Almost-Periodic Response Solutions for a Forced Quasi-Linear Airy Equation

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on $$\mathbb {T}^\infty $$
 
 
 T
 
 ∞
 
 .



Introduction
In this paper we study response solutions for almost-periodically forced quasilinear PDEs close to an elliptic fixed point.
The problem of response solutions for PDEs has been widely studied in many contexts, starting from the papers [24,25], where the Author considers a periodically forced PDE with dissipation. In the presence of dissipation, of course there is no small divisors problem. However as soon as the dissipation is removed, small divisors appear even in the easiest possible case of a periodic forcing when the spacial variable is one dimensional.
The first results of this type in absence of dissipation were obtained by means of a KAM approach [16,17,18,28,22,19]. However, a more functional approach, via a combination of a Ljapunov-Schmidt reduction and a Newton scheme, in the spirit of [24,25], was proposed by Craig-Wayne [14], and then generalized in many ways by Bourgain; see for instance [5,6,7] to mention a few. All the results mentioned above concern semi-linear PDEs and the forcing is quasi-periodic.
In more recent times, the Craig-Wayne-Bourgain approach has been fruitfully used and generalized in order to cover quasi-linear and fully nonlinear PDEs, again in the quasi-periodic case; see for instance [1,15,12,2].
Regarding the almost-periodic case, most of the classical results are obtained via a KAMlike approach; see for instance [10,23,9]. A notable exception is [8], where the Craig-Wayne-Bourgain method is used. More recently there have been results such as [26,27,20], which use a KAM approach. We mention also [29,11,3,4] which however are tailored for an autonomous PDE.
All the aforementioned results, concern semi-linear PDEs, with no derivative in the nonlinearity. Moreover they require a very strong analyticity condition on the forcing term. Indeed the difficulty of proving the existence of almost-periodic response solution is strongly related to the regularity of the forcing, since one can see an almost periodic function as the limit of quasiperiodic ones with an increasing number of frequencies. If such limit is reached sufficiently fast, the most direct strategy would be to iteratively find approximate quasi-periodic response solutions and then take the limit. This is the overall strategy of [23] and [26,27,20]. However this procedure works if one considers a sufficiently regular forcing term and a bounded nonlinearity, but becomes very delicate in the case of unbounded nonlinearities.
In the present paper we study the existence of almost-periodic response solutions, for a quasi-linear PDE on T. To the best of our knowledge this is the first result of this type.
Specifically we consider a quasi-linear Airy equation ∂ t u + ∂ xxx u + Q(u, u x , u xx , u xxx ) + f(t, x) = 0, x ∈ T := (R/(2πZ)) (1.1) where Q is a Hamiltonian, quadratic nonlinearity and f is an analytic forcing term with zero average w.r.t.
x. We assume f to be "almost-periodic" with frequency ω ∈ ℓ ∞ , in the sense of Definition 1.1. We mention that in the context of reducibility of linear PDEs a problem of this kind has been solved in [21]. Our aim is to provide a link between the linear techniques of [21] and the nonlinear Craig-Wayne-Bourgain method. Note that such a link is nontrivial, and requires a delicate handling; see below.
The overall setting we use is the one of [1]. However their strategy is taylored for Sobolev regularity; the quasi-periodic analytic case has been covered in [13]. Unfortunately the ideas of [13] cannot be directly applied in the almost-periodic case. Roughly, it is well known that the regularity and the small-divisor problem conflict. Thus, in the almost-periodic case one expect this issue to be even more dramatic. Specifically, we were not able to define a "Sobolev" norm for almost-periodic functions, satisfying the interpolation estimates needed in the Nash-Moser scheme; this is why we cannot use the theorem of [13].
