KAM theory for the Hamiltonian derivative wave equation
aa r X i v : . [ m a t h . A P ] S e p KAM theory for theHamiltonian derivative wave equation
Massimiliano Berti, Luca Biasco, Michela Procesi
Abstract:
We prove an infinite dimensional KAM theorem which implies the existence of Cantorfamilies of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative waveequations. : 37K55, 35L05.
In the last years many progresses have been done concerning KAM theory for nonlinear HamiltonianPDEs. The first existence results were given by Kuksin [18] and Wayne [29] for semilinear wave (NLW)and Schr¨odinger equations (NLS) in one space dimension (1 d ) under Dirichlet boundary conditions,see [24]-[25] and [21] for further developments. The approach of these papers consists in generatingiteratively a sequence of symplectic changes of variables which bring the Hamiltonian into a constantcoefficients (=reducible) normal form with an elliptic (=linearly stable) invariant torus at the origin.Such a torus is filled by quasi-periodic solutions with zero Lyapunov exponents. This procedurerequires to solve, at each step, constant-coefficients linear “homological equations” by imposing the“second order Melnikov” non-resonance conditions. Unfortunately these (infinitely many) conditionsare violated already for periodic boundary conditions.In this case, existence of quasi-periodic solutions for semilinear 1 d -NLW and NLS equations, wasfirst proved by Bourgain [3] by extending the Newton approach introduced by Craig-Wayne [9] forperiodic solutions. Its main advantage is to require only the “first order Melnikov” non-resonanceconditions (the minimal assumptions) for solving the homological equations. Actually, developing thisperspective, Bourgain was able to prove in [4], [6] also the existence of quasi-periodic solutions forNLW and NLS (with Fourier multipliers) in higher space dimensions, see also the recent extensions in[1], [28]. The main drawback of this approach is that the homological equations are linear PDEs withnon-constant coefficients. Translated in the KAM language this implies a non-reducible normal formaround the torus and then a lack of informations about the stability of the quasi-periodic solutions.Later on, existence of reducible elliptic tori was proved by Chierchia-You [7] for semilinear 1 d -NLW, and, more recently, by Eliasson-Kuksin [12] for NLS (with Fourier multipliers) in any spacedimension, see also Procesi-Xu [27], Geng-Xu-You [14].An important problem concerns the study of PDEs where the nonlinearity involves derivatives. Acomprehension of this situation is of major importance since most of the models coming from Physicsare of this kind.In this direction KAM theory has been extended to deal with KdV equations by Kuksin [19]-[20],Kappeler-P¨oschel [17], and, for the 1 d -derivative NLS (DNLS) and Benjiamin-Ono equations, by Liu-Yuan [22]. The key idea of these results is again to provide only a non-reducible normal form aroundthe torus. However, in this cases, the homological equations with non-constant coefficients are only scalar (not an infinite system as in the Craig-Wayne-Bourgain approach). We remark that the KAMproof is more delicate for DNLS and Benjiamin-Ono, because these equations are less “dispersive” thanKdV, i.e. the eigenvalues of the principal part of the differential operator grow only quadratically atinfinity, and not cubically as for KdV. As a consequence of this difficulty, the quasi-periodic solutions1n [19], [17] are analytic, in [22], only C ∞ . Actually, for the applicability of these KAM schemes, themore dispersive the equation is, the more derivatives in the nonlinearity can be supported. The limitcase of the derivative nonlinear wave equation (DNLW) -which is not dispersive at all- is excluded bythese approaches.In the paper [3] (which proves the existence of quasi-periodic solutions for semilinear 1 d -NLSand NLW), Bourgain claims, in the last remark, that his analysis works also for the Hamiltonian“derivation” wave equation y tt − y xx + g ( x ) y = (cid:16) − d dx (cid:17) / F ( x, y ) , see also [5], page 81. Unfortunately no details are given. However, Bourgain [5] provided a detailedproof of the existence of periodic solutions for the non-Hamiltonian equation y tt − y xx + m y + y t = 0 , m = 0 . These kind of problems have been then reconsidered by Craig in [8] for more general Hamiltonianderivative wave equations like y tt − y xx + g ( x ) y + f ( x, D β y ) = 0 , x ∈ T , where g ( x ) ≥ D is the first order pseudo-differential operator D := p − ∂ xx + g ( x ). Theperturbative analysis of Craig-Wayne [9] for the search of periodic solutions works when β <
1. Themain reason is that the wave equation vector field gains one derivative and then the nonlinear term f ( D β u ) has a strictly weaker effect on the dynamics for β <
1. The case β = 1 is left as an openproblem. Actually, in this case, the small divisors problem for periodic solutions has the same levelof difficulty of quasi-periodic solutions with 2 frequencies.The goal of this paper is to extend KAM theory to deal with the Hamiltonian derivative waveequation y tt − y xx + m y + f ( Dy ) = 0 , m > , D := p − ∂ xx + m , x ∈ T , (1.1)with real analytic nonlinearities (see Remark 7.1) f ( s ) = as + X k ≥ f k s k , a = 0 . (1.2)We write equation (1.1) as the infinite dimensional Hamiltonian system u t = − i ∂ ¯ u H , ¯ u t = i ∂ u H , with Hamiltonian H ( u, ¯ u ) := Z T ¯ uDu + F (cid:16) u + ¯ u √ (cid:17) dx , F ( s ) := Z s f , (1.3)in the complex unknown u := 1 √ Dy + i y t ) , ¯ u := 1 √ Dy − i y t ) , i := √− . Setting u = X j ∈ Z u j e i jx (similarly for ¯ u ), we obtain the Hamiltonian in infinitely many coordinates H = X j ∈ Z λ j u j ¯ u j + Z T F (cid:16) √ X j ∈ Z ( u j e i jx + ¯ u j e − i jx ) (cid:17) dx (1.4)2here λ j := p j + m (1.5)are the eigenvalues of the diagonal operator D . Note that the nonlinearity in (1.1) is x -independent im-plying, for (1.3), the conservation of the momentum − i Z T ¯ u∂ x u dx . This symmetry allows to simplifysomehow the KAM proof (a similar idea was used by Geng-You [13]).For every choice of the tangential sites I := { j , . . . , j n } ⊂ Z , n ≥
2, the integrable Hamiltonian X j ∈ Z λ j u j ¯ u j has the invariant tori { u j ¯ u j = ξ j , for j ∈ I , u j = ¯ u j = 0 for j
6∈ I} parametrized by theactions ξ = ( ξ j ) j ∈I ∈ R n . The next KAM result states the existence of nearby invariant tori for thecomplete Hamiltonian H in (1.4). Theorem 1.1.
The equation (1.1) - (1.2) admits Cantor families of small-amplitude, analytic, quasi-periodic solutions with zero Lyapunov exponents and whose linearized equation is reducible to constantcoefficients. Such Cantor families have asymptotically full measure at the origin in the set of param-eters. The proof of Theorem 1.1 is based on the abstract KAM Theorem 4.1, which provides a reduciblenormal form (see (4.12)) around the elliptic invariant torus, and on the measure estimates Theorem4.2. The key point in proving Theorem 4.2 is the asymptotic bound (4.9) on the perturbed normalfrequencies Ω ∞ ( ξ ) after the KAM iteration. This allows to prove that the second order Melnikovnon-resonance conditions (4.11) are fulfilled for an asymptotically full measure set of parameters(see (4.16)). The estimate (4.9), in turn, is achieved by exploiting the quasi-T¨oplitz property of theperturbation. This notion has been introduced by Procesi-Xu [27] in the context of NLS in higherspace dimensions and it is similar, in spirit, to the T¨oplitz-Lipschitz property in Eliasson-Kuksin[12]. The precise formulation of quasi-T¨oplitz functions, adapted to the DNLW setting, is given inDefinition 3.4 below.Let us roughly explain the main ideas and techniques for proving Theorems 4.1, 4.2. Thesetheorems concern, as usual, a parameter dependent family of analytic Hamiltonians of the form H = ω ( ξ ) · y + Ω( ξ ) · z ¯ z + P ( x, y, z, ¯ z ; ξ ) (1.6)where ( x, y ) ∈ T n × R n , z, ¯ z are infinitely many variables, ω ( ξ ) ∈ R n , Ω( ξ ) ∈ R ∞ and ξ ∈ R n . Thefrequencies Ω j ( ξ ) are close to the unperturbed frequencies λ j in (1.5).As well known, the main difficulty of the KAM iteration which provides a reducible KAM normalform like (4.12) is to fulfill, at each iterative step, the second order Melnikov non-resonance conditions.Actually, following the formulation of the KAM theorem given in [2], it is sufficient to verify | ω ∞ ( ξ ) · k + Ω ∞ i ( ξ ) − Ω ∞ j ( ξ ) | ≥ γ | k | τ , γ > , (1.7)only for the “final” frequencies ω ∞ ( ξ ) and Ω ∞ ( ξ ), see (4.11), and not along the inductive iteration.The application of the usual KAM theory (see e.g. [18], [24]-[25]), to the DNLW equation providesonly the asymptotic decay estimateΩ ∞ j ( ξ ) = j + O (1) for j → + ∞ . (1.8)Such a bound is not enough: the set of parameters ξ satisfying (1.7) could be empty. Note that forthe semilinear NLW equation (see e.g. [24]) the frequencies decay asymptotically faster, namely likeΩ ∞ j ( ξ ) = j + O (1 /j ).The key idea for verifying the second order Melnikov non-resonance conditions (1.7) for DNLW isto prove the higher order asymptotic decay estimate (see (4.9), (4.2))Ω ∞ j ( ξ ) = j + a + ( ξ ) + m2 j + O ( γ / j ) for j ≥ O ( γ − / ) (1.9)3here a + ( ξ ) is a constant independent of j (an analogous expansion holds for j → −∞ with apossibly different limit constant a − ( ξ )). In this way infinitely many conditions in (1.7) are verifiedby imposing only first order Melnikov conditions like | ω ∞ ( ξ ) · k + h | ≥ γ / / | k | τ , h ∈ Z . Indeed, for i > j > O ( | k | τ γ − / ), we get | ω ∞ ( ξ ) · k + Ω ∞ i ( ξ ) − Ω ∞ j ( ξ ) | = | ω ∞ ( ξ ) · k + i − j + m( i − j )2 ij + O ( γ / /j ) |≥ γ / | k | − τ − O ( | k | /j ) − O ( γ / /j ) ≥ γ / | k | − τ noting that i − j is integer and | i − j | = O ( | k | ) (otherwise no small divisors occur). We refer to section6 for the precise arguments, see in particular Lemma 6.2.The asymptotic decay (4.9) for the perturbed frequencies Ω ∞ ( ξ ) is achieved thanks to the “quasi-T¨oplitz” property of the perturbation (Definition 3.4). Let us roughly explain this notion. The newnormal frequencies after each KAM step are Ω + j = Ω j + P j where the corrections P j are the coefficientsof the quadratic form P z ¯ z := X j P j z j ¯ z j , P j := Z T n ( ∂ z j ¯ z j P )( x, , , ξ ) dx . We say that a quadratic form P is quasi-T¨oplitz if it has the form P = T + R where T is a T¨oplitz matrix (i.e. constant on the diagonals) and R is a “small” remainder satisfying R jj = O (1 /j ) (see Lemma 5.2). Then (1.9) follows with a := T jj which is independent of j .Since the quadratic perturbation P along the KAM iteration does not depend only on thequadratic perturbation at the previous steps, we need to extend the notion of quasi-T¨oplitz to general(non-quadratic) analytic functions.The preservation of the quasi-T¨oplitz property of the perturbations P at each KAM step (withjust slightly modified parameters) holds in view of the following key facts:1. the Poisson bracket of two quasi-T¨oplitz functions is quasi-T¨oplitz (Proposition 3.1),2. the hamiltonian flow generated by a quasi-T¨oplitz function preserves the quasi-T¨oplitz property(Proposition 3.2),3. the solution of the homological equation with a quasi-T¨oplitz perturbation is quasi-T¨oplitz(Proposition 5.1).We note that, in [12], the analogous properties 1 (and therefore 2) for T¨oplitz-Lipschitz functions isproved only when one of them is quadratic.The definition of quasi-T¨oplitz functions heavily relies on properties of projections. However, foran analytic function in infinitely many variables, such projections may not be well defined unless theTaylor-Fourier series (see (2.28)) is absolutely convergent. For such reason, instead of the sup-norm,we use the majorant norm (see (2.12), (2.54)), for which the bounds (2.14) and (2.55) on projectionshold (see also Remark 2.4).We underline that the majorant norm of a vector field introduced in (2.54) is very different fromthe weighted norm introduced by P¨oschel in [23]-Appendix C, which works only in finite dimension,see comments in [23] after Lemma C.2 and Remark 2.3. As far as we know this majorant norm ofvector fields is new. In Section 2 we show its properties, in particular the key estimate of the majorantnorm of the commutator of two vector fields (see Lemma 2.15).Before concluding this introduction we also mention the recent KAM theorem of Greb´ert-Thomann[16] for the quantum harmonic oscillator with semilinear nonlinearity. Also here the eigenvalues grow4o infinity only linearly. We quote the normal form results of Delort-Szeftel [10], Delort [11], forquasi-linear wave equations, where only finitely many steps of normal form can be performed. Finallywe mention also the recent work by G´erard-Grellier [15] on Birkhoff normal form for a degenerate“half-wave” equation.The paper is organized as follows: • In section 2 we define the majorant norm of formal power series of scalar functions (Defini-tion 2.2) and vector fields (Definition 2.6) and we investigate the relations with the notion ofanaliticity, see Lemmata 2.1, 2.2, 2.3, 2.11 and Corollary 2.1. Then we prove Lemma 2.15 oncommutators. • In section 3 we define the T¨oplitz (Definition 3.3) and Quasi-T¨oplitz functions (Definition 3.4).Then we prove that this class of functions is closed under Poisson brackets (Proposition 3.1) andcomposition with the Hamiltonian flow (Proposition 3.2). • In section 4 we state the abstract KAM Theorem 4.1. The first part of Theorem 4.1 follows bythe KAM Theorem 5.1 in [2]. The main novelty is part II, in particular the asymptotic estimate(4.9) of the normal frequencies. • In section 5 we prove the abstract KAM Theorem 4.1.We first perform (as in Theorem 5.1 in [2]) a first normal form step, which makes Theorem 4.1suitable for the direct application to the wave equation.In Proposition 5.1 we prove that the solution of the homological equation with a quasi-T¨oplitzperturbation is quasi-T¨oplitz. Then the main results of the KAM step concerns the asymptoticestimates of the perturbed frequencies (section 5.2.3) and the T¨oplitz estimates of the newperturbation (section 5.2.4). • In section 6 we prove Theorem 4.2: the second order Melnikov non-resonance conditions arefulfilled for a set of parameters with large measure, see (4.16). We use the conservation ofmomentum to avoid the presence of double eigenvalues. • In section 7 we finally apply the abstract KAM Theorem 4.1 to the DNLW equation (1.1)-(1.2),proving Theorem 1.1. We first verify that the Hamiltonian (1.4) is quasi-T¨oplitz (Lemma 7.1),as well as the Birkhoff normal form Hamiltonian (7.8) of Proposition 7.1. The main technicaldifficulties concern the proof in Lemma 7.4 that the generating function (7.17) of the Birkhoffsymplectic transformation is also quasi-T¨oplitz (and the small divisors Lemma 7.2). In section7.2 we prove that the perturbation, obtained after the introduction of the action-angle variables,is still quasi-T¨oplitz (Proposition 7.2). Finally in section 7.3 we prove Theorem 1.1 applyingTheorems 4.1 and 4.2. Acknowledgments :
We thank Benoit Gr´ebert for pointing out a technical mistake in the previousversion.
Given a finite subset
I ⊂ Z (possibly empty), a ≥ , p > /
2, we define the Hilbert space ℓ a,p I := n z = { z j } j ∈ Z \I , z j ∈ C : k z k a,p := X j ∈ Z \I | z j | e a | j | h j i p < ∞ o . When I = ∅ we denote ℓ a,p := ℓ a,p I . We consider the direct product E := C n × C n × ℓ a,p I × ℓ a,p I (2.1)5here n is the cardinality of I . We endow the space E with the ( s, r )-weighted norm v = ( x, y, z, ¯ z ) ∈ E , k v k E := k v k E,s,r = | x | ∞ s + | y | r + k z k a,p r + k ¯ z k a,p r (2.2)where, 0 < s, r <
1, and | x | ∞ := max h =1 ,...,n | x h | , | y | := n X h =1 | y h | . Note that, for all s ′ ≤ s , r ′ ≤ r , k v k E,s ′ ,r ′ ≤ max { s/s ′ , ( r/r ′ ) }k v k E,s,r . (2.3)We shall also use the notations z + j = z j , z − j = ¯ z j . We identify a vector v ∈ E with the sequence { v ( j ) } j ∈J with indices in J := ( j = ( j , j ) , j ∈ { , , , } , j ∈ ( { , . . . , n } if j = 1 , Z \ I if j = 3 , ) (2.4)and components v (1 ,j ) := x j , v (2 ,j ) := y j (1 ≤ j ≤ n ) , v (3 ,j ) := z j , v (4 ,j ) := ¯ z j ( j ∈ Z \ I ) , more compactly v (1 , · ) := x , v (2 , · ) := y, , v (3 , · ) := z, , v (4 , · ) := ¯ z . We denote by { e j } j ∈J the orthogonal basis of the Hilbert space E , where e j is the sequence withall zeros, except the j -th entry of its j -th components, which is 1. Then every v ∈ E writes v = X j ∈J v ( j ) e j , v ( j ) ∈ C . We also define the toroidal domain D ( s, r ) := T ns × D ( r ) := T ns × B r × B r × B r ⊂ E (2.5)where D ( r ) := B r × B r × B r , T ns := n x ∈ C n : max h =1 ,...,n | Im x h | < s o , B r := n y ∈ C n : | y | < r o (2.6)and B r ⊂ ℓ a,p I is the open ball of radius r centered at zero. We think T n as the n -dimensional torus T n := 2 π R n / Z n , namely f : D ( s, r ) → C means that f is 2 π -periodic in each x h -variable, h = 1 , . . . , n . Remark 2.1. If n = 0 then D ( s, r ) ≡ B r × B r ⊂ ℓ a,p × ℓ a,p . We consider formal power series with infinitely many variables f ( v ) = f ( x, y, z, ¯ z ) = X ( k,i,α,β ) ∈ I f k,i,α,β e i k · x y i z α ¯ z β (2.7)with coefficients f k,i,α,β ∈ C and multi-indices in I := Z n × N n × N ( Z \I ) × N ( Z \I ) (2.8)where N ( Z \I ) := n α := ( α j ) j ∈ Z \I ∈ N Z with | α | := X j ∈ Z \I α j < + ∞ o . (2.9)In (2.7) we use the standard multi-indices notation z α ¯ z β := Π j ∈ Z \I z α j j ¯ z β j j . We denote the monomials m k,i,α,β ( v ) = m k,i,α,β ( x, y, z, ¯ z ) := e i k · x y i z α ¯ z β . (2.10)6 emark 2.2. If n = 0 the set I reduces to N Z × N Z and the formal series to f ( z, ¯ z ) = X ( α,β ) ∈ I f α,β z α ¯ z β . We define the “majorant” of f as (cid:0) M f (cid:1) ( v ) = (cid:0) M f (cid:1) ( x, y, z, ¯ z ) := X ( k,i,α,β ) ∈ I | f k,i,α,β | e i k · x y i z α ¯ z β . (2.11)We now discuss the convergence of formal series. Definition 2.1.
A series X ( k,i,α,β ) ∈ I c k,i,α,β , c k,i,α,β ∈ C , is absolutely convergent if the function I ∋ ( k, i, α, β ) c k,i,α,β ∈ C is in L ( I , µ ) where µ is thecounting measure of I . Then we set X ( k,i,α,β ) ∈ I c k,i,α,β := Z I c k,i,α,β dµ . By the properties of the Lebesgue integral, given any sequence { I l } l ≥ of finite subsets I l ⊂ I with I l ⊂ I l +1 and ∪ l ≥ I l = I , the absolutely convergent series X k,i,α,β c k,i,α,β := X ( k,i,α,β ) ∈ I c k,i,α,β = lim l →∞ X ( k,i,α,β ) ∈ I l c k,i,α,β . Definition 2.2. (Majorant-norm: scalar functions)
The majorant-norm of a formal power series (2.7) is k f k s,r := sup ( y,z, ¯ z ) ∈ D ( r ) X k,i,α,β | f k,i,α,β | e | k | s | y i || z α || ¯ z β | (2.12) where | k | := | k | := | k | + . . . + | k n | . By (2.7) and (2.12) we clearly have k f k s,r = k M f k s,r .For every subset of indices I ⊂ I , we define the projection(Π I f )( x, y, z, ¯ z ) := X ( k,i,α,β ) ∈ I f k,i,α,β e i k · x y i z α ¯ z β (2.13)of the formal power series f in (2.7). Clearly k Π I f k s,r ≤ k f k s,r (2.14)and, for any I, I ′ ⊂ I , it results Π I Π I ′ = Π I ∩ I ′ = Π I ′ Π I . (2.15)Property (2.14) is one of the main advantages of the majorant-norm with respect to the usual sup-norm | f | s,r := sup v ∈ D ( s,r ) | f ( v ) | . (2.16)We now define useful projectors on the time Fourier indices. Definition 2.3.
Given ς = ( ς , . . . , ς n ) ∈ { + , −} n we define f ς := Π ς f := Π Z nς × N n × N ( Z \I ) × N ( Z \I ) f = X k ∈ Z nς ,i,α,β f k,i,α,β e i k · x y i z α ¯ z β (2.17) where Z nς := n k ∈ Z n with ( k h ≥ ς h = + k h < ς h = − ∀ ≤ h ≤ n o . (2.18)7hen any formal series f can be decomposed as f = X ς ∈{ + , −} n Π ς f (2.19)and (2.14) implies k Π ς f k s,r ≤ k f k s,r . We now investigate the relations between formal power series with finite majorant norm andanalytic functions. We recall that a function f : D ( s, r ) → C is • analytic , if f ∈ C ( D ( s, r ) , C ), namely the Fr´echet differential D ( s, r ) ∋ v df ( v ) ∈ L ( E, C )is continuous, • weakly analytic , if ∀ v ∈ D ( s, r ), v ′ ∈ E \ { } , there exists ε > { ξ ∈ C , | ξ | < ε } 7→ f ( v + ξv ′ ) ∈ C is analytic in the usual sense of one complex variable.A well known result (see e.g. Theorem 1, page 133 of [26]) states that a function f isanalytic ⇐⇒ weakly analytic and locally bounded . (2.20) Lemma 2.1.
Suppose that the formal power series (2.7) is absolutely convergent for all v ∈ D ( s, r ) .Then f ( v ) and M f ( v ) , defined in (2.7) and (2.11) , are well defined and weakly analytic in D ( s, r ) .If, moreover, the sup-norm | f | s,r < ∞ , resp. | M f | s,r < ∞ , then f , resp. M f , is analytic in D ( s, r ) . Proof . Since the series (2.7) is absolutely convergent the functions f , M f , and, for all ς ∈ { + , −} n , f ς := Π ς f , M f ς (see (2.17)) are well defined (also the series in (2.17) is absolutely convergent).We now prove that each M f ς is weakly analytic, namely ∀ v ∈ D ( s, r ), v ′ ∈ E \ { } ,M f ς ( v + ξv ′ ) = X k ∈ Z nς ,i,α,β | f k,i,α,β | m k,i,α,β ( v + ξv ′ ) (2.21)is analytic in {| ξ | < ε } , for ε small enough (recall the notation (2.10)). Since each ξ m k,i,α,β ( v + ξv ′ ) is entire, the analyticity of M f ς ( v + ξv ′ ) follows once we prove that the series (2.21) is totallyconvergent, namely X k ∈ Z nς ,i,α,β | f k,i,α,β | sup | ξ | <ε | m k,i,α,β ( v + ξv ′ ) | < + ∞ . (2.22)Let us prove (2.22). We claim that, for ε small enough, there is v ς ∈ D ( s, r ) such thatsup | ξ | <ε (cid:12)(cid:12) m k,i,α,β ( v + ξv ′ ) (cid:12)(cid:12) ≤ m k,i,α,β ( v ς ) , ∀ k ∈ Z nς , i, α, β . (2.23)Therefore (2.22) follows by X k ∈ Z nς ,i,α,β | f k,i,α,β | sup | ξ | <ε | m k,i,α,β ( v + ξv ′ ) | ≤ X k ∈ Z nς ,i,α,β | f k,i,α,β | m k,i,α,β ( v ς )= M f ς ( v ς ) < + ∞ . Let us construct v ς ∈ D ( s, r ) satisfying (2.23). Since v = ( x, y, z, ¯ z ) ∈ D ( s, r ) we have x ∈ T ns and,since T ns is open, there is 0 < s ′ < s such that | Im( x h ) | < s ′ , ∀ ≤ h ≤ n . Hence, for ε small enough,sup | ξ | <ε (cid:12)(cid:12) Im( x + ξx ′ ) h (cid:12)(cid:12) ≤ s ′ < s , ∀ ≤ h ≤ n . (2.24)8he vector v ς := ( x ς , y ς , z ς , ¯ z ς ) with components x ςh := − i ς h s ′ , y ςh := | y h | + ε | y ′ h | , ≤ h ≤ n ,z ςh := | z h | + ε | z ′ h | , ¯ z ςh := | ¯ z h | + ε | ¯ z ′ h | , h ∈ Z , (2.25)belongs to D ( s, r ) because | Im x ςh | = s ′ < s , ∀ ≤ h ≤ n , and also ( y ς , z ς , ¯ z ς ) ∈ D ( r ) for ε smallenough, because ( y, z, ¯ z ) ∈ D ( r ) and D ( r ) is open. Moreover, ∀ k ∈ Z nς , by (2.24), (2.18) and (2.25),sup | ξ | <ε (cid:12)(cid:12) e i k · ( x + ξx ′ ) (cid:12)(cid:12) ≤ e | k | s ′ = e i k · x ς . (2.26)By (2.10), (2.25), (2.26), we get (2.23). Hence each M f ς is weakly analytic and, by the decomposition(2.19), also f and M f are weakly analytic. The final statement follows by (2.20).
