Quantum singular complete integrability

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Quantum singular complete integrability

Thierry Paul, Laurent Stolovitch

To cite this version:

Thierry Paul, Laurent Stolovitch. Quantum singular complete integrability. Journal of Functional Analysis, Elsevier, 2016, 271, pp.1377-1443. �hal-00945409v2�

THIERRY PAUL AND LAURENT STOLOVITCH

Abstract. We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schr¨odinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diagonalised by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uniform in the Planck constant near ~ “ 0. The unperturbed frequencies satisfy a small divisors condition and we explicitly estimate how this condition can be released when the family tends to the unperturbed one. In the case where the number of operators is equal to the number of degrees of freedom - i.e. full integrability - our construction provides convergent normal forms for general perturbations of linear systems.

Contents

1. Introduction 2

Notations 7

2. Strategy of the proofs 7

3. The cohomological equation: the formal construction 9

3.1. First order 10

3.2. Higher orders 11

3.3. Toward estimating 11

4. Norms 12

5. Weyl quantization and first estimates 14

5.1. Weyl quantization, matrix elements and first estimates 14

5.2. Fundamental estimates 16

6. Fundamental iterative estimates: Brjuno condition case 22 7. Fundamental iterative estimates: Diophantine condition case 29

8. Strategy of the KAM iteration 30

9. Proof of the convergence of the KAM iteration 31

9.1. Convergence of the KAM iteration I: constraints on ω 31

9.2. Convergence of the KAM iteration II: general ω 36

9.3. Proof of Theorem 29 37

9.4. Convergence of the KAM iteration III: the classical limit 42 9.5. Convergence of the KAM iteration IV: the Diophantine case 47 9.6. Bound on the Brjuno constant insuring integrability 47

10. The case m “ l 48

References 48

Research of L. Stolovitch was supported by ANR grant “ANR-10-BLAN 0102” for the project DynPDE .

1. Introduction

Perturbation theory belongs to the history of quantum mechanics, and even to its pre-history, as it was used before the works of Heisenberg and Schr¨odinger in 1925/1926. The goal at that time was to understand what should be the Bohr-Sommerfeld quantum condi-tions for systems nearly integrable [MB], by quantizing the perturbation series provided by celestial mechanics [HP]. After (or rather during its establishment) the functional analysis point of view was settled for quantum mechanics, the “modern” perturbation theory took place, mostly by using the Neumann expansion of the perturbed resolvent, providing effi-cient and rigorous ways of establishing the validity of the Rayleigh-Schr¨odinger expansion and leading to great success of this method, in particular the convergence under a simple argument of size of the perturbation in the topology of operators on Hilbert spaces [TK], and Borel summability for (some) unbounded perturbations [GG, BS]. On the other hand, by relying on the comparison between the size of the perturbation and the distance between consecutive unperturbed eigenvalues, the method has two inconveniences: it remains local in the spectrum in the (usual in dimension larger than one) case of spectra accumulating at infinity and is even inefficient in the case of dense point unperturbed spectra which can be the case in the present article.

In the present article, we consider some commuting families of operators on L2_pTd_{q close}

to a commuting family of unperturbed Hamiltonians whose spectra are pure point and might be dense for all values of ~. As already emphasized, standard (Neumann series expansion) perturbation theory does not apply in this context. Nevertheless, we prove that the pure point property is preserved and moreover, we show that the perturbed spectra are analytic functions of the unperturbed ones. All these results are obtained using a method inspired by classical local dynamics, namely the analysis of quantum Birkhoff forms. Let us first recall some known fact of (classical) Birkhoff normal forms.

In the framework of (classical) local dynamics, R¨ussman proved in [Ru] (see also [Bru]) the remarkable result which says that, when the Birkhoff normal form (BNF), at any order, depends only on the unperturbed Hamiltonian, then it converges provided that the small divisors of the unperturbed Hamiltonian do not accumulate the origin too fast (we refer to [Ar2] for an introduction to this subject). This leads to the integrability of the perturbed system. On the other hand, Vey proved two theorems about the holomorphic normalization of families of l_{´ 1 (resp. l) of commuting germs of holomorphic vector fields,} volume preserving (resp. Hamiltonian) in a neighborhood of the origin of Cl _{(resp. C}2l₎

(and vanishing at the origin) with diagonal and independent 1-jets [JV1, JV2].

These results were extended by one of us in [LS1, LS2], in the framework of general local dynamics of a families of of 1_{ď m ď l commuting germs of holomorphic vector fields near} a fixed point. It is proved that under an assumption on the formal (Poincar´e) normal form of the family and and under a generalized Brjuno type condition of the family of linear parts, there exists an holomorphic transformation of the family to a normal form. This fills up therefore the gap between R¨ussman-Brjuno and the complete integrability of Vey.

In these directions, we should also mention works by H. Ito [It] and N.-T. Zung [Zu] in the analytic case and H. Eliasson [El] in the smooth case, and Kuksin-Perelman [KP] for a specific infinite dimensional version.

In [GP] one of us (the other) gave with S. Graffi a quantum version of the R¨ussmann theorem in the framework of perturbation theory of the quantization of linear vector fields on the torusTl_{. Moreover, in this setting, it is possible to read on the original perturbation}

if the R¨ussman condition is satisfied and the results are uniform in the Planck constant belonging to_{r0, 1s. The method seats in the framework of Lie method perturbation theory} initiated in classical mechanics in [De, Ho] and uses the quantum setting established in [BGP].

The goal of the present paper is to provide a full spectral resolution for certain families of commuting quantum Hamiltonians, not treatable by standard methods due to possi-ble spectral accumulation, through the convergence of quantum normal Birkhoff forms and underlying unitary transformations. These families generalize the quantum version of R¨usmman theorem treated in [GP], to the quantum version of “singular complete inte-grability” treated in [LS1]. The methods use the quantum version of the Lie perturbative algorithm together with a newton type scheme in order to overcome the difficulty created by small divisors.

Let m _{ď l P N}˚_{. For ω “ pω}

iqi“1...m with ωi “ pωjiqj“1...l P Rl, let us denote by

Lω “ pLωiqi“1...m, the operator valued vector of components

Lωi “ ´i~ωi.∇x “ ´i~ l ÿ j“1 ωij B Bxj , i_{“ 1 . . . m} on L2_pTl_q.

We define the operator valued vector H _{“ pH}iqi_“1...m by

H _{“ L}ω` V, (1.1)

where V is a bounded operator valued vector on L2_pTl_{q whose action is defined after a}

function V : _{px, ξ,}~_{q P T}˚_Tl_{ˆ r0, 1s ÞÑ Vpx, ξ,~q PR}m _{by the formula (Weyl quantization)}

pV fqpxq “ ż

Rl_ˆRl

V_{ppx ` yq{2, ξ,}~_qeiξpx´yq~ fpyq dydξ

p2π~_ql, (1.2)

where in the integral f_{p¨q and Vppx`¨q{2, ξ,}~_{q are extended to}Rl _{by periodicity (see Section}

5.1 for details). We make the following assumptions.

Main assumptions

8

ÿ

l“1

log M2k

2k ă `8 where MM :“ min_{1ďiďm0‰|q|ďM}max |xωi, qy|

´1_{. (1.3)}

We will sometimes impose to ω tu fulfill the strongest collective Diophantine condition: there exist γ ą 0, τ ě l such that

@q PZl, q_{‰ 0, min}

1_ďiďm|xωi, qy|

´1 _{ď γ|q|}τ_.

(1.4)

Remark : usually, 1

MM is denoted by ωM in the literature [Bru, LS1]

(A2) V takes the form, for some V1 _{: pΞ, x,~q P R}m

ˆTl

ˆ r0, 1s ÞÑ V1_{pΞ, x,~q P R}m_,

analytic in _{pΞ, xq and kth times differentiable in} ~,

V_{px, ξ,}~_{q “ V}1_pω1.ξ, . . . , ωm.ξ, x,~q, (1.5)

(A3) The family H satisfies

rHi, Hjs “ 0, 1ď i, j ď m, 0 ď~ď 1. (1.6)

Moreover we will suppose that the vectors ωj, j “ 1 . . . m are independent over R and we

define ω:_“ m ÿ j“1 |ωj| “ m ÿ j“1 ˜ _l ÿ i“1 pωi jq 2 ¸1{2 (1.7)

Let us define for ρ_{ą 0, k P t0uYN and V}1 _{: pΞ, x,~q P R}m_ˆTl_{ˆr0, 1s ÞÑ V}1_{pΞ, x,~q PR}m

}V1}ρ,ω,k “ m ÿ j“1 k ÿ r“0 }Br ~Vbej1}L1 ρ,ω,rpRmˆZlqbL8pr0,1sq and }∇V 1_} ρ,ω,k“ max i_“1...l m ÿ j“1 k ÿ r“0 }Br ~BΞjV 1 i}ρ,ω,k,

where be¨ denotes the Fourier transform on SpRm

ˆTl q and L1 ρ,ω,kpR m ˆZl q is the L1 _space

equipped with the weighted norm ř

q_PZl

ş

Rm

|fpp, qq|p1 ` |ω ¨ p| ` |q|qr2eρpω|p|`|q|qdp(See Section

4).

Let us remark that }V1_}

ρ,ω,k ă 8 implies that V1 is analytic in a complex strip ℑxă ρ,

ℑξ_{ă ρω and k-times differentiable in} ~_{P r0, 1s.}

We will denote V1_{pΞq :“} 1 p2πql

ş

TlV

1_{pΞ, xqdx.}

Our assumptions are shown to be non empty in Remark 5 and the relevance of assumption (A2) is discussed in Remark 6, both at the end of Section 2 below.

Our main result reads (see Theorems 29, 30 and 39 for more precise and explicit state-ments):

Theorem 1. Let k_{P N Y t0u and ρ ą 0 be fixed. Let H satisfy the Main Assumption above} and _}V1_}

ρ,ω,k,}∇V1}ρ,ω,k be small enough.

Then there exists a family of vector-valued functions B~

8p¨q, B j

~B ~

8p¨q being holomorphic

in _t|ℑzi| ă ρ₂, i “ 1 . . . mu uniformly with respect to ~P r0, 1s and 0 ď j ď k, such that the

family H is jointly unitary conjugated to B~

8pLωq and therefore the spectrum of each Hi is

pure point and equals the set _tpB~

8qipω ¨ nq, n PZlu where ω ¨ n “ pă ωi, nąqi_“1...m.

Note that the use of Brjuno condition necessitates the intermediary result Theorem 29 involving an extra condition on ω removed by a scaling argument in Theorem 30, as explained in Section 8.