Let us now present our main result in a more detailed way. First of all we note that (1.1) is an Hamiltonian PDE whose Hamiltonian is given by where G(u, u x ) is a cubic Hamiltonian density of the form G(u, u x ) := c 3 u 3 x + c 2 uu 2 x + c 1 u 2 u x + c 0 u 3 , c 0 , . . . , c 3 ∈ R (1. 3) and the symplectic structure is given by J = ∂ x . The Hamiltonian nonlinearity Q(u, . . . , u xxx ) is therefore given by Q(u, u x , u xx , u xxx ) = ∂ xx (∂ ux G(u, u x )) − ∂ x (∂ u G(u, u x )) (1.4) and the Hamilton equations are ∂ t u = ∂ x ∇ u H(u) .
We look for an almost-periodic solution to (1.1) with frequency ω in the sense below. For η > 0, define the set of infinite integer vectors with finite support as Z ∞ * := ℓ ∈ Z N : |ℓ| η := i∈N i η |ℓ i | < ∞ . (1.5) Note that ℓ i = 0 only for finitely many indices i ∈ N. In particular Z ∞ * does not depend on η.
Definition 1.1. Given ω ∈ [1,2] N with rationally independent components 1 and a Banach space (X, | · | X ), we say that F (t) : R → X is almost-periodic in time with frequency ω and analytic in the strip σ > 0 if we may write it in totally convergent Fourier series We shall be particularly interested in almost-periodic functions where X = H 0 (T σ ) is the space of analytic, real on real functions T s → C with zero-average, where T s := {ϕ ∈ C : Re(ϕ) ∈ T, |Im(ϕ)| ≤ s} is the thickened torus. We recall that a function u : T s → C is real on real if for any x ∈ T, u(x) ∈ R.
Of course we need some kind of Diophantine condition on ω. We give the following, taken from [9,21]. Definition 1.2. Given γ ∈ (0, 1), we denote by D γ the set of Diophantine frequencies We are now ready to state our main result.
then there is a Cantor-like set O (∞) ⊆ D γ with positive Lebesgue measure, and for all ω ∈ O (∞) a solution to (1.1) which is almost-periodic in time with frequency ω and analytic in a strip s (both in time and space).
Remark 1.4. Of course the same result holds verbatim if we replace the quadratic polynomial Q by a polinomial of arbitrary degree. We could also assume that the coefficients c j appearing in (1.4) depend on x and ωt. In that case Theorem 1.3 holds provided we further require a condition of the type sup j |∂ 2 x c j | S ≤ C. Actually one could also take Q to be an analytic function with a zero of order two. However this leads to a number of long and non particularly enlightening calculations.
To prove Theorem 1.3 we proceed as follows. First of all we regard (1.1) as a functional Implicit Function Problem on some appropriate space of functions defined on an infinite dimensional torus; see Definition 2.1 below. Then in Section 3 we prove an iterative "Nash-Moser-KAM" scheme to produce the solution of such Implicit Function Problem. It is well known that an iterative rapidly converging scheme heavily relies on a careful control on the invertibility of the linearized operator at any approximate solution. Of course, in the case of a quasi-linear PDE this amounts to study an unbounded non-constant coefficients operator. To deal with this problem, at each step we introduce a change of variables T n which diagonalizes the highest order terms of the linearized operator. An interesting feature is that T n preserves the PDE structure. As in [13] and differently from the classical papers, at each step we apply the change of variables T n to the whole nonlinear operator. This is not a merely technical issue. Indeed, the norms we use are strongly coordinate-depending, and the change of variable T n that we need to apply are not close-to-identity, in the sense that T n − Id is not a bounded operator small in size.
In Section 4 we show how to construct the change of variables T n satisfying the properties above. Then in order to prove the invertibility of the linearized operator after the change of variables T n is applied, one needs to perform a reducibility scheme: this is done in Section 5. For a more detailed description of the technical aspects see Remark 3.2. Acknowledgements. Riccardo Montalto is supported by INDAM-GNFM.

