Benoît Grébert

Eric Paturel

KAM FOR THE KLEIN GORDON EQUATION ON

Keywords: 2000 Mathematics Subject Classification. - KAM theory, Regularizing PDE, Klein Gordon equation. 1

Recently the KAM theory has been extended to multidimensional PDEs. Nevertheless all these recent results concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the structure of the resonant sets is quite simple. In the present paper, we consider an important physical example that do not fit in this context: the Klein Gordon equation on S d . Our abstract KAM theorem also allow to prove the reducibility of the corresponding linear operator with time quasiperiodic potentials.

Introduction.

If the KAM theorem is now well documented for nonlinear Hamiltonian PDEs in 1dimensional context (see [START_REF] Kuksin | Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum[END_REF][START_REF] Kuksin | Nearly integrable infinite-dimensional Hamiltonian systems[END_REF][START_REF] Pöschel | A KAM-theorem for some nonlinear partial differential equations[END_REF]) only few results exist for multidimensional PDEs.

Existence of quasi-periodic solutions of space-multidimensional PDE were first proved in [START_REF] Bourgain | Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equation[END_REF] (see also [START_REF] Bourgain | Green's function estimates for lattice Schrödinger operators and applications[END_REF]) but with a technique based on the Nash-Moser thorem that does not allow to analyze the linear stability of the obtained solutions. Some KAM-theorems for smallamplitude solutions of multidimensional beam equations (see (3.6) above) with typical m were obtained in [START_REF] Geng | A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces[END_REF][START_REF] Geng | KAM tori for higher dimensional beam equations with constant potentials[END_REF]. Both works treat equations with a constant-coefficient nonlinearity g(x, u) = g(u), which is significantly easier than the general case. The first complete KAM theorem for space-multidimensional PDE was obtained in [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF]. Also see [START_REF] Berti | Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential[END_REF][START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]. The techniques developed by Eliasson-Kuksin have been improved in [START_REF] Eliasson | KAM for the nonlinear beam equation 2 : a normal form theorem[END_REF][START_REF] Eliasson | KAM for the nonlinear beam equation 1: small amplitude solutions[END_REF] to allow a KAM result without external parameters. In these two papers the authors prove the existence of small amplitude quasi-periodic solutions of the beam equation on the d-dimensional torus. They further investigate the stability of these solutions and give explicit examples where the solution is linearly unstable and thus exhibits hyperbolic features (a sort of whiskered torus).

All these examples concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the structure of the resonant sets is the same for NLS, NLW or beam equation. In the present paper, adapting the technics in [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF], we consider an important example that do not fit in the Fourier analysis: the Klein Gordon equation on the sphere S d . Notice that existence of quasi-periodic solutions for NLW and NLS on compact Lie groups via Nash Moser technics (and without linear stability) has been proved recently in [START_REF] Berti | Procesi Nonlinear wave and Schrödinger equations on compact Lie groups and Homogeneous spaces[END_REF][START_REF] Berti | Procesi An Abstract Nash-Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds[END_REF].

To understand the new difficulties, let us start with a brief overview of the method developed in [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF]. Consider the nonlinear Schrödinger equation on T d iu t = -∆u + nonlinear terms, x ∈ T d , t ∈ R.

In Fourier variables it reads (1) i uk = |k| 2 u k + nonlinear terms, k ∈ Z d .

So two Fourier modes indexed by k, j ∈ Z d are (linearly) resonant when |k| 2 = |j| 2 . For the beam equation on the torus, the resonance relation is the same. The resonant sets 2 } define a natural clustering of Z d . All the modes in the block E k have the same energy, and we can expect that the interactions between different blocks are small, but the interactions inside a block could be of order one. With this idea in mind, the principal step of the KAM technique, i.e. the resolution of the so called homological equation, leads to the inversion of an infinite matrix which is block-diagonal with respect to this clustering. It turns out that these blocks have cardinality growing with |k| making harder the control of the inverse of this matrix. As a consequence we lose regularity each time we solve the homological equation. Of course, this is not acceptable for an infinite induction. The very nice idea in [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF] consists in considering a sub-clustering constructed as the equivalence classes of the equivalence relation on Z d generated by the pre-equivalence relation

E k = {j ∈ Z d | |j| 2 = |k|
a ∼ b ⇐⇒ |a| = |b| |a -b| ≤ ∆
Let [a] ∆ denote the equivalence class of a. The crucial fact (proved in [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF]) is that the blocks are finite with a maximal "diameter" max

[a] ∆ =[b] ∆ |a -b| ≤ C d ∆ (d+1)! 2
depending only on ∆. With such a clustering, we do not lose regularity when we solve the homological equation. Furthermore, working in a phase space of analytic functions u or equivalently, exponentially decreasing Fourier coefficients u k , it turns out that the homological equation is "almost" block diagonal relatively to this clustering. Then we let the parameter ∆ grow at each step of the KAM iteration.

Unfortunately, this estimate of the diameter of a block [a] ∆ by a constant independent of |a| is a sort of miracle that does not persist in other cases. For instance if we consider the nonlinear Klein Gordon equation on the sphere S 2 , (∂ 2 t -∆ + m)u = nonlinear terms, t ∈ R, x ∈ S 2 then the linear part diagonalizes in the harmonic basis Ψ j,ℓ (see Section 3) and the natural clustering is given by the resonant sets {(j, ℓ) ∈ N 2 | ℓ = -j, • • • , j}. We can easily convince ourself that there is no simple construction of a sub-clustering compatible with the equation, in such a way that the size of the blocks does no more depend on the energy. So we have to invent a new way to proceed. First we consider a phase space Y s with polynomial decay on the Fourier coefficient (corresponding to Sobolev regularity for u) instead of exponential decay and we use a different norm on the Hessian matrix that takes into account the polynomial decrease of the off-diagonal blocks:

(1.1) |M | β,s = sup j,k∈N M [j] [k] (kj) β min(j, k) + |j 2 -k 2 | min(j, k) s/2
where [j] = {(n, m) ∈ N 2 | n + m = j} is the block of energy j, M

[j]

[k] is the interaction matrix M reduced to the eigenspace of energy j and of energy k, and • is the operator norm in ℓ 2 . This norm was suggested by our study of the Birkhoff normal form in [START_REF] Bambusi | Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF] and [START_REF] Grébert | Normal forms for semilinear quantum harmonic oscillators[END_REF]. This technical changes make disappear the loss of regularity in the resolution of the homological equation. Nevertheless this is not the end of the story, since this Sobolev structure of the phase space T s,β (see Section 2) is not stable by Poisson bracket and thus is not adapted to an iterative scheme. So the second ingredient consists in a trick previously used in [START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF]: we take advantage of the regularizing effect of the homological equation to obtain a solution in a slightly more regular space T s,β+ and then we verify that {T s,β , T s,β+ } ∈ T s,β (see Section 4) which enables an iterative procedure. The last problem is to check that the non linear term, say P , belongs to the class T s,β which imposes a decreasing condition on the operator norm of the blocks of the Hessian of P . It turns out that this condition is satisfied for the Klein Gordon equation on spheres (and also on Zoll manifold, see Remark 3.3). A similar condition is also satisfied for the quantum harmonic oscillator on R d i u t = -∆u + |x| 2 u + nonlinear terms, x ∈ R d .

But unfortunately, in order to belong in the class T s,β , the gradient of the nonlinear term has to be regularizing, a fact that is not true for the quantum harmonic oscillator, and thus our KAM theorem does not apply in this case. Nevertheless, this last condition is not required when P is quadratic and thus this method allows to obtain a reducibility result for the quantum harmonic oscillator with time quasi periodic potential. This is detailed in our forthcoming paper [START_REF] Grébert | On reducibility of the quantum harmonic oscillator with quasiperiodic in time potential on R d[END_REF]. In this paper we only consider PDEs with external parameters (similar to a convolution potential in the case of NLS on the torus). Following [START_REF] Eliasson | KAM for the nonlinear beam equation 1: small amplitude solutions[END_REF] we could expect to remove these external parameters (and to use only internal parameters) but the technical cost would be very high.

We now state our result for the Klein Gordon equation on the sphere. Denote by ∆ the Laplace-Beltrami operator on the sphere S d , m > 0 and let Λ 0 = (-∆ + m) 1/2 . The spectrum of Λ 0 equals { j(j + d -1) + m | j ≥ 0}. For each j ≥ 1 let E j be the associated eigenspace, its dimension is d j = O(j d-1 ). We denote by Ψ j,l the harmonic function of degree j and order ℓ so that we have

E j = Span{Ψ j,l , l = 1, • • • , d j }. We denote E := {(j, ℓ) ∈ N × Z | j ≥ 0 and ℓ = 1, • • • , d j } in such a way that {Ψ a , a ∈ E} is a basis of L 2 (S d , C).
We introduce the harmonic multiplier M ρ defined on the basis (Ψ a ) a∈E of L 2 (S d ) by

(1.2) M ρ Ψ a = ρ a Ψ a for a ∈ E
where (ρ a ) a∈E is a bounded sequence of nonnegative real numbers. Let g be a real analytic function on S d × R such that g vanishes at least at order 2 in the second variable at the origin. We consider the following nonlinear Klein Gordon equation

(1.3) (∂ 2 t -∆ + m + δM ρ )u + εg(x, u) = 0, t ∈ R, x ∈ S d where δ > 0 and ε > 0 are small parameters. Introducing Λ = (-∆ + m + δM ρ ) 1/2 and v = -u t ≡ -u, (1.3) reads u = -v, v = Λ 2 u + εg(x, u). Defining ψ = 1 √ 2 (Λ 1/2 u + iΛ -1/2 v) we get 1 i ψ = Λψ + ε √ 2 Λ -1/2 g x, Λ -1/2 ψ + ψ √ 2 .
Thus, if we endow the space L 2 (S d , C) with the standard real symplectic structure given by the two-form -idψ ∧ d ψ then equation (1.3) becomes a Hamiltonian system

ψ = i ∂H ∂ ψ
with the hamiltonian function

H(ψ, ψ) = S d (Λψ) ψdx + ε S d G x, Λ -1/2 ψ + ψ √ 2 dx.
where G is a primitive of g with respect to the variable u:

g = ∂ u G.
The linear operator Λ is diagonal in the basis {Ψ a , a ∈ E}:

ΛΨ a = λ a Ψ a , λ a = w a (w a + d -1) + m + δρ a , ∀ a ∈ E
where we set w (j,ℓ) = j ∀ (j, ℓ) ∈ E.

Let us decompose ψ and ψ in the basis {Ψ a , a ∈ E}:

ψ = a∈E ξ a Ψ a , ψ = a∈E η a Ψ a .
On P C := ℓ 2 (E, C) × ℓ 2 (E, C) endowed with the complex symplectic structure -i s dξ s ∧ dη s we consider the Hamiltonian system

(1.4) ξa = i ∂H ∂ηa ηa = -i ∂H ∂ξa a ∈ E
where the Hamiltonian function H is given by

(1.5) H = a∈E λ a ξ a η a + ε S d G x, a∈E (ξ a + η a )Ψ a √ 2 λ 1/2 a dx.
The Klein Gordon equation (1.3) is then equivalent to the Hamiltonian system (1.4) restricted to the real subspace

P R := {(ξ, η) ∈ ℓ 2 (E, C) × ℓ 2 (E, C) | η a = ξa , a ∈ E}. Definition 1.1. -Let A ⊂ E a finite subset of cardinal n.
This set is admissible if and only if

(1.6) A ∋ (j 1 , ℓ 1 ) = (j 2 , ℓ 2 ) ∈ A ⇒ j 1 = j 2 .
We fix I a ∈ [1, 2] for a ∈ A, the initial n actions, and we write the modes A in action-angle variables:

ξ a = I a + r a e iθa , η a = I a + r a e -iθa . We define L = E \ A and, to simplify the presentation, we assume that ρ j,l = ρ j for (j, ℓ) ∈ A ; ρ j,l = 0 for (j, ℓ) ∈ L.

Set

w j,ℓ = j for (j, ℓ) ∈ E,

λ j,ℓ = j(j + d -1) + m for (j, ℓ) ∈ L, (ω 0 ) j,ℓ (ρ) = j(j + d -1) + m + δρ j for (j, ℓ) ∈ A, ζ = (ξ a , η a ) a∈L . (1.7)
With this notation H reads (up to a constant)

H(r, θ, ζ) = ω 0 (ρ), r + a∈L λ a ξ a η a + εf (r, θ, ζ) where f (r, θ, ζ) = S d G (x, û(r, θ, ζ)(x)) dx and (1.8) û(r, θ, ζ)(x) = a∈A 2(I a + r a ) cos θ a λ 1/2 a Ψ a (x) + a∈L (ξ a + η a ) √ 2 λ 1/2 a Ψ a (x).
Let us set u 1 (θ, x) = û(0, θ; 0)(x). Then for any I ∈ [1, 2] n and θ 0 ∈ T n the function (t, x) → u 1 (θ 0 + tω, x) is a quasi-periodic solution of (1.3) with ε = 0. Our main theorem states that for most external parameter ρ this quasi-periodic solution persists (but is slightly deformed) when we turn on the nonlinearity: Theorem 1.2. -Fix n the cardinality of an admissible set A, s > 1 the Sobolev regularity and g the nonlinearity. There exists an exponent υ(d) > 0 such that, for ε sufficiently small (depending on n, s and g) and satisfying

ε ≤ δ υ(d) ,
there exists a Borel subset D ′ , positive constants α and C with

D ′ ⊂ [1, 2] n , meas([1, 2] n \ D ′ ) ≤ Cε α , such that for ρ ∈ D ′ , there is a function u(θ, x), analytic in θ ∈ T n σ 2 and smooth in x ∈ S d , satisfying sup |ℑθ|< σ 2 u(θ, •) -u 1 (θ, •) H s (S d ) ≤ ε 11/12 ,
and there is a mapping

ω ′ : D ′ → R n , ω ′ -ω C 1 (D ′ ) ≤ ε,
such that for any ρ ∈ D ′ the function

u(t, x) = u(θ + tω ′ (ρ), x)
is a solution of the Klein Gordon equation (1.3). Furthermore this solution is linearly stable. The positive constant α depends only on n while C also depends on g and s.

Notice that in this work we did not try to optimize the exponents. In particular 11/12 could be replaced by any number strictly less than 1 and the choice of υ(d) obtained by inserting (3.1) in (6.6) is far from optimal. Actually we could expect that ε ≪ δ is sufficient but the technical cost would be very high. This effort is justified when we try to prove a KAM result without external parameters (see [START_REF] Kuksin | Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation[END_REF] where the authors obtained a condition of the form ε ≪ δ in the context of the NLS equation; see also [START_REF] Eliasson | KAM for the nonlinear beam equation 2 : a normal form theorem[END_REF], [START_REF] Eliasson | KAM for the nonlinear beam equation 1: small amplitude solutions[END_REF] for the beam equation and [START_REF] Bouthelja | KAM pour l'équation des ondes nonlinéaire sur le cercle[END_REF] for the 1d wave equation where the authors obtained a condition of the form ε ≪ δ 1+α for suitable α > 0 ).

We will deduce Theorem 1.2 from an abstract KAM result stated in Section 2 and proved in Section 6. The application to the Klein Gordon equation is detailed in Section 3. Roughly speaking, our abstract theorem applies to any multidimensional PDE with regularizing nonlinearity and which satisfies the second Melnikov condition (see Hypothesis A3). For instance, it doesn't apply to nonlinear Schrödinger on any compact manifold since we have no regularizing effect in that case. On the contrary, it applies to the beam equation on the torus T d (see Remark 3.4). Unfortunately it doesn't apply to the nonlinear wave equation on T d (see Remark 3.5), since in that case the second Melnikov condition is not satisfied. In Section 4 we study the Hamiltonian flows generated by Hamiltonian functions in T s,β . In Section 5 we detail the resolution of the homological equation. In both Sections 4 and 5 we use techniques and proofs that were developed in [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF] and [START_REF] Eliasson | KAM for the nonlinear beam equation 2 : a normal form theorem[END_REF]. The novelty lies in the use of different norms (see (1.1)) and the use of two different classes of Hamiltonians: T s,β and T s,β+ which, of course, complicate the technical arguments. For convenience of the reader we repeat most of the proofs. We point out that, for the resolution of the homological equation (Section 5), we use a variant of a Lemma due to Delort-Szeftel [START_REF] Delort | Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres[END_REF], whose proof is given in Appendix A.

Acknowledgement:

The authors acknowledge the support from the project ANAE (ANR-13-BS01-0010-03) of the Agence Nationale de la Recherche.