Our results being uniform in~ we get as a partial bi-product of the preceding result the following global version of [LS1]:

Theorem 2. Let ρ_{ą 0 be fixed. Let H be a family of m ď l Poisson commuting classical} Hamiltonians _pHiqi_“1...m on T˚Tl of the form H “ H0 ` V, H0px, ξq “ ω.ξ, ω and V

satisfying assumption _{pA1q and V on the form Vpx, ξq “ V}1_{pω1.ξ, . . . , ωm.ξ, xq. Let finally}

}V1_}

ρ,ω,}∇V1}ρ,ω,0 be small enough (here we consider V1 as a function constant in ~).

Then H is (globally) symplectomorphically and holomorphically conjugated to B0 8pH0q.

Once again let us mention that our results are much more explicit, precise and complete (in particular concerning radii of convergence and unitary/symplectic conjugations) as expressed in Theorems 29, 30 and 39 and Corollary 35.

Moreover it appears in the proofs that the statement in Theorem 1, as well as in Theorems 29, 30 and 39 and Corollary 35, is valid for fixed value of the Planck constant ~ under the Main Assumption lowed down by restricting (1.6) to ~ fixed. More precisely under the Main Assumption with (A3) restricted to, e.g., ~ _{“ 1, the Theorem 1 is still valid by}

putting in the statement k _{“ 0 and} ~ _{“ 1. Let us mention also that, as in the original}

formulations in [Ru]-[LS1], one easily sees that condition (A2) can be replaced by the fact that the quantum Birkhoff normal form (see section 2 below for the precise definition) at each order is a function of_pL1, . . . , Lmq only.

Let us emphasize the two extreme cases, that is m_{“ l and m “ 1.}

Corollary 1 (Quantum Vey theorem). Assume that the ωj P Rl, j “ 1, . . . , l, are

inde-pendent over R. Assume that the Hi “ Lωi` Vi, i“ 1, . . . , l are pairwise commuting. Let

the perturbation Vi be the quantization of any small enough analytic function Vi. Then the

family H is jointly unitary conjugated to B~

8pLωq as defined in theorem 1.

We emphasize that this last result do not require neither a small divisors condition nor a condition on the perturbation, see Section 10. This correspond to full quantum integrability. Quantum integrability is a huge subject - see the seminal articles [CdV1, CdV2] to quote only two. The difference that provides our construction is the fact that our results gives convergent result even at ~_{“ 1 is the case of perturbations of linear systems.}

Corollary 2 (consolidated Graffi-Paul theorem). Assume that ω _{P R}l _{satisfies Brjuno}

and V_{pξ, xq “ V}1_{pω.ξ, xq. Then H is unitary conjugated to B}~

8pLωq as defined in theorem

1.

The main difference between this last result and the main result of [GP] is the small divisors condition used (a Siegel type condition with constraints).

Le us finally mention a by-product of our resul, a kind of inverse result, obtained thanks to the fact that we carefully took care of the precise estimations and constants all a long the proofs. This result is motivated by the remark that, though a small divisors condition is necessary to obtain the perturbed integrability (and Brjuno condition is sufficient), such a condition should disappear when the perturbation vanishes, as the Hamiltonian H0 _is

always integrable, whatever the frequencies ω are. Our last result quantifies this remark. Let us define, for ω satisfying (1.4) and α _{ă 2 log 2,}

Bαpγ, τq “ 2 log

” 2τ_γ

p_eατ qτı

(note that Bαpγ, τq Ñ 8 as γ and/or τ Ñ 8).

The next Theorem shows that, in the Diophantine case, the small divisors condition can be released as Bαpγ, τq diverging logarithmically as the perturbation vanishes.

Theorem 3. Let k _{P N Y t0u and ρ ą 0 be fixed. Let ω and V satisfy (A1) (Diophantine} case), (A2) and (A3), and let 0 _{ă ω´ ď ω ď ω` ă 8 and }V}1_}

ρ,ω_{`</sub>,k, }∇V1}ρ,ω<sub>`},k be small

enough (depending only on k).

Then there exist a constant Cω_´ such that the conclusions of Theorems 1 hold as soon

as, for some α_{ă ρ{2, α ă 2 log 2,} Bαpγ, τq ă 1 3log ˜ 1 }V }ρ,ω<sub>`</sub>,k ¸ ` Cω_´.

See Corollary 40 for details and the Remark after on the case of the Brjuno condition. Let us remark that an equivalent result for Theorem 2 is straightforwardly obtainable.

Let us finish this section by mentioning three comments and remarks concerning our results.

First of all, as mentioned earlier, no hypothesis on the minimal distance between two consecutive unperturbed eigenvalues is required in our article. More, the spectra of our unperturbed operators Lωi might be dense for all value of ~ (actually in the Diophantine

case for m “ 1, l ą 1 they are) so the Neumann series expansion is not possible. For m _{ą 1 the non degeneracy of the unperturbed eigenvalues is not even insured by the} arithmetical property of ω because it relies on the minimum over i _{ď m of the inverse} of the small denominator of the vector ωi. In fact, for a resonant ωj the operator Hj

will have an eigenvalue with infinite degeneracy, so the projection of the perturbation Vj

on the corresponding and infinite dimensional eigenspace, which leads to the first order perturbation correction to the unperturbed eigenvalue, might have continuous spectrum. Nevertheless our results show that the perturbed spectra are analytic functions of the spectra of the Lωi’s.

Secondly, because of the fact that non degeneracy of some of the unperturbed spectra is not even guaranteed by our assumptions, the standard argument on existence of a common eigenbasis of commuting operators with simple spectra cannot be involved here. This existence is a bi-product of our results.

Finally let us mention that, as it was the case in [GP], though our hypothesis on the perturbations are restrictive, our results, compared with the usual construction of quasi-modes [Ra, CdV3, PU, Po1, Po2], have the property of being global in the spectra (full diagonalization), and exact (no smoothing or O_p~8_{q remainder), together of course with}

sharing the property of being uniform in the Planck constant.

Let us point out that this paper has been written in order to be self-contained

Notations

Function valued vectors inRn_{will be denoted in general in calligraphic style, and operator}

valued vectors by capital letters, e.g. V _{“ pV}lql_“1...m or V “ pVlql_“1...m.

For i, j _PZn _{we will denote by ¨}

ij or¨,ij when ¨ has already an index, the matrix element

of an (vector) operator in the basis tej, ejpxq “ eij.x{p2πq

l

2, θP Tlu, namely

Vij “ pVl,ijql“1...m“ ppei, VlejqL2

pTn_qql“1...m,

and by V the diagonal part of V :

Vij “ Viiδij, together with V _{“ p2πq}´l ż Tl Vdx.

We will denote by _{| ¨ | the Euclidean norm on} Rm _{(or C}m_),

|Z|2 “ m ř i_“1|Z i|2, and by } ¨ }L2 pTl_qÑL2

pTl_q the operator norm on the Hilbert space L2pTlq.

Finally for ω_{“ pω}i PRlqi“1...m and ξ PRl, p PRm, qPZl we will denote

ω_{¨ ξ “ pă ω}i, ξąRlqi“1...m PRm, (1.8) p.ω_“ ˜_m ÿ i“1 piωij ¸ j“1...l PRl (1.9) and p.ω.q _“ m ÿ i“1 l ÿ j“1 piωjiqj “ xp ¨ ω, qyRl. (1.10)

2. Strategy of the proofs

The general idea in proving Theorem 1 will be to construct a Newton-type iteration procedure consisting in constructing a family of unitary operators Ur such that (norms will

be defined later) U_r´1_pB~ rpLωq ` VrqUr “ B ~ r`1pLωq ` Vr<sub>`1</sub>, (2.1) with _}Vr`1}r`1 ď Dr`1}Vr}2r and B ~ 0pLωq “ Lω, V0 “ V .

Ur will be chosen of the form

Ur “ ei

Wr

~ , W

r self-adjoint. (2.2)

It is easy to realize that (2.2) implies (2.1) if Wr satisfies the (approximate) cohomological

equation 1 i~rB ~ rpLωq, Wrs ` Vr “ Dr<sub>`1</sub>pLωq ` Op}Vr}2rq, (2.3) or equivalently 1 i~rB ~ rpLωq, Wrs ` Vrco“ Dr<sub>`1</sub>pLωq ` Op}Vr}2rq, (2.4) for any Vco

r such that }Vrco´ Vr}r “ Op}Vr}2rq.

We will solve for each r the equation (2.4) where Vco

r will be obtained by a suitable

“cut-off” in order to have to solve (2.4) with only small denominators of finite order (see Brjuno condition (1.3)).

In fact we will see in Section 3 that we can find a (scalar) solution of the (vector) equation (2.4) satisfying 1 i~rB ~ rpLωq, Wrs ` Vrco“ B ~ r<sub>`1</sub>pLωq ` Rr, (2.5)

where_}Rr}k`1 “ Op}Vr}2rq. To do this we will remark that since the components of B

~

rpLωq`

Vr commute with each other (since the ones of Lω` V do) we have that

rpB~

rpLωqql,pVrql1s ´ rpB_r~pL_ωqq_l1,pV_rq_ls “ rpV_rq_l1,pV_rq_ls “ OpV_r2q (2.6)

which is an almost compatibility condition (see Section 3 for details).

Summarizing, the solution Wr of (2.4) will provide a unitary operator Ur such that (2.1)

will hold with B~

r_`1 “ B

~

r ` Dr`1 and Vr`1 being the sum of three terms:

‚ V1 r_{`1</sub> “ Ur´1pB ~ rpLωq ` VrqUr´ pBr~pLωq ` Vrq ´ <sub>i</sub>1~rB ~ rpLωq, Wrs ‚ V2 r<sub>`1} “ Vr´ Vrco ‚ V3 r`1 “ Rr

The choice of the family of norms _{} ¨ }}r will be made in order to have that

}Vr`1}r`1“ }Vr1<sub>`1</sub>` V 2 r<sub>`1</sub>` V 3 r<sub>`1</sub>}r`1 ď Dr`1}Vr}2r with Dr satisfying R ź r“1 D2<sub>r</sub>R´r <sub>ď C</sub>2R. Hence, we have }VR`1}R`1 ď pC}V0}0q 2R ,

so that _}VR_{`1</sub>}R<sub>`1} Ñ 0 as R Ñ 8 if }V0 “ V }0 ă C´1 and } ¨ }₈ exists.

