g i .
We estimate

w(q) 1 2 Υ r t ∂h r L 2 1 1 + t w(q) 1 2 ∂h L 2 ε 2 √ 1 + t , (3.17)
where we have used the estimate (3.13) of the previous section for h. This yields

d dt w(q) 1 2 ∂ g i L 2 ≤ ε 2 √ 1 + t . So w(q) 1 2 ∂ g i L 2 ≤ ε 2 √ 1 + t,
which is precisely the behaviour we are trying to avoid with this decomposition ! However we have not been able to exploit all the structure in (3.17). In order to do so, we will use dierent weight functions for g i and for h. If we set

w(q) = (1 + |q|) 1+2µ w(q), with 0 < µ ≤ 1
4 and we assume that we have w(q)

1 2 ∂h L 2 ε 2 √ 1 + t, then we can estimate w(q) 1 2 Υ r t ∂h r L 2 1 1 + t w(q) 1 2 Υ r t ∂h (1 + |q|) 1 2 +µ L 2
.

We write

| ∂h| 1 1 + s |Zh| 1 (1 + s) 1 2 +µ (1 + |q|) 1 2 -µ |Zh|, so we obtain w(q) 1 2 Υ r t ∂h r L 2 1 (1 + t) 3 2 +µ w(q) 1 2 Zh 1 + |q| L 2 1 (1 + t) 3 2 +µ w(q) 1 2 ∂Zh L 2 ,
where we used the weighted Hardy inequality. Consequently, the energy inequality for g i yields

d dt w(q) 1 2 ∂ g i L 2 ε 2 (1 + t) 1+µ , and therefore w(q) 1 2 ∂ g i L 2 ε 2 .
Recall that the weighted energy inequality forbids weights of the form (1 + |q|) α with α > 0 in the region q < 0. Therefore we are forced to make the following choice in the region q < 0

w(q) = O(1), w(q) = 1 (1 + |q|) 1+2µ .
Thus, for g i , the √ t loss has been replaced by a loss in (1 + |q|)

1 2 +µ .

The quasilinear structure

In this section we will discuss the challenges of the quasilinear structure. We will take as an example the equation for φ, g φ = 0. Following Remark 2.2, we can focus on the terms of the form g LL ∂ 2 q φ.

The wave coordinates condition yields ∂g LL ∼ ∂g.

If g satised g = 0, the L ∞ estimates for g given by Corollary 1.6 for suitable initial data would imply

|∂g LL | ≤ ε (1 + s) 3 2 1 + |q| ,
We would like to keep this decay in

1 (1+s) 3 2
after integrating with respect to q. However, we are not in the range of application of Lemma 1.11. To this end, we will assume more decay on the initial data. As stated in Theorem 1.12, we take (g,

∂ t g) ∈ H N +1 δ × H N δ+1 with 1 2 < δ < 1.
Then, with the weight w 0 stated in Theorem 1.12, the weighted energy inequality yields w 0 (q)∂Zg L 2 ε, and consequently, for q > 0, the weighted Klainerman-Sobolev inequality yields

|∂Zg| ε √ 1 + s(1 + |q|) 3 2 +δ
.

If we integrate from t = 0, we obtain for q > 0

|Zg| ε √ 1 + s(1 + |q|) 1 2 +δ
.

By writing | ∂g| 1 1+s |Zg|, we obtain |∂g LL | ε (1 + s) 3 2 (1 + |q|) 1 2 , f or q < 0, |∂g LL | ε (1 + s) 3 2 (1 + |q|) 1 2 +δ
, f or q > 0.

Since δ > 1 2 we can apply Lemma 1.11, which yields

|g LL | ε 1 + |q| (1 + s) 3 2
, f or q < 0, |g LL | ε

(1 + s) 3 2 (1 + |q|) δ-1 2
, f or q > 0.

Consequently we easily estimate

w 1 2 g LL ∂ 2 q Z I φ L 2 ε (1 + t) 3 2 w 1 2 ∂ q Z I+1 φ L 2 .
This strong decay in the region q > 0 is also needed when estimating

w 1 2 0 Z I g LL ∂ 2 q φ L 2 .
The idea will be rst to use the weighted Hardy inequality to derive

w 1 2 0 Z I g LL ∂ 2 q φ L 2 ε √ 1 + t w 1 2 0 (1 + |q|) 2 Z I g LL L 2 ε √ 1 + t w 1 2 0 (1 + |q|) ∂Z I g LL L 2
.

Then we rely on the wave coordinates condition, which yields

|∂Z I g LL | | ∂Z I g| 1 1 + s |Z I+1 g|,
and then use the weighted Hardy inequality again. However, one has to be careful when using the weighted Hardy inequality. In the region q > 0 the weight must be suciently large to allow to perform it twice. This is an other reason why we work with initial data in H N δ with δ > 1 2 , which is more than the decay which is necessary to prove the global well posedness of a semi linear wave equation with null structure.

Interaction with the metric g b

In this section we want to discuss the inuence of the crossed terms between g b and φ, g. We will take as an example the equation for φ, g φ = 0. As discussed in Section 2.5, we can focus on the term (2.13). We may look at the following model problem φ = ε r χ(q) ∂φ.

If we perform the weighted energy estimate, we obtain

d dt w 0 (q) 1 2 ∂Z I φ 2 + w 0 (q) 1 2 ∂Z I φ 2 L 2 ε 1 + t w 1 2 0 ∂Z I φ 2 L 2 . Therefore w 0 (q) 1 2 ∂Z I φ L 2 ≤ C 0 ε(1 + t) Cε
and for all σ > 0

T 0 1 (1 + t) σ w 0 (q) 1 2 ∂Z I φ 2 L 2 dt ε 2 . (3.18)
To avoid this logarithmic loss, we need to exploit more the structure of the equation. To this end we introduce the weight modulator

α(q) = 1 (1+|q|) σ , q > 0, α(q) = 1, q < 0, for 0 < σ < 1 2 . Then the energy inequality yields d dt αw 0 (q) 1 2 ∂Z I φ 2 L 2 ≤ ε 1 q>0 αw 1 2 0 1 + s ∂Z I φ L 2 αw 0 (q) 1 2 ∂Z I φ L 2 .
We estimate, for q > 0 α(q) 1 + s

1 (1 + t) 1 2 +σ (1 + |q|) 1 2
.

And therefore, we obtain

d dt αw 0 (q) 1 2 ∂Z I φ 2 L 2 ε (1 + t) 1 2 +σ 1 q>0 w 1 2 0 1 + |q| ∂Z I φ L 2 αw 0 (q) 1 2 ∂Z I φ L 2 ε (1 + t) σ w 0 (q) 1 2 ∂Z I φ L 2 + ε (1 + t) 1+σ αw 0 (q) 1 2 ∂Z I φ L 2 .
and consequently in view of (3.18) we obtain αw 0 (q)

1 2 ∂Z I φ L 2 ≤ C 0 ε + Cε 2 .
With this technique, the logarithmic loss in t has been replaced by a small loss in q.

4 Bootstrap assumptions and proof of Theorem 1.12 4.1 Bootstrap assumptions Let 1 2 < δ < 1. In view of the assumptions of Theorem 1.12, the initial data (φ 0 , φ 1 ) for φ are given in

H N +1 δ (R 2 ) × H N δ+1 (R 2 ). For b ∈ W 2,N such that S 1 bdθ = S 1 b cos(θ)dθ =
(θ) = b(θ) + b 0 + b 1 cos(θ) + b 2 sin(θ) (4.1) with b 0 , b 2 , b 2 , J(θ)
given by Theorem 1.3, the generalized wave coordinate condition given by H b is satised at t = 0.

The system 2.1 being a standard quasilinear system of wave equations, we know that there exists a solution until a time T . Moreover with our conditions on the initial data, our solution (g, φ) is solution of the Einstein equations (1.1), and the wave coordinate condition is satised for t ≤ T (see Appendix C). Remark 4.1. Our choice of generalized wave coordinates does not change the hyperbolic structure because H b does not contain derivatives of g.

We take three parameters ρ, σ, µ such that

ε ρ σ µ δ, (4.2)
σ + ρ < δ - 1 2 . (4.3)
We consider a time T > 0 such that there exists b(θ) ∈ W N,2 (S 1 ) and a solution (φ, g) of (2.4) on [0, T ], associated to initial data for g. We assume that on [0, T ], the following estimates hold.

Bootstrap assumptions for b

∂ I θ Πb(θ) + Π Σ T,θ (∂ q φ) 2 rdq L 2 (S 1 ) ≤ B ε 2 √ T , f or I ≤ N -4 (4.4) ∂ I θ b(θ) L 2 (S 1 ) ≤ Bε 2 , f or I ≤ N (4.5)
where Π is the projection dened by (3.16), Σ T,θ is dened by (1.15) and B is a constant depending on ρ, σ, µ, δ, N .

We introduce four decomposition of the metric g

g =g b + Υ r t h 0 dq 2 + g 1 , (4.6)
g =g b + Υ r t (h 0 + h)dq 2 + g 2 , (4.7) g =g b + Υ r t hdq 2 + g 3 , (4.8)
g =g b + Υ r t hdq 2 + Υ r t krdqdθ + g 4 , (4.9)
where h 0 is the solution of the transport equation

∂ q h 0 = -2r(∂ q φ) 2 -2b(θ)∂ 2 q (χ(q)q), h 0 | t=0 = 0, (4.10)
h is solution of the linear wave equation

h = Υ r t h 0 + Υ r t g LL ∂ 2 q h 0 + 2Υ r t (∂ q φ) 2 -2(R b ) qq + Υ r t Q LL (h 0 , g), ( h, ∂ t h)| t=0 = (0, 0), (4.11)
where

Q LL (h 0 , g) = ∂ L g LL ∂ L h 0 + ∂ L (g b ) U U ∂ L g LL . (4.12) g h = -2(∂ q φ) 2 + 2(R b ) qq + Q LL (h, g), (k, ∂ t k)| t=0 = (0, 0), (4.13) and k is the solution of g k = ∂ U g LL ∂ q h, (h, ∂ t h)| t=0 = (0, 0). (4.14)
L ∞ -based bootstrap assumptions For I ≤ N -14 we assume

|Z I φ| ≤ 2C 0 ε √ 1 + s(1 + |q|) 1 2 -4ρ , (4.15
)

|Z I g 1 | ≤ 2C 0 ε (1 + s) 1 2 -ρ , (4.16)
where here and in the following, C 0 is a constant depending on ρ, σ, µ, δ, N such that the inequalities are satised at t = 0 with 2C 0 replaced by C 0 . For I ≤ N -12 we assume

|Z I φ| ≤ 2C 0 ε (1 + s) 1 2 -2ρ , (4.17)
|Z I g 1 | ≤ 2C 0 ε (1 + s) 1 2 -2ρ . (4.18)
We assume the following estimate for h 0 for I ≤ N -7 and q < 0

|Z I h 0 | ≤ 2C 0 ε (1 + s) 1 2 + 2C 0 ε (1 + |q|) 1-4ρ . (4.19) and for q > 0 |Z I h 0 | ≤ 2C 0 ε (1 + |q|) 2+2(δ-σ) . (4.20)
We also assume the following for h and I ≤ N -7

|Z I h| ≤ 2C 0 ε (1 + |q|) 1 2 -ρ . (4.21)
L 2 -based bootstrap assumptions We introduce four weight functions

w 0 (q) = (1 + |q|) 2+2δ , q > 0, w 0 (q) = 1 + 1 (1+|q|) 2µ , q < 0, w 1 (q) = (1 + |q|) 2+2δ , q > 0, w 1 (q) = 1 (1+|q|) 1 2
, q < 0,

w 2 (q) = (1 + |q|) 2+2δ , q > 0, w 2 (q) = 1 (1+|q|) 1+2µ , q < 0, w 3 (q) = (1 + |q|) 3+2δ , q > 0, w 3 (q) = 1 + 1 (1+|q|) 2µ , q < 0.
We also introduce weight modulators

α(q) = 1 (1+|q|) σ , q > 0, α(q) = 1, q < 0, (4.22) α 2 (q) = 1 (1+|q|) 2σ , q > 0, α 2 (q) = 1, q < 0.
All the L 2 norms are taken with respect to the Lebesgue measure. We assume the following estimate for I ≤ N α 2 w 0 (q)

1 2 ∂Z I φ L 2 + α 2 w 2 (q) 1 2 ∂Z I g 4 L 2 + 1 √ 1 + t α 2 (q) 1 2 ∂Z I h L 2 + 1 √ 1 + t α 2 w 3 (q) 1 2 ∂Z I k L 2 ≤ 2C 0 ε(1 + t) ρ . (4.23) for I ≤ N -1 w 0 (q) 1 2 ∂Z I φ L 2 + w 2 (q) 1 2 ∂Z I g 3 L 2 + 1 √ 1 + t w 3 (q) 1 2 ∂Z I h L 2 ≤ 2C 0 ε(1 + t) ρ (4.24)
and for I ≤ N -2 α(q)w 0 (q)

1 2 ∂Z I φ L 2 + α(q)w 2 (q) 1 2 ∂Z I g 3 L 2 + 1 √ 1 + t α(q)w 3 (q) 1 2 ∂Z I h L 2 ≤ 2C 0 ε. (4.25)
In addition, for I ≤ N -8 we assume

w 1 (q) 1 2 ∂Z I g 2 L 2 ≤ 2C 0 ε(1 + t) ρ , α(q)w 1 (q) 1 2 ∂Z I g 2 L 2 ≤ 2C 0 ε (4.26)
and for I ≤ N -9 we assume

w 0 (q) 1 2 ∂Z I g 2 L 2 ≤ 2C 0 ε(1 + t) ρ , α(q)w 0 (q) 1 2 ∂Z I g 2 L 2 ≤ 2C 0 ε. (4.27)
Let us do two remarks to justify our dierent decompositions of the metric, and our dierent weight functions.

Remark 4.2. We use the decomposition (4.6) instead of (4.7) to avoid a logarithmic loss when we want to improve (4.16) and (4.18) with the L ∞ -L ∞ estimate. This loss would have been due to the terms coming from the non commutation of the wave operator with the null decomposition (4.7). However, we use the decomposition (4.7) instead of (4.6) to avoid a logarithmic loss in the energy estimate due to the term Q LL . When h 0 is a good approximation for h, we use the decomposition (4.7) instead of (4.8) in the energy estimate. This allows us to have a better control on the terms coming from the non commutation of the wave operator with the null decomposition. When h 0 is no longer a good approximation for h, we use the decomposition (4.8). Finally, the decomposition (4.9) allows us to isolate the term Z N ∂ U g LL ∂ L g LL on which we do not have a good control.

Remark 4.3. The weight w 2 is introduced to deal with the non commutation of the wave operator with the null decomposition (see Section 3.3). The weight w 1 is a transition weight between w 0 and w 2 . The weight w 3 allows us to compensate the loss in √ 1 + t for g LL by an additional decay in 1 + |q| in the exterior region. The weight modulators α 1 and α 2 are introduced to transform the logarithmic loss due to the interaction with the metric g b in a small loss in q (see Section 3.5).

Proof of Theorem 1.12

We have the following improvement for the bootstrap assumptions. The constant C will denote a constant depending only on ρ, σ, µ, δ, N . The proof of Proposition 4.4 is the object of Section 7.

Proposition 4.4. Let I ≤ N -5. We have the estimates

|Z I h 0 | ≤ Cε 2 √ 1 + s + Cε 2 (1 + |q|) 1-4ρ , f or q < 0, |Z I h 0 | ≤ Cε 2 (1 + |q|) 2+2(δ-σ) , f or q > 0.
Let I ≤ N -7. We have the estimate

|Z I h| ≤ Cε 2 (1 + s) 1 2 -ρ
.

The proof of Proposition 4.5 is the object of Section 8.

Proposition 4.5. Let I ≤ N -14. We have the estimates

|Z I g 1 | ≤ C 0 ε + Cε 2 (1 + s) 1 2 -ρ , |Z I φ| ≤ C 0 ε + Cε 2 √ 1 + s(1 + |q|) 1 2 -4ρ
.

Let I ≤ N -12. We have the estimates

|Z I φ| ≤ C 0 ε + Cε 2 (1 + s) 1 2 -2ρ , |Z I g 1 | C 0 ε + Cε 2 (1 + s) 1 2 -2ρ
.

The proof of Proposition 4.6 is the object of Section 10.

Proposition 4.6. We have the estimates for

I ≤ N α 2 w 0 (q) 1 2 ∂Z I φ L 2 + α 2 w 2 (q) 1 2 ∂Z I g 4 L 2 ≤ (C 0 ε + ε)(1 + t) C √ ε , α 2 (q) 1 2 ∂Z I h L 2 + α 2 w 3 (q) 1 2 ∂Z I k L 2 ≤ Cε 2 (1 + t) 1 2 +C √ ε , for I ≤ N -1 w 0 (q) 1 2 ∂Z I φ L 2 + w 2 (q) 1 2 ∂Z I g 3 L 2 ≤ (C 0 ε + ε)(1 + t) C √ ε , w 3 (q) 1 2 ∂Z I h L 2 ≤ Cε 2 (1 + t) 1 2 +C √ ε , for I ≤ N -2 α(q)w 0 (q) 1 2 ∂Z I φ L 2 + α(q)w 2 (q) 1 2 ∂Z I g 3 L 2 ≤ C 0 ε + Cε 5 4 , α(q)w 3 (q) 1 2 ∂Z I h L 2 ≤ Cε 3 2 ,
for I ≤ N -7

w 1 (q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε(1 + t) C √ ε + ε, α(q)w 1 (q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε + Cε 5 4 ,
and for I ≤ N -8

w 0 (q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε(1 + t) C √ ε + ε, α(q)w 0 (q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε + Cε 5 4 .
The proof of Proposition 4.7 is the object of Section 11 Proposition 4.7. We assume that the time T satises

T ≤ exp C √ ε .
There exists b (2) (θ) ∈ W N,2 (S 1 ) and (φ(2), g (2) ) solution of (1.1) in the generalized wave coordinates H b (2) , such that, if we write g (2) = g b 2 + g, then (φ(2), g (2) ) satises the same estimate as (φ, g), and we have the estimates for b (2)

∂ I θ Πb (2) (θ) + Π Σ T,θ (∂ q φ) 2 rdq L 2 ≤ C ε 4 √ T , f or I ≤ N -4, ∂ I θ b(θ) L 2 ≤ 2C 2 0 ε 2 , f or I ≤ N.
We may now prove Theorem 1.12.

Proof of Theorem 1.12. We may choose C 0 such that C 0 ≥ 2, and B such that B ≥ 4C 2 0 . We take ε small enough so that

Cε 1 4 ≤ C 0 2 , C √ ε ≤ ρ, Cε ≤ B 2 .
Then Propositions 4.4, 4.5, 4.6 imply that the bootstrap assumptions for (φ, g) are true with the constant 2C 0 replaced by 3C 0 2 . Moreover Proposition 4.7 yields the existence of b (2) and φ (2) , g (2) = g b (2) + g (2) solution of (1.1), such that the bootstrap assumptions are satised by (φ (2) , g (2) ) with the constant 2C 0 replaced by 3C 0 2 , and b (2) satisfy

∂ I θ Πb (2) (θ) + Π Σ T,θ (∂ q φ (2) ) 2 rdq L 2 ≤ B ε 2 2 √ T , f or I ≤ N -4, ∂ I θ b(θ) L 2 ≤ B 2 ε 2 , f or I ≤ N.
This concludes the proof of Theorem 1.12.

Let us note that the only place where we use the assumption T ≤ exp C √ ε is in the proof of Proposition 4.7.

First consequences of the bootstrap assumptions

Thanks to the weighted Klainerman-Sobolev inequality the bootstrap assumptions immediately imply the following proposition.

Proposition 4.8. We assume I ≤ N -4 we have the estimates, for q < 0

|∂Z I φ(t, x)| ε 1 + |q| √ 1 + s , (4.28) |∂Z I g 3 (t, x)| ε(1 + |q|) µ √ 1 + s , (4.29) |∂Z I h| ε 1 + |q| , (4.30)
and for q > 0

|∂Z I φ(t, x)| ε (1 + |q|) 3 2 +δ-σ √ 1 + s , (4.31) |∂Z I g 3 (t, x)| ε (1 + |q|) 3 2 +δ-σ √ 1 + s , (4.32) |∂Z I h| ε (1 + |q|) 2+δ-σ . (4.33)
Moreover, for I ≤ N -11 we have for q < 0 Corollary 4.9. We assume I ≤ N -4 we have the estimates, for q < 0

|∂Z I g 2 (t, x)| ε 1 + |q| √ 1 + s .
|Z I φ(t, x)| ε 1 + |q| √ 1 + s , (4.35) |Z I g 3 (t, x)| ε(1 + |q|) 1+µ √ 1 + s , (4.36)
|Z I h| ε 1 + |q|. and for q > 0

|Z I φ(t, x)| ε (1 + |q|) 1 2 +δ-σ √ 1 + s , (4.38) |Z I g 3 (t, x)| ε (1 + |q|) 1 2 +δ-σ √ 1 + s , (4.39) |Z I h| ε (1 + |q|) 1+δ-σ . (4.40)
Moreover, for I ≤ N -11 we have for q < 0

|Z I g 2 (t, x)| ≤ ε 1 + |q| √ 1 + s . (4.41)
The following remark allow us to compare the dierent decompositions of the metric g.

Remark 4.10. We have the following relations

g T T = ( g 1 ) T T = ( g 2 ) T T = ( g 3 ) T T = ( g 4 ) T T , g LL = ( g 1 ) LL = ( g 2 ) LL = ( g 3 ) LL = ( g 4 ) LL , g U L = ( g 1 ) U L = ( g 2 ) U L = ( g 3 ) U L .
The following corollary allow us to estimate g, independently of the chosen decomposition (4.6), (4.7), (4.8) or (4.9).

Corollary 4.11. We have the following estimates

|Z I g| ε (1 + |q|) 1 2 -ρ , f or I ≤ N -14, (4.42)
|Z I g| ε

(1 + |q|) 1 2 -2ρ
, f or I ≤ N -12, Moreover, for q > 0 we have the following estimate The rest of the paper is as followed In Section 5, we use the wave coordinates condition to obtain better decay on the coecients g T T of the metric. The strategy is similar to the one introduced in [START_REF] Lindblad | The global stability of Minkowski space-time in harmonic gauge[END_REF].

|Z I g| ε(1 + |q|) 1 2 +µ , |∂Z I g| ε(1 + |q|) -1 2 +µ , f or I ≤ N -4.
|Z I g| ε (1 + |q|) 1+δ-σ , f or I ≤ N -4.
In Section 6, we obtain the missing estimates for the angle and linear momentum, namely the three rst Fourier coecient of b which correspond to b -Πb, in order to get b(θ)

+ Σ T,θ (∂ q φ(q, s = T, θ)) 2 rdq ε 2 T 1 2
, by relying in particular on the constraint equations.

In Section 7, we improve the estimates for h 0 , and show that it is indeed a good approximation for the coecient g LL . We also obtain estimates for h. We prove Proposition 4.4.

In Section 8 we prove Proposition 4.5 thanks to the L ∞ -L ∞ estimate.

In Section 9 we derive a weighted energy estimate for an equation of the form g u = f , where g satises the bootstrap assumptions.

In Section 10, we prove Proposition 4.6 thanks to the weighted energy estimate.

In Section 11, we prove Proposition 4.7 by picking the right b = Πb.

The wave coordinates condition

The wave coordinates condition yields better decay properties in t for some components of the metric. Since far from a conical neighborhoud of the light cone, we have |q| ∼ s, this condition will only be relevant near the light cone. It is given by

H α b = - 1 |det(g)| ∂ µ (g µα | det(g)|).
Proposition 5.1. We have the following estimate, in the region t 2 ≤ r ≤ 2t,

|∂ q Z I g LL | J≤I | ∂Z J g LL | + | ∂Z J g T T | + 1 1 + s J≤I |Z I g LL | + |Z I g T T | .
Proof. The wave coordinate condition implies

-L α H α b = L α 1 | det(g)| ∂ µ (g µα det(g)) = g µα det(|g|) L α ∂ µ det(g) + ∂ µ (L α g µα ) -g µα ∂ µ (L α ) = g Lµ det(|g|) ∂ µ det(g) + ∂ µ (g Lµ ) - 1 r g U U = g LL det(|g|) ∂ L det(g) + g LT det(|g|) ∂ T det(g) + ∂ L g LL + ∂ U g LU + ∂ L g LL + 1 r g LR - 1 r g U U ,
where we have denoted by R the vector eld ∂ r , and used the following calculations

g µα ∂ µ (L α ) = -g µα ∂ µ (R α ) = -g 11 ∂ 1 cos(θ) -g 12 (∂ 2 cos(θ) -∂ 1 sin(θ)) -g 22 ∂ 2 sin(θ) = - g U U r , ∂ µ g Lµ =∂ 0 g L0 + ∂ 1 g L1 + ∂ 2 g L2 =∂ 0 g L0 + ∂ R g LR + ∂ U g LU + g LR (∂ 1 cos(θ) + ∂ 2 sin(θ)) + g LU (-∂ 1 sin(θ) + ∂ 2 cos(θ)) =∂ L g LL + ∂ U g LU + ∂ L g LL + g LR r .
Consequently

∂ L g LL = -L α Hα b + F α - g LL det(|g|) ∂ L det(g) - g LT det(|g|) ∂ T det(g) -∂ U g LU -∂ L g LL - 1 r g LR - 1 r g U U , (5.1)
where we have used (1.9). Also we have

det(g) = g LL (g LL g U U -(g U L ) 2 ) -g LL (g LL g U U -g LU g LU ) + g LU (g LL g U L -g LL g LU ). Therefore | det(g) -det(g b )| | g LL | + | g T T |.
We can express

g LL = 1 det(g) (g LL g U U -(g U L ) 2 ) = - 1 4 g LL + O( g T T )O(g), g LU = 1 det(g) (g LL g LU -g U L g LL ) = 1 2 g LU + O( g T T )O(g), g LL = 1 det(g) (g LL g U U -g U L g U L ) = 1 4 (g b ) U U g LL + O( g T T ),
where we have used the notation O(g) = O(g -m) where m is the Minkowski metric. Since in (5.1), by denition of Hα (see (1.10)) the terms involving only g b compensate, we have

|∂ q g LL | (| ∂ g LL | + | ∂ g T T |) + 1 1 + s (| g LL | + | g T T |) + s.t..
where s.t denotes similar terms (here these terms are quadratic terms with a better or similar decay), and we have used the fact that in the region

t 2 ≤ r ≤ 2t, we have r ∼ s. Since [Z, ∂ q ] ∼ ∂ q and [Z, ∂] ∼ ∂ we have |∂ q Z I g LL | J≤I-1 |Z J g LL | + | ∂Z I g LL | + | ∂Z I g T T | + 1 1 + s J≤I (|Z J g LL | + |Z J g T T |).
This concludes the proof of Proposition 5.1.