2. Setting and abstract KAM theorem.

Notations. In this section we state a KAM result for a Hamiltonian H = h + f of the following form

H = ω(ρ), r + 1 2 ζ, A(ρ)ζ + f (r, θ, ζ; ρ)
where ω ∈ R n is the frequencies vector corresponding to the internal modes in action-angle variables (r,

θ) ∈ R n + × T n . -ζ = (ζ s ) s∈L are the external modes: L is an infinite set of indices, ζ s = (p s , q s ) ∈ R 2
and R2 is endowed with the standard symplectic structure dq ∧ dp. -A is a linear operator acting on the external modes, typically A is diagonal.

f is a perturbative Hamiltonian depending on all the modes and is of order ε where ε is a small parameter. -ρ is an external parameter in D a compact subset of R p with p ≥ n. We now detail the structures behind these objects and the hypothesis needed for the KAM result.

Cluster structure on L. Let L be a set of indices and w : L → N \ {0} be an "energy" function (2) on L. We consider the clustering of L given by L = ∪ a∈L [a] associated to equivalence relation b ∼ a ⇐⇒ w a = w b . We denote L = L/ ∼. We assume that the cardinal of each energy level is finite and that there exist C b > 0 and d * > 0 two constants such that the cardinality of [a] is controlled by

C b w d a : (2.1) d a = d [a] = card{b ∈ L | w b = w a } ≤ C b w d * a .
Linear space. Let s ≥ 0, we consider the complex weighted ℓ 2 -space

Y s = {ζ = (ζ a ∈ C 2 , a ∈ L) | ζ s < ∞} where (3) ζ 2 s = a∈L |ζ a | 2 w 2s a .
In the spaces Y s acts the linear operator J, where ∇f is the gradient with respect to the scalar product in Y .

J : {ζ a } → {σ 2 ζ a }, with σ 2 = 0 -1 1
Infinite matrices. We denote by M s,β the set of infinite matrices A : L × L → M 2×2 (R) with value in the space of real 2 × 2 matrices that are symmetric

A b a = t A a b , ∀a, b ∈ L and satisfy |A| s,β := sup a,b∈L (w a w b ) β A [b] [a] w(a, b) + |w 2 a -w 2 b | w(a, b) s/2 < ∞
where A

C n × C n × Y s we define the norm (z, r, ζ) s = max(|z|, |r|, ζ s ). For σ > 0 we denote T n σ = {z ∈ C n : |ℑz| < σ}/2πZ n . For σ, µ ∈ (0, 1] and s ≥ 0 we set O s (σ, µ) = T n σ × {r ∈ C n : |r| < µ 2 } × {ζ ∈ Y s : ζ s < µ}
We will denote points in O s (σ, µ) as x = (θ, r, ζ). A function defined on a domain O s (σ, µ), is called real if it gives real values to real arguments. Let D = {ρ} ⊂ R p be a compact set of positive Lebesgue measure. This is the set of parameters upon which will depend our objects. Differentiability of functions on D is understood in the sense of Whitney.

So f ∈ C 1 (D) if it may be extended to a C 1 -smooth function f on R p , and |f | C 1 (D) is the infimum of | f | C 1 (R p ) , taken over all C 1 -extensions f of f . If (z,
O s (σ, µ) ∋ x → ∇ ζ f (x, ρ) ∈ Y s+β and O s (σ, µ) ∋ x → ∇ 2 ζ f (x, ρ) ∈ M s,β are
real holomorphic functions. We denote this set of functions by T s,β (σ, µ, D). We notice that for β > 0, both the gradient and the hessian of f ∈ T s,β (σ, µ, D) have a regularizing effect. For a function f ∈ T s,β (σ, µ, D) we define the norm

[f ] s,β σ,µ,D through sup max(|∂ j ρ f (x, ρ)|, µ ∂ j ρ ∇ ζ f (x, ρ) s+β , µ 2 |∂ j ρ ∇ 2 ζ f (x, ρ)| s,β )
, where the supremum is taken over all

j = 0, 1, x ∈ O s (σ, µ), ρ ∈ D.
In the case β = 0 we denote T s (σ, µ, D) = T s,0 (σ, µ, D) and

[f ]
∈ N F if (i) A is real valued, (ii) A is symmetric, i.e. A a b = t A b a , (iii) A satisfies ΠA = A, (iii) A is block diagonal, i.e.
q(ζ) = 1 2 a,b∈L ζ a , A b a ζ b .
In the complex variables, z a = (ξ a , η a ), a ∈ L, where

ξ a = 1 √ 2 (p a + iq a ), η a = 1 √ 2 (p a -iq a ),
we have

q(ζ) = 1 2 ξ, ∇ 2 ξ q ξ + 1 2 η, ∇ 2 η q η + ξ, ∇ ξ ∇ η q η . The matrices ∇ 2
ξ q and ∇ 2 η q are symmetric and complex conjugate of each other while

∇ ξ ∇ η q is Hermitian. If A ∈ M s,β then (2.2) sup a,b (∇ ξ ∇ η q) [b] [a] ≤ |A| s,β (w a w b ) β (1 + |w a -w b |) s .
We note that if A is on normal form, then the associated quadratic form q(ζ) = 1 2 ζ, Aζ reads in complex variables

(2.3) q(ζ) = ξ, Qη where Q : L × L → C is (i) Hermitian, i.e. Q a b = Q b a , (ii) Block-diagonal.
In other words, when A is on normal form, the associated quadatic form reads

q(ζ) = 1 2 p, A 1 p + p, A 2 q + 1 2 p, A 1 q with Q = A 1 + iA 2 Hermitian.
By extension we will say that a Hamiltonian is on normal form if it reads

(2.4) h = ω(ρ), r + 1 2 ζ, A(ρ)ζ
with ω(ρ) ∈ R n a frequency vector and A(ρ) on normal form for all ρ.

2.1. Hypothesis on the spectrum of A 0 .-We assume that A 0 is a real diagonal matrix whose diagonal elements λ a > 0, a ∈ L are C 1 . Our hypothesis depend on two constants 1 > δ 0 > 0 and c 0 > 0 fixed once for all. Hypothesis A1 -Asymptotics. We assume that there exist γ ≥ 1 such that Hypothesis A2 -Non resonances. There exists a δ 0 > 0 such that for all C 1 -functions

(2.5) c 0 w γ a ≤ λ a ≤
ω : D → R n , |ω -ω 0 | C 1 (D) < δ 0 ,
the following holds for each k ∈ Z n \ 0: either we have the following properties :

       | k, ω(ρ) | ≥ δ 0 for all ρ ∈ D, | k, ω(ρ) + λ a | ≥ δ 0 w a for all ρ ∈ D and a ∈ L, | k, ω(ρ) + λ a + λ b | ≥ δ 0 (w a + w b ) for all ρ ∈ D anda, b ∈ L, | k, ω(ρ) + λ a -λ b | ≥ δ 0 (1 + |w a -w b |) for all ρ ∈ D and a, b ∈ L,
or there exists a unit vector z ∈ R p such that

(∇ ρ • z)( k, ω ) ≥ δ 0
for all ρ ∈ D. The first term of the alternative will be used in order to control the small divisors for large k, and the second one is featured to control them for small k.

The last assumption above will be used to bound from below divisors | k, ω(ρ) + λ a (ρ)λ b (ρ)| with w a , w b ∼ 1. To control the (infinitely many) divisors with max(w a , w b ) ≫ 1 we need another assumption: Hypothesis A3 -Second Melnikov condition in measure. There exist absolute constants α 1 > 0, α 2 > 0 and C > 0 such that for all C 1 -functions

ω : D → R n , |ω -ω 0 | C 1 (D) < δ 0 ,
the following holds: for each κ > 0 and N ≥ 1 there exists a closed subset

D ′ = D ′ (ω 0 , κ, N ) ⊂ D satisfying (2.7) meas(D \ D ′ ) ≤ CN α 1 ( κ δ 0 ) α 2 (α 1 , α 2 ≥ 0)
such that for all ρ ∈ D ′ , all 0 < |k| ≤ N and all a, b ∈ L we have

(2.8) | k, ω(ρ) + λ a -λ b | ≥ κ(1 + |w a -w b |).
2.2. The abstract KAM Theorem.-We are now in position to state our abstract KAM result.

Theorem 2.2. -Assume that (2.9)

h 0 = ω 0 (ρ), r + 1 2 ζ, A 0 ζ
with the spectrum of A 0 satisfying Hypothesis A1, A2, A3 and let f ∈ T s,β (D, σ, µ) with β > 0, s > 0. There exists ε 0 > 0 (depending on n, d, s, β, σ, µ, on A, c 0 and sup |∇ ρ ω|), α > 0 (depending on n, d * , s, β, α 1 , α 2 ) and υ(β, d * ) > 0 such that (4) if

[f ] s,β σ,µ,D = ε < min ε 0 , δ υ(β,d * ) 0
there is a D ′ ⊂ D with meas(D \ D ′ ) ≤ ε α such that for all ρ ∈ D ′ the following holds: There are a real analytic symplectic diffeomorphism

Φ : O s (σ/2, µ/2) → O s (σ, µ)
and a vector ω = ω(ρ) such that

(h 0 + f ) • Φ = ω(ρ), r + 1 2 ζ, A(ρ)ζ + f (r, θ, ζ; ρ)
where

∂ ζ f = ∂ r f = ∂ 2 ζζ f = 0 for ζ = r = 0 and A : L × L → M 2×2 (R) is on normal form, i.e. A is real symmetric and block diagonal: A b a = 0 for all w a = w b . Moreover Φ satisfies Φ -Id s ≤ ε 11/12 ,
4. An explicit choice of υ is given in (6.6) but is surely far from optimality.

for all (r, θ, ζ) ∈ O s (σ/2, µ/2), and

|A(ρ) -A 0 | β ≤ ε, |ω(ρ) -ω 0 (ρ)| C 1 ≤ ε for all ρ ∈ D ′ .
This normal form result has dynamical consequences. For ρ ∈ D ′ , the torus {0} × T n × {0} is invariant by the flow of (h 0 + f ) • Φ and the dynamics of the Hamiltonian vector field of

h 0 + f on the Φ({0} × T n × {0}) is the same as that of ω(ρ), r + 1 2 ζ, A(ρ)ζ .
The Hamiltonian vector field on the torus {ζ = r = 0} is

   ζ = 0 θ = ω ṙ = 0,
and the flow on the torus is linear:

t → θ(t) = θ 0 + tω.
Moreover, the linearized equation on this torus reads

   ζ = JAζ + J∂ 2 rζ f (0, θ 0 + ωt, 0) • r θ = ∂ 2 rζ f (0, θ 0 + ωt, 0) • ζ + ∂ 2 rr f (0, θ 0 + ωt, 0) • r ṙ = 0.
Since A is on normal form (and in particular real symmetric and block diagonal) the eigenvalues of the ζ-linear part are purely imaginary: ±i λa , a ∈ L. Therefore the invariant torus is linearly stable in the classical sense (all the eigenvalues of the linearized system are purely imaginary). Furthermore if the λa are non-resonant with respect to the frequency vector ω (a property which can be guaranteed restricting the set D ′ arbitrarily little) then the linearized equation is reducible to constant coefficients. Then the ζ-component (and of course also the r-component) will have only quasi-periodic (in particular bounded) solutions.

Applications to Klein Gordon on S d

In this section we prove Theorem 1.2 as a corollary of Theorem 2.2. We use notations introduced in the introduction (see in particular (1.7)). Then the Klein Gordon Hamiltonian H reads (up to a constant)

H(r, θ, ζ) = ω 0 (ρ), r + a∈L λ a ξ a η a + εf (r, θ, ζ) where f (r, θ, ζ) = S d G (x, û(r, θ, ζ)(x)) dx.
δ 0 = δ 2 √ 2 + d + m max(w a , a ∈ A) 3 .
Proof. -Hypothesis A1 is clearly satisfied with c 0 = 1/2 and γ = 1. The control of the cardinality of the clusters (2.1) is given with

d * = d -1. On the other hand choosing z ≡ z k = k |k| we have (3.2) (∇ ρ • z)( k, ω ) ≥ δ 2 max((ω 0 ) a , a ∈ A) |k| ≥ δ √ 2 + d + m max(w a , a ∈ A) |k| for all k = 0 while (3.3) (∇ ρ • z)λ a = 0 for all a ∈ L.
Then for all k = 0 the second part of the alternative in Hypothesis A2 is satisfied choosing

δ 0 ≤ δ * := δ 2 max(w a , a ∈ A) √ 2 + d + m .
It remains to verify A3. Without loss of generality we can assume w a ≤ w b . First denoting

F κ (k, a, b) := {ρ ∈ D | | ω, k + λ a -λ b | ≤ κ}, we have using (3.2) that meas F κ (k, a, b) ≤ C(k, a, b) κ δ * .
On the other hand, defining

G ν (k, e) := {ρ ∈ D | | ω, k + e| ≤ 2ν}, we have, using again (3.2) that meas G ν (k, e) ≤ C ν δ * . Further | ω, k + e| ≤ 1 can occur only if |e| ≤ C|k| and thus G ν = 0<|k|≤N e∈Z G ν (k, e)
has a Lebesgue measure less than CN n+1 ν δ * . Now we remark that

|j + d -1 2 -j(j + d -1) + m| ≤ C ′ m,d j where C ′
m,d only depends on m and d, from which we deduce

|λ a -λ b -(w a -w b )| ≤ 2C ′ m,d w a . Therefore for ρ ∈ D \ G ν and w a ≥ 2C ′ m,d ν we have for all 0 < |k| ≤ N | ω, k + λ a -λ b | ≥ ν. Finally w a ≤ 2C ′ m,d ν and | ω, k + λ a -λ b | ≤ 1 leads to w b ≤ 2C ′ m,d ν + CN and thus, if we restrict ρ to D ′ = D \ G ν ∪ 0<|k|≤N wa,w b ≤ 2C ′ m,d ν +CN F κ (k, a, b) we get | ω, k + λ a -λ b | ≥ min(κ, ν), 0 < |k| ≤ N, a, b ∈ L. Further meas D \ D ′ ≤ CN n+1 ν δ * + 2C ′ m,d ν + CN 2 N n κ δ * .
Then choosing ν = κ 1/3 and δ 0 = δ 3 * , this measure is controlled by

CN n+2 κ δ 0 1/3
and we have

| ω, k + λ a -λ b | ≥ κ, for ρ ∈ D ′ , 0 < |k| ≤ N and a, b ∈ L. Now we remark that for |λ a -λ b | ≥ 2| ω, k |, | ω, k + λ a -λ b | ≥ 1 2 |λ a -λ b | ≥ 1 4 (1 + |w a -w b |) ≥ κ(1 + |w a -w b |) if we assume κ ≤ 1 4 . On the other hand, when |λ a -λ b | ≤ 2| ω, k | ≤ CN , | ω, k + λ a -λ b | ≥ κ(1 + |w a -w b |) where κ = κ 1+CN . Thus we get | ω, k + λ a -λ b | ≥ κ(1 + |w a -w b |), for ρ ∈ D ′ , 0 < |k| ≤ N and a, b ∈ L with meas D \ D ′ ≤ CN n+3 κ δ 0 1/3 . Lemma 3.2. -Assume that (x, u) → g(x, u) is real analytic on S d × R and s > 1 then there exist σ > 0, µ > 0 such that the maping O s (σ, µ) × D ∋ (r, θ, ζ; ρ) → f (r, θ, ζ; ρ) := S d G(x, û(r, θ, ζ)(x))dx,
where û is defined in (1.8), belongs to T s,1/2 (σ, µ, D) for any s of the form 2N -1 2 with N ∈ N and N > d.