Remark 4. [Propagation of assumptions (A2)-(A3)] It is clear (and it will be explicit in the body of the proofs of the main Theorem) that Condition (A2) will be satisfied by the solution of equations (2.3),(2.4) as soon as Vr and Vrco do. This last condition can be easily

seen to be propagated from the decomposition Vr_{`1</sub> “ Vr1<sub>`1}` Vr2<sub>`1</sub>` Vr3<sub>`1</sub> given before by

considering that Ur“ ei

Wr

~ by (2.2) and W

r satisfying (A2). (A3) is obviously propagated

Remark 5. [Non emptiness of the hypothesis] Consider a family of operators of the form Lω ` B~pLωq for B~ : Rm Ñ Rm with }B~}ρ,ω,k ă `8. Then for each bounded

self-adjoint operator W whose Weyl symbol W satisfies (A2) and _{}W }}ρ1_,ω,k ă `8 for some

ρ1 _{ą ρ, consider the family e}iW~ pL

ω` B~pLωqqe´i

W

~ :“ L

ω` V :“ pHiqi_“1...m. Obviously the

family pHiqi_“1...m satisfies (A3). By the same argument as the one in Remark 4 one sees

easily that the Weyl symbol V of V satisfies (A2) for some V1. Finally estimates (5.11) and (5.12) in Proposition 16 below show that the expansion eiW

~pL ω ` B~pLωqqe´i W ~ “ Lω`B~pLωq`rLω`B~pLωq,iW~ s` 1 2rrLω`B ~ pLωq,iW~ s, iW

~ s`. . . is actually convergent. This

implies that _}V}ρ,ω,k is bounded. Therefore the family Lω` V satisfies all the assumptions

of Section 1.

Remark 6. [Relevance of assumption (A2)] Let us recall some classical facts from dynamical systems. Let H0 “

n

ř

i_“1

λipx2i ` y2iq be a quadratic Hamiltonian on R2n. Any analytic

higher order perturbation H _{“ H}0` higher order terms is formally conjugate to a formal

Birkhoff normal form ˆH_px2

1 ` y12, . . . , x2n` yn2q. R¨ussman-Brjuno’s theorem asserts that, if

p˚q Hˆ _{“ ˆ}F_pH0q (i.e. ˆH is a function of that peculiar linear combination n

ř

i“1

λipx2i ` yi2q

and contains no other terms), for some formal power series ˆF of one variable and if a ”small divisors” condition is satisfied, then the transformation to the Birkhoff normal form is analytic in a neighborhood of the origin. Condition _{p˚q is known as Brjuno’s condition} A (cf. [Bru]). It is a sharp condition for the analycity of the transformation to Birkhoff normal form in the following sense : if a normal form NF doesn’t satisfy it, then it is possible to perturb H in such way that the analytic perturbation ˜H still has NF as normal form and the transformation from ˜H to NF is a divergent power series. In our quantum version, we only focus on the sufficiency of the analogue condition. The linear combination ř

jωjξj in our article plays the rˆole of ”quantum analogue” of

ř

iλipx2i ` yi2q

3. The cohomological equation: the formal construction

In this section we want to show how it is possible to construct the solution of the equation 1

i~rB ~

pLωq, W s ` V “ DpLωq ` OpV2q, (3.1)

where we denote by Lω, ω “ pωi P Rlqi_“1...m, the operator valued vector of components

(with a slight abuse of notation) Lωi “ ´i~ωi.∇x, i “ 1 . . . m on L

2_pTl_{q and V is a}

“cut-off”ed.

Vij “ 0 for |i ´ j| ą M.

We will present the strategy only in the case of the Brjuno condition, the Diophantine case being very close.

Let us recall also that equation (3.1) is in fact a system of m equations and that it might seem surprising at the first glance that the same W solves (3.1) for all ℓ_{“ 1 . . . m.}

3.1. First order. At the first order the cohomological equation is rLωℓ, Ws

i~ ` Vℓ “ DℓpLωq, l “ 1 . . . m (3.2)

solved on the eigenbasis of any Lωℓ by DℓpLωq “ diagpVℓq and

Wij “ ´pV

ℓ´ Dℓqij

iωℓ¨ pi ´ jq

. (3.3)

Indeed, since Lωl is selfadjoint, we have

ă ej,rLωl, Wsei ą “ ă ej, LωlW ei´ W Lωlei ą“ă Lωlej, W ei ą ´ ă ej, W Lωlei ą

“ iωl.pj ´ iq ă ej, W ei ą

In (3.3) we will picked up, for every ij such that |i ´ j| ď M, an index ℓ “ ℓi_´j which

minimize the quantity

|xωℓq, qy|´1 :“ min 1_ďiďm|xωi, qy| ´1 _{ď M} M. (3.4) We define W by Wij “ ´ pV ℓi´jqij iωℓi´j ¨ pi ´ jq , i_{´ j ‰ 0} (3.5)

Since_rHℓ, Hℓ1s “ 0, then we have that rL_ℓ1V_ℓs ` rV_ℓ1, L_ℓs “ ´rV_ℓ, V_ℓ1s. Therefore, evaluating

the operators on ej and taking the scalar product with ei, leads to

ωℓ1¨ pi ´ jqpV_ℓq_ij “ ω_ℓ¨ pi ´ jqpV_ℓ1q_ij ´ prV_ℓ, V_ℓ1sq_ij (3.6) that is pVℓqij ωℓ¨ pi ´ jq “ pVℓ1q_ij ωℓ1 ¨ pi ´ jq ´ prVℓ, Vℓ1sq_ij ωℓ¨ pi ´ jqωℓ1¨ pi ´ jq

(note that when ωℓ1¨ pi ´ jq “ 0 on has pV_ℓ1q_ij “ ´prVℓi´j

,V_ℓ1sqij

ω_{ℓi´j¨pi´jq} ).

Let us remark that, though _rVℓ, Vℓ1s is quadratic in V , it has the same cut-off property

as V , namely _prVℓ, Vℓ1sq_ij “ 0 if |i ´ j| ą M as seen clearly by (3.6).

This means that W defined by (3.5) satisfies rLω, Ws i~ ` V “ DpLωq ` bV , where p bVℓqij “ prV ℓ, Vℓi´jsqij i~ωℓi´j ¨ pi ´ jq . (3.7)

3.2. Higher orders. The cohomological equation at order r will follow the same way, at the exception that Lω has to be replaced by B~rpLωq.

The corresponding cohomological equation is therefore of the form rB~ rpLωq, Wrs i~ ` Vr “ OppVrq 2 q, (3.8) equivalent to B~ rp~ω¨ iq ´ B ~ rp~ω¨ jq i~ pWrqij ` pVrqij “ OppVrq 2 q. (3.9) Lemma 7. For B~

r close enough to the identity there exists a mˆ m matrix Arpi, jq such

that B~ rp~ω¨ iq ´ B ~ rp~ω¨ jq i~ “ pI ` A r pi, jqq ω.pi ´ jq, (3.10) where I is the m_{ˆ m identity matrix and ω.pi ´ jq “ pω}l.pi ´ jqql_“1...m. Moreover

}Arpi, jq}Cm_ÑCm ď }∇pB ~ r ´ B ~ 0q}pCm_ÑCm_qbL8_pRm_q ď max j“1...m m ÿ i“1 }∇jpBr~´ B ~ 0qi}L8_pRm_q. (3.11) Proof. We have B~ rp~ω¨ iq ´ B ~ rp~ω¨ jq i~ “ ω.pi ´ jq ` ż1 0 B t“pB ~ r ´ B ~ 0qpt~ω.i` p1 ´ tq~ω.jq ‰ dt ~ “ ω.pi ´ jq ` ż1 0 “∇pB~ r ´ B ~ 0qpt~ω.i` p1 ´ tq~ω.jq‰ ¨ rω.pi ´ jqsdt so Ar_{pi, jq “}ş1 0∇pB ~ r ´ B ~

0qpt~ω.i` p1 ´ tq~ω.jqdt and the first part of (3.11) follows. The

second part is a standard estimate of the operator norm.  Plugging (7.5) in (3.9) we get that W must solve

ω._{pi ´ jqW}ij “ pI ` Arpi, jqq´1“´pVrqij ` OppVrq2q‰ , (3.12)

and we are reduced to the first order case with VrÑ eVr where

e

Vijr :“ pI ` Arpi, jqq´1pVrqij. (3.13)

3.3. Toward estimating. We will first have to estimate eVr_{: this will be done out of its}

matrix coefficients given by (3.13) by the method developed in Section 5.1. We will estimate pI ` Ar

pi, jqq´1Ver _{in section 6 by using the formula}

pI ` Ar

pi, jqq´1 “ ř8

k“0p´A r

pi, jqqk _and

a bound of the norm of_p´Ar_{pi, jqq}k_Ver _{of the form|C|}k _{times the norm of eV}r _{leading to a}

bound of _{pI ` A}r

pi, jqq´1Ver of the form 1

1´|C| times the norm of eV

r_{, by summation of the}

geometric series ř8

k“0

Ck_{, possible at the condition that|C| ă 1.}

We will then have to estimate W defined through

Wij “ ´

p eV_ℓr_i´jqij

iωℓi´j ¨ pi ´ jq

with again _{p e}V_ℓr_i´jqij “ 0 for |i ´ j| ą M. We get

|Wij| ď MM|p eVℓi´jqij|,

and we will get an estimate of W , _{}W } ď M}M} eVr}, for a norm } ¨ } to be specified later.

Finally we will have to estimate

p bV_lr_qij “

pr eVr

l , eVℓri´jsqij

i~ωℓi´j ¨ pi ´ jq

. (3.15)

We will get immediately _{} b}Vr

l } ď MM}P }, Pij “ pr eVr

l, eV_ℓi´jr sqij

i~ and the estimate of the

commutator will be done by the method developed in Section 5.

In the two next sections we will define the norms and the Weyl quantization procedure used in order to precise the results of this section,

4. Norms

Let m, l be positive integers. For F _{P C}8_pRm_ˆTl_{ˆ r0, 1s;Cq we will use the following}

normalization for the Fourier transform.

Definition 8 (Fourier transforms). Let p_P Rm _{and q}

PZl b F_{pp, x,}~_{q “} 1 p2πqm ż Rm F_{pξ, x,}~_qe´ixp,ξydξ (4.1) e F_{pξ, q;}~_{q “} 1 p2πql ż Tl F_{pξ, x;}~_qe´ixq,xydx (4.2) be F_{pp, q,}~_{q “} 1 p2πqm`l ż Rm<sub>ˆ</sub>Tl F<sub>pξ, x,</sub>~<sub>qe</sub>´ixp,ξy´ixq,xydξdx (4.3) “ 1 p2πqm ż Rm e F<sub>pξ, q,</sub>~<sub>qe</sub>´ixp,ξydξ (4.4) “ 1 p2πql ż Tl b F<sub>pp, x,</sub>~<sub>qe</sub>´ixq,xydx (4.5) Note that F<sub>pξ, x,</sub>~<sub>q “</sub> ż Rm b F<sub>pp, x,</sub>~<sub>qe</sub>ixp,ξydp (4.6) “ ÿ qPZl e F<sub>pξ, q;</sub>~<sub>qe</sub>ixq,xy (4.7) “ ÿ qPZl ż Rm be F<sub>pp, q,</sub>~<sub>qe</sub>ixp,ξy`ixq,xy_{dp (4.8)}

Set now for k_P N_{Y t0u and p ¨ ω “ p} ř

j_“1...m

pj.ωjiqi“1...l:

µkpp, qq :“ p1 ` |p ¨ ω|2` |q|2q

k

(note that µrpp ´ p1, q´ q1q ď 2

k 2µ

rpp, qqµrpp1, q1q because |x ´ x1|2 ď 2p|x|2` |x1|2q and that

|p ¨ ω| Ñ 8 as |p| Ñ 8 because the vectors pωi

q1“1...l are independent over R).

Definition 9 (Norms I). For ρ_{ą 0, F P C}8_pRm_ˆTl_{ˆ r0, 1s;Cq we introduce the weighted}

norms }F}: ρ“ }F}:ρ,ω :“ max ~_Pr0,1s ż Rm ÿ q_PZl |Fbe_{pp, q,}~_{q| e}ρpω|p|`|q|qdp. (4.10) }F}:ρ,ω,k “ }F}:ρ,ω,k :“ max ~<sub>Pr0,1s</sub> k ÿ j<sub>“0</sub> ż Rm ÿ q<sub>P</sub>Zl µk<sub>´j</sub>pp, qqB~j|Fbepp, q,~q| e ρ<sub>pω|p|`|q|qdp.</sub> (4.11)

Note that ω is given by (1.7) and _{} ¨ }}:_{ρ;0 “ } ¨ }}: σ.

Definition 10 (Norms II). Let Oω be the set of functions F : Rl ˆTlˆ r0, 1s Ñ C such

that F_{pξ, x;}~_{q “ F}1_{pω ¨ ξ, x,~q for some F}1 _{: R}m

ˆTl

ˆ r0, 1s ÑC. Define, for F _{P O}ω:

}F}ρ,ω,k :“ }F1}:ρ,ω,k. (4.12)

We will also need the following definition for F _{P O}ω:

}F}~ ρ,ω,k :“ k ÿ j_“0 ż Rm ÿ qPZl µk´jpp, qqB~j|Fce1pp, q,~q| e ρpω|p|`|q|q_{dp. (4.13)}

Let us note that, obviously, _{} ¨ }}~

ρ,ω,k ď } ¨ }ρ,ω,k.

We will need an extension of the previous definition to the vector case. Consider now F _{P C}8_pRm

ˆTl

ˆ r0, 1s;Cm

q and G P C8_pRm

ˆ r0, 1s;Cm

q. The definition of the Fourier transform is defined as usual, component by component.

Definition 11. [Norms III] Let F _{“ pF}iqi“1...m P C8pRmˆTlˆ r0, 1s;Cmq. We define

(1) }F}:ρ,ω,k “ m ÿ i“1 }Fi}:ρ,ω,k (4.14) (2) Let Om_{ω “} F “ pF_iqi“1...m:Rm_ˆTl_{ˆ r0, 1s Ñ}Cm_{{ F}i P Oω, i“ 1 . . . m ( (4.15) Let F _{P O}m ω. We define: }F}ρ,ω,k “ m ÿ i_“1 }Fi}ρ,ω,k (4.16) Let Om_ωˆm _{“ F “ pFijqi,j“1...m :}Rm_ˆTl_{ˆ r0, 1s Ñ}Cm_{{ F}ij P Oω, i, j “ 1 . . . m ( (4.17) Let F _{P O}m_ˆm ω . We define: }F}ρ,ω,k “ sup i_“1...m ÿ j“1...m }Fij}ρ,ω,k. (4.18)

(3) Finally we denote F the Weyl quantization of F recalled in Section 5 and }F }ρ,ω,k “ }F}ρ,ω,k (4.19) J_k:_{pρ, ωq “ tF | }F}}:_{ρ,ω,k ă 8u,} (4.20) J_k:_{pρ, ωq “ tF | F P Jk}:_{pρ, ωqu,} (4.21) J_{kpρ, ωq “ tF P Oω| }F}ρ,ω,k ă 8u, (4.22)} J_{kpρ, ωq “ tF | F P Jkpρ, ωqu.} (4.23) J~ kpρ, ωq “ tF P Oω| }F} ~ ρ,ω,k ă 8u, (4.24) J~ kpρ, ωq “ tF | F P J ~ kpρ, ωqu. (4.25) J_km_{pρ, ωq “ tF P Oω}m_{| }F}ρ,ω,k ă 8u,} (4.26) J_km_{pρ, ωq “ tF | F P Jk}m_{pρ, ωqu.} (4.27) J_kmˆm_{pρ, ωq “ tF P Oω}mˆm_{| }F}ρ,ω,k ă 8u, (4.28)} J_kmˆm_{pρ, ωq “ tF | F P Jk}mˆm_{pρ, ωqu} (4.29) and J@_{pρ, ωq “ J}@ k“0pρ, ωq, J@pρ, ωq “ Jk@“0pρ, ωq @@ P t:, m, m ˆ mu.

When there will be no confusion we will forget about the subscript ω in the

label of the norms and also denote by J@

k pρq “ J @ k pρ, ωq.

5. Weyl quantization and first estimates

We express the definitions and results of this section in case of scalar operators and symbols. The extension to the vector case is trivial component by component. The reader only interested by explicit expression can skip the beginning of the next paragraph and go directly to Definition 5.4.

5.1. Weyl quantization, matrix elements and first estimates. In this section we recall briefly the definition of the Weyl quantization of T˚Tl_{. The reader is referred to [GP]}

for more details (see also e.g. [Fo]).

Let us recall that the Heisenberg group over T˚Tl

ˆR, denoted by H_lpRl_ˆZl_{ˆRq, is}

(the subgroup of the standard Heisenberg groupH_lpRl_ˆRl_{ˆRq) topologically equivalent to}

Rl

ˆZl

ˆRwith group law_{pu, tq¨pv, sq “ pu`v, t`s`}1

2Ωpu, vqq. Here u :“ pp, qq, p P R l_{, q}

PZl_,

t_PR and Ω_{pu, vq is the canonical 2´form on} Rl_ˆZl_{: Ωpu, vq :“ xu}

1, v2y ´ xv1, u2y.

The unitary representations of H_lpRl_ˆZl _{ˆRq in L}2_pTl_{q are defined for any ~ ‰ 0 as}

follows

pU~pp, q, tqfqpxq :“ ei

~t_`ixq,xy`~_xp.qy{2

f_{px `}~p_q (5.1) Consider now a family of smooth phase-space functions indexed by ~, Apξ, x,~_{q :} Rl _ˆ

Tl_{ˆ r0, 1s ÑC, written under its Fourier representation}

A_{pξ, x,}~_{q “} ż Rl ÿ qPZl be A_{pp, q;}~_qeipxp.ξy`xq,xyq_{dp (5.2)}

Definition 12 (Weyl quantization I). By analogy with the usual Weyl quantization on T˚Rl_{[Fo], the (Weyl) quantization of A is the operator Ap~q defined as}

A_p~_{q :“ p2πq}l ż Rl ÿ q_PZl be A_{pp, q;}~_qU~pp, q, 0q dp (5.3)

(note that the factor _p2πql _{in (5.3) is due to the (convenient for us) normalization of the}

Fourier transform in Definition 8).

It is a straightforward computation to show that, considering f _{P L}2

pTl

q and Vppx`¨q{2q as periodic functions on Rl_{, we get the equivalent definition}

Definition 13 (Weyl quantization II).

pAp~_{qfqpxq :“}

ż

Rl

ξˆRly

A_{ppx ` yq{2, ξ,}~_qeiξpx´yq~ fpyq dξdy

p2π~_ql (5.4)

Remark 14. The expression (13) is exactly the same as the definition of Weyl quantization on T˚_Rl _{except the fact that f is periodic. Note that A}

p~_{qf is periodic thanks to the fact}

that Apx, ξ,~_{q is periodic:}

ş Appx ` 2π ` yq{2, ξqeiξpx`2π´yq~ fpyqdξdy

~l “ş Appx ` 2π ` y ` 2πq{2, ξqe

iξpx´yq~ fpy ` 2πqdξdy

~l “

ş Appx ` yq{2 ` 2π, ξqeiξpx´yq~ fpyqdξdy

~l “ş Appx ` yq{2, ξqe

iξpx´yq~ fpyqdξdy

~l “ pAp~qqfpxq.

The first results concerning this definition are contained in the following Proposition. Proposition 15. Let A_p~_{q be defined by the expression (5.4). Then:}

(1) _{@ρ ą 0, @ k ě 0 we have:} }Ap~_q}B_pL2 pTl_qq ď }A}_ρ,k (5.5) and, if A_{pξ, x,}~_{q “ A}1_{pω ¨ ξ, x;~q} }Ap~_q}B_pL2 pTl_qq ď }A1}ρ,k. (5.6) (2) Let, for n_PZl_{, e} npxq “ e inx

p2πql. Then for all m, n in Zl,

xem, Ap~qenyL2

pTl_q “ eAppm ` nq~{2, m ´ n,~q (5.7)

(3) Reciprocally, let A_p~_{q be an operator whose matrix elements satisfy (5.7) for some}

A _{belonging to J}@_,@

P t:, m, m ˆ mu. Then Ap~_{q is the Weyl quantization of A.}

Proof. (5.7) is obtained by a simple computation. It also implies that }Ap~_qem}2 L2 pRl_q“ ÿ qPZl | eA_p~_{pm ` qq{2, m ´ q,}~_q|2 ď sup ξ_PRl ÿ qPZl | eA_{pξ, q,}~_q|2_. So that }Ap~_qÿ Zl cmem}2L2 pRl_q ď ÿ Zl |cm|2sup ξ_PRl ÿ qPZl | eA_{pξ, q,}~_q|2 ď pÿ Zl |cm|2q ¨ ˝ ÿ qPZl sup ξ_PRl| e A_{pξ, q,}~_q| ˛ ‚ 2 .