The other two contractions of the wave condition yield better decay on a conical neighbourhood of the light cone for g U L and g U U .

Proposition 5.2. We have the following property

|∂ q Z I g U L | J≤I |∂Z J g T V | + 1 1 + s J≤I |Z J g T V |, |∂ q Z I g U U | J≤I |∂Z J g| + 1 1 + s J≤I |Z J g|.
Proof. To obtain the rst estimate, we contract the wave coordinate condition with the vector eld

U . -U α H α b = 1 | det(g)| U α ∂ µ (g µα ) det(g) = g µα | det(g)| U α ∂ µ | det(g)| + ∂ µ (U α g µα ) + g µα ∂ µ (U α ) = g U µ | det(g)| ∂ µ | det(g)| + ∂ µ (g U µ ) + 1 r g U R = g U L | det(g)| ∂ L | det(g)| + g U T | det(g)| ∂ T | det(g)| + ∂ L g U L + ∂ U g U U + ∂ L g U L + 1 r g U R . Therefore ∂ L g U L = -U α H α b - g U L | det(g)| ∂ L | det(g)| - g U T | det(g)| ∂ T | det(g)| -∂ U g U U -∂ L g U L - 1 r g U R ,
and arguing as in Proposition 5.1 we infer

|∂ q g U L | | ∂ g T V | + 1 1 + s | g T V | + s.t.
Commuting with the vector elds Z as before, we obtain the desired estimate. To obtain the second one, we contract the wave coordinate condition with L

L α H α b = 1 | det g| L α ∂ µ (g µα ) | det(g)|. = 1 | det g| ∂ L | det(g)|g LL + 1 | det g| ∂ T | det(g)|g LT -g µα ∂ µ (L α ).
(5.2)

We note that

| det(g)|g LL = 1 | det(g)| (g LL g U U -g U L g U L ) = g LL g U U g 2 LL g U U + O( g T T )O(g) + O( g T T )O(g) = √ g U U + +O( g T T )O(g). Therefore (5.2) yields |∂ q g U U | | ∂ g| + 1 1 + s | g|.
We commute with the vector elds Z to conclude.

Thanks to the bootstrap assumptions, we obtain the following corollary.

Corollary 5.3. We have the estimates for q < 0

|∂Z I g U U | ε (1 + s) 3 2 -ρ , |∂Z I g LT | ε (1 + s)(1 + |q|) 1 2 -ρ , f or I ≤ N -15, (5.3)
|∂ Z I g LT | ε (1 + s) 3 2 -2ρ , |∂Z I g U U | ε (1 + s)(1 + |q|) 1 2 -2ρ
, f or I ≤ N -13,

(5.4)

|∂Z I g LT | ε 1 + |q| (1 + s) 3 2 , |∂Z I g U U | ε 1 + s , f or I ≤ N -12, (5.5
)

|∂Z I g LT | ε(1 + |q|) 1+µ (1 + s) 3 2 , |∂Z I g U U | ε(1 + |q|) 1 2 +µ 1 + s , f or I ≤ N -5, (5.6)
and for q > 0

|∂Z I g LT | ε (1 + |q|) 1 2 +δ-σ (1 + s) 3 2 , |∂Z I g U U | ε (1 + |q|) 1+δ-σ (1 + s) , f or I ≤ N -5.
Proof. As mentioned in Remark 4.10, the metric coecients g VT do not depend on the choice of decomposition between (4.6), (4.7) and (4.8). Thanks to Proposition 5.1 and 5.2, and the fact that

| ∂u| ≤ 1 1 + s |Zu|,
we may write

|∂Z I g LT | 1 1 + s |Z I+1 g T V |.
(5.7)

The bootstrap assumptions (4.16) and (4.18) in the region q < 0 yield

|Z J g T V | ε (1 + s) 1 2 -ρ , f or J ≤ N -14, |Z J g T V | ε (1 + s) 1 2 -2ρ
, f or J ≤ N -12.

Therefore we obtain, in view of (5.7)

|∂Z I g LT | ε (1 + s) 3 2 -ρ , f or I ≤ N -15, |∂Z I g LT | ε (1 + s) 3 2 -2ρ
f or I ≤ N -13.

Corollary 4.9 yields the following estimate for q < 0

|Z J g T V | ε 1 + |q| √ 1 + s , f or J ≤ N -11, |Z J g T V | ε(1 + |q|) 1+µ √ 1 + s , f or J ≤ N -4.
Therefore we obtain in view of (5.7)

|∂Z I g LT | ε 1 + |q| (1 + s) 3 2 f or I ≤ N -12, |∂Z I g LT | ε(1 + |q|) 1+µ (1 + s) 3 2
f or I ≤ N -5.

For q > 0 and I ≤ N -4, we have in view of Corollary 4.9

|Z I g T V | ε √ 1 + s(1 + |q|) 1 2 +δ-σ
which together with (5.7) yields

|∂ q Z I g LT | ε (1 + |q|) 1 2 +δ-σ (1 + s) 3 2
f or I ≤ N -5.

We now estimate Z I g U U . As for Z I g LT , Proposition 5.2 yields

|∂Z I g U U | 1 1 + s |Z I+1 g|.
Therefore, the estimates of Corollary 5.3 are a direct consequence of the estimates of Corollary 4.11.

Thanks to Lemma 1.11, since δ -σ > 1 2 we obtain the following corollary Corollary 5.4. We have the estimates for q < 0

|Z I g LT | ε(1 + |q|) (1 + s) 3 2 -ρ , |Z I g U U | ε(1 + |q|) 1 2 +ρ 1 + s , f or I ≤ N -15, (5.8
)

|Z I g LT | ε(1 + |q|) (1 + s) 3 2 -2ρ |Z I g U U | ε(1 + |q|) 1 2 +2ρ
1 + s , f or I ≤ N -13,

(5.9)

|Z I g LT | ε(1 + |q|) 3 2
(1 + s)

3 2 |Z I g U U | ε(1 + |q|) 1 + s , f or I ≤ N -12,
(5.10)

|Z I g LT | ε(1 + |q|) 2+µ (1 + s) 3 2 |Z I g U U | ε(1 + |q|) 3 2 +µ 1 + s , f or I ≤ N -5, (5.11)
and for q > 0

|Z I g LT | ε(1 + |q|) 1 2 +σ-δ (1 + s) 3 2 , |Z I g U U | ε (1 + s)(1 + |q|) δ-σ , f or I ≤ N -5.
(5.12)

Angle and linear momentum

We call angle and linear momentum the three rst coecients of b, b 0 , b 1 , b 2 . These coecients can not be prescribed arbitrarily, they are given by the resolution of the constraint equations (see Theorem 1.3). We need b to satisfy Proposition 6.1. We have

∂ I θ b(θ) + Σ T,θ (∂ q φ) 2 rdq L 2 ε 2 √ T , f or I ≤ N -4. ( 6
b(θ)dθ + 1 2 R 2 (∂ t φ) 2 + |∇φ| 2 (t, x)dx ε 2 √ 1 + t , b(θ) cos(θ)dθ - R 2 (∂ t φ∂ 1 φ) (t, x)dx ε 2 √ 1 + t , b(θ) sin(θ)dθ - R 2 (∂ t φ∂ 2 φ) (t, x)dx ε 2 √ 1 + t .
To prove this proposition, we need the following lemma.

Lemma 6.2. The equation for g µν can be written under the form

g µν = -2∂ µ φ∂ ν φ -2b(θ) ∂ 2 q (χ(q)q) r M µν + O ε 2 (1 + t) 3 2 (1 + |q|) 3 2 -2ρ , (6.2)
where the tensor M µν corresponds to dq 2 .

Proof of Lemma 6.2. We recall the quasilinear equation for g µν (see (2.4))

g αβ ∂ α ∂ β g µν -H ρ b ∂ ρ g µν = -2∂ µ φ∂ ν φ + 2(R b ) µν + P µν (∂ g, ∂ g) + P µν ( g, g b ).
The worst term in

g αβ ∂ α ∂ β g µν -g µν is, according to Remark 2.2, g LL ∂ 2 q g µν .
We distinguish two kinds of contributions :

g LL ∂ 2 q g 1 and g LL ∂ 2 q h 0 .

To estimate the rst term, we use (5.8) of Corollary 5.4, which gives

|g LL | ε(1 + |q|) (1 + s) 3 2 -ρ
.

We estimate then

|∂ 2 q g 1 | ≤ 1 (1 + |q|) 2 I≤2 |Z I g 1 |,
and we use the bootstrap assumption (4.16) for I ≤ N -14

|Z I g 1 | ≤ ε (1 + s) 1 2 -ρ , to obtain |g LL ∂ 2 q g 1 | ε 2 (1 + s) 2-2ρ (1 + |q|) . (6.3)
We now estimate the second term. To estimate ∂ 2 q h 0 , we recall (4.19) for I ≤ N -6

|Z I h 0 | ε √ 1 + s + ε (1 + |q|) 1-4ρ . Consequently |∂ 2 q h 0 | ε (1 + |q|) 2 √ 1 + s + ε (1 + |q|) 3-4ρ .
The rst contribution can be estimated like 6.3. To tackle the second contribution we need to use the estimate for g LL which gives the most decay in s : we use (5.10) of Corollary 5.4, which yields

|g LL | ε(1 + |q|) 3 2 (1 + s) 3 2
.

This, together with the estimate (6.3), yields

|g LL ∂ 2 q h 0 | ε 2 (1 + s) 3 2 (1 + |q|) 3 2 -4ρ + ε 2 (1 + s) 2-2ρ (1 + |q|) . (6.4)
The semi linear terms P µν (∂ g, ∂ g) are estimated similarly. We now turn to the crossed terms.

Thanks to Section 2.5, the worst contribution is (2.11), which gives a contribution of the form ε r ∂ g LL in the region q > 0. We estimate thanks to (4.32) of Corollary 4.9 in the region q > 0

|∂ g LL | ε (1 + s) 1 2 (1 + |q|) 3 2 +δ-σ
.

Therefore we obtain

1 q>0 ε r ∂ g LL ε (1 + s) 2 (1 + |q|) 3 2 +δ-σ . ( 6.5)
We now estimate (R b ) µν . Thanks to (1.7) and (1.8), we may write

(R b ) µν = - b(θ)∂ 2 q (qχ(q)) r M µν + O 1 1≤q≤2 ε 2 (1 + r) 2 . (6.6)
Thanks to (6.3), (6.4), (6.5) and (6.6) we conclude the proof of Lemma 6.2.

Proof of Proposition 6.1. We want to integrate equation (6.2) for (µ, ν) = 0, 0 over the space-like hypersurfaces of t constant. To deal with the term ∂ 2 t g 00 , we use the wave coordinate condition

g αβ ∂ β g α0 = 1 2 g αβ ∂ t g αβ + (H b ) 0 .
We can rewrite it, by denition of (H b ) 0

(g αβ -(g b ) αβ )∂ β g α0 + g αβ b (∂ β g α0 -∂ β (g b ) α0 ) = 1 2 (g αβ -g αβ b )∂ t g αβ + 1 2 g αβ b (∂ t g αβ -∂ t (g b ) αβ ) + F 0 .
By denition, F contains only terms of the form g∂ U g b , so we can estimate

|ZF | ε1 q>0 (1 + |q|) r 2 |Z g| ε 2 (1 + s) 2 (1 + |q|) δ-σ , (6.7)
where we have used (4.47) to estimate |Z g|. We note

m αβ ∂ β g α0 - 1 2 m αβ ∂ t g αβ = 1 2 (-∂ t g 00 -∂ t g 11 -∂ t g 22 ) + ∂ 1 g 01 + ∂ 2 g 02 ,
and we estimate

(g αβ -(g b ) αβ )∂ β g α0 = (g LL -m LL )∂ L g L0 + f 1 , 1 2 (g αβ -g αβ b )∂ t g αβ = (g LL -m LL )∂ t g LL + f 2 , (m αβ -g αβ b )∂ β g α0 = f 4 (m αβ -g αβ b )∂ t g αβ = f 5 ,
where the f i contain terms of the form

g LL ∂ g VV , g VV ∂ T g T V , bχ(q) r ∂ U g U V , ...
They satisfy the following estimate

|Zf i | ε 2 (1 + s) 3 2 (1 + |q|) 1 2 -2ρ . (6.8) We note 2∂ t g LL = ∂ L g LL + ∂ L g LL and 2g L0 = g LL + g LL . Consequently (g LL -m LL )(∂ L g L0 -∂ t g LL ) = O g LL ∂ L g LL + g LL ∂ L g LL
satises the same estimate (6.8) than the f i . Therefore the wave coordinate condition gives

1 2 (-∂ t g 00 -∂ t g 11 -∂ t g 22 ) + ∂ 1 g 01 + ∂ 2 g 02 = f 5
where f 5 satises (6.8). Therefore, dierentiating this equation with respect to t, and using (6.2) for (µ, ν) = (0, 0), (1, 1), (2, 2) we obtain

∆ g 00 + ∆ g 11 + ∆ g 22 -2∂ 1 ∂ t g 01 -2∂ 2 ∂ t g 02 = -2((∂ 0 φ) 2 + (∂ 1 φ) 2 + (∂ 2 φ) 2 ) -4b(θ) ∂ 2 q (χ(q)q) r + O ε 2 (1 + s) 3 2 (1 + |q|) 3 2 -2ρ
.

Integrating on the space-like hypersurface t constant we obtain, since

∞ 0 ∂ 2 (qχ(q))dr = 1, - 1 2 (∂ t φ) 2 + |∇φ| 2 dx = b(θ)dθ + O ε 2 √ 1 + t . (6.9)
To obtain the next relation we do the same reasoning but with (6.2) for (µ, ν) = (0, 1) and (µ, ν) = (0, 2). We only detail the case (µ, ν) = (0, 1) as the other one is treated in the same way. Recall the wave coordinates condition

g αβ ∂ β g α1 = 1 2 g αβ ∂ 1 g αβ + (H b ) 1 .
We can rewrite it, by denition of (H b ) 1

(g αβ -(g b ) αβ )∂ β g α1 + g αβ b (∂ β g α1 -∂ β (g b ) α1 ) = 1 2 (g αβ -g αβ b )∂ 1 g αβ + 1 2 g αβ b (∂ 1 g αβ -∂ 1 (g b ) αβ ) + F 1
We note

m αβ ∂ β g α1 - 1 2 m αβ ∂ 1 g αβ = -∂ t g 01 + ∂ 1 g 11 + ∂ 2 g 12 - 1 2 m αβ ∂ 1 g αβ and we estimate (g αβ -(g b ) αβ )∂ β g α1 = (g LL -m LL )∂ L g L1 + f 6 , 1 2 (g αβ -g αβ b )∂ 1 g αβ = (g LL -m LL )∂ 1 g LL + f 7 , (m αβ -g αβ b )∂ β g α1 = f 8 , (m αβ -g αβ b )∂ 1 g αβ = f 9 ,
where the quantities f i satisfy (6.8). We note 2∂ 1 g LL = -cos(θ)∂ L g LL + ∂ g LL and 2∂ L g L1 = -∂ L (cos(θ)g LL ) + g LT . Therefore we obtain

-∂ t g 01 + ∂ 1 g 11 + ∂ 2 g 12 - 1 2 m αβ ∂ 1 g αβ = f 10 ,
where f 10 satises (6.8). Dierentiating with respect to t and using (6.2) for (µ, ν) = (0, 1) we obtain

∆ g 01 + ∂ 1 ∂ t g 11 + ∂ 2 ∂ t g 12 - 1 2 m αβ ∂ 1 ∂ t g αβ = -2∂ t φ∂ 1 φ + 2b(θ) cos(θ) ∂ 2 q (χ(q)q) r + O ε 2 (1 + s) 3 2 (1 + |q|) 3 2ρ
.

Integrating on the space-like hypersurface t constant we obtain

∂ t φ∂ 1 φdx = b(θ) cos(θ)dθ + O ε 2 √ 1 + t , (6.10)
and similarly

∂ t φ∂ 2 φdx = b(θ) sin(θ)dθ + O ε 2 √ 1 + t . (6.11)
Estimates (6.9), (6.10) and (6.11) conclude the proof of Proposition 6.1 Corollary 6.3. We have the estimates

b(θ)dθ + Σ T (∂ q φ) 2 rdrdθ ε 2 √ T , b(θ) cos(θ)dθ + Σ T cos(θ)(∂ q φ) 2 rdrdθ ε 2 √ T , b(θ) sin(θ)dθ + Σ T sin(θ)(∂ q φ) 2 rdrdθ ε 2 √ T .
Proof. We may write

∂ t φ = -∂ q φ + ∂ s φ, ∂ 1 φ = cos(θ)∂ q φ + cos(θ)∂ s φ -sin(θ)∂ U φ, ∂ 1 φ = sin(θ)∂ q φ + sin(θ)∂ s φ + cos(θ)∂ U φ.
Moreover, thanks to the bootstrap assumption (4.15)

|∂φ ∂φ| 1 (1 + |q|)(1 + s) |Zφ| 2 ε 2 (1 + s) 2 (1 + |q|) 2-8ρ , and consequently ∂φ ∂φ (t, x)dx ε 2 1 + t . Therefore 2 Σ T (∂ q φ) 2 dx - Σ T ((∂ t φ) 2 + |∇φ| 2 )dx ε 2 1 + T , Σ T cos(θ)(∂ q φ) 2 dx + Σ T ∂ t φ∂ 1 φdx ε 2 1 + T , Σ T sin(θ)(∂ q φ) 2 dx + Σ T ∂ t φ∂ 2 φdx ε 2 1 + T .
This concludes the proof of Corollary 6.3.

Corollary 6.3 and the bootstrap assumption 4.4 directly imply the following corollary.

Corollary 6.4. We have, for I ≤ N -4

∂ I θ b(θ) + Σ T,θ (∂ q φ) 2 rdq ε 2 √ T .
7 The transport equation (4.10) In this section we will estimate h 0 , h 0 and h.

Estimations on h 0

We recall the equation (4.10)

∂ q h 0 = -2r(∂ q φ) 2 -2b(θ)∂ 2 q (χ(q)q), h 0 | t=0 = 0.
The solution of this equation is

h 0 (s, Q, θ) = Q s -2(∂ q φ) 2 -2 b(θ)∂ 2 q (qχ(q)) r rdq. (7.1)
All the estimates we will perform in this section take place in the region r > t 2 since we will always apply the cut-o function Υ r t to h 0 .

Proposition 7.1. In the region r > t 2 we have the estimates on h 0 , for q < 0

|∂ s h 0 | ε 2 (1 + s) 3 2 , |h 0 | ε 2 √ 1 + s + ε 2 (1 + |q|) 2-8ρ
and for q > 0

|∂ s h 0 | ε 2 (1 + s) 3 2 (1 + |q|) 3 2 +2(δ-σ) , |h 0 | ε 2 (1 + |q|) 2+2(δ-σ) .
Proof. We write the wave operator in coordinates (s, q, θ)

= 4∂ s ∂ q + 1 r (∂ s + ∂ q ) + 1 r 2 ∂ 2 θ . (7.2)
We calculate

∂ s ∂ q h 0 = ∂ s (-2r(∂ q φ) 2 ) = -r∂ q φ 4∂ s ∂ q φ + 1 r ∂ q φ = -r∂ q φ φ - 1 r ∂ s φ - 1 r 2 ∂ 2 θ φ , (7.3)
where we have used ∂ s (-2b(θ)∂ 2 q (qχ(q))) = 0.

Therefore we have

∂ s h 0 = Q s -φ + 1 r ∂ s φ + 1 r 2 ∂ 2 θ φ ∂ q φrdq + O ε 2 (1 + s) 3+2δ , (7.4)
where we have used

∂ s h 0 | t=0 = -∂ q h 0 | t=0 = 2r(∂ q φ) 2 + 2b(θ)∂ 2 q (χ(q)q) | t=0 = O ε 2 (1 + s) 3+2δ .
The bootstrap assumption (4.15) gives

1 r ∂ s φ + 1 r 2 ∂ 2 θ φ 1 (1 + s) 2 |Z 2 φ| ε (1 + s) 5 2 (1 + |q|) 1 2 -4ρ
, and

|∂ q φ| 1 1 + |q| |Zφ| ε (1 + s) 1 2 (1 + |q|) 3 2 -4ρ . 1 r ∂ s φ + 1 r 2 ∂ 2 θ φ ∂ q φr ε 2 (1 + s) 2 (1 + |q|) 2-8ρ . (7.5)
To estimate φ we write φ = (g )φ. Thanks to Remark 2.2, in the region q < 0 it is sucient to estimate g LL ∂ 2 q φ. We start with the region q < 0. To obtain all the possible decay in s, we use the estimate (5.10) of Corollary 5.4 for I ≤ N -11, which gives, for q < 0

|g LL | ε(1 + |q|) 3 2 (1 + s) 3 2
.

The bootstrap assumption (4.15) imply

|∂ 2 q φ| ε (1 + |q|) 5 2 -4ρ √ 1 + s , therefore |g LL ∂ 2 q φ∂ q φ| ε 3 (1 + |q|) 3 2
(1 + s)

5 2 (1 + |q|) 4-8ρ
, and we obtain

|( φ)∂ q φr| ε 3 (1 + s) 3 2 (1 + |q|) 5 2 -8ρ . ( 7.6)
Thanks to (7.5) and (7.6), in the region q < 0 we have

-φ + 1 r ∂ s φ + 1 r 2 ∂ 2 θ φ ∂ q φr ε 3 (1 + s) 3 2 (1 + |q|) 5 2 -8ρ . (7.7)
We now estimate the integrand in the region q > 0. Estimate (4.38) yields, for q > 0 and I ≤ N -3

|Z I φ| ε √ 1 + s(1 + |q|) 1 2 +δ-σ
, and estimate (5.12) yields for q > 0

|g LL | (1 + |q|) 1 2 +σ-δ (1 + s) 3 2
.

In the region q > 0, φg φ contains also terms of the form εχ(q) r ∂φ (see (2.13) in the discussion of Section 2.5). We can neglect them since we already take into account terms of the form

1 r ∂ s φ+ 1 r 2 ∂ 2 θ φ in (7.3). Consequently for q > 0 -φ + 1 r ∂ s φ + 1 r 2 ∂ 2 θ φ ∂ q φr ε 2 (1 + s) 3 2 (1 + |q|) 5 2 +2δ-2σ . (7.8)
Therefore, (7.4) and (7.8) yield for q > 0

|∂ s h 0 | ε 2 (1 + s) 3 2 (1 + |q|) 3 2 +2(δ-σ) , (7.9)
and (7.4), (7.7) and (7.8) yield for q < 0, since

1 (1+|q|) 5 2 -8ρ is integrable, |∂ s h 0 | ε 2 (1 + s) 3 2
.

(7.10)

+ Σ T,θ (∂ q φ) 2 rdr ε 2 T 1 2
.

Moreover ∂ r h 0 = ∂ q h 0 + ∂ s h 0 and therefore (7.10) and (7.9) yield

∂ r h 0 = -2r(∂ q φ) 2 -2b(θ)∂ 2 r (χ(q)q) + O ε 2 (1 + s) 3 2
.

(7.11)

Therefore, on the line t = T , with xed θ we obtain the following estimate for h 0 in the region r < t by integrating (7.11)

h 0 (T, R, θ) = - ∞ R -2r(∂ q φ) 2 + O ε 2 (r + T ) 3 2 + 2b(θ) = R 0 2r(∂ q φ) 2 dr + O ε 2 √ 1 + T =O ε 2 (1 + T ) 1 2 + O ε 2 (1 + q) 2-8ρ .
To estimate h 0 elsewhere in the region r < t, we can integrate the estimate (7.10), at xed q, as shown in the left of the gure 7.1. To estimate h 0 in the region r > t we integrate the transport equation from t = 0, as shown in the right of the gure 7.1 : we rely on formula (7.1) and the estimate for q > 0

|∂ q φ| ε √ 1 + s(1 + |q|) 3 2 +δ-σ
.