Functional setting
As it is habitual in the theory of quasi-periodic functions we shall study almost periodic functions in the context of analytic functions on an infinite dimensional torus. To this purpose, for η, s > 0, we define the thickened infinite dimensional torus T ∞ s as Given a Banach space (X, | · | X ) we consider the space F of pointwise absolutely convergent formal Fourier series T ∞ s → X and define the analytic functions as follows.
Definition 2.1. Given a Banach space (X, | · | X ) and s > 0, we define the space of analytic functions T ∞ s → X as the subspace We denote by H s the subspace of H(T ∞ s , H 0 (T s )) of the functions which are real on real. Moreover, we denote by H(T ∞ s × T s ), the space of analytic functions T ∞ s × T s → C which are real on real. The space H s can be identified with the subspace of zero-average functions of Throughout the algorithm we shall need to control the Lipschitz variation w.r.t. ω of functions in some H(T ∞ s , X), which are defined for ω in some Cantor set. Thus, for O ⊂ O (0) we introduce the following norm.
Linear operators. For any σ > 0, m ∈ R we define the class of linear operators of order m (densely defined on L 2 (T)) B σ,m as B σ,m := R : (2.4) and for T ∈ H(T ∞ σ , B σ,m ) we set In particular we shall denote by · Ω σ,m the corresponding Lipshitz norm. Moreover if m = 0 we shall drop it, and write simply · σ or · Ω σ .

The iterative scheme
Let us rewrite (1.1) as where we f(t, x) = f (ωt, x) and , as custumary the unknown u is a function of (ϕ, x) ∈ T ∞ × T. We introduce the (Taylor) notation where the constant C depends clearly on |c 0 |, . . . , |c 3 |. In particular, this implies that for all u ∈ H s one has the following.
1. There exist a sequence of Cantor sets O (n) ⊆ O (n−1) , n ≥ 1 such that 2. For n ≥ 1, there exists a sequence of linear, invertible, bounded and symplectic changes of variables defined for ω ∈ O (n−1) , of the form
3. For n ≥ 0, there exists a sequence of functionals F n (u) ≡ F n (ω, u(ω)), defined for ω ∈ O (n−1) , of the form F n (u) = f n + L n u + Q n (u), (3.14) such that (a) L n is invertible for ω ∈ O (n) and setting there exists r n = r n (ϕ) ∈ H(T ∞ s n−1 −3σ n−1 ) such that is a given function satisfying (c) L n is a linear operator of the form and for n ≥ 1 (3.20) with the coefficients q and

Remark 3.2.
Let us spend few words on the strategy of the algorithm. At each step we apply an affine change of variables translating the approximate solution to zero; the translation is not particularly relevant and we perform it only to simplify the notation. On the other hand the linear change of variables is crucial.
In (3.14) we denote by f n the "constant term", by L n is the "linearized" term and by Q n the "quadratic" part. In this way the approximate solution at the n-th step is h n = −L −1 n f n . In a classical KAM algorithm, in order to invert L n one typically applies a linear change of variables that diagonalizes L n ; this, together with the translation by h n is the affine change of variables mentioned above, at least in the classical KAM scheme.
Unfortunately, in the case of unbounded nonlinearities this cannot be done. Indeed in order to diagonalize L n in the unbounded case, one needs it to be a pseudo-differential operator. On the other hand, after the diagonalization is performed, one loses the pseudo-differential structure for the subsequent step. Thus we chose the operators T n in (3.12) in such a way that we preserve the PDE structure and at the same time we diagonalize the highest order terms.
In the [1]-like algorithm the Authors do not apply any change of variables, but they use the reducibility of L n only in order to deduce the estimates. However such a procedure works only in Sobolev class. Indeed in the analytic case, at each iterative step one needs to lose some analyticity, due to the small divisors. Since we are studying almost-periodic solutions, we need the analytic setting to deal with the small divisors. As usual, the problem is that the loss of the analyticity is related to the size of the perturbation; in the present case, at each step L n is a diagonal term plus a perturbation O(ε 0 ) with the same ε 0 for all n.
A more refined approach is to consider L n as a small variation of L n−1 ; however the problem is that such small variation is unbounded. As a consequence, the operators T n are not "close-toidentity". However, since F n is a differential operator, then the effect of applying T n is simply a slight modification of the coefficients; see (3.20) and (3.22). Hence there is a strong motivation for applying the operators T n . In principle we could have also diagonalized the terms up to order −k for any k ≥ 0; however the latter change of variables are close to the identity and they introduce pseudo-differential terms.