And therefore, since by (4.6)-(4.7)-(4.8) eA_{pξ, q,}~_{q “}ş RlAbepp, q,~qeiăξ,pądpso that| eApξ, q,~q| ďş Rl|Abepp, q,~q|dp, }Ap~_q}B_pL2 pTl_qqď ÿ qPZl sup ξ_PRl| e A_{pξ, q,}~_{q| ď} ż Rl ÿ qPZl |Abe_{pp, q,}~_{q|dp ď }A}ρ,k},_{@ρ ą 0, k ě 0.} (5.8) In the case A_{pξ, x,}~_{q “ A}1_{pω ¨ ξ, x;~q we get, @ρ ą 0, k ě 0:}

}Ap~_q}B_pL2 pTl_qqď ÿ q_PZl sup ξPRl | eA_{pξ, q,}~_{q| “} ÿ q_PZl sup Y_PRm| e A1_{pY, q,}~_{q| ď} ż Rm ÿ q_PZl | bA1_{pp, q,}~_{q|dp ď }A}1_}ρ,k. (3) is obvious. 

5.2. Fundamental estimates. This section contains the fundamental estimates which will be the blocks of the estimates needed in the proofs of our main results. These primary estimates are contained in the following Proposition. We shall omit to write the subscript ω in the norms.

Proposition 16. We have: (1) For F, G_{P J}1

kpρq, F G P Jk1pρq and fulfills the estimate

}F G}ρ,k ď pk ` 1q8k}F }ρ,k¨ }G}ρ,k (5.9)

(2) There exists a positive constant C1 such that for F P Jm

k pρq and for G P Jk1pρq, we have, _@δ1 ą 0, δ ě 0, ρ ą δ ` δ1, › › › › rF, Gs i~ › › › › ρ´δ´δ1,k ď 2pk ` 1q8 k e2_δ 1pδ ` δ1q}F } ρ,k}G}ρ´δ,k, (5.10) 1 d!}rG, . . . rGloooomoooon d times , F<sub>s ¨ ¨ ¨ s{pi</sub>~<sub>q</sub>d<sub>}</sub>ρ<sub>´δ,k</sub> ď 1 2π ˆ 2p1 ` kq8k δ2 ˙d }F }ρ,k}G}dρ,k, (5.11) and › › › › rLω, Gs i~ › › › › ρ´δ,k ď _eδω}G}ρ,k (5.12) (3) For F , G _{P J}1 kpρq, FG P Jk1pρq and }FG}ρ,kď pk ` 1q4k}F}ρ,k¨ }G}ρ,k. (5.13)

(4) Let V _{“ pV}lql_“1...mP Jkmpρq and let W be defined by xem, W eny “ xe

m,V_lm´neny

ω_{ℓm´n¨pm´nq} where,

@m, n P Z, the index lm_´n is such that |xωℓm´n, m´ ny|´1 :“ min

1ďiďm|xωi, m´ ny| ´1 _. Then }W }ρ´d,k ď γ ττ pedqτ}V }ρ,k (5.14)

in the Diophantine case and (obviously) when _|xem, Vlm´neny| “ 0 for |m ´ n| ą M,

}W }ρ,k ď MM}V }ρ,k (5.15)

(5) Let finally V “ pVlql_“1...mP Jkmpρq and let P be defined by pPlqij “

prVl,V_ℓi´jsqij

i~ for any

choice of _{pi, jq Ñ ℓ}i_´j. Then P “ pPlql_“1...m P Jkmpρ ´ δq, @δ1 ě 0, δ ą 0, ρ ą δ ` δ1

and }P }ρ_´δ´δ1,k ď 2<sub>pk ` 1q8</sub>k e2<sub>δ</sub> 1pδ ` δ1q}V } ρ,k}V }ρ_´δ,k (5.16) (6) Moreover let F : ξ _P Rm _{ÞÑ Fpξq PR}m _{be in J}m

k pρq. Let us define ∇F the matrix

pp∇Fqijqi,j“1...m with

p∇Fqij :“ BξiFj. (5.17)

Then, for all δ _{ą 0, ∇F P J}mˆm

k pρ ´ δq and

}∇F}ρ´δ,kď

1

eδ}F}ρ,k. (5.18)

Let us remark that, as the proof will show, Proposition 16 remains valid when the norm } ¨ }ρ,k is replaced by the norm } ¨ }

~

ρ,k

Proof. Items (1) and (2) are simple extension to the multidimensional case of the cor-responding results for m _{“ 1 proven in [GP]. For sake of completeness we give here an} alternative proof in the case m_{“ 1. The proof will use the three elementary inequalities,}

µkpp ` p1, q` q1q ď 2 k 2µ kpp, qqµkpp1, q1q (5.19) |pp.ω.q1´ p1ω.q_q{2|k _{ď µ}kpp, qqµkpp1, q1q (5.20) ˇ ˇ ˇ ˇB k ~ sin x~ ~ ˇ ˇ ˇ ˇ ď |x| k_`1 (5.21) |p ¨ ω ¨ q| ď ω max_j ´1...m|pj||q| ď ω|p||q| (5.22)

where we have used the notation (1.10) and the definition (1.7).

(in order to prove (5.19), (5.20), (5.21) and (5.22) just use _{|X ` X</sub>1<sub>|</sub>2 <sub>ď 2p|X|</sub>2 <sub>`}

|X1_|2 q for all X, X1 _{P R}2l_, |pp.ω.q1 _{´ p}1_ω.qq{2|2 ď p|p ¨ ω|2 ` |q|2 qp|p ¨ ω1<sub>|</sub>2 ` |q1_|2 q, sin x~ ~ “ x ş 0 cos_ps~_{qds and |p ¨ ω ¨ q| ď} m ř j“1|p j|| l ř i“1 ωi jqi| “ m ř j“1|p j||ωj ¨ q| ď m ř j“1|p j||ωj||q| by Cauchy-Schwarz, respectively).

We start with (5.9). Since F, G _{P J}1

pρq we know that there exist two functions F1_{, G}1

have that pF Gqmn “ ÿ q1_PZl Fmq1G_q1_n “ ÿ q1_PZl e Fˆ m ` q 1 2 ~, m´ q 1 ˙ e Gˆ q 1 <sub>` n</sub> 2 ~, q 1_{´ n} ˙ “ ÿ q1_PZl e Fˆ m ` n ` q 1 2 ~, m´ n ´ q 1 ˙ e Gˆ q 1_{` 2n</sub> 2 ~, q 1 ˙ “ ÿ q1<sub>P</sub>Zl e F1 ˆ ω<sub>¨</sub> m` n ` q 1 2 ~, m´ n ´ q 1 ˙ e G1 ˆ ω<sub>¨</sub>q 1<sub>` 2n} 2 ~, q 1 ˙ “ ÿ q1_PZl e F1 ˆ ω_¨ m` n ` q 1 2 ~, m´ n ´ q 1 ˙ e G1 ˆ ω_¨q 1_{` m ` n ´ pm ´ nq} 2 ~, q 1 ˙ . (5.23)

Calling P the symbol of F G we have that, by (5.7) again,_{pF Gq}mn“ ePpξ, qq with ξ “ m`n₂ ~

and q“ m ´ n. Therefore e P_{pξ, qq “} ÿ q1_PZl e F1 ˆ ω.ξ_{` ω ¨</sub>q 1 2~, q´ q 1 ˙ e G1 ˆ ω.ξ<sub>` ω ¨} q 1_{´ q} 2 ~, q 1 ˙ , (5.24)

so we see that P_{pξ, ¨q depends only on ω.ξ: Ppξ, xq “ P}1_{pω.ξ, xq. Moreover, since by}

(4.3) cPe1_{pp, ¨q “} 1 p2πqm

ş

RmfP1pΞ, ¨qe

´iăΞ,pą_{dΞ we get easily by simple changes of integration}

variables and the fact that the Fourier transform of a product is a convolution,

c_e P1_{pp, qq “} ż Rm ÿ q1_PZl ˆc_{Fe1pp ´ p}1_{, q´ q}1_qei~ 2pp´p 1_q.ω.q1˙ ˆ beG 1_pp1_{, q}1_qei~ 2p 1_.ω.pq1_´qqq˙ dp1. (5.25)

Therefore_{}F G}}ρ,k is equal to the maximum over ~ P r0, 1s of

k ÿ γ_“0 ż R2m ÿ pq,q1_qPZ2l µk_´γpp, qq|B~γ„cFe1pp ´ p1, q´ q1qe i~ 2ppp´p1q.ω.q1´p1.ω.pq´q1qqGbe1pp1, q1q  |eρ_{pω|p|`|q|qdpdp1.} (5.26)

Writing, by (5.20), that |B~γ„cFe1pp ´ p1, q´ q1qe i~ 2ppp´p1q.ω.q1´p1.ω.pq´q1qqGbe1pp1, q1q  | ď γ ÿ µ_“0 ˆγ µ ˙γ´µ ÿ ν_“0 ˆγ ´ µ ν ˙ |B~γ´µ´νFce1pp ´ p1, q´ q1q||B ν ~e i~ 2ppp´p 1 q.ω.q1´p1.ω._pq´q1_qq ||B~µGbe1pp1, q1q| ď γ ÿ µ“0 ˆγ µ ˙γ´µ ÿ ν“0 ˆγ ´ µ ν ˙ |B~γ´µ´νFce1pp ´ p1, q´ q1q||ppp ´ p1q.ω.q1´ p1.ω.pq ´ q1qq{2| ν |B~µGbe1pp1, q1q| “: PpF1, G1_q (5.27) ď γ ÿ µ“0 ˆγ µ ˙γ´µ ÿ ν“0 ˆγ ´ µ ν ˙ µνpp ´ p1, q´ q1qµνpp1, q1q|Bγ~´µ´νcFe1pp ´ p1, q´ q1q||B µ ~Geb1pp1, q1q| pchanging µ Ñ γ1, ν _{Ñ ν}1 :_{“ γ ´ γ}1_{´ νq} ď γ ÿ γ1_“0 ˆ γ γ1 ˙γ´γ1 ÿ ν1_“0 ˆγ ´ γ1 ν ˙ µγ_´γ1_´ν1pp ´ p1, q´ q1qµ_γ´γ1_´ν1pp1, q1q|Bν 1 ~ c_e F1_{pp ´ p}1_{, q´ q}1_q||B~γ1Gbe1_pp1_{, q}1_q| psince ˆm_n ˙ ď 2m, γ _{ď k, γ ´ γ}1 _{ď kq} ď k ÿ γ1_“0 2k k ÿ ν1_“0 2kµγ´γ1_´ν1pp ´ p1, q´ q1qµ_γ´γ1_´ν1pp1, q1q|Bν 1 ~ c_e F1_{pp ´ p}1_{, q´ q}1_q||Bγ1 ~Gbe1pp1, q1q| (5.28)

using (5.19) under the form

µkpp, qq ď 2

k 2µ

kpp ´ p1, q´ q1qµkpp1, q1q

together with the fact that µkpp, qq is increasing in k and µkµk1 “ µ_k`k1.