Corollary 2.1. If k f k s,r < + ∞ then f and M f are analytic and | f | s,r , | M f | s,r ≤ k f k s,r . (2.27) Proof . For all v = ( x, y, z, ¯ z ) ∈ T ns × D ( r ), we have | e i k · x | ≤ e | k | s and | f ( v ) | , | M f ( v ) | ≤ X k,i,α,β | f k,i,α,β | e | k | s | y i || z α || ¯ z β | (2.12) ≤ k f k s,r < + ∞ by assumption. Lemma 2.1 implies that f , M f are analytic.Now, we associate to any analytic function f : D ( s, r ) → C the formal Taylor-Fourier power series f ( v ) := X ( k,i,α,β ) ∈ I f k,i,α,β e i k · x y i z α ¯ z β (2.28)(as (2.7)) with Taylor-Fourier coefficients f k,i,α,β := 1(2 π ) n Z T n e − i k · x i ! α ! β ! ( ∂ iy ∂ αz ∂ β ¯ z f )( x, , , dx (2.29)where ∂ iy ∂ αz ∂ β ¯ z f are the partial derivatives .What is the relation between f and its formal Taylor-Fourier series f ? Lemma 2.2.
Let f : D ( s, r ) → C be analytic. If its associated Taylor-Fourier power series (2.28) - (2.29) is absolutely convergent in D ( s, r ) , and the sup-norm (cid:12)(cid:12)(cid:12) X k,i,α,β f k,i,α,β e i k · x y i z α ¯ z β (cid:12)(cid:12)(cid:12) s,r < ∞ , (2.31) then f = f , ∀ v ∈ D ( s, r ) . Proof . Since the Taylor-Fourier series (2.28)-(2.29) is absolutely convergent and (2.31) holds, byLemma 2.1 the function f : D ( s, r ) → C is analytic. The functions f = f are equal if the Taylor-Fouriercoefficients f k,i,α,β = f k,i,α,β , ∀ k, i, α, β , (2.32) For a multi-index α = X ≤ j ≤ k e i j , | α | = k , the partial derivative is ∂ αz f ( x, y, z, ¯ z ) := ∂ k ∂τ . . . ∂τ k | τ =0 f ( x, y, z + τ e i + . . . + τ k e i k , ¯ z ) . (2.30) f k,i,α,β are defined from f as in (2.29). Let us prove (2.32). Indeed, for example, f , ,e h , = 1(2 π ) n Z T n ddξ | ξ =0 X k ∈ Z n , m ∈ N f k, ,me h , e i k · x ξ m (2.33)= X k ∈ Z n , m ∈ N π ) n Z T n ddξ | ξ =0 f k, ,me h , e i k · x ξ m = f , ,e h , , using that the above series totally converge for r ′ < r , namely X k ∈ Z n , m ∈ N sup x ∈ R , | ξ |≤ r ′ | f k, ,me h , e i k · x ξ m | ≤ X k ∈ Z n , m ∈ N | f k, ,me h , | ( r ′ ) m ≤ X k,i,α,β | f k,i,α,β m k,i,α,β (0 , , r ′ e h , | < ∞ recall (2.10). For the others k, i, α, β in (2.32) is analogous.The above arguments also show the unicity of the Taylor-Fourier expansion. Lemma 2.3.
If an analytic function f : D ( s, r ) → C equals an absolutely convergent formal series,i.e. f ( v ) = X k,i,α,β ˜ f k,i,α,β e i k · x y i z α ¯ z β , then its Taylor-Fourier coefficients (2.29) are f k,i,α,β = ˜ f k,i,α,β . The majorant norm of f is equivalent to the sup-norm of its majorant M f . Lemma 2.4. | M f | s,r ≤ k f k s,r ≤ n | M f | s,r . (2.34) Proof . The first inequality in (2.34) is (2.27). The second one follows by k Π ς f k s,r ≤ | M f | s,r , ∀ ς ∈ { + , −} n , (2.35)where Π ς f is defined in (2.17). Let us prove (2.35). Let D + ( r ) := n ( y, z, ¯ z ) ∈ D ( r ) : y h ≥ , ∀ ≤ h ≤ n , z l , ¯ z l ≥ , ∀ l ∈ Z \ I o . For any 0 ≤ σ < s , we have | M f | s,r = sup ( x,y,z, ¯ z ) ∈ D ( s,r ) (cid:12)(cid:12)(cid:12) X k,i,α,β | f k,i,α,β | e i k · x y i z α ¯ z β (cid:12)(cid:12)(cid:12) ≥ sup x = − i ς σ,...,x n = − i ς n σ, ( y,z, ¯ z ) ∈ D + ( r ) (cid:12)(cid:12)(cid:12) X k,i,α,β | f k,i,α,β | e i k · x y i z α ¯ z β (cid:12)(cid:12)(cid:12) (2.18) ≥ sup ( y,z, ¯ z ) ∈ D + ( r ) X k ∈ Z nς ,i,α,β | f k,i,α,β | e | k | σ | y i || z α || ¯ z β | = sup ( y,z, ¯ z ) ∈ D ( r ) X k ∈ Z nς ,i,α,β | f k,i,α,β | e | k | σ | y i || z α || ¯ z β | = k Π ς f k σ,r . Then (2.35) follows since for every function g we have sup ≤ σ
Given formal power series f = X k,i,α,β f k,i,α,β e i k · x y i z α ¯ z β , g = X k,i,α,β g k,i,α,β e i k · x y i z α ¯ z β , with g k,i,α,β ∈ R + , we say that f ≺ g if | f k,i,α,β | ≤ g k,i,α,β , ∀ k, i, α, β . (2.36)10ote that, by the definition (2.11) of majorant series, f ≺ g ⇐⇒ f ≺ M f ≺ g . (2.37)Moreover, if k g k s,r < + ∞ , then f ≺ g = ⇒ k f k s,r ≤ k g k s,r .For any ς ∈ { + , −} n define q ς := ( q ( j ) ς ) j ∈J as q ( j ) ς := ( − ς h i if j = (1 , h ) , ≤ h ≤ n , . (2.38) Lemma 2.5.
Assume k f k s,r , k g k s,r < + ∞ . Then f + g ≺ M f + M g , f · g ≺ M f · M g (2.39) and M (cid:0) ∂ j (Π ς f ) (cid:1) = q ( j ) ς ∂ j (cid:0) M (Π ς f ) (cid:1) , j ∈ J , (2.40) where ∂ j is short for ∂ v ( j ) and q ( j ) ς are defined in (2.38) . Proof . Since the series which define f and g are absolutely convergent, the bounds (2.39) followby summing and multiplying the series term by term. Next (2.40) follows by differentiating the seriesterm by term.An immediate consequence of (2.39) is k f + g k s,r ≤ k f k s,r + k g k s,r , k f g k s,r ≤ k f k s,r k g k s,r . (2.41)The next lemma extends property (2.39) for infinite series. Lemma 2.6.
Assume that f ( j ) , g ( j ) are formal power series satisfying1. f ( j ) ≺ g ( j ) , ∀ j ∈ J ,2. k g ( j ) k s,r < ∞ , ∀ j ∈ J ,3. X j ∈J | g ( j ) ( v ) | < ∞ , ∀ v ∈ D ( s, r ) ,4. g ( v ) := X j ∈J g ( j ) ( v ) is bounded in D ( s, r ) , namely | g | s,r < ∞ . Then the function g : D ( s, r ) → C is analytic, its Taylor-Fourier coefficients (defined as in (2.29) ) are g k,i,α,β = X j ∈J g ( j ) k,i,α,β ≥ , ∀ ( k, i, α, β ) ∈ I , (2.42) and k g k s,r < ∞ . Moreover1. X j ∈J | f ( j ) ( v ) | < ∞ , ∀ v ∈ D ( s, r ) , f ( v ) := X j ∈J f ( j ) ( v ) is analytic in D ( s, r ) ,3. f ≺ g and k f k s,r ≤ k g k s,r < ∞ . roof . For each monomial m k,i,α,β ( v ) (see (2.10)) and v = ( x, y, z, ¯ z ) ∈ D ( s, r ), we have | m k,i,α,β ( v ) | = m k,i,α,β ( v + ) , (2.43)where v + := (i Im x, | y | , | z | , | ¯ z | ) ∈ D ( s, r ) with | y | := ( | y | , . . . , | y n | ) and | z | , | ¯ z | are similarly defined.Since k g ( j ) k s,r < ∞ (and f ( j ) ≺ g ( j ) ) the series g ( j ) ( v ) := X k,i,α,β g ( j ) k,i,α,β m k,i,α,β ( v ) , g ( j ) k,i,α,β ≥ v ∈ D ( s, r ) we prove that X j ∈J X k,i,α,β | g ( j ) k,i,α,β m k,i,α,β ( v ) | (2.44) , (2.43) = X j ∈J X k,i,α,β g ( j ) k,i,α,β m k,i,α,β ( v + ) (2.44) = X j ∈J g ( j ) ( v + ) = g ( v + ) < ∞ (2.45)by assumption 3. Therefore, by Fubini’s theorem, we exchange the order of the series g ( v ) = X j ∈J X k,i,α,β g ( j ) k,i,α,β m k,i,α,β ( v ) = X k,i,α,β (cid:16) X j ∈J g ( j ) k,i,α,β (cid:17) m k,i,α,β ( v ) (2.46)proving that g is equal to an absolutely convergent series. Lemma 2.1 and the assumption | g | s,r < ∞ imply that g is analytic in D ( s, r ). Moreover (2.46) and Lemma 2.3 imply (2.42). The g k,i,α,β ≥ g ( j ) k,i,α,β ≥
0, see (2.44). Therefore
M g = g , and, by (2.34) and the assumption | g | s,r < ∞ , wededuce k g k s,r < ∞ . Concerning f we have X j ∈J | f ( j ) ( v ) | ≤ X j ∈J X k,i,α,β (cid:12)(cid:12)(cid:12) f ( j ) k,i,α,β m k,i,α,β ( v ) (cid:12)(cid:12)(cid:12) ≤ X j ∈J X k,i,α,β g ( j ) k,i,α,β | m k,i,α,β ( v ) | (2.45) < ∞ and, arguing as for g , its Taylor-Fourier coefficients are f k,i,α,β = X j ∈J f ( j ) k,i,α,β , ∀ ( k, i, α, β ) ∈ I . Then | f k,i,α,β | ≤ X j ∈J | f ( j ) k,i,α,β | ≤ X j ∈J g ( j ) k,i,α,β (2.42) = g k,i,α,β . Hence f ≺ g and k f k s,r ≤ k g k s,r < ∞ . Finally f is analytic by Lemma 2.1. Lemma 2.7.
Let k f k s,r < ∞ . Then, ∀ < s ′ < s , < r ′ < r , we have k ∂ j f k s ′ ,r ′ < ∞ . Proof . It is enough to prove the lemma for each f ς = Π ς f defined in (2.17). By k f k s,r < ∞ andCorollary 2.1 the functions f ς , M f ς are analytic and k ∂ j f ς k s ′ ,r ′ (2.34) ≤ n | M ( ∂ j f ς ) | s ′ ,r ′ (2.40) = 2 n | ∂ j ( M f ς ) | s ′ ,r ′ ≤ c | M f ς | s,r (2.34) ≤ c k f ς k s,r for a suitable c := c ( n, s, s ′ , r, r ′ ), having used the Cauchy estimate (in one variable).We conclude this subsection with a simple result on representation of differentials. Lemma 2.8.
Let f : D ( s, r ) → C be Fr´echet differentiable at v . Then df ( v )[ v ] = X j ∈J ∂ j f ( v ) v ( j ) , ∀ v = X j ∈J v ( j ) e j ∈ E , (2.47) and X j ∈J | ∂ j f ( v ) v ( j ) | ≤ k df ( v ) k L ( E, C ) k v k E . (2.48)12 roof . (2.47) follows by the continuity of the differential df ( v ) ∈ L ( E, C ). Next, consider a vector˜ v = (˜ v ( j ) ) j ∈J ∈ E such that | ˜ v j | = | v j | and˜ v ( j ) ( ∂ j f )( v ) = | ( ∂ j f )( v ) v ( j ) | , ∀ j ∈ J . Hence df ( v )[˜ v ] = X j ∈J ˜ v ( j ) ( ∂ j f )( v ) = X j ∈J | ( ∂ j f )( v ) v ( j ) | which gives (2.48) because k ˜ v k E = k v k E . We now consider a formal vector field X ( v ) := (cid:16) X ( j ) ( v ) (cid:17) j ∈J (2.49)where each component X ( j ) is a formal power series X ( j ) ( v ) = X ( j ) ( x, y, z, ¯ z ) = X k,i,α,β X ( j ) k,i,α,β e i k · x y i z α ¯ z β (2.50)as in (2.7). We define its “majorant” vector field componentwise, namely M X ( v ) := (cid:16) ( M X ) ( j ) ( v ) (cid:17) j ∈J := (cid:16) M X ( j ) ( v ) (cid:17) j ∈J . (2.51)We consider vector fields X : D ( s, r ) ⊂ E → E , see (2.1). Definition 2.5.
The vector field X is absolutely convergent at v if every component X ( j ) ( v ) , j ∈ J ,is absolutely convergent (see Definition 2.1) and (cid:13)(cid:13)(cid:13)(cid:0) X ( j ) ( v ) (cid:1) j ∈J (cid:13)(cid:13)(cid:13) E < + ∞ . The properties of the space E in (2.1) (as target space), that we will use are:1. E is a separable Hilbert space times a finite dimensional space,2. the “monotonicity property” of the norm v , v ∈ E with | v ( j )0 | ≤ | v ( j )1 | , ∀ j ∈ J = ⇒ k v k E ≤ k v k E . (2.52)For X : D ( s, r ) → E we define the sup-norm | X | s,r := sup v ∈ D ( s,r ) k X ( v ) k E,s,r . (2.53) Definition 2.6. (Majorant-norm: vector field)
The majorant norm of a formal vector field X as in (2.49) is k X k s,r := sup ( y,z, ¯ z ) ∈ D ( r ) (cid:13)(cid:13)(cid:13)(cid:16) X k,i,α,β | X ( j ) k,i,α,β | e | k | s | y i || z α || ¯ z β | (cid:17) j ∈J (cid:13)(cid:13)(cid:13) E,s,r = sup ( y,z, ¯ z ) ∈ D ( r ) (cid:13)(cid:13)(cid:13) X k,i,α,β | X k,i,α,β | e | k | s | y i || z α || ¯ z β | (cid:13)(cid:13)(cid:13) E,s,r (2.54) where X k,i,α,β := (cid:0) X ( j ) k,i,α,β (cid:1) j ∈J and | X k,i,α,β | := (cid:0) | X ( j ) k,i,α,β | (cid:1) j ∈J . emark 2.3. The stronger norm (see [24]) || X || s,r := (cid:13)(cid:13)(cid:13)(cid:16) sup ( y,z, ¯ z ) ∈ D ( r ) X k,i,α,β | X ( j ) k,i,α,β | e | k | s | y i || z α || ¯ z β | (cid:17) j ∈J (cid:13)(cid:13)(cid:13) E,s,r is not suited for infinite dimensional systems: for X = Id we have || X || s,r = + ∞ . By (2.54) and (2.51) we get k X k s,r = k M X k s,r . For a subset of indices I ⊂ I we define the projection(Π I X )( x, y, z, ¯ z ) := X ( k,i,α,β ) ∈ I X k,i,α,β e i k · x y i z α ¯ z β . Lemma 2.9. (Projection) ∀ I ⊂ I , k Π I X k s,r ≤ k X k s,r . (2.55) Proof . See (2.54).
Remark 2.4.
The estimate (2.55) may fail for the sup-norm | | s,r and suitable I . Let us define the “ultraviolet” reps. infrared projections(Π | k |≥ K X )( x, y, z, ¯ z ) := X | k |≥ K,i,α,β X k,i,α,β e i k · x y i z α ¯ z β , Π | k | Assume k X k s,r < + ∞ . (2.61) Then the series in (2.49) - (2.50) , resp. (2.51) , absolutely converge to the analytic vector field X ( v ) ,resp. M X ( v ) , for every v ∈ D ( s, r ) . Moreover the sup-norm defined in (2.53) satisfies | X | s,r , | M X | s,r ≤ k X k s,r . (2.62) Proof . By (2.61) and Definition 2.6, for each j ∈ J , we havesup ( y,z, ¯ z ) ∈ D ( r ) X k,i,α,β | X ( j ) k,i,α,β | e | k | s | y i || z α || ¯ z β | < + ∞ X ( j ) , ( M X ) ( j ) : D ( s, r ) → C is analytic. Moreover (2.62) follows applying (2.27) componentwise. By (2.61) the maps X , M X : D ( s, r ) → E are bounded. Since E is a separable Hilbert space (times a finite dimensional space), Theorem 3-Appendix A in [26], implies that X , M X : D ( s, r ) → E are analytic.Viceversa, we associate to an analytic vector field X : D ( s, r ) → E a formal Taylor-Fourier vectorfield (2.49)-(2.50) developing each component X ( j ) as in (2.28)-(2.29). Definition 2.7. (Order relation: vector fields) Given formal vector fields X , Y , we say that X ≺ Y if each coordinate X ( j ) ≺ Y ( j ) , j ∈ J , according to Definition 2.4. If k Y k s,r < + ∞ and X ≺ Y = ⇒ k X k s,r ≤ k Y k s,r . (2.63)Applying Lemma 2.5 component-wise we get Lemma 2.12. If k X k s,r , k Y k s,r < ∞ then X + Y ≺ M X + M Y and k X + Y k s,r ≤ k X k s,r + k Y k s,r . Lemma 2.13. | M X | s,r ≤ k X k s,r ≤ n | M X | s,r . (2.64) Proof . As for Lemma 2.4 with f X , | X k,i,α,β | k X k,i,α,β k E and using (2.52).We define the space of analytic vector fields V s,r := V s,r,E := n X : D ( s, r ) → E with norm k X k s,r < + ∞ o . By Lemma 2.11 if X ∈ V s,r then X is analytic, namely the Fr´echet differential D ( s, r ) ∋ v dX ( v ) ∈L ( E, E ) is continuous. The next lemma bounds its operator norm from ( E, s, r ) := ( E, k k E,s,r ) to( E, s ′ , r ′ ), see (2.2). Lemma 2.14. (Cauchy estimate) Let X ∈ V s,r . Then, for s/ ≤ s ′ < s , r/ ≤ r ′ < r , sup v ∈ D ( s ′ ,r ′ ) k dX ( v ) k L (( E,s,r ) , ( E,s ′ ,r ′ )) ≤ δ − | X | s,r (2.65) where the sup-norm | X | s,r is defined in (2.53) and δ := min n − s ′ s , − r ′ r o . (2.66) Proof . In the Appendix.The commutator of two vector fields X, Y : D ( s, r ) → E is[ X, Y ]( v ) := dX ( v )[ Y ( v )] − dY ( v )[ X ( v )] , ∀ v ∈ D ( s, r ) . (2.67)The next lemma is the fundamental result of this section. Lemma 2.15. (Commutator) Let X, Y ∈ V s,r . Then, for r/ ≤ r ′ < r , s/ ≤ s ′ < s , k [ X, Y ] k s ′ ,r ′ ≤ n +3 δ − k X k s,r k Y k s,r (2.68) where δ is defined in (2.66) . roof . The lemma follows by k dX [ Y ] k s ′ ,r ′ ≤ n +2 δ − k X k s,r k Y k s,r , (2.69)the analogous estimate for dY [ X ] and (2.67).We claim that, for each ς ∈ { + , −} n , the vector field X ς defined in (2.59) satisfies k dX ς [ Y ] k s ′ ,r ′ ≤ n +2 δ − k X ς k s,r k Y k s,r (2.70)which implies (2.69) because k dX [ Y ] k s ′ ,r ′ (2.58) ≤ X ς ∈{ + , −} n k dX ς [ Y ] k s ′ ,r ′ (2.70) ≤ X ς ∈{ + , −} n n +2 δ − k X ς k s,r k Y k s,r (2.60) ≤ X ς ∈{ + , −} n n +2 δ − k X k s,r k Y k s,r ≤ n +2 δ − k X k s,r k Y k s,r . Let us prove (2.70). First note that, since k X ς k s,r (2.60) ≤ k X k s,r < + ∞ and k Y k s,r < + ∞ byassumption, Lemma 2.11 implies that the vector fields X ς , M X ς , Y, M Y : D ( s, r ) → E , ∀ ς ∈ { + , −} n , (2.71)are analytic, as well as each component X ( i ) ς , M X ( i ) ς , Y ( i ) , M Y ( i ) : D ( s, r ) → C , i ∈ J .The key for proving the lemma is the following chain of inequalities: dX ς [ Y ] ( i ) ≺ M ( dX ς [ Y ]) ( i ) (2.47) = M (cid:16) X j ∈J ( ∂ j X ( i ) ς ) Y ( j ) (cid:17) Lemma . ≺ X j ∈J M ( ∂ j X ( i ) ς ) M Y ( j ) (2.72) (2.40) = X j ∈J q ( j ) ς ∂ j (cid:0) M X ( i ) ς (cid:1) M Y ( j ) (2.47) = d (cid:0) M X ( i ) ς (cid:1)(cid:2) ˜ Y q ]where ˜ Y q := ( ˜ Y ( j ) q ) j ∈J := ( q ( j ) ς M Y ( j ) ) j ∈J ∈ E . (2.73)Actually, since | q ( j ) ς | = 1 (see (2.38)), then k ˜ Y q ( v ) k E = k M Y ( v ) k E (2.71) < + ∞ , ∀ v ∈ D ( s, r ) . (2.74)In (2.72) above we applied Lemma 2.6 with s s ′ , r r ′ , f ( j ) ( ∂ j X ( i ) ς ) Y ( j ) , g ( j ) M ( ∂ j X ( i ) ς ) M Y ( j ) . (2.75)Let us verify that the hypotheses of Lemma 2.6 hold:1. f ( j ) ≺ g ( j ) follows by (2.39) and since k f ( j ) k s ′ ,r ′ , k g ( j ) k s ′ ,r ′ < + ∞ because k X ( i ) ς k s,r ≤ k X k s,r < + ∞ , k Y ( j ) k s,r ≤ k Y k s,r < + ∞ , and Lemma 2.7.2. k g ( j ) k s ′ ,r ′ < ∞ is proved above. 