We obtain

h 0 = O ε 2 (1 + q) 2+2(δ-σ) , q > 0, h 0 = O ε 2 (1 + s) 1 2
+ O ε 2 (1 + q) 2-8ρ , q < 0.

Next we derive an estimate for Z I h 0 . Proposition 7.2. Let I ≤ N -5. We have the estimate for q < 0

|Z I h 0 | ε 2 √ 1 + s + ε 2 (1 + |q|) 1-4ρ , |∂ s Z I h 0 | ε 2 (1 + s) 3 2 , |∂ q Z I h 0 | ε 2 (1 + |q|) 2-4ρ ,
and for q > 0

|Z I h 0 | ε 2 (1 + |q|) 2+2(δ-σ) , |∂ s Z I h 0 | ε 2 (1 + s) 3 2 (1 + |q) 1+2(δ-σ) . Observe that S = s∂ s +q∂ q , Ω 12 = ∂ θ , Ω 01 = cos(θ)(s∂ s -q∂ q )- t r sin(θ)∂ θ , Ω 02 = sin(θ)(s∂ s -q∂ q )+ t r cos(θ)∂ θ .
Hence Proposition 7.2 is an immediate consequence of Proposition (7.3).

Proposition 7.3. We assume Let j + k + l ≤ N -5 then in the region r > t 2 we have the estimates on h 0 , for q < 0, if j, k ≥ 1

|∂ j s ∂ k q ∂ l θ h 0 | ε 2 s j+ 1 2 (1 + |q|) k+1-4ρ
and

|∂ k q ∂ l θ h 0 | ε 2 (1 + |q|) k 1 √ 1 + s + 1 (1 + |q|) 1-4ρ , |∂ j s ∂ l θ h 0 | ε 2 (1 + s) 3 2 +j
.

For q > 0 we have, with j ≥ 1

|∂ j s ∂ k q ∂ l θ h 0 | ε 2 s j+ 1 2 (1 + |q|) k+ 3 2 +2(δ-σ) , |∂ k q ∂ l θ h 0 | ε 2 (1 + |q|) k+2+2(δ-σ) .
Proof. We assume rst j = 0 and k ≥ 1. We assume l + k ≤ N -3. Then we can write

∂ k q ∂ l θ h 0 = -2∂ k-1 q ∂ l θ r(∂ q φ) 2 + ∂ 2 q (qχ(q))b(θ) .
Therefore we can estimate

|∂ k q ∂ l θ h 0 | r (1 + |q|) k-1 J≤k+l-1 |Z J (∂ q φ) 2 | + 1 (1 + |q|) k+1 |∂ l θ b|.
The terms in Z J (∂ q φ) 2 are of the form ∂ q Z J 1 φ∂ q Z J 2 φ, where J 1 ≤ k+l 2 ≤ N -15 therefore we can estimate, thanks to the bootstrap assumption (4.15)

|∂ q Z J 1 φ| ε

(1 + |q|) 3 2 -4ρ √ 1 + s ,
and we estimate ∂ q Z J 2 φ thanks to (4.28) of Proposition 4.8 since

J 2 ≤ l + k -1 ≤ N -4 |∂ q Z J 2 φ| ε 1 + |q| √ 1 + s .
Consequently we have shown that for k + l ≤ N -3, k ≥ 1

|∂ k q ∂ l θ h 0 | ε 2 (1 + |q|) k+1-4ρ . (7.12)
In the region q > 0 we have the better estimate for i = 1, 2 thanks to (4.31) of Proposition 4.8

|∂ q Z J i φ| ε (1 + |q|) 3 2 +δ-σ √ 1 + s , so |∂ k q ∂ l θ h 0 | ε 2 (1 + |q|) k+2+2δ-2σ . (7.13) We now assume k ≥ 1, j ≥ 1 and estimate ∂ j s ∂ k q ∂ l θ h 0 , for j + k + l ≤ N -4. Thanks to (7.3),
we can write

∂ j s ∂ k q ∂ l θ h 0 = -∂ j-1 s ∂ k-1 q ∂ l θ r∂ q φ φ - 1 r ∂ s φ - 1 r 2 ∂ 2 θ φ .
We estimate

∂ j-1 s ∂ k-1 q ∂ l θ r∂ q φ 1 r ∂ s φ + 1 r 2 ∂ 2 θ φ 1 (1 + |q|) k-1 (1 + |s|) j-1 Z j+k+l-2 r∂ q φ 1 r ∂ s φ + 1 r 2 ∂ 2 θ φ 1 (1 + |q|) k (1 + |s|) j J 1 +J 2 ≤j+k+l-2 |Z J 1 +2 φZ J 2 +1 φ|
We can assume J 1 ≤ j+k+l-2 2 . In the region q < 0, (4.15) and (4.35) yield

|Z J 1 +2 φ| ε √ 1 + s(1 + |q|) 1 2 -4ρ , |Z J 2 φ| ε 1 + |q| √ 1 + s .
Consequently, for q < 0

∂ j-1 s ∂ k-1 q ∂ l θ r∂ q φ 1 r ∂ s φ + 1 r 2 ∂ 2 θ φ ε 2 (1 + |q|) k-4ρ (1 + s) j+1 . (7.14)
To estimate the contribution of φ, we write as before φ = (g )φ. Following Remark 2.2, it is sucient to estimate

∂ j-1 s ∂ k-1 q ∂ l θ rg LL ∂ q φ∂ 2 q φ 1 (1 + |q|) k-1 (1 + |s|) j-1 Z k+j+l-2 rg LL ∂ q φ∂ 2 q φ 1 (1 + |q|) k (1 + |s|) j-2 J 1 +J 2 +J 3 ≤j+k+l-2 |Z J 1 g LL ∂ q Z J 2 +1 φ∂ q Z J 3 φ|. We have J 1 + J 2 + J 3 ≤ j + k + l -2 ≤ N -5. We separate in two cases J 1 ≤ N
2 -2 and J 2 ≤ N 2 -2 : then we have thanks to (5.10), (4.15) and (4.28)

|Z J 1 g LL | ε(1 + |q|) 3 2 (1 + s) 3 2
,

|∂ q Z J 2 +1 φ| ε √ 1 + s(1 + |q|) 3 2 -4ρ
, |∂ q Z J 3 φ| ε

1 + |q| √ 1 + s .
The case J 1 ≤ N 2 -2 and J 3 ≤ N 2 -2 can be treated in the same way.

J 2 ≤ N 2 -2 and J 3 ≤ N 2 -2 then, since |J 1 | ≤ j + k + l -2 ≤ N -4
we have thanks to (5.11) and (4.15)

|Z J 1 g LL | ε(1 + |q|) 2+µ (1 + s) 3 2 , |∂ q Z J φ| ε √ 1 + s(1 + |q|) 3 2 -4ρ
, f or J = J 2 + 1, J 3 .

In the rst case we obtain

|Z J 1 g LL ∂ q Z J 2 +1 φ∂ q Z J 3 φ| ε 3 (1 + s) 5 2 (1 + |q|) 1 2 -4ρ , (7.15)
and in the last case we obtain

|Z J 1 g LL ∂ q Z J 2 +1 φ∂ q Z J 3 φ| ε 3 (1 + s) 5 2 (1 + |q|) 1-8ρ-µ
.

We have µ + 4ρ ≤ 1 2 . Consequently, we have in the region q < 0

∂ j-1 s ∂ k-1 q ∂ l θ rg LL ∂ q φ∂ 2 q φ ε 3 (1 + |q|) k+ 1 2 -4ρ (1 + |s|) j-2
.

(7.16) Estimates (7.14) and (7.16) yield, in the region q < 0 for j + k + l ≤ N -4, j, k ≥ 1

|∂ j s ∂ k q ∂ l θ h 0 | ε 2 (1 + s) j+ 1 2 (1 + |q|) k+ 1 2 -4ρ
.

(7.17)

In the region q > 0, thanks to (4.31) and (5.12) we have the better estimate, for J ≤ N -5

|∂ q Z J φ| ≤ ε √ 1 + s(1 + |q|) 3 2 +δ-σ , |Z J g LL | ≤ ε(1 + |q|) 1 2 -δ+σ (1 + s) 3 2
, so we have

|∂ j s ∂ k q ∂ l θ h 0 | ε 2 (1 + s) j+ 1 2 (1 + |q|) k+ 3 2 +2(δ-σ) . (7.18)
We now assume k = 0 and j ≥ 1. We obtain an estimate on ∂ j s ∂ l θ h 0 for q > 0 by integrating (7.18) for k = 1 with respect to q, from the hypersurface t = 0. We obtain for j + l ≤ N -4, j ≥ 1, q > 0

|∂ j s ∂ l θ h 0 | ε 2 (1 + s) j+ 1 2 (1 + |q|) 3 2 +2(δ-σ) . (7.19)
For q < 0, we integrate (7.17) from q = 0. We obtain for j + l ≤ N -4, j ≥ 1,

|∂ j s ∂ l θ h 0 | ε 2 (1 + s) j+ 1 2 . ( 7.20)
We now estimate ∂ l θ h 0 for l ≤ N -5. Recall from Corollary 6.4 that

∂ l θ b(θ) + Σ T,θ (∂ q φ) 2 rdr ε 2 T 1 2
.

Moreover, we can write, thanks to the estimate (7.20)

∂ r ∂ l θ h 0 = ∂ q ∂ l θ h 0 + ∂ s ∂ l θ h 0 = ∂ l θ -2r(∂ q φ) 2 -2∂ 2 q (qχ(q))∂ l θ b + O ε 2 (1 + s) 3 2
.

Therefore, by integrating this on the line t = T , we have

∂ l θ h 0 (T, R, θ) = R 0 ∂ q ∂ l θ h 0 dr + O ε 2 √ 1 + T ,
and consequently, thanks to (7.12) we have the estimate, for l ≤ N -4 and q < 0

|∂ l θ h 0 (T, r, θ)| ε 2 √ 1 + T + ε 2 (1 + |q|) 1-4ρ .
To have an estimate everywhere, we integrate (7.20) for j = 0 with respect to s, as shown in the gure 7.1. We obtain, for l ≤ N -5

|∂ l θ h 0 | ε 2 √ 1 + s + ε 2 (1 + |q|) 1-4ρ . (7.21)
In the region q > 0, we just integrate (7.19) from t = 0, and we obtain

|∂ l θ h 0 | ε 2 (1 + |q|) 2+2(δ-σ) . (7.22)
In view of (7.17), (7.18), (7.12), (7.13), (7.20)

Z I Υ r t h 0 -Υ r t -2(∂ q φ) 2 -2 b(θ)∂ 2 q (qχ(q)) r ε 2 (1 + s) 3 2 (1 + |q|)
, and for q > 0

Z I Υ r t h 0 -Υ r t -2(∂ q φ) 2 -2 b(θ)∂ 2 q (qχ(q)) r ε 2 (1 + s) 3 2 (1 + |q|) 2+2(δ-σ)
.

Proof. We have in view of (4.10), (7.2) and (7.3),

Υ r t h 0 =Υ r t 4∂ s ∂ q h 0 + 1 r (∂ s h 0 + ∂ q h 0 ) + 1 r 2 ∂ 2 θ h 0 + ∇Υ r t .∇h 0 + h 0 Υ r t =Υ r t -4r∂ q φ φ - 1 r ∂ s φ - 1 r 2 ∂ 2 θ φ + 1 r -2(∂ q φ) 2 r -2b(θ)∂ 2 q (qχ(q)) + Υ r t 1 r ∂ s h 0 + 1 r 2 ∂ 2 θ h 0 + ∇Υ r t .∇h 0 + h 0 Υ r t =Υ r t -2(∂ q φ) 2 -4 b(θ)∂ 2 q (qχ(q)) r -4rΥ r t ∂ q φ φ + f (s, q, θ), where f (s, q, θ) = Υ r t 4∂ q φ ∂ s φ + ∂ 2 θ φ r + 1 r ∂ s h 0 + 1 r 2 ∂ 2 θ h 0 + ∇Υ r t .∇h 0 + h 0 Υ r t .
We can estimate Z I f , noticing that when Υ r t = 0 we have r ∼ t ∼ |q|. We obtain

|Z I f | 1 (1 + s) 2 J≤I+2 |Z I h 0 | + 1 1 + s I 1 +I 2 ≤I |Z I 1 +2 φ||∂ q Z I 2 φ|. Proposition 7.2 yields, for I ≤ N -7 1 (1 + s) 2 J≤I+2 |Z I h 0 | ε 2 (1 + s) 2 1 + |q| ,
and as usual we may estimate, thanks to (4.28) and (4.15),

|Z I 1 +2 φ∂ q Z I 2 φ| ε 2 (1 + s)(1 + |q|) 1-4ρ , therefore we obtain |Z I f | ε 2 (1 + s) 2 1 + |q| .
In the region q > 0, we have the better estimate

|Z I f | ε 2 (1 + s) 2 (1 + |q|) 2+2(δ-σ) .
To estimate φ we write, as before φ = φg φ.

It is sucient to estimate a term of the form g LL ∂ 2 q φ. Therefore we write, like in estimate (7.15),

|Z I (rg LL ∂ q φ∂ 2 q φ)| r 1 + |q| J 1 +J 2 +J 3 ≤I |Z J 1 g LL ||∂Z J 2 +1 φ||∂Z J 3 φ| ε 3 (1 + s) 3 2 (1 + |q|) 3 2 -4ρ
.

In the region q > 0, we have the better estimate

|Z I (r∂ q φ φ)| ε 3 (1 + s) 3 2 (1 + |q|) 2+2(δ-σ)
.

This concludes the proof of Proposition 7.4.

Estimation on h

We recall that h satises the equation

h = Υ r t h 0 + Υ r t g LL ∂ 2 q h 0 + 2Υ r t (∂ q φ) 2 -2(R b ) qq + Υ r t Q LL (h 0 , g), ( h, ∂ t h)| t=0 = (0, 0),
where Q LL is dened by (4.12). Proposition 7.5. h satises, for I ≤ N -7

|Z I h| ε 2 (1 + s) 1 2 -ρ .
Proof. Proposition 7.4 gives for I ≤ N -7 and q < 0

Z I Υ r t h 0 + 2Υ r t (∂ q φ) 2 -2(R b ) qq ε 2 (1 + s) 3 2 (1 + |q|) , (7.23)
where we have used that thanks to (1.7)

(R b ) qq -2 b(θ)∂ 2 q (qχ(q)) r 1 1≤q≤2 ε 2 (1 + s) 2 .
To estimate Z I (g LL ∂ 2 q h 0 ) we use the transport equation for h 0

g LL ∂ 2 q h 0 = g LL ∂ q (-2r(∂ q φ) 2 -2b(θ)∂ 2 q (qχ(q))
We estimate the rst term as in the proof of Proposition 7.4.

|Z I (rg LL ∂ q φ∂ 2 q φ)| ε 2 (1 + s) 3 2 (1 + |q|) 3 2 -4ρ
.

To estimate the second term, we note that the terms of the form χ (j) (q) decay faster than any power of q, so thanks to (5.11),

|Z I g LL b(θ)∂ 2 q (qχ(q)) | ε 2 s 3 2 (1 + |q|) 3
.

Consequently we have proved

Z i Υ r t g LL ∂ 2 q h 0 ε 2 (1 + s) 3 2 (1 + |q|) 3 2 -4ρ . ( 7

.24)

We now estimate Q LL (h 0 , g). We note than in the region q < 0 the only term is ∂ L g LL ∂ L h 0 . We use again the transport equation for h 0 ∂ q g LL ∂ q h 0 = ∂ q g LL (-2r(∂ q φ) 2 -2b(θ)∂ 2 q (qχ(q)).

Consequently, for similar reasons than for (7.24), we obtain in the region q < 0

Z i Υ r t ∂ q g LL ∂ q h 0 ε 2 (1 + s) 3 2 (1 + |q|) 3 2 -4ρ . (7.25)
Thanks to (7.23), (7.24) and (7.25), we have in the region q < 0 for I ≤ N -7

| Z I h| ε 2 (1 + s) 3 2 (1 + |q|) . ( 7

.26)

In the region q > 0, we have to estimate in Q LL (h 0 , g) the term ∂ L (g b ) U U ∂ L g LL , which is of the form χ(q)b(θ) r ∂ q g LL . Thanks to (4.32) we have

Z I χ(q)b(θ) r ∂ q g LL ε 2
(1 + s) The other terms give contributions similar to the one of Proposition 7.4. Consequently, for q > 0 we have the better estimate for I ≤ N -7

| Z I h| ε 2 (1 + s) 3 2 (1 + |q|) 3 2 +δ-σ . (7.28)
We now use lemma 7.6, whose proof is given at the end of this section, to conclude.

Lemma 7.6. Let β, α ≥ 0, such that β -α ≥ ρ > 0. Let u be such that

| u| 1 (1 + s) 3 2 -α (1 + |q|) , f or q < 0 | u| 1 (1 + s) 3 2 -α (1 + |q|) 1+β
, f or q > 0, and (u, ∂ t u)| t=0 = 0. Then we have the estimate

|u| (1 + t) α+ρ √ 1 + s .
Thanks to (7.26) and (7.28), the conditions of Lemma 7.6 are satised with α = 0 and β = 1 2 + δ -σ. Moreover, the initial data for Z I h are given by the right-hand side of (4.11) (i.e. they are quadratic), therefore, for I ≤ N -7 at t = 0 we have

|Z I h| + (1 + r)|∂ t Z I h| ε 2 (1 + r) 1+δ .
Consequently, Lemma 7.6 and Proposition 1.5 yield for I ≤ N -7

|Z I h| ε 2 (1 + t) ρ √ 1 + s .
This concludes the proof of Lemma 7.5.

Proof of Lemma 7.6. Let t 0 > 0. We consider times t ≤ t 0 . In the region r ≤ 2t we have |q| ≤ t ≤ t 0 and s ≤ 3t ≤ 3t 0 . Therefore

| u| (1 + t 0 ) α+ρ (1 + |q|) 1+ ρ 2 (1 + s) 3 2 + ρ 2 .
In the region r ≤ 2t, we have r 2 ≤ |q| ≤ r and r ≤ s ≤ 3r 2 , therefore

| u| 1 (1 + r) 3 2 -α+1+β (1 + t 0 ) α+ρ (1 + r) 5 2 -α+β (1 + t 0 ) α+ρ (1 + |q|) 1+ ρ 2 (1 + s) 3 2 + ρ 2 , provided 5 2 + ρ ≤ 5 2 + β -α, i.e. β -α ≥ ρ. Consequently, the L ∞ -L ∞ estimate yields, for t ≤ t 0 |u| (1 + t 0 ) α+ρ √ 1 + s .
If we take t = t 0 we have proved

|u| (1 + t) α+ρ √ 1 + s ,
which concludes the proof of Lemma 7.6.

8 Commutation with the vector elds and L ∞ estimates 8.1 Estimates for I ≤ N -14

Proposition 8.1. We have the estimates for for I ≤ N -14

|Z I g 1 | ≤ C 0 ε + Cε 2 (1 + s) 1 2 -ρ , |Z I φ| ≤ C 0 ε + Cε 2 √ 1 + s(1 + |q|) 1 2 -4ρ
.

This proposition is a consequence of L ∞ -L ∞ estimates and the following propositions.

Proposition 8.2. We have the estimate for I ≤ N -14

| Z I φ| ε 2 (1 + s) 2-3ρ (1 + |q|)
, q < 0,

| Z I φ| ε 2 (1 + s) 2 (1 + |q|) 1+δ-σ , q > 0.
Proposition 8.3. We have the estimate for I ≤ N -14

| Z I g 1 | ε 2 (1 + s) 3 2 (1 + |q|)
, q < 0,

| Z I g 1 | ε 2 (1 + s) 3 2 (1 + |q|) 3 2 +δ-σ
, q > 0.

We rst assume Proposition 8.2 and 8.3, and prove Proposition 8.1.

Proof of Proposition 8.1. We have

| Z I φ| ε 2 (1 + s) 2-3ρ (1 + |q|) ε 2 (1 + s) 2-4ρ (1 + |q|) 1+ρ ,
therefore the L ∞ -L ∞ estimate, combined with Proposition 1.5 for the contribution of the initial data yields

|Z I φ| ≤ C 0 ε √ 1 + s 1 + |q| + Cε 2 √ 1 + s(1 + |q|) 1 2 -4ρ
, where C is a constant depending on ρ.

The estimate g 1 follows from Lemma 7.6 with α = 0, β = 3 2 + δ -σ combined with Proposition 1.5 Proof of Proposition 8.2. We rst estimate Z I φ in the region q < 0 Z I φ = Z I ( φg φ) .

|Z I g 1 | ≤ C 0 ε √ 1 + s 1 + |q| + Cε 2 (1 + s)
In the region q < 0, thanks to Remark 2.2, it is sucient to estimate Z

I g LL ∂ 2 q φ |Z I-J g LL ∂ 2 q Z J φ| 1 (1 + |q|) 2 |Z I-J g LL ||Z J+2 φ|. If J ≤ N -14 2 we have J + 2 ≤ N -14 2 + 2 ≤ N -14 so, thanks to (4.15) |Z J+2 φ| ε (1 + s) 1 2 (1 + |q|) 1 2 -4ρ
, and since I -J ≤ N -14 we have thanks to (5.9)

|Z I-J g LL | ε(1 + |q|) (1 + s) 3 2 -2ρ
.

Therefore |Z I-J g LL ∂ 2 q Z J φ| ε 2 (1 + s) 2-2ρ (1 + |q|) 3 2 -4ρ
.

If I -J ≤ N -14

2

≤ N -15 we have thanks to (5.8)

|Z I-J g LL | ε(1 + |q|) (1 + s) 3 2 -ρ
, and since J + 2 ≤ N -12 we have thanks to (4.17)

|Z J+2 φ| ε

(1 + s) 1 2 -2ρ
.

In the two cases, we have for q < 0 |Z I-J g LL ∂ 2 q Z J φ| ε 2 (1 + s) 2-3ρ (1 + |q|) .

(8.1)

In the region q > 0 we have the better estimate thanks to (5.12) and (4.32)

|Z I-J g LL ∂ 2 q Z J φ| ε 2 (1 + s) 2 (1 + |q|) 2+2(δ-σ) . (8.2)
In the region q > 0 we also have to take into account the crossed term. These terms are described by (2.13) in Section 2.5. It is sucient to estimate

Z I b(θ) χ(q) r ∂ s φ .
Since they occur only in the region q > 0, we can estimate, thanks to (4.38) Proof of Proposition 8.3. We write the equation for g 1 . We have, thanks to (2.4) and (2.7)

|Z I φ| ε √ 1 + s(1 + |q|) 1 2 +δ-σ . Therefore Z I b(θ) ∂ q (qχ(q)) r ∂ s φ ε 2 (1 + s) 5 2 (1 + |q|) 1 2 +δ-σ ε 2 (1 + s) 2 (1 + |q|) 1+δ-σ .
( g 1 ) µν = -2∂ µ φ∂ ν φ + 2(R b ) µν + (dq 2 ) µν Υ r t h 0 + Υ r t 1 r 2 u 1 µν (θ)h 0 + u 2 µν (θ)∂ θ h 0 + P µν (g)(∂ g, ∂ g) + P µν ( g, g b ), (8.4)
and therefore

Z I ( g 1 ) µν = f µν , where the terms in f µν are of the forms the quasilinear terms : thanks to Remark 2.2 it is sucient to study Z I (g LL ∂ 2 q g 1 ),

the terms coming from the non commutation of the wave operator with the null decomposition: they are calculated in (2.7) and they are of the form Υ( r t ) 1 r 2 ∂ θ Z I h 0 , the semi-linear terms: following section the worst term is the term Z I ∂ L g LL ∂ L g LL appearing in Z I P LL (see (2.9)).

the crossed terms with the background metric g b : the worst term is the term

Z I ∂ L (g b ) U U ∂ L g LL appearing in Z I P LL (see (2.11)).
The quasilinear terms We estimate

Z I g LL ∂ 2 q g 1 = J≤I Z I-J g LL Z J ∂ 2 q g 1 .
We have

|Z I-J g LL ∂ 2 q Z J g 1 | 1 (1 + |q|) 2 ||Z I-J g LL ||Z J+2 g 1 |.
If J ≤ N -14 2 we have J + 2 ≤ N -14 2 + 2 ≤ N -14 so thanks to (4.16)

|Z J+2 g 1 | ε (1 + s) 1 2 -ρ
, and since I -J ≤ N -14 we have thanks to (5.9)

|Z I-J g LL | ε(1 + |q|) (1 + s) 3 2 -2ρ
.