We find that µk´γpp, qqPpF1, G1q ď 4k2k2 k ÿ γ1_“0 k ÿ ν1_“0 µk_{´γ`γ´γ</sub>1<sub>´ν</sub>1pp ´ p1, q´ q1qµ<sub>k´γ`γ´γ}1_´ν1pp1, q1q|Bν 1 ~Fce1pp ´ p1, q´ q1q||B γ1 ~ Gbe1pp1, q1q|

preplacing 2k2 by 2k to avoid heavy notations and since k´ γ1´ ν1 ď k ´ γ1, k´ ν1q

ď 8k k ÿ γ1_“0 k ÿ ν1_“0 µk_´ν1pp ´ p1, q´ q1qµ_k´γ1pp1, q1q|Bν 1 ~ c_e F1_{pp ´ p}1_{, q´ q}1_q||B~γ1Gbe1_pp1_{, q}1_{q|. (5.29)}

Note that γ disappeared from (5.29) so the

k

ř

γ“0

in (5.26) gives a fractor<sub>p1`kq. We get that</sub> p1 ` kq´1₈´k_{}F G}}

ρ,k is majored by the maximum over ~ P r0, 1s (note the change ν1 Ñ γ)

ÿ q,q1_PZl ż R2m k ÿ γ,γ1_“0 µk_´γpp, qq|B~γcFe1pp, qq|µk_´γ1pp1, q1q|Bγ 1 ~Gbe1pp1, q1q|e ρpω|p|`ω|p1<sub>|`|q|`|q</sub>1_|q dpdp1(5.30) which is equal to }F1}ρ,k}G1}ρ,k.

The proof of (5.10) follows the same lines, except that it is easy to see that, in (5.25), ei~

2ppp´p

1_q.ω.q1_´p1_.ω.pq´q1_qq

has to be replaced by 2 sin`~

2ppp ´ p1q.ω.q1´ p1.ω.pq ´ q1qq˘, since (5.23) becomes ˆ rF, Gs i~ ˙ mn “ ÿ q1_PZl Fmq1G_q1_n´ G_mq1F_q1_n i~ “ _i1_~ ÿ q1_PZl „ e Fˆ m ` n ` q 1 2 ~, m´ n ´ q 1 ˙ e Gˆ m ` n ` q 1 _{´ pm ´ nq} 2 ~, q 1 ˙ ´ eGˆ m ` n ` q 1 2 ~, m´ n ´ q 1 ˙ e Gˆ m ` n ` q 1_{´ pm ´ nq} 2 ~, q 1 ˙ (5.31)

It generates in (5.26) the replacement of _{|ppp ´ p}1_q.ω.q1_{´ p}1_{.ω.pq ´ q}1_qq{2|ν_{by the term}

2_{|ppp ´ p}1_q.ω.q1_{´ p}1.ω._{pq ´ q}1_qq{2|ν`1 _{ď µ}νpp ´ p1, q´ q1qµνpp1, q1qp|p ¨ ω ¨ q1´ p1 ¨ ω ¨ q|q

thanks to (5.20), and we get by a discussion verbatim the same than the one contained in equations (5.27)-(5.30) that }rF, Gs{i~_}ρ´δ´δ1,k ď p1 ` kq8 k ÿ q,q1<sub>P</sub>Zl ż R2m k ÿ γ,γ1<sub>“0</sub> µk<sub>´γ</sub>pp, qqµk<sub>´γ</sub>1pp1, q1qQdpdp1, (5.32) where thanks to (5.22), Q “ |Bγ ~Fce1pp, qq|p|p ¨ ω ¨ q1| ` |p1 ¨ ωq|q|B γ1 ~Gbe1pp1, q1q|epρ´δ´δ 1qpω|p|`|q|`ω|p1|`|q1|q ď „ |B~γcFe1pp, qq||B γ1 ~Gbe1pp1, q1q|e ρ<sub>pω|p|`|q|q`pρ´δqpω|p</sub>1<sub>|`|q</sub>1_|q ˆ ˆpω|p||q1| ` ω|p1||q|qe´pδ`δ1qpω|p|`|q|q´δ1pω|p1|`|q1|qq ď _e2_δ 2 1pδ ` δ1q|B γ ~Fce1pp, qq||B γ1 ~Gbe1pp1, q1q|e ρpω|p|`|q|q`pρ´δqpω|p1<sub>|`|q</sub>1_|q (5.33) because (e´x _{ď 1, x ě 0 and)} sup xPR` xe´αx _“ 1 eα. (5.34)

The proof of (5.12) follows also the same line and is obtained thanks to the remark (5.34): indeed since ˆ rLω, Ws i~ ˙ mn “ ´iω ¨ pm ´ nqWmn,

we see, again by (5.7), that the symbol Q_{pξ, xq of rL}ω, Ws{i~ is given trough the formula

e

Q_{pξ, qq “ prLω}, W_s{i~_qmn for ξ _“ m` n

2 ~ and q“ m ´ n. Therefore eQ_{pξ, qq “ p´iω ¨ qq f}W_{pξ, qq, so Qpξ, xq “ Q}1_{pω.ξ, xq with}

f Q1_{pω.ξ, qq “ ´iω.q fW}1_{pω.ξ, qq.} We get immediatly c f Q1_{pp, qqe}pρ´δqpω|p|`|q|q <sub>ď</sub> ω eδ c f W1<sub>pp, qqe</sub>ρpω|p|`|q|q and (5.12) follows.

(5.11) is easily obtained by iteration of (5.10) and the Stirling formula: consider the finite sequence of numbers δs “ d´s_d δ. We have δ0 “ δ, δd “ 0 and δs´1 ´ δs “ δ_d. Let us

define G0 :“ F and Gs_`1 :“ _i1~rG, Gss, for 0 ď s ď d ´ 1. According to (5.10), we have

}Gs}ρ_´δd´s,k ď ck e2_δ d´spδdq }G}ρ,k}Gs_´1}ρ_´δd´s`1,k, where ck :“ 2pk ` 1q8k.

Hence, by induction, we obtain 1 d!}Gd}ρ´δ0,k ď cd_k´1 d!e2pd´1q_{δ0¨ ¨ ¨ δd} ´2pδdqd´1 }G}d_´1 ρ,k }G1}ρ_´δd´1,k (5.35) ď c d k d!e2d_δ 0¨ ¨ ¨ δd´1δd´1pδdqd´1 }G}dρ,k}F }ρ,k ď c d k d!e2d_d!pδ dq d_δ d_´1pδ_dqd´1}G} d ρ,k}F }ρ,k ď 1 2π ˆ ckd2 e2_δ2 ˙d 1 pd ´ 1q!d!}G} d ρ,k}F }ρ,k “ 1 2π ´c_k δ2 ¯d ˜? 2πddd_e´d d! ¸2 }G}d ρ,k}F }ρ,k ď _2π1 ´_δck2 ¯d }G}dρ}F }ρ,k since _? d!

2πde´d_dd ě 1. This weel know inequality can be seen from Binet’s second expression

for the log Γ_{pzq [WW][p. 251] :} log ˜ n! `_n e ˘n? 2πn ¸ “ 2 ż8 0 arctan_pt{nq e2πt_{´ 1} dtě 0.

Finally (5.13) is obtained by noticing that _}FG}ρ,k has the same expression as }F G}ρ,k

after removing the term ei~

2pp.ω.q

1_´p1_.ω.qq

in (5.25).

To prove (4) it is enough to notice that by (5.7) the symbol of W satisfies fW_{pξ, q,}~_{q “}

e V_ℓqpξ,q,~_q ω_ℓq¨q , so that cfWpp, q,~q “ b e V 1_ℓqpp,q,~_q

ω_ℓq¨q and therefore, for all rP N,

|Br ~ c f W_{pp, q,}~_{q| ď γ|q|}τ _sup l_“1...m|B r ~beVlpp, q,~q| ď γ|q| τ m ÿ l“1 |Br ~beVlpp, q,~q|

out of which we deduce (5.14) by standard arguments (xτ_e´δx _{ď p}τ eδq

τ_{, x ą 0) in the}

Diophantine case, and

|Br ~ c f W_{pp, q,}~_{q| ď MM} sup l_“1...m|B r ~beVlpp, q,~q| ď MM m ÿ l“1 |Br ~beVlpp, q,~q|

from which (5.15) follows.

To prove (5.18) we just notice that _B\^ξiFjpp, q,~q “ pi

c f F_{jpp, q,}~_{q. So} |B\^ξiFjpp, q,~q| ď |pi c f F_{jpp, q,}~_{q| ď |}cfF_{jpp, q,}~_q||p|. Therefore_|Br ~ \ ^ BξiFjpp, q,~q|epρ´δq|p| ď 1 eδ|B r ~ \ ^ BξiFjpp, q,~q|e ρ|p| _{and (5.18) follows.}

(5) is an easy extension of (5.10). Indeed we find immediately, by (5.7) and the fact that li´j depends only on i ´ j, that the Fourier transform of the symbol of Pl is

be

P_{pp, q,}~_{q “} Xbe_{lqpp, q,}~_{q where} Xbe_lq is the Fourier transform of the symbol of the oper-ator Xlq “ rVl,Vlqs i~ . Therefore |B r ~Xbelqpp, q,~q| ď max l“1...m|B r ~Xbelpp, q,~q|, @r ě 0, q P Z l_{. So} }Pl}ρ_´δ,k ď max l1_“1...m}rVl, Vl1s{i~}ρ´δ,k and }P }ρ´δ,kď max l,l1_“1...m}rVl, Vl1s{i~}ρ´δ,k ď m ÿ l,l1_“1 }rVl, Vl1s{i~}_ρ´δ,k

and we conclude by using (5.10). _

6. Fundamental iterative estimates: Brjuno condition case In all this section the norm subscripts ω and k are omitted.

Let us recall from Sections 2 and 3 that we want to find Wr such that

eiWr~ pH

r` Vrqe´i

Wr

~ “ H

r`1` Vr`1 (6.1)

where Hr<sub>`1</sub> “ Hr` hr_{`1</sub> and Hr“ BrpLωq, hr<sub>`1} “ Vr “ DrpLωq and, for 0 ă δ ă ρ ă 8,

}hr`1}ρ “ }Vr}ρď }Vr}ρ, }Vr`1}ρ´δ ď Dr}Vr}2ρ, (6.2)

and that we look at Wr solving:

1

i~rHr, Wrs ` V

co,r

“ Vco,r_{` bV}r

with

Vco,r _{“ V}r´ VMr.