16. We have X j ∈J | g ( j ) ( v ) | < ∞ , for all v ∈ D ( s ′ , r ′ ), because X j ∈J | g ( j ) ( v ) | (2.75) = X j ∈J | M ( ∂ j X ( i ) ς )( v ) M Y ( j ) ( v ) | (2.40) = X j ∈J | q ( j ) ς ∂ j (cid:0) M X ( i ) ς (cid:1) ( v ) M Y ( j ) ( v ) | (2.38) = X j ∈J | ∂ j (cid:0) M X ( i ) ς (cid:1) ( v ) M Y ( j ) ( v ) | (2.48) ≤ k dM X ( i ) ς ( v ) k L ( E, C ) k M Y ( v ) k E < + ∞ by (2.71), (2.74). Actually we also proved that g ( j ) = q ( j ) ς ∂ j (cid:0) M X ( i ) ς (cid:1) M Y ( j ) .4. The function g ( v ) := X j ∈J g ( j ) ( v ) = X j ∈J q ( j ) ς ∂ j (cid:0) M X ( i ) ς (cid:1) M Y ( j ) (2.47) = d (cid:0) M X ( i ) ς (cid:1)(cid:2) ˜ Y q ]since M X ( i ) ς is differentiable (see (2.71)) and ˜ Y q ∈ E (see (2.74)).Moreover the bound | g | s ′ ,r ′ < ∞ follows by | g | s ′ ,r ′ = | d (cid:0) M X ( i ) ς (cid:1)(cid:2) ˜ Y q ] | s ′ ,r ′ ≤ | d (cid:0) M X ς (cid:1)(cid:2) ˜ Y q ] | s ′ ,r ′ and | d (cid:0) M X ς (cid:1)(cid:2) ˜ Y q ] | s ′ ,r ′ (2.53) = sup v ∈ D ( s ′ ,r ′ ) (cid:13)(cid:13)(cid:13) d (cid:0) M X ς (cid:1) ( v ) (cid:2) ˜ Y q ( v ) (cid:3)(cid:13)(cid:13)(cid:13) E,s ′ ,r ′ ≤ sup v ∈ D ( s ′ ,r ′ ) (cid:13)(cid:13)(cid:13) d (cid:0) M X ς (cid:1) ( v ) (cid:13)(cid:13)(cid:13) L (( E,s,r ) , ( E,s ′ ,r ′ )) k ˜ Y q ( v ) k E,s,r (2.65) ≤ δ − | M X ς | s,r sup v ∈ D ( s ′ ,r ′ ) k ˜ Y q ( v ) k E,s,r (2.62) , (2.74) ≤ δ − k X ς k s,r sup v ∈ D ( s ′ ,r ′ ) k (cid:0) M Y (cid:1) ( v ) k E,s,r (2.53) ≤ δ − k X ς k s,r | M Y | s,r (2.64) ≤ δ − k X ς k s,r k Y k s,r < + ∞ (2.76)because k Y k s,r < + ∞ and k X ς k s,r ≤ k X k s,r < + ∞ by assumption.Hence Lemma 2.6 implies dX ( i ) ς [ Y ] (2.47) = X j ( ∂ j X ( i ) ς ) Y ( j ) =: f Lemma . ≺ g := d (cid:0) M X ( i ) ς (cid:1)(cid:2) ˜ Y q ] , ∀ i ∈ J , namely, by (2.37) and Definition 2.7, dX ς [ Y ] ≺ M ( dX ς [ Y ]) ≺ d (cid:0) M X ς (cid:1)(cid:2) ˜ Y q ] . (2.77)Hence (2.73) is fully justified. By (2.77) and (2.63) we get k dX ς [ Y ] k s ′ ,r ′ ≤ k d (cid:0) M X ς (cid:1)(cid:2) ˜ Y q ] k s ′ ,r ′ (2.64) ≤ n (cid:12)(cid:12)(cid:12) M (cid:16) d (cid:0) M X ς (cid:1)(cid:2) ˜ Y q ] (cid:17)(cid:12)(cid:12)(cid:12) s ′ ,r ′ = 2 n (cid:12)(cid:12)(cid:12) d (cid:0) M X ς (cid:1)(cid:2) ˜ Y q ] (cid:12)(cid:12)(cid:12) s ′ ,r ′ (2.78)because d (cid:0) M X ς (cid:1)(cid:2) ˜ Y q ] coincides with its majorant by (2.77). Finally (2.70) follows by (2.78), (2.76).17 .2 Hamiltonian formalism Given a function H : D ( s, r ) ⊂ E → C we define the associated Hamiltonian vector field X H := ( ∂ y H, − ∂ x H, − i ∂ ¯ z H, i ∂ z H ) (2.79)where the partial derivatives are defined as in (2.30).For a subset of indices I ⊂ I , the bound (2.55) implies k X Π I H k s,r ≤ k X H k s,r . (2.80)The Poisson brackets are defined by { H, K } := { H, K } x,y + { H, K } z, ¯ z := (cid:16) ∂ x H · ∂ y K − ∂ x K · ∂ y H (cid:17) + i (cid:16) ∂ z H · ∂ ¯ z K − ∂ ¯ z H · ∂ z K (cid:17) = ∂ x H · ∂ y K − ∂ x K · ∂ y H + i ∂ z + H · ∂ z − K − i ∂ z − H · ∂ z + K = ∂ x H · ∂ y K − ∂ x K · ∂ y H + i X σ = ± , j ∈ Z \I σ∂ z σj H ∂ z − σj K (2.81)where “ · ” denotes the standard pairing a · b := X j a j b j . We recall the Jacobi identity {{ K, G } , H } + {{ G, H } , K } + {{ H, K } , G } = 0 . (2.82)Along this paper we shall use the Lie algebra notationsad F := { , F } , e ad F := ∞ X k =0 ad kF k ! . (2.83)Given a set of indices I := { j , . . . , j n } ⊂ Z , (2.84)we define the momentum M := M I := n X l =1 j l y l + X j ∈ Z \I jz j ¯ z j = n X l =1 j l y l + X j ∈ Z \I jz + j z − j . We say that a function H satisfies momentum conservation if { H, M} = 0.By (2.81), any monomial e i k · x y i z α ¯ z β is an eigenvector of the operator ad M , namely { e i k · x y i z α ¯ z β , M} = π ( k, α, β ) e i k · x y i z α ¯ z β (2.85)where π ( k, α, β ) := n X l =1 j l k l + X j ∈ Z \I j ( α j − β j ) . (2.86)We refer to π ( k, α, β ) as the momentum of the monomial e i k · x y i z α ¯ z β . A monomial satisfies momentumconservation if and only if π ( k, α, β ) = 0 . Moreover, a power series (2.7) with k f k s,r < + ∞ satisfiesmomentum conservation if and only if all its monomials have zero momentum.Let O ⊂ R n be a subset of parameters , and f : D ( s, r ) × O → C with X f : D ( s, r ) × O → E . (2.87)18or λ > 0, we consider | X f | λs,r, O := | X f | λs,r := sup O | X f | s,r + λ | X f | lip s,r (2.88):= sup ξ ∈O | X f ( ξ ) | s,r + λ sup ξ,η ∈O , ξ = η | X f ( ξ ) − X f ( η ) | s,r | ξ − η | . Note that | · | λs,r is only a semi-norm on spaces of functions f because the Hamiltonian vector field X f = 0 when f is constant. Definition 2.8. A function f as in (2.87) is called • regular , if the sup-norm | X f | s,r, O := sup O | X f | s,r < ∞ , see (2.53) . • M-regular , if the majorant norm k X f k s,r, O := sup O k X f k s,r < ∞ , see (2.54) . • λ -regular , if the Lipschitz semi-norm | X f | λs,r, O < ∞ , see (2.88) .We denote by H s,r the space of M-regular Hamiltonians and by H null s,r its subspace of functions satisfyingmomentum conservation.When I = ∅ (namely there are no ( x, y ) -variables) we denote the space of M-regular functionssimply by H r , similarly H null r , and we drop s form the norms, i.e. | · | r , k · k r , | · | r, O , etc. Note that, by (2.62) and (2.88), we haveM − regular = ⇒ regular ⇐ = λ − regular . (2.89)If H , F satisfy momentum conservation, the same holds for { H, K } . Indeed by the Jacobi identity(2.82), {M , H } = 0 and {M , K } = 0 = ⇒ {M , { H, K }} = 0 . (2.90)For H, K ∈ H s,r we have X { H,K } = dX H [ X K ] − dX K [ X H ] = [ X H , X K ] (2.91)and the commutator Lemma 2.15 implies the fundamental lemma below. Lemma 2.16. Let H, K ∈ H s,r . Then, for all r/ ≤ r ′ < r , s/ ≤ s ′ < s k X { H,K } k s ′ ,r ′ = k [ X H , X K ] k s ′ ,r ′ ≤ n +3 δ − k X H k s,r k X K k s,r (2.92) where δ is defined in (2.66) . Unlike the sup-norm, the majorant norm of a function is very sensitive to coordinate transforma-tions. For our purposes, we only need to consider close to identity canonical transformations that aregenerated by an M -regular Hamiltonian flow. We show below that the M -regular functions are closedunder this group and we estimate the majorant norm of the transformed Hamiltonian vector field. Lemma 2.17. (Hamiltonian flow) Let r/ ≤ r ′ < r , s/ ≤ s ′ < s , and F ∈ H s,r with k X F k s,r < η := δ/ (2 n +5 e ) (2.93) with δ defined in (2.66) . Then the time -hamiltonian flow Φ F : D ( s ′ , r ′ ) → D ( s, r ) is well defined, analytic, symplectic, and, ∀ H ∈ H s,r , we have H ◦ Φ F ∈ H s ′ ,r ′ and k X H ◦ Φ F k s ′ ,r ′ ≤ k X H k s,r − η − k X F k s,r . (2.94) Finally if F, H ∈ H null s,r then H ◦ Φ F ∈ H null s ′ ,r ′ . roof . We estimate by Lie series the Hamiltonian vector field of H ′ = H ◦ Φ F = e ad F H = ∞ X k =0 ad kF Hk ! = ∞ X k =0 H ( k ) k ! , i . e . X H ′ = ∞ X k =0 X H ( k ) k ! , (2.95)where H ( i ) := ad iF ( H ) = ad F ( H ( i − ), H (0) := H .For each k ≥ 0, divide the intervals [ s ′ , s ] and [ r ′ , r ] into k equal segments and set s i := s − i s − s ′ k , r i := r − i r − r ′ k , i = 0 , . . . , k . By (2.92) we have k X H ( i ) k s i ,r i = k [ X F , X H ( i − ] k s i ,r i ≤ n +3 δ − i k X H ( i − k s i − ,r i − k X F k s i − ,r i − (2.96)where δ i := min (cid:26) − s i s i − , − r i r i − (cid:27) ≥ δk . (2.97)By (2.96)-(2.97) we deduce k X H ( i ) k s i ,r i ≤ n +3 kδ − k X H ( i − k s i − ,r i − k X F k s i − ,r i − , i = 1 , . . . , k . Iterating k -times, and using k X F k s i − ,r i − ≤ k X F k s,r (see (2.3)) k X H ( k ) k s ′ ,r ′ ≤ (2 n +5 kδ − ) k k X H k s,r k X F k ks,r . (2.98)By (2.95), using k k ≤ e k k ! and recalling the definition of η in (2.93), we estimate k X H ′ k s ′ ,r ′ (2.95) ≤ ∞ X k =0 k X H ( k ) k s ′ ,r ′ k ! (2.98) ≤ k X H k s,r ∞ X k =0 (2 n +5 kδ − k X F k s,r ) k k ! ≤ k X H k s,r ∞ X k =0 ( η − k X F k s,r ) k (2.93) = k X H k s,r − η − k X F k s,r proving (2.94).Finally, if F and H satisfy momentum conservation then each ad kF H , k ≥ 1, satisfy momentumconservation. For k = 1 it is proved in (2.90) and, for k > 1, it follows by induction and the Jacobiidentity (2.82). By (2.95) we conclude that also H ◦ Φ F satisfies momentum conservation.We conclude this section with two simple lemmata. Lemma 2.18. Let P = X | k |≤ K,i,α,β P k,i,α,β e i k · x y i z α ¯ z β and | ∆ k,i,α,β | ≥ γ h k i − τ , ∀| k | ≤ K, i, α, β . Then F := X | k |≤ K,i,α,β P k,i,α,β ∆ k,i,α,β e i k · x y i z α ¯ z β satisfies k X F k s,r ≤ γ − K τ k X P k s,r . Proof . By Definition 2.6 and | ∆ k,i,α,β | ≥ γK − τ for all | k | ≤ K . Lemma 2.19. Let P = X j ∈ Z \I P j z j ¯ z j with k X P k r < ∞ . Then | P j | ≤ k X P k r . Proof . By (2.79) and Definition 2.6 we have k X P k r = 2 sup k z k a,p 2. 20 Quasi-T¨oplitz functions Let N ∈ N , θ, µ ∈ R be parameters such that1 < θ, µ < , N L − + 2 κN b − < , κ := max ≤ l ≤ n | j l | , (3.1)(the j l are defined in (2.84)) where 0 < b < L < . (3.2)For N ≥ N , we decompose ℓ a,p I × ℓ a,p I = ℓ a,pL ⊕ ℓ a,pR ⊕ ℓ a,pH (3.3)where ℓ a,pL := ℓ a,pL ( N ) := n w = ( z + , z − ) ∈ ℓ a,p I × ℓ a,p I : z σj = 0 , σ = ± , ∀| j | ≥ N L o ℓ a,pR := ℓ a,pR ( N ) := n w = ( z + , z − ) ∈ ℓ a,p I × ℓ a,p I : z σj = 0 , σ = ± , unless 6 N L < | j | < N o ℓ a,pH := ℓ a,pH ( N ) := n w = ( z + , z − ) ∈ ℓ a,p I × ℓ a,p I : z σj = 0 , σ = ± , ∀| j | ≤ N o . Note that by (3.1)-(3.2) the subspaces ℓ a,pL ∩ ℓ a,pH = 0 and ℓ a,pR = 0. Accordingly we decompose any w ∈ ℓ a,p × ℓ a,p as w = w L + w R + w H and we call w L ∈ ℓ a,pL the “low momentum variables” and w H ∈ ℓ a,pH the “high momentum variables”.We split the Poisson brackets in (2.81) as {· , ·} = {· , ·} x,y + {· , ·} L + {· , ·} R + {· , ·} H where { H, K } H := i X σ = ± , | j | >cN σ∂ z σj H ∂ z − σj K . (3.4)The other Poisson brackets {· , ·} L , {· , ·} R are defined analogously with respect to the splitting (3.3). Lemma 3.1. Consider two monomials m = c k,i,α,β e i k · x y i z α ¯ z β and m ′ = c ′ k ′ ,i ′ ,α ′ ,β ′ e i k ′ · x y i ′ z α ′ ¯ z β ′ .The momentum of mm ′ , { m , m ′ } , { m , m ′ } x,y , { m , m ′ } L , { m , m ′ } R , { m , m ′ } H , equals the sum of themomenta of each monomial m , m ′ . Proof . By (2.86), (2.81), and π ( k + k ′ , α + α ′ , β + β ′ ) = π ( k, α, β ) + π ( k ′ , α ′ , β ′ ) = π ( k, α − e j , β ) + π ( k ′ , α ′ , β ′ − e j ) , for any j ∈ Z .We now define subspaces of H s,r (recall Definition 2.8). Definition 3.1. (Low-momentum) A monomial e i k · x y i z α ¯ z β is ( N, µ ) -low momentum if X j ∈ Z \I | j | ( α j + β j ) < µN L , | k | < N b . (3.5) We denote by L s,r ( N, µ ) ⊂ H s,r the subspace of functions g = X g k,i,α,β e i k · x y i z α ¯ z β ∈ H s,r (3.6)21 hose monomials are ( N, µ ) -low momentum. The corresponding projection Π LN,µ : H s,r → L s,r ( N, µ ) (3.7) is defined as Π LN,µ := Π I (see (2.13) ) where I is the subset of I (see (2.8) ) satisfying (3.5) . Finally,given h ∈ Z , we denote by L s,r ( N, µ, h ) ⊂ L s,r ( N, µ ) the subspace of functions whose monomials satisfy π ( k, α, β ) + h = 0 . (3.8)By (3.5), (3.1)-(3.2), any function in L s,r ( N, µ ), 1 < µ < 6, only depends on x, y, w L and therefore g, g ′ ∈ L s,r ( N, µ ) = ⇒ gg ′ , { g, g ′ } x,y , { g, g ′ } L do not depend on w H . (3.9)Moreover, by (2.86), (3.1), (3.5), if | h | ≥ µN L + κN b = ⇒ L s,r ( N, µ, h ) = ∅ . (3.10) Definition 3.2. ( ( N, θ, µ ) -bilinear) We denote by B s,r ( N, θ, µ ) ⊂ H null s,r the subspace of the ( N, θ, µ ) -bilinear functions defined as f := X | m | , | n | >θN,σ,σ ′ = ± f σ,σ ′ m,n ( x, y, w L ) z σm z σ ′ n with f σ,σ ′ m,n ∈ L s,r ( N, µ, σm + σ ′ n ) (3.11) and we denote the projection Π N,θ,µ : H s,r → B s,r ( N, θ, µ ) . Explicitely, for g ∈ H s,r as in (3.6) , the coefficients in (3.11) of f := Π N,θ,µ g are f σ,σ ′ m,n ( x, y, w L ) := X ( k,i,α,β ) s . t . (3.5) holdsand π ( k,α,β )= − σm − σ ′ n f σ,σ ′ k,i,α,β,m,n e i k · x y i z α ¯ z β (3.12) where f + , + k,i,α,β,m,n := (2 − δ mn ) − g k,i,α + e m + e n ,β , f + , − k,i,α,β,m,n := g k,i,α + e m ,β + e n ,f − , − k,i,α,β,m,n := (2 − δ mn ) − g k,i,α,β + e m + e n , f − , + k,i,α,β,m,n := g k,i,α + e n ,β + e m . (3.13)For parameters 1 < θ < θ ′ , 6 > µ > µ ′ , we have B s,r ( N, θ ′ , µ ′ ) ⊂ B s,r ( N, θ, µ ) . Remark 3.1. The projection Π N,θ,µ can be written in the form Π I , see (2.13) , for a suitable I ⊂ I . The representation in (3.11) is not unique. It becomes unique if we impose the “symmetric” conditions f σ,σ ′ m,n = f σ ′ ,σn,m . (3.14) Note that the coefficients in (3.12) - (3.13) satisfy (3.14) . .1 T¨oplitz functions Let N ≥ N . Definition 3.3. (T¨oplitz) A function f ∈ B s,r ( N, θ, µ ) is ( N, θ, µ ) - T¨oplitz if the coefficients in (3.11) have the form f σ,σ ′ m,n = f σ,σ ′ (cid:0) s ( m ) , σm + σ ′ n (cid:1) for some f σ,σ ′ ( ς, h ) ∈ L s,r ( N, µ, h ) , (3.15) with s ( m ) := sign( m ) , ς = + , − and h ∈ Z . We denote by T s,r := T s,r ( N, θ, µ ) ⊂ B s,r ( N, θ, µ ) the space of the ( N, θ, µ ) -T¨oplitz functions. For parameters N ′ ≥ N , θ ′ ≥ θ , µ ′ ≤ µ , r ′ ≤ r , s ′ ≤ s we have T s,r ( N, θ, µ ) ⊆ T s ′ ,r ′ ( N ′ , θ ′ , µ ′ ) . (3.16) Lemma 3.2. Consider f, g ∈ T s,r ( N, θ, µ ) and p ∈ L s,r ( N, µ , with < µ, µ < . For all < s ′ < s , < r ′ < r and θ ′ ≥ θ, µ ′ ≤ µ one has Π N,θ ′ ,µ ′ { f, p } L , Π N,θ ′ ,µ ′ { f, p } x,y ∈ T s ′ ,r ′ ( N, θ ′ , µ ′ ) . (3.17) If moreover µN L + κN b < ( θ ′ − θ ) N (3.18) then Π N,θ ′ ,µ ′ { f, g } H ∈ T s ′ ,r ′ ( N, θ ′ , µ ′ ) . (3.19) Proof . Write f ∈ T s,r ( N, θ, µ ) as in (3.11) where f σ,σ ′ m,n satisfy (3.15) and (3.14), namely f σ,σ ′ m,n = f σ ′ ,σn,m = f σ,σ ′ ( s ( m ) , σm + σ ′ n ) ∈ L s,r ( N, µ, σm + σ ′ n ) , (3.20)similarly for g . Proof of (3.17) . Since the variables z σm , z σ ′ n , | m | , | n | > θN , are high momentum, { f σ,σ ′ m,n z σm z σ ′ n , p } L = { f σ,σ ′ m,n , p } L z σm z σ ′ n and { f σ,σ ′ m,n , p } L does not depend on w H by (3.9) (recall that f σ,σ ′ m,n , p ∈ L s,r ( N, µ )). The coefficientof z σm z σ ′ n in Π N,θ ′ ,µ ′ { f, p } L isΠ LN,µ ′ { f σ,σ ′ m,n , p } L (3.20) = Π LN,µ ′ { f σ,σ ′ ( s ( m ) , σm + σ ′ n ) , p } L ∈ L s ′ ,r ′ ( N, µ ′ , σm + σ ′ n )using Lemma 3.1 (recall that p has zero momentum). The proof that Π N,θ ′ ,µ ′ { f, p } x,y ∈ T s ′ ,r ′ ( N, θ ′ , µ ′ )is analogous. Proof of (3.19) . A direct computation, using (3.4), gives { f, g } H = X | m | , | n | >θN,σ,σ ′ = ± p σ,σ ′ m,n z σm z σ ′ n with p σ,σ ′ m,n = 2i X | l | >θN , σ = ± σ (cid:16) f σ,σ m,l g − σ ,σ ′ l,n + f σ ′ ,σ n,l g − σ ,σl,m (cid:17) . (3.21)23y (3.9) the coefficient p σ,σ ′ m,n does not depend on w H . ThereforeΠ N,θ ′ ,µ ′ { f, g } H = X | m | , | n | >θ ′ N, σ,σ ′ = ± q σ,σ ′ m,n z σm z σ ′ n with q σ,σ ′ m,n := Π LN,µ ′ p σ,σ ′ m,n (3.22)(recall (3.7)). It results q σ,σ ′ m,n ∈ L s ′ ,r ′ ( N, µ ′ , σm + σ ′ n ) by (3.22), (3.21), and Lemma 3.1 since, i.e., f σ,σ m,l ∈ L s,r ( N, µ, σm + σ l ) and g − σ ,σ ′ l,n ∈ L s,r ( N, µ, − σ l + σ ′ n ) . Hence the ( N, θ ′ , µ ′ )-bilinear function Π N,θ ′ ,µ ′ { f, g } H in (3.22) is written in the form (3.11). It remainsto prove that it is ( N, θ ′ , µ ′ )-T¨oplitz, namely that for all | m | , | n | > θ ′ N , σ, σ ′ = ± , q σ,σ ′ m,n = q σ,σ ′ (cid:0) s ( m ) , σm + σ ′ n (cid:1) for some q σ,σ ′ ( ς, h ) ∈ L s,r ( N, µ ′ , h ) . (3.23)Let us consider in (3.21)-(3.22) the term (with m, n, σ, σ ′ , σ fixed)Π LN,µ ′ X | l | >θN f σ,σ m,l g − σ ,σ ′ l,n (3.24)(the other is analogous). Since f, g ∈ T s,r ( N, θ, µ ) we have f σ,σ m,l = f σ,σ (cid:0) s ( m ) , σm + σ l (cid:1) ∈ L s,r ( N, µ, σm + σ l ) (3.25) g − σ ,σ ′ l,n = g − σ ,σ ′ (cid:0) s ( l ) , − σ l + σ ′ n (cid:1) ∈ L s,r ( N, µ, − σ l + σ ′ n ) . (3.26)By (3.10), (3.25), (3.26), if the coefficients f σ,σ m,l , g − σ ,σ ′ l,n are not zero then | σm + σ l | , | − σ l + σ ′ n | < µN L + κN b . (3.27)By (3.27), (3.1), we get cN > | σm + σ l | = | σσ s ( m ) | m | + s ( l ) | l || , which implies, since | m | > θ ′ N > N (see (3.22)), that the sign s ( l ) = − σσ s ( m ) . (3.28)Moreover | l | ≥ | m | − | σm + σ l | (3.27) > θ ′ N − µN L − κN b (3.18) > θN . This shows that the restriction | l | > θN in the sum (3.24) is automatically met. ThenΠ LN,µ ′ X | l | >θN f σ,σ m,l g − σ ,σ ′ l,n (3.26) = Π LN,µ ′ X l ∈ Z f σ,σ (cid:0) s ( m ) , σm + σ l (cid:1) g − σ ,σ ′ (cid:0) s ( l ) , − σ l + σ ′ n (cid:1) = Π LN,µ ′ X j ∈ Z f σ,σ (cid:0) s ( m ) , j (cid:1) g − σ ,σ ′ (cid:0) s ( l ) , σm + σ ′ n − j (cid:1) (3.28) = Π LN,µ ′ X j ∈ Z f σ,σ (cid:0) s ( m ) , j (cid:1) g − σ ,σ ′ (cid:0) − σσ s ( m ) , σm + σ ′ n − j (cid:1) depends only on s ( m ) and σm + σ ′ n , i.e. (3.23). Given f ∈ H s,r and ˜ f ∈ T s,r ( N, θ, µ ) we setˆ f := N (Π N,θ,µ f − ˜ f ) . (3.29)All the functions f ∈ H s,r below possibly depend on parameters ξ ∈ O , see (2.87). For simplicity weshall often omit this dependence and denote k k s,r, O = k k s,r .24 efinition 3.4. (Quasi-T¨oplitz) A function f ∈ H null s,r is called ( N , θ, µ ) -quasi-T¨oplitz if the quasi-T¨oplitz semi-norm k f k Ts,r := k f k Ts,r,N ,θ,µ := sup N ≥ N h inf ˜ f ∈T s,r ( N,θ,µ ) (cid:16) max {k X f k s,r , k X ˜ f k s,r , k X ˆ f k s,r } (cid:17)i (3.30) is finite. We define Q Ts,r := Q Ts,r ( N , θ, µ ) := n f ∈ H null s,r : k f k Ts,r,N ,θ,µ < ∞ o . In other words, a function f is ( N , θ, µ )-quasi-T¨oplitz with semi-norm k f k Ts,r if, for all N ≥ N , ∀ ε > 0, there is ˜ f ∈ T s,r ( N, θ, µ ) such thatΠ N,θ,µ f = ˜ f + N − ˆ f and k X f k s,r , k X ˜ f k s,r , k X ˆ f k s,r ≤ k f k Ts,r + ε . (3.31)We call ˜ f ∈ T s,r ( N, θ, µ ) a “ T¨oplitz approximation ” of f and ˆ f the “ T¨oplitz-defect ”. Note that, byDefinition 3.3 and (3.29) Π N,θ,µ ˜ f = ˜ f , Π N,θ,µ ˆ f = ˆ f . By the definition (3.30) we get k X f k s,r ≤ k f k Ts,r (3.32)and we complete (2.89) noting thatquasi-T¨oplitz = ⇒ M − regular = ⇒ regular ⇐ = λ − regular . (3.33)Clearly, if f is ( N , θ, µ )-T¨oplitz then f is ( N , θ, µ )-quasi-T¨oplitz and k f k Ts,r,N ,θ,µ = k X f k s,r . (3.34)Then we have the following inclusions T s,r ⊂ Q Ts,r , B s,r ⊂ H null s,r ⊂ H s,r . Note that neither B s,r ⊆ Q Ts,r nor B s,r ⊇ Q Ts,r . Lemma 3.3. For parameters N ≥ N , µ ≤ µ , θ ≥ θ , r ≤ r , s ≤ s , we have Q Ts,r ( N , θ, µ ) ⊂ Q Ts ,r ( N , θ , µ ) and k