Large KAM Tori for Quasi-linear Perturbations of KdV

Abstract

In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question of whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.

1 Introduction

The Korteweg–de Vries (KdV) equation

$$\begin{aligned} \partial _t u = - \partial _x^3u + 6 u \partial _x u \end{aligned}$$

(1.1)

is one of the most important model equations for dispersive phenomena with numerous applications in physics. The seminal discovery in the late sixties that (1.1) admits infinitely many conservation laws [28, 30], and the development of the inverse scattering transform method [18], led to the modern theory of infinite dimensional integrable systems (for example [13, 16] and references therein).

One of the most distinguished features of (1.1) is the existence of sharply localized traveling waves of arbitrarily large amplitudes and particle like properties. Kruskal and Zabusky, who discovered them in numerical experiments in the early sixties, both on the real line and in the periodic setup (cf. [24]), coined the name solitons for them. More generally, they found solutions, which are localized near finitely many points in space. In the periodic setup, these solutions are referred to as periodic multi-solitons or finite gap solutions. Due to their importance in applications, various stability aspects, in particular long time asymptotics, have been extensively studied. A major question concerns the persistence of the multi-solitons under perturbations. In the last thirty years, KAM methods pioneered by Kolmogorov, Arnold, and Moser to treat perturbations of integrable systems of finite dimension, were developed for PDEs. Most of the work focused on small amplitude solutions or semilinear perturbations. It has been a longstanding question from experts in PDEs and in infinite dimensional dynamical systems whether KAM results hold also for solutions of arbitrary size under quasi-linear perturbations, called strongly nonlinear in [26], of integrable PDEs.

The aim of this paper is to prove the first persistence result of periodic multi-solitons of KdV of arbitrary size under strongly nonlinear perturbations—see Theorem 1.1 below. Note that in this case, it was not even known if there exist solutions of the perturbed equation which are global in time.

To describe the class of perturbations of the KdV equation considered, we recall that (1.1), with space periodic variable x in ${\mathbb {T}}_1 \,{:=}\, {\mathbb {R}}/ {\mathbb {Z}}$, can be written in Hamiltonian form

$$\begin{aligned} \partial _t u = \partial _x \nabla {H}^{kdv}(u) , \quad {H}^{kdv}(u) \,{:=}\, \int _{{\mathbb {T}}_1} \frac{1}{2} (\partial _x u)^2(x) + u^3 (x) \, \hbox {d}x , \end{aligned}$$

(1.2)

where $\nabla H^{kdv}$ denotes the $L^2$-gradient of $H^{kdv}$ and $\partial _x$ is the Poisson structure, corresponding to the Poisson bracket, defined for functionals F, G by $\{ F , G \} \,{:=}\, \int _{{\mathbb {T}}_1} \nabla F \, \partial _x \nabla G \, \hbox {d}x$.

We consider quasi-linear perturbations of (1.1) of the form

$$\begin{aligned} \partial _t u = - \partial _x^3 u + 6 u \partial _x u + \varepsilon a(x, u(x), \partial _x u(x)) \partial _x^3u + \cdots , \end{aligned}$$

(1.3)

where $0< \varepsilon < 1$ is a small parameter and $\cdots$ comprises terms which are $\varepsilon$-small and contain x-derivatives of u up to second order. We assume that the perturbation is Hamiltonian, namely $\varepsilon a \partial _x^3u + \cdots = \varepsilon \partial _x \nabla P$, where $\nabla P$ is the $L^2$-gradient of a functional of the form

$$\begin{aligned} P (u) \,{:=}\! \int _{{\mathbb {T}}_1}\! f(x, u(x), \partial _x u(x))\, \hbox {d}x , \quad f\!: {\mathbb {T}}_1 \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\quad {\mathcal {C}}^\infty \text {-smooth}.\quad \end{aligned}$$

(1.4)

Note that the nonlinear vector field

$$\begin{aligned} \partial _x \nabla P ( u ) = - (\partial ^2_{u_x} f ) (x, u(x), \partial _x u(x) ) \partial _x^3 u + \cdots , \quad u_x \,{:=}\, \partial _x u \end{aligned}$$

(1.5)

has the same order as the linear one $\partial _x^3 u$ in (1.1). When written as a Hamiltonian PDE, (1.3) takes the form

$$\begin{aligned} \partial _t u = \partial _x \nabla {H}_\varepsilon (u) \end{aligned}$$

(1.6)

with Hamiltonian

$$\begin{aligned} H_\varepsilon (u) \,{:=}\, H^{kdv}(u) + \varepsilon P(u). \end{aligned}$$

(1.7)

To state our main result, we first need to introduce some more notation. Note that the mean $u \mapsto \int _{{\mathbb {T}}_1} u(x)\, \hbox {d}x$ is a prime integral for (1.6). We restrict our attention to functions with zero average (cf. Remark (R1) below) and choose as phase spaces for (1.6) the scale of Sobolev spaces $H_0^s ({\mathbb {T}}_1)$, $s \ge 0$,

$$\begin{aligned} H_0^s ({\mathbb {T}}_1) \,{:=}\, \Big \{ u \in H^s ({\mathbb {T}}_1) : \int _{{\mathbb {T}}_1} u(x)\, \hbox {d}x = 0 \Big \} , \quad L_0^2 ({\mathbb {T}}_1) \equiv H_0^0 ({\mathbb {T}}_1) , \end{aligned}$$

where

$$\begin{aligned} H^s({\mathbb {T}}_1) \,{:=}\, \Big \{ u(x) = \sum _{n \in {\mathbb {Z}}} u_n e^{\mathrm{i} 2 \pi n x} \, : \, \Vert u \Vert _{H^s_x} \,{:=}\, \left( \sum _{n \in {\mathbb {Z}}} \langle n \rangle ^{2 s} |u_n|^2 \right) ^{\frac{1}{2}} < \infty , \ u_{-n} = \overline{u_{n}}\quad \forall n \in {\mathbb {Z}}\Big \} \end{aligned}$$

(1.8)

and $\langle n \rangle \,{:=}\, \mathrm{max}\{ 1, |n| \}$ for any $n \in {\mathbb {Z}}$. We also write $L^2 ({\mathbb {T}}_1)$ for $H^0 ({\mathbb {T}}_1)$. The symplectic form on $L^2_0 ({\mathbb {T}}_1)$ is given by

$$\begin{aligned} {{\mathcal {W}}}_{L^2_0} (u, v) \,{:=}\! \int _{{\mathbb {T}}_1} \!(\partial _x^{- 1} u ) v\, \hbox {d}x , \quad \! \partial _x^{- 1} u = \sum _{n \ne 0} \frac{1}{\mathrm{i} n} u_n e^{\mathrm{i} 2 \pi n x}, \quad \! \forall u, v \in L^2_0({\mathbb {T}}_1) \end{aligned}$$

(1.9)

Note that the Hamiltonian vector field $X_H (u) = \partial _x \nabla H (u)$, associated with a Hamiltonian H, is determined by $\hbox {d} H (u)[ \cdot ] = {{\mathcal {W}}}_{L^2_0} ( X_H , \cdot )$.

${{\mathbb {S}}}_+$-gap potentials According to [21], the KdV equation (1.1) on ${\mathbb {T}}_1$ is an integrable PDE in the strongest possible sense, meaning that it admits globally defined canonical coordinates on $H^0_0({\mathbb {T}}_1)$, so that (1.1) can be solved by quadrature, see Theorem 3.1 in Section 3 for a precise statement. These coordinates, referred to as Birkhoff coordinates, are particularly suited to describe the finite gap solutions of KdV. Each of these solutions is contained in a finite dimensional integrable subsystem ${\mathcal {M}}_{{{\mathbb {S}}}_+}$, of dimension $2 |{{\mathbb {S}}}_+|$, with ${{\mathbb {S}}}_+$ being a finite subset of ${\mathbb {N}}_+ \,{:=}\, \{1, 2, \ldots \}$. The integrable subsystem ${\mathcal {M}}_{{{\mathbb {S}}}_+}$ can be described in terms of action angle coordinates $\theta \,{:=}\, (\theta _n)_{n \in {{\mathbb {S}}}_+}$, $I \,{:=}\, (I_n)_{n \in {{\mathbb {S}}}_+}$ as follows: there exists a real analytic, canonical diffeomorphism

$$\begin{aligned} \Psi _{{{\mathbb {S}}}_+} : {\mathbb {T}}^{{{\mathbb {S}}}_+}\! \times {\mathbb {R}}^{{{\mathbb {S}}}_+}_{> 0} \rightarrow {\mathcal {M}}_{{{\mathbb {S}}}_+}, (\theta , I) \mapsto q(\theta , \cdot ; I) , \quad {\mathbb {T}}^{{{\mathbb {S}}}_+} : = ({\mathbb {R}}/ 2\pi {\mathbb {Z}})^{{{\mathbb {S}}}_+}, \end{aligned}$$

(1.10)

(cf. (3.5)) so that the pull-back of the KdV Hamiltonian, $H^{kdv} \circ \Psi _{{{\mathbb {S}}}_+}$, is a real analytic function of the actions I alone. Elements in ${\mathcal {M}}_{{{\mathbb {S}}}_+}$ are referred to as ${{\mathbb {S}}}_+$-gap potentials. The function $q(\theta , x) \equiv q(\theta , x; I)$ is real analytic. In action angle coordinates, any solution of (1.1) on ${\mathcal {M}}_{{{\mathbb {S}}}_+}$ is given by

$$\begin{aligned} \theta (t) = \theta ^{(0)} - \omega ^{kdv} (\nu ) t , \quad I(t) = \nu , \end{aligned}$$

where $\theta ^{(0)} \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$ denotes the initial angles, $\nu \in {\mathbb {R}}^{{{\mathbb {S}}}_+}_{>0}$ the initial actions, and $\omega ^{kdv} (\nu )$ the frequency vector

$$\begin{aligned} \omega ^{kdv} (\nu ) \,{:=}\, \partial _I (H^{kdv} \circ \Psi _{{{\mathbb {S}}}_+}) (0, \nu ) \in {\mathbb {R}}^{{{\mathbb {S}}}_+} . \end{aligned}$$

(1.11)

(Cf. Section 3.1 for more details.) The corresponding finite-gap solution of (1.1) on ${\mathcal {M}}_{{{\mathbb {S}}}_+}$ is then given by

$$\begin{aligned} t \mapsto q\big ( \theta ^{(0)} - \omega ^{kdv}(\nu )t,\, x; \, \nu \big ) \end{aligned}$$

(1.12)

and hence is quasi-periodic in time. The map

$$\begin{aligned} {\mathbb {R}}^{{\mathbb {S}}_+}_{> 0} \rightarrow {\mathbb {R}}^{{{\mathbb {S}}}_+} , \ \nu \mapsto \omega ^{kdv} (\nu ) , \end{aligned}$$

(1.13)

is a local diffeomorphism (see Remark 3.10). In the entire paper, $\Xi \subset {\mathbb {R}}^{{{\mathbb {S}}}_+}_{> 0}$ will always denote an open, nonempty set with the property that the map $\Xi \rightarrow {\mathbb {R}}^{{{\mathbb {S}}}_+},\, \nu \mapsto \omega ^{kdv}(\nu )$, defined by (1.11), is a diffeomorphism onto its image and that its closure is a compact subset of ${\mathbb {R}}^{{{\mathbb {S}}}_+}_{>0}$. Then, for some $\delta > 0$ small enough,

$$\begin{aligned} \Xi \, + \, B_{{\mathbb {S}}_+}(\delta ) \, \subseteq \, {\mathbb {R}}^{{\mathbb {S}}_+}_{> 0} , \end{aligned}$$