Proof. -First we notice that f does not depend on the parameter ρ. Due to the analyticity of g and the fact that (5) s > d/2, there exist positive σ and µ such that

f : O(σ, µ) × D → C is a C 1 -function, analytic in the first variables (r, θ, ζ), whose gradient in ζ analytically maps Y s to Y -s . Further we have ∂f ∂ξ a = ∂f ∂η a = 1 2λ 1/2 a S d G(x, û(x))Ψ a (x) dx. Since x → G(x, û(x)) ∈ H s (S d ), we deduce that ∇ ζ f ∈ Y s+1/2 . It remains to verify that ∇ 2 ζ f (r, θ, ζ; ρ) ∈ M s,1/2 . We have (3.4) ∂ 2 f ∂ξ a ξ b = ∂ 2 f ∂η a η b = ∂ 2 f ∂ξ a η b = 1 2λ 1/2 a λ 1/2 b S d G(x, û(x))Ψ a Ψ b dx.
We note that for s > d/2 and (r,

θ, ζ) ∈ O s (σ, µ), x → û(x) is bounded on S d .
It remains to prove that the infinite matrix M defined by

M b a = 1 λ 1/2 a λ 1/2 b S d G(x, û)Ψ a Ψ b dx belongs to M s,1/2 , i.e. sup a,b∈L w 1/2 a w 1/2 b M [b] [a] w(a, b) + |w 2 a -w 2 b | w(a, b) s/2+1/4 < ∞ where we recall that w(a, b) = min(w a , w b ). The case w a = w b is straightforward, since λ a ∼ w a , x → û(x) is bounded on S d , and Φ a , Φ b are normalized in L 2 (S d ). If w a = w b , first we notice that M [b] [a] = sup u , v =1 | M [b] [a] u, v | = 1 λ 1/2 a λ 1/2 b sup Φa∈E [a] , Φa =1 Φ b ∈E [b] , Φ b =1 S d G(x, û)Φ a Φ b dx , where E [a] (resp. E [b]
) is the eigenspace of -∆ associated to the cluster [a] (resp. [b]). Then we follow arguments developed in [2, Proposition 2]. The basic idea lies in the following commutator lemma: Let A be a linear operator which maps H s (S d ) into itself and define the sequence of operators

A N := [-∆, A N -1 ], A 0 := A 5. s > d/2 is needed to insure that Ys is an algebra.
where ∆ denotes the Laplace Beltrami operator on S d , then with [2, Lemma 7], we have for any a, b ∈ L with w a = w b and any N ≥ 0

| AΦ a , Φ b | ≤ 1 |w 2 a -w 2 b | N | A N Φ a , Φ b |.
Let A be the operator given by the multiplication by the function

Φ(x) = G(x, û(r, θ, ζ)(x)).
We note that Φ ∈ H s+1/2 for (r, θ, ζ) ∈ O s (σ, µ). Then, by an induction argument, Lemma 8]). Therefore one gets

A N = 0≤|α|≤N C α,N D α where C α,N = 0≤|β|≤2N -|α| V α,β,N (x)D β Φ and V α,β,N are C ∞ functions (cf. [2,
| S d Φ a Φ b Φdx| ≤ 1 |w 2 a -w 2 b | N A N Φ a L 2 ≤ C 1 |w 2 a -w 2 b | N 0≤|α|≤N 0≤|β|≤2N -|α| ||D β ΦD α Ψ a || L 2 ≤ C 1 |w 2 a -w 2 b | N 0≤|α|≤N/2 0≤|β|≤2N -|α| Φ a |α|+ν 0 ||Φ|| |β| + N/2<|α|≤N 0≤|β|≤2N -|α| Φ a |α| ||Φ|| |β|+ν 0 ≤ C 1 |w 2 a -w 2 b | N Φ a N ||Φ|| 2N where we used ∀ν 0 > d/2 f g L 2 ≤ C f ν 0 g L 2 . On the other hand since -∆Φ a = w a (w a + d -1)Φ a (3.5) Φ a N ≤ Cw N a . Therefore choosing N = 1 2 (s + 1 2 ) | S d Φ a Φ b Φdx| ≤ C w a |w 2 a -w 2 b | s/2+1/4 ≤ 2 s/2+1/4 C w a w(a, b) + |w 2 a -w 2 b | s/2+1/4
.

Clearly the same estimate remains true when interchanging a and b.

So Main Theorem applies (for any choice of vector I ∈ [1, 2] A ) and Theorem 1.2 is proved.

Remark 3.3. -Theorem 1.2 still holds true when we consider the Klein Gordon equation on a Zoll manifold. This technical extension follows from results and computations in [START_REF] Delort | Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres[END_REF] and [START_REF] Bambusi | Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF]. We prefer to focus on the sphere in order to simplify the presentation.

Remark 3.4. -We can also consider the Beam equation on the torus T d with convolution potential in a Sobolev-like phase space:

(3.6)

u tt + ∆ 2 u + mu + V ⋆ u + ε∂ u G(x, u) = 0, x ∈ T d .
Here m is the mass, G is a real analytic function on T d × R vanishing at least of order 3 at the origin. The convolution potential V : T d → R is supposed to be analytic with real positive Fourier coefficients V (a), a ∈ Z d . The same equation, but in an analytic phase space, were considered in [START_REF] Eliasson | KAM for the nonlinear beam equation 2 : a normal form theorem[END_REF][START_REF] Eliasson | KAM for the nonlinear beam equation 1: small amplitude solutions[END_REF]. Actually following [START_REF] Eliasson | KAM for the nonlinear beam equation 2 : a normal form theorem[END_REF] and the proof of Lemma 3.2, in order to apply our abstract KAM theorem, it remains to control the | • | s,1/2 -norm of the infinite matrix (6) M

b a = 1 λ 1/2 a λ 1/2 b T d ∂ 2 u G(x, u)Ψ a Ψ b dx
restricted to the block defined by [a] = {b ∈ Z d | |a| = |b|}. This is achieved in the same way as in Lemma 3.2.

Remark 3.5. -Notice that our theorem does not apply to the nonlinear wave equation:

(3.7)

u tt + ∆u + mu + V ⋆ u + ε∂ u G(x, u) = 0, x ∈ T d
since in that case the second Melnikov condition is not satisfied.

Poisson brackets and Hamiltonian flows.

It turns out that the space T s,β (σ, µ, D) is not stable by Poisson brackets. Therefore, in this section, we first define a new space T s,β+ (σ, µ, D) ⊂ T s,β (σ, µ, D) and then we prove a structural stability which is essentially contained in the claim

{T s,β+ (σ, µ, D) , T s,β (σ, µ, D)} ∈ T s,β (σ, µ, D).
We will also study the hamiltonian flows generated by hamiltonian functions in T s,β+ (σ, µ, D). In this section, all constants C will depend only on s, β and n.

4.1. New Hamiltonian space. -We introduce T s,β+ (σ, µ, D) defined by

T s,β+ (σ, µ, D) = {f ∈ T s,β (σ, µ, D) | ∂ j ρ ∇ 2 ζ f ∈ M + s,β , j = 0, 1} where M + s,β = {M ∈ M s,β | |M | s,β+ < ∞} and |M | s,β+ = sup a,b∈L (1 + |w a -w b |) w(a, b) + |w 2 a -w 2 b | w(a, b) s 2 (w a w b ) β M [b] [a] .
We endow T s,β+ (σ, µ, D) with the norm

[f ] s,β+ σ,µ,D = [f ] s,β σ,µ,D + sup j=0,1 µ 2 |∂ j ρ ∇ 2 ζ f | s,β+ . Lemma 4.1. -Let 0 < β ≤ 1 and s > d/2 there exists a constant C ≡ C(β, s) > 0 such that (i) Let A ∈ M s,β and B ∈ M + s,β then AB and BA belong to M s,β and |AB| s,β , |BA| s,β ≤ C|A| s,β |B| s,β+ . (ii) Let A, B ∈ M + s,β then AB and BA belong to M + s,β and |AB| s,β+ , |BA| s,β+ ≤ C|A| s,β+ |B| s,β+ . (iii) Let A ∈ M + s,β then A ∈ L(Y s , Y s+β ) and Aζ s+β ≤ C|A| s,β+ ζ s . (iv) Let X ∈ Y s and Y ∈ Y s and denote A = X ⊗ Y then A and t A belong to L(Y s ) and A L(Ys) , t A L(Ys) ≤ C X s Y s . (v) Let X ∈ Y s+β and Y ∈ Y s+β then A = X ⊗ Y ∈ M s,β and A s,β ≤ C X s+β Y s+β .
6. Here λa = |a| 4 + m and Ψa(x) = e ia•x , a ∈ Z d .

Proof.

-(i) Let a, b ∈ L (AB) [b] [a] ≤ c∈ L A [c] [a] B [b] [c] ≤ |A| β+ |B| β (w a w b ) β w(a, b) w(a, b) + |w 2 a -w 2 b | s 2 c∈ L 1 w 2β c (1 + |w a -w c |) ≤ C |A| β+ |B| β (w a w b ) β w(a, b) w(a, b) + |w 2 a -w 2 b | s 2
where we used that by Lemma A.1

w(a, b) w(a, b) + |w 2 a -w 2 b | ≥ w(a, c) w(a, c) + |w 2 a -w 2 c | w(c, b) w(c, b) + |w 2 c -w 2 b | and that by Lemma A.2, c∈ L 1 w 2β c (1+|wa-wc|)
≤ C where C only depends on β.

(ii) Similarly let a, b ∈ L and assume without loss of generality that w a ≤ w b (AB)

[b] [a] ≤ c∈ L A [c] [a] B [b] [c] ≤ |A| β+ |B| β+ (w a w b ) β w(a, b) w(a, b) + |w 2 a -w 2 b | s 2 c∈ L 1 w 2β c (1 + |w a -w c |)(1 + |w b -w c |) ≤ 2|A| β+ |B| β+ (w a w b ) β (1 + |w a -w b |) w(a, b) w(a, b) + |w 2 a -w 2 b | s 2 c∈ L wc≤ 1 2 (wa+w b ) 1 w 2β c (1 + |w a -w c |) + c∈ L wc≥ 1 2 (wa+w b ) 1 w 2β c (1 + |w b -w c |) ≤ C |A| β+ |B| β+ (w a w b ) β (1 + |w a -w b |) w(a, b) w(a, b) + |w 2 a -w 2 b | s 2 . (iii) Let ζ ∈ Y s we have Aζ 2 s+β ≤ a∈ L w 2s+2β a b∈ L A [b] [a] ζ [b] 2 ≤ |A| 2 s,β+ a∈ L b∈ L w s a w s b ζ [b] w s+β b (1 + |w a -w b |) w(a, b) w(a, b) + |w 2 a -w 2 b | s 2 2 ≤ 2 2s+1 |A| 2 s,β+ a∈ L b∈ L wa≤2w b ) w s b ζ [b] w β b (1 + |w a -w b |) + b∈ L wa≥2w b w s b ζ [b] w(a, b) s 2 w s+β b (1 + |w a -w b |) 2 ≤ 2 2s+1 |A| 2 s,β+ a∈ L b∈ L w s b ζ [b] w β b (1 + |w a -w b |) 2 ≤ C|A| 2 s,β+ ζ 2 s
where we used that the convolution between the ℓ p sequence, p < 2,

w s-β b ζ [b] and the ℓ q sequence, q = 2p 3p-2 > 1, 1 (1+|w b |) is a ℓ 2 sequence in a whose norm is bounded by C ζ s . (iv) Let u ∈ Y s , we have Au s = | Y, u | X s ≤ X s Y s u s . (v) Let a, b ∈ L A [b] [a] = X [a] Y [b] ≤ (w a w b ) -s-β X s+β Y s+β ≤ (w a w b ) -β 1 (1 + |w 2 a -w 2 b |) s/2 X s+β Y s+β ≤ (w a w b ) -β w(a, b) (w(a, b) + |w 2 a -w 2 b |) s/2 X s+β Y s+β .
4.2. Jets of functions.-For any function h ∈ T s (σ, µ, D) we define its jet h T = h T (x, ρ) as the following Taylor polynomial of h at r = 0 and ζ = 0:

h T =h θ + h r , r + h ζ , ζ + 1 2 h ζζ ζ, ζ =h(θ, 0, ρ) + ∇ r h(θ, 0, ρ), r + ∇ ζ h(θ, 0, ρ), ζ + 1 2 ∇ 2 ζζ h(θ, 0, ρ)ζ, ζ (4.1)
Functions of the form h T will be called jet-functions.

Directly from the definition of the norm [h] s,β σ,µ,D we get that

|h θ (θ, ρ)| ≤ [h] s σ,µ,D , |h r (θ, ρ)| ≤ µ -2 [h] s σ,µ,D , h ζ (θ, ρ) s+β ≤ µ -1 [h] s,β σ,µ,D , |h ζζ (θ, ρ)| s,β ≤ µ -2 [h] s,β σ,µ,D , (4.2)
for any θ ∈ T n σ and any ρ ∈ D. Moreover, the first derivative with respect to ρ will satisfy the same estimates. We also notice that by Cauchy estimates we have that for

x ∈ O(σ, µ ′ ) (4.3) ∇ 2 ζ h(x) L(Ys,Y s+β ) ≤ sup y∈O(σ,µ) ∇ ζ h(y) s µ -µ ′ .
Thus h ζζ is a linear continuous operator from Y s to Y s+β and (4.4)

h ζζ (θ, ρ) L(Ys,Y s+β ) ≤ µ -2 [h] s σ,µ,D
for any θ ∈ T n σ and any ρ ∈ D.

Proposition 4.2. -For any h ∈ T s,β (σ, µ, D) we have h T ∈ T s,β (σ, µ, D),

[h T ] s,β σ,µ,D ≤ C[h] s,β σ,µ,D , and, for any 0 < µ ′ < µ, [h -h T ] s,β σ,µ ′ ,D ≤ C µ ′ µ 3 [h] s,β σ,µ,D ,
where C is an absolute constant.

Proof. -We start with the second statement. Consider first the hessian

∇ 2 ζζ (h -h T )(x) for x = (θ, r, ζ) ∈ O s (σ, µ ′ ). Let us denote m = µ ′ /µ. Then for z ∈ D 1 = {z ∈ C : |z| ≤ 1} we have (θ, (z/m) 2 r, (z/m)ζ) ∈ O s (σ, µ). Consider the function f : D 1 × O s (σ, µ ′ ) → M β , (z, x) → ∇ 2 ζζ h(θ, (z/m) 2 r, (z/m)ζ) = h 0 (x) + h 1 (x)z + . . . .
It is holomorphic and its norm is bounded by µ -2 [h] s,β σ,µ,D . So, by the Cauchy estimate,

|h j (x)| s,β ≤ µ -2 [h] s,β σ,µ,D for j = 1, 2, . . . and x ∈ O s (σ, µ ′ ). Since ∇ 2 ζζ (h -h T )(x) = h 1 (x)m + h 2 (x)m 2 + • • • , then ∇ 2 ζζ (h -h T ) is holomorphic in x ∈ O s (σ, µ), and
|∇ 2 ζζ (h -h T )(x)| s,β ≤ µ -2 [h] s,β σ,µ,D (m + m 2 + . . . ) ≤ µ -2 [h] s,β σ,µ,D m 1 -m . So ∇ 2 ζζ (h -h T ) satisfies the required estimate with C = 2, if µ ′ ≤ µ/2. Same argument applies to bound the norms of ∂ ρ ∇ 2 ζζ (h -h T ), h -h T and ∇ ζ (h -h T ) if µ ′ ≤ µ/2,
and to prove the analyticity of these mappings. Now we turn to the first statement and write

h T as h -(h -h T ). This implies that h T , ∇ ζ h T and ∇ 2 ζζ h T are analytic on O s (σ, 1 2 µ) and that [h T ] s,β σ, 1 2 µ,D ≤ C 1 [h] s,β σ,µ,D .
Since h T is a quadratic polynomial, then the mappings h T , ∇ ζ h T and ∇ 2 ζζ h T are as well analytic on O s (σ, µ), and the norm [h T ] s,β σ,µ,D satisfies the same estimate, modulo another constant factor, for any 0 < µ ′ ≤ µ.