f k Ts ,r ,N ,θ ,µ ≤ max { s/s , ( r/r ) }k f k Ts,r,N ,θ,µ . (3.35) Proof . By (3.31), for all N ≥ N ≥ N (since θ ≥ θ , µ ≤ µ )Π N,θ ,µ f = Π N,θ ,µ Π N,θ,µ f = Π N,θ ,µ ˜ f + N − Π N,θ ,µ ˆ f . The function Π N,θ ,µ ˜ f ∈ T s ,r ( N, θ , µ ) and k X Π N,θ ,µ ˜ f k s ,r (2.80) ≤ k X ˜ f k s ,r (3.31) ≤ k f k Ts ,r + ε , k X Π N,θ ,µ ˆ f k s ,r (2.80) ≤ k X ˆ f k s ,r (3.31) ≤ k f k Ts ,r + ε . Hence, ∀ N ≥ N , inf ˜ f ∈T s ,r ( N,θ ,µ ) (cid:16) max {k X f k s ,r , k X ˜ f k s ,r , k X ˆ f k s ,r } (cid:17) ≤ k f k Ts ,r + ε , ε > f ∈ H s,r we define its homogeneous component of degree l ∈ N , f ( l ) := Π ( l ) f := X k ∈ Z n , | i | + | α | + | β | = l f k,i,α,β e i k · x y i z α ¯ z β , (3.36)and the projections f K := Π | k |≤ K f := X | k |≤ K,i,α,β f k,i,α,β e i k · x y i z α ¯ z β , Π >K f := f − Π | k |≤ K f . (3.37)We also set f ≤ K := Π | k |≤ K f ≤ , f ≤ := f (0) + f (1) + f (2) . (3.38)The above projectors Π ( l ) , Π | k |≤ K , Π >K have the form Π I , see (2.13), for suitable subsets I ⊂ I . Lemma 3.4. (Projections) Let f ∈ Q Ts,r ( N , θ, µ ) . Then, for all l ∈ N , K ∈ N , k Π ( l ) f k Ts,r,N ,θ,µ ≤ k f k Ts,r,N ,θ,µ (3.39) k f ≤ k Ts,r,N ,θ,µ , k f − f ≤ K k Ts,r,N ,θ,µ ≤ k f k Ts,r,N ,θ,µ (3.40) k Π | k |≤ K f k Ts,r,N ,θ,µ ≤ k f k Ts,r,N ,θ,µ (3.41) k Π k =0 Π | α | = | β | =1 Π (2) f k Tr,N ,θ,µ ≤ k Π (2) f k Ts,r,N ,θ,µ (3.42) and, ∀ < s ′ < s , k Π >K f k Ts ′ ,r,N ,θ,µ ≤ e − K ( s − s ′ ) ss ′ k f k Ts,r,N ,θ,µ . (3.43) Proof . We first note that by (2.15) (recall also Remark 3.1) we haveΠ ( l ) Π N,θ,µ g = Π N,θ,µ Π ( l ) g , ∀ g ∈ H s,r . (3.44)Then, applying Π ( l ) in (3.31), we deduce that, ∀ N ≥ N , ∀ ε > 0, there is ˜ f ∈ T s,r ( N, θ, µ ) such thatΠ ( l ) Π N,θ,µ f = Π N,θ,µ Π ( l ) f = Π ( l ) ˜ f + N − Π ( l ) ˆ f (3.45)and, by (2.80), (3.31), k X Π ( l ) f k s,r , k X Π ( l ) ˜ f k s,r , k X Π ( l ) ˆ f k s,r ≤ k f k Ts,r + ε . (3.46)We claim that Π ( l ) ˜ f ∈ T s,r ( N, θ, µ ), ∀ l ≥ 0. Hence (3.45)-(3.46) imply Π ( l ) f ∈ Q Ts,r ( N , θ, µ ) and k Π ( l ) f k Ts,r ≤ k f k Ts,r + ε , i.e. (3.39). Let us prove our claim. For l = 0 , ( l ) ˜ f = 0 because ˜ f ∈ T s,r ( N, θ, µ )is bilinear. For l ≥ 2, write ˜ f in the form (3.11) with coefficients ˜ f σ,σ ′ m,n satisfying (3.15). Then also g := Π ( l ) ˜ f has the form (3.11) with coefficients g σ,σ ′ m,n = Π ( l − ˜ f σ,σ ′ m,n which satisfy (3.15) noting that Π ( l ) L s,r ( N, µ, h ) ⊂ L s,r ( N, µ, h ). Hence g ∈ T s,r ( N, θ, µ ), ∀ l ≥ emma 3.5. Assume that, ∀ N ≥ N ∗ , we have the decomposition G = G ′ N + G ′′ N with k G ′ N k Ts,r,N,θ,µ ≤ K , N k X Π N,θ,µ G ′′ N k s,r ≤ K . (3.47) Then k G k Ts,r,N ∗ ,θ,µ ≤ max {k X G k s,r , K + K } . Proof . By assumption, ∀ N ≥ N ∗ , we have k G ′ N k Ts,r,N,θ,µ ≤ K . Then, ∀ ε > 0, there exist˜ G ′ N ∈ T s,r ( N, θ, µ ), ˆ G ′ N , such thatΠ N,θ,µ G ′ N = ˜ G ′ N + N − ˆ G ′ N and k X ˜ G ′ N k s,r , k X ˆ G ′ N k s,r ≤ K + ε . (3.48)Therefore, ∀ N ≥ N ∗ ,Π N,θ,µ G = ˜ G N + N − ˆ G N , ˜ G N := ˜ G ′ N , ˆ G N := ˆ G ′ N + N Π N,θ,µ G ′′ N where ˜ G N ∈ T s,r ( N, θ, µ ) and k X ˜ G N k s,r = k X ˜ G ′ N k s,r (3.48) ≤ K + ε, (3.49) k X ˆ G N k s,r ≤ k X ˆ G ′ N k s,r + N k X Π N,θ,µ G ′′ N k s,r (3.48) , (3.47) ≤ K + ε + K . (3.50)Then G ∈ Q Ts,r,N ∗ ,θ,µ and k G k Ts,r,N ∗ ,θ,µ ≤ sup N ≥ N ∗ max (cid:8) k X G k s,r , k X ˜ G N k s,r , k X ˆ G N k s,r (cid:9) (3.49) , (3.50) ≤ max {k X G k s,r , K + K + ε } . Since ε > Proposition 3.1. (Poisson bracket) Assume that f (1) , f (2) ∈ Q Ts,r ( N , θ, µ ) and N ≥ N , µ ≤ µ , θ ≥ θ , s/ ≤ s < s , r/ ≤ r < r satisfy κN b − L < µ − µ , µN L − + κN b − < θ − θ, N e − N b s − s < , b ( s − s ) N b > . (3.51) Then { f (1) , f (2) } ∈ Q Ts ,r ( N , θ , µ ) and k{ f (1) , f (2) }k Ts ,r ,N ,θ ,µ ≤ C ( n ) δ − k f (1) k Ts,r,N ,θ,µ k f (2) k Ts,r,N ,θ,µ (3.52) where C ( n ) ≥ and δ := min n − s s , − r r o . (3.53)The proof is based on the following splitting Lemma for the Poisson brackets. Lemma 3.6. (Splitting lemma) Let f (1) , f (2) ∈ Q Ts,r ( N , θ, µ ) and (3.51) hold. Then, for all N ≥ N , Π N,θ ,µ { f (1) , f (2) } =Π N,θ ,µ (cid:16)n Π N,θ,µ f (1) , Π N,θ,µ f (2) o H + n Π N,θ,µ f (1) , Π LN, µ f (2) o L + n Π LN, µ f (1) , Π N,θ,µ f (2) o L + n Π N,θ,µ f (1) , Π LN,µ f (2) o x,y + n Π LN,µ f (1) , Π N,θ,µ f (2) o x,y + n Π | k |≥ N b f (1) , f (2) o + n Π | k | 2, with single monomials (with zero momentum) andwe analyze under which conditions the projectionΠ N,θ ,µ n e i k (1) · x y i (1) z α (1) ¯ z β (1) , e i k (2) · x y i (2) z α (2) ¯ z β (2) o , | k (1) | , | k (2) | < N b , is not zero. By direct inspection, recalling the Definition 3.2 of Π N,θ ,µ and the expression (2.81)of the Poisson brackets { , } = { , } x,y + { , } z, ¯ z , one of the following situations (apart from a trivialpermutation of the indexes 1 , 2) must hold:1. one has z α (1) ¯ z β (1) = z ˜ α (1) ¯ z ˜ β (1) z σm z σ j and z α (2) ¯ z β (2) = z ˜ α (2) ¯ z ˜ β (2) z σ ′ n z − σ j where | m | , | n | ≥ θ N , σ, σ , σ ′ = ± , and z ˜ α (1) ¯ z ˜ β (1) z ˜ α (2) ¯ z ˜ β (2) is of ( N, µ )-low momentum. We consider the Poissonbracket { , } z, ¯ z (in the variables ( z + j , z − j )) of the monomials.2. one has z α (1) ¯ z β (1) = z ˜ α (1) ¯ z ˜ β (1) z σm z σ ′ n z σ j and z α (2) ¯ z β (2) = z ˜ α (2) ¯ z ˜ β (2) z − σ j where | m | , | n | ≥ θ N and z ˜ α (1) ¯ z ˜ β (1) z ˜ α (2) ¯ z ˜ β (2) is of ( N, µ )–low momentum. We consider the Poisson bracket { , } z, ¯ z .3. one has z α (1) ¯ z β (1) = z ˜ α (1) ¯ z ˜ β (1) z σm z σ ′ n and z α (2) ¯ z β (2) = z ˜ α (2) ¯ z ˜ β (2) , where | m | , | n | ≥ θ N and z ˜ α (1) ¯ z ˜ β (1) z ˜ α (2) ¯ z ˜ β (2) is of ( N, µ )-low momentum. We consider the Poisson bracket { , } x,y , i.e. inthe variables ( x, y ).Note that when we consider the { , } x,y Poisson bracket, the case z α (1) ¯ z β (1) = z ˜ α (1) ¯ z ˜ β (1) z σm and z α (2) ¯ z β (2) = z ˜ α (2) ¯ z ˜ β (2) z σ ′ n , | m | , | n | ≥ θ N , and z ˜ α (1) ¯ z ˜ β (1) z ˜ α (2) ¯ z ˜ β (2) is of ( N, µ )-low momentum, does not appear. Indeed, the momentum conser-vation − σm = π (˜ α (1) , ˜ β (1) , k (1) ), (2.86) and | k (1) | < N b , give θ N < | m | ≤ X l ∈ Z \I | l | ( | ˜ α (1) l | + | ˜ β (1) l | ) + κN b ≤ µ N L + κN b , which contradicts (3.1). Case 1. The momentum conservation of each monomial gives σ j = − σm − π (˜ α (1) , ˜ β (1) , k (1) ) = σ ′ n + π (˜ α (2) , ˜ β (2) , k (2) ) . (3.56)Since z ˜ α (1) ¯ z ˜ β (1) z ˜ α (2) ¯ z ˜ β (2) is of ( N, µ )-low momentum (Definition 3.1), X l ∈ Z \I | l | (˜ α (1) l + ˜ β (1) l + ˜ α (2) l + ˜ β (2) l ) ≤ µ N L = ⇒ X l ∈ Z \I | l | (˜ α ( i ) l + ˜ β ( i ) l ) ≤ µ N L , i = 1 , , which implies, by (3.56), (2.86), | k (1) | < N b , | j | ≥ θ N − µ N L − κN b > θN by (3.51). Hence | m | , | n | , | j | > θN . Then e i k ( h ) · x y i ( h ) z α ( h ) ¯ z β ( h ) , h = 1 , 2, are ( N, θ, µ )-bilinear. Moreover the ( z j , ¯ z j )are high momentum variables, namely { , } z, ¯ z = { , } H , see (3.4). As m, n run over all Z \ I with | m | , | n | ≥ θ N , we obtain the first term in formula (3.54). Case 2. The momentum conservation of the second monomial reads − σ j = − π (˜ α (2) , ˜ β (2) , k (2) ) . (3.57)28hen, using also (2.86), | k (2) | < N b , that z ˜ α (1) ¯ z ˜ β (1) z ˜ α (2) ¯ z ˜ β (2) is of ( N, µ )-low momentum, | j | + X l ∈ Z \I | l | (˜ α (1) l + ˜ β (1) l ) (3.57) = | π (˜ α (2) , ˜ β (2) , k (2) ) | + X l ∈ Z \I | l | (˜ α (1) l + ˜ β (1) l ) ≤ X l ∈ Z \I | l | (˜ α (1) l + ˜ β (1) l + ˜ α (2) l + ˜ β (2) l ) + κN b ≤ µ N L + κN b (3.51) < µN L . Then z ˜ α (1) ¯ z ˜ β (1) z σ j is of ( N, µ )-low momentum and the first monomial e i k (1) · x y i (1) z α (1) ¯ z β (1) = e i k (1) · x y i (1) z ˜ α (1) ¯ z ˜ β (1) z σ j z σm z σ ′ n is ( N, θ, µ )-bilinear ( µ ≤ µ ). The second monomial e i k (2) · x y i (2) z α (2) ¯ z β (2) = e i k (2) · x y i (2) z ˜ α (2) ¯ z ˜ β (2) z − σ j is ( N, µ )-low-momentum because, arguing as above, | j | + X l | l | (˜ α (2) l + ˜ β (2) l ) (3.57) = | π (˜ α (2) , ˜ β (2) , k (2) ) | + X l | l | (˜ α (2) l + ˜ β (2) l ) ≤ µ N L + κN b (3.51) < µN L . The ( z j , ¯ z j ) are low momentum variables, namely { , } z, ¯ z = { , } L , and we obtain the second and thirdcontribution in formula (3.54). Case 3. We have, for i = 1 , 2, that X l | l | (˜ α ( i ) l + ˜ β ( i ) l ) ≤ X l | l | (˜ α (1) l + ˜ β (1) l + ˜ α (2) l + ˜ β (2) l ) ≤ µ N L ≤ µN L . Then e i k (1) · x y i (1) z α (1) ¯ z β (1) is ( N, θ, µ )-bilinear and e i k (2) · x y i (2) z α (2) ¯ z β (2) is ( N, µ )-low-momentum. Weobtain the fourth and fifth contribution in formula (3.54). Proof of Proposition 3.1. Since f ( i ) ∈ Q Ts,r ( N , θ, µ ), i = 1 , 2, for all N ≥ N ≥ N there exist˜ f ( i ) ∈ T s,r ( N, θ, µ ) and ˆ f ( i ) such that (see (3.31))Π N,θ,µ f ( i ) = ˜ f ( i ) + N − ˆ f ( i ) , i = 1 , , (3.58)and k X f ( i ) k s,r , k X ˜ f ( i ) k s,r , k X ˆ f ( i ) k s,r ≤ k f ( i ) k Ts,r . (3.59)In order to show that { f (1) , f (2) } ∈ Q Ts ,r ( N , θ , µ ) and prove (3.52) we have to provide a decom-position Π N,θ ,µ { f (1) , f (2) } = ˜ f (1 , + N − ˆ f (1 , , ∀ N ≥ N , so that ˜ f (1 , ∈ T s ,r ( N, θ , µ ) and k X { f (1) ,f (2) } k s ,r , k X ˜ f (1 , k s ,r , k X ˆ f (1 , k s ,r < C ( n ) δ − k f (1) k Ts,r k f (2) k Ts,r (3.60)(for brevity we omit the indices N , θ , µ , N , θ, µ ). By (2.92) we have ( δ is defined in (3.53)) k X { f (1) ,f (2) } k s ,r ≤ n +3 δ − k X f (1) k s,r k X f (2) k s,r . f (1 , := Π N,θ ,µ (cid:16)n ˜ f (1) , ˜ f (2) o H + n ˜ f (1) , Π LN, µ f (2) o L + n Π LN, µ f (1) , ˜ f (2) o L + n ˜ f (1) , Π LN,µ f (2) o x,y + n Π LN,µ f (1) , ˜ f (2) o x,y (3.61)and T¨oplitz-defect ˆ f (1 , := N (cid:16) Π N,θ ,µ { f (1) , f (2) } − ˜ f (1 , (cid:17) . (3.62)Lemma 3.2 and (3.51) imply that ˜ f (1 , ∈ T s ,r ( N, θ , µ ). The estimate (3.60) for ˜ f (1 , follows by(3.61), (2.92), (2.80), (3.59). Nextˆ f (1 , = Π N,θ ,µ (cid:16)n ˜ f (1) , ˆ f (2) o H + n ˆ f (1) , ˜ f (2) o H + N − n ˆ f (1) , ˆ f (2) o H + n ˆ f (1) , Π LN, µ f (2) o L + n Π LN, µ f (1) , ˆ f (2) o L + n ˆ f (1) , Π LN,µ f (2) o x,y + n Π LN,µ f (1) , ˆ f (2) o x,y + N n Π | k |≥ N b f (1) , f (2) o + N n Π | k | 2, where σ := s − s . Since (cid:16) − s s + σ/ (cid:17) − ≤ (cid:16) − s s (cid:17) − ≤ δ − with the δ in (3.53), by (2.92) we get k X g k s ,r ≤ C ( n ) δ − N k X Π | k |≥ Nb f (1) k s + σ/ ,r k X f (2) k s,r (2.57) ≤ C ( n ) δ − N ss e − N b ( s − s ) / k X f (1) k s,r k X f (2) k s,r (3.51) ≤ C ( n ) δ − k X f (1) k s,r k X f (2) k s,r , for every N ≥ N . The proof of Proposition 3.1 is complete.The quasi-T¨oplitz character of a function is preserved under the flow generated by a quasi-T¨oplitzHamiltonian. Proposition 3.2. (Lie transform) Let f, g ∈ Q Ts,r ( N , θ, µ ) and let s/ ≤ s ′ < s , r/ ≤ r ′ < r .There is c ( n ) > such that, if k f k Ts,r,N ,θ,µ ≤ c ( n ) δ , (3.63) with δ defined in (2.66) , then the hamiltonian flow of f at time t = 1 , Φ f : D ( s ′ , r ′ ) → D ( s, r ) is welldefined, analytic and symplectic, and, for N ′ ≥ max { N , ¯ N } , ¯ N := exp (cid:16) max n b , L − b , − L , o(cid:17) , (3.64)( recall (3.2)) , µ ′ < µ , θ ′ > θ , satisfying κ ( N ′ ) b − L ln N ′ ≤ µ − µ ′ , (6 + κ )( N ′ ) L − ln N ′ ≤ θ ′ − θ , N ′ ) − b ln N ′ ≤ b ( s − s ′ ) , (3.65) we have e ad f g ∈ Q Ts ′ ,r ′ ( N ′ , θ ′ , µ ′ ) and k e ad f g k Ts ′ ,r ′ ,N ′ ,θ ′ ,µ ′ ≤ k g k Ts,r,N ,θ,µ . (3.66)30 oreover, for h = 0 , , , and coefficients ≤ b j ≤ /j ! , j ∈ N , (cid:13)(cid:13)(cid:13) X j ≥ h b j ad jf ( g ) (cid:13)(cid:13)(cid:13) Ts ′ ,r ′ ,N ′ ,θ ′ ,µ ′ ≤ Cδ − k f k Ts,r,N ,θ,µ ) h k g k Ts,r,N ,θ,µ . (3.67)Note that (3.66) is (3.67) with h = 0, b j := 1 /j ! Proof . Let us prove (3.67). We define G (0) := g , G ( j ) := ad jf ( g ) := ad f ( G ( j − ) = { f, G ( j − } , j ≥ , and we split, for h = 0 , , G ≥ h := X j ≥ h b j G ( j ) = J − X j = h b j G ( j ) + X j ≥ J b j G ( j ) =: G ≥ h 2, we get k X G ≥ h k s ′ ,r ′ ≤ η h k X g k s,r . (3.72)For any N ≥ N ′ we choose J := J ( N ) := ln N , (3.73)and we set G ′ N := G ≥ h 1. We define, ∀ i = 0 , . . . , j , µ i := µ − i µ − µ ′ j , θ i := θ + i θ ′ − θj , r i := r − i r − r ′ j , s i := s − i s − s ′ j , (3.77)and we prove inductively that, for all i = 0 , . . . , j , k ad if ( g ) k Ts i ,r i ,N,θ i ,µ i ≤ ( C ′ jδ − k f k Ts,r ) i k g k Ts,r , (3.78)which, for i = j , gives (3.76). For i = 0, formula (3.78) follows because g ∈ Q Ts,r ( N , θ, µ ) and Lemma3.3.Now assume that (3.78) holds for i and prove it for i + 1. We want to apply Proposition 3.1 to thefunctions f and ad if ( g ) with N N , s s i , s s i +1 , θ θ i , θ θ i +1 , etc. We have to verifyconditions (3.51) that reads κN b − L < µ i − µ i +1 , µ i N L − + κN b − < θ i +1 − θ i , (3.79)2 N e − N b si − si +12 < , b ( s i − s i +1 ) N b > . (3.80)Since, by (3.77), µ i − µ i +1 = µ − µ ′ j , θ i +1 − θ i = θ − θ ′ j , s i − s i +1 = s − s ′ j and j < J = ln N (see (3.73)), 0 < b < L < µ ′ ≤ µ ≤ 6, the above conditions(3.79)-(3.80) are implied by κN b − L ln N < µ − µ ′ , (6 + κ ) N L − ln N < θ ′ − θ , N e − N b ( s − s ′ ) / N < , b ( s − s ′ ) N b > N . (3.81)The last two conditions (3.81) are implied by b ( s − s ′ ) N b > N and since N ≥ e / − b (recall(3.64)). Recollecting we have to verify κN b − L ln N ≤ µ − µ ′ , (6 + κ ) N L − ln N ≤ θ ′ − θ , N − b ln N ≤ b ( s − s ′ ) . (3.82)Since the function N N − γ ln N is decreasing for N ≥ e /γ , we have that (3.82) follows by (3.64)-(3.65). Therefore Proposition 3.1 implies that ad i +1 f ( g ) ∈ Q Ts i +1 ,r i +1 ( N, θ i +1 , µ i +1 ) and, by (3.52),(3.35), we get k ad i +1 f ( g ) k Ts i +1 ,r i +1 ,N,θ i +1 ,µ i +1 ≤ C ′ δ − i k f k Ts,r k ad if ( g ) k Ts i ,r i ,N,θ i ,µ i (3.83)where δ i := min (cid:26) − s i +1 s i , − r i +1 r i (cid:27) ≥ δj (3.84)and δ is defined in (2.66). Then k ad i +1 f ( g ) k Ts i +1 ,r i +1 ,N,θ i +1 ,µ i +1 (3.83) , (3.84) ≤ C ′ jδ − k f k Ts,r,N ,θ,µ k ad if ( g ) k Ts i ,r i ,N,θ i ,µ i (3.78) ≤ ( C ′ jδ − k f k Ts,r ) i +1 k g k Ts,r proving (3.78) by induction. 32 An abstract KAM theorem We consider a family of integrable Hamiltonians N := N ( x, y, z, ¯ z ; ξ ) := e ( ξ ) + ω ( ξ ) · y + Ω( ξ ) · z ¯ z (4.1)defined on T ns × C n × ℓ a,p I × ℓ a,p I , where I is defined in (2.84), the tangential frequencies ω := ( ω , . . . , ω n )and the normal frequencies Ω := (Ω j ) j ∈ Z \I depend on n -parameters ξ ∈ O ⊂ R n . For each ξ there is an invariant n -torus T = T n × { } × { } × { } with frequency ω ( ξ ). In itsnormal space, the origin ( z, ¯ z ) = 0 is an elliptic fixed point with proper frequencies Ω( ξ ). The aim isto prove the persistence of a large portion of this family of linearly stable tori under small analyticperturbations H = N + P .( A1 ) Parameter dependence . The map ω : O → R n , ξ ω ( ξ ), is Lipschitz continuous.With in mind the application to NLW we assume( A2 ) Frequency asymptotics . We haveΩ j ( ξ ) = p j + m + a ( ξ ) ∈ R , j ∈ Z \ I , (4.2)for some Lipschitz continuous functions a ( ξ ) ∈ R .By (A1) and (A2), the Lipschitz semi-norms of the frequency maps satisfy, for some 1 ≤ M < ∞ , | ω | lip , | Ω | lip ∞ ≤ M (4.3)where the Lipschitz semi-norm is | Ω | lip ∞ := | Ω | lip ∞ , O := sup ξ,η ∈O ,ξ = η | Ω( ξ ) − Ω( η ) | ∞ | ξ − η | (4.4)and | z | ∞ := sup j ∈ Z \I | z j | . Note that by the Kirszbraun theorem (see e.g. [21]) applied componentwisewe can extend ω, Ω on the whole R n with the same bound (4.3).