The zero-th step
we have the following result.
Then there exists a unique solution h 0 ∈ H s of (3.27) . Moreover one has for some constant τ = τ (η) > 0.
Remark 3.4. Note that from Lemma 3.3 above it follows that there is C 0 such that a solution h 0 of (3.27) actually satisfies where we recall that by (3.6), µ > max{1, 1 η }. Of course the constant C 0 is correlated with the correction to the exponent 1 η .

From Lemma 3.3 and (3.27) it follows that
proving item 4. for |f 0 | S small enough, which is true by (3.7).

The n + 1-th step
Assume now that we iterated the procedure above up to n ≥ 0 times. This means that we arrived at a quadratic equation By the inductive hypothesis (3.22) we deduce that for all 0 < s − σ < s n−1 − 3σ n−1 one has Moreover, again by the inductive hypothesis, we can invert L n and define h n by (3.15). Now we set and r n+1 are to be chosen in order to ensure that L n+1 := F ′ n+1 (0) has the form (3.18) with n n + 1.
Of course by Taylor expansion we can identify (3.34) Remark 3.5. Note that the last equality in (3.34) follows from the fact that the nonlinearity Q in (1.1) is quadratic. In the general case, the last term is controlled by the second derivative, and thus one has to assume a bound of the type (3.31) for Q ′′ .
In section 4 we prove the following such that (3.33) is well defined and symplectic as well as its inverse, and moreover and (3.19) and (3.20) hold with n n + 1.
The assumption (3.35) follows from (3.10), provided that we choose the constants τ, C and ǫ 0 appropriately.
We now prove (3.21) and (3.22) for n n + 1, namely the following result.
Lemma 3.7. One has (3.39) Proof. By construction Now we first note that Hence the terms ∂ i x T n+1 v are of the form In order to obtain the bound (3.43) we used the first line of (3.22) to control the sums appearing in (3.42). Finally , since the bound follows.
In Section 5 we prove the existence of a Cantor set O (n+1) where item 3.(a) of the iterative lemma holds with n n + 1.
such that setting The assumption (3.45) follows from (3.10), provided that we choose the constants τ, C and ǫ 0 appropriately. Remark 3.9. Note that in the context of [13] Proposition 3.8 is much simpler to prove, because in order to diagonalize the linearized operator one uses tame estimates coming from the Sobolev regularity on the boundary of the domain. Then the smallness conditions are much simpler to handle. Here we have to strongly rely on the fact that L n+1 is a "small" unbounded perturbation of L n in order to show that the operators M (n) and M (n+1) are close to each other. This is a very delicate issue; see Lemma 5.2 and Section 5.3, which are probably the more technical parts of this paper.
Proof. The result follows simply by using the definition of O (n+1) and applying Lemma A.7.
Of course from Lemma 3.10 it follows that, Now we want to show inductively that for ε 0 small enough. By the definition of ε n in (3.6), (3.54) is equivalent to Since the r.h.s. of (3.55) admits a positive minimum, we can regard it as a smallness condition on ε 0 , which is precisely (3.10).
Proof. The proof is identical to the one for Lemma 6.2 in [21], simply replacing j 2 with j 3 .
We now the convergence of the scheme. Precisely we show that the series (3.25) converges totally in H s . Note that Thuse, using (3.60) into (3.25) we get we deduce that u ∞ ∈ H s . Finally by continuity so the assertion follows since (recall s := s ∞ − n≥1 σ n and (3.62)) We finally conclude the proof of Proposition 3.1 by showing that (3.26) holds.
First of all, reasoning as in Lemma 3.12 and using Lemma A.8, we see that so that (3.26) follows by (3.11).