Finally, the estimate for [h

-h T ] s,β σ,µ ′ ,D when µ/2 ≤ µ ′ ≤ µ, with a suitable constant C, follows from the estimate for [h T ] s,β σ,µ,D since [h -h T ] s,β σ,µ ′ ,D ≤ [h T ] s,β σ,µ,D + [h] s,β σ,µ,D .
4.3. Poisson brackets and flows. -The Poisson brackets of functions is defined by

(4.5) {f, g} = ∇ r f • ∇ θ g -∇ θ f • ∇ r g + J∇ ζ f, ∇ ζ g . Lemma 4.3. -Let s ≥ 1.
Let f ∈ T s,β+ (σ, µ, D) and g ∈ T s,β (σ, µ, D) be two jet functions then for any 0 < σ ′ < σ we have {f, g} ∈ T s,β (σ ′ , µ, D) and

[{f, g}] s,β σ ′ ,µ,D ≤ C(σ -σ ′ ) -1 µ -2 [f ] s,β+ σ,µ,D [g] s,β σ,µ,D . Proof.
-Let denote by h 1 , h 2 , h 3 the three terms on the right hand side of (4.5). Since ∇ r f (θ, r, ζ, ρ) = f r (θ, ρ) and ∇ r g(θ, r, ζ, ρ) = g r (θ, ρ) are independent of r and ζ, the control of h 1 and h 2 is straightforward by Cauchy estimates and (4.2). We focus on the third term in formula:

h 3 = J∇ ζ f, ∇ ζ g . As, from (4.1), we have ∇ ζ f = f ζ + f ζζ ζ and similarly for ∇ ζ g, we obtain h 3 = Jf ζ , g ζ -ζ, f ζζ Jg ζ + g ζζ Jf ζ , ζ + g ζζ Jf ζζ ζ, ζ .
Using (4.2), (4.4) and ζ s ≤ µ, we get

|h 3 (x, •)| ≤ Cµ -2 [f ] s,β σ,µ,D [g] s,β σ,µ,D , for any x ∈ O(σ, µ) and ρ ∈ D. Since ∇ ζ h 3 = -f ζζ Jg ζ + g ζζ Jf ζ + g ζζ Jf ζζ ζ -f ζζ Jg ζζ ζ
, then, using (4.4) and Lemma 4.1, we get that for x ∈ O s (σ, µ) and ρ ∈ D

∇ ζ h 3 (x, •) s+β ≤ Cµ -3 [f ] s,β σ,µ,D [g] s,β σ,µ,D . Finally, as ∇ 2 h 3 = g ζζ Jf ζζ -f ζζ Jg ζζ , then, using again Lemma 4.1 we get that for x ∈ O s (σ, µ) and ρ ∈ D |∇ 2 h 3 (x, •)| s,β ≤ Cµ -4 [f ] s,β+ σ,µ,D [g] s,β σ,µ,D .
   ṙ = ∇ θ f (r, θ, ζ; ρ), θ = -∇ r f (r, θ, ζ; ρ), ζ = J∇ ζ f (r, θ, ζ; ρ).
and denote by Φ t f ≡ Φ t , t ∈ R, the corresponding flow map (if it exists). Now let f ≡ f T be a jet-function

(4.7) f = f θ (θ; ρ) + f r (θ; ρ) • r + f ζ (θ; ρ), ζ + 1 2 f ζζ (θ; ρ)ζ, ζ .
Then Hamilton equations (4.6) take the form (7) (4.8)

   ṙ = -∇ θ f (r, θ, ζ), θ = f r (θ), ζ = J (f ζ (θ) + f ζζ (θ)ζ) .
Denote by

V f = (V r f , V θ f , V ζ f ) the corresponding vector field. It is analytic on any domain O s (σ -2η, µ -2ν) =: O 2η,2ν
, where 0 < 2η < σ, 0 < 2ν < µ. The flow maps Φ t f of V f on O 2η,2ν are analytic as long as they exist. We will study them as long as they map O 2η,2ν to O η,ν . Assume that

(4.9) [f ] s σ,µ,D ≤ 1 2 ν 2 η.
Then for x = (r, θ, ζ) ∈ O 2η,2ν by the Cauchy estimate (8) and (4.4) we have

|V r f | C n ≤ (2η) -1 [f ] s σ,µ,D ≤ ν 2 , |V θ f | C n ≤ (4ν 2 ) -1 [f ] s σ,µ,D ≤ η, V ζ f s ≤ µ -1 + µ -2 µ [f ] s σ,µ,D ≤ ν.
Noting that the distance from O 2η,2ν to ∂O η,ν in the r-direction is 2νµ -3ν 2 > ν 2 , in the θ-direction is η and in the ζ-direction is ν, we see that the flow maps

(4.10) Φ t f : O s (σ -2η, µ -2ν) → O s (σ -η, µ -ν), 0 ≤ t ≤ 1,
are well defined and analytic.

For

x ∈ O 2s,2ν denote Φ t f (x) = (r(t), θ(t), ζ(t)). Since V θ f is independent from r and ζ, then θ(t) = K(θ; t), where K is analytic in both arguments. As V ζ f = Jf ζ + Jf ζζ ζ
, where the non autonomous linear operator Jf ζζ (θ(t)) is bounded in the space Y s and both the operator and the curve Jf ζ (θ(t)) analytically depend on θ (through θ(t) = K(θ; t)), then ζ(t) = T (θ, t) + U (θ; t)ζ, where U (θ; t) is a bounded linear operator, both U and T analytic in θ. Similar since V ζ f is a quadratic polynomial in ζ and an affine function of r, then r(t) = L(θ, ζ; t) + S(θ; t)r, where S is an n × n matrix and L is a quadratic polynomial in ζ, both analytic in θ.

The vector field V f is real for real arguments, and so behaves its flow map. Since the vector field is hamiltonian, then the flow maps are symplectic (e.g., see [START_REF] Kuksin | Nearly integrable infinite-dimensional Hamiltonian systems[END_REF]). We have proven Lemma 4.4. -Let 0 < 2η < σ, 0 < 2ν < µ and f = f T ∈ T s (σ, µ, D) satisfy (4.9). Then for 0 ≤ t ≤ 1 the flow maps Φ t f of equation (4.8) define analytic mappings (4.10) and define symplectomorphisms from O s (σ -2η, µ -2ν) to O s (ση, µν). They have the form

(4.11) Φ t f :   r θ ζ   →   L(θ, ζ; t) + S(θ; t)r K(θ; t) T (θ; t) + U (θ; t)ζ   ,
where L(θ, ζ; t) is quadratic in ζ, while U (θ; t) and S(θ; t) are bounded linear operators in corresponding spaces.

Our next result specifies the flow maps Φ t f and their representation (4.11) when f ∈ T s,β+ (σ, µ, D):

Lemma 4.5. -Let 0 < 2η < σ ≤ 1, 0 < 2ν < µ ≤ 1 and f = f T ∈ T s,β+ (σ, µ, D) satisfy (4.12) [f ] s,β+ σ,µ,D ≤ 1 2 ν 2 η 7.
Here and below we often suppress the argument ρ. 8. Notice that the distance from O s (σ -2η, µ -2ν) to ∂O s (σ, µ) in the r-direction is 4νµ -4ν 2 > 4ν 2 .

Then:

1) Mapping L is analytic in (θ, ζ) ∈ T σ-2η × O µ (Y s ).
Mappings K, T and operators S and U analytically depend on θ ∈ T σ-2η ; their norms and operator-norms satisfy

S(θ; t) L(C n ,C n ) , t U (θ; t) -I L(Ys,Y s+β ) , U (θ; t) -I L(Ys,Y s+β ) , |U (θ; t) -I| s,β+ ≤ 2, (4.13)
while for any component L j of L and any (θ, r, ζ) ∈ O s (σ -2η, µ -2ν) we have

∇ ζ L j (θ, ζ; t) s+β ≤ Cη -1 µ -1 [f ] s,β+ σ,µ,D , |∇ 2 ζ L j (θ, ζ; t)| s,β+ ≤ Cη -1 µ -2 [f ] s,β+ σ,µ,D . (4.14)
2) The flow maps Φ t f analytically extend to mappings

C n × T n σ-2η × Y s ∋ x 0 = (r 0 , θ 0 , ζ 0 ) → x(t) ∈ C n × T n σ × Y s , x(t) = (r(t), θ(t), ζ(t)), which satisfy |r(t) -r 0 | ≤ 4η -1 1 + µ -1 ζ 0 s + µ -2 |r 0 | + µ -2 ||ζ 0 || 2 s [f ] s,β+ σ,µ,D , |θ(t) -θ 0 | ≤ µ -2 [f ] s,β+ σ,µ,D , ζ(t) -ζ 0 s+β ≤ µ -2 ζ 0 s + µ -1 [f ] s,β+ σ,µ,D , (4.15)
Moreover, the ρ-derivative of the mapping x 0 → x(t) satisfies the same estimates as the increments x(t)x 0 .

Proof. -Consider the equation for ζ(t) in (4.8):

(4.16) ζ(t) = a(t) + B(t)ζ(t), ζ(0) = ζ 0 ∈ O µ-2ν (Y s ),
where

a(t) = Jf ζ (θ(t)) is an analytic curve [0, 1] → Y γ and B(t) = Jf ζζ (θ(t)) is an analytic curve [0, 1] → M.
Both analytically depend on θ 0 . By the hypotheses and using (4.3) (4.17)

||a(t)|| s ≤ µ -1 [f ] s σ,µ,D , B L(Ys,Ys) ≤ µ -2 [f ] s σ,µ,D ≤ 1 2 ν ≤ 1 2 .
On the other hand by Lemma 4.

ζ(t) = a ∞ (t) + (I + B ∞ (t))ζ 0 ,
where

a ∞ (t) = t 0 a(t 1 )dt 1 + k≥2 t 0 t 1 0 • • • t k-1 0 k-1 j=1 B(t j )a(t k )dt k • • • dt 2 dt 1 , and
B ∞ (t) = k≥1 t 0 t 1 0 • • • t k-1 0 k j=1 B(t j )dt k • • • dt 2 dt 1 .
Due to (4.12), (4.17) and (4.18), for each k and for 0 ≤ t k ≤ . . . t 1 ≤ 1 we have that

B(t 1 ) . . . B(t k ) L(Ys,Y s+β ) ≤ ( 1 2 ) k-1 µ -2 [f ] s,β+ σ,µ,D .
By this relation and (4.17) we get that a ∞ and B ∞ are well defined for t ∈ [0, 1] and satisfy

B ∞ (t) L(Ys,Y s+β ) ≤ µ -2 [f ] s,β+ σ,µ,D , a ∞ (t) s+β ≤ µ -3 ([f ] s,β+ σ,µ,D ) 2 ≤ µ -1 [f ]
ṙ(t) = -α(t) -Λ(t)r(t), r(0) = r 0 ∈ O (µ-2ν) 2 (C n )
where Λ(t) = ∇ θ f r (θ(t)) and

(4.22) α(t) = ∇ θ f θ (θ(t)) + ∇ θ f ζ (θ(t)), ζ(t) + 1 2 ∇ θ f ζζ (θ(t))ζ(t), ζ(t) .
The curve of matrices Λ(t) and the curve of vectors α(t) analytically depend on θ 0 ∈ T n σ-2η . Besides, α(t) analytically depends on ζ 0 ∈ Y s , while Λ is ζ 0 -independent.

By the Cauchy estimate and (4.12), for any θ(t) ∈ T n σ-η we have

|Λ(t)| L(C n ,C n ) ≤ η -1 µ -2 [f ] s σ,µ,D ≤ 1 2 , |α(t)| ≤ 2η -1 [f ] s σ,µ,D (1 + µ -1 ζ 0 s + µ -2 ζ 0 2 s ) (4.23)
where for the second estimate we used that

ζ(t) -ζ 0 s ≤ 1 + ζ 0 s . Since ∇ ζ(t) α(t) = ∇ θ f ζ (θ(t)) + ∇ θ f ζζ (θ(t))ζ(t) and ∇ ζ 0 = t U (θ; t)∇ ζ ,
∇ ζ 0 α(t) s+β ≤ 4η -1 µ -1 [f ] s,β+ σ,µ,D (1 + µ -1 ζ 0 s ). Since ∇ 2 ζ 0 α(t) = t U ∇ 2 ζ(t) α(t)U = t U ∇ θ f ζζ (θ(t))U ,
r(t) = -α ∞ (t) + (1 -Λ ∞ (t))r 0 , where (4.27) α ∞ (t) = t 0 α(t 1 )dt 1 + k≥2 t 0 t 1 0 • • • t k-1 0 k-1 j=1 Λ(t j )α(t k )dt k • • • dt 2 dt 1 , and
(4.28) Λ ∞ (t) = k≥1 t 0 t 1 0 • • • t k-1 0 k j=1 Λ(t j )dt k • • • dt 2 dt 1 .
Using (4.23) we get that

|Λ ∞ (t)| L(C n ×C n ) ≤ 1 2 , |α ∞ (t)| C n ≤ 2η -1 1 + µ -1 ζ 0 s + µ -2 ζ 0 2 s [f ] s σ,µ,D .
Since in (4.11) S(θ; 1) = I -Λ ∞ (t), then the first estimate in (4.13) follows. Since Λ ∞ (t) in (4.26) is ζ 0 -independent, then L(θ, ζ; t) = -α ∞ (t). This is a quadratic in ζ 0 expression, and the estimates (4.14) follow from (4.24)-(4.25) and in view of the estimate for Λ ∞ above. Finally using the estimates for Λ ∞ and α ∞ we get from (4.26) that r = r(t) satisfies (4.15) 1 , as (4.15) 2 directly comes from (4.8) and (4.2).

Next we study how the flow maps Φ t f transform functions from T s,β (σ, µ, D).

Lemma 4.6. -Let 0 < 2η < σ ≤ 1, 0 < 2ν < µ ≤ 1. Assume that f = f T ∈ T s,β+ (σ, µ, D) satisfies (4.12). Let h ∈ T s,β (σ, µ, D) and denote for 0 ≤ t ≤ 1

h t (x; ρ) = h(Φ t f (x; ρ); ρ). Then h t ∈ T s,β (σ -2η, µ -2ν, D) and [h t ] s,β σ-2η,µ-2ν,D ≤ C µ ν [h] s,β σ,µ,D
where C is an absolute constant.

Proof. -Let us write the flow map Φ t f as

x 0 = (r 0 , θ 0 , ζ 0 ) → x(t) = (r(t), θ(t), ζ(t)). By Lemma 4.5, h t (x 0 ) is analytic in x 0 ∈ O(σ -2η, µ -2ν). Clearly |h t (x 0 , •)| ≤ [h] s,β
σ,µ,D for x 0 ∈ O(σ -2s, µ -2ν) and ρ ∈ D. So it remains to estimate the gradient and hessian of h(x 0 ). 1) Estimating the gradient. Since θ(t) does not depend on ζ 0 , we have

∂h t ∂ζ 0 = n k=1 ∂h(x(t)) ∂r k ∂r k (t) ∂ζ 0 + b∈L ∂h(x(t)) ∂ζ b (t) ∂ζ b (t) ∂ζ 0 = Σ 1 + Σ 2 . i) Since x(t) ∈ O(σ -η, µ -ν), we get by the Cauchy estimate that ∂h(x(t)) ∂r k ≤ 1 3ν 2 [h] s σ,µ,D .
As ∇ ζ 0 r k (t) was estimated in (4.14), then using (4.12) we get

Σ 1 s+β ≤ Cν -2 [h] s,β σ,µ,D η -1 µ -1 [f ] s,β σ,µ,D ≤ Cµ -1 [h] s,β σ,µ,D . ii) Noting that Σ 2 (r, θ, ζ) = t U (θ; t)∇ ζ h,
∂ ρ ∇ ζ 0 h t s+β ≤ Cµ -1 [h] s,β σ,µ,D . 2) Estimating the hessian. Since θ(t) does not depend on ζ 0 and since ζ(t) is affine in ζ 0 , then ∂ 2 h t ∂ζ 0 a ∂ζ 0 b (x) = ∂ 2 h(x(t)) ∂ζ∂ζ ∂ζ(t) ∂ζ 0 a ∂ζ(t) ∂ζ 0 b + ∂ 2 h(x(t)) ∂r 2 ∂r(t) ∂ζ 0 a ∂r(t) ∂ζ 0 b + ∂ 2 h(x(t)) ∂r∂ζ ∂r(t) ∂ζ 0 a ∂ζ(t) ∂ζ 0 b + ∂h(x(t)) ∂r ∂ 2 r(t) ∂ζ 0 a ∂ζ 0 b =: ∆ 1 + ∆ 2 + ∆ 3 + ∆ 4 . (4.29) i) We have |∂ 2 h/∂ζ a ∂ζ b | β ≤ Cµ -2 [h] s,β σ,µ,D .
Using this estimate jointly with (4.13) and Lemma 4.1 we see that

|∆ 1 | β ≤ Cµ -2 [h] s,β σ,µ,D . ii) Since for x 0 ∈ O s (σ -2s, µ -2ν) by (4.14) we have ∇ ζ r j s+β ≤ Cη -1 µ -1 [f ] s,β+ σ,µ,D ,
and since by Cauchy estimate |d 2 r h| ≤ Cν -4 [h] s,β σ,µ,D , we get using Lemma 4.1(v) and (4.12)

|∆ 2 | β ≤ Cν -4 [h] s,β σ,µ,D η -2 µ -2 ([f ] s,β+ σ,µ,D ) 2 ≤ Cµ -2 [h] s,β σ,µ,D .
iii) For any j we have by the Cauchy estimate that ∂ ∂r j ∇ ζ h s+β ≤ Cν -3 [h] s,β σ,µ,D . Therefore by (4.13)

a ′ ∂ 2 h ∂r j ∂ζ a ′ ∂ζ a ′ ∂ζ 0 a s+β ≤ Cν -3 [h] s,β σ,µ,D .
Since

∇ ζ 0 r j s+β ≤ Cη -1 µ -1 [f ] s,β
σ,µ,D ≤ Cν 2 µ -1 by (4.14), then using Lemma 4.1 (v) we find that

|∆ 3 | β ≤ Cν -1 µ -1 [h] s,β σ,µ,D . iv) As |∂h/∂r(x(t))| ≤ ν -2 [h] s,β σ,µ,D and
∂ 2 r ∂ζ 0 a ∂ζ 0 b β ≤ Cη -1 µ -2 [f ] s,β+ σ,µ,D
by (4.14), then

|∆ 4 | β ≤ Cµ -2 [h] s,β σ,µ,D .
The ρ-gradient of the hessian leads to estimates similar to the above. So the lemma is proven.