If I -J ≤ N -14 2
≤ N -15 we have thanks to (5.8)

|Z I-J g LL | ε(1 + |q|) (1 + s) 3 2 -ρ
, and since J + 2 ≤ N -12 we have thanks to (4.18)

|Z J+2 g 1 | ε (1 + s) 1 2 -2ρ
In the two cases, we have

|Z I-J g LL ∂ 2 q Z J g 1 | ε 2 (1 + s) 2-3ρ (1 + |q|) . (8.5)
The term coming from the non commutation of the wave operator with the null structure We have to estimate

Υ r t ∂ θ Z I h 0 r 2 . Since I ≤ N -14, we have I + 1 ≤ N -5 so thanks to Proposition 7.2 Υ r t ∂ θ Z I h 0 r 2 ε 2 (1 + s) 2 1 + |q| ε 2 (1 + s) 3 2 (1 + |q|) . (8.6)
The semi-linear terms We estimate Z I ∂ L g LL ∂ L g LL . For this, we have to estimate, using the decomposition (4.6)

Z I ∂ L h 0 ∂ L g LL and Z I ∂ L g 1 ∂ L g LL
The rst term has been estimated in (7.23). For the second term, we write

|Z I ∂ L g 1 ∂ L g LL | 1 1 + |q| J≤I |Z J+1 g 1 ||∂Z I-J g LL |,
and we estimate if J ≤ N -14 2 thanks to (4.16) and (5.4)

|Z J+1 g 1 | ε (1 + s) 1 2 -ρ and |∂Z I-J g LL | ε (1 + s) 3 2 -2ρ
.

If I -J ≤ N -14 2
thanks to (4.18) and ( 5.3) we have

|Z J+1 g 1 | ε (1 + s) 1 2 -2ρ
and |∂Z I-J g LL | ε

(1 + s) 3 2 -ρ
.

In the two cases we have

|Z I ∂ L g 1 ∂ L g LL | ε 2 (1 + s) 2-3ρ (1 + |q|)
.

This estimate and (7.23) yields for I ≤ N -14

|Z I ∂ L g 1 ∂ L g LL | ε 2 (1 + s) 3 2 (1 + |q|) 3 2 -4ρ . (8.7)
We have now estimated Z I ( g 1 ) µν in the region q < 0. Thanks to (8.5), (8.6) and (8.7) we have, for q < 0 and I ≤ N -14

| Z I g 1 | ε 2 (1 + s) 2-3ρ (1 + |q|) (8.8)
The crossed terms The crossed term are only present in the region q > 0. The estimate of (7.27). The other terms give better contributions in the region q > 0 (see Remark 2.3). Therefore we have for q < 0 and I ≤ N -4

Z I ∂ q (g b ) U U ∂ q g LL is done in
| Z I g 1 | ε 2 (1 + s) 3 2 (1 + |q|) 3 2 +δ-σ . ( 8.9)
The estimates (8.8) and (8.9) conclude the proof of Proposition 8.3.

Estimates for I ≤ N -12

Proposition 8.4. We have the estimates for I ≤ N -12

|Z I φ| ≤ C 0 ε + Cε 2 (1 + s) 1 2 -2ρ
,

|Z I g 1 | C 0 ε + Cε 2 (1 + s) 1 2 -2ρ
.

This proposition is a straightforward consequence of Lemma 7.6, Proposition 1.5 and the following propositions.

Proposition 8.5. We have the estimate for I ≤ N -12

| Z I φ| ε 2 (1 + s) 3 2 -ρ (1 + |q|)
, q < 0,

| Z I φ| ε 2 (1 + s) 2 (1 + |q|) 1+δ-σ , q > 0.
Proposition 8.6. We have the estimate for I ≤ N -12

| Z I g 1 | ε 2 (1 + s) 3 2 -ρ (1 + |q|)
, q < 0,

| Z I g 1 | ε 2 (1 + s) 3 2 (1 + |q|) 3 2 +δ-σ
, q > 0.

Proof of Proposition 8.5. We rst estimate φ Z I φ = Z I ( φg φ) .

In the region q < 0, it is sucient to estimate

Z I g LL ∂ 2 q φ |Z I-J g LL ∂ 2 q Z J φ| 1 1 + |q| |Z I-J g LL ||∂ q Z J+1 φ| If J ≤ N -12 2
we have J + 1 ≤ N -14 so thanks to (4.15)

|∂Z J+1 φ| ε

(1 + s) 1 2 (1 + |q|) 3 2 -4ρ
, and since I -J ≤ N -12 we have thanks to (5.10)

|Z I-J g LL | ε(1 + |q|) 1 + s . Therefore |Z I-J g LL ∂ 2 q Z J φ| ε 2 (1 + s) 3 2 (1 + |q|) 3 2 -4ρ
.

If I -J ≤ N -12

2

≤ N -15 we have thanks to (5.8)

|Z I-J g LL | ε(1 + |q|) (1 + s) 3 2 -ρ
, and since J + 1 ≤ N -12 ≤ N -4 we have thanks to (4.28)

|∂Z J+1 φ| ε √ 1 + s (1 + |q|) .
In the two cases, we have

|Z I-J g LL ∂ 2 q Z J φ| ε 2 (1 + s) 3 2 -ρ (1 + |q|)
.

The main contribution in the region q > 0 is like (8.3) in the proof of Proposition 8.2. This concludes the proof of Proposition 8.5.

Proof of Proposition 8.6. We estimate g 1 . We only deal with the quasilinear and semilinear terms in the region q < 0, as the control obtained in the proof of Proposition 8.3 is sucient to deal with the others (see (8.6) and (7.27)).

The semi-linear terms We estimate Z I ∂ L g LL ∂ L g LL . For this, we have to estimate

Z I ∂ L h 0 ∂ L g LL and Z I ∂ L g 1 ∂ L g LL
The rst term has been estimated in (7.23). For the second term, we write

|Z I ∂ L g 1 ∂ L g LL | 1 1 + |q| J≤I |Z J+1 g 1 ||∂Z I-J g LL |,
and we estimate if J ≤ N -12 2 thanks to (4.16) and (5.5)

|Z J+1 g 1 | ε (1 + s) 1 2 -ρ and |∂Z I-J g LL | ε 1 + |q| (1 + s) 3 2
.

If I -J ≤ N -12 2
thanks to (4.44) and (5.3) we have

|Z J+1 g 1 | ε and |∂Z I-J g LL | ε (1 + s) 3 2 -ρ .
In the two cases we have

|Z I ∂ L g 1 ∂ L g LL | ε 2 (1 + s) 3 2 -ρ (1 + |q|)
.

This estimate and (7.23) yields for I ≤ N -12

|Z I ∂ L g 1 ∂ L g LL | ε 2 (1 + s) 3 2 -ρ (1 + |q|) . (8.10)
The quasilinear terms We estimate Z I g LL ∂ 2 q g 1 . We have

|Z I-J g LL ∂ 2 q Z J g 1 | 1 1 + |q| |Z I-J g LL ||∂ q Z J+1 g 1 |. If J ≤ N -12 2 we have J + 2 ≤ N -12 2
+ 2 ≤ N -14 so thanks to (4.16)

|∂ q Z J+1 g 1 | ε (1 + s) 1 2 -ρ (1 + |q|)
, and since |I -J| ≤ N -12 we have thanks to (5.10)

|Z I-J g LL | ε(1 + |q|) 1 + s . If |I -J| ≤ N -12 2
≤ N -15 we have thanks to (5.8)

|Z I-J g LL | ε(1 + |q|) (1 + s)
|∂ q Z J+1 g 1 | ε 1 + |q| .
In the two cases, we have

|Z I-J g LL ∂ 2 q Z J g 1 | ε 2 (1 + s) 3 2 -ρ (1 + |q|) . (8.11)
The equation (8.10) and (8.11), together with (8.6) proved during the proof of Proposition 8.3 conclude the proof of Proposition 8.6 for q < 0. The estimate for Z I g 1 in the region q > 0 is given by (8.9). This concludes the proof of Proposition 8.6 for q > 0.

Weighted energy estimate

We consider the equation

g u = f,
where g = g b + g is our space-time metric, satisfying the bootstrap assumptions. We introduce the energy-momentum tensor associated to

g Q αβ = ∂ α u∂ β u - 1 2 g αβ g µν ∂ µ u∂ ν u.
We have

D α Q αβ = f ∂ β u.
We also note T = ∂ t , and introduce the deformation tensor of T

π αβ = D α T β + D β T α
where D is the covariant derivative. We have

D α (Q αβ T β ) = f ∂ t u + Q αβ π αβ . (9.1)
We remark that

Q T T = 1 2 (∂ t u) 2 + |∇u| 2 + O(ε(∂u) 2 ).
Proposition 9.1. Let w be any of our weight functions. We have the following weighted energy estimate for u

d dt Q T T w(q)dx +C w (q) (∂ s u) 2 + ∂ θ u r 2 dx ε 1 + t w(q)(∂u) 2 dx+ w(q)|f ∂ t u|dx.
Moreover, if we use the weight modulator α dened in (4.22), we obtain

d dt Q T T α 2 w(q)dx + C α 2 w (q) (∂ s u) 2 + ∂ θ u r 2 dx ε (1 + t) 1+2σ w(q)(∂u) 2 dx + α 2 w(q)|f ∂ t u|dx.
Proof. We multiply (9.1) by w(q) and integrate it on an hypersurface of constant t. We obtain

- d dt Q T T w(q)dx = w(q) f ∂ t u + Q αβ π αβ dx + Q T α D α wdx. (9.2)
We have

Q T α D α u = -2w (q)g αL Q T α = w (q)Q T L + g LT w (q)(∂u) 2 .
We calculate

Q T L =∂ t u(∂ t u + ∂ r u) - 1 2 (-(∂ t u) 2 + |∇u| 2 ) + g LL (∂ q u) 2 + g LL (∂ s u) 2 + s.t. = 1 2 (∂ s u) 2 + ∂ θ u r 2 + g LL (∂ q u) 2 + g LL (∂ s u)
Q T α D α u = (∂ s u) 2 + ∂ θ u r 2 (1 + O(ε))w (q) + O εw (q)(1 + |q|) (1 + t) 3 2 -ρ (∂u) 2 ,
and since |w (q)| w(q) 1+|q|

Q T α D α u = (∂ s u) 2 + ∂ θ u r 2 (1 + O(ε))w (q) + O εw(q)
(1 + t) We now estimate the deformation tensor of T . We have

π αβ = L T g αβ = ∂ t g αβ .
We obtain

π LL = ∂ T g LL = O ε (1 + t) 3 2 -ρ , π U L = ∂ T g U L = O ε (1 + t) 3 2 -ρ , π LL = ∂ T g LL = O ε (1 + t) 1 2 -ρ (1 + |q|) , π U L = ∂ T g U L = O ε (1 + t) 1 2 -ρ (1 + |q|) , π LL = ∂ T g LL = O ε (1 + |q|) 3 2 -ρ , π U U = ∂ T g U U = ∂ q (qχ(q))b(θ) r + O ε (1 + s)(1 + |q|) 1 2 -ρ , Consequently, the terms Q LL π LL and Q U L Q U L give contributions of the form ε (1 + t) 3 2 -ρ (∂u) 2 . (9.4)
We can calculate

Q LL = ∂ L u∂ L u - 1 2 g LL 2g LL ∂ L u∂ L u + (∂ U u) 2 + g T T (∂u) 2 + s.t. = (∂ U u) 2 + g T T (∂u) 2 + s.t.
Consequently the term Q LL π LL gives contributions of the form ε

(1 + |q|)(1 + t) 1 2 -ρ ( ∂u) 2 .
(9.5)

The terms Q LL π LL and Q LU π LU give contributions of the form ε

(1 + |q|) 3 2 -ρ ( ∂u) 2 , (9.6)
and the term Q U U π U U gives contributions of the form ∂ q (qχ(q))b(θ) r ∂u∂u, ε

(1 + s)(1 + |q|) 1 2 -ρ
∂u∂u.

(9.7)

Thanks to (9.2), (9.3), (9.4), (9.5), (9.6) and ( 9.7) what we obtain is

d dt Q T T w(q)dx + 1 2 w (q) (∂ s u) 2 + ∂ θ u r 2 dx ε (1 + t) 3 2 -ρ w(q)(∂u) 2 dx + ε w(q) (1 + |q|) 3 2 -ρ ( ∂u) 2 dx + ε w(q)
1 q>1 r |∂u ∂u|dx + w(q)|f ∂ t u|dx.

(9.8)

In the region q > 1, we have 1 r ≤ 1 t+1 . Moreover, all our weight functions satisfy w(q)

(1 + |q|) 3 2 -ρ w (q),
therefore, for ε small enough, we can subtract from our inequality the term ε w(q)

(1 + |q|) 3 2 -ρ ( ∂u) 2 ,
and we obtain

d dt Q T T w(q)dx +C w (q) (∂ s u) 2 + ∂ θ u r 2 dx ε 1 + t w(q)(∂u) 2 dx+ w(q)|f ∂ t u|dx.
This concludes the rst part of the proof of Proposition 9.1.

Next, we perform the estimate with the weight modulator α. If we replace w by α 2 w in (9.8), and we absorb as before the term ε

α 2 w(q) (1+|q|) 3 2 -ρ ( ∂u) 2 dx we obtain d dt Q T T α 2 w(q)dx + 1 2 (α 2 w) (q) (∂ s u) 2 + ∂ θ u r 2 dx ε (1 + t) 3 2 -ρ α 2 w(q)(∂u) 2 dx + α 2 w(q)1 q>1 r |∂u ∂u|dx + α 2 w(q)|f ∂ t u|dx.
We write

1 q>1 r ≤ 1 q>1 (1 + |q|) σ (1 + t) 1 2 +σ (1 + |q|) 1 2
, and so we estimate, since in the region q > 1 we have α(q) = (1 + |q|) -σ ε α 2 (q)w(q)1 q>1 r |∂u ∂u|dx ≤ε α(q)w(q)1 q>1

(1 + t)

1 2 +σ (1 + |q|) 1 2 |∂u ∂u|dx ≤ ε (1 + t) 1+2σ 1 q>1 w(q)(∂u) 2 dx + ε 1 q>1 α 2 (q)w(q) 1 + |q| ( ∂u) 2 dx. Moreover 1 q>1 α 2 (q)w(q) 1+|q| (α 2 w) . Therefore d dt Q T T α 2 w(q)dx + C (α 2 w) (q) (∂ s u) 2 + ∂ θ u r 2 dx ε (1 + t) 3 2 -ρ α 2 w(q)(∂u) 2 dx + ε (1 + t) 1+2σ 1 q>1 w(q)(∂u) 2 dx + ε (wα 2 ) (q)( ∂u) 2 dx + α 2 w(q)|f ∂ t u|dx.
We note that with our weight functions and the denition of α, we have α 2 w ∼ (α 2 w) . For ε small enough, we can absorb the term ε w (q)α 2 (q)( ∂u) 2 to obtain We note for J < N

d dt Q T T α 2 w(q)dx + C α 2 w (q) (∂ s u) 2 + ∂ θ u r 2 dx ε (1 + t) 3 2 -ρ α 2 w(q)(∂u) 2 dx + ε (1 + t) 1+2σ 1 q>1 w(q)(∂u) 2 dx + α 2 w(q)|f ∂ t u|dx,
E J = I≤J w 0 (q) 1 2 ∂Z I φ 2 L 2 + w 2 (q) 1 2 ∂Z I g 3 2 L 2 + 1 ε(1 + t) w 3 (q) 1 2 ∂Z I h 2 L 2 and E N = I≤N α 2 w 0 (q) 1 2 ∂Z I φ 2 L 2 + α 2 w 2 (q) 1 2 ∂Z I g 4 2 L 2 + 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂Z I h 2 L 2 + 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂Z I k 2 L 2 .
We also note for J < N

A J = I≤J w 0 (q) 1 2 ∂Z I φ 2 L 2 + + w 2 (q) 1 2 ∂Z I g 3 2 L 2 + 1 ε(1 + t) w 3 (q) 1 2 ∂Z I h 2 L 2 and A N = I≤J α 2 w 0 (q) 1 2 ∂Z I φ 2 L 2 + + α 2 w 2 (q) 1 2 ∂Z I g 4 2 L 2 + 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂Z I h 2 L 2 + 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂Z I k 2 L 2 .
Remark 10.1. Because of the decompositions (4.8) and (4.9) for the metric, and the non commutation of the wave operator with the null decomposition, we have to deal with terms of the form

∂ θ h
r 2 in the equation for g 4 or g 3 . Written like this, these terms are not quadratic. However, since we choose for h zero initial data, and since the equation for h is quadratic, h in itself is quadratic.

To carry this information along the proof, we may divide in the energies E I the norms involving h and k by ε. Since the initial data for h and k are zero, we have

E I (0) ≤ C 2 0 ε 2 . (10.1)
Proposition 10.2. We have the estimates for I ≤ N ,

E I ≤ (C 2 0 ε 2 + ε 2 )(1 + t) C √ ε ,
and for κ ε

t 0 1 (1 + τ ) κ A I dτ ε 2 .
This is a straightforward consequence of the following proposition. Proposition 10.3. We have the inequality, up to some negligible terms dened in Lemmas 10.4, 10.5 and 10.6 for

I ≤ N d dt E I + A I √ ε 1 + t E I + ε 5 2

+ t .

We rst prove Proposition 10.2, admitting Proposition 10.3.

Proof of Proposition 10.2. We have proved

d dt E I ≤ C √ ε 1 + t E I + C ε 5 2 1 + t , therefore, if we note E I = G(1 + t) C √ ε , we have d dt G ≤ C ε 5 2 (1 + t) 1+C √ ε .
After integrating, we obtain

G(t) ≤ G(0) + ε 2 - ε 2 (1 + t) C √ ε ,
and hence

E I ≤ (E I (0) + ε 2 )(1 + t) C √ ε ≤ (C 2 0 ε 2 + ε 2 )(1 + t) C √ ε .
Moreover, we have

d dt E I + A I ≤ C √ ε 1 + t E I + ε 5 2

+ t ,

therefore if we multiply this inequality by

1 (1+t) κ we obtain d dt E I (1 + t) κ + A I (1 + t) κ ≤ 1 (1 + t) κ d dt E I + A I ≤ C √ ε (1 + t) 1+κ E I + Cε 5 2 (1 + t) 1+κ ≤ Cε 5 2 (1 + t) 1+κ-C √ ε .
Therefore, if C √ ε < κ, the right-hand side is integrable and so

1 (1 + τ ) κ A I dτ ε 2 .
This concludes the proof of Proposition 10.2. Proposition 10.3 is a direct consequence of the three following lemmas. Lemma 10.4. We have the inequality,

d dt α 2 w(q) 1 2 ∂ Z N φ 2 L 2 + α 2 w (q) 1 2 ∂ Z N φ 2 L 2 ε 1 + t E N + ε α 2 w 2 (q) 1 2 ∂Z N g 4 2 L 2 + ε 3 1 + t where Z N φ -Z N φ is composed of terms of the form χ(q)q∂φ∂ N -1 θ b g U U ,
and we have

α 2 w 1 2 0 ∂( Z N φ -Z N φ) L 2 ε 2 .
For I < N we have

d dt w 0 (q) 1 2 ∂Z I φ 2 L 2 + w 0 (q) 1 2 ∂Z I φ 2 L 2 ε 1 + t E I + ε w 2 (q) 1 2 ∂Z I g 4 2 L 2 .
Lemma 10.5. We have the inequality,

d dt 1 εt α 2 w 3 (q) 1 2 ∂ Z N h 2 L 2 + 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂ Z N h 2 L 2 √ ε 1 + t E N + √ ε α 2 w 1 2 2 (q) ∂ Z N g 4 2 L 2 + ε 5 2 1 + t ,
where

Z N h -Z N h is composed of terms of the form χ(q)q∂h∂ N -1 θ b g U U ,
and we have

α 2 w 1 2 0 ∂( Z N h -Z N h) L 2 ε 3 √ 1 + t.
We have a similar estimate for k d dt

1 εt α 2 w 3 (q) 1 2 ∂Z N k 2 L 2 + 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂Z N k 2 L 2 √ ε 1 + t E N + √ ε α 2 w 1 2 2 (q) ∂ Z I g 4 2 L 2 .
Moreover for I < N d dt

1 εt w 3 (q) 1 2 ∂Z I h 2 L 2 + 1 ε(1 + t) w 3 (q) 1 2 ∂Z I h 2 L 2 √ ε 1 + t E I + √ ε w 1 2 2 (q) ∂ Z I g 4 2 L 2 + ε 5 2 1 + t ,
Lemma 10.6. We have the estimate

d dt α 2 w 2 (q) 1 2 ∂ Z N g 4 2 L 2 + α 2 w 2 (q) 1 2 ∂ Z N g 4 2 L 2 √ ε 1 + t E N + √ ε 1 ε(1 + t) ( α 2 w 3 (q) 1 2 ∂Z N h 2 L 2 + α 2 w 3 (q) 1 2 ∂Z N k 2 L 2 )
where Z N g 4 -Z N g 4 is composed of terms of the form

- χ(q)q∂ g 4 ∂ N -1 θ b g U U , hZ N g LL dqds
and we have

α 2 w 1 2 0 ∂( Z N g 4 -Z N g 4 ) L 2 ε 2 + ε α 2 w 1 2 2 Z N g LL L 2 .
For I < N , we have

d dt w 2 (q) 1 2 ∂Z I g 3 2 L 2 + w 2 (q) 1 2 ∂Z I g 3 2 L 2 √ ε 1 + t E I + √ ε 1 ε(1 + t) w 3 (q) 1 2 ∂Z I h 2 L 2 .
We prove Proposition 10.3.

Proof of Proposition 10.3. Therefore, if we combine Lemmas 10.4, 10.5 and 10.6 we obtain

d dt E I + A I √ ε 1 + t E I + √ εA I + ε 5 2

+ t ,

and therefore

d dt E I + (1 -C √ ε)A I √ ε 1 + t E I + ε 5 2 1 + t . If ε is small enough, we have 1 -C √ ε ≥ 1
2 , which concludes the proof of Proposition 10.3.

It is sucient to prove these three lemmas for I = N . For I < N everything work in the same way. The weight modulator α 2 is only needed to estimate a particular term for I = N and is no longer needed for I < N . Proof of Lemma 10.4. We start with the estimates for φ. We use the weighted energy estimate for the equation

g Z N φ = I+J≤N J≤N -1 Z I g αβ Z J ∂ α ∂ β φ + I+J≤N J≤N -1 Z I H ρ b Z J ∂ ρ φ. (10.2) It yields d dt α 2 w 0 (q) 1 2 ∂Z N φ 2 L 2 + α 2 w 0 (q) 1 2 ∂Z N φ 2 L 2 α 2 w 0 g Z N φ L 2 α 2 w 0 (q) 1 2 ∂Z N φ L 2 + ε 1 + t α 2 w 0 (q) 1 2 ∂Z N φ 2 L 2 .
Estimate of the rst term Thanks to Remark 2.2, it is sucient to estimate

I+J≤N J≤N -1 Z I g LL ∂ 2 q Z J φ 1 (1 + |q|) I+J≤N J≤N -1 |Z I g LL ∂ q Z J+1 φ|. If I ≤ N 2 ≤ N -15
, we can estimate thanks to (5.8)

|Z I g LL | ε(1 + |q|) (1 + t) 3 2 -ρ so α 2 w 1 2 0 (1 + |q|) Z I g LL ∂ q Z J φ L 2 ε (1 + t) 3 2 -ρ α 2 w 1 2 0 ∂ q Z J φ L 2 .
(10.3)

If J ≤ N 2 , we can estimate |∂ q φ| ε (1 + |q|) 3 2 -4ρ √ 1 + t . Therefore, α 2 w 1 2 0 (1 + |q|) Z I g LL ∂ q Z J φ L 2 ε √ 1 + t α 2 w 0 (q) 1 2 (1 + |q|) 5 2 -4ρ Z I g LL L 2 ε √ 1 + t α 2 v(q) 1 2 1 + |q| Z I g LL L 2
, where

   v(q) = 1 (1+|q|) 5
2 -4ρ f or q < 0, v(q) = w 0 (q) (1+|q|) = (1 + |q|) 1+2δ f or q > 0.

We do not keep all the decay in q in the region q > 0 in order to be in the range of application of the weighted Hardy inequality and we obtain

α 2 w 1 2 0 (1 + |q|) Z I g LL ∂ q Z J φ L 2 ε √ 1 + t α 2 v(q) 1 2 ∂ q Z I g LL L 2 .
We use Proposition 5.1, which gives

∂ q Z N g LL ∼ ∂Z N ( g LL + g T T ). (10.4)
Consequently, thanks to Remark 4.10, we have ∂ q Z N g LL ∼ ∂Z N g 4 . Moreover, we calculate

w 2 (q) = 1+2µ (1+|q|) 2+2µ f or q < 0, w 2 (q) = (2 + 2δ)(1 + |q|) 1+2δ f or q > 0.
Therefore, v w 2 and we obtain

α 2 w 1 2 0 (1 + |q|) Z I g LL ∂ q Z J φ L 2 α 2 w 0 (q) 1 2 ∂Z N φ L 2 ε 1 + t α 2 w 0 (q) 1 2 ∂Z N φ 2 L 2 +ε α 2 w 2 (q) 1 2 ∂ Z N g 4 2 L 2 .
(10.5)

Estimate of the second term The second term contains only the crossed term, which occur only in the region q > 0. Thanks to the discussion of Section 2.5, it is sucient to estimate (2.13), which gives a contribution of the form

Z N (∂(g b ) U U ∂φ) . For I ≤ N -2 we have |Z I ∂(g b ) U U | ε1 q>0 r
and consequently

α 2 w 1 2 0 ∂Z I (g b ) U U ∂Z N -I φ L 2 ε 1 + t α 2 w 1 2 ∂Z N -I φ L 2 .
(10.6)

In ∂Z I (g b ) U U ∂ ρ Z N -I φ with I ≥ N -2,
we have to note the presence of terms of the form

χ(q)q∂ N +1 θ b(θ) r 2 ∂φ, (10.7)
which require a special treatment since ∂ N +1 θ b(θ) does not belong to L 2 . To deal with these terms we write

g χ(q)q∂ N -1 θ b g U U ∂φ - χ(q)q∂ N +1 θ b(θ) r 2 ∂φ χ(q) 1 + s   I≤2 |∂Z I φ|   |∂ N θ b| + |∂ N -1 θ b| + s.t.
We can estimate, thanks to the estimate (4.31) for ∂φ, w

1 2 0 ∂ χ(q)q∂φ∂ N -1 θ b L 2 ε √ 1 + s(1 + |q|) 1 2 +2σ-σ ∂ N θ b L 2 ε 2 . (10.8)
Therefore, we may perform the energy estimate for Z N φ = Z N φ -

χ(q)q∂φ∂ N -1 θ b g U U instead of Z N φ.
We are reduced to estimate

α 2 w 1 2 0 χ(q) 1 + s   I≤2 |∂Z I φ|   |∂ N θ b| + |∂ N -1 θ b| L 2 ε (1 + s) 3 2 (1 + |q|) 1 2 +σ |∂ N θ b| + |∂ N -1 θ b| L 2 ε 3 1 + t .
(10.9)

The other terms in ∂Z I (g b ) U U ∂Z N -I φ with I ≥ N -2, give contributions similar to (10.9).