(1.14)

where $B_{{\mathbb {S}}_+}(\delta )$ denotes the ball in ${\mathbb {R}}^{{\mathbb {S}}_+}$ of radius $\delta$, centered at the origin. Furthermore we introduce the Sobolev spaces of periodic, real valued functions $H^s \equiv H^s ( {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1, {\mathbb {R}})$,

$$\begin{aligned} \begin{aligned} H^s \,{:=}\, \Big \{&g = \sum _{(\ell ,j) \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} \times {\mathbb {Z}}} g_{\ell , j} \, e^{\mathrm{i} (\ell \cdot \varphi + 2 \pi jx)} \, : \, \ \overline{g_{\ell , j}} = g_{-(\ell , j)} , \ \forall \, (\ell ,j) \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} \times {\mathbb {Z}}, \\&\Vert g \Vert _s \,{:=}\, \Big ( \sum _{(\ell ,j) \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} \times {\mathbb {Z}}} | g_{\ell , j}|^2 \langle \ell ,j \rangle ^{2s} \Big )^{1/2} < + \infty \, , \quad \langle \ell ,j \rangle \,{:=}\, \max \{ 1, |\ell |, |j| \} \, \Big \} . \end{aligned}\nonumber \\ \end{aligned}$$

(1.15)

Note that by the Sobolev embedding theorem, $H^s \subset {\mathcal {C}}^0 ( {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1, {\mathbb {R}})$ for any $s > (|{{\mathbb {S}}}_+| + 1)/2$ where ${\mathcal {C}}^0 ( {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1, {\mathbb {R}})$ denotes the Banach space of continuous functions endowed with the supremum norm.

The main result of this paper is Theorem 1.1 below. It says that for $\varepsilon$ small enough and for $\nu$ in a subset $\Xi _\varepsilon$ of $\Xi$ of asymptotically full Lebesgue measure, there is a quasi-periodic solution of equation (1.6) close to the finite gap solution $q( - \omega ^{kdv} (\nu ) t, x; \nu )$ of (1.1). More precisely, the following holds:

Theorem 1.1

Let f be a function in ${{{\mathcal {C}}}}^{\infty }({\mathbb {T}}_1 \times {\mathbb {R}}\times {\mathbb {R}}, {\mathbb {R}})$, ${{\mathbb {S}}}_+$ a finite subset of ${\mathbb {N}}_+$, and ${{\mathfrak {b}}}$ a real number in (0, 1) . Then there exist ${\bar{s}} > (|{{{\mathbb {S}}}_+}| +1) / 2$, and $0< \varepsilon _0 < 1$ so that the following holds: there exists a decreasing family of measurable subsets $\Xi _\varepsilon \subseteq \Xi$, $0 < \varepsilon \le \varepsilon _0$, with asymptotically full measure, that is

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} |\Xi \setminus \Xi _\varepsilon | = 0 , \end{aligned}$$

(1.16)

with the property that for any $\nu \in \Xi _\varepsilon$, the perturbed KdV equation (1.6) admits a quasi-periodic solution $t \mapsto u_\varepsilon ( \omega _\varepsilon ( \nu ) t, x; \nu )$ with frequency vector $\omega _\varepsilon (\nu ) = - \omega ^{kdv}( \nu ) \in {\mathbb {R}}^{{{\mathbb {S}}}_+}$, where $u_\varepsilon (\cdot , \cdot \, ; \nu ) \in H^{{\bar{s}}}({\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1, {\mathbb {R}})$ and

$$\begin{aligned} \Vert u_\varepsilon (\cdot , \cdot \, ; \nu ) - q ( \cdot , \cdot \, ; \nu ) \Vert _{{\bar{s}}} \lesssim \varepsilon ^{1 - {{\mathfrak {b}}}} . \end{aligned}$$

(1.17)

Here, $q ( \theta , x; \nu )$, $\theta \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$, is the ${{\mathbb {S}}}_+$-gap potential in ${\mathcal {M}}_{{{\mathbb {S}}}_+}$, defined in (1.10). The quasi-periodic solution $t \mapsto u_\varepsilon ( \omega _\varepsilon (\nu ) t, x; \nu )$ is linearly stable.

We make the following remarks:

• (R0) Since the Hamiltonian vector field in (1.6) is autonomous, any translate $u_\varepsilon ( \omega _\varepsilon ( \nu ) t + \theta ^{(0)}, x; \nu )$, $\theta ^{(0)} \in {\mathbb {T}}^{{\mathbb {S}}_+}$, of $u_\varepsilon ( \omega _\varepsilon ( \nu ) t, x; \nu )$ is also a solution of the perturbed KdV equation (1.6).

• (R1) The result of Theorem 1.1 holds for any density f in ${{{\mathcal {C}}}}^{s_*}({\mathbb {T}}_1 \times {\mathbb {R}}\times {\mathbb {R}}, {\mathbb {R}})$ with $s_*$ large enough and for any family of ${{\mathbb {S}}}_+$-gap solutions of KdV with average c (cf. [21, page 112]). We assume in this paper that f is ${{{\mathcal {C}}}}^{\infty }$ and $c=0$ merely to simplify the exposition.

• (R2) The methods developed to prove Theorem 1.1 are quite general. We expect that analogous results can also be proved for equations in the KdV hierarchy as well as for the defocusing NLS and equations in the NLS hierarchy such as the defocusing mKdV equation.

Theorem 1.1 is proved at the end of the paper in Section 8.3. It is deduced from Theorem 4.1 (Section 4), which is proved by applying a Nash–Moser iteration scheme (Section 8.1) and by establishing the measure estimates of Section 8.2. Before describing the main ideas of the proof in detail, we first comment on the novelty of our result.

• 1. The first KAM results for (1.1) were proved by Kuksin [25] (cf. also [26]) and Kappeler–Pöschel [21] for finite gap solutions of arbitrary size for semilinear perturbations of the KdV equation. This means that the density f of (1.4) does not depend on $u_x$, and hence $\partial _x \nabla { P}(u) = \partial _u^2 f(x, u(x))u_x + \cdots$ depends only on u and $u_x$. (Note that in addition, the dependence on $u_x$ is linear.) Subsequently, Liu–Yuan [29] proved KAM results for semilinear perturbations of small amplitude solutions of the derivative NLS and the Benjamin–Ono equations whereas Zhang–Gao–Yuan [32] proved analogous results for the reversible derivative NLS equation. More recently, Berti–Biasco–Procesi [6]–[7] proved existence of small quasi-periodic solutions of derivative Klein-Gordon equations. For the NLS and the beam equations in higher space dimension, KAM results were obtained by Eliasson–Kuksin [15] and, respectively, Eliasson-Grébert–Kuksin [14]. In all of these works, the perturbations are required to be semilinear. On the other hand, the results in Baldi–Berti–Montalto [3, 4], for quasi-linear perturbations of the KdV and mKdV equations concern only small amplitude solutions. The proofs of these results make use of pseudo-differential calculus and rely in a decisive manner on the differential nature of these equations. The latter property cannot be read off in the action-angle coordinates outside a neighborhood of the origin. Furthermore, also the results of Giuliani [19] for the generalized KdV equation, the ones of Feola–Procesi [17] for the NLS equation, and the ones of Berti–Montalto [10] and Baldi–Berti–Haus–Montalto [1] for water waves concern small amplitude solutions. Thus the challenging problem of the persistence of the finite gap solutions of (1.1) of arbitrary size under strongly nonlinear perturbations (1.5) remained completely open.

• 2. In [9], we used the “one-smoothing property” of the Birkhoff coordinates of the defocusing NLS equation on ${\mathbb {T}}_1$, established in [23], to prove a KAM result for semilinear perturbations. This property is used to deal with the difficulties related to the double “asymptotic multiplicity” of the frequencies. For the KdV equation, a “one-smoothing property” has been proved near the equilibrium in [27] and then in general in [22], however it is not sufficient for dealing with the quasi-linear perturbations (1.5).

• 3. The proof of Theorem 1.1 uses the canonical coordinates constructed in [20] near any given compact family of ${{\mathbb {S}}}_+$-gap potentials in ${\mathcal {M}}_{{{\mathbb {S}}}_+}$, reviewed in Section 3.2. These coordinates admit an expansion in terms of pseudo-differential operators up to a remainder of arbitrary negative order. Due to its length, this part of the proof of Theorem 1.1 has been published in a separate paper [20]. In Section 3.3 we show that the linearization of the Hamiltonian vector field $X_{H_\varepsilon }$, when expressed in these coordinates, admits an expansion in terms of pseudo-differential operators. This property is one of the key ingredients for implementing the Nash–Moser iteration scheme as explained in the subsequent paragraph.

Ideas of the Proof Theorem 1.1 is proved by means of a Nash–Moser iterative scheme to construct, for any $\nu$ belonging to a suitable subset $\Xi _\varepsilon$ of $\Xi$, a quasi-periodic solution of (1.6) with frequency vector $\omega = - \omega ^{kdv}(\nu )$ near the ${{\mathbb {S}}}_+$-gap solution $t \mapsto q\big ( - \omega ^{kdv}(\nu )t,\, x; \, \nu \big )$ of the KdV equation (1.1) (cf. (1.12)), which evolves on the torus ${\mathcal {T}}_\nu \,{:=}\, \Psi _{{{\mathbb {S}}}_+}({\mathbb {T}}^{{{\mathbb {S}}}_+} \times \{\nu \})$ (cf. (1.10), (3.5)). The subset $\Xi _\varepsilon$ is obtained by imposing along the iterative scheme suitable non-resonance conditions. In particular we will always assume $\omega$ to be diophantine, meaning that there exist positive constants $0< \gamma < 1$ and $\tau > |{{\mathbb {S}}}_+| -1$ so that $|\omega \cdot \ell | \geqq \gamma | \ell |^{-\tau }$ for any $\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} \setminus \{0\}$.