We summarize the results of this section into a proposition.

Proposition 4.7. -Let 0 < σ ′ < σ ≤ 1, 0 < µ ′ < µ ≤ 1.There exists an absolute constant C ≥ 1 such that (i) if f = f T ∈ T s,β (σ, µ, D) and (4.30) [f ] s,β σ,µ,D ≤ 1 2 (µ -µ ′ ) 2 (σ -σ ′ ),
then for all 0 ≤ t ≤ 1, the Hamiltonian flow map

Φ t f is a C 1 -map O s (σ ′ , µ ′ ) × D → O s (σ, µ);
real holomorphic and symplectic for any fixed ρ ∈ D. Moreover,

||Φ t f (x, •) -x|| s,D ≤ C 1 σ -σ ′ + 1 µ 2 [f ] s,β σ,µ,D for any x ∈ O s (σ ′ , µ ′ ). (ii) if f = f T ∈ T s,β+ (σ, µ, D) and (4.31) [f ] s,β+ σ,µ,D ≤ 1 2 (µ -µ ′ ) 2 (σ -σ ′ ),
then for all 0 ≤ t ≤ 1 and all h ∈ T s,β (σ, µ, D), the function h t (x; ρ) = h(Φ t f (x, ρ); ρ) belongs to T s,β (σ ′ , µ ′ , D) and

[h t ] s,β σ ′ ,µ ′ ,D ≤ C µ (µ -µ ′ ) [h] s,β σ,µ,D .
Proof. -Take σ ′ = σ -2s and µ ′ = µ -2ν and apply Lemmas 4.5 and 4.6.

Homological equation

Let us first recall the KAM strategy. Let h 0 be the normal form Hamiltonian given by (2.9)

h 0 (r, ζ, ρ) = ω 0 (ρ), r + 1 2 ζ, A 0 ζ
satisfying Hypotheses A1-A3. Let f be a perturbation and

f T = f θ + f r , r + f ζ , ζ + 1 2 f ζζ ζ, ζ
be its jet (see (4.1)). If f T were zero, then {ζ = r = 0} would be an invariant n-dimensional torus for the Hamiltonian h 0 + f . In general we only know that f is small, say f = O(ε), and thus f T = O(ε). In order to decrease the error term we search for a hamiltonian jet S = S T = O(ε) such that its time-one flow map Φ S = Φ 1 S transforms the Hamiltonian

h 0 + f to (h 0 + f ) • Φ S = h + f + ,
where h is a new normal form, ε-close to h 0 , and the new perturbation f + is such that its jet is much smaller than f T . More precisely,

h = h 0 + h, h = c(ρ) + χ(ρ), r + 1 2 ζ, B(ρ)ζ = O(ε),
and (f

+ ) T = O(ε 2 ).
As a consequence of the Hamiltonian structure we have (at least formally) that

(h 0 + f ) • Φ S = h 0 + {h 0 , S} + f T + O(ε 2 ).
So to achieve the goal above we should solve the homological equation:

(5.1) {h 0 , S} = h -f T + O(ε 2 ).
Repeating iteratively the same procedure with h instead of h 0 etc., we will be forced to solve the homological equation, not only for the normal form Hamiltonian (2.9), but for more general normal form Hamiltonians (2.4) with ω close to ω 0 and A close to A 0 .

In this section we will consider a homological equation (5.1) with f in T s,β (σ, µ, D) and we will build a solution S in T s,β+ (σ, µ, D). In this section, constants C may take different values, but will only depend on s, β, n, d * , γ, c 0 , α 1 and α 2 given in Hypothesis A1, A2 and A3.

Four components of the homological equation. -Let h be a normal form Hamil

- tonian (2.4), h(r, ζ, ρ) = ω(ρ), r + 1 2 ζ, A(ρ)ζ ,
and let us write a jet-function S as

S(θ, r, ζ) = S θ (θ) + S r (θ), r + S ζ (θ), ζ + 1 2 S ζζ (θ)ζ, ζ .
Therefore the Poisson bracket of h and S equals {h,

S} = (∇ θ • ω)S θ + (∇ θ • ω)S r , r + (∇ θ • ω)S ζ , ζ + 1 2 (∇ θ • ω)S ζζ , ζ -AJS ζ , ζ + S ζζ JAζ, ζ .
Accordingly the homological equation (5.1) with h 0 replaced by h decomposes into four linear equations. The first two are

∇ θ S θ , ω = -f θ + c + O(ε 2 ), (5.2) ∇ θ S r , ω = -f r + χ + O(ε 2 ). (5.3)
In these equations, we are forced to choose

c(ρ) = [ [ f θ (•, ρ) ] ] and χ(ρ) = [ [ f r (•, ρ) ] ]
where

[ [ f ]
] denotes averaging of a function f in θ ∈ T n , to get that the space mean-value of the r.h.s. vanishes. The other two equations are

∇ θ S ζ , ω -AJS ζ = -f ζ + O(ε 2 ), (5.4) ∇ θ S ζζ , ω -AJS ζζ + S ζζ JA = -f ζζ + B + O(ε 2 ) , (5.5)
where the operator B will be chosen later. The most delicate, involving the small divisors (see (2.8)), is the last equation.

∇ θ ϕ(θ, ρ), ω(ρ) = ψ(θ, ρ) with [ [ ψ ] ] = 0. Here ω : D → R n is C 1 and verifies |ω -ω 0 | C 1 (D) ≤ δ 0 .
Expanding ϕ and ψ in Fourier series,

ϕ = k∈Z n \{0} φ(k)e ik•θ , ψ = k∈Z n \{0} ψ(k)e ik•θ ,
we solve eq. ( 5.6) by choosing

φ(k) = - i ω, k ψ(k), k ∈ Z n \ {0}; φ(0) = 0.
Using Assumption A2 we have, for each k = 0, either that

| ω(ρ), k | ≥ δ 0 ∀ρ or that (∇ ρ • z)( k, ω(ρ) ) ≥ δ 0 ∀ρ
for a suitable choice of a unit vector z. The second case implies that

| ω(ρ), k | ≥ κ ,
where κ ≤ δ 0 , for all ρ outside some open set

F k ≡ F k (ω) of Lebesgue measure ≤ δ -1 0 κ. Let D 1 = D \ 0<|k|≤N F k .
Then

| φ(k)| ≤ 1 κ | ψ(k)| . Setting ϕ(θ, ρ) = 0<|k|≤N φ(k, ρ)e ik•θ , we get that (5.7) ∇ θ ϕ(θ, ρ), ω(ρ) = ψ(θ, ρ) + R(θ, ρ).
Hence ϕ is an approximate solution of eq. (5.6) with the error term R(θ, ρ) = -|k|>N ψ(k, ρ)e ik•θ . We obtain by a classical argument that for (θ, ρ) ∈ T n σ ′ × D 1 , 0 < σ ′ < σ, and j = 0, 1

|ϕ(θ, ρ)| ≤ C κ(σ -σ ′ ) n sup |ℑθ|<σ |ψ(θ, ρ)|, |∂ j ρ R(θ, ρ)| ≤ C e -1 2 (σ-σ ′ )N (σ -σ ′ ) n sup |ℑθ|<σ |∂ j ρ ψ(θ, ρ)| , (5.8)
where C only depends on n. If ψ is a real function, then so are ϕ and R.

Differentiating in ρ the definition of φ(k) gives (9) ∂

ρ φ(k) = χ |k|≤N (k) - i ω, k ∂ ρ ψ(k) + i ω, k 2 ∂ ρ ω, k ψ(k) .
From this we derive that

|∂ ρ ϕ(θ, ρ)| ≤ C(|ω 0 (ρ)| C 1 + 1)N κ 2 (σ -σ ′ ) n sup |ℑθ|<σ |ψ(θ, ρ)| + sup |ℑθ|<σ |∂ ρ ψ(θ, ρ)| ,
where we estimated the derivative of ω by |ω 0 (ρ

)| C 1 + δ 0 ≤ |ω 0 (ρ)| C 1 + 1.
Applying this construction to (5.2) and ( 5.

ω 0 | C 1 (D) ≤ δ 0 . Let f ∈ T s (σ, µ, D) and let δ 0 ≥ κ > 0, N ≥ 1.
Then there exists a closed set

D 1 = D 1 (ω, κ, N ) ⊂ D, satisfying meas(D \ D 1 ) ≤ CN n κ δ 0 , and
(i) there exist real C 1 -functions S θ and R θ on T n σ × D 1 → C, analytic in θ, such that ∇ θ S θ (θ, ρ), ω(ρ) = -f θ (θ, ρ) + [ [ f θ (•, ρ) ] ] + R θ (θ, ρ)
and for all (θ, ρ) ∈ T n σ ′ × D 1 , σ ′ < σ, and j = 0, 1

|∂ j ρ S θ (θ, ρ)| ≤ CN κ 2 (σ -σ ′ ) n [f ] s σ,µ,D 1 , |∂ j ρ R θ (θ, ρ)| ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n [f ] s σ,µ,D 1 .
(ii) there exist real C 1 vector-functions S r and R r on

T n σ × D 1 , analytic in θ, such that ∇ θ S r (θ, ρ), ω(ρ) = -f r (θ, ρ) + [ [ f r (•; ρ) ] ] + R r (θ, ρ),
and for all (θ, ρ) ∈ T n σ ′ × D 1 , σ ′ < σ, and j = 0, 1

|∂ j ρ S r (θ, ρ)| ≤ C κ 2 (σ -σ ′ ) n [f ] s σ,µ,D 1 , |∂ j ρ R r (θ, ρ)| ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n [f ] s σ,µ,D 1 .
The constant C only depends on |ω 0 | C 1 (D) .

The third equation. -

To begin with, we recall a result proved in the appendix of [START_REF] Eliasson | KAM for the nonlinear Schrödinger equation[END_REF].

Lemma 5.2. -Let A(t) be a real diagonal N × N -matrix with diagonal components a j which are C 1 on I =] -1, 1[, satisfying for all j = 1, . . . , N and all t ∈ I a ′ j (t) ≥ δ 0 . Let B(t) be a Hermitian N × N -matrix of class C 1 on I such that (10) B ′ (t) ≤ δ 0 /2, for all t ∈ I. Then

(A(t) + B(t)) -1 ≤ 1 ε outside a set of t ∈ I of Lebesgue measure ≤ CN εδ -1
0 , where C is a numerical constant. Concerning the third component (5.4) of the homological equation we have

ω 0 | C 1 (D) ≤ δ 0 . Let D ∋ ρ → A(ρ) ∈ N F ∩ M 0 be C 1 and verifying (5.9) ∂ j ρ (A(ρ) -A 0 ) [a] ≤ 1 2 δ 0 for j = 0, 1, a ∈ L and ρ ∈ D. Let f ∈ T s (σ, µ, D), 0 < κ ≤ min( δ 0 2 , c 0
∇ θ S ζ (θ, ρ), ω(ρ) -A(ρ)J S ζ (θ, ρ) = -f ζ (θ, ρ) + R ζ (θ, ρ)
and for all (θ, ρ) ∈ T n σ ′ × D 2 , σ ′ < σ, and j = 0, 1

µ ∂ j ρ S ζ (θ, ρ) s+1 ≤ CN κ 2 (σ -σ ′ ) 2n [f ] s,β σ,µ,D , µ ∂ j ρ R ζ (θ, ρ) s ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n [f ] s,β σ,µ,D .
The exponent exp only depends on d * , n, γ while the constant C also depends on

|ω 0 | C 1 (D) .
Proof. -It is more convenient to deal with the hamiltonian operator JA than with operator AJ. Therefore we multiply eq. ( 5.10) by J and obtain for JS ζ the equation

(5.11) ∇ θ (JS ζ )(θ, ρ), ω(ρ) -JA(ρ)(JS ζ )(θ, ρ) = -Jf ζ (θ, ρ) + JR ζ (θ, ρ)
Let us re-write (5.11) in the complex variables t (ξ, η). For a ∈ L (5.12)

ζ a = p a q a = U a ξ a η a , U a = 1 √ 2 1 1 -i i .
The symplectic operator U a transforms the quadratic form (λ a /2) ζ a , ζ a to iλ a ξ a η a . Therefore, if we denote by U the direct product of the operators diag (U a , a ∈ L) then it transforms (1/2) ζ, A 0 ζ to a∈L iλ a ξ a η a . So it transforms the hamiltonian matrix JA 0 to the diagonal hamiltonian matrix diag {iλ a -1 0 0 1 , a ∈ L}.

Then we make in (5.11) the substitution

JS ζ = U S, JR ζ = U R and -Jf ζ = U F ζ
, where S = t (S ξ , S η ), etc. In this notation eq. (5.10) decouples into two equations

∇ θ S ξ , ω -i t QS ξ = F ξ + R ξ , ∇ θ S η , ω + iQS η = F η + R η . (5.13)
Here Q : L × L → C is the scalar valued matrix associated to A via the formula (2.3), i.e.

Q = diag{λ a , a ∈ L} + B,
where B is Hermitian and block diagonal.

Written in the Fourier variables, eq. (5.13) becomes

i( k, ω -t Q) Ŝξ (k) = F ξ (k) + Rξ (k), k ∈ Z n , i( k, ω + Q) Ŝη (k) = F η (k) + Rη (k), k ∈ Z n . (5.14)
The two equations in (5.14) are similar, so let us consider (for example) the second one, and let us decompose it into its "components" over the blocks [a]:

(5.15) i( k, ω(ρ) + Q(ρ) [a] ) Ŝ[a] (k) = F[a] (k, ρ) + R[a] (k)
where the matrix Let α(ρ) denote an eigenvalue of the matrix Q [a] (ρ), a ∈ L. It follows from (5.9) that

Q [a] is the restriction of Q to [a] × [a]
|α(ρ) -λ a | ≤ δ 0 2 ≤ c 0 2 for some appropriate a ∈ [a], which implies that |α(ρ)| ≥ c 0 2 w γ a ≥ 2κw a by (2.5). Hence, L(0, [a], ρ) -1 ≤ κw a -1 ∀ρ, ∀a. Assume that 0 < |k| ≤ N . Since | k, ω(ρ) | ≤ CN it follows from (2.5) that | k, ω(ρ) + α(ρ)| ≥ c 0 4 w γ a ≥ κ w a whenever w a ≥ ( 4CN c 0 ) 1 γ .
Hence for these a's we get

(5.16) L(k, [a], ρ) -1 ≤ κ w a -1 ∀ρ. Now let w a ≤ ( 4CN c 0 ) 1 γ
. By Hypothesis A2 we have either

| k, ω(ρ) + λ a | ≥ δ 0 w a ∀ρ, ∀a
or we have a unit vector z such that

(∇ ρ • z)( k, ω(ρ) + λ a ) ≥ δ 0 ∀ρ, ∀a.
The first case clearly implies (5.16), so let us consider the second case. By (5.9) it follows that

(∇ ρ • z)H [a] (ρ) ≤ δ 0 2 .
The Hermitian matrix ( k, ω(ρ)

+ Q(ρ) [a]
) is of dimension w d * a (see (2.1)) therefore, by Lemma 5.2, we conclude that (5.16) holds for all ρ outside a suitable set F a,k of measure

w d * +1 a κδ -1 0 . Let D 2 = D \ |k|≤N wa≤( 4CN c 0 ) 1 γ F a,k .
Then we get

meas(D \ D 2 ) ≤ CN n N c 0 d * +2 γ
(5.17) Ŝ[a] (k, ρ) = χ |k|≤N (k)L(k, [a], ρ) -1 F[a] (k, ρ), a ∈ L , and
(5.18) R[a] (k, ρ) = χ |k|>N (k) F[a] (k, ρ), a ∈ L .
Using (5.16), we have for ρ ∈ D 2

S [a] (θ, ρ) ≤ C κ w a (σ -σ ′ ) n sup |ℑθ|<σ F [a] (θ, ρ) , R [a] (θ, ρ) ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n sup |ℑθ|<σ |F [a] (θ, ρ)|. for θ ∈ T d σ ′ , see (5.8). Since S 2 s = a∈L w 2s a |S a | 2 = a∈ L w 2s a S [a] 2 these estimates imply that S(θ, ρ) s+1 ≤ C κ(σ -σ ′ ) n sup |ℑθ|<σ F (θ, ρ) s , R(θ, ρ) s ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n sup |ℑθ|<σ F (θ, ρ) s ,
for any σ ′ ≤ σ. The estimates of the derivatives with respect to ρ are obtained by differentiating (5.15) to obtain

L(k, [a], ρ)[∂ ρ Ŝ[a] (k)] = -[∂ ρ L(k, [a], ρ)] Ŝ[a] (k) + [∂ ρ F[a] (k, ρ)] + [∂ ρ R[a] (k)]
which is an equation of the same type as (5.15

) for ∂ ρ Ŝ[a] (k) and ∂ ρ R[a] (k) where -[∂ ρ L(k, [a], ρ)] Ŝ[a] (k) + [∂ ρ F[a] (k, ρ)] := B [a] (k, ρ) plays the role of F[a] (k, ρ).
We solve this equation as in (5.17)-(5.18) and we note that

χ |k|>N (k)B [a] (k, ρ) = χ |k|>N (k)[∂ ρ F[a] (k, ρ)]
and thus

R(θ, ρ) s ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n sup |ℑθ|<σ F (θ, ρ) s .
On the other hand

B [a] (k, ρ) s ≤ CN κ(σ -σ ′ ) n sup |ℑθ|<σ F (θ, ρ) s + sup |ℑθ|<σ ∂ ρ F (θ, ρ) s
and therefore we get

∂ ρ S(θ, ρ) s+1 ≤ CN µ -1 κ 2 (σ -σ ′ ) 2n [f ] s,β σ,µ,D .
The functions F and R are complex, and the constructed solution S ζ may also be complex. Instead of proving that it is real, we replace S ζ , θ ∈ T n , by its real part and then analytically extend it to T n σ ′ , using the relation

ℜS ζ (θ, ρ) := 1 2 (S ζ (θ, ρ) + Sζ ( θ, ρ)).
Thus we obtain a real solution which obeys the same estimates.