Proof of Proposition 3.6
In order to prove Proposition 3.6, we start by dropping the index n, i.e. we set L ≡ L n (see (3.18)) and Q ≡ Q ′ n (h n ) (see (3.34)). More generally, we consider a Hamiltonian operator of the form defined for all ω ∈ Ω ⊆ D γ and λ 3 , a 0 , a 1 , d 0 . . . , d 3 satisfy the following properties.
1. There is δ 0 small enough such that 2. There is ρ > 0 such that a i ∈ H(T ∞ ρ × T ρ ) and and moreover i.e. it does not depend on ϕ.
Let us now choose ζ such that 0 < ζ ≪ ρ and for some τ ′ > 0. We shall conjugate L (0) to a new operator 1 r L + with r = r(ϕ) an explicit function with with the coefficients satisfying |λ + 3 − λ 3 | Ω δ (4.8) and This will allow us to conclude the proof of Proposition 3.6.

Elimination of the x-dependence from the highest order term
Consider an analytic function α(ϕ, x) (to be determined) and let We choose α(ϕ, x) and m 3 (ϕ) in such a way that which implies (4.11) By (4.2), (4.5) and Lemma A.5 one has Note that for any 0 < ζ ≪ ρ such that δζ −1 ≪ 1, by Lemma A.1, x → x + α(ϕ, x) is invertible and the inverse is given by y → y + α(ϕ, y) with A direct calculations shows that A −1 u(ϕ, y) = u(ϕ, y + α(ϕ, y)) , T −1 and the following conjugation rules hold: Clearly one can get similar conjugation formulae for higher order derivatives, having expression similar to (3.41). In conclusion (4.17)

Elimination of the ϕ-dependence from the highest order term
We now consider a quasi periodic reparametrization of time of the form where β : T ∞ ρ−ζ → R is an analytic function to be determined. Precisely we choose λ + 3 ∈ R and β(ϕ) in such a way that obtaining thus where we recall the definition A.3. By the estimates (4.12) and by Lemma 3.3, one obtains that By Lemma A.1 and (4.6) we see that ϕ → ϕ + ωβ(ϕ) is invertible and the inverse is given by The inverse of the operator T 2 is then given by

Time dependent traslation of the space variable
Let p : T ∞ ρ−2ζ → R be an analytic function to be determined and let Computing explicitly and by (4.4) one has (4.29) We want to choose p(ϕ) in such a way that the x-average of d 1 is constant. To this purpose we define where for any a : T ∞ σ × T σ → C, a ϕ,x is defined by

Proof of Proposition 3.8
In order to prove Proposition 3.8, we start by considering a linear Hamiltonian operator defined for ω ∈ O ⊆ D γ of the form We want to show that, for any choice of the coefficients λ 3 , a 1 , a 0 satisfying some hypotheses (see below), it is possible to reduce L to constant coefficients. Moreover we want to show that such reduction is "Lipshitz" w.r.t. the parameters λ 3 , a 1 , a 0 , in a sense that will be clarified below.

Reduction of the first order term
We consider an operator L of the form (5.1) satisfying the hypotheses above. We start by showing that it is possible to reduce it to constant coefficients up to a bounded reminder, and that such reduction is "Lipshitz" w.r.t. the parameters λ 3 , a 1 , a 0 .
We now consider a "small modification" of the operator L in the following sense. We consider an operator Of course we can apply Lemma 5.1 and conjugate L + to with R + 0 a bounded operator. We want to show that L + 0 is "close" to L 0 , namely the following result.