VMr _{is given by}

VMr

ij “ pVrqij if |i ´ j| ą Mr, VijMr “ 0 otherwise. (6.4)

(note that Vco,r _{“ V}r_).

b Vr _{“ p bV}r l ql“1...m is given by p bVlrqij “ pr eVr l , eVlrpi´jqsqij i~ωl_pi´jq¨ pi ´ jq , Veijr :“ pI ` Arpi, jqq´1Vijco, Vco,r “ Vr´ VMr, (6.5)

where Ar_{pi, jq is the matrix given by Lemma 7, that is:}

B_rp~ω_{¨ iq ´ B}rp~ω¨ jq i~ “ pI ` A r pi, jqq ω.pi ´ jq. Let Zk“ 2pk ` 1q8k. (6.6)

Let us denote adW the operator H ÞÑ rW, Hs. The l.h.s. of (6.1) is then:

Hr` Vr` 1 i~rHr, Wrs ` 8 ÿ j<sub>“1</sub> 1 p´i~<sub>q</sub>j<sub>j!</sub>ad j WrpVrq ` 8 ÿ j_“2 1 p´i~_qj_j!ad j WrpHrq that is Hr` Vco,r` bVr` Vr´ Vco,r` 8 ÿ j_“1 1 p´i~_qj_j!ad j WrpVrq ` 8 ÿ j<sub>“2</sub> 1 p´i~<sub>q</sub>j<sub>j!</sub>ad j WrpHrq. or Hr` hr`1` pVr´ Vco,rq ` bVr` R1 ` R2 (6.7) Let us set Vr`1 :“ pVr´ Vco,rq ` bVr` R1` R2. (6.8)

We want to estimate Vr`1. We first prove the following proposition.

Proposition 17. Let W be in Jkpρq and 0 ă δ ă ρ. Then

}rHr, Ws{i~}ρ´δď 1 eδpω ` Zk}∇pB ~ r ´ B ~ 0q}ρq}W }ρ. (6.9) and for d<sub>ě 2,</sub> 1 d!}rHr, Wloomoons, . . . d times s{pi~<sub>q</sub>d }ρ´δď δω 2πZkp1 ` Z k}∇pB ~ r´ B ~ 0q}ρq ˆ Zk δ2 r ˙d }W }d ρ (6.10)

Let now Wr be the (scalar) solution of (6.3). Then, we have

}Wr}ρď M_Mr 1_{´ Z}k}DpBr~´ B ~ 0q}ρ}V r}ρ, (6.11)

and therefore for d_{ě 2,} 1 d!}rHr, Wlo omo onrs, . . . d times s{pi~_qd_}ρ_´δ ď ω 1_{` Z}k}DpBr~´ B ~ 0q}ρ 2πZk{δ ˆ ZkMMr{δ 2 r 1_{´ Z}k}DpB~r ´ B ~ 0q}ρ}V r}ρ ˙d (6.12) (MM is defined in (1.3) and }DB}ρ is meant for max

i_“1...m

ř

j“1...m}∇

iBj}ρ).

Proof. We first prove (6.9). Note that the proof is somehow close to the proof of Proposition 16, items (1) and (2). Since B0 pLωq “ Lω, (5.12) reads }rH0, Wrs{i~}ρ´δ ď ω eδ}Wr}ρ. (6.13) Note that _prHr´ H0, Wrs{~qij “ Gr pω.i~_q´Gr_pω.j~_q ~ Wij where G r pY q “ B~ rpY q ´ Y, Y P Rm

(note that Gr _{has an explicit dependence in~that we omit to avoid heaviness of notations).}

Indeed, since each Lωi is self-adjoint on L

2_pTl_{q, B}~

rpLωq can be defined by the spectral

theorem. Hence, we have

rB~ rpLωq, W sij “ pei,rB ~ rpLωq, W sejq “ pei, B ~ rpLωqW ej ´ W B ~ rpLωqeiq “ pB~ rpω.i~q ´ B ~ rpω.j~qqpei, W ejq.

Using (5.13) we get that

}rHr´ H0, Wrs{i~}ρ_´δ ď }Xr}ρ_´δ. (6.14)

where Xr is defined throughpXrqij “

Gr_pω.i~q´Gr_pω.j~q

~ pWrqij.

In order to estimate the norm of Xr _{we need to express its symbol X}

r. This is done

thanks to formula (5.7) and the fact that we know the matrix elements of Xr.

Expressing _pXrqij as a function ofppi ` jq~{2, i ´ jq through i, j “ i`j₂ ˘i´j₂ and using (5.7)

we get that f X_{rpξ, q,}~_{q “} G r pω.ξ ` ω.q~_{{2q ´ G}r pω.ξ ´ ω.q~_{2q ~ Wfr1pω.ξ, q,~q :“ fXr1pω.ξ, q,~q,

so that, using (remember that we denote p.ω.q_“ ř

j_“1...m ř i_“1...l pjωjiqi) ż RmpG r

pΞ ` ω.q~_{{2q ´ G}r_{pΞ ´ ω.q}~_{2qqe´iăΞ,pądp_{“ 2 sin rp.ω.q}~_{2s

ż Rm Gr_pΞqe´iăΞ,pądp, c f X1 rpp, q,~q “ ż Rm b G_ir_{pp ´ p}1_qsinrpp ´ p1q.ω.q~{2s ~ c f W1_rpp1_{, q,~qdp.} Therefore_}Xr}ρ´δ is equal to

m ÿ i_“1 ÿ qPZl ż R2m dpdp1_|2 k ÿ γ“1 µk´γpp, qqBγ~ „ b G_ir_{pp ´ p}1_qsinrpp ´ p 1_q.ω.q~{2s ~ c_f W_rpp1, q,~_q  |epρ´δqpω|p|`|q|q ď m ÿ i<sub>“1</sub> ÿ qPZl ż k ÿ γ<sub>“1</sub> µk´γpp, qq γ ÿ µ<sub>“1</sub> γ<sub>´µ</sub> ÿ ν<sub>“1</sub> ˆγ µ ˙ˆγ ´ µ ν ˙ ˆ ˆ2|Bγ~´µ´νGb r ipp ´ p1q||B ν ~ sin<sub>ppp ´ p</sub>1<sub>q.ω.q~{2q</sub> ~ ||B µ ~Wcfrpp1, q,~q|epρ´δqpω|p|`|q|qdpdp1 (6.15) (6.16) Using now the inequalities (5.21) and (5.22), we get,

| k ÿ γ_“1 µk_´γpp, qq γ ÿ µ_“1 γ_´µ ÿ ν_“1 ˆγ µ ˙ˆγ ´ µ ν ˙ B~γ´µ´νGb r ipp ´ p1qB ν ~ sinppp ´ p1_q.ω.q~q ~ B µ ~ c_f W_rpp1, q,~_q| ď ω max_j “1...m|pj´p 1 j||q| k ÿ γ“1 µk´γpp, qq γ ÿ µ“1 γ´µ ÿ ν“1 ˆγ µ ˙ˆγ ´ µ ν ˙ |Bγ~´µ´νGb r ipp´p1q||pp´p1q.ω.q|ν|B µ ~ c_f W_rpp1, q,~_q|.

Therefore we notice (after changing q_{Ø q}1_{) that}X}

r}ρ´δ is majored by the maximum over

~_{P r0, 1s of} k ÿ γ_“0 ż R2m ÿ pq,q1_qPZ2l µk_´γpp, qqω max j_“1...m|pj ´ p 1 j|PpG r i, Wrqepρ´δqpω|p|`|q|qdpdp1 (6.17)

where P is defined in (5.27) and

G_ir_{pΞ, xq “ Gi}r_{pΞq so that} cfGr

ipp, qq “ cG r

ippqδq_“0.

Therefore we can verbatim use the argument contained between formulas (5.27)-(5.29) and we arrive, in analogy with (5.30), to the fact that p1 ` kq´1₈´k_}X

r}ρ_´δ is majored by the maximum over ~ _{P r0, 1s of} ÿ q,q1_PZl ż R2m ω max j“1...m|pj||q 1_| k ÿ γ,γ1_“0 µk´γpp, qq|B~γ c f Gr ipp, qq|µk´γ1pp1, q1q|Bγ 1 ~ c f W_rpp1_{, q}1_q|epρ´δqpω|p|`ω|p1|`|q|`|q1|q<sub>dpdp</sub>1 “ ÿ q1<sub>P</sub>Zl ż R2m ω max j“1...m|pj||q 1<sub>|</sub> k ÿ γ,γ1<sub>“0</sub> µk´γpp, 0q|B~γGcirppq|µk´γ1pp1, q1q|Bγ 1 ~ c f W<sub>rpp</sub>1, q1<sub>q|e</sub>pρ´δqpω|p|`ω|p1|`|q1|qdpdp1 Since_{| b}Gr

ippq||pj| “ | bGirppqpj| “ |∇\jGirppq|, we get that (use ρ´δ ď ρ and again |q1|e´δ|q

1_| ď 1 eδ) }Xr}ρ´δ ď p1 ` kq8 k_ω eδ }∇G r }ρ}Wr}ρď Zkω eδ }∇pB ~ r ´ B ~ 0q}ρ}Wr}ρ. (6.18) Here _}∇pB~ r ´ B ~

0q}ρ is understood in the sense of (5.17)-(4.18).

We will prove (6.10) by the same argument as in the proof of Proposition 16. Take (5.35) with G1 :“ _i1~rWr, Hrs, Gs`1 “ 1 i~rWr, Gss and γs “ d´s d δ for 1 ď s ď d ´ 1, γ0 “ δ, γd<sub>´1</sub> “ δ<sub>d</sub>. We get 1 d!}Gd}ρ´γ0 ď Z<sub>k</sub>d´1 d!e2pd´1q<sub>γ0¨ ¨ ¨ γd</sub> ´2pδdqd´1 }Wr}dρ´1}G1}ρ<sub>´γ</sub>d´1 ď Z d<sub>´1</sub> k d!d!e2pd´1q<sub>p</sub>δ dq2d´2 }Wr}dρ´1}G1}ρ<sub>´δ{d</sub> ď 1` Zk}∇pB ~ r ´ B ~ 0q}ρ d!d!e2d_´1pδ dq2d´1 Z_kd´1_}Wr}dρ ď _2πd2δ_e´1_Zkp1 ` Zk}∇pB ~ r ´ B ~ 0q}ρq ˜ dd_e´d?_2πd d! ¸2 ˆ Zk δ2 ˙d }Wr}dρ and we get (6.10) by dd_e´d?_2πd d! ď 1 and d 2_e´1 ě 1 if d ě 2, and setting ρ “ ρr, γ0 “ γr.