ω 0 | C 1 (D) ≤ δ 0 . Let D ∋ ρ → A(ρ) ∈ N F ∩ M s,β be C 1 and verifying (5.19) ∂ j ρ (A(ρ) -A 0 ) s,β ≤ δ 0 4 for j = 0, 1 and ρ ∈ D. Let f ∈ T s,β (σ, µ, D), 0 < κ ≤ δ 0 2 and N ≥ 1. Then there exists a subset D 3 = D 3 (h, κ, N ) ⊂ D, satisfying meas(D \ D 3 ) ≤ C N c 0 exp κ δ 0 exp ′ ,
and there exist real C 1 -functions B :

D 3 → M s,β ∩ N F , S ζζ (•; ρ) : D 3 → M + s,β and R ζζ (•; ρ) : T n σ × D 3 → M s,β , analytic in θ, such that (5.20) ∇ θ S ζζ (θ, ρ), ω(ρ) -A(ρ)JS ζζ (θ, ρ) + S ζζ (θ, ρ)JA(ρ) = -f ζζ (θ, ρ) + B(ρ) + R ζζ (θ, ρ)
and for all (θ, ρ) ∈ T n σ ′ × D 3 , σ ′ < σ, and j = 0, 1

µ 2 ∂ j ρ R ζζ (θ, ρ) s,β ≤ C e -1 2 (σ-σ ′ )N (σ -σ ′ ) n [f ] s,β σ,µ,D , (5.21) µ 2 ∂ j ρ S ζζ (θ, ρ) s,β+ ≤ C N 1+d * /γ κ 2+d * /2β (σ -σ ′ ) n [f ] s,β σ,µ,D , (5.22)
µ 2 ∂ j ρ B(ρ) s,β ≤ C [f ] s,β σ,µ,D . (5.23)
The two exponents exp and exp ′ are positive numbers depending on n, γ, d * , α 1 , α 2 , β. The constant C also depends on |ω 0 | C 1 (D) .

Proof. -As in the previous section, and using the same notation, we re-write (5.20) in complex variables. So we introduce ) ηη ) a,b∈L . We use similar notations for R, B and F . In this notation (5.20) decouples into three equations (11) ∇ θ S ξξ , ω

S = t U S ζ,ζ U , R = t U R ζζ U and F = t U f ζζ U . By construction, S b a ∈ M 2×2 for
+ iQS ξξ + iS ξξ t Q = B ξξ -F ξξ + R ξξ , ∇ θ S ηη , ω -i t QS ηη -iS ηη Q = B ηη -F ηη + R ηη , ∇ θ S ξη , ω + iQS ξη -iS ξη Q = B ξη -F ξη + R ξη ,
where we recall that Q is the scalar valued matrix associated to A via the formula (2.3). The first and the second equations are of the same type, so we focus on the resolution of the second and the third equations. Written in Fourier variables, they read

i( k, ω -t Q) Ŝηη (k) -i Ŝηη (k)Q = δ k,0 B ηη -F ηη (k) + Rηη (k), k ∈ Z n , (5.24) i( k, ω + Q) Ŝξη (k) -i Ŝξη (k)Q = δ k,0 B ξη -F ξη (k) + Rξη (k), k ∈ Z n , (5.25)
where δ k,j denotes the Kronecker symbol.

Equation (5.24). We chose B ηη = 0 and decompose the equation into "components" on each product block [a] × [b]:

(5.26) L Ŝ[b] [a] (k) = i F [b] [a] (k, ρ)-i R[b]
[a] (k) where we have suppressed the upper index ηη and the operator L := L(k, [a], [b], ρ) is the linear Hermitian operator, acting in the space of complex [a] × [b]-matrices defined by

L M = k, ω(ρ) -t Q [a] (ρ) M -M Q [b] (ρ).
The matrix Q [a] can be diagonalized in an orthonormal basis:

t P [a] Q [a] P [a] = D [a] .
Therefore denoting Ŝ′ [b] [a]

= t P [a] S [b] [a] P [b] , F ′ [b] [a] = t P [a] F [b] [a] P [b] and R′ [b] [a] = t P [a] R [b] [a] P [b] the homological equation (5.26) reads (5.27) ( k, ω + D [a] ) Ŝ′ [b] [a] (k) -S ′ [b] [a] (k)D [b] = i F ′ [b] [a] (k)-i R′ [b]
[a] (k). This equation can be solved term by term:

(5.28) R′ jℓ (k) = F ′ jℓ (k), j ∈ [a], ℓ ∈ [b], |k| > N
(k) = i k, ω(ρ) -α j (ρ) -β ℓ (ρ) F ′ jℓ (k), j ∈ [a], ℓ ∈ [b], |k| ≤ N
where α j (ρ) and β ℓ (ρ) denote eigenvalues of Q [a] (ρ) and Q [b] (ρ), respectively. First notice that by (5.28) one has

|R(θ)| s,β = |R ′ (θ)| s,β ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n sup |ℑθ|<σ |F (θ)| s,β .
To estimate S we want to use Lemma A.3 below. As

Q [a] = diag{λ a : a ∈ [a]} + B [a]
with B Hermitian, using hypothesis (5. [START_REF] Grébert | On reducibility of the quantum harmonic oscillator with quasiperiodic in time potential on R d[END_REF]) we get that (5.30)

|(α j (ρ) + β ℓ (ρ)) -(λ a + λ b )| ≤ δ 0 4 + δ 0 4 1 (w a w b ) β ≤ δ 0 2(w a w b ) β .
Moreover, in order to apply Lemma A.3 we have to estimate |α j (ρ)λ a | and |β l (r)λ b |, this is done thanks to assumption (5.19) :

|α j -λ a | ≤ Q [a] (ρ) -λ [a] Id ≤ 1 w 2β [a] |A(ρ) -A 0 | s,β ≤ δ 0 4w 2β [a]
and the corresponding estimate holds for |β l (r)λ b |.

It follows as in the proof of Proposition 5.3, using Lemma 5.2, relation (2.5), Assumption A2 and (5.19), that there exists a subset Therefore we obtain a solution S satisfying for any

D 2 = D 2 (h, κ, N ) ⊂ D, satisfying meas(D \ D 2 ) ≤ C N c 0 exp κ δ 0 , such that | k, ω(ρ) -α j (ρ) -β ℓ (ρ)| ≥ κ(1 + |w a + w b |),
|ℑθ| < σ ′ |S(θ)| s,β+ ≤ CN d * /(2γ) κ 1+d * /(4β) (σ -σ ′ ) n sup |ℑθ|<σ |F (θ)| s,β .
The estimates for the derivatives with respect to ρ are obtained by differentiating (5.26) which leads to (here we drop all the indices to simply the formula)

L(∂ ρ Ŝ[b] [a] (k, ρ)) = -(∂ ρ L) Ŝ[b] [a] (k, ρ) + i∂ ρ F [b] [a] (k, ρ) -i∂ ρ R[b] [a] (k, ρ) ,
which is an equation of the same type as (5.26

) for ∂ ρ Ŝ[b] [a] (k, ρ) and ∂ ρ R[b] [a] (k, ρ) where i F [b] [a] (k, ρ) is replaced by B [b] [a] (k, ρ) = -(∂ ρ L) Ŝ[b] [a] (k, ρ) + i∂ ρ F [b]
[a] (k, ρ). This equation is solved by defining

∂ ρ Ŝ[b] [a] (k, ρ) =χ |k|≤N (k)L(k, [a], [b], ρ) -1 B [b] [a] (k, ρ), ∂ ρ R[b] [a] (k, ρ) = -iχ |k|>N (k)B [b] [a] (k, ρ) = χ |k|>N (k)∂ ρ F [b] [a] (k, ρ) . Since |(∂ ρ L) Ŝ(k, ρ)| s,β ≤ C(N (|∂ ρ ω 0 | + δ 0 ) + 2δ 0 )| Ŝ(k, ρ)| s,β ≤ CN | Ŝ(k, ρ)| s,β , we obtain |B(k, ρ)| s,β ≤ CN κ -1-d * /(4β N d * /2γ | F (k)| s,β
and thus following the same strategy as in the resolution of (5.26) we get

µ 2 |∂ ρ S(θ)| s,β+ ≤ CN 1+d * /γ κ 2+d * /(2β) (σ -σ ′ ) n [f ] s,β σ,µ,D , µ 2 |∂ ρ R(θ)| s,β ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n [f ] s,β σ,µ,D .
Equation (5.25). It remains to consider (5.25) which decomposes into the "components" over the product blocks [a] × [b] (we have suppressed the upper index ξη):

k, ω(ρ) Ŝ[b] [a] (k) + Q [a] (ρ) Ŝ[b] [a] (k) - Ŝ[b] [a] (k)Q [b] (ρ) = -iδ k,0 B [b] [a] + i F [b] [a] (k, ρ) -i R[b]
[a] (k).

(5.31)

First we solve this case when k = 0 and w a = w b by defining

Ŝ[a] [a] (0) = 0, R[a] [a] (0) = 0 and B [a] [a] = F [a]
[a] (0).

Then we impose B

[b]

[a] = 0 for w a = w b in such a way B ∈ M s,β ∩ N F and satisfies

|B| s,β ≤ | F (0)| s,β .
The estimates of the derivatives with respect to ρ are obtained by differentiating the expressions for B.

Then, when k = 0 or w a = w b , with the same definition of S ′ , F ′ as in (5.27) we obtain

(5.32) ( k, ω + D [a] ) Ŝ′ [b] [a] (k) -S ′ [b] [a] (k)D [b] = i F ′ [b] [a] (k)-i R′ [b]
[a] (k). This equation can be solved term by term:

(5.33) R′ jℓ (k) = F ′ jℓ (k), j ∈ [a], ℓ ∈ [b], |k| > N and (5.34) Ŝ′ jℓ (k) = i k, ω(ρ) -α j (ρ) -β ℓ (ρ) F ′ jℓ (k), j ∈ [a], ℓ ∈ [b], |k| ≤ N
where α j (ρ) and β ℓ (ρ) denote eigenvalues of Q [a] (ρ) and Q [b] (ρ), respectively. First notice that by (5.33) one has

|R(θ)| s,β = |R ′ (θ)| s,β ≤ Ce -1 2 (σ-σ ′ )N (σ -σ ′ ) n sup |ℑθ|<σ |F (θ)| s,β .
To solve (5.34) we face the small divisors

(5.35) k, ω(ρ) + α j (ρ) -β ℓ (ρ), j ∈ [a], ℓ ∈ [b].
To estimate them, we have to distinguish between the case k = 0 and k = 0.

The case k = 0. In that case we know that w a = w b and we use (5.19) and (2.6) to get

|α j (ρ) -β ℓ (ρ)| ≥ c 0 |w a -w b | - δ 0 4w 2β a - δ 0 4w 2β b ≥ κ(1 + |w a -w b |).
This last estimate allows us to use Lemma A.3 to conclude that

| Ŝ(0)| β+ ≤ C κ d * +1 | F (0)| β .
The case k = 0. If k = 0 we face the small divisors (5.35) with non-trivial k, ω . Using Hypothesis A3, there is a set

D ′ 2 = D(ω, 2η, N ), meas(D \ D ′ 2 ) ≤ CN α 1 ( η δ 0 ) α 2 ,
such that for all ρ ∈ D ′ 2 and 0

< |k| ≤ N | k, ω(ρ) -λ a + λ b | ≥ 2η(1 + |w a -w b |).
By (5.19) this implies

| k, ω(ρ) -α j (ρ) + β ℓ (ρ)| ≥ 2η(1 + |w a -w b |) - δ 0 4w 2β a - δ 0 4w 2β b ≥ η(1 + |w a -w b |) if w b ≥ w a ≥ δ 0 2η 1 2β
.

Let now w a ≤ ( δ 0 2η )

1 2β . We note that | k, ω(ρ) -λ a + λ b | ≤ 1 implies that w δ b ≤ ( δ 0 2η ) δ 2β + C|k| ≤ ( δ 0 2η ) δ 2β + N .
As in Section 5.3, we obtain that

(5.36) | k, ω(ρ) + α j (ρ) -β ℓ (ρ)| ≥ κ(1 + |w a -w b |) ∀j ∈ [a], ∀ℓ ∈ [b]
holds outside a set F If F is the union of

F [a],[b],k for |k| ≤ N , [a], [b] ∈ L such that w a ≤ ( δ 0 2η ) 1 2β and w δ b ≤ ( δ 0 2η ) δ 2β + N respectively, we have meas(F ) ≤ C( δ 0 2η ) d * +1 2β ( δ 0 2η ) δ 2β + N (d * +2)/δ κ δ 0 N n ≤ CN n+(d * +2)/δ ( δ 0 η ) 2d * +3 2β κ δ 0 .
Now we choose η so that

( η δ 0 ) α 2 = ( δ 0 η ) 2d * +3 2β κ δ 0 i.e. η δ 0 = κ δ 0 2β 2d * +3+2βα 2 .
Then, as β ≤ 1, η ≤ κ and δ ≥ 1, we have

meas(F ) ≤ CN n+d * +2 κ δ 0 2βα 2 2d * +3+2βα 2 . Let D 3 = D 2 ∩ D ′ 2 \ F , we have meas(D \ D 3 ) ≤ CN exp κ δ 0 2βα 2 2d * +3+2βα 2
and by construction for all ρ ∈ D 3 , 0

< |k| ≤ N , a, b ∈ L and j ∈ [a], ℓ ∈ [b] we have | k, ω(ρ) -α j (ρ) + β ℓ (ρ)| ≥ κ(1 + |w a -w b |).
Hence using Lemma A.3 once again we obtain from (5.32) that Ŝ′ (k) ∈ M + s,β and

| Ŝ′ (k)| s,β+ ≤ Cκ -1-d * /2δ N d * /2γ | F ′ (k)| s,β .
Therefore we obtain a solution S satisfying for any

|ℑθ| < σ ′ |S(θ)| s,β+ ≤ CN d * /2γ κ 1+d * /2δ (σ -σ ′ ) n sup |ℑθ|<σ |F (θ)| s,β ,
The estimates of the derivatives with respect to ρ are obtained by differentiating (5.31) and proceeding as at the end of the resolution of equation (5.24).

In this way we have constructed a solution S ζζ , R ζζ , B of the fourth component of the homological equation which satisfies all required estimates. To guarantee that it is real, as at the end of Section 5.3 we replace S ζζ , R ζζ , B by their real parts and extend it analytically to T n σ ′ (e.g, replace S ζζ (θ, ρ) by 1 2 (S ζζ (θ, ρ) + Sζζ ( θ, ρ))).