The starting point of our proof is to express the perturbed KdV equation (1.6) in the canonical coordinates $(\theta , y, w) \in {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times L^2_\bot ({\mathbb {T}}_1)$, constructed in [20], in a neighborhood of a torus ${\mathcal {T}}_\nu$. Here

$$\begin{aligned} L^2_\bot ({\mathbb {T}}_1) \,{:=}\, \Big \{ w = \sum _{n \in {{\mathbb {S}}}^\bot } w_n e^{\mathrm{i} 2\pi n x} \in L^2_0 ({\mathbb {T}}_1) \Big \}, \quad {{\mathbb {S}}}^\bot \,{:=}\, {\mathbb {Z}}\setminus \big ( {{\mathbb {S}}}_+\cup (-{{\mathbb {S}}}_+) \cup \{ 0 \} \big ). \end{aligned}$$

(1.18)

To be more precise, denote by $B_\bot (\delta )$ the open ball in $L^2_\bot ({\mathbb {T}}_1)$, centered at 0, of radius $\delta$, and by $B_{{{\mathbb {S}}}_+}(\delta )$ the one in ${\mathbb {R}}^{{{\mathbb {S}}}_+}$ (cf. (1.14)). According to [20], for $\nu \in \Xi$ and $0< \delta < 1$ sufficiently small, there exists a canonical coordinate chart

$$\begin{aligned} \Psi _\nu : {{{\mathcal {V}}}}(\delta ) \rightarrow L^2_0({\mathbb {T}}_1) , \ \,(\theta , y, w) \mapsto \Psi _\nu (\theta , y, w) , \quad {{{\mathcal {V}}}}(\delta ) \,{:=}\, {\mathbb {T}}^{{{\mathbb {S}}}_+} \times B_{{{\mathbb {S}}}_+}(\delta ) \times B_\bot (\delta ) , \end{aligned}$$

so that the following key properties hold (cf. Theorem 3.2):

• (P1) $\Psi _\nu (\theta , y, 0) = \Psi _{{{\mathbb {S}}}_+}(\theta , \nu + y)$ for any $(\theta , y) \in {\mathbb {T}}^{{{\mathbb {S}}}_+} \times B_{{{\mathbb {S}}}_+}(\delta )$ (with $\Psi _{{{\mathbb {S}}}_+}$ as in (1.10), (3.5));

• (P2) $\Psi _\nu (\theta , y, w) \in H^s_0({\mathbb {T}}_1)$ for any $(\theta , y, w) \in {\mathbb {T}}^{{{\mathbb {S}}}_+} \times B_{{{\mathbb {S}}}_+}(\delta ) \times \big ( B_\bot (\delta ) \cap H^s({\mathbb {T}}_1)\big )$ and $s \in {\mathbb {N}}$;

• (P3) equation (1.6) takes the form $({{\dot{\theta }}}, \dot{y}, \dot{w}) = X_{{\mathcal {H}}_\varepsilon }$ where the Hamiltonian vector field $X_{{\mathcal {H}}_\varepsilon }$ is given by

  $$\begin{aligned} X_{{\mathcal {H}}_\varepsilon } = ( - \nabla _y {\mathcal {H}}_\varepsilon , \nabla _\theta {\mathcal {H}}_\varepsilon , \partial _x \nabla _w {\mathcal {H}}_\varepsilon ), \quad {\mathcal {H}}_\varepsilon \,{:=}\, H_\varepsilon \circ \Psi _\nu ; \end{aligned}$$

  for $\varepsilon = 0$, the manifold $\{ w = 0 \}$ is invariant for the constant vector field $X_{{\mathcal {H}}_0} = ( - \omega ^{kdv}(\nu ), 0 ,0)$;

• (P4) $\Psi _\nu$ admits an expansion in terms of pseudo-differential operators, up to regularizing operators satisfying tame estimates, as stated in Theorem 3.2-(AE1) (note that in the estimates Theorem 3.2-(Est1), the dependence with respect to the highest Sobolev norm is linear);

• (P5) the linearization of $({{\dot{\theta }}}, \dot{y}, \dot{w}) = X_{{\mathcal {H}}_0}$ along the manifold $\{ y= 0, w = 0\}$ is in diagonal form with coefficients depending only on $\nu$; more specifically $\partial _t {{{\widehat{\theta }}}} = - \Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) {\widehat{y}}$, $\partial _t {\widehat{y}} = 0$, $\partial _t {\widehat{w}} = \partial _x \Omega ^{kdv}(D ;\nu ) {\widehat{w}}$, (cf. the normal form Hamiltonian (3.12)).

As a consequence of (P1)–(P3), for $\varepsilon =0,$ the curve $t \mapsto (- \omega ^{kdv} (\nu ) t, 0, 0)$ is a solution of $({{\dot{\theta }}}, \dot{y}, \dot{w}) = X_{{\mathcal {H}}_0}$, evolving on the torus ${\mathbb {T}}^{{{\mathbb {S}}}_+} \times \{ 0 \} \times \{ 0 \}$, which is invariant under the flow of $X_{{\mathcal {H}}_0}$. We look for a quasi-periodic solution of $({{\dot{\theta }}}, \dot{y}, \dot{w}) = X_{{\mathcal {H}}_\varepsilon }$ near the torus ${\mathbb {T}}^{{{\mathbb {S}}}_+} \times \{ 0 \} \times \{ 0 \}$, with frequency vector $\omega = - \omega ^{kdv} (\nu )$, of the form $\breve{\iota }(\omega t )$ where

$$\begin{aligned} \breve{\iota } : {\mathbb {R}}^{{{\mathbb {S}}}_+} \rightarrow {\mathbb {R}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times H^s_\bot ({\mathbb {T}}_1) , \quad H^s_\bot ({\mathbb {T}}_1) \,{:=}\, H^s ({\mathbb {T}}_1) \cap L^2_\bot ({\mathbb {T}}_1) , \end{aligned}$$

with s sufficiently large, is the lift

$$\begin{aligned} \breve{\iota }(\varphi ) = (\varphi , 0, 0) + \iota (\varphi ) , \quad \iota (\varphi ) = ( {\theta } (\varphi ) - \varphi , y(\varphi ), w(\varphi )) \end{aligned}$$

of a torus embedding and $\iota$ is $(2 \pi {\mathbb {Z}})^{{{\mathbb {S}}}_+}$-periodic. Thus the unknown function $\iota$ satisfies (cf. (4.6))

$$\begin{aligned} {\mathcal {F}}_\omega ( \iota ) = 0 , \quad {\mathcal {F}}_\omega ( \iota ) \,{:=}\, \omega \cdot \partial _{\varphi } \breve{\iota } (\varphi ) - X_{{\mathcal {H}}_\varepsilon }(\breve{\iota } (\varphi )) \end{aligned}$$

and the map $t \mapsto \Psi _\nu (\breve{\iota }(\omega t)) \in H^s_0 ({\mathbb {T}}_1)$ is a quasi-periodic solution of (1.6). The equation ${\mathcal {F}}_\omega ( \iota ) = 0$ is solved by a Nash–Moser iteration scheme. The core of this scheme is the construction of an approximate right inverse of the linearized operator $d {{\mathcal {F}}}_\omega$ at an embedding $\breve{\iota } (\varphi ) = (\theta (\varphi ), y(\varphi ), w(\varphi ))$, near $\breve{\iota }_0(\varphi ) = (\varphi , 0 , 0)$, and the proof that it satisfies suitable tame estimates, cf. Theorem 5.7. One of the main issues is to construct an approximate inverse of the linear operator, acting on $L^2_\bot ({\mathbb {T}}_1)$,

$$\begin{aligned} {{{\mathcal {L}}}}^{(0)}_\omega = \omega \cdot \partial _\varphi - \partial _x d_\bot \nabla _w {\mathcal {H}}_\varepsilon ( \breve{\iota } (\varphi ))\, , \end{aligned}$$

where $d_\bot$ denotes the differential with respect to w. We achieve this goal by reducing ${{{\mathcal {L}}}}^{(0)}_\omega$ to a linear diagonal operator with constant coefficients. Using properties (P4) and (P5) we prove that

• (P6) the linearized Hamiltonian operator $\partial _x d_\bot \nabla _w {\mathcal {H}}_\varepsilon$ in a neighborhood of a ${{\mathbb {S}}}_+$-gap potential is close to $\partial _x \Omega ^{kdv}(D ;\nu )$ (acting in $H^s_{\bot } ({\mathbb {T}}_1)$) and it admits an expansion in terms of classical pseudo-differential operators, up to smoothing remainders which satisfy tame estimates in $H^s_{\bot } ({\mathbb {T}}_1)$ – see Lemmata 3.5 and 3.7 in Section 3.3.

Property (P6) allows us to use pseudo-differential techniques, developed in [1, 3, 10], to reduce ${{{\mathcal {L}}}}^{(0)}_\omega$ to a diagonal one with constant coefficients up to smoothing remainders. Actually, using (P6) we prove that the operator ${{{\mathcal {L}}}}^{(0)}_\omega$ has the form (cf. Lemma  6.2)

$$\begin{aligned} {{{\mathcal {L}}}}^{(0)}_\omega = \omega \cdot \partial _\varphi - \Pi _{\bot } \Big ( a_3^{(0)} \partial _x^3 + 2 (a_3^{(0)})_x \partial _x^2 + a_1^{(0)} \partial _x + \sum _{k=0}^{M} a_{-k}^{(0)} \partial _x^{-k} + Q_{-1}^{kdv} (D ; \omega ) \Big ) + {{\mathcal {R}}_M^{(0)}},\nonumber \\ \end{aligned}$$

(1.19)

where $\Pi _\bot$ is the $L^2$-orthogonal projector onto the subspace $L^2_\bot ({\mathbb {T}}_1)$, the coefficients $a_{-k}^{(0)} (\varphi , x)$, $k = - 3, \ldots , M$, are real valued functions, $a_3^{(0)} \sim - 1$, and ${{\mathcal {R}}_M^{(0)}}$ is a $\varphi$-dependent regularizing operator which satisfies tame estimates in the Sobolev spaces $H^s ( {\mathbb {T}}^{{{\mathbb {S}}}_+}_\varphi \times {\mathbb {T}}_1 )$. The order M of regularization has to be sufficiently large to ensure the convergence of the KAM iterative reducibility scheme, carried out in Section 7; M is fixed in (7.6) and depends on the cardinality $|{{\mathbb {S}}}_+|$ of ${{\mathbb {S}}}_+$ and on the diophantine exponent $\tau$ of the frequency vector $\omega$. We point out that the term $Q_{-1}^{kdv} (D ; \omega )$ in (1.19) is not $\varepsilon$-small, since the finite gap solutions considered might have large amplitudes. More precisely, $Q_{-1}^{kdv} (D ; \omega )$ is the Fourier multiplier, acting on $L^2_\bot ({\mathbb {T}}_1)$, with symbol $\omega ^{kdv}_n - (2 \pi n)^3$ (cf. (3.62)), which takes into account the difference between the KdV-frequencies and the frequencies $(2 \pi n)^3$, $n \in {\mathbb {Z}}$, of the Airy equation $\partial _t v = - \partial ^3_x v$. We also mention that the pseudo-differential operator $\sum _{k=0}^{M} a_{-k}^{(0)} \partial _x^{-k}$ is not present in [3], since in the latter paper only small amplitude finite gap solutions are considered.

In order to show that the regularizing operator ${{\mathcal {R}}_M^{(0)}}$ in (1.19) is tame (which is a key property for the convergence of a Nash–Moser iterative scheme), we prove in Section 3.2 novel results of independent interest concerning the extensions of the differential of the canonical coordinates of [20] to Sobolev spaces $H^s ({\mathbb {T}}_1)$, of negative order $s < 0$ (cf. Corollaries 3.3 and 3.4).