Remark 10.7. We introduce the weight modulator α 2 to deal with the term (10.7) which is only present for I = N . It is no longer needed for I < N . To see this, let us estimate χ(q)q∂ N θ b(θ) r 2 ∂φ which is the analogue of (10.7) for I = N -1.

w 1 2 0 χ(q)q∂ N θ b(θ) r 2 ∂φ L 2 I≤2 w 1 2 0 ∂Z I φ L 2 χ(q)q∂ N θ b r 2 √ 1 + s 1 + |q| L 2 1 1 + t E 2 ∂ N θ b L 2 (S 1 ) ,
where we have used the weighted Klainerman-Sobolev inequality

|w 1 2 0 ∂φ| 1 √ 1 + s 1 + |q| I≤2 w 1 2 0 ∂Z I φ L 2 ,
and consequently

w 1 2 0 χ(q)q∂ N θ b(θ) r 2 ∂φ L 2 ε 2 1 + t E 2 .
(10.10)

Thanks to (10.3), (10.5), (10.6), (10.9) we obtain

d dt α 2 w 0 (q) 1 2 ∂ Z N φ 2 L 2 + αw (q) 1 2 ∂ Z N φ 2 L 2 ε 1 + t E N + ε α 2 w 2 (q) 1 2 ∂ g 4 2 L 2 + ε 3 1 + t
, (10.11) which, with the estimate (10.8) for Z N φ -Z N φ concludes the proof of Lemma 10.4.

Proof of Lemma 10.5. We now estimate h. The equation for Z N h writes

g Z I h = I+J≤N J≤N -1 Z I g αβ Z J ∂ α ∂ β h + I+J≤N J≤N -1 Z I H ρ Z J ∂ ρ h + Z I ((∂ q φ) 2 + (R b ) qq + Q LL (h, g)).
(10.12)

Estimate of the rst term Following Remark 2.2, it is sucient to estimate Z I g LL ∂ 2 q Z J h. For I ≤ N 2 , similarly than (10.3) we have

α 2 w 3 (q) 1 2 (1 + |q|) Z I g LL ∂ q Z J h L 2 ε (1 + t) 3 2 -ρ α 2 w 3 (q) 1 2 ∂ q Z J h L 2 .
(10.13)

For J ≤ N 2 , we have the estimate, thanks to (4.42),

|∂ q Z J h| ε

(1 + |q|) 3 2 -ρ , so α 2 w 3 (q) 1 2 (1 + |q|) Z I g LL ∂ q Z J h L 2 α 2 1 + |q| w 3 (q) (1 + |q|) 3-2ρ 1 2 Z I g LL L 2
.

We have 3-2ρ f or q < 0, (1 + |q|) 2δ-2ρ ≤ (1 + |q|) 1+2δ f or q > 0.

w 3 (q) (1 + |q|) 3-2ρ ≤ 1 (1+|q|)
This yields

w 3 (q) (1 + |q|) 3-2ρ w 2 (q).
Therefore the weighted Hardy inequality and the wave coordinate condition give, similarly than for (10.5),

α 2 w 3 (q) 1 2 (1 + |q|) Z I g LL ∂ q Z J h L 2 ε α 2 w 2 (q) 1 2 ∂ q Z I g LL L 2 ε α 2 w 1 2 2 (q) ∂ g 4 L 2 . (10.14)
Estimate of the second term The second term contains crossed terms, which can be studied exactly in the same way than for φ. Similarly than (10.6), we have for I ≤ N -2

α 2 w 1 2 3 ∂Z I (g b ) U U ∂Z N -I h L 2 ε 1 + t α 2 w 1 2 ∂Z N -I h L 2 .
(10.15)

Like for φ the following term require a special treatment :

χ(q)q∂ N +1 θ b(θ) r 2
∂h.

(10.16)

We have

g χ(q)q∂φ∂ N -1 θ b g U U ∂h - χ(q)q∂ N +1 θ b(θ) r 2 ∂h χ(q) 1 + s   I≤2 |∂Z I h|   |∂ N θ b| + |∂ N -1 θ b| + s.t.
We can estimate, thanks to the estimate (4.33) for ∂h,

α 2 w 1 2 3 ∂ χ(q)q∂h∂ N -1 θ b L 2 ε (1 + |q|) 1+2σ-σ ∂ N θ b L 2 ε 2 √ 1 + t. (10.17)
Therefore, we may perform the energy estimate for

Z N h = Z N h - χ(q)q∂h∂ N -1 θ b g U U instead of Z N h.
We are reduced to estimate

α 2 w 1 2 3 χ(q) 1 + s   I≤2 |∂Z I h|   |∂ N θ b| + |∂ N -1 θ b| L 2 ε (1 + s)(1 + |q|) 1+σ |∂ N θ b| + |∂ N -1 θ b| L 2 ε 3 √ 1 + t . (10.18)
The other terms in ∂Z I (g b ) U U ∂Z N -I h with I ≥ N -2, give contributions similar to (10.18).

Estimate of Z

N (∂ q φ) 2 We have α 2 w 3 (q) 1 2 Z N (∂ q φ) 2 L 2 I+J≤N α 2 w 3 (q) 1 2 ∂ q Z I φ∂ q Z J φ L 2 .
We can assume I ≤ N 2 and estimate thanks to (4.28)

|∂ q Z I φ| ε

1 + |q| √ 1 + t .
Then, since

w 1 2 3 1 + |q| ≤ w 1 2 0 ,
we obtain

α 2 w 3 (q) 1 2 Z N (∂ q φ) 2 L 2 ε √ 1 + t I≤N α 2 w 0 (q) 1 2 ∂ q Z J φ L 2 . (10.19) Estimate of Z N (R b ) qq Thanks to (1.7), the main contribution in (R b ) qq is ∂ 2 q (qχ(q))b(θ) r ,
which is supported in 1 ≤ q ≤ 2. We estimate

α 2 w 3 (q) 1 2 Z N b(θ)∂ 2 q (qχ(q)) r L 2 1 √ 1 + t I≤N ∂ I θ b L 2 (S 1 ) ε 2 √ 1 + t . (10.20) Estimate of Z N Q LL (h, g) We recall from (2.12) that Q LL (h, g) = ∂ L g LL ∂ L h + ∂ L g LL ∂ L h + ∂ L (g b ) U U ∂ L g LL .
The terms Z N (∂ L g LL ∂ L h) and Z N (∂ L g LL ∂ L h) may be treated in a similar way than the quasilinear term, giving contributions similar to (10.13) and (10.14). The term

Z N (∂ L (g b ) U U ∂ L g LL ) is a crossed term, hence it is supported only in the region q > 0. It is sucient to estimate ∂ L (g b ) U U ∂ L Z N g LL .
We have

|∂ q (g b ) U U | ε1 q>0 r ε1 q>0 √ 1 + t 1 + |q| .
so we can estimate

α 2 w 1 2 3 ∂ q (g b ) U U ∂Z N g LL L 2 ε √ 1 + t α 2 w 1 2 3 1 q>0 1 + |q| ∂Z N g LL L 2 ε √ 1 + t α 2 w 1 2 2 ∂Z N g LL L 2
, and consequently, since g LL = ( g 4 ) LL we have

α 2 w 1 2 3 ∂ q (g b ) U U ∂Z N g LL L 2 ε √ 1 + t α 2 w 1 2 2 ∂Z N g 4 L 2 . ( 10.21)
In view of (10.13), (10.14), (10.15), (10.18), (10. [START_REF] Lindblad | Global solutions of quasilinear wave equations[END_REF]), (10.20), (10.21), the energy inequality

yields d dt α 2 w 3 (q) 1 2 ∂ Z N h 2 L 2 + α 2 w 3 (q) 1 2 ∂ Z N h 2 L 2 ε 1 + t α 2 w 1 2 3 ∂Z N h L 2 + ε α 2 w 2 (q) 1 2 ∂Z N g 4 L 2 + ε √ 1 + t α 2 w 1 2 0 ∂ q Z J φ L 2 + α 2 w 1 2 2 ∂ q Z J g 4 L 2 + ε 2 √ 1 + t α 2 w 1 2 3 ∂ q Z N h L 2 + s.t.
We note that d dt

1 ε(1 + t) α 2 w 1 2 3 ∂Z N h 2 L 2 ≤ 1 ε(1 + t) d dt α 2 w 1 2 3 ∂Z N h 2 L 2
and we calculate

ε ε(1 + t) 2 α 2 w 1 2 3 ∂ q Z N h 2 L 2 ≤ ε 1 + t α 2 w 1 2 3 ∂ q Z J h 2 L 2 ε(1 + t) , ε ε(1 + t) α 2 w 1 2 2 (q) ∂Z N g 4 L 2 α 2 w 1 2 3 ∂ q Z N h L 2 ≤ √ ε α 2 w 1 2 2 (q) ∂ Z I g 4 2 L 2 + 1 √ ε(1 + t) 2 α 2 w 1 2 3 ∂ q Z N h 2 L 2 , ε ε(1 + t) 3 2 α 2 w 1 2 0 ∂ q Z J φ L 2 α 2 w 1 2 3 ∂ q Z N h L 2 ≤ √ ε 1 + t α 2 w 1 2 ∂ q Z J φ 2 L 2 + 1 √ ε(1 + t) 2 α 2 w 1 2 3 ∂ q Z N h 2 L 2 .
This yields d dt

1 ε(1 + t) α 2 w 3 (q) 1 2 ∂ Z N h 2 L 2 + 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂ Z N h 2 L 2 √ ε 1 + t E N + √ ε α 2 w 1 2 2 (q) ∂ Z I g 4 2 L 2 + ε 5 2
1 + t . The estimate for Z N k is totally similar. This, with the estimate (10.17) concludes the proof of Lemma 10.5.

Proof of Lemma 10.6. We now go to the estimate for Z N g 4 . We write g Z N g 4 = f µν . The energy estimate writes

d dt α 2 w 2 (q) 1 2 ∂Z N g 4 2 L 2 + α 2 w 2 (q) 1 2 ∂Z N g 4 2 L 2 α 2 w 2 (q) 1 2 f µν L 2 α 2 w 2 (q) 1 2 ∂Z N g 4 L 2 + ε (1 + t) α 2 w 2 (q) 1 2 ∂Z N g 4 2 L 2
We recall that the terms in f µν consist of the quasilinear terms, the terms coming from the non commutation of the wave operator with the null decomposition: it will be sucient to study the term Υ( r t ) 1 r 2 ∂ θ Z N h, the semi-linear terms: it is sucient to study the term Z N (g LL ∂ L g LL ∂ L h). We note that thanks to our decomposition, the term

Z N (∂ U g LL ∂ L h) is absent,
The crossed terms: their analysis is similar to the one for φ.

The quasilinear terms We consider

I+J≤N J≤N -1 Z I g LL ∂ 2 q Z J g 4 1 (1 + |q|) I+J≤N J≤N -1 Z I g LL ∂ q Z J+1 g 4 |. If I ≤ N 2 , we can estimate |Z I g LL | ε(1 + |q|) (1 + t) 3 2 -ρ , so α 2 w 1 2 2 (1 + |q|) Z I g LL ∂ q Z J g 4 L 2 ε (1 + t) 3 2 -ρ α 2 w 1 2 2 ∂ q Z J g 4 L 2 .
(10.23)

If J ≤ N 2 , we can estimate, thanks to Proposition 4.8 and since the dierence between g 4 and g 3 is contained in g LU , which is equal to ( g 3 ) LU ,

|∂ q Z J g 4 | ε 1 + |q| √ 1 + t . (10.24)
Therefore, if we apply Hardy inequality we obtain

α 2 w 1 2 2 (1 + |q|) Z I g LL ∂ q Z J g 4 L 2 ε √ 1 + t α 2 w 1 2 2 (1 + |q|) 1 2 ∂ q Z I g LL L 2
.

Thanks to (10.4) and the fact that

w 1 2 2 (1+|q|) 1 2 w 2 (q) 1 2 we obtain α 2 w 1 2 2 (1 + |q|) Z I g LL ∂ q Z J g 4 L 2 α 2 w 2 (q) 1 2 ∂Z N g 4 L 2 ε 1 + t w 2 (q) 1 2 ∂Z N g 4 2 L 2 + ε α 2 w 2 1 2 ∂ g 4 2 L 2 .
(10.25)

The term coming from the non commutation of the wave operator with the null decomposition We note that ∂ θ h r is a tangential derivative ∂h. Therefore

α 2 w 1 2 2 (q)Υ r t 1 r 2 ∂ θ Z N h L 2 1 1 + t α 2 w 1 2 2 ∂Z N h L 2
We calculate

w 3 (q) = 2µ 1 (1+|q|) 1+2µ f or q < 0, w 3 (q) = (3 + 2δ)(1 + |q|) 2+2δ f or q > 0.
Therefore w 2 w 3 and we obtain w

1 2 2 (q)Υ r t 1 r 2 ∂ θ Z N h L 2 1 1 + t α 2 w 3 (q) 1 2 ∂Z N h L 2 .
This yields

α 2 w 1 2 2 (q)Υ r t 1 r 2 ∂ θ Z N h L 2 α 2 w 2 (q)∂Z N g 4 L 2 1 √ ε(1 + t) α 2 w 3 (q) 1 2 ∂Z N h 2 L 2 + √ ε 1 + t α 2 w 2 (q) 1 2 ∂Z N g 4 2 L 2 .
(10.26)

The semi-linear terms We now estimate Z N (g LL ∂ L g LL ∂ L h). We rst estimate w 2 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 for I 1 + I 2 ≤ N and I 1 ≤ N -1. If I 1 ≤ N 2 we estimate | ∂Z I 1 g LL | 1 1 + s |Z I 1 +1 g LL | ε(1 + |q|) (1 + s) 5 2 -ρ (1 + |q|) ρ (1 + t) 3 2
. Therefore α 2 w 2 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε (1 + t) 3 2 α 2 w 2 (q) 1 2 (1 + |q|) ρ ∂Z I 2 h L 2 ε (1 + t) 3 2 α 2 w 3 (q) 1 2 ∂Z I 2 h L 2 .
If I 2 ≤ N 2 we estimate, thanks to (4.42)

|∂Z I 2 h| ≤ ε (1 + |q|) 3 2 -ρ , therefore α 2 w 2 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε α 2 w 1 2 2 (1 + |q|) 3 2 -ρ ∂Z I 1 g LL L 2 ε 1 + t α 2 w 1 2 2 (1 + |q|) 3 2 -ρ Z I 1 +1 g LL L 2 ε 1 + t α 2 w 1 2 2 (1 + |q|) 1 2 -ρ ∂Z I 1 +1 g LL L 2
where in the third inequality we have used the weighted Hardy inequality. Consequently α 2 w 2 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε 1 + t α 2 w 2 (q) 1 2 ∂Z I 1 +1 g 4 L 2 .
(10.27)

It is not possible to do the same reasoning for I 1 = N . To treat the term g LL ∂ L Z N g LL ∂ L h, which appears only in P LL we will write

g (hZ N g LL ) = D α D α (hZ N g LL ) = h g Z N g LL + Z N (g LL ) g h + g αβ ∂ α h∂ β Z N g LL . We estimate w 2 (q) 1 2 ∂(hZ N g LL ) L 2 ε w 2 (q) 1 2 ∂Z N g LL L 2 ,
(10.28) therefore, we can perform the energy estimate for

Z N g 4 = Z N g 4 -hZ N g LL - χ(q)q∂ g 4 ∂ N -1 θ b g U U instead of Z N g 4 ,
where the last term is here to deal with the troublesome crossed term which is the equivalent of (10.7). We have now to estimate h g Z N g LL + Z N (g LL ) g h + ∂Z N g LL ∂h. We estimate rst

α 2 w 2 (q) 1 2 ∂Z N g LL ∂h L 2 ε 1 + t w 2 (q) 1 2 ∂Z N g LL L 2 .
(10.29)

We have

g h = -2(∂ q φ) 2 + ∂ q h∂ q g LL + ... therefore | g h| ε 2 (1 + t)(1 + |q|) and α 2 w 2 (q) 1 2 Z N g LL g h L 2 ε 1 + t α 2 w 2 (q) 1 2 (1 + |q|) Z N g LL ε 1 + t w 2 (q) 1 2 ∂Z N g 4 L 2 .
(10.30)

To estimate the last term, we have to note that since g LL ∂ L Z N g LL ∂ L h appears only in P LL , it is absent from g Z N g LL . However, we have terms appearing from the non commutation of the wave operator with the null decomposition. They are of the form

1 r h ∂Z N g LL . We estimate α 2 w 2 (q) 1 2 1 r h ∂Z N g LL L 2 ε 1 + t α 2 w 2 (q) 1 2 ∂Z N g 4 L 2 (10.31)
The other terms in g Z N g LL have already been estimated.

Remark 10.8. This reasoning would not have been possible to treat terms of the form

∂ U g LL ∂ q h.
It is why we have introduced the function k, which is allowed to decay less.

Thanks to (10.23), (10.25), (10.26), (10.27), (10.29), (10.30), (10.31) the energy estimate yields

d dt α 2 w 2 (q) 1 2 ∂ Z N g 4 2 L 2 + α 2 w 2 (q) 1 2 ∂ Z N g 4 2 L 2 √ ε 1 + t α 2 w 2 (q) 1 2 ∂Z N g 4 2 L 2 + 1 ε(1 + t) α 2 w 1 2 3 ∂h 2 L 2 + √ ε 1 ε(1 + t) α 2 w 3 (q) 1 2 ∂Z N h 2 L 2 + α 2 w 2 (q) 1 2 ∂Z N g 4 2 L 2 + s.t. (10.32)
This, together with the estimates (10.28) concludes the proof of Lemma 10.6.

Estimates for

I ≤ N -2
Proposition 10.9. Let I ≤ N -2. We have the estimates

αw 0 (q) 1 2 ∂Z I φ L 2 ≤ C 0 ε + Cε 3 2 , αw(q) 1 2 ∂Z I h L 2 ≤ Cε 3 2 (1 + t), αw 2 (q) 1 2 ∂Z I g 3 L 2 ≤ C 0 ε + Cε 5 4 . Moreover t 0 αw 2 (q) 1 2 ∂Z I g 3 2 L 2 ε 2 .
We prove the proposition by using the energy estimate for φ, h and g 3 .

Proposition 10.10. Let I ≤ N -2. We have

d dt J≤I αw 0 (q) 1 2 ∂Z J φ 2 L 2 + J≤I αw 0 (q) 1 2 ∂Z J φ 2 L 2 ε 3 1 (1 + t) 1+σ-C √ ε + ε (1 + t) σ J≤I w 0 (q) 1 2 ∂Z J φ 2 L 2 .
Proposition 10.11. Let I ≤ N -2. We have the estimate

d dt J≤I αw 3 (q) 1 2 ∂Z J h 2 L 2 + J≤I αw 3 (q) ∂Z J h 2 L 2 ε 3 .
Proposition 10.12. Let I ≤ N -2. We have

d dt J≤I αw 2 (q) 1 2 ∂Z J g 3 2 L 2 + J≤I αw 2 (q) 1 2 ∂Z J g 3 2 L 2 ε 5 2 (1 + t) 1+σ + ε (1 + t) σ w 2 (q) 1 2 ∂Z I+1 g 3 2 L 2 .
We admit for the moment Propositions 10.10, 10.11 and 10.12 and prove Proposition 10.9.

Proof of Proposition 10.9. We estimate φ. Since σ > C √ ε for ε > 0 small enough, by integrating the inequality of Proposition 10.10 with respect to t we obtain

J≤I αw 0 (q) 1 2 ∂Z J φ 2 L 2 + t 0 J≤I αw 0 (q) 1 2 ∂Z J φ 2 L 2 dτ ≤ J≤I αw 0 (q) 1 2 ∂Z I φ(0) 2 L 2 + Cε 3 + C t 0 ε (1 + τ ) σ J≤I w 0 (q) 1 2 ∂Z J φ 2 L 2 dτ.
Thanks to Proposition 10.2, we have

t 0 ε (1 + τ ) σ J≤I w 0 (q) 1 2 ∂Z J φ 2 L 2 dτ ε 2 ,
and therefore

J≤I αw 0 (q) 1 2 ∂Z J φ 2 L 2 C 2 0 ε 2 + Cε 3 .
We now estimate h. We integrate the inequality of Proposition 10.11 with respect to t. We obtain, since we take zero initial data for h, and therefore, initial data for Z I h of size ε 2

J≤I αw 3 (q) 1 2 ∂Z J h 2 L 2 + t 0 J≤I αw 3 (q) ∂Z J h 2 L 2 dτ ε 3 (1 + t).
We now integrate the inequality of Proposition 10.12 to estimate g 3 . We obtain

J≤I αw 2 (q) 1 2 ∂Z J g 3 2 L 2 + t 0 J≤I αw 2 (q) 1 2 ∂Z J g 3 2 L 2 dτ ≤ J≤I αw 2 (q) 1 2 ∂Z J g 3 (0) 2 L 2 + Cε 5 2 + t 0 ε (1 + τ ) σ αw 2 (q) 1 2 ∂Z I+1 g 4 2 L 2 dτ. Proposition 10.2 yields t 0 1 (1 + τ ) σ w 2 (q) 1 2 ∂Z I+1 g 4 2 L 2 dτ ε 2 , Therefore I≤N -1 αw 2 (q) 1 2 ∂Z I g 3 2 L 2 + t 0 I≤N -1 αw 2 (q) 1 2 ∂Z I g 3 2 L 2 dτ ≤ C 2 0 ε 2 + Cε 5 2 .
This concludes the proof of Proposition 10.9.

Proof of Proposition 10.10. We follow the proof of Lemma 10.4. Let I ≤ N -1. We use the weighted energy estimate for the equation (10.2). It yields

d dt αw(q) 1 2 ∂Z I φ 2 L 2 + αw (q) 1 2 ∂Z I φ 2 L 2 g Z I φ L 2 αw(q) 1 2 ∂Z I φ L 2 + ε (1 + t) 1+σ w 1 2 ∂Z I φ 2 L 2 .
(10.33)

We rst estimate

I 1 +I 2 ≤I I 2 ≤I-1 Z I 1 g LL ∂ 2 q Z I 2 φ 1 (1 + |q|) I 1 +I 2 ≤I |Z I 1 g LL ∂ q Z I 2 φ|. If I 1 ≤ N 2 , we can estimate |Z I 1 g LL | ε(1 + |q|) (1 + t) 3 2 -ρ so αw 1 2 0 (1 + |q|) Z I 1 g LL ∂ q Z I 2 φ L 2 ε (1 + t) 3 2 -ρ αw 0 (q) 1 2 ∂ q Z I 2 φ L 2 .
(10.34)

If I 2 ≤ N 2 , we can estimate |∂ q Z I 2 φ| ε (1 + |q|) 3 2 -4ρ √ 1 + t , f or q < 0, |∂ q Z I 2 φ| ε (1 + |q|) 3 2 +δ-σ √ 1 + t , f or q > 0.
We apply the weighted Hardy inequality, but in order to be in its range, we cannot keep all the decay in q in the region q > 0.

αw 0 (q)

1 2

(1 + |q|) Z I 1 g LL ∂ q Z I 2 φ L 2 ε √ 1 + t v(q) 1 2 (1 + |q|) Z I g LL L 2 ε √ 1 + t v(q) 1 2 ∂ q Z I 1 g LL L 2
where v(q) = 1 (1+|q|) 3-4ρ f or q < 0, v(q) = w 0 (q) (1+|q|) 1+δ = (1 + |q|) 1+δ f or q > 0.