Summing up. -Let

h = ω(ρ) • r + 1 2 ζ, A(ρ)ζ
where ρ → ω(ρ) and ρ → A(ρ) are C 1 on D and A is on normal form.

Proposition 5.5. -Assume (5.37)

|∂ j ρ (A(ρ) -A 0 )| s,β ≤ δ 0 4 , |∂ j ρ (ω -ω 0 )| ≤ δ 0 for j = 0, 1 and ρ ∈ D. Let f ∈ T s,β (σ, µ, D), 0 < κ ≤ δ 0 2 and N ≥ 1. Then there exists a subset D ′ = D ′ (h, κ, N ) ⊂ D, satisfying meas(D \ D ′ ) ≤ CN exp κ δ 0 exp ′ ,
and there exist real jet-functions S ∈ T s,β+ (σ ′ , µ, D ′ ) , R ∈ T s,β (σ ′ , µ, D ′ ) and a normal form

ĥ = [ [ f (•, 0; ρ) ] ] + [ [ ∇ r f (•, 0; ρ) ] ] • r + 1 2 ζ, B(ρ)ζ , such that {h, S} + f T = ĥ + R.
Furthermore, for all 0 ≤ σ ′ < σ

(5.38) |∂ j ρ B(ρ)| s,β ≤ C [f ] s,β σ,µ,D ′ , j = 0, 1 and ρ ∈ D ′ (5.39) [S] s,β+ σ ′ ,µ,D ′ ≤ C N 1+d * /γ κ 2+d * /2β (σ -σ ′ ) n [f ] s,β σ,µ,D ′ (5.40) [R] s,β σ ′ ,µ,D ′ ≤ C e -1 2 (σ-σ ′ )N (σ -σ ′ ) n [f ] s,β σ,µ,D ′ .
The two exponents exp and exp ′ are positive numbers depending on c 0 , n, d * , α 1 , a 2 , γ, β.

The constant C also depends on |ω 0 | C 1 (D) .

Proof. -We define S by

S(θ, r, ζ) = S θ (θ) + S r (θ), r + S ζ (θ), ζ + 1 2 S ζζ (θ)ζ, ζ .
where S θ , S r , S ζ and S ζζ are constructed in Propositions 5.1, 5. 6. Proof of the KAM Theorem.

The Theorem 2.2 is proved by an iterative KAM procedure. We first describe the general step of this KAM procedure.

= ω • r + 1 2 ζ, A(ω)ζ
with A on normal form, A-A 0 ∈ M β and satisfying (5.37). Let f ∈ T s,β (σ, µ, D) be a (small) Hamiltonian perturbation. Let S = S T ∈ T s,β+ (σ ′ , µ, D ′ ) be the solution of the homological equation (6.1) {h, S} + f T = ĥ + R.

defined in Proposition 5.5. Then defining

h + := h + ĥ, we get h • Φ 1 S = h + + f + with (6.2) f + = R + (f -f T ) • Φ 1 S + 1 0 {(1 -t)( ĥ + R) + tf T , S} • Φ t S dt.
The following Lemma gives an estimation of the new perturbation:

Lemma 6.1. -Let κ > 0, N ≥ 1, 0 < σ ′ < σ ≤ 1 and 0 < 2µ ′ < µ ≤ 1. Assume that D ′ ⊂ D, that f ∈ T s,β (σ, µ, D)
, that R satisfies (5.40) and that S = S T belongs to T s,β+ (σ ′′ , µ, D ′ ) with σ ′′ = σ+σ ′ 2 and satisfies

(6.3) [S] s,β+ σ ′′ ,µ,D ′ ≤ 1 16 µ 2 (σ -σ ′ ).
Then the function f + given by formula (6.2) belongs to T s,β (σ ′ , µ ′ , D ′ ) and

[f + ] s,β σ ′ ,µ ′ ,D ′ ≤ M e -1 2 (σ-σ ′ )N (σ -σ ′ ) n + µ ′ µ 3 + N 1+d * /γ κ 2+d * /2β µ 2 (σ -σ ′ ) n+1 [f ] s,β σ,µ,D [f ] s,β σ,µ,D (6.4)
where M is a constant depending on n, d * , α 1 , α 2 , c 0 , γ and β.

Proof. -Let us denote the three terms in the r.h.s. of (6.2) by f + 1 , f + 2 and f + 3 . In view of (5.40), we have that [f + 1 ] s,β σ ′ ,µ ′ ,D ′ is controlled by the first term in r.h.s. of (6.4). By Proposition 4.2, we get

[f -f T ] s,β σ,2µ ′ ,D ′ ≤ C µ ′ µ 3 [f ] s,β σ,µ,D .
By hypothesis S = S T belongs to T s,β+ (σ ′ , µ, D ′ ) and satisfies (6.3) which implies

[S] s,β+ σ ′′ ,µ,D ′ ≤ 1 2 (µ-µ ′ ) 2 (σ ′′ -σ ′ ) since 2µ ′ < µ. Therefore by Lemma 4.6 and since 2µ ′ ≤ 2(µ-µ ′ ), [f + 2 ] s,β σ ′ ,µ ′ ,D ′
is controlled by the second term in r.h.s. of (6.4).

It remains to control [f

+ 3 ] s,β σ ′ ,µ ′ ,D ′ .
To begin with, g t := (1t)( ĥ + R) + tf T is a jet function in T s,β (σ ′ , µ, D). Furthermore, defining for j = 1, 2, σ j = σ ′ + j σσ ′ 3 and using (5.40) we get (for N large enough)

[g t ] s,β σ 2 ,µ,D ′ ≤ C 1 + 3 n e -(σ-σ ′ )N/6 (σ -σ ′ ) n [f ] s,β σ,µ,D ≤ C[f ] s,β σ,µ,D .
On the other hand S ∈ T s,β+ (σ 2 , µ, D ′ ) is also a jet function and satisfies

[S] s,β+ σ 2 ,µ,D ′ ≤ CN 1+d * /γ κ 2+d * /2β (σ -σ ′ ) n [f ] s,β σ,µ,D .
Then using Lemma 4.3 we have

[{g t , S}] s,β σ 1 ,µ,D ′ ≤ C N 1+d * /γ κ 2+d * /2β µ 2 (σ -σ ′ ) n+1 ([f ] s,β σ,µ,D ) 2 .
We conclude the proof by Proposition 4.6.

6.2. Choice of parameters. -To prove the main theorem we construct the transformation Φ as the composition of infinitely many transformations S as in Theorem 5.5, i.e. for all k ≥ 1 we construct iteratively S k , h k , f k following the general scheme (6.1)-(6.2) as follows :

(h + f ) • Φ 1 S 1 • • • • • Φ 1 S k = h k + f k . At each step f k ∈ T s,β (σ k , µ k , D k ) with [f k ] s,β σ k ,µ k ,D k ≤ ε k , h k = ω k , r + 1 2 ζ, A k ζ is
on normal form, the Fourier series are truncated at order N k and the small divisors are controlled by κ k . In this section we specify the choice of all the parameters for k ≥ 1. First we fix

κ 0 = ε 1 24(2+d * /2β) .
We define ε 0 = ε, σ 0 = σ, µ 0 = µ and for j ≥ 1 we choose

σ j-1 -σ j =C * σ 0 j -2 , N j =2(σ j -σ j+1 ) -1 ln ε -1 j , κ j =ε 1 24(2+d * /2β) j µ j = ε j (2M ) j ε 6/5 1 3
, where M is the absolute constant defined in (6.4) and (C * ) -1 = 2 j≥1 1 j 2 , and (6.5)

ε j = (ε j-1 ) 5 4
.

Observe that with this choice, (µ j ) satisfies 2µ j+1 ≤ µ j . Then the only unfixed parameter is ε = ε 0 , that will be fixed next section. Nevertheless, ε will be small enough to ensure the property κ j ≤ δ 0 2 that is necessary to apply Proposition 5.5. This is guaranteed if

(6.6) ε 1 24(2+d * /2β) ≤ δ 0 2 . 6.3. Iterative lemma. -Let set D 0 = D, h 0 = ω 0 (ρ), r + 1 2 ζ, A 0 ζ and f 0 = f in such a way [f 0 ] s,β σ 0 ,µ 0 ,D 0 ≤ ε 0 . For k ≥ 0 let us denote O k = O s (σ k , µ k ).
Lemma 6.2. -For ε sufficiently small depending on µ 0 , σ 0 , n,s, β and |ω 0 | C 1 (D) we have the following: For all k ≥ 1 there exist

D k ⊂ D k-1 , S k ∈ T s,β+ (σ k , µ k , D k ), h k = ω k , r + 1 2 ζ, A k ζ on normal form and f k ∈ T s,β (σ k , µ k , D k ) such that (i) The mapping (6.7) Φ k (•, ρ) = Φ 1 S k : O k → O k-1 , ρ ∈ D k , k = 1, 2, • •
• is an analytic symplectomorphism linking the hamiltonian at step k -1 and the hamiltonian at the step k, i.e.

(h k-1 + f k-1 ) • Φ k = h k + f k .
(ii) we have the estimates

meas(D k-1 \ D k ) ≤ ε α k-1 , [h k -h k-1 ] s,β σ k ,µ k ,D k ≤ Cε k-1 , [f k ] s,β σ k ,µ k ,D k ≤ ε k , Φ k (x, ρ) -x s ≤ ε 4/5 .ε 1/4 k-1 , for x ∈ O k , ρ ∈ D k .
The exponents α is a positive number depending on n, d * , α 1 , a 2 , γ, β. The constant C also depends on |ω 0 | C 1 (D) .

Proof. -At step 1, h 0 = ω 0 (ρ), r + 1 2 ζ, A 0 ζ and thus hypothesis (5.37) is trivially satisfied and we can apply Proposition 5.5 to construct S 1 , R 0 , B 0 and D 1 such that for ρ ∈ D 1 {h 0 , S 0 } + f T 0 = ĥ0 + R 0 . Then we see that, using (5.39) and defining σ 1/2 = σ 0 +σ 1 2 , we have

[S 1 ] s,β+ σ 1/2 ,µ 0 ,D 1 ≤ C ε 0 N 1+d * /γ 0 κ 2+d * /2β 0 (σ 0 -σ 1/2 ) n ≤ 1 16 µ 2 0 (σ 0 -σ 1 )
for ε = ε 0 small enough in view of our choice of parameters. Therefore both Proposition 4.7 and Lemma 6.2 apply and thus for any

ρ ∈ D 1 , Φ 1 (•, ρ) = Φ 1 S 1 : O 1 → O 0 is an analytic symplectomorphism such that (h 0 + f 0 ) • Φ 1 = h 1 + f 1 with h 1 , f 1 , D 1 and Φ 1 satisfying the estimates (ii) k=1 . In particular we have Φ 1 (x) -x s ≤ C σ 0 µ 2 0 [S 1 ] s,β+ σ 1/2 ,µ 0 ,D 1 ≤ CN 1+d * /γ 0 σ n+1 0 µ 2 0 κ 2+d * /2β 0 ε 0 ≤ C(ln ε 0 ) 1+d * /γ σ n+2+d * /γ 0 µ 2 0 ε 23/24 0 ≤ 1 2 ε 11/12 0
for ε 0 small enough.

Now assume that we have completed the iteration up to step j. We want to perform the step j + 1. We first note that by construction (see Proposition 5.5)

A j = A 0 + B 0 + • • • + B j-1
and by (5.38)

|A j | β ≤ ε 0 + • • • + ε j-1 ≤ 2ε 0 ≤ 1 4 δ 0
for ε 0 small enough. Similarly

ω j = ω 0 + [ [ ∇ r f 0 (•, 0; ρ) ] ] + • • • + [ [ ∇ r f j-1 (•, 0; ρ) ] ]
and thus |∂ j r (ω jω 0 )| ≤ δ 0 for ε 0 small enough. Therefore (5.37) is satisfied at rank j and we can apply Proposition 5.5 in order to construct S j+1 , B j , R j and D j .

Then we construct f j+1 as in (6.2), i.e.

f j+1 = R j + (f j -f T j ) • Φ 1 S j+1 + 1 0
{(1t)( ĥj + R j ) + tf T j , S j+1 } • Φ t S j+1 dt.

To control f j+1 we may apply Lemma 6.1 since, defining σ j+1/2 = Therefore we can apply Lemma 6.1 and, using the preceding choice of parameters, we may bound all the terms of the r.h.s. of (6.4). Let us start with the second term:

(6.8) M µ j+1 µ j 3 ε j = 1 2 ε j+1 .
The third term may be computed as (6.9) M 2(j + 1) 2 ln(ε -1 j ) C * σ 0 1+d * /γ (j + 1) 2 C * σ 0 n+1 ε 2-1/24 j µ 2 j = C(j+1) 2n+3+2d * /γ (2M ) 2j/3 ε 4/5 (ε j ) 1/24 ε j+1 and there exists ε1 > 0 such that for 0 < ε ≤ ε1 we have for any j ≥ 1 C(j + 1) 2n+3+2d * /γ (2M ) 2j/3 (ε)

Let us set

Q l = O s (σ/ℓ, µ/ℓ), Z s = T n σ × C n × Y s where ℓ ≥ 2, and recall that • s denotes the natural norm on C n × C n × Y s . It defines the distance on Z s . We used the notations introduced in Lemma 6.2. By Proposition 4.5 assertion 2 and since σ k > σ/2, for each ρ ∈ D ′ and k ≥ 2, the map Φ k extends to Q 2 and satisfies on Q 2 the same estimate as on O k : (6.12)

Φ k : Q 2 → Z s , Φ k -Id s ≤ Cµ -2 k (σ k-1 -σ k ) -1 ε k ≤ k 2 C * σ 0 (2M ) 2k/3 ε 1/3 k ε 4/5 .
Now for 0 ≤ j ≤ N let us denote Φ j N = Φ j+1 • • • • • Φ N . Due to (6.7), it maps O N to O j . Again using Proposition 4.5, this map extends analytically to a map Φ j N : Q 2 → Z s , and by (6.12), for M > N , Φ j N -Φ j M s ≤ Cε Here

ω N = ω + [ [ ∇ r f 0 (•, 0; ρ) ] ] + • • • + [ [ ∇ r f N -1 (•, 0; ρ) ] ]
and

A N = A 0 + B 0 + • • • + B N -1
where B k is built from ∇ 2 ζζ f k (•, 0) as in the proof of Proposition 5.4. Clearly, ω N → ω ′ and A N → A where the vector ω ′ ≡ ω ′ (ρ) and the operator A ≡ A(ρ) satisfy the assertions of Theorem 2.2. Let us denote Φ = Φ 0 ∞ , consider the limiting hamiltonian H ′ = H 0 • Φ and write it as

H ′ = ω ′ , r + 1 2 ζ, A(ρ)ζ + f ′ .
The function f ′ is analytic in the domain

Q 2 . Since H ′ = H k • Φ k ∞ , we have ∇H ′ (x) = DΦ k ∞ (x) • ∇H k (Φ k ∞ (x)). As [f k ] s,β σ k ,µ k ,D k ≤ ε k , we deduce ∇ r H k (Φ k ∞ (θ, 0, 0)) = ω k + O(ε 1/4 k ) θ ∈ T n σ 3 .
Since the map Φ k ∞ satisfies (6.14), then ∇ r H ′ (θ, 0, 0) = ω ′ + O(ε

1/ 4
k ) for all k ≥ 1 and θ ∈ T n σ

3

.

Hence, ∇ r H ′ (θ, 0, 0) = ω ′ and thus ∇ r f ′ (θ, 0, 0) ≡ 0 for θ ∈ T n σ

3

.

Similar arguments lead to

∇ ζa f ′ (θ, 0, 0) ≡ 0 and ∇ ζa ∇ r f ′ (θ, 0, 0) ≡ 0 for θ ∈ T n σ This concludes the proof of Theorem 2.2.

Step 2 : suppose A ∈ M s,β ∩ Υ 2 s,β (k 1 , k 2 ). The only nonzero blocks A

[b]

[a] correspond to weights w a and w b such that max(w a , w b ) ≤ k 1 min(w a , w b ) and max(w a , w b ) > k 2 .

Notice that these two conditions imply that min(w a , w b ) ≥ k 2 k 1 .

We define the square matrix D[a] = λ a 1 [a] , where 1 [a] is the identity matrix. Then This allows to write (A.13) as (A. [START_REF] Geng | A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces[END_REF])

B [b] [a] -L 2 [a]×[b] -1 K [a]×[b] B [b] [a] = L 2 [a]×[b] -1 A [b] [a] ,
where K ). We have, thanks to (A.3), in operator norm, (A.17

B [b] [a] = Id -(L 2 [a]×[b] ) -1 K [a]×[b] -1 L 2 [a]×[b] -1 A [b] [a] ≤ 2 L 2 [a]×[b] -1 A [b] [a]
,

Hence in this case

(A.19) |B| s,β+ ≤ 2 κ |A| s,β
Step 3 : suppose A ∈ M s,β ∩ Υ 3 s,β (k 1 , k 2 ). The only nonzero blocks A

0 .