The special form (1.19) allows to find preliminary transformations which diagonalize ${{{\mathcal {L}}}}_\omega ^{(0) }$ up to a pseudo-differential operator of order zero plus a regularizing remainder (see Section 6). More precisely we conjugate ${{{\mathcal {L}}}}_\omega ^{(0) }$ to the Hamiltonian operator (cf. (6.69))

$$\begin{aligned} {{\mathcal {L}}}_\omega ^{(4)} = \omega \cdot \partial _\varphi - \big ( m_3 \partial _x^3 + m_1 \partial _x + \mathrm{Op}( r_0^{(4)}) + Q_{-1}^{kdv} ( D; \omega )\big ) + {{\mathcal {R}}_M^{(4)}}, \end{aligned}$$

(1.20)

where $m_3 + 1$ and $m_1$ are real constants, which are $\varepsilon$-small, $\mathrm{Op}( r_0^{(4)})$ is a pseudo-differential operator of order 0 and ${{\mathcal {R}}}_M^{(4)}$ is a regularizing operator satisfying tame estimates. The map which conjugates ${{\mathcal {L}}}_\omega ^{(0)}$ to ${{\mathcal {L}}}_\omega ^{(4)}$ is obtained by the composition of the transformations introduced in Sections 6.2–6.5. These transformations, inspired by [3], are Fourier integral operators given by symplectic flows of linear Hamiltonian transport PDEs or pseudo-differential maps. In particular, we point out that in order to conjugate the pseudo-differential terms $a_{-k}^{(0)} \partial _x^{-k}$ under the transport flow used in Section 6.3, we need a quantitative version of the Egorov theorem, which is stated and proved in Section 2.5. We remark that in contrast to [3], we implement in Section 6.2 the time-quasi-periodic reparametrization before the conjugation with the transport flow to avoid a technical difficulty in the conjugation of the remainders obtained in the Egorov theorem. Furthermore, we mention that related transformations have been developed in [5] for proving upper bounds for the growth of Sobolev norms for certain classes of PDEs.

At this point, using properties of the KdV frequencies that are recorded in Section 3.4, we are able to perform in Section 7 a KAM reducibility scheme to complete the diagonalization of the operator ${{{\mathcal {L}}}}^{(4)}_\omega$ in (1.20) for most values of $\nu$. Since the variable coefficients term $- \mathrm{Op}( r_0^{(4)}) + {{\mathcal {R}}_M^{(4)}}$ in (1.20) (which is renamed ${\mathtt {R}}_0$ in (7.3)) is proven to satisfy the tame estimates of Lemma 7.1, such a KAM reducibility scheme can be implemented along the lines developed in Berti-Montalto [10]. See Theorem 7.3 for details.

Finally in Section 8 we implement a standard Nash–Moser iterative scheme to construct a solution of ${{{\mathcal {F}}}}_\omega ( \iota ) = 0$ for all frequency vectors $\omega$ satisfying “Melnikov nonresonance conditions”, cf. (8.37). By the results of Section 8.2, using properties of the KdV frequencies, we prove that the set of such non-resonant frequencies has asymptotically full measure as $\varepsilon \rightarrow 0$.

Notation. We denote by ${\mathbb {N}}\,{:=}\, \{0, 1, 2, \ldots \}$ the natural numbers and set ${\mathbb {N}}_+ \,{:=}\, \{ 1, 2, \ldots \}$. Given a Banach space X with norm $\Vert \cdot \Vert _X$, we denote by $H^s_\varphi X = H^s ({\mathbb {T}}^{{{\mathbb {S}}}_+}, X)$, $s \in {\mathbb {N}}$, the Sobolev space of functions $f : {\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow X$ equipped with the norm

$$\begin{aligned} \Vert f \Vert _{H^s_\varphi X} \,{:=}\, \Vert f \Vert _{L^2_\varphi X} + \max _{|\beta |=s} \Vert \partial _\varphi ^\beta f \Vert _{L^2_\varphi X} . \end{aligned}$$

In case $s=0$, we often write $L^2_\varphi X$ instead of $H^0_\varphi X$. Occasionally, we denote the Sobolev space $H^s ({\mathbb {T}}_1) \equiv H^s ({\mathbb {T}}_1, {\mathbb {R}})$ in (1.8) by $H^s_x$ and write $L^2_x$ for $H^0_x$. The space $L^2_x$ is endowed with the standard $L^2$-inner product $\big ( f, g \big )_{L^2_x}$ given by

$$\begin{aligned} \big ( f, g \big )_{L^2_x} \,{:=}\, \int _{{\mathbb {T}}_1} f(x) g(x) \, \hbox {d}x . \end{aligned}$$

(1.21)

Note that the Sobolev space $H^s \equiv H^s ( {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1, {\mathbb {R}})$ defined in (1.15) is an algebra for the product of functions if $s \ge s_0$ where throughout the paper, $s_0$ is defined as

$$\begin{aligned} s_0 \,{:=}\, \left[ \frac{|{{\mathbb {S}}}_+| + 1}{2} \right] + 1 , \end{aligned}$$

(1.22)

where $[ \ ]$ denotes the integer part. For any $s \geqq 0$, denote by $h^s_0$ the sequence space

$$\begin{aligned} h^s_0&\,{:=}\, \big \{ z = (z_n)_{n \in {\mathbb {Z}}} \in h^s : z_0 = 0 \big \} , \nonumber \\ h^s&\,{:=}\, \big \{ z = (z_n)_{n \in {\mathbb {Z}}} , \ z_n \in {\mathbb {C}}\, : \, \Vert z \Vert _s^2 < \infty , \ \overline{z_n} = z_{- n} , \,\, \forall n \in {\mathbb {Z}}\big \}, \end{aligned}$$

(1.23)

where $\Vert z \Vert _s^2 \,{:=}\, \sum _{n \in {\mathbb {Z}}} \langle n \rangle ^{2s} |z_n|^2 < \infty$. By ${{{\mathcal {F}}}}$ we denote the Fourier transform, ${{{\mathcal {F}}}} : L^2({\mathbb {T}}_1) \rightarrow h^0$, $u \mapsto (u_n)_{n \in {\mathbb {Z}}}$, where $u_n \,{:=}\, \int _{{\mathbb {T}}_1} u(x) e^{- \mathrm{i} 2 \pi n x}\, \hbox {d}x$ for any $n \in {\mathbb {Z}}$ and by ${{{\mathcal {F}}}}^{- 1} : h^0 \rightarrow L^2({\mathbb {T}}_1)$ its inverse. Furthermore, we denote by $\Pi _0^\bot$ the $L^2$-orthogonal projector onto the subspace of functions with zero average $L^2_0({\mathbb {T}}_1)$. We set

$$\begin{aligned}&H^s_\bot ({\mathbb {T}}_1) \,{:=}\, H^s({\mathbb {T}}_1) \cap L^2_\bot ({\mathbb {T}}_1), \nonumber \\&H^s_\bot \equiv H^s_\bot ({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1) \,{:=}\, \big \{ u \in H^s({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1) : u(\varphi , \cdot ) \in L^2_\bot ({\mathbb {T}}_1)\big \}\qquad \end{aligned}$$

(1.24)

where $L^2_\bot ({\mathbb {T}}_1)$ is defined in (1.18). Often we write $L^2_\bot$ for $H^0_\bot$. The space $H_\bot ^0({\mathbb {T}}_1)$ is also denoted by $L^2_\bot ({\mathbb {T}}_1)$. By $\Pi _\bot$ we denote the $L^2$-orthogonal projector onto $L^2_\bot ({\mathbb {T}}_1)$, $\Pi _\bot : L^2({\mathbb {T}}_1) \rightarrow L^2_\bot ({\mathbb {T}}_1)$. Let

$$\begin{aligned} {{\mathcal {E}}}_s:= & {} {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times H^s_\bot ({\mathbb {T}}_1), \quad {{\mathcal {E}}} \equiv {{\mathcal {E}}}_0,\nonumber \\ E_s:= & {} {\mathbb {R}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times H^s_\bot ({\mathbb {T}}_1), \quad E \equiv E_0. \end{aligned}$$

(1.25)

Elements of ${{\mathcal {E}}}$ are denoted by $\mathfrak {x} = (\theta , y , w)$ and the ones of its tangent space E by $\widehat{ \mathfrak {x}} = ({{\widehat{\theta }}}, {\widehat{y}},{\widehat{w}})$. For $s < 0$, we consider the Sobolev space $H^s_\bot ({\mathbb {T}}_1)$ of distributions, and the spaces ${{\mathcal {E}}}_s$ and $E_s$ are defined in a similar way as in (1.25). Note that $H^{-s}_\bot ({\mathbb {T}}_1)$ is the dual space of $H^s_\bot ({\mathbb {T}}_1)$. On E, we denote by $\langle \cdot , \cdot \rangle$ the inner product, defined by

$$\begin{aligned} \big \langle ({{\widehat{\theta }}}_1, {\widehat{y}}_1, {\widehat{w}}_1), ({{\widehat{\theta }}}_2, {\widehat{y}}_2, {\widehat{w}}_2) \big \rangle \,{:=}\, {{\widehat{\theta }}}_1 \cdot {{\widehat{\theta }}}_2 + {\widehat{y}}_1 \cdot {\widehat{y}}_2 + \big ( {\widehat{w}}_1, {\widehat{w}}_2 \big )_{L^2_x} . \end{aligned}$$

(1.26)

By a slight abuse of notation, $\Pi _\bot$ also denotes the projector of $E_s$ onto its third component, $\Pi _\bot : E_s \rightarrow H^s_\bot ({\mathbb {T}}_1)$, $({\widehat{\theta }}, {\widehat{y}}, {\widehat{w}}) \mapsto {\widehat{w}}$. Furthermore, we denote by $d_\bot$ the differential with respect to w of any map, defined on an open set of ${\mathcal {E}}_s$, taking values in some Banach space.

For any $0< \delta < 1$, we denote $B_{{{\mathbb {S}}}_+}(\delta )$ the open ball in ${\mathbb {R}}^{{{\mathbb {S}}}_+}$ of radius $\delta$ centered at 0 and by $B_\bot ^s(\delta )$, $s \ge 0$, the corresponding one in $H^s_\bot ({\mathbb {T}}_1)$ where we also write $B_\bot (\delta )$ for $B^0_\bot (\delta )$. These balls are used to define the following open neighborhoods in ${\mathcal {E}}_s$, $s \in {\mathbb {N}}$,

$$\begin{aligned} {{{\mathcal {V}}}}^s(\delta ) \,{:=}\, {\mathbb {T}}^{{{\mathbb {S}}}_+}_1 \times B_{{{\mathbb {S}}}_+}(\delta ) \times B_\bot ^s(\delta ) , \quad {{{\mathcal {V}}}}(\delta ) \equiv {{{\mathcal {V}}}}^0(\delta ) , \quad 0< \delta < 1\, . \end{aligned}$$

(1.27)

The space of bounded linear operators between Banach spaces $X_1, X_2$ is denoted by ${\mathcal {B}} (X_1,X_2)$ and endowed with the operator norm. For two linear operators A, B we denote by [A, B] their commutator, $[ A, B] \,{:=}\, A B - B A$ and for a linear operator A, acting on an Hilbert space H, by $A^\top$ the transpose of A with respect to the scalar product of H. In case A is invertible, the transpose of the inverse $A^{-1}$ of A is denoted by $A^{-\top }$.

Throughout the paper, $\mathtt {\Omega } \subseteq {\mathbb {R}}^{{\mathbb {S}}_+}$ denotes a parameter set of frequency vectors. Given any function $f : \mathtt {\Omega } \rightarrow X$, we denote by $\Delta _\omega f$ the difference function $\Delta _\omega f: \mathtt {\Omega } \times \mathtt {\Omega } \rightarrow X$, $(\omega _1, \omega _2) \mapsto f(\omega _1) - f (\omega _2)$.