( A3 ) Regularity . The perturbation P : D ( s, r ) × O → C is λ -regular (see Definition 2.8).In order to obtain the asymptotic expansion (4.9) for the perturbed frequencies we also assume( A4 ) Quasi-T¨oplitz . The perturbation P (preserves momentum and) is quasi-T¨oplitz (see Defini-tion 3.4).Thanks to the conservation of momentum we restrict to the set of indices I := n ( k, l ) ∈ Z n × Z ∞ , ( k, l ) = (0 , , | l | ≤ , where (4.5)or l = 0 , k · j = 0 , or l = σe m , m ∈ Z \ I , k · j + σm = 0 , or l = σe m + σ ′ e n , m, n ∈ Z \ I , k · j + σm + σ ′ n = 0 o . Let P = P ( x ) + ¯ P ( x, y, z, ¯ z ) where ¯ P ( x, , , 0) = 0 . (4.6)33 heorem 4.1. (KAM theorem) Suppose that H = N + P satisfies (A1) - (A4) with s, r > , < θ, µ < , N > . Let γ > be a small parameter and set ε := max n γ − / | X P | λs,r , γ − / k X P k s,r , γ − | X ¯ P | λs,r , γ − k ¯ P k Ts,r,N,θ,µ o , λ := γ/M . (4.7) If ε is small enough, then there exist: • (Frequencies) Lipschitz functions ω ∞ : R n → R n , Ω ∞ : R n → ℓ ∞ , a ∞± : R n → R , such that | ω ∞ − ω | + λ | ω ∞ − ω | lip , | Ω ∞ − Ω | ∞ + λ | Ω ∞ − Ω | lip ∞ ≤ Cγε , | a ∞± | ≤ Cγε , (4.8)sup ξ ∈ R n | Ω ∞ j ( ξ ) − Ω j ( ξ ) − a ∞ s ( j ) ( ξ ) | ≤ γ / ε C | j | , ∀| j | ≥ C ⋆ γ − / . (4.9) • (KAM normal form) A Lipschitz family of analytic symplectic maps Φ : D ( s/ , r/ × O ∞ ∋ ( x ∞ , y ∞ , w ∞ ; ξ ) ( x, y, w ) ∈ D ( s, r ) (4.10) close to the identity where O ∞ := n ξ ∈ O : | ω ∞ ( ξ ) · k + Ω ∞ ( ξ ) · l | ≥ γ | k | τ , ∀ ( k, l ) ∈ I defined in (4.5) , | ω ∞ ( ξ ) · k + p | ≥ γ / | k | τ , ∀ k ∈ Z n , p ∈ Z , ( k, p ) = (0 , , τ > /b see (3.2) , | ω ( ξ ) · k | ≥ γ / | k | n , ∀ < | k | < γ − / (7 n ) o (4.11) such that, ∀ ξ ∈ O ∞ : H ∞ ( · ; ξ ) := H ◦ Φ( · ; ξ ) = ω ∞ ( ξ ) · y ∞ + Ω ∞ ( ξ ) · z ∞ ¯ z ∞ + P ∞ has P ∞≤ = 0 . (4.12) Then, ∀ ξ ∈ O ∞ , the map x ∞ Φ( x ∞ , , ξ ) is a real analytic embedding of an elliptic, n -dimensionaltorus with frequency ω ∞ ( ξ ) for the system with Hamiltonian H . The main novelty of Theorem 4.1 is the asymptotic decay (4.9) of the perturbed frequencies. Inorder to prove (4.9) we use the quasi-T¨oplitz property (A4) of the perturbation. The reason forintroducing in (4.7) conditions for both the Lipschitz-sup and the T¨oplitz-norms is the following. Forthe measure estimates, we need the usual Lipschitz dependence of the perturbed frequencies withrespect to the parameters, see (4.8). This is derived as in [24] and [2]. On the other hand, we do notneed (in section 6) a Lipschitz estimate on a ∞± (that, in any case, could be obtained). For this reason,we do not introduce the Lipschitz dependence in the T¨oplitz norm.In the next Theorem 4.2 we verify the second order Melnikov non-resonance conditions thanks to1. the asymptotic decay (4.9) of the perturbed frequencies,2. the restriction to indices ( k, l ) ∈ I in (4.11) which follows by momentum conservation, see (A4).As in [2], the Cantor set of “good” parameters O ∞ in (4.11), is expressed in terms of the finalfrequencies ω ∞ ( ξ ), Ω ∞ ( ξ ) (and of the initial tangential frequencies ω ( ξ )) and not inductively as, forexample, in [24]. This simplifies the measure estimates. Theorem 4.2. (Measure estimate) Let O := [ ρ/ , ρ ] n , ρ > . Suppose ω ( ξ ) = ¯ ω + Aξ , ¯ ω ∈ R n , A ∈ Mat( n × n ) , Ω j ( ξ ) = p j + m + ~a · ξ , a ∈ R n (4.13)34 nd assume the non-degeneracy condition: A invertible and 2( A − ) T ~a / ∈ Z n \ { } . (4.14) Then, the Cantor like set O ∞ defined in (4.11) , with exponent τ > max { n + 1 , /b } (4.15) ( b is fixed in (3.2) ), satisfies |O \ O ∞ | ≤ C ( τ ) ρ n − γ / . (4.16)Theorem 4.2 is proved in section 6. The asymptotic estimate (4.9) is used for the key inclusion (6.12). In the following by a ⋖ b we mean that there exists c > n, m , κ such that a ≤ cb . We perform a preliminary change of variables to improve the smallness conditions. For all ξ in O ∗ := n ξ ∈ O : | ω ( ξ ) · k | ≥ γ / | k | n , ∀ < | k | < γ − / (7 n ) o (5.1)we consider the solution F ( x ) := X < | k | <γ − / (7 n ) P ,k i ω ( ξ ) · k e i k · x (5.2)of the homological equation − ad N F + Π | k | <γ − / (7 n ) P ( x ) = h P i . (5.3)Here P is defined in (4.6) and h·i denotes the mean value on the angles. Note that for any function F ( x ) we have k F k Ts,r = k X F k s,r , see Definition 3.4. We want to apply Proposition 3.2 with s, r, s ′ , r ′ s/ , r/ , s/ , r/ 2. The condition (3.63) is verified because k F k T s/ ,r = k X F k s/ ,r (5.2) , (5.1) , (2.55) ≤ C ( n, s ) γ − / k X P k s,r (4.7) ≤ C ( n, s ) ε and ε is sufficiently small. Hence the time–one flowΦ := Φ F : D ( s , r ) × O ∗ → D ( s, r ) with s := s/ , r := r/ , (5.4)is well defined, analytic, symplectic. Let µ < µ , θ > θ , N > N large enough, so that (3.65) issatisfied with s, r, N , θ, µ, s, r, N, θ, µ and s ′ , r ′ , N ′ , θ ′ , µ ′ s , r , N , θ , µ . Note that here N isindependent of γ . Hence (3.66) implies k e ad F ¯ P k Ts ,r ,N ,θ ,µ ≤ k ¯ P k Ts,r,N,θ,µ . (5.5)Noting that e ad F P = P and e ad F N = N + ad F N the new Hamiltonian is H := e ad F H = e ad F N + e ad F P + e ad F ¯ P = N + ad F N + P + e ad F ¯ P (5.3) = (cid:0) h P i + N (cid:1) + (cid:16) Π | k |≥ γ − / (7 n ) P + e ad F ¯ P (cid:17) =: N + P . (5.6)35y (2.57) (and since P ( x ) depends only on x ) (cid:13)(cid:13)(cid:13) Π | k |≥ γ − / (7 n ) P (cid:13)(cid:13)(cid:13) T s/ ,r ≤ e − sγ − / (7 n ) / k X P k s,r (4.7) ≤ γ / e − sγ − / (7 n ) / ε ≤ γε , (5.7)for γ small. By (5.7), (5.5) and (4.7) we get k P k Ts ,r ,N ,θ ,µ < γε . (5.8)In the same way, since | X F | λ s/ ,r ≤ C ( n, s ) γ − / | X P | λs,r , we also obtain the Lipschitz estimate | X P | λs ,r < γε . (5.9) We now consider the generic KAM step for an Hamiltonian H = N + P = N + P ≤ K + ( P − P ≤ K ) (5.10)where P ≤ K are defined as in (3.38). Assume that | Ω j − p j + m − a s ( j ) | ≤ γ | j | , ∀ | j | ≥ j ∗ , (5.11) for some a + , a − ∈ R . Let ∆ k,m,n := ω · k + Ω m − Ω n , ˜∆ k,m,n := ω · k + | m | − | n | .If | m | , | n | ≥ max { j ∗ , √ m } and s ( m ) = s ( n ) , then | ∆ k,m,n − ˜∆ k,m,n | ≤ m2 | m − n || n || m | + γ (cid:16) | m | + 1 | n | (cid:17) + m (cid:18) | m | + 1 | n | (cid:19) . (5.12) Proof . For 0 ≤ x ≤ |√ x − − x/ | ≤ x / 2. Setting x := m /n (which is ≤ 1) andusing (5.11), we get (cid:12)(cid:12)(cid:12)(cid:12) Ω n − | n | − m2 | n | − a s ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ γ | n | + m | n | . An analogous estimates holds for Ω m . Since | ∆ k,m,n − ˜∆ k,m,n | = | Ω m − | m | − Ω n + | n || the estimate(5.12) follows noting that a s ( m ) = a s ( n ) .For a monomial m k,i,α,β := e i k · x y i z α ¯ z β we set[ m k,i,α,β ] := ( m k,i,α,β if k = 0 , α = β . (5.13)The following key proposition proves that the solution of the homological equation with a quasi-T¨oplitzdatum is quasi-T¨oplitz. Proposition 5.1. (Homological equation) Let K ∈ N . For all ξ ∈ O such that | ω ( ξ ) · k + Ω( ξ ) · l | ≥ γ h k i τ , ∀ ( k, l ) ∈ I ( see (4.5)) , | k | ≤ K , (5.14) then ∀ P ( h ) K ∈ H null s,r , h = 0 , , see (3.36) , (3.37)) , the homological equations − ad N F ( h ) K + P ( h ) K = [ P ( h ) K ] , h = 0 , , , (5.15)36 ave a unique solution of the same form F ( h ) K ∈ H null s,r with [ F ( h ) K ] = 0 and k X F ( h ) K k s,r < γ − K τ k X P ( h ) K k s,r , | X F ( h ) K | λs,r ⋖ γ − K τ +1 | X P ( h ) K | λs,r (5.16) where γλ − ≥ | ω | lip , | Ω | lip ∞ . In particular F ≤ K := F (0) K + F (1) K + F (2) K solves − ad N F ≤ K + P ≤ K = [ P ≤ K ] . (5.17) Assume now that P ( h ) K ∈ Q Ts,r ( N , θ, µ ) and Ω( ξ ) satisfies (5.11) for all | j | ≥ θN ∗ where N ∗ := max n N , ˆ cγ − / K τ +1 o (5.18) for a constant ˆ c := ˆ c (m , κ ) ≥ . Then, ∀ ξ ∈ O such that | ω ( ξ ) · k + p | ≥ γ / h k i τ , ∀| k | ≤ K, p ∈ Z , (5.19) we have F ( h ) K ∈ Q Ts,r ( N ∗ , θ, µ ) , h = 0 , , , and k F ( h ) K k Ts,r,N ∗ ,θ,µ ≤ cγ − K τ k P ( h ) K k Ts,r,N ,θ,µ . (5.20) Proof . The solution of the homological equation (5.15) is F ( h ) K := − i X | k |≤ K, ( k,i,α,β ) =(0 ,i,α,α )2 i + | α | + | β | = h P k,i,α,β ∆ k,i,α,β e i k · x y i z α ¯ z β , ∆ k,i,α,β := ω ( ξ ) · k + Ω( ξ ) · ( α − β ) . The divisors ∆ k,i,α,β = 0 , ∀ ( k, i, α, β ) = (0 , i, α, α ), because ( k, i, α, β ) = (0 , i, α, α ) is equivalent to( k, α − β ) ∈ I , and the bounds (5.14) hold. Then the first estimates in (5.16) follows by Lemma2.18. The Lipsichtz estimate in (5.16) is standard, see e.g. Lemma 1 (and the first comment afterthe statement) of [24]. We just note that the Melnikov condition used in [24] follows by (5.14) andmomentum consevation, e.g. | ω · k + Ω m − Ω n | (5.14) ≥ γ h k i τ (5.22) = γ | m − n || j · k |h k i τ ≥ γ | m − n | κ h k i τ +1 . For the T¨oplitz estimate notice that the cases h = 0 , N,θ,µ F ≤ K = 0. When h = 2we first consider the subtlest case when P (2) K contains only the monomials with i = 0, | α | = | β | = 1(see (3.36)), namely P := P (2) K = X | k |≤ K,m,n ∈ Z \I P k,m,n e i k · x z m ¯ z n , (5.21)and, because of the conservation of momentum, the indices k, m, n in (5.21) are restricted to j · k + m − n = 0 . (5.22)The unique solution F (2) K of (5.15) with [ F (2) K ] = 0 is F := F (2) K := − i X | k |≤ K, ( k,m,n ) =(0 ,m,m ) P k,m,n ∆ k,m,n e i k · x z m ¯ z n , ∆ k,m,n := ω ( ξ ) · k + Ω m ( ξ ) − Ω n ( ξ ) (5.23)Note that by (5.14) and (5.22) we have ∆ k,m,n = 0 if and only if ( k, m, n ) = (0 , m, m ).37et us prove (5.20). For all N ≥ N ∗ Π N,θ,µ F = − i X | k |≤ K, | m | , | n | >θN P k,m,n ∆ k,m,n e i k · x z m ¯ z n , (5.24)and note that e i k · x is ( N, µ )-low momentum since | k | ≤ K < ( N ∗ ) b ≤ N b by (5.18) and τ > /b . Byassumption P ∈ Q Ts,r,N ,θ,µ and so, recalling formula (3.45), we may write, ∀ N ≥ N ∗ ≥ N ,Π N,θ,µ P = ˜ P + N − ˆ P with ˜ P := X | k |≤ K, | m | , | n | >θN ˜ P k,m − n e i k · x z m ¯ z n ∈ T s,r ( N, θ, µ ) (5.25)and k X P k s,r , k X ˜ P k s,r , k X ˆ P k s,r ≤ kPk Ts,r . (5.26)We now prove that˜ F := X | k |≤ K, | m | , | n | >θN ˜ P k,m − n ˜∆ k,m,n e i k · x z m ¯ z n , ˜∆ k,m,n := ω ( ξ ) · k + | m | − | n | , (5.27)is a T¨oplitz approximation of F . Since | m | , | n | > θN ≥ θN ∗ > N ∗ > κ K ≥ | j · k | by (3.1), wededuce by (5.22) that m, n have the same sign. Then˜∆ k,m,n = ω ( ξ ) · k + | m | − | n | = ω ( ξ ) · k + s ( m )( m − n ) , s ( m ) := sign( m ) , and ˜ F in (5.27) is ( N, θ, µ )-T¨oplitz (see (3.15)). Moreover, since | m | − | n | ∈ Z , by (5.19), we get | ˜∆ k,m,n | ≥ γ / h k i − τ , ∀| k | ≤ K, m, n, (5.28)and Lemma 2.18 and (5.27) imply k X ˜ F k s,r ≤ γ − / K τ k X ˜ P k s,r . (5.29)The T¨oplitz defect is N − ˆ F := Π N,θ,µ F − ˜ F (5.30) (5.24) , (5.27) = X | k |≤ K, | m | , | n | >θN (cid:16) P k,m,n ∆ k,m,n − ˜ P k,m − n ˜∆ k,m,n (cid:17) e i k · x z m ¯ z n = X | k |≤ K, | m | , | n | >θN h(cid:16) P k,m,n ∆ k,m,n − P k,m,n ˜∆ k,m,n (cid:17) + (cid:16) P k,m,n − ˜ P k,m − n ˜∆ k,m,n (cid:17)i e i k · x z m ¯ z n (5.25) = X | k |≤ K, | m | , | n | >θN h P k,m,n (cid:16) ˜∆ k,m,n − ∆ k,m,n ∆ k,m,n ˜∆ k,m,n (cid:17) + N − ˆ P k,m,n ˜∆ k,m,n i e i k · x z m ¯ z n . By (5.12), | m | , | n | ≥ θN ≥ N , and | m − n | ≤ κK (see (5.22)) we get, taking ˆ c large enough, | ˜∆ k,m,n − ∆ k,m,n | ≤ m κK N + 2 γN + m N ≤ ˆ c N (cid:18) KN + γ (cid:19) (5.18) ≤ min (cid:26) ˆ cγ / N , γ / K τ (cid:27) . (5.31)Hence | ∆ k,m,n | ≥ | ˜∆ k,m,n | − | ˜∆ k,m,n − ∆ k,m,n | (5.28) , (5.31) ≥ γ / h k i τ − γ / K τ ≥ γ / h k i τ . (5.32)38herefore (5.31), (5.28), (5.32) imply | ˜∆ k,m,n − ∆ k,m,n || ∆ k,m,n || ˜∆ k,m,n | ≤ ˆ cγ / N h k i τ γ / h k i τ γ / ≤ ˆ cN γ K τ and (5.30), (5.28), and Lemma 2.18, imply k X ˆ F k s,r ≤ ˆ cγ − K τ k X P k s,r + γ − / K τ k X ˆ P k s,r (5.26) ≤ cγ − K τ kPk Ts,r . (5.33)In conclusion (5.16), (5.29), (5.33) prove (5.20) for F .Let us briefly discuss the case when h = 2 and P (2) K contains only the monomials with i = 0, | α | = 2, | β | = 0 or viceversa (see (3.36)). Denoting P := P (2) K := X | k |≤ K,m,n ∈ Z \I P k,m,n e i k · x z m z n , (5.34)we have Π N,θ,µ F = − i X | k |≤ K, | m | , | n | >θN P k,m,n ω · k + Ω m + Ω n e i k · x z m z n where | ω · k + Ω m + Ω n | > ( | m | + | n | ) / > θN/ | m | , | n | > θN and | k | ≤ K < N b . In this casewe may take as T¨oplitz approximation ˜ F = 0. H + Let F = F ≤ K be the solution of the homological equation (5.17). If, for s/ ≤ s + < s , r/ ≤ r + < r ,the condition k F k Ts,r,N ∗ ,θ,µ ≤ c ( n ) δ + , δ + := min n − s + s , − r + r o (5.35)holds (see (3.63)), then Proposition 3.2 (with s ′ s + , r ′ r + , N N ∗ defined in (5.18)) impliesthat the Hamiltonian flow Φ F : D ( s + , r + ) → D ( s, r ) is well defined, analytic and symplectic. Wetransform the Hamiltonian H in (5.10), obtaining H + := e ad F H (2.83) = H + ad F ( H ) + X j ≥ j ! ad jF ( H ) (5.10) = N + P ≤ K + ( P − P ≤ K ) + ad F N + ad F P + X j ≥ j ! ad jF ( H ) (5.17) = N + [ P ≤ K ] + P − P ≤ K + ad F P + X j ≥ j ! ad jF ( H ) := N + + P + with new normal form N + := N + ˆ N , ˆ N := [ P ≤ K ] = ˆ e + ˆ ω · y + ˆΩ z · ¯ z ˆ ω i := ∂ y i | y =0 ,z =¯ z =0 h P i , i = 1 , . . . n , ˆΩ := ( ˆΩ j ) j ∈ Z \I , ˆΩ j := [ P ] j := ∂ z j ¯ z j | y =0 ,z =¯ z =0 h P i (5.36)(the h i denotes the average with respect to the angles x ) and new perturbation P + := P − P ≤ K + ad F P ≤ + ad F P ≥ + X j ≥ j ! ad jF ( H ) (5.37)having decomposed P = P ≤ + P ≥ with P ≥ := X h ≥ P ( h ) , see (3.36).39 .2.3 The new normal form N + Lemma 5.2. Let P ∈ Q Ts,r ( N , θ, µ ) with < θ, µ < , N ≥ . Then | ˆ ω | , | ˆΩ | ∞ ≤ k P (2) k Ts,r,N ,θ,µ (5.38) and there exist ˆ a ± ∈ R satisfying | ˆ a ± | ≤ k P (2) k Ts,r,N ,θ,µ such that | ˆΩ j − ˆ a s ( j ) | ≤ | j | k P (2) k Ts,r,N ,θ,µ , ∀ | j | ≥ N + 1) . (5.39) Moreover | ˆ ω | lip , | ˆΩ | lip ∞ ⋖ | X P (2) | lip s,r . Lemma 5.2 is based on the following elementary Lemma, whose proof is postponed. Lemma 5.3. Suppose that, ∀ N ≥ N ≥ , j ≥ θN , Ω j = a N + b N,j N − with a N , b N,j ∈ R , | a N | ≤ c , | b N,j | ≤ c , (5.40) for some c > (independent of j ). Then there exists a ∈ R , satisfying | a | ≤ c , such that | Ω j − a | ≤ c | j | , ∀ | j | ≥ N + 1) . (5.41) proof of Lemma 5.2. The estimate on ˆ ω is trivial. Regarding ˆΩ we set (recall (3.36), (3.42)) P (2)0 := Π k =0 Π | α | = | β | =1 Π (2) P = X j [ P ] j z j ¯ z j since, by the momentum conservation (2.86), all the monomials in P (2)0 have α = β = e j . Note that[ P ] j is defined in (5.36). By Lemma 2.19 | [ P ] j | ≤ k X P (2)0 k r (3.30) ≤ k P (2)0 k Tr (3.42) ≤ k P (2) k Ts,r . (5.42)We now prove (5.39) for j > j < P (2)0 ∈ Q Tr ( N, θ, µ ), for all N ≥ N ,we may write Π N,θ,µ P (2)0 = ˜ P (2)0 ,N + N − ˆ P (2)0 ,N with˜ P (2)0 ,N := X j>θN ˜ P j z j ¯ z j ∈ T r ( N, θ, µ ) , ˆ P (2)0 ,N := X j>θN ˆ P j z j ¯ z j and k X P (2)0 k r , k X ˜ P (2)0 ,N k r , k X ˆ P (2)0 ,N k r ≤ k P (2)0 k Tr ≤ k P (2) k Ts,r . (5.43)For | j | > θN , since all the quadratic forms in (5.43) are diagonal, we haveˆΩ j = [ P ] j = ˜ P j + N − ˆ P j := a N, + + N − b N,j where a N, + := ˜ P j is independent of j > P (2)0 ,N ∈ T r ( N, θ, µ ) (see (3.15)). Applying Lemma2.19 to ˜ P (2)0 ,N and ˆ P (2)0 ,N , we obtain | a N, + | ≤ k X ˜ P (2)0 ,N k s,r (5.43) ≤ k P (2) k Ts,r , | b N,j | = | ˆ P j | ≤ k X ˆ P (2)0 ,N k r (5.43) ≤ k P (2) k Tr . Hence the assumptions of Lemma 5.3 are satisfied with c = 2 k P (2) k Ts,r and (5.39) follows.40he final Lipschitz estimate is standard, see e.g. [2], [24]. Proof of Lemma 5.3. For all N > N ≥ N , j ≥ θN we get, by (5.40), | a N − a N | = | b N ,j N − − b N,j N − | ≤ c N − . (5.44)Therefore a N is a Cauchy sequence. Let a := lim N → + ∞ a N be its limit. Since | a N | ≤ c we have | a | ≤ c .Moreover, letting N → + ∞ in (5.44), we derive | a − a N | ≤ c N − , ∀ N ≥ N , and, using also (5.40), | Ω j − a | ≤ | Ω j − a N | + | a N − a | ≤ c N − , ∀ N ≥ N , j ≥ N . (5.45)For all j ≥ N + 1) let N := [ j/ 6] (where [ · ] denotes the integer part). Since N ≥ N , j ≥ N , | Ω j − a | (5.45) ≤ c [ j/ ≤ c ( j/ − ≤ c j (cid:16) N (cid:17) ≤ c j for all j ≥ N + 1). P + We introduce, for h = 0 , , ε ( h ) := γ − max n k P ( h ) k Ts,r,N ,θ,µ , | X P ( h ) | λs,r o , ¯ ε := X h =0 ε ( h ) , (5.46)Θ := γ − max n k P k Ts,r,N ,θ,µ , | X P | λs,r o , ( λ defined in (4.7)) and the corresponding quantities for P + with indices r + , s + , N +0 , θ + , µ + . The P ( h ) denote the homogeneous components of P of degree h (see (3.36)). Proposition 5.2. (KAM step) Suppose ( s, r, N , θ, µ ) , ( s + , r + , N +0 , θ + , µ + ) satisfy s/ ≤ s + < s , r/ ≤ r + < r , N +0 > max { N ∗ , ¯ N } (recall (5.18) , (3.64) ) , N +0 ) − b ln N +0 ≤ b ( s − s + ) , (5.47) κ ( N +0 ) b − L ln N +0 ≤ µ − µ + , (6 + κ )( N +0 ) L − ln N +0 ≤ θ + − θ . (5.48) Assume that ¯ εK ¯ τ δ − ≤ c small enough , Θ ≤ , (5.49) where ¯ τ := 2 τ + n + 1 and δ + is defined in (5.35) . Suppose also that (5.11) holds for | j | ≥ θN ∗ .Then, for all ξ ∈ O satisfying (5.14) , (5.19) , denoting by F := F ≤ K the solution of the homologicalequation (5.17) , the Hamiltonian flow Φ F : D ( s + , r + ) → D ( s, r ) , and the transformed Hamiltonian H + := H ◦ Φ F = e ad F H = N + + P + satisfies ε (0)+ ⋖ δ − K τ ¯ ε + ε (0) e − ( s − s + ) K ε (1)+ ⋖ δ − K τ (cid:0) ε (0) + ¯ ε (cid:1) + ε (1) e − ( s − s + ) K ε (2)+ ⋖ δ − K τ (cid:0) ε (0) + ε (1) + ¯ ε (cid:1) + ε (2) e − ( s − s + ) K (5.50)Θ + ≤ Θ(1 + Cδ − K τ ¯ ε ) . (5.51)41e focus on the quasi-T¨oplitz estimates, the Lipschitz ones follow formally in the same way. Theproof splits in several lemmas where we analyze each term of P + in (5.37). We note first that k P ≤ K k Ts,r,N ,θ,µ (3.41) ≤ k P ≤ k Ts,r,N ,θ,µ (3.38) , (5.46) ≤ γ ¯ ε . (5.52)Moreover, the solution F = F (0) + F (1) + F (2) of the homological equation (5.17) (for brevity F ( h ) ≡ F ( h ) K and F ≡ F ≤ K ) satisfies, by (5.20) (with N ∗ defined in (5.18)), (3.41), (5.46), k F ( h ) k Ts,r,N ∗ ,θ,µ ⋖ K ¯ τ ε ( h ) , h = 0 , , , k F k Ts,r,N ∗ ,θ,µ ⋖ K ¯ τ ¯ ε . (5.53)Hence (5.49) and (5.53) imply condition (5.35) and therefore Φ F : D ( s + , r + ) → D ( s, r ) is well defined.We now estimate the terms of the new perturbation P + in (5.37). Lemma 5.4. (cid:13)(cid:13)(cid:13) ad F ( P ≤ ) (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + + (cid:13)(cid:13)(cid:13) X j ≥ j ! ad jF ( H ) (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + ⋖ δ − γK τ ¯ ε . Proof . We have X j ≥ j ! ad jF ( H ) = X j ≥ j ! ad jF ( N + P ) = X j ≥ j ! ad j − F (ad F N ) + X j ≥ j ! ad jF ( P ) (5.17) = X j ≥ j ! ad j − F ([ P ≤ K ] − P ≤ K ) + X j ≥ j ! ad jF ( P ) . By (5.47), (5.48) and (5.35) we can apply Proposition 3.2 with N , N ′ , s ′ , r ′ , θ ′ , µ ′ , δ N ∗ , N +0 , s + ,r + , θ + , µ + , δ + . We get (recall N ∗ ≥ N ) (cid:13)(cid:13)(cid:13) X j ≥ j ! ad jF ( P ) (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + (3.67) , (3.35) ⋖ (cid:16) δ − k F k Ts,r,N ∗ ,θ,µ (cid:17) k P k Ts,r,N ,θ,µ (5.53) , (5.46) ⋖ δ − K τ ¯ ε γ Θ (5.54)and, similarly, (cid:13)(cid:13)(cid:13) X j ≥ j ! ad j − F ( P ≤ K ) (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + = (cid:13)(cid:13)(cid:13) X j ≥ j + 1)! ad jF ( P ≤ K ) (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + (3.67) ⋖ δ − k F k Ts,r,N ∗ ,θ,µ k P ≤ K k Ts,r,N ,θ,µ (5.53) , (5.52) ⋖ δ − K ¯ τ γ ¯ ε . (5.55)Finally, by Proposition 3.1, applied with N , N , s , r , θ , µ , δ N ∗ , N +0 , s + , r + , θ + , µ + , δ + , (5.56)we get (cid:13)(cid:13)(cid:13) ad F ( P ≤ ) (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + (3.52) ⋖ δ − k F k Ts,r,N ∗ ,θ,µ k P ≤ k Ts,r,N ,θ,µ (5.53) , (5.52) ⋖ δ − K ¯ τ γ ¯ ε . (5.57)The bounds (5.54), (5.55), (5.57), and Θ ≤ emma 5.5. (5.51) holds. Proof . By Proposition 3.1 (applied with (5.56)) we have (cid:13)(cid:13)(cid:13) ad F ( P ≥ ) (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + ⋖ δ − k F k Ts,r,N ∗ ,θ,µ k P ≥ k Ts,r,N ,θ,µ (5.53) , (3.40) , (5.46) ⋖ δ − K ¯ τ γ ¯ ε Θ , (5.58)and (5.51) follows by (5.37), (3.40), (3.35), (5.46) (5.58), Lemma 5.4 and ¯ ε ≤ 3Θ (which follows by(5.46) and (3.39)).We now consider P ( h )+ , h = 0 , , 2. The term ad F P ≥ in (5.37) does not contribute to P (0)+ . Onthe contrary, its contribution to P (1)+ is { F (0) , P (3) } (5.59)and to P (2)+ is { F (1) , P (3) } + { F (0) , P (4) } . (5.60) Lemma 5.6. k{ F (0) , P (3) }k Ts + ,r + ,N +0 ,θ + ,µ + ⋖ δ − γK ¯ τ ε (0) Θ and (cid:13)(cid:13)(cid:13) { F (1) , P (3) } + { F (0) , P (4) } (cid:13)(cid:13)(cid:13) Ts + ,r + ,N +0 ,θ + ,µ + ⋖ δ − K ¯ τ γ ( ε (0) + ε (1) )Θ . Proof . By (3.52) (applied with (5.56)), (5.53), (5.46) and (3.39).The contribution of P − P ≤ K in (5.37) to P ( h )+ , h = 0 , , 2, is P ( h ) >K . Lemma 5.7. k P ( h ) >K k Ts + ,r + ,N +0 ,θ + ,µ + ≤ e − K ( s − s + ) γε ( h ) Proof . By (3.43) and (5.46). Proof of Proposition 5.2 completed. Finally, (5.50) follows by (5.37), Lemmata 5.4, 5.6(and (5.59)-(5.60)), Lemma 5.7 and Θ ≤ Lemma 5.8. Suppose that ε (0) i , ε (1) i , ε (2) i ≥ , i = 0 , . . . , ν , satisfy ε (0) i +1 ≤ C ∗ K i ¯ ε i + C ∗ ε (0) i e − K ∗ i (5.61) ε (1) i +1 ≤ C ∗ K i (cid:0) ε (0) i + ¯ ε i (cid:1) + C ∗ ε (1) i e − K ∗ i ε (2) i +1 ≤ C ∗ K i (cid:0) ε (0) i + ε (1) i + ¯ ε i (cid:1) + C ∗ ε (2) i e − K ∗ i , i = 0 , . . . , ν − , where ¯ ε i := ε (0) i + ε (1) i + ε (2) i , for some K , C ∗ , K ∗ > . Then there exist ¯ ε ⋆ < , C ⋆ > , χ ∈ (1 , ,depending only on K , C ∗ , K ∗ (and not on ν and satisfying ≤ C ⋆ e − K ∗ ), such that, if ¯ ε ≤ ¯ ε ⋆ = ⇒ ¯ ε i ≤ C ⋆ ¯ ε e − K ∗ χ i , ∀ i = 0 , . . . , ν . (5.62) Proof . Iterating three times (5.61) we get¯ ε j +3 ≤ c C c ∗ K c j (cid:16) ε (0) j +2 + ε (1) j +2 + ¯ ε j +2 + ¯ ε j +2 e − K ∗ j +2 (cid:17) ≤ c C c ∗ K c j (cid:16) ε (0) j +1 + ¯ ε j +1 + ¯ ε j +1 + ¯ ε j +1 e − K ∗ j +1 (cid:17) ≤ c C c ∗ K c j (cid:16) ¯ ε j + ¯ ε j + ¯ ε j e − K ∗ j (cid:17) , ∀ ≤ j ≤ ν − , (5.63)43or suitable constants 1 < c < c < c . We first claim that (5.62) holds with χ := 6 / i = 3 j ≤ ν. Setting a j := ¯ ε j , we prove thatthere exist C ⋆ large and ¯ ε ⋆ small (as in the statement) such that if a ≤ ¯ ε ⋆ then( S ) j a j ≤ c j +14 a e − K ∗ ˜ χ j , ∀ ≤ j ≤ ν/ c = c ( K , C ∗ , K ∗ ) ≥ χ < / , e.g. ˜ χ := 5 / 4. We proceed byinduction. The statement ( S ) is trivial. Now suppose ( S ) j holds true. Note that a j ≤ ε ⋆ ≤ min j ≥ e K ∗ ˜ χ j /c j +14 . Then ( S ) j +1 follows by a j +1 = ¯ ε j +3 (5.63) ≤ c C c ∗ K c j (cid:16) a j + a j + a j e − K ∗ j (cid:17) a j ≤ ≤ c C c ∗ K c j (cid:16) a j + a j e − K ∗ j (cid:17) ( S ) j ≤ c C c ∗ K c j (cid:16) ( c j +14 a e − K ∗ ˜ χ j ) + ( c j +14 a e − K ∗ ˜ χ j ) e − K ∗ j (cid:17) ≤ c j +24 a e − K ∗ ˜ χ j +3 since 4 c C c ∗ K c j ( c j +14 a e − K ∗ ˜ χ j ) e − K ∗ j ≤ c j +24 a e − K ∗ ˜ χ j +3 taking c large enough (use ˜ χ < 2) and4 c C c ∗ K c j ( c j +14 a e − K ∗ ˜ χ j ) ≤ c j +24 a e − K ∗ ˜ χ j +3 taking a ≤ ¯ ε ⋆ small enough. We have proved inductively ( S ) j . Then (5.62) for i = 3 j follows since6 / χ < ˜ χ := 5 / C ⋆ large enough. The cases i = 3 j + 1 and i = 3 j + 2 followanalogously noting that ¯ ε , ¯ ε can be made small by (5.61) taking ¯ ε ⋆ small.For ν ∈ N , we define • s ν +1 := s ν − s − ν − ց s , r ν +1 := r ν − r − ν − ց r , D ν := D ( s ν , r ν ) , • K ν := K ν , N ν := N νρ with N := ˆ cγ − / K τ +10 , ρ := max n τ + 1) , L − b , − L o , • µ ν +1 := µ ν − µ − ν − ց µ , θ ν +1 := θ ν + θ − ν − ր θ . (5.64)We consider H = N + P : D × O ∗ → C with N := e + ω (0) ( ξ ) · y + Ω (0) ( ξ ) · z ¯ z . We supposethat ω (0) and Ω (0) are defined on the whole R n (using in case the Kirszbraun extension theorem),that Ω (0) satisfies (4.2) and | ω (0) | lip , | Ω (0) | lip ∞ ≤ M on R n . Let O ⊆ { ξ ∈ O ∗ : B γ/M ( ξ ) ⊂ O ∗ } where O ∗ is defined in (5.1) and B r ( ξ ) denotes the open ball in R n of center ξ and radius r > Lemma 5.9. (Iterative lemma) Let H be as above and let ¯ ε , Θ be defined as in (5.46) for P .Then there are K > large enough, ǫ > small enough, such that, if ¯ ε , Θ ≤ ǫ , (5.65) then ( S1 ) ν ∀ ≤ i ≤ ν , there exist ω ( i ) , Ω ( i ) , a ( i ) ± defined for all ξ ∈ R n , satisfying | ω ( i ) − ω (0) | + λ | ω ( i ) − ω (0) | lip , | Ω ( i ) − Ω (0) | ∞ + λ | Ω ( i ) − Ω (0) | lip ∞ ≤ C (1 − − i ) γ ¯ ε (5.66) | a ( i ) ± | ≤ C (1 − − i ) γ ¯ ε , | ω ( i ) | lip , | Ω ( i ) | lip ∞ ≤ (2 − − i ) M . (5.67) There exists H i := N i + P i : D i × O i → C with N i := e i + ω ( i ) ( ξ ) · y + Ω ( i ) ( ξ ) · z ¯ z in normal form,where, for i > , O i := n ξ ∈ O i − : | ω ( i − ( ξ ) · k + Ω ( i − ( ξ ) · l | ≥ (1 − − i ) 2 γ | k | τ , ∀ ( k, l ) ∈ I , | k | ≤ K i − , | ω ( i − ( ξ ) · k + p | ≥ (1 − − i ) 2 γ / | k | τ , ∀ ( k, p ) = (0 , , | k | ≤ K i − , p ∈ Z o . (5.68)44 oreover, ∀ ≤ i ≤ ν , H i = H i − ◦ Φ i where Φ i : D i × O i → D i − is a (Lipschitz) family (in ξ ∈ O i )of close-to-the-identity analytic symplectic maps. Setting, for h = 0 , , , ε ( h ) i := γ − max n k P ( h ) i k Ts i ,r i ,N i ,θ i ,µ i , | X P ( h ) i | λs i ,r i o , ¯ ε i := X h =0 ε ( h ) i , (5.69)Θ i := γ − max n k P i k Ts i ,r i ,N i ,θ i ,µ i , | X P i | λs i ,r i o , ∀ ≤ i ≤ ν and ∀ ξ ∈ R n | ω ( i ) ( ξ ) − ω ( i − ( ξ ) | , | Ω ( i ) ( ξ ) − Ω ( i − ( ξ ) | ∞ , | a ( i ) ± ( ξ ) − a ( i − ± ( ξ ) | ≤ γ ¯ ε i − , | Ω ( i ) j ( ξ ) − a ( i ) s ( j ) ( ξ ) − Ω ( i − j ( ξ ) + a ( i − s ( j ) ( ξ ) | ≤ γ ¯ ε i − | j | , ∀| j | ≥ N i − + 1) . (5.70)( S2 ) ν ∀ ≤ i ≤ ν − , the ε (0) i , ε (1) i , ε (2) i satisfy (5.61) with K = 4 τ +1 , ¯ τ := 2 τ + n + 1 , C ∗ = 4 K τ , K ∗ = s K / . ( S3 ) ν ∀ ≤ i ≤ ν , we have ¯ ε i ≤ C ⋆ ¯ ε e − K ∗ χ i and Θ i ≤ (recall that C ⋆ e − K ∗ ≥ , see Lemma 5.8). Proof . The statement ( S1 ) follows by the hypotheses setting a (0) ± ( ξ ) := 0, ∀ ξ ∈ R n . ( S2 ) isempty. ( S3 ) is trivial. We then proceed by induction.( S1 ) ν +1 . We denote ˆ ω ( ν ) := ∇ y h P ν ( ξ ) i| y =0 ,z =¯ z =0 and ˆΩ ( ν ) j ( ξ ) := ∂ z j ¯ z j | y =0 ,z =¯ z =0 h P ν ( ξ ) i , see (5.36),for all ξ ∈ O ν if ν ≥ ξ ∈ O ∗ (see (5.1)) if ν = 0. By Lemma 5.2 and (5.69) there exist constantsˆ a ( ν ) ± ( ξ ) ∈ R such that | ˆ ω ( ν ) ( ξ ) | , | ˆΩ ( ν ) ( ξ ) | ∞ , | ˆ a ( ν ) ± ( ξ ) | ≤ γ ¯ ε ν , | ˆΩ ( ν ) j ( ξ ) − ˆ a ( ν ) s ( j ) ( ξ ) | ≤ γ ¯ ε ν | j | , ∀| j | ≥ N ν + 1) , (5.71)uniformly in ξ ∈ O ν (resp. O ∗ if ν = 0), and | ˆ ω ( ν ) | lip , | ˆΩ ( ν ) | lip ∞ ≤ C ¯ ε ν . (5.72)Let η := λ = γ/M , η ν := γ/ (2 ν +3 M K τ +1 ν − ) , ν ≥ . (5.73)We claim that, for ν ≥ 1, the η ν -neighborhood of O ν +1 ˜ O ν +1 := [ ξ ∈O ν +1 n ˜ ξ ∈ R n : ˜ ξ = ξ + ˆ ξ , | ˆ ξ | < η ν o ⊆ O ν . (5.74)Note that the definitions of O , O in (5.68), and (5.73) imply ˜ O ⊂ O ∗ . Recalling (5.68), we have toprove that for ν ≥ 1, for every ˜ ξ = ξ + ˆ ξ , ξ ∈ O ν +1 , | ˆ ξ | ≤ η ν , we have | ω ( ν − ( ˜ ξ ) · k + Ω ( ν − ( ˜ ξ ) · l | ≥ (1 − − ν ) 2 γ | k | τ , ∀ ( k, l ) ∈ I , | k | ≤ K ν − , (5.75)and the analogous estimate for | ω ( ν − ( ˜ ξ ) · k + p | . By the expression (5.77) (at the previous step) for ω ( ν ) , Ω ( ν ) , and since χ ν − ∈ [0 , | ω ( ν − ( ˜ ξ ) · k + Ω ( ν − ( ˜ ξ ) · l | ≥ | ω ( ν ) ( ˜ ξ ) · k + Ω ( ν ) ( ˜ ξ ) · l | − | χ ν − ( ˜ ξ ) || ˆ ω ( ν − ( ˜ ξ ) · k + ˆΩ ( ν − ( ˜ ξ ) · l | (5.71) ≥ | ω ( ν ) ( ξ ) · k + Ω ( ν ) ( ξ ) · l | − (cid:12)(cid:12)(cid:12) ( ω ( ν ) ( ˜ ξ ) − ω ( ν ) ( ξ )) · k + (Ω ( ν ) ( ˜ ξ ) − Ω ( ν ) ( ξ )) · l (cid:12)(cid:12)(cid:12) − γ ¯ ε ν − ( K ν − + 2) ξ ∈O ν +1 , (5.68) , ( S ν ≥ (1 − − ν − ) 2 γ | k | τ − ( K ν − + 2)2 M η ν − γ ¯ ε ν − ( K ν − + 2) (5.73) , ( S ν ≥ (1 − − ν ) 2 γ | k | τ ǫ small enough, and (5.75) follows. The estimate for | ω ( ν − ( ˜ ξ ) · k + p | follows similarly.We define a smooth cut-off function χ ν : R n → [0 , 1] which takes value 1 on O ν +1 and value 0outside ˜ O ν +1 . Thanks to (5.74) and recalling (5.73) we can construct χ ν , ν ≥ 0, in such a way that | χ ν | lip ⋖ γ − M ν K τ +1 ν − (5.76)where K − := 1. We extend ˆ ω ( ν ) , ˆΩ ( ν ) , ˆ a ( ν ) ± to zero outside O ν for ν ≥ ν = 0 outside O ⋆ .Then we define on the whole R n ω ( ν +1) := ω ( ν ) + χ ν ˆ ω ( ν ) , Ω ( ν +1) := Ω ( ν ) + χ ν ˆΩ ( ν ) , a ( ν +1) ± := a ( ν ) ± + χ ν ˆ a ( ν ) ± . (5.77)By (5.76) , (5.72) , (5.71), we get | ω ( ν +1) − ω ( ν ) | lip ≤ | χ ν | lip | ˆ ω ( ν ) | + | χ ν || ˆ ω ( ν ) | lip ≤ CK τ +1 ν − M ¯ ε ν + C ¯ ε ν ≤ − ν − M by ( S3 ) ν and ¯ ε small enough. Similarly for | Ω ( ν +1) − Ω ( ν ) | lip ∞ . Recalling also (5.71), we get (5.66) and(5.67) with i = ν + 1. Moreover (5.71)-(5.77) imply (5.70) for i = ν + 1 and ∀| j | > N ν + 1).We wish to apply the KAM step Proposition 5.2 with N = N ν , P = P ν , N = N ν , θ = θ ν . . . and N +0 = N ν +1 , θ + = θ ν +1 , . . . Our definitions in (5.64) (and τ > /b ) imply that the conditions (5.47)-(5.48) are satisfied, for all ν ∈ N , taking K large enough. Moreover, since δ + = δ ν +1 := min n − s ν +1 s ν , − r ν +1 r ν o so that 2 − ν − ≤ δ ν +1 ≤ − ν − , (5.78)and ( S ν the condition (5.49) is satisfied, for ¯ ε ≤ ǫ small enough, ∀ ν ∈ N . By (5.70), the condition(5.11) holds for | j | ≥ θ ν N ν , and (5.14) and (5.19) hold for all ξ ∈ O ν +1 (it is the definition of O ν +1 ,see (5.68)). Hence Proposition 5.2 applies. For all ξ ∈ O ν +1 the Hamiltonian flow Φ ν +1 := Φ F ν : D ν +1 × O ν +1 → D ν and we define H ν +1 := H ν ◦ Φ ν +1 = e ad F ν H ν = N ν +1 + P ν +1 : D ν +1 × O ν +1 → C . ( S2 ) ν +1 follows by (5.50) and (5.64).( S3 ) ν +1 . By ( S ν we can apply Lemma 5.8 and (5.62) implies ¯ ε ν +1 ≤ C ⋆ ¯ ε e − K ∗ χ ν +1 . Moreover, for ǫ small enough, Θ ν +1 (5.51) ≤ Θ Π νi =0 (cid:16) Cδ − i +1 K τi ¯ ε i (cid:17) (5.78) , ( S ν ≤ . Proof of the KAM Theorem 4.1 completed. We apply the iterative Lemma 5.9 to the Hamil-tonian H in (5.6) where ω (0) = ω and Ω (0) = Ω are defined in (4.1). We choose O := n ξ ∈ O : | ω ( ξ ) · k | ≥ γ / | k | n , ∀ < | k | < γ − / (7 n ) o (5.79)so that O ⊆ { ξ ∈ O ∗ : B γ/M ( ξ ) ⊂ O ∗ } , see (5.1) and (4.3). The smallness assumption (5.65) holdsby (5.8)-(5.9) (use also Lemma 3.4) and ε small enough. Then the iterative Lemma 5.9 applies. Letus define ω ∞ := lim ν →∞ ω ( ν ) , Ω ∞ := lim ν →∞ Ω ( ν ) , a ∞± := lim ν →∞ a ( ν ) ± . It could happen that O ν = ∅ for some ν . In such a case O ∞ = ∅ and the iterative process stops afterfinitely many steps. However, we can always set ω ( ν ) := ω ( ν ) , Ω ( ν ) := Ω ( ν ) , a ( ν ) ± := a ( ν ) ± , ∀ ν ≥ ν ,and ω ∞ , Ω ∞ , a ∞± are always well defined.The bounds (4.8) follow by (5.66) (with a different constant C ). We now prove (4.9). We considerthe case j > 0. For all ∀ ν ≥ j ≥ N ν + 1), we have (recall that a (0)+ = 0) | Ω ∞ j − Ω (0) j − a ∞ + | ≤ X ≤ i ≤ ν | Ω ( i +1) j − a ( i +1)+ − Ω ( i ) j + a ( i )+ | + X i>ν | Ω ( i +1) j − Ω ( i ) j | + | a ( i +1)+ − a ( i )+ | (5.70) ≤ γ X ≤ i ≤ ν ¯ ε i j + 4 γ X i>ν ¯ ε i ( S3 ) ν ⋖ ¯ ε γj + γ X i>ν ¯ ε i . For example the first inequality in (5.47) reads N ν +1 ≥ max { N ν , ˆ cγ − / K τ +1 ν , ¯ N } . ∀ ν ≥ 0, 6( N ν + 1) ≤ j < N ν +1 + 1), | Ω ∞ j − Ω (0) j − a ∞ + | ⋖ ¯ ε γj + γ N ν +1 j X i>ν ¯ ε i (5.64) ⋖ ¯ ε γj + γj γ − / K τ +10 ρ ( ν +1) X i>ν ¯ ε i and (4.9) follows by ( S ν .The symplectic transformation Φ in (4.10) is defined byΦ := lim ν →∞ Φ ◦ Φ ◦ Φ ◦ · · · ◦ Φ ν with Φ defined in (5.4). We now verify that Φ is defined on O ∞ , see (4.11). Lemma 5.10. O ∞ ⊂ ∩ i O i (defined in (5.68) ). Proof . We have O ∞ ⊆ O by (4.11) and (5.79). For i ≥ 1, if ξ ∈ O ∞ then, for all | k | ≤ K i , | l | ≤ | ω ( i ) ( ξ ) · k + Ω ( i ) ( ξ ) · l |≥ | ω ∞ ( ξ ) · k + Ω ∞ ( ξ ) · l | − | k | X n ≥ i | ω ( n +1) ( ξ ) − ω ( n ) ( ξ ) | − X n ≥ i | Ω ( n +1) ( ξ ) − Ω ( n ) ( ξ ) | ∞ (4.11) , (5.70) ≥ γ | k | τ − K i γ X n ≥ i ¯ ε n − γ X n ≥ i ¯ ε n ≥ (1 − − i ) 2 γ | k | τ by the definition of K i in (5.64), ( S ν and ε small enough. The other estimate is analogous.Finally P ∞≤ = 0 (see (4.12)) follows by ¯ ε i → i → ∞ . This concludes the proof of Theorem 4.1. We have to estimate the measure of O \ O ∞ = [ ( k,l ) ∈ Λ ∪ Λ ∪ Λ +2 ∪ Λ − R kl ( γ ) [ ( k,p ) ∈ Z n +1 \{ } ˜ R kp ( γ / ) [ ( O \ O ) (6.1)where R kl ( γ ) := R τkl ( γ ) := n ξ ∈ O : | ω ∞ ( ξ ) · k + Ω ∞ ( ξ ) · l | < γ | k | τ o (6.2)˜ R kp ( γ / ) := ˜ R τkp ( γ / ) := n ξ ∈ O : | ω ∞ ( ξ ) · k + p | < γ / | k | τ o (6.3)and Λ h := n ( k, l ) ∈ I (see (4.5)) , | l | = h o , h = 0 , , , Λ = Λ +2 ∪ Λ − , (6.4)Λ +2 := n ( k, l ) ∈ Λ , l = ± ( e i + e j ) o , Λ − := n ( k, l ) ∈ Λ , l = e i − e j o . We first consider the most difficult case Λ − . Setting R k,i,j ( γ ) := R k,e i − e j ( γ ) we show that (cid:12)(cid:12)(cid:12) [ ( k,l ) ∈ Λ − R k,l ( γ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) [ ( k,i,j ) ∈ I R k,i,j ( γ ) (cid:12)(cid:12)(cid:12) ⋖ γ / ρ n − (6.5)where I := n ( k, i, j ) ∈ Z n × ( Z \ I ) : ( k, i, j ) = (0 , i, i ) , j · k + i − j = 0 o . (6.6)Note that the indices in I satisfy || i | − | j || ≤ κ | k | and k = 0 . (6.7)Since the matrix A in (4.13) is invertible, the bound (4.8) implies, for ε small enough, that ω ∞ : O → ω ∞ ( O ) is invertible and | ( ω ∞ ) − | lip ≤ k A − k . (6.8)47 emma 6.1. For ( k, i, j ) ∈ I , η ∈ (0 , , we have |R τk,i,j ( η ) | ⋖ ηρ n − | k | τ +1 . (6.9) Proof . By (4.8) and (4.13) ω ∞ ( ξ ) · k + Ω ∞ i ( ξ ) − Ω ∞ j ( ξ ) = ω ∞ ( ξ ) · k + p i + m − p j + m + r i,j ( ξ )where | r i,j ( ξ ) | = O ( εγ ) , | r i,j | lip = O ( ε ) . (6.10)We introduce the final frequencies ζ := ω ∞ ( ξ ) as parameters (see (6.8)), and we consider f k,i,j ( ζ ) := ζ · k + p i + m − p j + m + ˜ r i,j ( ζ )where also ˜ r i,j := r i,j ◦ ( ω ∞ ) − satisfies (6.10). In the direction ζ = sk | k | − + w , w · k = 0, the function˜ f k,i,j ( s ) := f k,i,j ( sk | k | − + w ) satisfies˜ f k,i,j ( s ) − ˜ f k,i,j ( s ) (6.10) ≥ ( s − s )( | k | − Cε ) ≥ ( s − s ) | k | / . Since | k | ≥ (cid:12)(cid:12)(cid:12)n ζ ∈ ω ∞ ( O ) : | f k,i,j ( ζ ) | ≤ η | k | τ o(cid:12)(cid:12)(cid:12) ⋖ ηρ n − | k | τ +1 . By (6.8) the bound (6.9) follows.We split I = I > ∪ I < where I > := n ( k, i, j ) ∈ I : min {| i | , | j |} > C ♯ γ − / (1 + | k | τ ) o (6.11)where C ♯ > C ⋆ in (4.9) and τ := n + 1. We set I < := I \ I > . Lemma 6.2. For all ( k, i, j ) ∈ I > we have R τ k,i,j ( γ / ) ⊂ R τ k,i ,j (2 γ / ) (6.12) (see (6.2) ), i , j ∈ Z \ I satisfy s ( i ) = s ( i ) , s ( j ) = s ( j ) , | i | − | j | = | i | − | j | (6.13) and min {| i | , | j |} = h C ♯ γ − / (1 + | k | τ ) i . (6.14) Proof . Since | j | ≥ γ − / C ⋆ , by (4.9) and (4.13) we have the frequency asymptoticΩ ∞ j ( ξ ) = | j | + m2 | j | + ~a · ξ + a ∞ s ( j ) ( ξ ) + O (cid:18) m | j | (cid:19) + O (cid:18) ε γ / | j | (cid:19) . (6.15)By (6.7) we have || i | − | j || = || i | − | j || ≤ C | k | , | k | ≥ 1. If ξ ∈ O \ R τ k,i ,j (2 γ / ), since | i | , | j | ≥ µ :=min {| i | , | j |} (recall (6.11) and (6.14)), we have | ω ∞ ( ξ ) · k + Ω ∞ i ( ξ ) − Ω ∞ j ( ξ ) | ≥ | ω ∞ ( ξ ) · k + Ω ∞ i ( ξ ) − Ω ∞ j ( ξ ) |−| Ω ∞ i ( ξ ) − Ω ∞ i ( ξ ) − Ω ∞ j ( ξ ) + Ω ∞ j ( ξ ) | (6.15) ≥ γ / | k | τ − || i | − | i | − | j | + | j ||−| a ∞ s ( i ) − a ∞ s ( i ) − a ∞ s ( j ) + a ∞ s ( j ) |− Cε γ / µ − C m µ − m2 || i | − | j ||| i | | j | − m2 || i | − | j ||| i | | j | (6.13) ≥ γ / | k | τ − Cε γ / µ − C | k | µ 20 (6.14) ≥ γ / | k | τ C ♯ in (6.14) large enough. Therefore ξ ∈ O \ R τ k,i,j ( γ / ) proving (6.12).As a corollary we deduce: Lemma 6.3. (cid:12)(cid:12)(cid:12) [ ( k,i,j ) ∈ I > R τk,i,j ( γ ) (cid:12)(cid:12)(cid:12) ⋖ γ / ρ n − . Proof . Since 0 < γ ≤ τ ≥ τ (see (4.15)), we have (see (6.2)) R τk,i,j ( γ ) ⊂ R τ k,i,j ( γ / ). ThenLemma 6.2 and (6.9) imply that, for each k ∈ Z n , p ∈ Z fixed (cid:12)(cid:12)(cid:12) [ ( k,i,j ) ∈ I > , | i |−| j | = p R τk,i,j ( γ ) (cid:12)(cid:12)(cid:12) ⋖ γ / ρ n − | k | τ +1 . Therefore (cid:12)(cid:12)(cid:12) [ ( k,i,j ) ∈ I > R τk,i,j ( γ ) (cid:12)(cid:12)(cid:12) ⋖ X k, | p |≤ C | k | γ / ρ n − | k | τ +1 ⋖ X k γ / ρ n − | k | τ proving the lemma. Lemma 6.4. (cid:12)(cid:12)(cid:12) [ ( k,i,j ) ∈ I < R τk,i,j ( γ ) (cid:12)(cid:12)(cid:12) ⋖ γ / ρ n − . Proof . For all ( k, i, j ) ∈ I < such that R τk,i,j ( γ ) = ∅ we have (see (6.6))min {| i | , | j |} < C ♯ γ − / (1 + | k | τ ) , j − i = k · j = ⇒ max {| i | , | j |} < C ′ γ − / (1 + | k | τ ) . Therefore, using also Lemma 6.1 and (6.7) (cid:12)(cid:12)(cid:12) [ ( k,i,j ) ∈ I < R τk,i,j ( γ ) (cid:12)(cid:12)(cid:12) ⋖ X k X | i |≤ C ′ γ − / | k | τ j = i + k · j γρ n − | k | τ +1 ⋖ X k γ / ρ n − | k | τ − τ +1 which, by (4.15), gives the lemma.Lemmata 6.3, 6.4 imply (6.5). This concludes the case ( k, l ) ∈ Λ − . Let consider the other cases.The analogue of Lemma 6.1 is Lemma 6.5. For ( k, l ) ∈ Λ ∪ Λ ∪ Λ +2 , η ∈ (0 , √ m / , we have |R kl ( η ) | ⋖ ηρ n − | k | τ . (6.16) Proof . We consider only the case ( k, l ) ∈ Λ +2 , l = e i + e j . By (4.8) and (4.13) f k,i,j ( ξ ) := ω ∞ ( ξ ) · k + Ω ∞ i ( ξ ) + Ω ∞ j ( ξ ) = ω ∞ ( ξ ) · k + p i + m + p j + m + 2 ~a · ξ + r i,j ( ξ )where | r i,j ( ξ ) | = O ( εγ ) , | r i,j | lip = O ( ε ). Changing variables ζ := ω ∞ ( ξ ) we find f k,i,j ( ζ ) := ζ · k + p i + m + p j + m + 2 ~a · A − ( ζ − ¯ ω ) + ˜ r i,j ( ζ ) (6.17)where also ˜ r i,j ( ζ ) = O ( εγ ) , | ˜ r i,j | lip = O ( ε ) . (6.18)If k = ~a = 0 then the function in (6.17) is bigger than √ m and R l ( η ) = ∅ , for 0 ≤ η ≤ √ m / a := A T k + 2 ~a = A T (cid:0) k + 2( A − ) T ~a (cid:1) satisfies | ˜ a | ≥ c = c ( A, ~a ) > , ∀ k = 0 . (6.19)49he function ˜ f k,i,j ( s ) := f k,i,j ( s ˜ a | ˜ a | − + w ), ˜ a · w = 0, satisfies ˜ f k,i,j ( s ) − ˜ f k,i,j ( s ) ≥ ( s − s )( | ˜ a | − Cε ) ≥ ( s − s ) | ˜ a | / (cid:12)(cid:12)(cid:12) [ ( k,l ) ∈ Λ ∪ Λ ∪ Λ +2 R kl ( γ ) (cid:12)(cid:12)(cid:12) ⋖ γρ n − , (cid:12)(cid:12)(cid:12) [ ( k,p ) ∈ Z n +1 \{ } ˜ R kp ( γ / ) (cid:12)(cid:12)(cid:12) , |O \ O | ⋖ γ / ρ n − . (6.20)Finally (6.1), (6.5), (6.20) imply (4.16). For ~ = ( j , . . . , j d ) ∈ Z d , ~σ = ( σ , . . . , σ d ) ∈ {±} d we denote ~σ · ~ := σ j + . . . + σ d j d , and, given( u j , ¯ u j ) j ∈ Z = ( u + j , u − j ) j ∈ Z , we define the monomial u ~σ~ := u σ j · · · u σ d j d (of degree d ). We now consider the Hamiltonian (1.4) when F ( s ) = s / H = X j ∈ Z λ j u + j u − j + X ~ ∈ Z ,~σ ∈{±} ,~σ · ~ =0 u ~σ~ =: N + G (7.1)= X j ∈ Z λ j u j ¯ u j + X | α | + | β | =4 , π ( α,β )=0 G α,β u α ¯ u β , G α,β := ( | α | + | β | )! α ! β ! = 4! α ! β ! , where ( u + , u − ) = ( u, ¯ u ) ∈ ℓ a,p × ℓ a,p for some a > p > / 2, and the momentum is (see (2.86)) π ( α, β ) = X j ∈ Z j ( α j − β j ) . Note that 0 ≤ G α,β ≤ 4! (recall α ! = Π i ∈ Z α i !) Lemma 7.1. For all R > , N satisfying (3.1) , the Hamiltonian G defined in (7.1) belongs to Q TR ( N , / , and k G k TR,N , / , = k X G k R ⋖ R . (7.2) Proof . The Hamiltonian vector field X G := ( − i ∂ ¯ u G, i ∂ u G ) has componentsi σ∂ u σl G = i σ X | α | + | β | =3 ,π ( α,β )= − σl G l,σα,β u α ¯ u β , σ = ± , l ∈ Z , where G l, + α,β = ( α l + 1) G α + e l ,β , G l, − α,β = ( β l + 1) G α,β + e l . Note that 0 ≤ G l,σα,β ≤ 5! By Definitions 2.6, 2.8 and (2.2) k X G k R = 1 R sup k u k a,p , k ¯ u k a,p For any I ⊂ Z and m > , there exists R > and areal analytic, symplectic change of variables Γ : B R/ × B R/ ⊂ ℓ a,p × ℓ a,p → B R × B R ⊂ ℓ a,p × ℓ a,p , < R < R , that takes the Hamiltonian H = N + G in (7.1) into H Birkhoff := H ◦ Γ = N + G + ˆ G + K (7.8) where G, ˆ G are defined in (7.6) and K := X ~ ∈ Z d, ~σ ∈{ + , −} d,d ≥ , ~σ · ~ =0 K ~,~σ u ~σ~ (7.9) satisfies, for N ′ := N ′ (m , I , L, b ) large enough, k K k TR/ ,N ′ , , ⋖ R . (7.10)51he rest of this subsection is devoted to the proof of Proposition 7.1. We start following thestrategy of [25]. By (2.81) the Poisson bracket { N, u ~σ~ } = − i ~σ · λ ~ u ~σ~ (7.11)where λ ~ := ( λ j , . . . , λ j d ) and λ j := λ j (m) := p j + m.The following lemma extends Lemma 4 of [25]. Lemma 7.2. (Small divisors) Let ~ ∈ Z , ~σ ∈ {±} be such that ~σ · ~ = 0 and (up to permutationof the indexes) ~ = 0 , X i =1 σ i = 0 , (7.12)or ~ = (0 , , q, q ) , q = 0 , σ = σ , (7.13)or ~ = ( p, p, − p, − p ) , p = 0 , σ = σ , (7.14)or ~ = ( p, p, q, q ) . (7.15) Then, there exists an absolute constant c ∗ > , such that, for every m ∈ (0 , ∞ ) , | ~σ · λ ~ (m) | ≥ c ∗ m( n + m) / > where n := min {h j i , h j i , h j i , h j i} . (7.16) Proof . In the Appendix.The map Γ := Φ F is obtained as the time-1 flow generated by the Hamiltonian F := − X ~ · ~σ =0 ,~σ · λ~ =0and ~/ ∈ ( I c )4 i ~σ · λ ~ u ~σ~ (7.17)We notice that the condition ~ · ~σ = 0 , ~σ · λ ~ = 0 is equivalent to requiring that ~ · ~σ = 0 and ~, ~σ satisfy(7.12)-(7.15). By Lemma 7.2 there is a constant ¯ c > I ) such that ~ · ~σ = 0 , ~σ · λ ~ = 0 and ~ / ∈ ( I c ) = ⇒ | ~σ · λ ~ | ≥ ¯ c > . (7.18)We have proved that the moduli of the small divisors in (7.17) are uniformly bounded away from zero.Hence F is well defined and, arguing as in Lemma 7.1, we get k X F k R ⋖ R . (7.19)Moreover F ∈ H null R because in (7.17) the sum is restricted to ~σ · ~ = 0 (see also (7.4)). Lemma 7.3. F in (7.17) solves the homological equation { N, F } + G = ad F ( N ) + G = G + ˆ G (7.20) where G , ˆ G are defined in (7.6) . Proof . We claim that the only ~ ∈ Z , ~σ ∈ {±} with ~ · ~σ = 0 which do not satisfy (7.12)-(7.15)have the form j = j , j = j , σ = − σ , σ = − σ (or permutations of the indexes) . (7.21)52ndeed:If ~ = 0, X i σ i = 0: the σ i are pairwise equal and (7.21) holds.If ~ = (0 , , q, q ), q = 0, and σ = − σ : by ~ · ~σ = 0 we have also σ = − σ and (7.21) holds.If ~ = ( p, p, − p, − p ), p = 0 and σ = − σ : by ~ · ~σ = 0 we have also σ = − σ and (7.21) holds.If j = j , j = j , j , j = 0, j = − j : Case 1: j = j . Then 0 = ~σ · ~ = ( σ + σ ) j + ( σ + σ ) j implies σ = − σ , σ = − σ . Case 2: j = j and so j = j = j = j = 0. Hence 0 = ( σ + σ + σ + σ ) j and (7.21) follows.By (7.17) and (7.11) all the monomials in { N, F } cancel the monomials of G in (7.1) except forthose in ˆ G (see (7.6)) and those of the form | u p | | u q | , p or q ∈ I , which contribute to G . Theexpression in (7.6) of G follows by counting the multiplicities.The Hamiltonian F ∈ H null R in (7.17) is quasi-T¨oplitz: Lemma 7.4. Let R > . If N := N (m , I , L, b ) is large enough, then F defined in (7.17) belongs to Q TR ( N , / , and k F k TR,N , / , ⋖ R . (7.22) Proof . We have to show that F ∈ H null R verifies Definition 3.4. For all N ≥ N , we compute, by(7.17) and Definition 3.2 (in particular (3.12)), the projectionΠ N, / , F = X | n | , | m | >CN/ ,σ,σ ′ = ± , | σm + σ ′ n | < NL F σ,σ ′ m,n ( w L ) u σm u σ ′ n (7.23)where F σ,σ ′ m,n ( w L ) := − X | i | + | j | < NL, i or j ∈I ,σii + σjj + σm + σ ′ n =0 , i = j if m = n u σ i i u σ j j σ i λ i + σ j λ j + σλ m + σ ′ λ n (7.24)= X P j | j | ( αj + βj ) < NL, P j ∈I ( αj + βj ) > ,σm + σ ′ n = − π ( α,β ) , | α | + | β | =2 , α = β if m = n F σ,σ ′ α,β,m,n u α ¯ u β (7.25)and F σ,σ ′ α,β,m,n := − α ! β ! 1 λ α,β + σλ m + σ ′ λ n , λ α,β := X h λ h ( α h − β h ) . (7.26)Notice that in (7.24) the restriction i = j if m = n is equivalent to requiring { ( i, j, m, n ) , ( σ i , σ j , σ, σ ′ ) } 6 = { ( i, i, m, m ) , ( σ i , − σ i , σ, − σ ) } , see Formula (7.17) and (7.21). Indeed if m = n , | i | + | j | < N L and | m | > CN/ σ = − σ ′ and hence | i | = | j | .We define the T¨oplitz approximation˜ F := X ˜ F σ,σ ′ m,n ( w L ) u σm u σ ′ n with ˜ F σ,σ ′ m,n ( w L ) := X ˜ F σ,σ ′ α,β,m,n u α ¯ u β (7.27)where the indexes in the two sums have the same restrictions as in (7.23), (7.25), respectively, andthe coefficients are ˜ F σ, − σα,β,m,n := − α ! β ! 1 λ α,β + σ | m | − σ | n | , ˜ F σ,σα,β,m,n := 0 . (7.28)The coefficients in (7.28) are well defined for N ≥ N large enough, because | λ α,β + σ | m | − σ | n || ≥ | λ α,β + σλ m − σλ n | − | λ m − | m || − | λ n − | n || (7.18) , (7.30) ≥ ¯ c − m2 (cid:18) | m | + 1 | n | (cid:19) ≥ ¯ c − 23 m N ≥ ¯ c , (7.29)53¯ c defined in (7.18)) having used the elementary inequality | p n + m − | n || ≤ / (2 | n | ) . (7.30)Then (7.27), (7.28), (7.29) imply, arguing as in the proof of Lemma 7.1, that k X ˜ F k R ⋖ R . (7.31)For proving that ˜ F ∈ T R ( N , / , 4) we have to show (3.15) (with f ˜ F ), namely˜ F σ,σ ′ α,β,m,n = ˜ F σ,σ ′ α,β ( s ( m ) , σm + σ ′ n ) (7.32)with ˜ F σ, − σα,β ( s, h ) := − α ! β ! 1 λ α,β + sh , ˜ F σ,σα,β ( s, h ) = 0 , s = ± , h ∈ Z . Recalling (7.28), this is obvious when σ ′ = σ . When σ ′ = − σ we first note that s ( m ) = s ( n ). Indeedthe restriction on the first sum in (7.27) is (recall (7.23)) | m | , | n | > N/ | σm − σn | < N L , whichimplies s ( m ) = s ( n ) by (3.1). Then σ | m | − σ | n | = σ s ( m ) m − σ s ( n ) n = s ( m )( σm − σn )and (7.32) follows. We have proved that ˜ F ∈ T R ( N , / , . The T¨oplitz defect, defined by (3.29), isˆ F := X ˆ F σ,σ ′ m,n ( w L ) u σm u σ ′ n with ˆ F σ,σ ′ m,n ( w L ) := X ˆ F σ,σ ′ α,β,m,n u α ¯ u β (7.33)where the indexes in the two sums have the same restrictions as in (7.23)-(7.25), andˆ F σ,σα,β,m,n = − α ! β ! Nλ α,β + σλ m + σλ n (7.34)ˆ F σ, − σα,β,m,n = − N α ! β ! (cid:18) λ α,β + σλ m − σλ n − λ α,β + σ | m | − σ | n | (cid:19) = 24i α ! β ! N σ ( λ m − | m | − λ n + | n | )( λ α,β + σλ m − σλ n )( λ α,β + σ | m | − σ | n | ) (7.35)We now proof that the coefficients in (7.34)-(7.35) are bounded by a constant independent of N .The coefficients in (7.34) are bounded because | λ α,β | ≤ X h λ h ( | α h | + | β h | ) ≤ X h | h | ( | α h | + | β h | ) + √ m X h ( | α h | + | β h | ) ≤ N L + 2 √ mby (7.26)-(7.25) (note that λ h ≤ | h | + √ m) and | λ α,β + σλ m + σλ n | ≥ | λ m + λ n | − | λ α,β | ≥ N − N L − √ m ≥ N/ N ≥ N large enough.The coefficients in (7.35) are bounded by (7.18), (7.29), and N | λ m − | m | − λ n + | n || (7.30) ≤ N m2 (cid:16) | m | + 1 | m | (cid:17) ≤ 23 m . Hence arguing as in the proof of Lemma 7.1 we get k X ˆ F k R ⋖ R . (7.36)54n conclusion, (7.19), (7.31), (7.36) imply (7.22) (recall (3.30)). Proof of Proposition 7.1 completed. We have e ad F H = e ad F N + e ad F G = N + { N, F } + X i ≥ i ! ad iF ( N ) + G + X i ≥ i ! ad iF ( G ) (7.20) = N + G + ˆ G + X i ≥ i + 1)! ad iF (cid:0) ad F ( N ) (cid:1) + X i ≥ i ! ad iF ( G )= N + G + ˆ G + K where, using again (7.20), K := X i ≥ i + 1)! ad iF ( G + ˆ G − G ) + X i ≥ i ! ad iF G =: K + K . (7.37) Proof of (7.9) . We claim that in the expansion of K in (7.37) there are only monomials u ~σ~ with ~ ∈ Z d , ~σ ∈ { + , −} d , d ≥ 3. Indeed F, G, G , ˆ G contain only monomials of degree four and, for anymonomial m , ad F ( m ) contains only monomials of degree equal to the deg( m ) + 2. The restriction ~σ · ~ = 0 follows by the Jacobi identity (2.82), since F, G, G, ˆ G preserve momentum, i.e. Poissoncommute with M . Proof of (7.10) . We apply Proposition 3.2 with (no ( x, y ) variables and) f F , g ( G + ˆ G − G for K ,G for K , r R , r ′ R/ , δ / ,θ / , θ ′ , µ , µ ′ ,N defined in Lemma 7.4 and N ′ ≥ N satisfying (3.64) and κ ( N ′ ) b − L ln N ′ ≤ , (6 + κ )( N ′ ) L − ln N ′ ≤ / . (7.38)Note that (3.65) follows by (7.38). By (7.22), the assumption (3.63) is verified for every 0 < R < R , with R small enough. Then Proposition 3.2 applies and (7.10) follows by (3.67) (with h We introduce action-angle variables on the tangential sites I := { j , . . . , j n } (see (7.5)) via the analyticand symplectic map Φ( x, y, z, ¯ z ; ξ ) := ( u, ¯ u ) (7.39)defined by u j l := p ξ l + y l e i x l , ¯ u j l := p ξ l + y l e − i x l , l = 1 , . . . , n , u j := z j , ¯ u j := ¯ z j , j ∈ Z \ I . (7.40)Let O ρ := n ξ ∈ R n : ρ ≤ ξ l ≤ ρ , l = 1 , . . . , n o . (7.41) Lemma 7.5. (Domains) Let r, R, ρ > satisfy r < ρ , ρ = C ∗ R with C − ∗ := 48 nκ p e s + aκ ) . (7.42) Then, for all ξ ∈ O ρ ∪ O ρ , the map Φ( · ; ξ ) : D ( s, r ) → D ( R/ 2) := B R/ × B R/ ⊂ ℓ a,p × ℓ a,p (7.43) is well defined and analytic ( D ( s, r ) is defined in (2.5) and κ in (3.1)) . roof . Note first that for ( x, y, z, ¯ z ) ∈ D ( s, r ) we have (see (2.6)) that | y l | < r < ρ/ < ξ l , ∀ ξ ∈ O ρ ∪ O ρ . Then the map y l p ξ l + y l is well defined and analytic. Moreover, for ξ l ≤ ρ , | j l | ≤ κ , x ∈ T ns , k z k a,p < r , we get k u ( x, y, z, ¯ z ; ξ ) k a,p (7.39) = n X l =1 ( ξ l + y l ) | e x l || j l | p e a | j l | + X j ∈ Z \I | z j | h j i p e a | j | ≤ n (cid:16) ρ + ρ (cid:17) e s κ p e aκ + 4 r < R / u is the same).Given a function F : D ( R/ → C , the previous Lemma shows that the composite map F ◦ Φ : D ( s, r ) → C . The main result of this section is Proposition 7.2: if F is quasi-T¨oplitz in the variables( u, ¯ u ) then the composite F ◦ Φ is quasi-T¨oplitz in the variables ( x, y, z, ¯ z ) (see Definition 3.4).We write F = X α,β F α,β m α,β , m α,β := ( u (1) ) α (1) (¯ u (1) ) β (1) ( u (2) ) α (2) (¯ u (2) ) β (2) , (7.44)where u = ( u (1) , u (2) ) , u (1) := { u j } j ∈I , u (2) := { u j } j ∈ Z \I , similarly for ¯ u , and( α, β ) = ( α (1) + α (2) , β (1) + β (2) ) , ( α (1) , β (1) ) := { α j , β j } j ∈I , ( α (2) , β (2) ) := { α j , β j } j ∈ Z \I . (7.45)We define H dR := n F ∈ H R : F = X | α (2) + β (2) |≥ d F α,β u α ¯ u β o . (7.46) Proposition 7.2. (Quasi–T¨oplitz) Let N , θ, µ, µ ′ satisfying (3.1) and ( µ ′ − µ ) N L > N b , N − Nb κ +1 < . (7.47) If F ∈ Q TR/ ( N , θ, µ ′ ) ∩ H dR/ with d = 0 , , then f := F ◦ Φ ∈ Q Ts,r ( N , θ, µ ) and k f k Ts,r,N ,θ,µ, O ρ ⋖ (8 r/R ) d − k F k TR/ ,N ,θ,µ ′ . (7.48)The rest of this section is devoted to the proof of Proposition 7.2. Introducing the action-anglevariables (7.40) in (7.44), and using the Taylor expansion(1 + t ) γ = X h ≥ (cid:18) γh (cid:19) t h , (cid:18) γ (cid:19) := 1 , (cid:18) γh (cid:19) := γ ( γ − . . . ( γ − h + 1) h ! , h ≥ , (7.49)we get f := F ◦ Φ = X k,i,α (2) ,β (2) f k,i,α (2) ,β (2) e i k · x y i z α (2) ¯ z β (2) (7.50)with Taylor–Fourier coefficients f k,i,α (2) ,β (2) := X α (1) − β (1) = k F α,β n Y l =1 ξ α (1) l + β (1) l − i l l (cid:18) α (1) l + β (1) l i l (cid:19) . (7.51)We need an upper bound on the binomial coefficients.56 emma 7.6. For | t | < / we have ( i ) X h ≥ | t | h (cid:12)(cid:12)(cid:12)(cid:18) k h (cid:19)(cid:12)(cid:12)(cid:12) ≤ k , ∀ k ≥ , ( ii ) X h ≥ | t | h (cid:12)(cid:12)(cid:12)(cid:18) k h (cid:19)(cid:12)(cid:12)(cid:12) ≤ k | t | , ∀ k ≥ . (7.52) Proof . By (7.49) and the definition of majorant (see (2.11)) we have X h ≥ (cid:12)(cid:12)(cid:12)(cid:18) k h (cid:19)(cid:12)(cid:12)(cid:12) t h = M (1 + t ) k (2.39) ≺ ( M (1 + t ) ) k = (cid:16) X h ≥ (cid:12)(cid:12)(cid:12)(cid:18) h (cid:19)(cid:12)(cid:12)(cid:12) t h (cid:17) k ≺ (cid:16) X h ≥ t h (cid:17) k (7.53)because (cid:12)(cid:12)(cid:12)(cid:18) h (cid:19)(cid:12)(cid:12)(cid:12) ≤ | t | < / i ). Ne X h ≥ | t | h (cid:12)(cid:12)(cid:12)(cid:18) k h (cid:19)(cid:12)(cid:12)(cid:12) ≤ | t | X h ≥ | t | h (cid:12)(cid:12)(cid:12)(cid:18) k h + 1 (cid:19)(cid:12)(cid:12)(cid:12) (7.49) = | t | X h ≥ | t | h (cid:12)(cid:12)(cid:12)(cid:18) k h (cid:19)(cid:12)(cid:12)(cid:12) | k − h | h + 1 ≤ k | t | X h ≥ | t | h (cid:12)(cid:12)(cid:12)(cid:18) k h (cid:19)(cid:12)(cid:12)(cid:12) (7.52) − ( i ) ≤ k k | t | which implies (7.52)-( ii ) for k ≥ Lemma 7.7. ( M -regularity) If F ∈ H dR/ then f := F ◦ Φ ∈ H s, r and k X f k s, r, O ρ ∪O ρ ⋖ (8 r/R ) d − k X F k R/ . (7.54) Moreover if F preserves momentum then so