 0 It provides the spaces Y s , s ≥ 0, with the symplectic structure Jdζ ∧ dζ. To any C 1 -smooth function defined on a domain O ⊂ Y s , corresponds the Hamiltonian equation ζ = J∇f (ζ),

  denotes the restriction of A to the block [a] × [b], w(a, b) = min(w a , w b ) and • denotes the operator norm induced by the Y 0 -norm. A class of regularizing Hamiltonian functions. Let us fix any n ∈ N. On the space

  r, ζ) are C 1 functions on D, then we define (z, r, ζ) s,D = max j=0,1 (|∂ j ρ z|, |∂ j ρ r|, ∂ j ρ ζ s ). Let f : O 0 (σ, µ) × D → C be a C 1 -function, real holomorphic in the first variable x, such that for all ρ ∈ D

Definition 2 . 1 .

 21 s σ,µ,D = [f ] s,0 σ,µ,D . Normal form: We introduce the orthogonal projection Π defined on the 2 × 2 complex matrices Π : M 2×2 (C) → CI + CJ where -A matrix A : L × L → M 2×2 (C) is on normal form and we denote A

A a b = 0

 0 for all w a = w b . To a real symmetric matrix A = (A b a ) ∈ M we associate in a unique way a real quadratic form on Y s ∋ (ζ a ) a∈L = (p a , q a ) a∈L

  for ρ ∈ D and a ∈ L and (2.6) |λ aλ b | ≥ c 0 |w aw b | for a, b ∈ L .

Lemma 3 . 1 .

 31 -Hypothesis A1, A2 and A3 hold true with D = [1, 2] n and (3.1)

4. 4 .

 4 Hamiltonian flows. -To any C 1 -function f on a domain O s (σ, µ) × D we associate the Hamilton equations (4.6)

  we get using (4.13):Σ 2 s+β ≤ 4µ -1 [h]s,β σ,µ,D . Estimating similarly ∂ ∂ρ ∂ht ∂ζ we see that for x ∈ O(σ -2η, µ -2ν)

5. 2 .

 2 The first two equations. -We begin with equations (5.2) and (5.3) which are both of the form(5.6)

  the closed set D 1 satisfies meas(D \ D 1 ) ≤ N n κ δ 0 , and | ω(ρ), k | ≥ κ for all ρ ∈ D 1 . Hence, for ρ ∈ D 1 and all 0 < |k| ≤ N we have

3 ) we get Proposition 5 . 1 .

 351 -Let ω : D → R n be C 1 and verifying |ω -

9 .

 9 Here and below χQ(k) stands for the characteristic function of a set Q ⊂ Z n . 10. Here • means the operator-norm of a matrix associated to the euclidean norm on C N . Proposition 5.3. -Let ω : D → R n be C 1 and verifying |ω -

2 )

 2 and N ≥ 1. Then there exists a closed set D 2 = D 2 (ω, A, κ, N ) ⊂ D, satisfying meas(D \ D 2 ) ≤ CN exp κ δ 0 , and there exist real C 1 -functions S ζ and R ζ from T n × D 2 to Y s , analytic in θ, such that (5.10)

  and the vector F[a] (k, ρ) is the restriction of F (k, ρ) to [a] -here we have suppressed the upper index η. Denoting by L(k, [a], ρ) the Hermitian operator in the left hand side of equation (5.15), we want to estimate the operator norm of L(k, [a], ρ) -1 , i.e. we look for a lower bound of the modulus of the eigenvalues of L(k, [a], ρ).

  16) holds for all ρ ∈ D 2 , all |k| ≤ N and all [a]. Equation (5.15) is now solved by

5. 4 .

 4 The last equation. -Concerning the fourth component of the homological equation, (5.5), we have the following result Proposition 5.4. -Let ω : D → R n be C 1 and verifying |ω -

  all a, b ∈ L. Let us denote S b a = (S b a ) ξξ (S b a ) ξη (S b a ) ξη (S b a ) ηη and then S ξξ = ((S b a ) ξξ ) a,b∈L , S ξη = ((S b a ) ξη ) a,b∈L , S ηη = ((S b a

holds for all ρ ∈ D 2 ,

 2 all |k| ≤ N , all j ∈ [a], ℓ ∈ [b] and all [a], [b] ∈ L. Thus for ρ ∈ D 2 we obtain by Lemma A.3 that Ŝ′ (k) ∈ M + s,β for all |k| ≤ N and | Ŝ′ (k)| s,β+ ≤ Cκ -1-d * /(4β) N d * /(2γ) | F ′ (k)| s,β .

  [a],[b],k of measure w d * a w d * b (1+|w a -w b |)κδ -1 0 . This can be done considering equation (5.34) as the multiplication of a vector of size d [a] d [b] called F ′ jl (k) by a real diagonal (hence hermitian) square d [a] d [b] × d [a] d [b] matrix, and using Hypothesis A2, Condition (5.19) and Lemma 5.2.

  3 and 5.4. Hamiltonians R and B are also constructed in these 3 propositions. Then all the statements in Proposition 5.5 are satisfied and in particular we notice that ∇ ζ S = S ζ + S ζζ ζ belongs to Y s+β as a consequence of Propositions 5.3, 5.4 and Lemma 4.1 (iii).

6. 1 .

 1 The KAM step. -Let h be a normal form Hamiltonian h

σ j +σ j+1 2 ,[

 2 S j+1 ] s,β+ σ j+1/2 ,µ j ,D j+1 ≤ C ε j N 1+d * /γ j κ 2+d * /2β j (σ jσ j+1 ) n ≤ 1 8 µ 2 j (σ jσ j+1 ).

5 4 1 M

 51 ) j ε j+1 , and there exists ε2 > 0 such that for 0 < ε ≤ ε2 we have for any j ≥ (j + 1) 2nC * σ 0 (ε)

3 4 .( 5 4 ) j ≤ 1 4 .≤

 354 Take ε 0 ≤ ε = min(ε 1 , ε2 ) > 0 and we conclude that(6.11) [f j+1 ] s,β σ j+1 ,µ j+1 ,D j+1 ≤ ε j+1 . On the other hand by Proposition 5.5 the domain D j+1 satisfies meas(D j \ D j+1 ) ≤ CN exp j α > 0 and for ε 0 = ε small enough. The estimate concerning h k+1h k follows from (5.38) and (6.11) for the infinite dimensional part, from (6.11) for the control of [ [ f j+1 (•, 0; ρ) ] ] and a straightforward Cauchy estimate for the control of the mean value [ [ ∇ r f j+1 (•, 0; ρ) ] ]. Concerning the flow, we have for j ≥ 1, Φ j+1 (x)x s ≤ C σ j µ 2 j [S j+1 ] s,β+ σ j+1/2 ,µ j ,D j+1 ≤ CN 1+d C ′ (ln ε j ) 1+d * /γ (2M ) 2j/3 j n+2+d * /γ ε 4/5 ε

6. 4 .

 4 Transition to the limit and proof of Theorem 2.2. -LetD ′ = ∩ k≥0 D k .In view of the iterative lemma, this is a Borel set satisfying meas(D \ D ′ ) ≤ 2ε α .

1 / 4 N ε 4 / 5 5 k≥j ε 1 / 4 k ≤ Cε 4/5 ε 1 / 4 j , ∀j ≥ 1 . 1 / 4 j

 144551414114 , i.e. (Φ j N ) N is a Cauchy sequence. Thus when N → ∞ the maps Φ j N converge to a limiting mapping Φ j ∞ : Q 2 → Z s . Furthermore we have(6.13) Φ j ∞ -Id s ≤ Cε 4/By the Cauchy estimate the linearized map satisfies (6.14) DΦ j ∞ (x) -Id L(Ys,Ys) ≤ Cε 4/5 ε , ∀x ∈ Q 3 , ∀j ≥ 1. By construction, the map Φ 0 N transforms the original hamiltonianH 0 = ω, r + 1 2 ζ, A 0 ζ + f into H N = ω N , r + 1 2 ζ, A N (ω)ζ + f N .

3 . 4 k 3 .

 343 Now consider ∇ ζa ∇ ζ b H ′ (x). To study this matrix let us write it in the form (4.29), with h = H k and x(1) = Φ k ∞ (x). Repeating the arguments used in the proof of Proposition 4.6 we get that∇ ζa ∇ ζ b H ′ (θ, 0, 0) = (A k ) ab + O(ε 1/) for all k ≥ 1 and θ ∈ T n σ Therefore ∇ ζa ∇ ζ b H ′ (θ, 0, 0) = A ab i.e. ∇ ζa ∇ ζ b f ′ (θ, 0, 0) = 0 for θ ∈ T n σ 3 .

1 +

 1 ε( D[a] -D [a] )B ( D[b] -D [b] ) = A [b] [a] ,where we denote byL 2 [a]×[b] the operator acting on matrices of size [a] × [b] such that (A.14) L 2 [a]×[b] B [b] [a] := ( k, ω(ρ) + ελ aλ b ) B [b] [a] .This dilation is invertible and (A.4) then gives, in operator norm, |w aw b |) .

  [a]×[b] B [b] [a] = ε( D[a] -D [a] )B ( D[b] -D [b]

k 2 ≥

 2 k 1 ( 2C µ κ ) 1/δ , the operator Id -(L 2 [a]×[b] ) -1 K [a]×[b]is invertible and from (A.16) we get

  correspond to weights w a and w b such that max(w a , w b ) ≤ k 1 min(w a , w b ) and max(w a , w b ) ≤ k 2 , hence there are only finitely many such blocks. In this case, for any j ∈ [a] and l ∈ [b] we have (A.20)|B l j | = i k, ω(ρ) + εµ jµ l |A l j | ≤ 1 κ(1 + |w aw b |) |A l j |

  Again, the curves a ∞ and B ∞ analytically depend on θ 0 . Inserting(4.20) in (4.19) we get that ζ = ζ(t) satisfies the third estimate of (4.15). On the other hand for all t ∈ [0, 1], B ∈ M + s,β and |B(t)| s,β+ ≤ µ -2 [f ] s,β+ σ,µ,D .

	Therefore using Lemma 4.1 we get
	σ,µ,D . |B ∞ (t)| s,β+ ≤ µ -2 [f ] s,β+ Since in (4.11) U (θ; t) = I + B ∞ (t), then the estimates on U in (4.13) follow from (4.20) and (4.21)
	(4.21).
	Now consider equation for r(t):

s,β+ σ,µ,D .

(4.20)

  [START_REF] Delort | Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres[END_REF]. Actually(5.20) decomposes into four scalar equations but the fourth one is the transpose of the third one.

	and
	(5.29)	Ŝ′ jℓ

We could replace the assumption that w takes integer values by {wa -w b | a, b ∈ L} accumulates on a discrete set.

We provide C 2 with the hermitian norm, |ζa| = |(pa, qa)| = |pa| 2 + |qa| 2 .

Appendix A Some calculus

Lemma A.1. -Let j, k, ℓ ∈ N \ {0} then (A. [START_REF] Arnold | Mathematical methods in classical mechanics[END_REF] min(j, k) min(j, k)

Proof. -Without lost of generality we can assume j ≤ ℓ.

If k ≤ j then |k 2ℓ 2 | ≥ |j 2ℓ 2 | and thus min(j, ℓ) min(j, ℓ)

for a constant C depending only on β > 0.

Proof. -We note that

where

We have that b ∈ ℓ p for any 1 < p ≤ +∞ and that a ∈ ℓ q for any 1 β < q ≤ +∞. Thus by Young inequality a ⋆ b ∈ ℓ r for r such that 1 p + 1 q = 1 + 1 r . In particular choosing q = 2 β and p = 2 2-β we conclude that a ⋆ b ∈ ℓ ∞ .

The following Lemma is a variant of Proposition 2.2.4 in [START_REF] Delort | Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres[END_REF].

where ε = ±1, (µ a ) a∈L is a sequence of real numbers satisfying

for a given C µ > 0 and δ > 0, and such that for all a, b ∈ L and all |k| ≤ N

Then B ∈ M + s,β and there exists a constant C > 0 depending only on C µ , A and δ such that

for all |k| ≤ N.

Proof. -We first remark that the claimed property only concerns the operator norms of the blocks B

[b]

[a] , which can be computed separately. Let k 1 and k 2 be positive integers that will be fixed later. We define the following decomposition in M s,β , according to the weights w a and w b :

, where

[a] = 0 if max(w a , w b ) > k 1 min(w a , w b ) or max(w a , w b ) > k 2 , and we prove the desired estimates according to this decomposition. Since we estimate the operator norm of B

[b]

[a] , we need to rewrite the definition (A.2) in a operator way : denoting by D [a] the diagonal (square) matrix with entries µ j , for j ∈ [a] and D ′

[a] the diagonal (square) matrix with entries k, ω(ρ) + εµ j , for j ∈ [a], equation (A.2) reads

Step 1

[a] correspond to weights w a and w b such that

is invertible and gives an upper bound for the operator norm of its inverse. Then (A.5) is equivalent to

We have obtained that, for k 1 ≥ max(C


ATMOSPHERIC EVOLUTION

Exosphere Ionization

Technology Objective

Europa EGM

A carbon nanotube electron gun (CNTEG) is constructed for the highly sensitive exploration of exospheres, i.e. extremely tenuous atmospheres (n < 10 8 cm -3 ). The CNTEG is based on the quantum principle of field emission 1 seeking to efficiently impact and therefore ionize diffuse neutrals known to be present around planetary bodies.

ł 2 ứ=-ρ/ϵ 0
3)Electron-impact ion production rate:

dn/dt=n A *n B <σv> AB Apurva V. Oza 1 , F. Leblanc 1 , J.J. Berthelier 1 , J. Becker 2 , R.Coulomb 2 , P.Gilbert 2 , S.Lee 3 , N.T Hong 4 , L.Vettier 2 . "

1 Sorbonne Universités, UPMC, CNRS, LATMOS, Paris, France 2 LATMOS, UVSQ, CNRS, Paris/Guyancourt, France. 3 Ajou University, Korea, 4 Center for High Technology, Ha Noi, Vietnam.

Carbon Nanotubes as Cold Cathodes

Results

Extended Exosphere Clouds are simulated, due to: § Jovian gravitational drag is evident. § Similar to sodium clouds at Io 5 § Escape rates could indicate an Enceladus-like hydrotorus. Perspectives from Surface-Exosphere inhomogeneities: § Sputtering may not be global 9 .

§ O+, S+ ions may not dominate 6 . § Water-product escape rates match previous studies 7 . § O2 is thermalized to T surface , speeds are not sufficient to populate upper exosphere.

A Carbon Nanotube Electron Gun <CNTEG> is developed to ionize neutral atmospheres for future space spectrometry missions.

EGM simulations of Europa's Hydro-Exosphere show stark atmospheric structures dominated principally by Jupiter's gravitation. Sub-jovian O2 is > 1000 x more dense than the anti-jovian observations.

Jupiter is unique source of upper atmospheric O2. à Atomic O I more likely.

O Europa's hill sphere extends to about 8 R EU beyond which Jovian gravity dominates.

Day-night asymmetry is apparent. Effect is less for leading hemisphere as anti-solar is close to anti-jove. Upper exospheric oxygen behavoir is identical to other waterproducts: H2, OH, H2O.

Fig. 1

 1 Fig. 1 CNTEG: • Electric 'field effect emission' generates current due to solid-state quantum tunneling 1 . • Moderate E-field. (E ~1MV/m) • Power-efficient. (P < 0.1 Watts) • Sufficiently powerful current (I ~ 200 μA +/-0.1 μA) • Very stable. dI /dt < 0.1 μA/s • Light-weight and robust.

Fig. 4

 4 Fig. 4 : CNTEG electron field emission measured at each electrode labeled above.• CNTEGs are emitting consistently at ~ 60% . • Anode emission is at 75%. • Emission > 100μA is achieved with a cathode-gate distance of d cg = 250μm

  Anode transmission. 58.8% Collector 1 transmission.

Fig. 3

 3 Fig.3 SIMION simulation of CNTEG electrodes' equipotentials and e - trajectories. *Grids are at 80% transparency. Collector 1. Anode Grid.* Gate 1 Grid.* CNT Voltage' source.

Fig. 2 Fig. 6 :Fig. 7 :

 267 Fig. 2 15 mm 2 CNT chip in the lens of the G1 grid grounded at +0V.