(10.35)

We use (10.4), which gives ∂ q Z I 1 g LL ∼ ∂Z I 1 g 4 so

|∂ q Z I 1 g LL | 1 1 + s |Z I 1 +1 g 4 | 1
(1 + t)

1 2 +σ (1 + |q|) 1 2 -σ |Z I 1 +1 g 4 |.
(10.36) Therefore, we obtain αw 0 (q)

1 2

(1 + |q|) Z I 1 g LL ∂ q Z I 2 φ L 2 ε (1 + t) 1+σ v(q) 1 2 (1 + |q|) 1 2 -σ Z I 1 +1 g 4 L 2 ε (1 + t) 1+σ v 1 2 (1 + |q|) 1 2 +σ ∂ q Z I 1 +1 g 4 L 2
where we have used again the weighted Hardy inequality. We calculate

v(q)(1 + |q|) 1+2σ = 1 (1+|q|) 2-4ρ-2σ f or q < 0, v(q)(1 + |q|) 1+2σ = (1 + |q|) 2+δ+2σ f or q > 0.
Therefore if 1 -4ρ -2σ ≥ µ and δ + 2σ < 2δ we have v(q)(1 + |q|) 1+2σ ≤ w 2 so we obtain, together with Proposition 10.2, αw 0 (q)

1 2 (1 + |q|) Z I 1 g LL ∂ q Z I 2 φ L 2 ε (1 + t) 1+σ w 2 (q) 1 2 ∂ q Z I 1 +1 g 4 L 2 ε 2 (1 + t) C √ ε (1 + t) 1+σ (10.37)
We now estimate the crossed terms, for which the weight modulator α has been introduced. They are of the form (2.13). It is sucient to estimate, for I ≤ N -1

ε1 q>0 r α(q)w 0 (q) 1 2 ∂Z I φ L 2
.

We obtain ε1 q>0 r α(q)w 0 (q)

1 2 ∂Z N -1 φ L 2 ε (1 + t) 1 2 +σ 1 q>0 (1 + |q|) 1 2 w 0 (q) 1 2 ∂Z N -1 φ L 2
. and consequently, since in the region q > 0 we have w 0 (q)

1 2 √ 1+|q| w 0 (q) 1 2 , ε1 q>0 r α(q)w 0 (q) 1 2 ∂Z N -1 φ L 2 αw 1 2 0 ∂Z I φ 2 L 2 ε (1 + t) 1+σ αw 1 2 0 ∂Z I φ 2 L 2 + ε (1 + t) σ w 0 (q) 1 2 ∂Z N -1 φ 2 L 2 .
(10.38)

The last term which appears in (10.33) can be estimated thanks to Proposition 10.2

ε (1 + t) 1+σ w 1 2 0 ∂Z I φ 2 L 2 ε 3 (1 + t) 1+σ-C √ ε . (10.39)
The estimates (10.33), (10.34), (10.37), (10.38) and (10.39), together with the bootstrap assumption (4.25) which imply αw 0 (q) 1 2 ∂Z I φ L 2 ε, conclude the proof of Proposition 10.10.

We now estimate h Proof of 10.11. The equation for Z I h is given by (10.12). We estimate rst αw

1 2 3 I 1 +I 2 ≤I I 2 ≤I-1 Z I 1 g αβ Z I 2 ∂ α ∂ β h L 2
.

As before, we can estimate, for I 1 ≤ N 2 , thanks to the bootstrap assumption (4.25), αw

(1 + |q|)

Z I 1 g LL ∂ q Z I 2 h L 2 ε (1 + t) 3 2 -ρ αw 1 2 3 ∂ q Z I 2 h L 2 ε 2 (1 + t) 1-ρ . (10.40) For I 2 ≤ N 2 , we have the estimate |∂ q Z I 2 h| ε (1 + |q|) 3 2 -ρ , f or q < 0, |∂ q Z I 2 h| ε (1 + |q|) 2+δ-σ , f or q < 0, so αw 1 2 3 (1 + |q|) Z I 1 g LL ∂ q Z I 2 h L 2 αv 1 2 (1 + |q|) Z I 1 g LL L 2
.

where v is dened by (10.35) and with Hardy inequality and the same reasoning than for φ αw

1 2 3 (1 + |q|) Z I 1 g LL ∂ q Z I 2 h L 2 ε v 1 2 ∂ q Z I 1 g LL L 2 ε 1 (1 + t) 1 2 +σ w 2 (q) 1 2 ∂Z I 1 +1 g 4 L 2 ,
and thanks to Proposition 10.2 we obtain αw 3 Z I (∂ q φ) 2

1 2 3 (1 + |q|) Z I 1 g LL ∂ q Z I 2 h L 2 ε 2 (1 + t) C √ ε (1 + t)
L 2 I 1 +I 2 ≤I αw 1 2 3 ∂ q Z I 1 φ∂ q Z I 2 φ L 2 ε √ 1 + t J≤I αw 1 2 0 ∂ q Z I φ L 2
so thanks to (4.25) we obtain αw 3 ∂Z I h

1 2 3 Z I (∂ q φ) 2 L 2 ε 2 √ 1 + t .
L 2 ε √ 1 + t αw 1 2 3 ∂Z I h ε 2 √ 1 + t .
3 ∂Z I h 2 L 2 + αw 3 (q) 1 2 ∂Z I h 2 L 2 ε 2 √ 1 + t αw 1 2 3 ∂ q Z I h L 2 + ε 1 + t αw 1 2 3 ∂ q Z I h 2 L 2 ε 3 ,
which concludes the proof of Proposition 10.11.

We now estimate g 3

Proof of Proposition 10.12. We write g Z I g 3 = f µν . The energy estimate yields

d dt αw 2 (q) 1 2 ∂Z I g 3 2 L 2 + αw (q) 1 2 ∂Z I g 3 2 L 2 αw 2 (q)f µν L 2 αw 2 (q)∂Z I g 3 L 2 + ε (1 + t) σ w 2 (q) 1 2 ∂Z I g 3 2 L 2 .
We recall that the terms in f µν are the quasilinear terms, the terms coming from the non commutation of the wave operator with the null decomposition: it will be sucient to study the term χ( r t ) 1 r 2 ∂ θ Z N h, the semi-linear terms: it is sucient to study the term Z I ∂ U g LL ∂ L h, the crossed term: their analysis is the same than for φ.

We rst estimate the quasilinear term.

I 1 +I 2 ≤I I 2 ≤I-1 Z I 1 g LL ∂ 2 q Z I 2 g 4 1 (1 + |q|) I 1 +I 2 ≤I |Z I 1 g LL ∂ q Z I 2 g 4 | If I 1 ≤ N 2 , we can estimate |Z I 1 g LL | ε(1 + |q|) (1 + t) 3 2 -ρ so αw 1 2 2 (1 + |q|) Z I 1 g LL ∂ q Z I 2 g 4 L 2 ε (1 + t) 3 2 -ρ αw 1 2 2 ∂ q Z I 2 g 4 L 2 (10.44) If I 2 ≤ N
2 , we can estimate, thanks (10.24) and (4.42)

|∂ q Z I 2 g 4 | ε √ 1 + t 1 + |q| 1 2 ε (1 + |q|) 3 2 -ρ 1 2 ε (1 + t) 1 4 (1 + |q|) 1-ρ 2 αw 1 2 2 (1 + |q|) Z I 1 g LL ∂ q Z I 2 g 4 L 2 ε t 1 4 αw 1 2 2 (1 + |q|) 2-ρ 2 Z I 1 g LL L 2 ε t 1 4 αw 1 2 2 (1 + |q|) 1-ρ 2 ∂Z I 1 g LL L 2
, where we have used the weighted Hardy inequality, noting that in the region q > 0 α 2 w 2 (1 + |q|) 2-ρ = (1 + |q|) 2δ-2σ-ρ , so the condition δ > σ + ρ + 1 2 ensure that we can apply the weighted Hardy inequality. We use the wave coordinate condition, (10.4) which gives ∂ q Z I 1 g LL ∼ ∂Z I 1 g 4 . We obtain

|∂ q Z I 1 g LL | 1 1 + s |Z I 1 +1 g 4 | 1
(1 + t)

3 4 +σ (1 + |q|) 1 4 -σ |Z I 1 +1 g 4 |.
It yields, by using Hardy inequality again αw 1 2

2

(1 + |q|)

Z I 1 g LL ∂ q Z I 2 g 4 L 2 ε (1 + t) 1+σ αw 1 2 2 (1 + |q|) 5 4 -ρ 2 -σ Z I 1 +1 g 4 L 2 ε (1 + t) 1+σ αw 1 2 2 (1 + |q|) 1 4 -ρ 2 -σ Z I 1 +1 g 4 L 2 .
Consequently, since

1 4 -ρ 2 -σ > 0 we have αw 1 2 2 (1 + |q|) Z I 1 g LL ∂ q Z I 2 g 4 L 2 ε (1 + t) 1+σ w 1 2 2 ∂Z I+1 g 4 L 2 ε 2 (1 + t) C √ ε (1 + t) 1+σ , (10.45)
where we have used Proposition 10.2. We now estimate the term coming from the non commutation with the wave operator αw

1 2 2 (q)Υ r t 1 r 2 ∂ θ Z I h L 2 .
On the support of Υ r t , we have r ∼ t and hence 1 r 2

1

(1 + t)

3 2 +σ (1 + |q|) 1 2 -σ . Therefore αw 1 2 2 (q)Υ r t 1 r 2 ∂ θ Z I h L 2 1 (1 + t) 3 2 +σ αw 1 2 2 (1 + |q|) 1 2 -σ Z I+1 h L 2 1 (1 + t) 3 2 +σ αw 1 2 2 (1 + |q|) 1 2 +σ ∂Z I+1 h L 2
, where we have applied the weighted Hardy inequality. We calculate α 2 w 2 (1 + |q|) 1+2σ = (1 + |q|) 2σ-2µ f or q < 0, (1 + |q|) 3+2δ f or q > 0.

If σ < µ we have α 2 w 2 (1 + |q|) 1+2σ ≤ w 3 and αw

1 2 2 (q)χ r t 1 r 2 ∂ θ Z I h L 2 1 (1 + t) 3 2 +σ w 3 (q) 1 2 ∂Z I+1 h L 2 ε 3 2 (1 + t) C √ ε (1 + t) 1+σ , (10.46)
where we have used Proposition 10.2 which yields, for I ≤ N -2

w 3 (q) 1 2 ∂Z I+1 h L 2 ε 3 2 (1 + t) 1 2 +C √ ε .
We now estimate Z I (∂ U g LL ∂ L h). We have

αw 2 (q) 1 2 Z I (∂ U g LL ∂ L h) L 2 I 1 +I 2 ≤I αw 2 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 . If I 1 ≤ N 2 we estimate | ∂Z I 1 g LL | 1 1 + s |Z I 1 +1 g LL | ε(1 + |q|) (1 + t) 5 2 -ρ (1 + |q|) ρ+σ (1 + t) 3 2 +σ
.

Therefore αw 2 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε (1 + t) 3 2 +σ αw 1 2 2 (1 + |q|) ρ+σ ∂Z I 2 h L 2 ε t 3 2 +σ αw 3 (q) 1 2 ∂Z I 2 h L 2 ,
and consequently αw 2 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε 2 t 1+σ . (10.47) If I 2 ≤ N 2 thanks to (4.42) we estimate |∂Z I 2 h| ≤ ε (1 + |q|) 3 2 -ρ , therefore αw 2 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε αw 1 2 2 (1 + |q|) 3 2 -ρ ∂Z I 1 g LL L 2 ε (1 + t) 1 2 +σ αw 1 2 2 (1 + |q|) 2-ρ-σ Z I 1 +1 g LL L 2 ε (1 + t) 1 2 +σ αw 1 2 2 (1 + |q|) 1-ρ-σ ∂Z I 1 +1 g LL L 2 ε (1 + t) 1 2 +σ αw 2 (q) 1 2 ∂Z I 1 +1 g 4 L 2
where in the last inequality we have used the wave coordinate condition. Therefore αw 2 (q) Proposition 10.13. We have for I ≤ N -8 and for I ≤ N -9

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 αw 2 (q) 1 2 ∂Z I g 3 ε (1 + t) σ αw 2 (q) 1 2 ∂Z I 1 +1 g 4 2 L 2 + ε 3 (1 + t) 1+σ .
w 1 (q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε(1 + t) Cε , (10.49) αw 1 (q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε + Cε
w(q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε(1 + t) Cε , (10.51) αw(q) 1 2 ∂Z I g 2 L 2 ≤ C 0 ε + Cε 3 2 . (10.52)
This is a consequence of the following two propositions. Proposition 10.14. We have for I ≤ N -8

d dt J≤I w 1 (q) 1 2 ∂Z J g 2 2 L 2 + J≤I w 1 (q) 1 2 ∂Z J g 2 2 L 2 ε 3 (1 + t) 1+σ + ε 1 + t J≤I w 1 (q) 1 2 ∂Z J g 2 2 L 2 , (10.53) and d dt J≤I αw 1 (q) 1 2 ∂Z J g 2 2 L 2 + J≤I αw 1 (q) 1 2 ∂Z J g 2 2 L 2 ε 3 (1 + t) 1+σ + ε αw 2 (q) 1 2 ∂Z I+1 g 3 2 L 2 .
(10.54) Proposition 10.15. We have for I ≤ N -9 Proof of Proposition 10.13. The inequalities (10.49) and (10.51) are straightforward consequences of (10.53) and (10.55). To prove (10.50), we integrate (10.54). We obtain

d dt J≤I w 0 (q) 1 2 ∂Z J g 2 2 L 2 + J≤I w 0 (q) 1 2 ∂Z J g 2 2 L 2 ε 3 (1 + t) 1+σ + + ε 1 + t J≤I w 0 (q) 1 2 ∂Z J g 2 2 L 2 , (10.55) and d dt J≤I αw 0 (q) 1 2 ∂Z J g 2 2 L 2 + J≤I αw 0 (q) 1 2 ∂Z J g 2 2 L 2 ε 3 (1 + t) 1+σ + ε αw 1 (q) 1 2 ∂Z I+1 g 2 2 L 2 .
J≤I αw 1 (q) 1 2 ∂Z J g 2 2 L 2 + t 0 J≤I αw 1 (q) 1 2 ∂Z J g 2 2 L 2 dτ ≤ J≤I αw 1 (q) 1 2 ∂Z J g 2 (0) 2 L 2 + Cε 3 + Cε t 0 αw 2 (q) 1 2 ∂Z I+1 g 3 2 L 2 dτ.
Thanks to Proposition 10.9, we have

t 0 αw 2 (q) 1 2 ∂Z I+1 g 3 2 L 2 ε 2 ,
and consequently

J≤I αw 1 (q) 1 2 ∂Z J g 2 2 L 2 + t 0 J≤I αw 1 (q) 1 2 ∂Z J g 2 2 L 2 dτ ≤ C 2 0 ε 2 + Cε 3 , (10.57)
which proves (10.50). To prove (10.52), we integrate (10.56)

J≤I αw 0 (q) 1 2 ∂Z J g 2 2 L 2 + J≤I αw 0 (q) 1 2 ∂Z J g 2 2 L 2 dτ ≤ J≤I αw 0 (q) 1 2 ∂Z J g 2 (0) 2 L 2 + Cε 3 + Cε t 0 αw 1 (q) 1 2 ∂Z I+1 g 2 2 L 2 dτ.
Thanks to (10.57), we have for I ≤ N -

9 t 0 αw 1 (q) 1 2 ∂Z I+1 g 2 2 L 2 dτ ε 2 ,
and consequently

J≤I αw 0 (q) 1 2 ∂Z J g 2 2 L 2 ≤ C 2 0 ε 2 + Cε 3 ,
which concludes the proof of Proposition 10.13.

Proof of Proposition 10.14. It is sucient to estimate the terms in the region q < 0, since in the region q > 0, we have w 0 = w 1 = w 2 so the estimates are strictly the same than in the previous section. Once again, the weight modulator α is used to tackle the crossed terms, which create a logarithmic loss in the estimates. However, in the region q < 0, since α = 1, we write everything with the weight w 1 , and do everything as if no terms were present in the region q > 0, since the inuence of these terms have already been tackled.

We rst estimate the term coming from the non commutation of the wave operator with the null decomposition, Υ r t 1 r 2 ∂ θ Z I (h 0 + h).

Since I + 1 ≤ N -7, we can use the Propositions 7.2 for Z I+1 h 0 and Proposition 7.5 for Z I+1 h.

We obtain (10.58) where we have used the calculation Υ r t

|Z I+1 (h 0 + h)| ε 2 (1 + |q|) 1 2 -ρ . Therefore Υ r t 1 r 2 ∂ θ Z I (h 0 + h) L 2 ε 2 (1 + t) 1+σ Υ r t 1 (1 + |q|) 1 2 -ρ r 1-σ L 2 ε 2 (1 + t) 1+σ ,
1 (1 + |q|) 1 2 -ρ r 1-σ 2 L 2 ≤ 2π Υ r t 2 1 (1 + |q|) 1-2ρ r 2-2σ rdr ≤ 2π dq (1 + |q|) 2-2ρ-2σ < +∞, if ρ + σ < 1 2 .
We now estimate Z I (∂ U g LL ∂ L h). We have

1 q<0 w 1 (q) 1 2 Z I (∂ U g LL ∂ L h) L 2 I 1 +I 2 ≤I 1 q<0 w 1 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 If I 1 ≤ N 2 we estimate | ∂Z I 1 g LL | 1 1 + s |Z I 1 +1 g LL | ε(1 + |q|) (1 + s) 5 2 -ρ (1 + |q|) ρ+σ (1 + t) 3 2 +σ
. Therefore 1 q<0 w 1 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε (1 + t) 3 2 +σ 1 q<0 (1 + |q|) ρ+σ (1 + |q|) 1 4 ∂Z I 2 h L 2 ε (1 + t) 3 2 +σ 1 q<0 w 3 (q) 1 2 ∂Z I 2 h L 2 if ρ + σ ≤ 1
4 , and consequently 1 q<0 w 1 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε 2 (1 + t) 1+σ . (10.59) If I 2 ≤ N 2 we estimate, thanks to (4.42) |∂Z I 2 h| ≤ ε (1 + |q|) 3 2 -ρ therefore 1 q<0 w 1 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε 1 q<0
(1 + |q|)

1 4 + 3 2 -ρ ∂Z I 1 g LL L 2
.

We estimate

| ∂Z I 1 g LL | 1 1 + s |Z I 1 +1 g LL | 1
(1 + t)

1 2 +σ (1 + |q|) 1 2 -σ |Z I 1 +1 g LL |.
We obtain 1 q<0 w 1 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε (1 + t) 1 2 +σ 1 q<0 (1 + |q|) 2+ 1 4 -ρ-σ Z I 1 +1 g LL L 2 ε (1 + t) 1 2 +σ 1 q<0 (1 + |q|) 1+ 1 4 -ρ-σ ∂Z I 1 +1 g LL L 2 ε (1 + t) 1 2 +σ 1 q<0 w 2 (q) 1 2 ∂Z I 1 +1 g 3 L 2
where in the last inequality we have used the wave coordinate condition, and the fact that, since for q < 0 w 2 (q) = 1 + 2µ (1 + |q|) 2+2µ , we have

1 (1 + |q|) 1+ 1 4 -ρ-σ ≤ w 2 (q) 1 2 if σ + ρ + µ ≤ 1 4 . Therefore 1 q<0 w 1 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 w 1 (q) 1 2 ∂Z I g 2 ε (1 + t) σ 1 q<0 w 2 (q) 1 2 ∂Z I 1 +1 g 3 2 L 2 + ε 3 (1 + t) 1+σ . (10.60)
In view of (10.58), (10.59) and (10.60), we conclude the proof of Proposition 10.14.

Proof of Proposition 10.15. We have already proved w(q)

1 2 Υ r t 1 r 2 ∂ θ Z I (h 0 + h) L 2 ε 2 (1 + t) 1+σ . (10.61) We now estimate Z I (∂ U g LL ∂ L h). We have w(q) 1 2 Z I (∂ U g LL ∂ L h) L 2 I 1 +I 2 ≤I w(q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 . If I 1 ≤ N 2 we use the estimate | ∂Z I 1 g LL | 1 1 + s |Z I 1 +1 g LL | ε(1 + |q|) (1 + s) 5 2 -ρ ε (1 + |q|) ρ+σ (1 + t) 3 2 +σ
.

Instead of estimating w(q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 we estimate w(q) 1 2 ∂Z I 1 g LL ∂Z I 2 (h 0 + h) L 2 and w(q) 1 2 ∂Z I 1 g LL ∂Z I 2 g 2 | L 2 .
We can also estimate since I 2 + 1 ≤ N -7, thanks to (4. [START_REF] Lindblad | Global solutions of quasilinear wave equations[END_REF]) and (4.21)

∂Z I 2 (h 0 + h) ε (1 + |q|) 3 2 -ρ . Therefore 1 q<0 w 0 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε 2 1 q<0 Υ r t (1 + |q|) ρ+σ (1 + s) 3 2 +σ (1 + |q|) 3 2 -ρ L 2 ε 2 (1 + t) 1+σ 1 q<0 1 √ 1 + s(1 + |q|) 3 2 -2ρ-σ L 2
and consequently w 0 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε 2 (1 + t) 1+σ . (10.62)
We estimate also w 0 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 g 2 L 2 ε (1 + t) 3 2 -ρ w 0 (q) 1 2 ∂Z I 2 g 2 L 2 ε 2
(1 + t) |∂Z I2 h| ≤ ε

(1 + |q|) therefore 1 q<0 w 0 (q) 1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 ε 1 q<0 (1 + |q|) 3 2 -ρ ∂Z I 1 g LL L 2 ε (1 + t) 1 2 +σ 1 q<0 (1 + |q|) 2-ρ-σ Z I 1 +1 g LL L 2 ε (1 + t) 1 2 +σ 1 q<0 (1 + |q|) 1-ρ-σ ∂Z I 1 +1 g LL L 2 ε (1 + t) 1 2 +σ w 1 (q) 1 2 ∂Z I 1 +1 g 2 L 2
where in the last inequality we have used the wave coordinate condition, and the fact that, since for q < 0 w 1 (q) = 1

2(1 + |q|) 3 2
, we have

1 (1 + |q|) 1-ρ-σ w 1 (q) 1 2 , if σ + ρ ≤ 1 4
. and q < 0. Therefore 1 q<0 w 0 (q)

1 2 ∂Z I 1 g LL ∂Z I 2 h L 2 w 0 (q) 1 2 ∂Z I g 2 ε 1 q<0 w 1 (q) 1 2 ∂Z I 1 +1 g 2 2 L 2 + ε 3 (1 + t) 1+2σ .

Improvement of the estimates for Πb

In order to conclude the proof of Theorem 1.12, it still remains to imrpove the bootstrap assumptions (4.4) and (4.5). To this end, we will set b (2) (θ) = Π Σ T,θ (∂ q φ) 2 rdq. There exists (φ (2) , g (2) ) solution of (1.1) in [0, T ] in the generalized wave coordinates H b [START_REF] Alinhac | An example of blowup at innity for a quasilinear wave equation[END_REF] , such that, if we write g (2) = g b 2 + g, then (φ (2) , g (2) ) satises the same estimate as (φ, g), and we have the estimates for b (2)

∂ I θ Πb (2) (θ) + Π Σ T,θ (∂ q φ (2) ) 2 rdq L 2 ≤ C ε 4 √ T , f or I ≤ N -4, ∂ I θ b(θ) L 2 ≤ 2C 2 0 ε 2 , f or I ≤ N.
The rest of this section is devoted to the proof of Proposition 11.1.

We solve the constraint equations with parameter b (2) . The initial data we obtain, constructed in Theorem 1.3 are of the form g = g b (2) + g (2) where we write

b (2) = b (2) + b (2) 0 + b (2) 1 cos(θ) + b (2) 2 sin(θ), with b (2) 0 , b (2)
1 , b

(2) 2 given by Theorem 1.3. We have the following estimates for the initial data at t = 0 g -g (2)

H N -3 δ + ∂ t g -∂ t g (2) H N -4 δ+1 b -b (2) W N -4,2 ≤ C 0 ε 2 √ T ,
thanks to (4.4), and g -g (2)

H N +1 δ + ∂ t g -∂ t g (2)
H N δ+1 ε 2 .

We solve, on an interval [0, T 2 ], the system (2.4) in generalized coordinates given by g b [START_REF] Alinhac | An example of blowup at innity for a quasilinear wave equation[END_REF] . We note (φ (2) , g (2) ) the solution.

We want to estimate the dierence between (φ (2) , g (2) ) and (φ, g). However, it will not be possible to estimate the dierence with the same norms than when we estimated φ and g. When we estimated h 0 we were able to use the condition

b + Π Σ T,θ (∂ q φ) 2 ε 2 √ T ,
to obtain decay in 1 √ 1+|q| for h 0 . However we want to keep the factor 1 √ T in the estimates of the dierence. To this end, we will loose the decay of h 0 -h

(2) 0 in 1 √ 1+|q|
and consequently in g -g (2) .

We will prove Proposition 11.1 with a bootstrap argument.

11.1 Bootstrap assumptions for φ (2) -φ and g (2) -g L ∞ estimates First some L ∞ estimates on φ -φ (2) .

|Z I (φ -φ (2) )| ≤ 2C 0 ε 2 √ T √ 1 + s(1 + q) 1 2 -2κ-5ρ
, f or I ≤ N -20, (11.2)

|Z I (φ -φ (2) )| ≤ 2C 0 ε 2 √ T (1 + s) 1 2 -2κ-2ρ
, f or I ≤ N -18.