2 Preliminaries

The goal of this section is to record analytical tools used throughout the paper. In Section 2.1 we introduce function spaces of functions of the variables $\varphi$, x, depending on a parameter $\omega$ in a Lipschitz continuous way, and state their main properties. In addition, we introduce the classes of $\varphi$-dependent linear operators used in the paper, and the subclasses of Hamiltonian and of symplectic ones. In Section 2.2 we review the notion of periodic pseudo-differential operators and basic elements of their calculus. They are a key tool in Section 6, for the reduction of the linearized operators obtained along the Nash–Moser iteration, to operators with constant coefficients, up to smoothing remainders. In Section 2.3 we discuss the notion of tame and modulo-tame operators, introduced in [10] as a technical tool in order to facilitate the derivations of tame estimates, mainly needed to setup the KAM reducibility scheme in Section 7. The results of Sections 2.4 and 2.5 are new. In Section 2.4 we prove tame estimates for compositions of functions and operators with a torus embedding $\breve{\iota } : {\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow \mathcal {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times H^s_\bot ({\mathbb {T}}_1)$, acting in spaces of functions of the variables $(\varphi ,x) \in {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1$. These tame estimates are at the heart of the convergence of the KAM reducibility scheme, proved in Section 7 and of the Nash–Moser iteration, proved in Section 8.1. In Section 2.5, we prove a version of the Egorov theorem with quantitative tame estimates needed in the reduction step of Section 6.3.

2.1 Function Spaces and Linear Operators

In the paper we consider real or complex functions $u( \varphi , x; \omega )$, $(\varphi ,x) \in {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1$, which are Lipschitz continuous with respect to the parameter $\omega \in \mathtt {\Omega }$, where ${ \mathtt {\Omega }}$ is a subset of ${\mathbb {R}}^{{{\mathbb {S}}}_+}$. In the sequel, we will often suppress $\omega$ in $u( \varphi , x; \omega )$ to make notation lighter. Given $0< \gamma < 1$ and $s \ge 0$, we define the norm

$$\begin{aligned} \begin{aligned}&\Vert u \Vert _s^\mathrm{{Lip}(\gamma )}\,{:=}\, \Vert u \Vert _{s,{ \mathtt {\Omega }}}^\mathrm{{Lip}(\gamma )}\,{:=}\, \Vert u \Vert _s^\mathrm{sup} + \gamma \Vert u \Vert _s^\mathrm{lip} ,\\&\Vert u \Vert ^\mathrm{sup} \,{:=}\, \sup _{\omega \in { \mathtt {\Omega }}} \Vert u(\omega ) \Vert _s , \quad \Vert u \Vert _s^\mathrm{lip} \,{:=}\, \sup _{\omega _1, \omega _2 \in { \mathtt {\Omega }} , \omega _1 \ne \omega _2} \frac{\Vert u(\omega _1) - u(\omega _2) \Vert _{s }}{|\omega _1 - \omega _2|} , \end{aligned} \end{aligned}$$

(2.1)

where $\Vert \ \Vert _s$ is the norm of the Sobolev space $H^s$, defined in (1.15), and $u(\omega _1) = u(\cdot , \cdot ; \omega _1)$. For a function $u: \mathtt {\Omega } \rightarrow {\mathbb {C}}$, the sup norm and the Lipschitz semi-norm are denoted by $|u |^\mathrm{sup}$ and, respectively, $|u|^\mathrm{lip}$. Correpondingly, we write $| u |^\mathrm{{Lip}(\gamma )}\,{:=}\, | u |^\mathrm{sup} + \gamma | u |^\mathrm{lip}$.

By $\Pi _N$, $N \in {\mathbb {N}}_+$, we denote by $\Pi _N$ the smoothing operators on $H^s$,

$$\begin{aligned}&(\Pi _N u)(\varphi ,x) \,{:=}\, \sum _{ |(\ell ,j )| \leqq N} u_{\ell , j} e^{\mathrm{i} (\ell \cdot \varphi + 2 \pi jx)} , \nonumber \\&\quad \quad u_{\ell , j} = \frac{1}{ (2\pi )^{|{{\mathbb {S}}}_+|}} \int _{{\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1} u(\varphi , x) e^{- \mathrm{i} (\ell \cdot \varphi + 2 \pi j x )}d \varphi \hbox {d} x , \end{aligned}$$

(2.2)

and let $\Pi ^\perp _N \,{:=}\, \mathrm{Id} - \Pi _N$. For any $\alpha \geqq 0$ and $s \in {\mathbb {R}}$, the operators $\Pi _N$ and $\Pi ^\bot _N$ satisfy the standard estimates

$$\begin{aligned} \Vert \Pi _N u \Vert _{s}^\mathrm{{Lip}(\gamma )}\leqq N^\alpha \Vert u \Vert _{s-\alpha }^\mathrm{{Lip}(\gamma )}, \quad \Vert \Pi _N^\bot u \Vert _{s}^\mathrm{{Lip}(\gamma )}\leqq N^{-\alpha } \Vert u \Vert _{s + \alpha }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.3)

Furthermore, the following interpolation inequalities hold: for any $0 \le s_1<s_2$ and $0< \theta < 1$,

$$\begin{aligned} \Vert u \Vert _{\theta s_1 + (1- \theta ) s_2}^\mathrm{{Lip}(\gamma )}\leqq 2 (\Vert u \Vert _{s_1}^\mathrm{{Lip}(\gamma )})^\theta (\Vert u \Vert _{s_2}^\mathrm{{Lip}(\gamma )})^{1 - \theta }. \end{aligned}$$

(2.4)

Multiplication and composition with Sobolev functions satisfy the following tame estimates:

Lemma 2.1

(Product and composition)

• (i) For any $s \geqq s_0 = [ (|{{\mathbb {S}}}_+| + 1)/2] + 1$ (cf. (1.22)),

  $$\begin{aligned} \Vert uv \Vert _{s}^\mathrm{{Lip}(\gamma )}&\leqq C(s) \Vert u \Vert _{s}^\mathrm{{Lip}(\gamma )}\Vert v \Vert _{s_0}^\mathrm{{Lip}(\gamma )}+ C(s_0) \Vert u \Vert _{s_0}^\mathrm{{Lip}(\gamma )}\Vert v \Vert _{s}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (2.5)

• (ii) Let $\beta (\cdot ,\cdot ; \omega ) : {\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1 \rightarrow {\mathbb {R}}$ with $\Vert \beta \Vert _{2s_0+2}^\mathrm{{Lip}(\gamma )}\leqq \delta (s_0)$ small enough. Then the composition operator ${\mathcal {B}}: u \mapsto {\mathcal {B}}u, \, ({\mathcal {B}}u)(\varphi ,x) \,{:=}\, u(\varphi , x + \beta (\varphi ,x))$ satisfies, for any $s \geqq s_0 + 1$,

  $$\begin{aligned} \Vert {{{\mathcal {B}}}} u \Vert _{s}^\mathrm{{Lip}(\gamma )}\lesssim _{s} \Vert u \Vert _{s+1}^\mathrm{{Lip}(\gamma )}+ \Vert \beta \Vert _{s}^\mathrm{{Lip}(\gamma )}\Vert u \Vert _{s_0+2}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (2.6)

  The function $\breve{\beta }$, obtained by solving $y = x + \beta (\varphi , x)$ for x,  $x = y + \breve{\beta } ( \varphi , y )$, satisfies

  $$\begin{aligned} \Vert \breve{\beta } \Vert _{s}^\mathrm{{Lip}(\gamma )}\lesssim _{s} \Vert \beta \Vert _{s+1}^\mathrm{{Lip}(\gamma )}\, , \quad \forall s \geqq s_0. \end{aligned}$$

  (2.7)

• (iii) Let $\alpha (\cdot ; \omega ) : {\mathbb {T}}^{{\mathbb {S}}_+} \rightarrow {\mathbb {R}}$ with $\Vert \alpha \Vert _{2s_0+2}^\mathrm{{Lip}(\gamma )}\leqq \delta (s_0)$ small enough. Then the composition operator ${{{\mathcal {A}}}} : u \mapsto {{{\mathcal {A}}}} u, \, ({{{\mathcal {A}}}} u)(\varphi ,x) \,{:=}\, u(\varphi + \alpha (\varphi )\omega , x)$ satisfies, for any $s \geqq s_0 + 1$,

  $$\begin{aligned} \Vert {{{\mathcal {A}}}} u \Vert _{s}^\mathrm{{Lip}(\gamma )}\lesssim _{s} \Vert u \Vert _{s+1}^\mathrm{{Lip}(\gamma )}+ \Vert \alpha \Vert _{s}^\mathrm{{Lip}(\gamma )}\Vert u \Vert _{s_0+2}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (2.8)

  The function $\breve{\alpha }$, obtained by solving $\vartheta = \varphi + \alpha (\varphi )\omega$ for $\varphi$, $\varphi = \vartheta + \breve{\alpha }( \vartheta )\omega$, satisfies

  $$\begin{aligned} \Vert \breve{\alpha } \Vert _{s}^\mathrm{{Lip}(\gamma )}\lesssim _{s} \Vert \alpha \Vert _{s+1}^\mathrm{{Lip}(\gamma )}, \quad \forall s \geqq s_0 . \end{aligned}$$

  (2.9)

Remark 2.2

Note that for $s > s_0$, the bound $C(s) \Vert u \Vert _{s}^\mathrm{{Lip}(\gamma )}\Vert v \Vert _{s_0}^\mathrm{{Lip}(\gamma )} + C(s_0) \Vert u \Vert _{s_0}^\mathrm{{Lip}(\gamma )}\Vert v \Vert _{s}^\mathrm{{Lip}(\gamma )}$ of $\Vert uv \Vert _{s}^\mathrm{{Lip}(\gamma )}$ in (2.5) is linear in $\Vert u \Vert _{s}^\mathrm{{Lip}(\gamma )}$ and $\Vert v \Vert _{s}^\mathrm{{Lip}(\gamma )}$. It is this property of the estimate which is referred to as “tame”.

Proof

Item (i) follows from (2.72) in [10] and (ii)–(iii) follow from [10, Lemma 2.30]. $\quad \square$

If the vector $\omega \in {\mathbb {R}}^{{{\mathbb {S}}}_+}$ is diophantine, that is $|\omega \cdot \ell | \geqq \gamma / | \ell |^{\tau }$ for any $\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} \setminus \{ 0 \}$, the equation $\omega \cdot \partial _\varphi v = u$ with $u(\varphi ,x)$ satisfying $u_{0, j} = 0$ for any $j \in {\mathbb {Z}}$, has the periodic solution

$$\begin{aligned} v = (\omega \cdot \partial _\varphi )^{-1} u = \sum _{j \in {\mathbb {Z}}, \ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} \setminus \{0\}} \frac{ u_{\ell , j} }{\mathrm{i} \omega \cdot \ell }e^{\mathrm{i} (\ell \cdot \varphi + 2 \pi j x )} , \end{aligned}$$

and it satisfies the standard estimate (cf. [9, Lemma 2.2])

$$\begin{aligned} \Vert (\omega \cdot \partial _\varphi )^{-1} u \Vert _{s}^\mathrm{{Lip}(\gamma )}\leqq C \gamma ^{-1}\Vert u \Vert _{s+ 2 \tau + 1}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.10)

We also record Moser’s tame estimate for the nonlinear composition operator

$$\begin{aligned} u(\varphi , x) \mapsto {\mathtt {f}}(u)(\varphi , x) \,{:=}\, f(\varphi , x, u(\varphi , x)) . \end{aligned}$$

Since the variables $\varphi$ and x play the same role, we state it for the Sobolev space $H^s ({\mathbb {T}}^d )$.