does F ◦ Φ . Proof . We first bound the majorant norm k f k s, r, O ρ ∪O ρ (7.50) , (7.46) := sup ξ ∈O ρ ∪O ρ sup ( y,z, ¯ z ) ∈ D (2 r ) X k,i, | α (2) + β (2) |≥ d | f k,i,α (2) ,β (2) | e | k | s | y i || z α (2) || ¯ z β (2) | . (7.55)Fix α (2) , β (2) . Since for all ξ ∈ O ρ ∪ O ρ , y ∈ B (2 r ) , we have | y l /ξ l | < / X k e | k | s X i | f k,i,α (2) ,β (2) || y | i (7.56) (7.51) ≤ X α (1) ,β (1) e s ( | α (1) | + | β (1) | ) | F α,β | ξ α (1)+ β (1)2 n Y l =1 X i l ≥ (cid:12)(cid:12)(cid:12)(cid:12) y l ξ l (cid:12)(cid:12)(cid:12)(cid:12) i l (cid:12)(cid:12)(cid:12)(cid:18) α (1) l + β (1) l i l (cid:19)(cid:12)(cid:12)(cid:12) (7.57) (7.52) ≤ X α (1) ,β (1) e s ( | α (1) | + | β (1) | ) | F α,β | ξ α (1)+ β (1)2 n Y l =1 α (1) l + β (1) l (7.58) ≤ X α (1) ,β (1) e s ( | α (1) | + | β (1) | ) | F α,β | (2 ρ ) | α (1) | + | β (1) | | α (1) | + | β (1) | = X α (1) ,β (1) (2 e s p ρ ) | α (1) | + | β (1) | | F α,β | . Then, substituting in (7.55), k f k s, r, O ρ ∪O ρ ≤ sup k z k a,p , k ¯ z k a,p < r G ( z, ¯ z ) where (7.59) G ( z, ¯ z ) := X | α (2) + β (2) |≥ d (2 e s p ρ ) | α (1) | + | β (1) | | F α,β || z α (2) || ¯ z β (2) | . (7.60)By (7.42), for all k z k a,p , k ¯ z k a,p < r , the vector ( u ∗ , ¯ u ∗ ) defined by u ∗ j = ¯ u ∗ j := 2 e s p ρ , j ∈ I , u ∗ j := ( R/ (8 r )) | z j | , ¯ u ∗ j := ( R/ (8 r )) | ¯ z j | , j ∈ Z \ I (7.61)57elongs to B R/ × B R/ . Then, by (7.60), recalling (2.11), Definition 2.2 (and since R/ (8 r ) > G ( z, ¯ z ) ≤ (8 r/R ) d ( M F )( u ∗ , ¯ u ∗ ) ≤ (8 r/R ) d k F k R/ , ∀ k z k a,p , k ¯ z k a,p < r . Hence by (7.59) k f k s, r, O ρ ∪O ρ ≤ (8 r/R ) d k F k R/ . (7.62)This shows that f is M -regular. Similarly we get k ∂ z f k s, r, O ρ ∪O ρ ≤ k ∂ u (2) F k R/ (8 r/R ) d − , same for ∂ ¯ z . (7.63)Moreover, by the chain rule, and (7.62) k ∂ x i f k s, r, O ρ ∪O ρ ≤ ( k ∂ u (1) i F k R/ + k ∂ ¯ u (1) i F k R/ ) p ρ + ρ/ e s (8 r/R ) d k ∂ y i f k s, r, O ρ ∪O ρ ≤ ( k ∂ u (1) i F k R/ + k ∂ ¯ u (1) i F k R/ ) e s p ρ/ − ρ/ r/R ) d . Then (7.54) follows by (7.42) (recalling (2.2)). Definition 7.1. For a monomial m α,β := ( u (1) ) α (1) (¯ u (1) ) β (1) ( u (2) ) α (2) (¯ u (2) ) β (2) ( as in (7.44)) we set p ( m α,β ) := n X l =1 h j l i ( α (1) j l + β (1) j l ) , h j i := max { , | j |} . (7.64) For any F as in (7.44) , K ∈ N , we define the projection Π p ≥ K F := X p ( m α,β ) ≥ K F α,β m α,β , Π p Let F ∈ H R/ . Then k X (Π p ≥ K F ) ◦ Φ k s,r, O ρ ≤ − K κ +1 k X F ◦ Φ k s, r, O ρ . (7.66) Proof . For each monomial m α,β as in (7.44) with p ( m α,β ) ≥ K we have | α (1) + β (1) | (7.45) = n X l =1 α (1) j l + β (1) j l (3.1) ≥ κ − n X l =1 h j l i ( α (1) j l + β (1) j l ) (7.64) = κ − p ( m α,β ) ≥ κ − K and then, ∀ ξ ∈ O ρ , y ∈ B r , | ( m α,β ◦ Φ)( x, y, z, ¯ z ; ξ ) | (7.40) = | ( ξ + y ) α (1)+ β (1)2 e i( α (1) − β (1) ) · x z α (2) ¯ z β (2) | (7.67)= 2 − | α (1)+ β (1) | | (2 ξ + 2 y ) α (1)+ β (1)2 e i( α (1) − β (1) ) · x z α (2) ¯ z β (2) |≤ − K κ | ( m α,β ◦ Φ)( x, y, z, ¯ z ; 2 ξ ) | . The bound (7.66) for the Hamiltonian vector field follows applying the above rescaling argument toeach component, and noting that the derivatives with respect to y in the vector field decrease thedegree in ξ by one.Let N , θ, µ, µ ′ be as in Proposition 7.2. For N ≥ N and F ∈ H R/ we set f ∗ := Π N,θ,µ (cid:16) ( F − Π N,θ,µ ′ F ) ◦ Φ (cid:17) . (7.68)Note that Π N,θ,µ ′ is the projection on the bilinear functions in the variables u, ¯ u , while Π N,θ,µ in thevariables x, y, z, ¯ z . 58 emma 7.9. We have k X f ∗ k s,r, O ρ ≤ − Nb κ +1 k X F ◦ Φ k s, r, O ρ . (7.69) Proof . We first claim that if F = m α,β is a monomial as in (7.44) with p ( m α,β ) < N b then f ∗ = 0. Case m α,β is ( N, θ, µ ′ )–bilinear, see Definition 3.2. Then Π N,θ,µ ′ m α,β = m α,β and f ∗ = 0, see(7.68). Case m α,β is not ( N, θ, µ ′ )–bilinear. Then Π N,θ,µ ′ m α,β = 0 and f ∗ = Π N,θ,µ ( m α,β ◦ Φ), see(7.68). We claim that m α,β ◦ Φ is not ( N, θ, µ )–bilinear, and so f ∗ = Π N,θ,µ ( m α,β ◦ Φ) = 0. Indeed, m α,β ◦ Φ = ( ξ + y ) α (1)+ β (1)2 e i( α (1) − β (1) ) · x z α (2) ¯ z β (2) (7.70)is ( N, θ, µ )–bilinear if and only if (see Definitions 3.2 and 3.1) z α (2) ¯ z β (2) = z ˜ α (2) ¯ z ˜ β (2) z σm z σ ′ n , X j ∈ Z \I | j | (˜ α (2) j + ˜ β (2) j ) < µN L , | m | , | n | > θN , | α (1) − β (1) | < N b . (7.71)We deduce the contradiction that m α,β = ( u (1) ) α (1) (¯ u (1) ) β (1) ( u (2) ) ˜ α (2) (¯ u (2) ) ˜ β (2) u σm u σ ′ n is ( N, θ, µ ′ )-bilinear because (recall that we suppose p ( m α,β ) < N b ) n X l =1 | j l | ( α (1) j l + β (1) j l ) + X j ∈ Z \I | j | (˜ α (2) j + ˜ β (2) j ) (7.64) , (7.71) < p ( m α,β ) + µN L < N b + µN L (7.47) < µ ′ N L . For the general case, we divide F = Π p Let F ∈ T R/ ( N, θ, µ ′ ) with Π p ≥ N b F = 0 . Then F ◦ Φ( · ; ξ ) ∈ T s, r ( N, θ, µ ′ ) , ∀ ξ ∈O ρ ∪ O ρ . Proof . Recalling Definition 3.3 we have F = X | m | , | n | >θN,σ,σ ′ = ± F σ,σ ′ ( s ( m ) , σm + σ ′ n ) u σm u σ ′ n with F σ,σ ′ ( ς, h ) ∈ L R/ ( N, µ ′ , h ) . Composing with the map Φ in (7.40), since m, n / ∈ I , we get F ◦ Φ = X σ,σ ′ = ± , | m | , | n | >θN F σ,σ ′ ( s ( m ) , σm + σ ′ n ) ◦ Φ z σm z σ ′ n . Each coefficient F σ,σ ′ ( s ( m ) , σm + σ ′ n ) ◦ Φ depends on n, m, σ, σ ′ only through s ( m ) , σm + σ ′ n, σ, σ ′ .Hence, in order to conclude that F ◦ Φ ∈ T s, r ( N, θ, µ ′ ) it remains only to prove that F σ,σ ′ ( s ( m ) , σm + σ ′ n ) ◦ Φ ∈ L s, r ( N, µ ′ , σm + σ ′ n ), see Definition 3.1. Each monomial m α,β of F σ,σ ′ ( s ( m ) , σm + σ ′ n ) ∈L R/ ( N, µ ′ , σm + σ ′ n ) satisfies n X l =1 ( α j l + β j l ) | j l | + X j ∈ Z \I ( α j + β j ) | j | < µ ′ N L and p ( m α,β ) < N b by the hypothesis Π p ≥ N b F = 0. Hence m α,β ◦ Φ (see (7.70)) is ( N, µ ′ )-low momentum, in particular | α (1) − β (1) | ≤ p ( m α,β ) < N b . 59 roof of Proposition 7.2 . Since F ∈ Q TR/ ( N , θ, µ ′ ) (see Definition 3.4), for all N ≥ N , thereis a T¨oplitz approximation ˜ F ∈ T R/ ( N, θ, µ ′ ) of F , namelyΠ N,θ,µ ′ F = ˜ F + N − ˆ F with k X F k R/ , k X ˜ F k R/ , k X ˆ F k R/ < k F k TR/ ,N ,θ,µ ′ . (7.72)In order to prove that f := F ◦ Φ ∈ Q Ts,r ( N , θ, µ ) we define its candidate T¨oplitz approximation˜ f := Π N,θ,µ ((Π p N f ∗ + N Π N,θ,µ (cid:0) (Π p ≥ N b ˜ F ) ◦ Φ (cid:1) . Using (2.80), Lemmata 7.8 and 7.9 imply that, since N − Nb κ +1 ≤ , ∀ N ≥ N by (7.47), k X ˆ f k s,r, O ρ ≤ k X ˆ F ◦ Φ k s,r, O ρ + N − Nb κ +1 ( k X F ◦ Φ k s, r, O ρ + k X ˜ F ◦ Φ k s, r, O ρ ) ⋖ k X ˆ F ◦ Φ k s, r, O ρ + k X F ◦ Φ k s, r, O ρ + k X ˜ F ◦ Φ k s, r, O ρ (7.54) ⋖ (8 r/R ) d − ( k X ˆ F k R/ + k X F k R/ + k X ˜ F k R/ ) (7.75) (7.72) ⋖ (8 r/R ) d − k F k TR/ ,N ,θ,µ ′ (7.76)(to get (7.75) we also note that F, ˆ F , ˜ F ∈ H dR/ with d = 0 , 1, unless are zero).The bound (7.48) follows by (7.54), (7.74), (7.76).We conclude this subsection with a lemma, similar to Lemma 7.7, used in Lemma 7.12 (see (7.90)). Lemma 7.11. Let F ∈ H R/ , f := F ◦ Φ and ˜ f ( x, y ) := f ( x, y, , − f ( x, , , . Then, assuming (7.42) , k X ˜ f k s, r, O ρ ∪O ρ ⋖ k X F k R/ . (7.77) Moreover if F preserves momentum then so does ˜ f . Proof . We proceed as in Lemma 7.7. The main difference is that here there are no ( z, ¯ z )-variablesand the sum in (7.56) runs over i = 0. Then in the product in (7.57) (at least) one of the sumsis on i l ≥ 1. Therefore we can use the second estimate in (7.52) gaining a factor r /ρ (since | y l | / | ξ l | ≤ r /ρ by (7.41)). Continuing as in the proof of Lemma 7.7 we get (recall (7.54) with d = 0) k X ˜ f k s, r, O ρ ∪O ρ ⋖ ( r /ρ )( r/R ) − k X F k R/ ⋖ k X F k R/ proving (7.77). Actually we have the constant 3 instead of 2 in (7.58) and 3 e s instead of 2 e s in (7.59) and (7.61). .3 Proof of Theorem 1.1 We now introduce the action-angle variables (7.40) (via the map (7.39)) in the Birkhoff normal formHamiltonian (7.8). Hence we obtain the parameter dependent family of Hamiltonians H ′ := H Birkhoff ◦ Φ = N + P (7.78)where (up to a constant), by (7.6), N := ω ( ξ ) · y + Ω( ξ ) z ¯ z , P := 12 Ay · y + By · z ¯ z + ˆ G ( z, ¯ z ) + K ′ ( x, y, z, ¯ z ; ξ ) , (7.79) ω ( ξ ) := ¯ ω + Aξ , ¯ ω := ( λ j , . . . , λ j n ) , Ω( ξ ) := ¯Ω + Bξ , ¯Ω := ( λ j ) j ∈ Z \I , (7.80) A = ( A lh ) ≤ l,h ≤ n , A lh := 12(2 − δ lh ) , B = ( B jl ) j ∈ Z \I , ≤ l ≤ n , B jl := 24 , K ′ := K ◦ Φ . (7.81)The parameters ξ stay in the set O ρ defined in (7.41) with ρ = C ∗ R as in (7.42). As in (4.6) wedecompose the perturbation P = P + ¯ P where P ( x ; ξ ) := K ′ ( x, , , ξ ) , ¯ P := P − P . (7.82) Lemma 7.12. Let s, r > as in (7.42) and N large enough (w.r.t. m , I , L, b ). Then k X P k s,r ⋖ R r − , k ¯ P k Ts,r,N, , ⋖ r + R r − (7.83) and, for λ > , | X P | λs,r ⋖ (1 + λ/ρ ) R r − , | X ¯ P | λs,r ⋖ (1 + λ/ρ )( r + R r − ) , (7.84) for ξ belonging to O ( ρ ) := n ξ ∈ R n : 23 ρ ≤ ξ l ≤ ρ , l = 1 , . . . , n o ⊂ O ρ . (7.85) Proof . By the definition (7.82) we have k X P k s,r (2.55) ≤ k X K ′ k s,r (3.32) ≤ k K ′ k Ts,r,N, , = k K ◦ Φ k Ts,r,N, , ⋖ (cid:16) rR (cid:17) − k K k TR/ ,N, , (7.86)(applying (7.48) with d N N , θ µ µ ′ 3) and taking N large enough so that(7.47) holds. Take also N ≥ N ′ defined in Proposition 7.1. Then by (7.86) we get k X P k s,r (3.35) ⋖ (cid:16) rR (cid:17) − k K k TR/ ,N ′ , , ⋖ (cid:16) rR (cid:17) − R ⋖ R r proving the first estimate in (7.83). Let us prove the second bound. By (7.82) and (7.79) we write¯ P = 12 Ay · y + By · z ¯ z + ˆ G ( z, ¯ z ) + K + K (7.87)where K := K ′ ( x, y, z, ¯ z ; ξ ) − K ′ ( x, y, , ξ ) , K := K ′ ( x, y, , ξ ) − K ′ ( x, , , ξ ) . Using (7.7) (note that r < R by (7.42)) for N ≥ N large enough to fulfill (3.1), we have by (3.35) (cid:13)(cid:13)(cid:13) Ay · y + By · z ¯ z + ˆ G ( z, ¯ z ) (cid:13)(cid:13)(cid:13) Ts,r,N, , ⋖ r . (7.88)61y (7.48) (with d N N , µ µ ′ N ≥ N (m , I , L, b ) large enough, we get k K k Ts,r,N, , ⋖ (cid:16) rR (cid:17) − R ⋖ R r . (7.89)Moreover, since K does not depend on ( z, ¯ z ), we have k K k Ts,r,N, , = k X K k s,r (7.77) ⋖ k X K k R/ ⋖ k K k TR/ ,N ′ , , ⋖ R . (7.90)In conclusion, (7.87), (7.88), (7.89), (7.90) imply the second estimate in (7.83): k ¯ P k Ts,r,N, , ⋖ r + R r + R ⋖ r + R r . Let us prove the estimates (7.84) for the Lipschitz norm defined in (2.88) (which involves only thesup-norm of the vector fields). First | X P | s,r (2.62) ≤ k X P k s,r (7.83) ⋖ R r − , | X ¯ P | s,r (2.62) ≤ k X ¯ P k s,r (3.32) ≤ k ¯ P k Ts,r,N, , ⋖ r + R r − Next, since the vector fields X P , X ¯ P are analytic in the parameters ξ ∈ O ρ , Cauchy estimates inthe domain O ( ρ ) ⊂ O ρ (see (7.85)) imply | X P | lip s,r, O ( ρ ) ⋖ ρ − | X P | s,r, O ρ ⋖ R r − , | X ¯ P | lip s,r, O ( ρ ) ⋖ ρ − | X ¯ P | s,r, O ρ ⋖ r + R r − and (7.84) are proved.All the assumptions of Theorems 4.1-4.2 are fulfilled by H ′ in (7.78) with parameters ξ ∈ O ( ρ )defined in (7.85). Note that the sets O = [ ρ/ , ρ ] n defined in Theorem 4.2 and O ( ρ ) defined in (7.85)are diffeomorphic through ξ i (7 ρ + 2 ξ i ) / 12. The hypothesis (A1)-(A2) follow from (7.80), (7.81)with a ( ξ ) = 24 X l =1 ,...,n ξ l , and M = 24 + k A k . Then (A3)-(A4) and the quantitative bound (4.7) follow by (7.83)-(7.84), choosing s = 1 , r = R , ρ = C ∗ R as in (7.42) , N as in Lemma 7 . , θ = 2 , µ = 2 , γ = R , (7.91)and taking R small enough . Hence Theorem 4.1 applies.Let us verify that also the assumptions of Theorem 4.2 are fulfilled. Indeed (4.13) follows by (7.80),(7.81) with ~a = 24(1 , . . . , ∈ R n . The matrix A defined in (7.81) is invertible and A − = ( A − lh ) ≤ l,h ≤ n , A − lh = 112 (cid:16) n − − δ lh (cid:17) . Finally the non-degeneracy assumption (4.14) is satisfied because A = A T and2 A − ~a = 42 n − , . . . , / ∈ Z n \ . We deduce that the Cantor set of parameters O ∞ ⊂ O in (4.11) has asymptotically full density because |O \ O ∞ ||O| (4.16) ⋖ ρ − γ / ⋖ R − R (3+ ) = R → . The proof of Theorem 1.1 is now completed. Remark 7.1. The terms X k ≥ f k s k in (1.2) contribute to the Hamiltonian (7.1) with monomials oforder 6 or more and (7.8) holds (with a possibly different K satisfying (7.10) ). On the contrary, theterm f s in (1.2) would add monomials of order to the Hamiltonian in (7.1) . Hence (7.10) holdswith R instead of R . This estimate is not sufficient. These -th order terms should be removed bya Birkhoff normal form. For simplicity, we did not pursue this point. Appendix Proof of Lemma 2.14. We need some notation: we write E = ⊕ j =1 E j , E := ( C n , | | ∞ ), E :=( C n , | | ), E := E := ℓ a,p I so that a vector v = ( x, y, z, ¯ z ) ∈ E can be expressed by its four components v ( j ) ∈ E j , v (1) := x, v (2) := y, v (3) := z, v (4) := ¯ z, and the norm (2.2) is k v k E,s,r := X j =1 | v ( j ) | E j ρ j , where ρ = s , ρ = r , ρ = ρ = r . (8.1)We are now ready to prove (2.65). By definition k dX ( v ) k L (( E,s,r );( E,s ′ ,r ′ )) := sup k Y k E,s,r ≤ k dX ( v )[ Y ] k E,s ′ ,r ′ (8.1) = sup k Y k E,s,r ≤ X i =1 | dX ( i ) ( v )[ Y ] | E i ρ ′ i = sup k Y k E,s,r ≤ X i =1 | P j =1 d v ( j ) X ( i ) ( v ) Y ( j ) | E i ρ ′ i ≤ sup k Y k E,s,r ≤ X i,j =1 | d v ( j ) X ( i ) ( v ) Y ( j ) | E i ρ ′ i ≤ sup k Y k E,s,r ≤ X i,j =1 ρ ′ i k d v ( j ) X ( i ) ( v ) k L ( E j ,E i ) | Y ( j ) | E j ≤ sup k Y k E,s,r ≤ sup ˜ v ∈ D ( s,r ) 4 X i,j =1 ρ ′ i | X ( i ) (˜ v ) | E i ( ρ j − ρ ′ j ) | Y ( j ) | E j by the Cauchy estimates in Banach spaces. Then k dX ( v ) k L (( E,s,r );( E,s ′ ,r ′ )) ≤ sup ˜ v ∈ D ( s,r ) 4 X i =1 ρ i ρ ′ i | X ( i ) (˜ v ) | E i ρ i sup k Y k E,s,r ≤ X j =1 (cid:16) − ρ ′ j ρ j (cid:17) − | Y ( j ) | E j ρ j (8.1) ≤ max i =1 ,..., ρ i ρ ′ i max j =1 ,..., (cid:16) − ρ ′ j ρ j (cid:17) − sup ˜ v ∈ D ( s,r ) k X (˜ v ) k E,s,r ≤ δ − | X | s,r by (2.53), (2.66). This proves (2.65). Proof of Lemma 7.2. We first extend Lemma 4 of [25] proving that: Lemma 8.1. If ≤ i ≤ j ≤ k ≤ l with i ± j ± k ± l = 0 for some combination of plus and minussigns and ( i, j, k, l ) = ( p, p, q, q ) for p, q ∈ N , then, there exists an absolute constant c > , such that | ± λ i (m) ± λ j (m) ± λ k (m) ± λ l (m) | ≥ c m( i + m) − / (8.2) for all possible combinations of plus and minus signs Proof . When i > i = 0. Then j ± k ± l = 0 for some combination of plus and minus signs. Since ( i, j, k, l ) = (0 , , q, q ),the only possibility is l = j + k with j ≥ i = j = 0 and k = l ). We have to study δ (m) := ± λ (m) ± λ j (m) ± λ k (m) ± λ l (m)for all possible combinations of plus and minus signs. To this end, we distinguish them according totheir number of plus and minus signs. To shorten notation we let, for example, δ ++ − + = λ + λ j − λ k + λ l , similarly for the other combinations. The only interesting cases are when there are one or63wo minus signs. The case when there are no (or four) minus signs is trivial. When there are 3 minussigns we reduce to the case with one minus sign by a global sign change. One minus sign. Since δ ++ − + , δ + − ++ , δ − +++ ≥ δ +++ − := δ we study only the last case. We have δ (0) = j + k − l = 0 , δ ′ (m) = 12 (cid:18) λ + 1 λ j + 1 λ k − λ l (cid:19) ≥ λ = 12 √ m . Therefore δ (m) ≥ √ m ≥ c m(1 + m) − / for an absolute constant c > Two minus signs. Now we have δ − + − + , δ −− ++ ≥ δ + −− + and all other cases reduce to these ones byinverting signs. So we consider only δ = δ + −− + . Since the function f ( t ) := p t + m is monotoneincreasing and convex for t ≥ 0, we have the estimate λ l − λ k ≥ λ l − p − λ k − p , ∀ ≤ p ≤ k . (8.3)Hence λ l − λ k ≥ λ j +1 − λ and λ j +1 − λ j ≥ λ − λ (using j = l − k ≥ δ = λ − λ j − λ k + λ l ≥ λ − λ j − λ + λ j +1 ≥ λ − λ + λ ≥ m(4 + m) − / . The last inequality follows since f ′′ ( t ) = m( t + m) − / is decreasing and λ − λ + λ = f (2) − f (1) + f (0) = f ′′ ( ξ ) ≥ f ′′ (2) for some ξ ∈ (0 , case (7.12) . Since X i σ i = 0 is even, (7.16) follows by | σ · λ ~ | = | X i σ i λ ~ | ≥ λ ~ = 2 √ m ≥ m(1 + m) − / . case (7.13) . By ~σ · ~ = ( σ + σ ) q = 0, q = 0, we deduce σ = − σ . Hence (7.16) follows by | σ · λ ~ | = | ( σ + σ ) λ | = 2 √ m ≥ m(1 + m) − / case (7.14) . Since ~ = ( p, p, − p, − p ) and σ = σ then ~σ · ~ = 0 implies σ = σ = σ and | σ · λ ~ | = | λ p | = 4 p p + m ≥ m( p + m) − / . case (7.15) . Set | j | =: i , | j | =: j , | j | =: k , | j | =: l . After reordering we can assume 0 ≤ i ≤ j ≤ k ≤ l. Since, by assumption, ~σ · ~ = 0, the following combination of plus and minus signs gives s ( j ) σ i + s ( j ) σ j + s ( j ) σ k + s ( j ) σ l = 0. Hence Lemma 8.1 implies (7.16) for every ~ exceptwhen | j | = | j | and | j | = | j | (in this case i = j and k = l and Lemma 8.1 does not apply). 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