In order to prove (6.11) we first estimate _{} e}Vr_}

ρr defined by (6.5) where A

r_{pi, jq is given}

by (7.5).

Lemma 18. Let V1 _{be defined by V}1

ij “ Arpi, jqVijco. Then }V1}ρď Zk}DpB ~ r´ B ~ 0q}ρ}Vco}ρ. (6.19)

Proof. The proof will actually be close to the one of (6.9). Ar

pi, jqω.pi´jq “ Grpω.j~q´Gpω´i~ ~q

so Ar_{pi, jq “} ż1 0 ∇Gr_{pp1 ´ tqω.j}~_{` tω.i</sub>~<sub>qdt.</sub> Therefore V1 <sub>“</sub> 1 ż 0 m ÿ n<sub>“1</sub> ∇<sub>n</sub>Gr<sub>pp1 ´ tqω.j</sub>~<sub>` tω.i}~_qVco n dt so }V1} ď sup 0_ďtď1 m ÿ n“1 }∇nGrpp1 ´ tqω.j~` tω.i~qVnco}. Let Xr n be defined through pXr nqij “ ∇nGrpp1 ´ tqω.j~` tω.i~qpVncoqij “ ∇nGrpω ¨ i_{` j} 2 ~´ p1 ´ 2tqpi ´ jq ~ 2qpV co n qij

By the argument as before, using (5.7), we get that the symbol of Xr _{satisfies fX}r

npξ, qq “ ∇_nGr_{pω ¨ ξ ´ p1 ´ 2tqq}~ 2q fV co n pξ, qq “pX^nrq1pω ¨ ξ, qq :“ ∇nGrpω ¨ ξ ´ p1 ´ 2tqq ~ 2q ^ pVco n q1pω ¨ ξ, qq since Vco

n has the same structure as V so there exists pVncoq1 such that Vncopξ, xq “ pVncoq1pω ¨

Taking now the Fourier transform of_pX^r

nq1pΞ, qq with respect to Ξ one gets by

translation-convolution \ ^ pXr nq1pp, q,~q “ ż Rm \ ∇_nGr_{pp ´ p}1_qeipp´p1q.ω.qp1´2tq~_{2Vcf_co n pp1, q,~qdp1. So, as before, |B~γ \ ^ pXr nq1pp, q,~q| ď ż γ ÿ µ_“1 γ´µ ÿ ν_“1 |B~γ´µ´ν∇\nGrpp ´ p1qB ν ~e i_pp´p1_{q.ω.qp1´2tq}~_{2 B~µ c f Vco n pp1, q,~q| ˆγ µ ˙ˆγ ´ µ ν ˙ dp1 ď ż Rm γ ÿ µ_“1 γ_´µ ÿ ν_“1 |B~γ´µ´ν∇\nGrpp ´ p1qp|p ´ p1||q|{2q ν B~µ c f Vco n pp1, q,~q| ˆγ µ ˙ˆγ ´ µ ν ˙ dp1

Following the same lines than in the proof of (6.9) we get that (remember that, by definition, }Xr_} ρ:“ m ř n_“1}X r n}ρ, }Xnr}ρ “ }pXnrq1}ρ by Definitions 11 and 10) }Xr}ρď Zk m ÿ i“1 m ÿ n“1 }∇nGir}ρ}Vnco}ρď Zkmax n }∇n Gr_}ρ m ÿ n“1 }Vnco}ρď Zk}∇Gr}ρ}Vco}ρ

and the Lemma is proved. 

Corollary 19. Let V2 _{defined by V}2

ij “ p1 ` Arpi, jqq´1Vijco. Then }V2}ρď 1 1_{´ Z}k}∇pBr~´ B ~ 0q}ρ}V co }ρ. (6.20)

(6.11) is now a consequence of (5.15) and the fact that _}Vco

}ρď }V }ρ.

(6.12) is obtained by putting (6.11) in (6.10). The proposition is proved. 

We need finally the following obvious Lemma: Lemma 20. Define VM_{px, ξq :“} ÿ |q|ěM e V_qpξqeiqx_{. (6.21)} Then }VM }ρ´δď e´δM}V}ρ (6.22)

Corollary 21. Let VM _{be defined by}

VijM “ Vij when |i ´ j| ě M

“ 0 when _{|i ´ j| ă M.} Then

Proof. Just notice that the symbol of VM_{, V}M_{, satisfies, by (5.7), eV}M

pξ, q,~_{q “ 0 when}

|q| ď M and apply Lemma 20. 

Let us define, for a decreasing positive sequence_pρrqr“0...8, ρr`1 “ ρr´ δr to be specified

later, Gr “ }DpB ~ r ´ B ~ 0q}ρr “ max i_“1...m ÿ j_“1...m }∇ipB ~ r ´ B ~ 0qj}ρr. (6.24)

We are now in position to derive the following fundamental estimates of the five terms in (6.7): }hr<sub>`1</sub>}ρr´δr ď }hr`1}ρr “ }V co,r }ρr “ }Vr}ρr (6.25) }Vr´ Vco,r}ρr´δr “ }V Mr }ρr´δr ď ˆ e´Mrδr }Vr_} ρr ˙ }Vr}2ρr (6.26) } bVr_}ρr´δr ď Zk M_Mr δ2 r p1 ´ ZkGrq2}V r}2ρr (6.27) }R1}ρr´δr ď Zk M_Mr δ2 rp1´ZkGrq 1_{´ Z}k M_Mr δ2 rp1´ZkGrq}V r_} ρr }Vr}2ρr (6.28) }R2}ρr´δr ď Zk M2 Mrωp1`ZkGrq δ3 rp1´ZkGrq2 1_{´ Z}k M_Mr δ2 rp1´ZkGrq}V r_} ρr }Vr}2ρr (6.29)

Indeed, _{p6.25q is obvious and p6.26q is nothing but Corollary 21.}

p6.27q is derived by using Proposition 16, item (5) equation (5.16), Lemma 19 and equation (6.5). Note that, as pointed out before, bVr _{is cut-offed as V}co,r _{thanks to (3.6).}

p6.28q is obtained through the definition R1 “

ř8

j_“1 1 p´i~_qj_j!ad

j

WrpVrq, the fact that, by

(5.11), _} 1 p´i~_qj_j!ad j WrpVrq}ρr´δr ď pZk{δ 2 rqj}Vr}ρr}Wr} j ρr and (6.11), so that }R1}ρr´δr ď 8 ÿ j_“1 pZk{δr2q j }Vr}ρr ˆ _M Mr 1_{´ Z}kGr}V r}ρr ˙j .

p6.29q is proven by the definition R2 “ 8 ř j“2 1 p´i~_qjj!ad j

WrpHrq and the fact that, by (6.12) we

have that_} 1 p´i~_qjj!ad j WrpHrq}ρr´δr ď ω 1`ZkGr Zk{δr ´_Z kMMr{δr2 1_´ZkGr }Vr}ρr ¯j .

Collecting all the preceding estimates together with the definition _{p6.8q :} Vr`1 :“ pVr´ Vco,rq ` bVr` R1` R2,

Proposition 22. For r_{“ 0, 1, . . . , we have} }Vr<sub>`1</sub>}ρr´δr ď Fr}Vr} 2 ρr ` e ´δrMr_}V r}ρr (6.30) with Fr “ M_MrZk δ2 rp1 ´ ZkGrq2 ˜ 1<sub>`</sub>p1 ´ ZkGrq ` M_Mr δr ωp1 ` ZkGrq 1_´ MMr 1_´ZkGr Zk δ2 r}Vr}ρr ¸ . (6.31)

7. Fundamental iterative estimates: Diophantine condition case In all this section also the norm subscripts ω and k are omitted.

Let 0_{ă δ ă ρ. Let us recall that we want to find W}r such that

eiWr~ pH

r` Vrqe´i

Wr

~ “ H

r`1` Vr`1 (7.1)

where Hr<sub>`1</sub> “ Hr` hr_{`1</sub> and Hr“ Br~pLωq, hr<sub>`1} “ Vr “ DrpLωq and

}hr_{`1</sub>}ρ “ }Vr}ρď }Vr}ρ, }Vr<sub>`1}}ρ_´δ ď Dr}Vr}2ρ. (7.2)

In the case where ω satisfies the Diophantine condition (1.4) we look at Wr solving:

1 i~rHr, Wrs ` Vr “ Vr` bV r (7.3) with bVr _{“ p b}V_lr_ql“1...m given by p bVlrqij “ pr eVr l , eVlrpi´jqsqij i~ωl_pi´jq¨ pi ´ jq , Veijr :“ pI ` Aεrpi, jqq´1pVrqij. (7.4)

Here Ar_{pi, jq is the matrix given by Lemma 7, that is:}

B~ rp~ω¨ iq ´ Brp~ω¨ jq i~ “ pI ` A r pi, jqq ω.pi ´ jq, (7.5) The l.h.s. of (7.1) is: Hr` Vr` 1 i~rHr, Wrs ` 8 ÿ j_“1 1 p´i~_qj_j!ad j WrpVrq ` 8 ÿ j<sub>“2</sub> 1 p´i~<sub>q</sub>j<sub>j!</sub>ad j WrpHrq that is Hr` Vr` bVr` 8 ÿ j“1 1 p´i~_qj_j!ad j WrpVrq ` 8 ÿ j“2 1 p´i~<sub>q</sub>j<sub>j!</sub>ad j WrpHrq. or Hr` hr<sub>`1</sub>` bVr` R1 ` R2 (7.6)

Proposition 23. Let Wr the (scalar) solution of (7.3).Then, for d ě 2, 0 ă δ ă ρ ă 8,

1 d!}rHr, Wlo omo onrs, . . . d times s{pi~_qd_{}ρ´δ ď} δω 2πZkp1 ` Z k}∇pB ~ r ´ B ~ 0q}ρq ˆ Zk δ2 ˙d }Wr}dρ_´δ (7.7)