Lemma 2.3

(Composition operator, [10, Lemma 2.31]) Let $f \in {{{\mathcal {C}}}}^{\infty }({\mathbb {T}}^d \times {\mathbb {R}}^n, {\mathbb {C}})$. If $v(\cdot ;\omega ) \in H^s({\mathbb {T}}^d, {\mathbb {R}}^n )$, $\omega \in { \mathtt {\Omega }}$, is a family of Sobolev functions satisfying $\Vert v \Vert _{s_0(d)}^\mathrm{{Lip}(\gamma )}\leqq 1$ where $s_0(d) > d/2$, then, for any $s \geqq s_0(d)$,

$$\begin{aligned} \Vert {\mathtt {f}}(v) \Vert _{s}^\mathrm{{Lip}(\gamma )}\leqq C(s, f ) ( 1 + \Vert v \Vert _{s}^\mathrm{{Lip}(\gamma )}) . \end{aligned}$$

(2.11)

Moreover, if $f(\varphi , x, 0) = 0$, then $\Vert {\mathtt {f}}(v) \Vert _{s}^\mathrm{{Lip}(\gamma )}\leqq C(s, f ) \Vert v \Vert _{s}^\mathrm{{Lip}(\gamma )}$.

Next we discuss classes of linear operators used in this paper. Throughout the paper we consider $\varphi$-dependent families of linear operators $A : {\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow {{{\mathcal {L}}}}( L^2({\mathbb {T}}_1, {\mathbb {C}}))$, $\varphi \mapsto A(\varphi )$, acting on complex valued functions u(x) of the space variable x. We also let A act on functions $u(\varphi , x)$ of space-time. In this way we get an element in ${{{\mathcal {L}}}}(L^2({\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1, {\mathbb {C}}) )$, again denoted by A, which is defined by

$$\begin{aligned} A [u] (\varphi , x) \equiv ( A u) (\varphi , x) \,{:=}\, (A(\varphi ) u(\varphi , \cdot ))(x) . \end{aligned}$$

(2.12)

We say that the linear operator A is real if it maps real valued functions to real valued functions. When u in (2.12) is expanded in its Fourier series,

$$\begin{aligned} u(\varphi , x ) = \sum _{j \in {\mathbb {Z}}} u_{j} (\varphi ) e^{2 \pi \mathrm{i} j x } = \sum _{j \in {\mathbb {Z}}, \ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} u_{\ell ,j} e^{\mathrm{i} (\ell \cdot \varphi + 2 \pi j x)}, \end{aligned}$$

(2.13)

one obtains

$$\begin{aligned} (A u) (\varphi , x)&= \sum _{j , j' \in {\mathbb {Z}}} A_j^{j'}(\varphi ) u_{j'}(\varphi ) e^{\mathrm{i} 2 \pi j x} \nonumber \\&\quad = \sum _{j \in {\mathbb {Z}}, \ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} \, \sum _{j' \in {\mathbb {Z}}, \ell ' \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} A_j^{j'}(\ell - \ell ') u_{\ell ', j'} e^{\mathrm{i} (\ell \cdot \varphi + 2 \pi j x)} . \end{aligned}$$

(2.14)

We shall identify an operator A with the matrix $\big ( A^{j'}_j (\ell - \ell ') \big )_{j, j' \in {\mathbb {Z}}, \ell , \ell ' \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} }$.

Definition 2.4

Given a linear operator A as in (2.14) we define the following operators:

• 1. |A| (majorant operator) whose matrix elements are $| A_j^{j'}(\ell - \ell ')|$.

• 2. $\Pi _N A$, $N \in {\mathbb {N}}_+,$ (smoothed operator) whose matrix elements are

  $$\begin{aligned} (\Pi _N A)^{j'}_j (\ell - \ell ') \,{:=}\, {\left\{ \begin{array}{ll} A^{j'}_j (\ell - \ell ') \quad \mathrm{if} \quad \langle \ell - \ell ' \rangle \leqq N \\ 0 \quad \quad \quad \mathrm{otherwise} . \end{array}\right. } \end{aligned}$$

  (2.15)

• 3. $\langle \partial _{\varphi } \rangle ^{b} A$, $b \in {\mathbb {R}}$, whose matrix elements are $\langle \ell - \ell ' \rangle ^b A_j^{j'}(\ell - \ell ')$.

• 4. $\partial _{\varphi _m} A(\varphi ) = [\partial _{\varphi _m}, \, A]$ (differentiated operator) whose matrix elements are $\mathrm{i} (\ell _m - \ell _m') A_{j}^{j'}(\ell - \ell ')$.

Hamiltonian and symplectic operators will play an important role in the reduction procedure of linearized operators, implemented in Sections 6 and 7. They are defined as follows.

Definition 2.5

(Hamiltonian and symplectic operators)

• (i) A $\varphi$-dependent family of linear operators $X(\varphi )$, $\varphi \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$, densely defined in $L_0^{2} ({\mathbb {T}}_1)$, is Hamiltonian if $X(\varphi ) = \partial _x G( \varphi )$ for some real linear operator $G(\varphi )$ which is self-adjoint with respect to the $L^2$-inner product. By a slight abuse of terminology, $\omega \cdot \partial _{\varphi } - \partial _x G( \varphi )$ is also said to be a Hamiltonian operator.

• (ii) A $\varphi$-dependent family of linear operators $A (\varphi ) : L_0^2 ({\mathbb {T}}_1) \rightarrow L_0^2 ({\mathbb {T}}_1)$, $\forall \varphi \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$, is symplectic if

  $$\begin{aligned} {{{\mathcal {W}}}}_{L^2_0}(A (\varphi ) u, A (\varphi ) v) = {{{\mathcal {W}}}}_{L^2_0} (u, v) , \quad \forall u,v \in L_0^2 ({\mathbb {T}}_1 ), \end{aligned}$$

  where the symplectic 2-form ${{{\mathcal {W}}}}_{L^2_0}$ is the one defined in (1.9).

Under a $\varphi$-dependent family of symplectic transformations $\Phi (\varphi )$, $\varphi \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$, the linear Hamiltonian operator $\omega \cdot \partial _{\varphi } - \partial _x G(\varphi )$ transforms into one which is again Hamiltonian. Self-adjoint operators and real ones are characterized in terms of their matrix elements as follows:

Lemma 2.6

A family of linear operators $R (\varphi )$, $\varphi \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$ with Fourier series $R (\varphi ) = \sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} R (\ell ) e^{\mathrm{i} \ell \cdot \varphi }$, is

• (i) self-adjoint if and only if $\overline{ R_j^{j'}( \ell ) } = R_{j'}^j(- \ell ) , \quad \forall j, j' \in {\mathbb {Z}}, \ \ell , \ell ' \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}$;

• (ii) real if and only if $\overline{R^j_{j'}(\ell )} = R^{-j}_{-j'}(-\ell ) , \quad \forall j, j' \in {\mathbb {Z}}, \ \ell , \ell ' \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}$;

• (iii) Real and self-adjoint if and only if $R_j^{j'}(\ell )\,{=}\,R_{- j'}^{- j}\,(\ell ),\forall j, j'\,{\in }\,{\mathbb {Z}}, \ell , \ell '\,{\in }\,{\mathbb {Z}}^{{{\mathbb {S}}}_+}$.

The next lemma describes the structure of specific linear Hamiltonian operators (cf. Definition 2.5), and will be used in Lemmata 6.2 and 6.5.

Lemma 2.7

Let $X : H^{s + 3}_0({\mathbb {T}}_1) \rightarrow H^s_0({\mathbb {T}}_1)$ be a linear Hamiltonian vector field of the form

$$\begin{aligned} X = { a_{3}(x) \partial _x^{3} + a_{2}(x) \partial _x^{2} + a_{1}(x) \partial _x} + \mathrm{bounded \ operator} \end{aligned}$$

(2.16)

where $a_{3}, a_{2}, a_{1} \in {{{\mathcal {C}}}}^\infty ({\mathbb {T}}_1, {\mathbb {R}})$. Then $a_2 = 2(a_3)_x$.

Proof

Since X is a linear Hamiltonian vector field it has the form $X = \partial _x {{{\mathcal {A}}}}$ where ${{{\mathcal {A}}}}$ is a densely defined operator on $L^2_0({\mathbb {T}}_1)$ satisfying ${{{\mathcal {A}}}} = {{{\mathcal {A}}}}^\top$. Since by (2.16), ${{{\mathcal {A}}}} = \partial _x^{-1} X = a_3 (x) \partial _{xx} + \big ( - (a_3)_x + a_2 \big ) \partial _x + \ldots$ and ${{{\mathcal {A}}}}^\top = - X^\top \partial _{x}^{-1} = a_3 (x) \partial _{xx} + \big ( 3 (a_3)_x - a_2 \big ) \partial _x + \ldots$, the identity ${{{\mathcal {A}}}} = {{{\mathcal {A}}}}^\top$ implies that $a_2 = 2 (a_3)_x$. $\quad \square$

2.2 Pseudo-differential Operators

In this section we introduce the class of pseudo-differential operators, acting on functions on ${\mathbb {T}}_1$, which are used in this paper, and discuss their basic calculus, following [10]. (Note however that in [10], the space variable x is in ${\mathbb {R}}/ (2 \pi {\mathbb {Z}})$ whereas in this paper it is in ${\mathbb {T}}_1$.)

Definition 2.8

(Pseudo-differential operators, symbols) We say that $a: {\mathbb {T}}_1 \times {\mathbb {R}}\rightarrow {\mathbb {C}}$ is a symbol of order $m \in {\mathbb {R}}$ if, for any $\alpha , \beta \in {\mathbb {N}}$,

$$\begin{aligned} \big | \partial _x^\alpha \partial _\xi ^\beta a (x,\xi ) \big | \leqq C_{\alpha ,\beta } \langle \xi \rangle ^{m - \beta } , \quad \forall (x, \xi ) \in {\mathbb {T}}_1 \times {\mathbb {R}}. \end{aligned}$$

(2.17)

The set of such symbols is denoted by $S^m$. Given $a \in S^m,$ we denote by A the operator, which maps a one periodic function $u(x) = \sum _{j \in {\mathbb {Z}}}u_j e^{ \mathrm{i} 2 \pi j x}$ to

$$\begin{aligned} A[u](x) \equiv (Au) (x) \,{:=}\, {\mathop \sum }_{j \in {\mathbb {Z}}} a(x,j) u_j e^{\mathrm{i} 2 \pi j x}. \end{aligned}$$

The operator A is referred to as the pseudo-differential operator ($\Psi \mathrm{DO}$) of order m, associated to the symbol a, and is also denoted by Op(a) or a(x, D) where $D = \frac{1}{\mathrm{i} } \partial _x$. Furthermore we denote by $OPS^m$ the set of pseudo-differential operators a(x, D) with $a(x, \xi ) \in S^m$ and set $OPS^{-\infty } \,{:=}\, \cap _{m \in {\mathbb {R}}} OPS^{m}$.