Reducibility
We now consider an operator L 0 of the form with P 0 a bounded operator and and we show that, under some smallness conditions specified below it is possible to reduce it to constant coefficients, and that the reduction is "Lipschitz" w.r.t. the parameters λ 1 , λ 3 , P 0 . In order to do so, we introduce three sequences 0 < . . . < ρ m < . . . < ρ 0 , 0 < . . . ≪ δ m ≪ . . . ≪ δ 0 and 1 ≪ N 0 ≪ N 1 ≪ · · · and we assume that setting and for some τ, C > 0. We have the following result.
For k = 0, . . . , m there is a sequence of sets E k ⊆ E k−1 and a sequence of symplectic maps Φ k defined for ω ∈ E k+1 such that setting L 0 as in (5.20) and for k ≥ 1, one has the following.
1. L k is of the form with r 0 (j) = 0 and for k ≥ 1, r k (j) is defined for ω ∈ E 0 = O and satisfies • The operator P k is such that The sets E k are defined as ∀(ℓ, j, j ′ ) = (0, j, j), |ℓ| η ≤ N k−1 .

(5.32)
Proof. The statement is trivial for k = 0 so we assume it to hold up to k < m and let us prove it for k + 1. For any Φ k := exp(Ψ k ) one has where the operator P k+1 is defined by Then we choose Ψ k in such a way that namely for ω ∈ E k+1 we set , ∀(ℓ, j, j ′ ) = (0, j, j), |ℓ| η ≤ N k , 0 otherwise.
and by applying Lemma A.6, using the induction estimate (5.30), one obtains for any ζ < ρ k . We now define the diagonal part D k+1 .
For any j ∈ Z\{0} and any ω ∈ E k one has |(P ≤ δ k . The Hamiltonian structure guarantees that P (k) k (0) j j is purely imaginary and by the Kiszbraun Theorem there exists a Lipschitz extension ω ∈ O → iz k (j) (with z k (j) real) of this function satisfying the bound |z k (j)| O δ k . Then, we define which is the estimate (5.29) at the step k + 1. We now estimate the remainder P k+1 in (5.34). Using (5.35) we see that . . , m .
. We estimate separately the four terms in the definition of P (k+1) k+1 in (5.42). By Lemma A.9-(ii), one has By applying (A.7) and the estimate of Lemma A.9-(iii), one obtains In conclusion we obtained where C ′ is an appropriate constant and the last summand is a bound for the term e −Ψ k P (k+1) k e Ψ k , which can be obtained reasoning as in (5.43). Thus we obtain which is of course follows from (5.23).
Now that we reduced L 0 to the form L m = ω · ∂ ϕ + D m + P m we can apply a "standard" KAM scheme to complete the diagonalization. This is a super-exponentially convergent iterative scheme based on iterating the following KAM step.
We now iterate the step of Lemma 5.4, using at each step a smaller loss of analyticity, namely at the p-th step we take ζ p with p≥m+1 ζ p = ζ, so that we obtain the following standard reducibility result; for a complete proof see [21].

Variations
We now consider an operator where L, λ 1 , λ 3 , P 0 are given in (5.21) and O + ⊆ O. In other words, L + 0 is a small variation of L 0 in (5.20) with also m m + 1. Of course we can apply Proposition 5.5 to L + 0 ; our aim is to compare the "final frequencies" of L + ∞ with those of L ∞ . To this aim, we first apply Lemma 5.3 with L 0 L + 0 and γ γ + < γ. In this way we obtain a sequence of sets E + k ⊆ E + k−1 and a sequence of symplectic maps Φ + k defined for ω ∈ E + k+1 such that setting L + 0 as in (5.60) and one has The sets E + k are defined as E + 0 := O + and for k ≥ 1 The following lemma holds.
For k = m + 1 the proof can be repeated word by word, the only difference being that Ψ m is defined in (5.52) while Ψ + m is defined in (5.36) with k = m.