(11.3)

We use the decompositions

g (2) = g b (2) + Υ r t (h (2) 0 + h (2) )dq 2 + g (2) 2 , (11.4)
where h

(2) 0 satises the transport equation

∂ q h (2) 0 = -2r ∂ q φ (2) 2 -2b (2) (θ)∂ 2 q (χ(q)q), h (2)
0 | t=0 = 0, and h (2) satises the linear wave equation

h (2) = h (2) 0 + g (2) LL ∂ 2 q h (2) 0 + 2 ∂ q φ (2) 2 -2(R b (2) ) qq + Q LL (h (2)
0 , g (2) ), ( h (2) , ∂ t h (2) )| t=0 = (0, 0),

We assume the following estimates on h 0 -h

(2) 0 for I ≤ N -12 |Z I (h 0 -h (2) 0 )| ≤ 2C 0 ε 2 √ T . (11.5)
We introduce the two weight modulators β 1 (q) = 1, q > 0, β 1 (q) = 1 (1+|q|) κ , q < 0, and β 2 (q) = 1, q > 0, β 2 (q) = 1 (1+|q|) 2κ , q < 0, with 0 < κ 1. We assume for I ≤ N -15

β 1 w 1 2 0 ∂Z I ( g 2 -g (2)
2 )

L 2 ≤ 2C 0 ε 2 √ T (1 + t) ρ (11.6) αβ 1 w 1 2 ∂Z I ( g 2 -g (2)
2 )

L 2 ≤ 2C 0 ε 2 √ T (11.7)
and

β 2 w 1 2 1 ∂Z N -14 g 2 -g (2) 2 L 2 ≤ 2C 0 ε 2 √ T (1 + t) ρ (11.8) αβ 2 w 1 2 1 ∂Z N -14 g 2 -g (2) 2 L 2 ≤ 2C 0 ε 2 √ T . (11.9)
We use the decomposition

g (2) = g b (2) + Υ r t h (2) dq 2 + g (2) 3 , (11.10)
where h (2) is the solution of 2) , g (2) ), (h (2) ∂ t h (2) )| t=0 = (0, 0).

g (2) h (2) = -2(∂ q φ (2) ) 2 + 2(R b (2) ) qq + Q LL (h (
We assume for I ≤ N -6

αβ 2 w 1 2 0 ∂Z I φ -φ (2) L 2 + αβ 2 w 1 2 2 ∂Z I g 3 -g (2) 3 L 2 + 1 √ 1 + t αβ 2 w 1 2 3 ∂Z I h -h (2) L 2 ≤ 2C 0 ε 2 √ T , (11.11)
and for I ≤ N -5

αβ 2 w 1 2 0 ∂Z I φ -φ (2) L 2 + αβ 2 w 1 2 2 ∂Z I g 3 -g (2) 3 L 2 + 1 √ 1 + t αβ 2 w 1 2 3 ∂Z I h -h (2) L 2 ≤ 2C 0 ε 2 √ T (1 + t) ρ .
(11.12)

We use the decomposition

g (2) = g b (2) + Υ r t h (2) dq 2 + Υ r t k (2) rdqdθ + g (2)
4 ,

where k (2) is the solution of

g k (2) = ∂ U g (2)
LL ∂ q h (2) , (h (2) , ∂ t h (2) )| t=0 = (0, 0).

We assume for I ≤ N -4

α 2 β 2 w 1 2 0 ∂Z I φ -φ (2) L 2 + α 2 β 2 w 1 2 2 ∂Z I g 3 -g (2) 3 L 2 + 1 √ 1 + t α 2 β 2 w 1 2 3 ∂Z I h -h (2) L 2 + 1 √ 1 + t α 2 β 2 w 1 2 3 ∂Z I k -k (2) L 2 ≤ 2C 0 ε 2 √ T (1 + t) ρ . (11.13)
To improve the estimates, we follow the same steps than when we imrpoved the bootstrap assumptions of Section 4. The dierence of our new bootstrap assumptions compared with the estimates of Section 4 is at worse a factor

ε √ 1+|q| √ T
in the region q < 0. In the region q > 0 the decay is the same and we have won a factor ε √ T

. Therefore we can restrict our study to the region q < 0: we will perform our estimates as if no term was present in the region q > 0. We will follow the same steps as before, but with much less details since the mechanisms are the same.

Remark 11.2. As long as the bootstrap estimates for φ (2) -φ and g (2) -g are satised, φ (2) and g (2) satisfy the same estimates as φ and g. L ∞ estimates using the weighted Klainerman-Sobolev inequality The following estimates are a direct consequence of the bootstrap assumptions and the weighted Klainerman-Sobolev inequality. For I ≤ N -8 we have

∂Z I φ (2) -φ ε 2 √ T √ 1 + t(1 + |q|) 1 2 -2κ , (11.14)
∂Z I g

(2)

3 -g 3

ε 2 (1 + |q|) 1 2 +µ+2κ √ T √ 1 + s , (11.15
)

∂Z I h (2) -h ε 2 √ T (1 + |q|) 1 2 -2κ , (11.16)
and for I ≤ N -17

∂Z I g (2) 2 -g 2 ε 2 (1 + |q|) κ √ T √ 1 + s 1 + |q| .
(11.17)

Improvement of the estimate of h

0 -h (2) 0 and h (2) -h 0 Estimate of h 0 -h (2) 0
The quantity h 0 -h

(2) 0 satises the transport equation

∂ q h (2) 0 -h 0 = -2r ∂ q φ (2) 2 -(∂ q φ) 2 -2 b (2) (θ) -b(θ) ∂ 2 q (χ(q)q), (h (2) 0 -h 0 )| t=0 = 0.
We write this equation under the form

∂ q h (2) 0 -h 0 = -2r ∂ q φ (2) + ∂ q φ ∂ q φ (2) -∂ q φ -2 b (2) (θ) -b(θ) ∂ 2 q (χ(q)q).
For k + l ≤ N -7, k ≥ 1, the equivalent of estimate (7.12), that we obtain using (11.

√ 1+|q| √ T . ∂ k q ∂ l θ h 0 -h (2) 0 ε 3 √ T (1 + |q|) k+ 1 2 -4ρ
.

We obtain the estimate for k = 0 by integrating the previous one with respect to q. We obtain, for l ≤ N -8

|∂ l θ h 0 | ε 3 √ T .
For k + l + j ≤ N -8, k ≥ 1 and j ≥ 1 the equivalent of (7.17) is

∂ j s ∂ k q ∂ l θ h 0 -h (2) 0 ε 3 √ T (1 + s) j+ 1 2 (1 + |q|) k-4ρ
.

Consequently we have proved that for I ≤ N -8 we have

Z I h 0 -h (2) 0 ε 3 √ T . (11.18)
Estimation of h (2) -h 0 The quantity h (2) -h 0 satises the linear equation

           h (2) -h = h (2) 0 -0 + 2 ∂ q φ (2) 2 -(∂ q φ) 2 -2(R b (2) ) qq + 2(R b ) qq + g (2) LL ∂ 2 q h (2) 0 -g LL ∂ 2 q h 0 + Q LL (h (2)
0 , g (2) ) -Q LL (h 0 , g), h (2) -h, ∂ t h (2) -h | t=0 = (0, 0).

Proceeding as for the estimate of (7.26), and in view of the bootstrap assumptions for φ -φ (2) and g -g (2) we obtain the analogue of (7.26) for Z I h (2) -Z I h , where the corresponding right-hand side gets multiplied by ε √ 1+|q| √ T

. We obtain, for I ≤ N -10 and q < 0

Z I h (2) -Z I h ε 3 √ T (1 + s) 3 2 (1 + |q|) 1 2
.

Therefore if we perform the weighted energy estimate we obtain

d dt w 1 2 ∂ Z I h (2) -Z I h L 2 ε 3 √ T (1 + s) 3 2 (1 + |q|) 1 2 L 2 ε 3 ln(1 + t) √ T (1 + t) ,
and therefore for I ≤ N -10 we have

w 1 2 0 ∂ Z I h (2) -Z I h L 2 ε 3 √ T (1 + t) ρ . (11.19)
The weighted Klainerman-Sobolev inequality yields, for I ≤ N -12

∂ Z I h (2) -Z I h ε 3 (1 + t) ρ √ T √ 1 + s 1 + |q| . (11.20)
11.3 Improvement of the L ∞ estimate for φ -φ (2) We write the equation satised by φ (2) -φ 2) .

g φ -φ (2) = g (2) αβ -g αβ ∂ α ∂ β φ (2) + (H b (2) -H b ) ρ ∂ ρ φ (
We limit ourselves to the region q < 0. We estimate for I + J ≤ N -20

Z I g (2)
LL -g LL Z J ∂ 2 φ.

With the wave coordinate condition and the estimate (11.17), we obtain, for I ≤ N -17

Z I g (2) LL -g LL ε 2 (1 + |q|) 3 2 +κ (1 + s) 3 2
.

(11.21)

Moreover we have, for J ≤ N -20 thanks to (4.15)

|Z J ∂ 2 φ| 1 (1 + |q|) 2 |Z J+2 φ| ε 2 √ T √ 1 + s(1 + |q|) 5 2 -4ρ . Consequently Z I g (2) LL -g LL Z J ∂ 2 φ ε 3 √ T (1 + s) 2 (1 + |q|) 1-4ρ-κ ε 3 √ T (1 + s) 2-5ρ-κ (1 + |q|) 1+ρ .
We now estimate for I + J ≤ N -20 2) .

Z I g LL Z J ∂ 2 φ -φ (
We have, thanks to (5.8) and (11.3)

|Z I g LL | ε(1 + |q|) (1 + s) 3 2 -ρ , Z J ∂ 2 φ -φ (2) 1 (1 + |q|) 2 Z J+2 φ -φ (2) ε 2 √ T (1 + s) 1 2 -2ρ-2κ (1 + |q|) 2 . Consequently Z I g LL Z J ∂ 2 φ -φ (2) ε 3 (1 + s) 2-5ρ-2κ (1 + |q|) 1+ρ
and the L ∞ -L ∞ estimate yields for I ≤ N -20, since the initial data for φ -φ (2) are zero :

Z I φ -φ (2) ≤ Cε 3 √ T (1 + s) 1 2 (1 + |q|) 1 2 -5ρ-2κ . ( 11
Z I g (2) LL -g LL Z J ∂ 2 φ ε 2 (1 + |q|) 3 2 +κ √ T (1 + s) 3 2 ε (1 + |q|) 5 2 -4ρ (1 + s) 1 2 ε 3 √ T (1 + s) 2-5ρ-κ (1 + |q|) 1+ρ , Z I g LL Z J ∂ 2 φ -φ (2) ε(1 + |q|) (1 + s) 3 2 -ρ ε 2 (1 + |q|) 2κ √ T (1 + |q|) 3 2 √ 1 + s ε 3 √ T (1 + s) 2-ρ (1 + |q|) 1 2 -2κ
.

Consequently, for I ≤ N -18 and q < 0 we have

Z I φ -φ (2) ε 3 √ T (1 + s) 3 2 -ρ-2κ (1 + |q|)
and Lemma 7.6 yields, for I ≤ N -18, since the initial data for φ -φ (2) are zero.

Z I φ -φ (2) ≤ Cε 3 √ T (1 + s) 1 2 -2ρ-2κ . ( 11
( g 2 ) µν -g (2) 2 µν = f µν ,
where the terms in f µν are the terms coming from the non commutation of the null decomposition with the wave operator: it is sucient to study the term Υ r t 1

r 2 ∂ θ h (2) 0 -h 0 + h (2) -h , the semi-linear terms: it is sucient to study ∂ L (h)∂ U g LL -g (2)
LL and ∂ U g LL ∂ L h (2) -h , the quasilinear terms: it is sucient to study the terms g LL ∂ 2 L g (2) -g and g

LL -g LL ∂ 2 L g, the crossed terms: they do not occur in the region q < 0.

We estimate the rst term. Thanks to (11.20) and (11.18) we have, for I ≤ N -15

∂ θ Z I h (2) 0 -h 0 + h (2) -h ε 3 (1 + |q|) ρ √ T .
Therefore, we can estimate in the region q < 0,

β 1 w 1 2 0 Υ r t 1 r 2 ∂ θ Z I h (2) 0 -h 0 + h (2) -h L 2 Υ r t ε 3 √ T (1 + s) 2 (1 + |q|) κ-ρ L 2
and consequently

β 1 w 1 2 0 Υ r t 1 r 2 ∂ θ Z I h (2) 0 -h 0 + h (2) -h L 2 ε 3 √ T (1 + t) 1+κ-ρ .
(11.24)

We now estimate the semi-linear terms. For I ≤ N -13, we have, thanks to (4.43)

|∂ L (Z I h)| ε (1 + |q|) 3 2 -2ρ
, Therefore we can estimate, for I + J ≤ N -15 in the region q < 0

β 1 w 1 2 0 Z I ∂ L (h)Z J ∂ U g LL -g (2) LL L 2 ε (1 + |q|) 3 2 -2ρ+κ (1 + s) Z J+1 g LL -g (2) LL L 2 ε (1 + |q|) 1 2 -2ρ+κ (1 + s) ∂Z J+1 g LL -g (2) LL L 2 ε (1 + t) 1 2 +σ 1 (1 + |q|) 1-2ρ+κ-σ ∂Z J+1 g 2 -g (2) 2 L 2
and consequently

β 1 w 1 2 0 Z I ∂ L (h)Z J ∂ U g LL -g (2) LL L 2 ε (1 + t) 1 2 +σ β 2 w 1 (q) 1 2 ∂Z J+1 g 2 -g (2) 2 L 2 , (11.25)
where we have used the wave coordinate condition and the fact that, for q < 0

β 2 w 1 (q) 1 2 = 1 4(1 + |q|) 34+2κ ≥ 1 (1 + |q|) 1-2ρ+κ-σ .
For I ≤ N -14 thanks to Proposition 8.6, we have

|Z I ∂ U g LL | ε(1 + |q|) (1 + s) 5 2 -2ρ
.

In order to estimate

β 1 w 1 2 0 ∂ U Z I g LL ∂ L Z J h (2) -h L 2
we will perform the estimates with h (2) -h replaced by h

(2) 0 -h 0 , h (2) -h and g

2 -g 2 . We estimate, in the region q < 0, thanks to (11.18),

β 1 w 1 2 0 ∂ U Z I g LL ∂ L Z J h (2) 0 -h 0 L 2 ε 3 √ T ε(1 + |q|) 1-κ (1 + s) 5 2 -2ρ (1 + |q|) L 2 ε 3 √ T (1 + t) 3 2
, thanks to (11.19)

β 1 w 1 2 0 ∂ U Z I g LL ∂ L Z J h (2) -h L 2 ε ε(1 + |q|) 1-κ (1 + s) 5 2 -2ρ ∂ L Z J h (2) -h L 2 ε 3 √ T (1 + t) 3 2 +κ-2ρ
, and thanks to (11.6)

β 1 w 1 2 0 ∂ U Z I g LL ∂ L Z J g (2) 2 -g 2 L 2 ε ε(1 + |q|) 1-κ (1 + s) 5 2 -2ρ ∂ L Z J g (2) -g L 2 ε 3 √ T (1 + t) 3 2 -ρ . β 1 w 1 2 0 ∂ U Z I g LL ∂ L Z J h (2) -h L 2 ε 3 √ T (1 + t) 3 2 -ρ . (11.26)
The other terms are similar to estimate. Thanks to (11.24), (11.25) and (11.26), the energy inequality yields for I ≤ N -15

d dt β 1 w 1 2 0 ∂Z I g (2) 2 -g 2 2 L 2 + β 1 w 0 (q) 1 2 ∂Z I g (2) 2 -g 2 2 L 2 ε 3 √ T (1 + t) 1+σ β 1 w 1 2 0 ∂Z I g (2) 2 -g 2 L 2 + ε (1 + t) σ β 2 w 1 (q) 1 2 ∂Z I+1 g 2 -g (2) 2 2 L 2 + ε (1 + t) 1+σ β 1 w 1 2 0 ∂Z I g (2) 2 -g 2 2 L 2 . (11.27) L 2 estimate for ∂Z I g (2)
2 -g 2 with I ≤ N -14. We follow the same steps as in the previous paragraph. First we still have

β 2 w 1 2 1 Υ r t 1 r 2 ∂ θ Z I h (2) 0 -h 0 + h (2) -h L 2 Υ r t ε 3 √ T (1 + s) 2 (1 + |q|) 1 4 +2κ-ρ L 2 ε 3 √ T (1 + t) 5 4
.

We estimate the second terms for I + J ≤ N -14

β 2 w 1 2 1 Z I ∂ L (h)Z J ∂ U g LL -g (2) LL L 2 ε (1 + |q|) 3 2 -2ρ+ 1 4 +2κ (1 + s) Z J+1 g LL -g (2) LL L 2 ε (1 + |q|) 3 4 -2ρ+2κ (1 + s) ∂Z J+1 g LL -g (2) LL L 2 ε (1 + t) 1 2 +σ 1 (1 + |q|) 5 4 -2ρ+2κ-σ ∂Z J+1 g 3 -g (2) 3 L 2 ε (1 + t) 1 2 +σ β 2 w 2 (q) 1 2 ∂Z J+1 g 3 -g (2) 3 L 2 ,
where we have used the wave coordinate condition and the fact that

β 2 w 2 (q) 1 2 = 1 (1 + |q|) 1+2κ+µ ≥ 1 (1 + |q|) 5 4 -2ρ+2κ-σ .
The other terms are similar to estimate than for I ≤ N -15. The energy inequality yields for I ≤ N -14

d dt β 2 w 1 2 1 ∂Z I g (2) 2 -g 2 2 L 2 + β 2 w 1 (q) 1 2 ∂Z I g (2) 2 -g 2 2 L 2 ε 3 √ T (1 + t) 1+σ β 2 w 1 2 1 ∂Z I g (2) 2 -g 2 L 2 + ε (1 + t) σ β 2 w 2 (q) 1 2 ∂Z I+1 g 3 -g (2) 3 2 L 2 + ε (1 + t) 1+σ β 2 w 1 2 1 ∂Z I g (2) 2 -g 2 2 L 2 .
(11.28) L 2 estimates for ∂Z I φ (2) -φ with I ≤ N -6. We estimate for I + J ≤ N -6, J ≤ N -7,

β 2 w 1 2 0 Z I g (2) LL -g LL ∂ 2 q Z J φ L 2 . If I ≤ N -7
2 we can estimate, thanks to (11.21)

Z I g (2) LL -g LL ε 2 (1 + |q|) 3 2 +κ √ T (1 + s) 3 2
, and therefore, if we restrict our quantities to q < 0

β 2 w 1 2 0 Z I g (2) LL -g LL ∂ 2 q Z J φ L 2 ε 2 (1 + |q|) 3 2 +κ √ T (1 + s) 3 2 (1 + |q|) 1+2κ ∂Z J+1 φ L 2 ε 2 √ T (1 + t) 1+κ w 1 2 ∂Z J+1 φ L 2 ε 3 √ T (1 + t) 1+κ .
The case J ≤ N -6

2 can be treated as in Section 10.2.

We now evaluate

β 2 w 1 2 0 Z I g LL ∂ 2 q Z J φ (2) -φ L 2
for I + J ≤ N -6 and J ≤ N -6

2 . We have, since

N -6 2 + 2 ≤ N -20 ∂ 2 q Z J φ (2) -φ ε 2 √ T √ 1 + s(1 + |q|) 5 2 -5ρ-2κ
.

Therefore we can estimate

β 2 w 1 2 0 Z I g LL ∂ 2 q Z J φ (2) -φ L 2 ε 2 √ T √ 1 + s(1 + |q|) 5 2 -5ρ Z I g LL L 2 ε 2 √ T √ 1 + s(1 + |q|) 3 2 -5ρ ∂Z I g LL L 2 ε 2 √ T (1 + s) 3 2 (1 + |q|) 3 2 -5ρ Z I+1 g 3 L 2 ε 2 √ T (1 + s) 3 2 (1 + |q|) 1 2 -5ρ ∂Z I+1 g 3 L 2 ε 2 √ T (1 + t) 3 2 -5ρ-µ w 1 2 2 ∂Z I+1 g 3 L 2 ε 3 √ T (1 + t)
0 ∂Z I φ (2) 2 -φ 2 L 2 + β 2 w 0 (q) 1 2 ∂Z I φ (2) 2 -φ 2 L 2 ε 3 √ T (1 + t) 1+κ β 2 w 1 2 ∂Z I φ (2) 2 -φ L 2 .
(11.29)

Consequently, since the initial data for φ

2 -φ are zero we have

β 2 w 1 2 0 ∂Z I φ (2) 2 -φ L 2 ε 3 √ T .
(11.30)

L 2 estimates for ∂Z I h (2) -h with I ≤ N -6. We write the equation satised by h 2) , g (2) ).

(2) -h g h -h (2) = 2(∂ q φ (2) ) 2 -2(∂ q φ) 2 + 2(R b ) qq -2(R b (2) ) qq + Q LL (h, g) -Q LL (h (
We rst estimate for I + J ≤ N -6 and I ≤ N -6

2 . We recall that we restrict all the quantities to q < 0 (therefore w 3 = w 0 ).

β 2 w 1 2 0 ∂ q Z I φ -φ (2) ∂ q Z J φ L 2 ε 2 √ T √ 1 + s(1 + |q|) 3 2 -5ρ ∂ q Z J φ L 2 ε 2 √ T √ 1 + t w 1 2 0 ∂ q Z J φ L 2 ε 3 √ T √ 1 + t .
We now estimate the quasilinear term

β 2 w 1 2 0 Z I g (2) LL -g LL ∂ 2 q Z J h L 2
for I + J ≤ N -6 and I ≤ N -6

2 . We have

β 2 w 1 2 0 Z I g (2) LL -g LL ∂ 2 q Z J h L 2 ε 2 (1 + |q|) 3 2 +κ √ T (1 + s) 3 2 (1 + |q|) 1+2κ ∂Z J+1 h L 2 ε 2 √ T (1 + t) 1+κ w 1 2 0 ∂Z J+1 h L 2 ε 3 √ T (1 + t) 1 2 +κ
.

The other terms can be treated as in the proof of Proposition 10.11. The energy inequality yields

d dt β 2 w 1 2 0 ∂Z I h (2) 2 -h 2 L 2 + β 2 w 0 (q) 1 2 ∂Z I h (2) 2 -h 2 L 2 ε 3 √ T (1 + t) 1 2 β 2 w 1 2 0 ∂Z I h (2) 2 -h L 2
.

Consequently, since the initial data for h

2 -h are zero, we have

β 2 w 1 2 0 ∂Z I h (2) 2 -h L 2 ε 3 √ 1 + t √ T . (11.31)
L 2 estimates for ∂Z I g (2) -g with I ≤ N -6. As usual we estimate the following contributions the terms coming from the non commutation of the null decomposition with the wave operator: it is sucient to study the term Υ r t 1

r 2 ∂ θ h (2) -h , the semi-linear terms: it is sucient to study ∂ L (h)∂ U g LL -g (2)
LL and ∂ U g LL ∂ L h (2) -h , the quasilinear terms: it is sucient to study the terms g LL ∂ 2 L g (2) -g and g

(2)

LL -g LL ∂ 2 L g.

We estimate the rst term. We recall that we restrict all the quantities to q < 0.

β 2 w 1 2 2 χ r t 1 r 2 ∂ θ Z I h (2) -h L 2 β 2 1 (1 + s) 2 (1 + |q|) 1 2 +µ Z I+1 h (2) -h L 2
β 2 1

(1 + s) .

We estimate the second term

β 2 w 1 2 1 Z I ∂ L (h)Z J ∂ U g LL -g (2) LL L 2
for I + J ≤ N -6 and J ≤ N -6

2 . We have, thanks to (11.21) Z J ∂ U g LL -g

(2) LL

1 1 + s Z J+1 g LL -g (2) LL ε 2 (1 + |q|) 3 2 +κ √ T (1 + s) 5 2
.

Therefore we can estimate

β 2 w 1 2 2 Z I ∂ L (h)Z J ∂ U g LL -g (2) LL L 2 ε 2 (1 + |q|) 3 2 +κ √ T (1 + s) 5 2 (1 + |q|) 1 2 +µ+2κ ∂ L Z I h L 2 ε 2 √ T (1 + s) 3 2 +µ+κ ∂ L Z I h L 2 ε 2 √ T (1 + t) 3 2 +µ+κ w 1 2 ∂ L Z I h L 2 ε 3 √ T (1 + t) 1+µ+κ .
The other terms are treated as in the proof of Proposition 10.12. We have proved, when we restrict ourselves to q < 0 d dt β 2 w 1 2

2 ∂Z I g

(2)

3 -g 3 2

L 2 + β 2 w 2 (q) 1 2 ∂Z I g (2)
3 -g 3 2

L 2 ε 3 √ T (1 + t) 1+σ + ε (1 + t) 1 2 +σ β 2 w 2 (q) 1 2 ∂Z J+1 g 4 -g (2) 4 L 2 + 1 (1 + t) 3 2 +σ β 2 w 1 2 ∂Z I+1 h (2) -h L 2 β 2 w 1 2
2 ∂Z I g

(2)

3 -g 3 L 2

.

(11.32)

L 2 estimates for I ≤ N -4 We can prove, following Section 10.1 that, since we do as if no quantity was present for q > 0,

d dt β 2 w 1 2 ∂Z I φ -φ (2) 2 L 2 + β 2 w 1 2
2 ∂Z I g 4 -g

(2) 4 2

L 2 + 1 ε(1 + t) β 2 w 1 2 3 ∂Z I h -h (2) 2 L 2 + 1 ε(1 + t) β 2 w 1 2 3 ∂Z I k -k (2) 2 L 2
+ β 2 w 0 (q) 1 2 ∂Z I φ -φ (2) 2 L 2 + β 2 w 2 (q) 1 2 (q) ∂Z I g 4 -g

(2) 4

2

L 2 + 1 ε(1 + t) β 2 w 3 (q) 1 2 ∂Z I h -h (2) 2 L 2 + 1 ε(1 + t) β 2 w 3 (q) 1 2 ∂Z I k -k (2) 2 L 2 . √ ε 1 + t β 2 w 1 2 ∂Z I φ -φ (2) 2 L 2 + β 2 w 1 2
2 ∂Z I g 4 -g

(2) 4

2

L 2 + 1 ε(1 + t) β 2 w 1 2 3 ∂Z I h -h (2) 2 L 2 + 1 ε(1 + t) β 2 w 1 2 3 ∂Z I k -k (2) 2 L 2 + O ε 9 2
T (1 + t) 2 ∂Z I g 3 -g

(2)

3 L 2 + 1 ε(1 + t) β 2 w 1 2 3 ∂Z I h -h (2) L 2 + 1 ε(1 + t) β 2 w 1 2 3 ∂Z I k -k (2) L 2 ≤ 1 √ T C 0 ε 2 + ε 2 (1 + t) C √ ε .
Therefore, if we had chosen C 0 ≥ 2 and C √ ε ≤ ρ we have imrpoved this estimate (11.13) and (11.12). Moreover we have 2 ) ∂Z J+1 g 4 -g T .

β 2 w 1 2 3 ∂Z I h -h (2) L 2 ε 5 2 √ T (1 + t)
Therefore, for Cε 1 2 ≤ C 0 2 , this, together with (11.31) and (11.30) improve the estimate (11.11). We proceed in the same way to imrpove the remaining estimates, using (11.28) and (11.27). Consequently, the solution (φ (2) , g (2) ) exists in [0, T ] and we have the following estimate for φ -φ (2)

|Z I (φ -φ (2) )| ≤ Cε 3 √ T √ 1 + s(1 + q) 1 2 -2κ-5ρ
, f or I ≤ N -20 We now go to the amelioration of the estimate for b. In view of the denition (11.1) of b (2) we have for I ≤ N -4.

∂ I θ b (2) (θ) -Π Σ T,θ (∂ q φ (2) ) 2 rdr =∂ I θ Π Σ T,θ (∂ q φ) 2 -(∂ q φ (2) ) 2 rdr = I 1 +I 2 ≤J Π Σ T,θ ∂ I 1 θ (∂ q φ + ∂ q φ (2) )∂ I 2 θ (∂ q φ -∂ q φ (2)
)rdr.

We estimate, for I 1 ≤ N 2 Σ T,θ ∂ I 1 θ (∂ q φ + ∂ q φ (2) )∂ I 2 θ (∂ q φ -∂ q φ (2) )rdr

L 2 (S 1 ) Σ T,θ ε √ 1 + s(1 + |q|) 3 2 -4ρ ∂ I 2 θ (∂ q φ -∂ q φ (2) ) L 2 (S 1 ) rdr ∞ 0 ε 2 (1 + s)(1 + |q|) 3-8ρ-4κ rdr 1 2 β 2 ∂ I 2 θ (∂ q φ -∂ q φ (2) ) L 2
Then the estimate (11.35), with the condition (1 + T ) C √ ε ≤ 1 yields for I ≤ N -4

∂ I θ b (2) (θ) -Π Σ T,θ (∂ q φ (2) ) 2 rdr L 2 (S 1 )
ε 4 √ T .

(11.36)

The case I 2 ≤ N 2 can be treated similarly thanks to (11.34). To conclude, we estimate

∂ I θ b (2) L 2 (S 2 ) = ∞ 0 I 1 +I 2 =I ∂ q ∂ I 1 θ φ∂ q ∂ I 2 θ φrdr L 2 (S 1 ) ≤ ∞ 0 C 0 ε √ 1 + s(1 + |q|) 3 2 -4ρ ∂ q ∂ I 2 θ φ L 2 (S 1 ) rdr ≤ C 2 0 ε 2 (1 + s)(1 + |q|) 3-8ρ rdr 1 2 ∂ q ∂ I 2 θ φ L 2 ≤ 2C 2 0 ε 2
where we have used again (1 + T ) C √ ε ≤ 1. This concludes the proof of Proposition 11.1, and the proof of Theorem 1.12.

A Reduction of the Einstein equations

We recall the form of the Einstein equations in the presence of a space-like translational Killing eld. We follow here the exposition in [START_REF] Choquet-Bruhat | General relativity and the Einstein equations[END_REF]. A metric (4) g on R 2 × R × R admitting ∂ 3 as a Killing eld can be written (4) g = g + e 2γ (dx

3 + A α dx α ) 2 ,
where g is a Lorentzian metric on R 1+2 , γ is a scalar function on R 1+2 , A is a 1-form on R 1+2 and x α , α = 0, 1, 2, are the coordinates on R 1+2 . Since ∂ 3 is a Killing eld, g, γ and A do not depend on x 3 . The polarized case consists in choosing A = 0. Let (4) R µν denote the Ricci tensor associated to (4) g. R αβ and D are respectively the Ricci tensor and the covariant derivative associated to g.

With this metric, the vacuum Einstein equations (4) R µν = 0, µ, ν = 0, 1, 2, 3 can be written in the basis (dx α , dx 3 + A α dx α ) (see [START_REF] Choquet-Bruhat | General relativity and the Einstein equations[END_REF] appendix VII) By setting φ = √ 2γ we obtain the system 1.1.

0 = (4) R αβ = R αβ -D α ∂ β γ -∂ α γ∂ β γ,

B Construction of the initial data

Theorem 1.3 is a consequence of the following result on the constraint equations, proved in [START_REF] Huneau | Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing eld in the asymptotically at case II[END_REF].

The method of solving is inspired from the conformal method in three dimension. We look for space-like metrics ḡ of the form ḡ = e 2λ δ. We introduce the traceless part of K, We note

H ij = K ij - 1 2 τ ḡij ,
ε 2 = φ2 + |∇φ| 2 .
We assume

φ2 H N -1 δ+2 + |∇φ| 2 H N -1 δ+2 + b W N,2 ε 2 .
Let B ∈ W N,2 (S 1 ). We assume

B W N,2 ε 4 .
Let Ψ ∈ H N +1 δ+1 be such that Ψ = 2π. If ε > 0 is small enough, there exist α, ρ, η, A, J, c 1 , c 2 in R, a scalar function λ ∈ H N +1 δ and a symmetric traceless tensor H ∈ H N δ+1 such that, if r, θ are the polar coordinates centered in c 1 , c 2 , and if we note Proof of Lemma B.2. The metric induced by g a on the space-like hypersurface t = 0 is r -2α δ. The shift is given by β θ = -J and the lapse is given by N = 1. Therefore we calculate

K ij = - 1 2N (∂ t ḡij -∂ i β j -∂ j β i ). D The L ∞ -L ∞ estimate
For the sake of completeness, we give here the proof of the L ∞ -L ∞ estimate by Kubo and Kubota (see [START_REF] Kubo | Scattering for systems of semilinear wave equations with dierent speeds of propagation[END_REF]).

Proposition D.1. Let u be a solution of Proof. By eventually rotating the axis, we can assume x = (r, 0) in (x 1 , x 2 ) coordinates. Therefore dθ.

We make the change of variable λ = (r 2 + ρ 2 + 2ρr cos(θ)) We exchange the integration in ρ with the integration in λ, noticing that 11 |ρ-r|≤λ≤ρ+r = 11 |λ-r|≤ρ≤λ+r , and we make the decomposition I = I 1 + I 2 , separating the region λ + r ≤ t -s from λ + r ≥ t -s. We make the change of variable α = s -λ, β = λ + s. We obtain

I 1 t-r 0 βdβ √ t -r -β(1 + β) µ t-r r-t dα √ t + r -α(1 + |α|) ν .
We estimate the rst factor. We note that if t -r ≤ 1, this factor is bounded. We assume therefore that t -r ≥ 1. where we have used in the last inequality the fact that ν > 1. We have proved I 1 (1 + |t -r|) [2-µ] + √ 1 + t + r |t -r| .

t-r 0 βdβ √ t -r -β(1 + β) µ = t-r 2 0 βdβ √ t -r -β(1 + β) µ + t-r t-r 2 βdβ √ t -r -β(1 + β) µ 1 √ t -r t-r 2 0 βdβ (1 + β) µ + (t -r)

D.2 Estimate of I 2

As in the estimate of I 1 , we write We make the same change of variable α = s -λ, β = λ + s. We obtain We estimate the rst factor. We rst assume t -r > 0. We estimate the second factor

dα √ t + r -α 1 √ t + r ,
where we have used the fact that ν > 1. We have proved therefore that

I 2 (1 + |t -r|) -1 2 +[2-µ] + √ 1 + t + r , so I ≤ I 1 + I 2 (1 + |t -r|) [2-µ] + √ 1 + t + r 1 + |t -r|
The proof of the L ∞ -L ∞ estimate is now complete.

  Hα b = g λβ b (Γ b ) α λβ (1.10)

y∈R 2 ( 1 +

 21 |y|) µ |φ 0 (y)| + (1 + |y|) µ+1 (|φ 1 (y)| + |∇φ 0 (y)|)

  and the quantities b(θ)dθ, b(θ) cos(θ)dθ, b(θ) sin(θ)dθ are imposed by the constraint equations for the initial data (see Theorem 1.3). The quantity b(θ)dθ is called decit angle, and the vector ( b(θ) cos(θ)dθ, b(θ) sin(θ)dθ) is called linear momentum.

1 2

 1 ∂Z I u L 2 ε.

S 1 b

 1 sin(θ)dθ = 0, and b W 2,N 2C 0 ε 2 , Theorem 1.3 allows us to nd initial data g and ∂ t g such that g ij , K ij satisfy the constraint equations, g and ∂ t g are compatible with the decomposition g = g b + g, where b

( 4 .

 4 34)Thanks to Lemma 1.11 we deduce the following corollary.

( 4 .

 4 43)|Z I g| ε, |∂Z I g| ε 1 + |q| , f or I ≤ N -11,(4.44)

( 4

 4 .47) Proof. Estimate (4.42) is obtained by using the decomposition (4.6) and taking the maximum of the bounds given by (4.19) and (4.16). Estimate (4.43) is obtained by using the decomposition (4.6) and taking the maximum of the bounds given by (4.19) and (4.18). Estimate (4.44) is obtained by using the decomposition (4.7) and taking the maximum of the bounds given by (4.19), (4.21) and (4.40). Estimate (4.45) is obtained by using the decomposition (4.8) and taking the maximum of the bounds given by (4.37) and (4.36). Estimate (4.47) is obtained by using the decomposition (4.8) and taking the maximum of the bounds given by (4.40) and (4.39).

Figure 1 :

 1 Figure 1: Integration of h 0

( 8 . 3 )

 83 Estimates (8.1), (8.2) and (8.3) conclude the proof of Proposition 8.2.

3 2

 2 -ρ and since J + 1 ≤ N -11 we have thanks to(4.44)

  which concludes the proof of Proposition 9.1. 10 Commutation with the vector elds and L 2 estimate 10.1 Estimation for I ≤ N

(

  10.42) The semi-linear term ∂ L g LL ∂ L h, appearing in Q LL can be estimated in the same way as the rst. The crossed term ∂ L (g b ) U U ∂ L g LL appearing in Q LL and the term (R b ) qq can be estimated in the same way than in the case I ≤ N . The crossed terms of H ρ b ∂ ρ h can be estimated in the following way

(10. 43 )

 43 Thanks to(10.40), (10.41),(10.42) and (10.43), and the bootstrap assumption (4.25), the energy inequality yields (we use here the rst inequality of Proposition 9.1)

  [START_REF] Hoshiga | The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space[END_REF].44),(10.45), (10.46), (10.47) and (10.48) conclude the proof of Proposition 10.12.10.3 Estimates forI ≤ N -8

(

  10.56) We assume Proposition 10.14 and 10.15 and prove Proposition 10.13.

2

  we estimate, thanks to(4.42)

(

  10.64) The estimates (10.61), (10.62), (10.63) and (10.64) conclude the proof of Proposition 10.15.

( 11 . 1 )

 111 Proposition 11.1. We assume that the time T satisesT ≤ exp C √ ε .

3 2 +σ ( 1 + 2 1( 1

 3121 |q|) µ-σ ∂Z I+1 h (2) -h L + t)

0

  ∂Z I+1 h (2) -h L 2

5 8 Z 18 Z. 1 2 2 , β 2 w 1 2 2 + β 2 w 1 2

 581812122 Conclusion of the proof of Proposition 11.1Estimate(11.18) gives us for I ≤ N -I h 0 -h .22) gives us for I ≤ N -I φ -φ(2) Estimate (11.23) gives us for I ≤ N -16Z I φ -φ (2) ≤ Cε 3 √ T (1 + s) -2ρ-2κ.Therefore, if Cε ≤ C 0 we have imrpoved the L ∞ estimates (11.5), (11.2) and(11.3). Estimate(11.33) implies, following the proof of Proposition 10.∂Z I φ -φ (2) L

0

  ∂Z I (φ -φ (2) ) ≤ Cε 3 (1 + t) C √ ε √ T , f or I ≤ N -4.

(A. 1 )0 = ( 4 )

 14 R 33 = -e -2γ g αβ ∂ α γ∂ β γ + g αβ D α ∂ β γ , (A.2)and the equation 0 = (4) R α3 is automatically satised. By doing the conformal change of metric g = e -2γ g, (A.1) and (A.2), yield the following system, g γ = 0, R αβ = 2∂ α γ∂ β γ α, β = 0, 1, 2.

1 b 1 b

 11 , H = e -λ H, τ = e λ τ.We also introduce the notationM θ = cos(2θ) sin(2θ) sin(2θ) -cos(2θ) , N θ = -sin(2θ) cos(2θ) cos(2θ) sin(2θ) . Theorem B.1. Let 0 < δ < 1. Let φ2 , |∇φ| 2 ∈ H N -1 δ+2 and b ∈ W N,2 (S 1) such that S (θ) cos(θ)dθ = S (θ) sin(θ)dθ = 0.

λ 2 + H H N δ+1 ε 2 .

 22 = -αχ(r) ln(r) + λ,H = -( b(θ) + ρ cos(θ -η)) χ(r) 2r M θ + e -λ χ(r) r 2 (J -(1 -α)B(θ))N θ -B (θ) 2 M θ + H,then λ, e λ H are solutions of the constraint equations withτ = ( b(θ) + ρ cos(θ -η)) χ(r) r + e -λ B (θ) cos(η) -c 1 sin(η)) + O(ε 4 ), A = -1 2π φr∂ r φ + 1 2π χ (r)rdr b(θ)dθ + O(ε 4 ),andλ H N +1 δWe will use the notation b(1) = ρ cos(θ -η) + b(θ).

(B. 1 )

 1 The end of this section is devoted to the proof of Theorem 1.3.Lemma B.2. The second fundamental form of the space-time metricg a = -dt 2 -2Jdtdθ + r -2α (dr 2 + (r -b (1) (θ)r α t) 2 dθ 2 ) -2B (θ)tdθ 2 + 4(1 -α)B(θ) t r drdθ (B.2)is given at t = 0 byK ij = H ij + 1 2 (g a ) ij τ, with τ = r α b (1) (θ) r + r 2α B (θ) r , H = -r -α b (1) (θ) χ(r) 2r M θ + (J -(1 -α)B(θ)) χ(r) r 2 N θ -B (θ) χ(r) 2r 2 M θ .

u 1 2 1 (t -s) 2 -|y| 2 F 1 (t -s) 2 -|y| 2 1 ( 1 +

 11221211 = F, (u, ∂ t u)| t=0 = (0, 0), The L ∞ -L ∞ estimate reads : for µ > 3 2 , ν > 1 |u(t, x)|(1 + t + |x|) ≤ C(µ, ν)M µ,ν (f )(1 + |t -|x|||) -1 2 +[2-µ] + ,whereM µ,ν (f ) = sup(1 + |y| + s) µ (1 + |s -|y||) ν |F (y, s)|,and we have the conventionA [0] + = ln(A).Proof. We write the solution u of u = F, (u, ∂ t u)| t=0 = (0, 0), (s, x -y)dyds. With M µ,ν (f ) = sup(1 + |y| + s) µ (1 + |s -|y||) ν |F (y, s)|, we can write |u(x, t)| ≤ M µ,ν (f ) |x -y| + s) µ (1 + |s -|x -y||) ν dyds.

2 =

 2 It is therefore sucient to study the quantityI(x, t) = |x -y| + s) µ (1 + |s -|x -y||) ν dyds.We begin with a lemma on spherical means. Lemma D.2. Let b ∈ C 0 (R 2 ). We have the following equality for ρ ≥ 0|ω|=1 b(|x + ρω|)dω = 4 ρ+r |ρ-r| λb(λ)h(λ, ρ, r)dλ,where we note r = |x| andh(λ, ρ, r) = λ 2 -(ρ -r) 2 -1 2 (ρ + r) 2 -λ 2 -1 (λ + r) 2 -ρ 2 -1 2 ρ 2 -(λ -r) 2 -1 2 .

b (r 2 +

 2 ρ 2 + 2rρ cos(θ)) 1 2

1 2 , 2 -r 2 -ρ 2 ) 2 ( 2 --λ 2 + r 2 + ρ 2 1 2 2ρr + λ 2 -ρ 2 -r 2 1 2 -s) 2 -ρ 2 |ω|=1 1 ( 1 +

 2222222222211 for θ ∈ [0, π[. Then we have (λ 2 -r 2 -ρ 2 ) dθ.We have therefore dθ = -2λh(λ, ρ, r)dλ, which concludes the proof of Lemma D.2.We use Lemma D.2 to calculate II(x, t) = |x + ρω| + s) µ (1 + |s -|x + ρω||) ν dωdρds λ + s) µ (1 + |s -λ|) ν λdλdρds.

  h(λ, ρ, r)(t -s) 2 -ρ 2 ρdρdλds, where z(s, λ) = (1 + λ + s) µ (1 + |s -λ|) ν . s) 2 -ρ 2 (λ + r) 2 -ρ 2 ρ 2 -(λ -r) with a = (λ -r) 2 , b = (λ + r) 2 and d = (t -s) 2 .Recall that in the integration region of I 1 , we have λ + r ≤ t -s so b ≤ d. This yields s) 2 -(λ + r) 2 (1 + λ + s) µ (1 + |s -λ|) ν dλds.

  with a = (λ -r) 2 , b = (λ + r) 2 and d = (t -s) 2 . In the region λ + r ≥ t -s, we have b ≥ d, h(λ, ρ, r) (t -s) 2 -ρ 2 ρdρ 1 (λ + r) 2 -(t -s) 2 . r) 2 -(t -s) 2 (1 + λ + s) µ (1 + |s -λ|) ν dλds.

  r -α(1 + |α|) ν .

( 1 + 3 2 1 +( 1 +

 1311 |t -r|) -µ ,where we have made consecutively the changes of variable ρ = β -|t -r| and u = ρ 1+|t-r| , and where we use in the last inequality the fact that(1+u) 1-µ √ u is integrable.We now assume t -r < -1. Then t+r0 βdβ β + |t -r|(1 + β) µ |tρ(1 + |t -r|ρ) µ dρ (1 + |t -r|) |t -r|) -1 2 +[2-µ] + ,where we have made the change of variable ρ = β |t-r| , and also used the fact that µ > 3 2 .

  Proposition 7.4. Let I ≤ N -7 We have the estimate for q < 0

	, (7.19), (7.21), (7.22) we conclude the proof of
	Proposition 7.3.
	7.2 Estimation of Υ( r t )h 0

  2) and(11.14) to estimate ∂(φ -φ(2)) and (4.15) and (4.28) to estimate ∂(φ + φ(2)) corresponds to(7.12)

	multiplied by	ε

  t) σ β 2 w 2 (q) 1 2 ∂Z I g 4 -gand consequently, estimate (11.32), together with the bootstrap assumption(11.11) yields

	Estimate (11.33) also implies
		0	t	1 (1 + (2) 4	2 L 2	ε 4 T
	d dt	β 2 w	1 2 2 ∂Z I g 3 -g 3 (2)	2 L 2	+ β 2 w 2 (q)	1 2 ∂Z I g 3 -g 3 (2)	2 L 2
	ε T (1 + t) 1+σ + 9 2	ε (1 + t) σ (β 2 w	1 2
								1 2 +C	√	ε .

-ρ

Acknowledgement The author would like to thank Jérémie Szeftel for his attentive reading of this paper and his supervising.

We infer

sin 2 (θ) -4(1 -α)B(θ) r 2 cos(θ) sin(θ) + 2J r 2 cos(θ) sin(θ) , (1) (θ) r + 2B (θ) r 2 cos 2 (θ) + 4(1 -α)B(θ) r 2 cos(θ) sin(θ) -2J r 2 cos(θ) sin(θ) ,

2r -α b (1) (θ) r + 2B (θ) r 2 cos(θ) sin(θ) + 4(1 -α)B(θ) 2r 2 (cos 2 (θ) -sin 2 (θ)) -2J r 2 (cos 2 (θ) -sin 2 (θ)) .

We calculate

so we obtain exactly

Lemma B.3. The metric g a , dened by (B.2) is isometric to g b + g (1) where at t = 0 we have g (1) = O 1 r 2 , ∂ t g (1) = O

and g b is dened by (1.6), where b(θ) = b (1) (F (θ)) 1 -α -b (1) (F (θ)) ,

Proof. During all the proof, the notation g ∼ g stands for g is isometric to g + g where g = O We perform some changes of variable in g as . First of all we introduce r such that r = r 1-α 1 -α , dr = r -α dr.

The metric g a becomes

where we keep writing r instead of r . We now make the change of variable

Since we will neglect the contributions to the metric decaying like

We keep also writing θ instead of θ . We infer

We choose

With this choice we obtain

We impose α = -b (1) (θ)dθ = -b(θ)dθ.

Therefore we can nd f (θ) such that

We perform the change of variable

We note F the inverse function of θ → θ + f (θ),

We set

Let us note that J is at the same level of regularity than b.

We are now ready to prove Theorem 1.3.

Proof of Theorem 1.3. We consider the map Π is the projection

satisfy the constraint equations, where we have noted K b the second fundamental form associated to g b . We complete the initial data as follow. We write our metric in the form g = g b + g. The initial data for g are the following g ij is given by g ij = ( g 0 ) ij , g 00 and g 0i are taken to be 0 1

∂ t g 00 and ∂ t g 0i are chosen such that the generalized wave coordinate condition is satised at t = 0.

Let us describe the last point. The generalized wave coordinate condition can be written

Therefore, if we write it for α = i we obtain a relation for ∂ t g 0i and if we write it for α = 0, we obtain a relation for ∂ t g 00 . However, if we write g = g b + g, the term

which do not belong to H N δ+1 because we are missing a derivative on b, since b ∈ W 2,N . Therefore, we will take F α as dened in (1.9). With this choice, the generalized wave coordinate condition imply that ∂ t g 00 and ∂ t g 0i are given by a sum of terms the form

With this choice, ∂ t g 0i and ∂ t g 00 belong to H N δ+1 .

1 The lapse ans shift are given by g b : we have N = 1 and βr = 0 and β θ = -J.

C The generalised wave coordinates

In a coordinate system, the Ricci tensor is given by

where the Γ λ αβ are the Christoel symbols given by

R µν is an operator of order two for g. In order to single out the hyperbolic part, we will write

which can also be written

We compute R µν in terms of g and H.

Proposition C.1. If the coupled system of equations

with F a function which may depend on φ, g, is satised on a time interval [0, T ] with T > 0, if the initial induced Riemannian metric and second fundamental form (ḡ, K) satisfy the constraint equations, and if the initial compatibility condition

is satised, then for all time, the equations (1.1) are satised on [0, T ], together with the wave coordinate condition

Proof. We use the twice contracted Bianchi Identity

with H dened by (C.3). Since in [0, T ], we have

Thanks to (C.4) we obtain

Consequently, since D µ R µν -1 2 Rg µν = 0 and D µ ∂ µ φ∂ ν φ -1 2 g µν ∂ α φ∂ α φ = 0 and we obtain the following equation on

Multiplying by g να we obtain

. This is an equation in hyperbolic form, therefore if the initial data (F α -H α )| t=0 and ∂ t (F α -H α )| t=0 are zero, then the solution is identically zero on [0, T ]. Since we assume (C.6), we only have to check

Since the constraint equations are satised, we have

Therefore, using once again equation (C.4) and (C.6) we obtain

This system can be written as

implies that the Einstein Equations (1.1) are satised.

E Hardy inequality with weight

Proposition E.1. Let α < 1 and β > 1. We have, with q = r -t,

where

Proof. We look rst at the region r > t. We can assume, by a density argument that u is compactly supported. We calculate

This condition is satised if

We have proved

We now look at the region r < t. We calculate Proof. We introduce the decomposition