Conclusion of the proof
To conclude the proof of Proposition 3.8 we star by noting that, setting O appearing in (5.2) as O (n) appearing in (3.11), the operator L n+1 appearing in (3.18) with of course n n + 1 is of the form (5.1) with where s k , σ k and ε k are defined in (3.6), so that L n+1 satisfies (5.2) with m = n. Thus, fixing the smallness conditions (5.3) follows by definition. Hence we can apply Lemma 5.1 to L n+1 obtaining an operator of the form (5.5). In particular the conjugating operator M satisfies We are now in the setting of Section 5.2 with for some τ 3 > 0. A direct calculation shows that the smallness conditions (5.22), (5.23), (5.24), (5.49) are satisfied provided we choose N k appropriately, so that we can apply Proposition 5.5.
Finally (3.47) follows from Lemmata 5.2 and 5.6 where L + has the role of L n+1 while L has the role of L n . This means that here we are taking m n − 1.
Lemma A.5 (Moser composition lemma). Let f : B R (0) → C be an holomorphic function defined in a neighbourhood of the origin B R (0) of the complex plane C. Then the composition operator F (u) := f • u is a well defined non linear map H(T ∞ σ × T σ ) → H(T ∞ σ × T σ ) and if |u| σ ≤ r < R, one has the estimate |F (u)| σ 1 + |u| σ . If f has a zero of order k at 0, then for any |u| σ ≤ r < R, one gets the estimate |F (u)| σ |u| k σ .
Lemma A.7. Given u ∈ H(T ∞ σ , X) for X some Banach space, let g be a pointwise absolutely convergent Formal Fourier series such that for some τ ′ > 0. Then for any 0 < ρ < σ, then g ∈ H(T ∞ σ−ρ , X) and satisfies Proof. Follows directly from Lemma A.6 and Definition 2.1. Proof. First of all note that for all ℓ ∈ Z * ∞ one has .
Then we recall that (see [9]) By iterating the estimate (i) of Lemma A.9, one has that for any n ≥ 1 for some constant C > 0.
(i) For any k ≥ 1, one has Proof. Proof of item (i). We prove the statement by induction on k. For k = 1, one has that Since for j < i one has that ρ j > ρ i and so all the terms in the above sum are analytic at least in the strip of width ρ i . By applying Lemma A.9-(i) one has for any 0 < ζ i < ρ i . . , n. Now we argue by induction. Assume that for some k ≥ 1, . . , n for any 0 < ζ i < ρ i . Of course this implies that for all j < i one has
Hence by applying Lemma A.9-(i) and using the induction hypothesis, one obtains Proof of (ii). One has Therefore the assertion follows.
Lemma A.11. Let {ρ n } n≥0 and {δ n } n≥0 as in Lemma A.10. Choose any n > 0 and consider Then the following holds.
Lemma A.12. Let A, A + , B, B + be bounded operators w.r.t. a norm · σ , and define Then the following holds.
(i) For any k ≥ 0, one has Proof. Proof of (i). We argue by induction. Of course the result is trivial for k = 0. Assume that the estimate holds for some k ≥ 1. Then Hence, by the induction hypothesis, using (A.11), (A.7) and Lemma A.9-(i), one obtains that for some C * > 0 large enough. Proof of (ii). It follows by item (i), using that Lemma A.13. Let A, A + , B, B + be linear operators satisfying A ρ,−1 , A + ρ,−1 , B ρ,1 , B + ρ,1 < C 0 .
Then the following holds. In order to estimate Ad k A + (B + )− Ad k A (B) = Ad k−1 A + Ad A + (B + )− Ad k−1 A Ad A (B) for any k ≥ 2, we apply Lemma A.12-(i) where we replace B + with Ad A + (B + ) and B with Ad A (B), together with the estimates (A.12), (A.13).
Proof. We only prove the item (i). The item (ii) can be proved by similar arguments. We compute (A.14) Hence