When the symbol a is independent of $\xi$, the operator $A = \mathrm{Op} (a)$ is the multiplication operator by the function a(x), that is, $A : u (x) \mapsto a ( x) u(x )$ and we also write a for A. If a is independent of x, the operator $A = \mathrm{Op} (a)$ is referred to as Fourier multiplier. In particular, $\langle D \rangle$ denotes the Fourier multiplier with symbol $\langle \xi \rangle \,{:=}\, \max \{1, | \xi | \}$

More generally, we consider symbols $a(\varphi , x, \xi ; \omega )$, depending in addition on the variable $\varphi \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$ and the parameter $\omega \in \Omega$, where a is ${{{\mathcal {C}}}}^\infty$ in $\varphi$ and Lipschitz continuous with respect to $\omega$. By a slight abuse of notation, we denote the class of such symbols of order m also by $S^m$. Alternatively, we denote A by $A(\varphi )$, $\mathrm{Op} (a(\varphi , \cdot ))$, or $a(\varphi , x, D; \omega )$.

Given an even cut-off function $\chi _0 \in {{{\mathcal {C}}}}^\infty ({\mathbb {R}}, {\mathbb {R}})$, satisfying

$$\begin{aligned} \begin{aligned}&0 \leqq \chi _0 \leqq 1 , \quad \quad \chi _0(\xi ) = 0 , \quad \forall |\xi | < \frac{1}{2} , \quad \quad \chi _0(\xi ) = 1 \, , \quad \forall |\xi | \geqq \frac{2}{3} , \end{aligned} \end{aligned}$$

(2.18)

we define, for any $m \in {\mathbb {Z}},$ $\partial _x^m = \mathrm{Op}(\chi _0(\xi ) (\mathrm{i} 2 \pi \xi )^{m})$, so that

$$\begin{aligned} \partial _x^{m}[e^{\mathrm{i} 2 \pi j x}] = (\mathrm{i} 2 \pi j)^m e^{\mathrm{i} 2 \pi j x}, \ \ j \in {\mathbb {Z}}\setminus \{ 0 \} , \quad \partial _x^{ m}[1] = 0. \end{aligned}$$

(2.19)

Note that $\partial ^0_x [u] (x) = u(x) - u_0$, hence $\partial _x^{0}$ is not the identity operator.

Following [10, Definition 2.11], we introduce for any $s \ge 0$ the norm of a symbol $a (\varphi , x, \xi ; \omega )$ in $S^m$, which controls the regularity in $(\varphi , x)$ and the decay in $\xi$ of a and its derivatives $\partial _\xi ^\beta a \in S^{m - \beta }$, $0 \leqq \beta \leqq \alpha$, in the Sobolev norm $\Vert \ \Vert _s$. By a slight abuse of terminology, we refer to it as the norm of the corresponding pseudo-differential operator. Unlike as in [10], we consider the difference quotient instead of the derivative with respect to $\omega$, and write $| \ |_{m,s,\alpha }^{\mathrm{{Lip}(\gamma )}}$ instead of $| \ |_{m,s,\alpha }^{1,\gamma }$.

Definition 2.9

(Norm of pseudo-differential operators) Let $A(\omega ) \,{:=}\, a(\varphi , x, D; \omega ) \in OPS^m$ be a family of pseudo-differential operators with symbols $a(\varphi , x, \xi ; \omega ) \in S^m$ of order $m \in {\mathbb {R}}$. For $\gamma \in (0,1)$, $\alpha \in {\mathbb {N}}$, $s \geqq 0$, we define the weighted $\Psi$ do norm of A as

$$\begin{aligned} | A |_{m, s, \alpha }^{\mathrm{{Lip}(\gamma )}} \,{:=}\, \sup _{\omega \in { \mathtt {\Omega }}} |A(\omega ) |_{m, s, \alpha } + \gamma \sup _{\begin{array}{c} \omega _1, \omega _2 \in { \mathtt \Omega } \\ \omega _1 \ne \omega _2 \end{array}} \frac{|A(\omega _1) - A(\omega _2)|_{m, s, \alpha }}{|\omega _1 - \omega _2|} \end{aligned}$$

where $|A(\omega ) |_{m, s, \alpha } \,{:=}\, \max _{0 \leqq \beta \leqq \alpha } \, \sup _{\xi \in {\mathbb {R}}} \Vert \partial _\xi ^\beta a(\cdot , \cdot , \xi ; \omega ) \Vert _{s} \langle \xi \rangle ^{-m + \beta }$.

The pseudo-differential norm $| \cdot |_{m, s, \alpha }^{\mathrm{{Lip}(\gamma )}}$ satisfies the following elementary properties: for any $s \leqq s'$, $\alpha \leqq \alpha '$, and $m \leqq m',$

$$\begin{aligned} | \cdot |_{m, s, \alpha }^\mathrm{{Lip}(\gamma )}\leqq \, | \cdot |_{m, s', \alpha }^{\mathrm{{Lip}(\gamma )}} , \quad | \cdot |_{m, s, \alpha }^\mathrm{{Lip}(\gamma )}\leqq \, | \cdot |_{m, s, \alpha '}^\mathrm{{Lip}(\gamma )}, \quad \ | \cdot |_{m', s, \alpha }^\mathrm{{Lip}(\gamma )}\leqq | \cdot |_{m, s, \alpha }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.20)

For a Fourier multiplier $g( D; \omega )$ with symbol $g \in S^m$, one has

$$\begin{aligned} | \mathrm{Op}(g) |_{m, s, \alpha }^\mathrm{{Lip}(\gamma )}= | \mathrm{Op}(g) |_{m, 0, \alpha }^\mathrm{{Lip}(\gamma )}\leqq C(m, \alpha , g) , \quad \forall s \geqq 0 , \end{aligned}$$

(2.21)

and for a function $a(\varphi , x; \omega )$,

$$\begin{aligned} |\mathrm{Op}(a)|_{0, s, \alpha }^\mathrm{{Lip}(\gamma )}= |\mathrm{Op}(a)|_{0, s, 0}^\mathrm{{Lip}(\gamma )}\lesssim \Vert a \Vert _s^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.22)

Composition. If $A = a(\varphi , x, D; \omega ) \in OPS^{m}$, $B = b(\varphi , x, D; \omega ) \in OPS^{m'}$, then the composition $A B \,{:=}\, A \circ B$ is a pseudo-differential operator with a symbol $\sigma _{AB} (\varphi , x, \xi ; \omega )$ in $S^{m+m'}$ which, for any $N \geqq 0$, admits the asymptotic expansion

$$\begin{aligned} \sigma _{AB} (\varphi , x, \xi ; \omega )= & {} \sum _{\beta =0}^{N} \frac{1}{\mathrm{i} ^\beta \beta !} \partial _\xi ^\beta a (\varphi , x, \xi ; \omega ) \, \partial _x^\beta b(\varphi , x, \xi ; \omega ) + r_N (\varphi , x, \xi ; \omega ) \end{aligned}$$

(2.23)

with remainder $r_N \in S^{m + m' - N -1}$. We record the following tame estimate for the composition of two pseudo-differential operators, proved in [10, Lemma 2.13]:

Lemma 2.10

(Composition) Let $A = a(\varphi , x, D; \omega )$, $B = b(\varphi , x, D; \omega )$ be pseudo-differential operators with symbols $a ( \varphi , x, \xi ; \omega ) \in S^m$, $b ( \varphi , x, \xi ; \omega ) \in S^{m'}$, $m , m' \in {\mathbb {R}}$. Then $A \circ B$ is the pseudo-differential operator of order $m + m'$, associated to the symbol $\sigma _{AB}(\varphi , x, \xi ; \omega )$ which satisfies, for any $\alpha \in {\mathbb {N}}$, $s \geqq s_0$,

$$\begin{aligned} | A B |_{m + m', s, \alpha }^\mathrm{{Lip}(\gamma )}&\lesssim _{m, \alpha } C(s) | A |_{m, s, \alpha }^\mathrm{{Lip}(\gamma )}| B |_{m', s_0 + \alpha + |m|, \alpha }^\mathrm{{Lip}(\gamma )}\nonumber \\&\quad + C(s_0) | A |_{m, s_0, \alpha }^\mathrm{{Lip}(\gamma )}| B |_{m', s + \alpha + |m|, \alpha }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.24)

Moreover, for any integer $N \geqq 1$, the remainder $R_N \,{:=}\, \mathrm{Op}(r_N)$ with $r_N$ as in (2.23) satisfies

$$\begin{aligned} | R_N |_{m+ m'- N -1, s, \alpha }^\mathrm{{Lip}(\gamma )}&\lesssim _{m, N, \alpha } C(s) | A |_{m, s, N + 1 + \alpha }^\mathrm{{Lip}(\gamma )}| B |_{m', s_0 + 2(N+1) + |m| + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}\nonumber \\&\quad + C(s_0)| A |_{m, s_0 , N + 1 + \alpha }^\mathrm{{Lip}(\gamma )}| B |_{m', s + 2(N+1) + |m| + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.25)

By (2.23) the commutator [A, B] of $A = a(x, D) \in OPS^{m}$ and $B = b (x, D) \in OPS^{m'}$ is a pseudo-differential operator of order $m +m' -1$, and Lemma 2.10 yields (cf. [10, Lemma 2.15]).

Lemma 2.11

(Commutator) If $A = a(\varphi , x, D; \omega ) \in OPS^m$ and $B = b(\varphi , x, D; \omega ) \in OPS^{m'}$, $m , m' \in {\mathbb {R}}$, then the commutator $[A, B] \,{:=}\, AB - B A$ is the pseudo-differential operator of order $m+m'-1$ associated to the symbol $\sigma _{AB} (\varphi , x, \xi ; \omega ) - \sigma _{BA}(\varphi , x, \xi ; \omega ) \in S^{m+m'-1}$ which for any $\alpha \in {\mathbb {N}}$ and $s \geqq s_0$ satisfies

$$\begin{aligned} \begin{aligned} | [A,B] |_{m+m' -1, s, \alpha }^\mathrm{{Lip}(\gamma )}&\lesssim _{m,m', \alpha } C(s) | A |_{m, s + 2 + |m' | + \alpha , \alpha + 1}^\mathrm{{Lip}(\gamma )}| B |_{m', s_0 + 2 + |m| + \alpha , \alpha + 1}^\mathrm{{Lip}(\gamma )}\\&\quad \quad + C(s_0) | A |_{m, s_0 + 2 + |m' | + \alpha , \alpha + 1}^\mathrm{{Lip}(\gamma )}| B |_{m', s + 2 + |m| + \alpha , \alpha + 1}^\mathrm{{Lip}(\gamma )}. \end{aligned} \end{aligned}$$

(2.26)

In the case of operators of the special form $a \partial _x^m$, Lemmas 2.10 and 2.11 simplify as follows:

Lemma 2.12

(Composition and commutator of homogeneous symbols) Let $A = a \partial _x^m$, $B = b \partial _x^{m'}$ where $m, m' \in {\mathbb {Z}}$ and $a(\varphi , x ; \omega )$, $b(\varphi , x ; \omega )$ are ${{{\mathcal {C}}}}^\infty$-smooth functions with respect to $(\varphi , x)$ and Lipschitz with respect to $\omega \in \mathtt \Omega$. Then there exist combinatorial constants $K_{n, m} \in {\mathbb {R}}$, $0 \le n \le N$, with $K_{0, m} = 1$ and $K_{1, m} = m$ so that the following holds:

• (i) For any $N \in {\mathbb {N}}$, the composition $A \circ B$ is in $OPS^{m+m'}$ and admits the asymptotic expansion

  $$\begin{aligned} A \circ B = \sum _{n = 0}^{N} K_{n, m} \, a\, (\partial _x^n b) \partial _x^{m + m' - n} + {{{\mathcal {R}}}}_N(a, b) \end{aligned}$$

  where the remainder ${{{\mathcal {R}}}}_N(a, b)$ is in $OPS^{m + m' - N - 1}$. Furthermore there is a constant $\sigma _N(m) > 0$ so that, for any $s \geqq s_0$, $\alpha \in {\mathbb {N}}$,

  $$\begin{aligned}&|{{{\mathcal {R}}}}_N(a, b)|_{m + m'- N - 1, s, \alpha }^\mathrm{{Lip}(\gamma )}\lesssim _{m, m', s, N, \alpha } \Vert a\Vert _{s + \sigma _N(m) }^\mathrm{{Lip}(\gamma )}\Vert b \Vert _{s_0 + \sigma _N(m)}^\mathrm{{Lip}(\gamma )}\\&+ \Vert a \Vert _{s_0 + \sigma _N(m)}^\mathrm{{Lip}(\gamma )}\Vert b \Vert _{s + \sigma _N(m)}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

• (ii) For any $N \in {\mathbb {N}}_+$, the commutator [A, B] is in $OPS^{m + m' - 1}$ and admits the asymptotic expansion

  $$\begin{aligned}{}[A, B] = \sum _{n = 1}^N \big ( K_{n, m}a (\partial _x^n b) - K_{ n, m'}(\partial _x^n a) b \big ) \partial _x^{m + m' - n} + {{{\mathcal {Q}}}}_N(a, b) \end{aligned}$$

  where the remainder ${{{\mathcal {Q}}}}_N(a, b)$ is in $OPS^{m + m' - N - 1}$. Furthermore, there is a constant $\sigma _N(m, m') > 0$ so that, for any $s \geqq s_0$, $\alpha \in {\mathbb {N}}$,

  $$\begin{aligned} |{{{\mathcal {Q}}}}_N(a, b)|_{m + m'- N - 1, s, \alpha }^\mathrm{{Lip}(\gamma )}&\lesssim _{m, m', s, N, \alpha } \Vert a\Vert _{s + \sigma _N(m, m') }^\mathrm{{Lip}(\gamma )}\Vert b \Vert _{s_0 + \sigma _N(m, m')}^\mathrm{{Lip}(\gamma )}\\&\quad + \Vert a \Vert _{s_0 + \sigma _N(m, m')}^\mathrm{{Lip}(\gamma )}\Vert b \Vert _{s + \sigma _N(m, m')}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

Proof

The results follow from the asymptotic expansion formula (2.23) and Lemma 2.10. $\quad \square$

Finally we give the following result on the exponential of a pseudo-differential operator of order 0.

Lemma 2.13

(Exponential map) If $A \,{:=}\, \mathrm{Op}(a(\varphi , x, \xi ; \omega ))$ is in $OPS^{0}$, then $\sum _{k \ge 0} \frac{1}{k!} \sigma _{A^k}(\varphi , x , \xi ; \omega )$ is a symbol of order 0 and hence the corresponding pseudo-differential operator, denoted by $\Phi = \exp (A)$, is in $OPS^0$. Furthermore, for any $s \geqq s_0$, $\alpha \in {\mathbb {N}}$, there is a constant $C(s, \alpha ) > 0$ so that

$$\begin{aligned} |\Phi - \mathrm{Id} |_{0, s, \alpha }^\mathrm{{Lip}(\gamma )}\leqq |A|_{0, s + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}\mathrm{exp} \big ( C(s, \alpha ) |A|_{0, s_0 + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}\big ) . \end{aligned}$$

(2.27)

Proof

Iterating (2.24), for any $s \geqq s_0$, $\alpha \in {\mathbb {N}}$, there is a constant $C(s, \alpha ) > 0$ such that

$$\begin{aligned} |A^k|_{0, s, \alpha }^\mathrm{{Lip}(\gamma )}\leqq C(s, \alpha )^{k - 1} (|A|_{0, s_0 + \alpha , \alpha }^\mathrm{{Lip}(\gamma )})^{k - 1} |A|_{0, s + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}, \quad \forall k \geqq 1 . \end{aligned}$$

(2.28)

Therefore

$$\begin{aligned} \begin{aligned} |\Phi - \mathrm{Id}|_{0, s, \alpha }^\mathrm{{Lip}(\gamma )}&\leqq \sum _{k \geqq 1} \frac{1}{k!} |A^k|_{0, s, \alpha }^\mathrm{{Lip}(\gamma )}{\mathop {\leqq }\limits ^{(2.28)}} |A|_{0, s + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}\sum _{k \geqq 1} \frac{1}{k!} C(s, \alpha )^{k - 1} (|A|_{0, s_0 + \alpha , \alpha }^\mathrm{{Lip}(\gamma )})^{k - 1} \\&\leqq |A|_{0, s + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}\mathrm{exp}\big ( C(s, \alpha ) |A|_{0, s_0 + \alpha , \alpha }^\mathrm{{Lip}(\gamma )}\big ) \end{aligned} . \end{aligned}$$

This shows that $\sum _{k \ge 0} \frac{1}{k!} \sigma _{A^k}(\varphi , x , \xi ; \omega )$ is a symbol in $S^0$ and that the estimate (2.27) holds. $\quad \square$

2.3 $\mathrm{{Lip}(\gamma )}$-Tame and Modulo-Tame Operators

In this section we review the notions and the main properties of $\mathrm{{Lip}(\gamma )}$-$\sigma$-tame and $\mathrm{{Lip}(\gamma )}$-modulo-tame operators, introduced in [10, Section 2.2]. (Again, unlike [10], we consider difference quotients instead of first order derivatives with respect to $\omega$.)

Definition 2.14

($\mathrm{{Lip}(\gamma )}$-$\sigma$-tame) Let $\sigma \ge 0$ and $0< \gamma < 1$. A linear operator $A = A(\omega )$ as in (2.12) is said to be $\mathrm{{Lip}(\gamma )}$-$\sigma$-tame if there exist numbers $s_1$, S with $s_0 \le s_1 < S$ and a non-decreasing function $[s_1, S] \rightarrow [0, + \infty )$, $s \mapsto {{\mathfrak {M}}}_A(s)$, so that, for any $s_1 \leqq s \leqq S$ and $u \in H^{s+\sigma }$,

$$\begin{aligned}&\sup _{\omega \in { \mathtt {\Omega }}} \Vert A(\omega ) u \Vert _s + \gamma \sup _{\begin{array}{c} \omega _1, \omega _2 \in {\mathtt {\Omega }} \\ \omega _1 \ne \omega _2 \end{array}} \Big \Vert \frac{A(\omega _1) - A(\omega _2)}{|\omega _1- \omega _2|} u \Big \Vert _{s} \nonumber \\&\quad \leqq {{\mathfrak {M}}}_A(s_1) \Vert u \Vert _{s+\sigma } + {{\mathfrak {M}}}_A (s) \Vert u \Vert _{s_1+\sigma } . \end{aligned}$$

(2.29)

When $\sigma$ is zero, we simply write $\mathrm{{Lip}(\gamma )}$-tame instead of $\mathrm{{Lip}(\gamma )}$-0-tame. We say that ${{\mathfrak {M}}}_{A}(s)$ is a tame constant of the operator A. Note that ${{\mathfrak {M}}}_A(s)$ is not uniquely determined and that it may also depend on $\sigma$, referred to as loss of derivatives. We will not explicitly record this dependence.

Remark 2.15

In the sequel, often we will not explicitly record the domain of definition $[s_1,S]$ of the $\mathrm{{Lip}(\gamma )}$-$\sigma$-tame constant ${{\mathfrak {M}}}_A(s)$ in order to make the statements lighter. Similarly, we will always assume that $0< \gamma < 1$, without stating it explicitly.

Representing the operator A by its matrix elements $\big (A_j^{j'} (\ell - \ell ') \big )_{\ell , \ell ' \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}, j, j' \in {\mathbb {Z}}}$ as in (2.14), we have, for any $j' \in {\mathbb {Z}}$, $\ell ' \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}$ and any $\omega _1, \omega _2 \in { \mathtt \Omega }$, $\omega _1 \ne \omega _2$,

$$\begin{aligned} \begin{aligned}&{\mathop \sum }_{\ell , j} \langle \ell , j \rangle ^{2 s_1} \Big ( \big | A_j^{j'}(\ell - \ell ') \big |^2 + \gamma ^{2} \Big | \frac{\Delta _\omega A_j^{j'}(\ell - \ell ')}{|\omega _1 - \omega _2|} \Big |^2 \Big ) \\&\quad \lesssim \big ({{\mathfrak {M}}}_A(s_1) \big )^2 \langle \ell ', j' \rangle ^{2 (s_1+\sigma )} \end{aligned} \end{aligned}$$

(2.30)

where we recall that $\Delta _\omega f = f(\omega _1)- f (\omega _2)$.

Lemma 2.16

(Composition, [10, Lemma 2.20]) Let A, B be a $\mathrm{{Lip}(\gamma )}$-$\sigma _A$-tame and respectively, a $\mathrm{{Lip}(\gamma )}$-$\sigma _B$-tame operator with tame constants ${{\mathfrak {M}}}_A (s)$ and ${{\mathfrak {M}}}_B (s)$. Then the composition $A \circ B$ is $\mathrm{{Lip}(\gamma )}$-$(\sigma _A + \sigma _B)$-tame with a tame constant satisfying

$$\begin{aligned} {{\mathfrak {M}}}_{A B} (s) \lesssim {{\mathfrak {M}}}_{A}(s) {{\mathfrak {M}}}_{B} (s_1 + \sigma _A) + {{\mathfrak {M}}}_{A} (s_1) {{\mathfrak {M}}}_{B} (s + \sigma _A) . \end{aligned}$$

We now discuss the action of a $\mathrm{{Lip}(\gamma )}$-$\sigma$-tame operator $A(\omega )$ on a family of Sobolev functions $u (\omega ) \in H^s$.

Lemma 2.17

(Action on $H^s,$ [10, Lemma 2.22]) Let $A \,{:=}\, A(\omega )$ be a $\mathrm{{Lip}(\gamma )}$-$\sigma$-tame operator with tame constant ${{\mathfrak {M}}}_A(s)$. Then for any $s \in [s_1,S]$, for any family of Sobolev functions $u \,{:=}\, u(\omega ) \in H^{s+\sigma }$, Lipschitz continuous with respect to $\omega$, the following tame estimates hold:

$$\begin{aligned} \Vert A u \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim {{\mathfrak {M}}}_A(s_1) \Vert u \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ {{\mathfrak {M}}}_A(s) \Vert u \Vert _{s_1 + \sigma }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

Pseudo-differential operators are tame operators. We will use, in particular, the following lemma:

Lemma 2.18

Let $a( \varphi , x, \xi ; \omega ) \in S^0$ be a family of symbols that are Lipschitz continuous with respect to $\omega$. If $A = a( \varphi , x, D; \omega )$ satisfies $| A |_{0, s, 0}^\mathrm{{Lip}(\gamma )}< + \infty$ for any $s \geqq s_0$, then A is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

$$\begin{aligned} {{\mathfrak {M}}}_A(s) \leqq C(s) | A |_{0, s, 0}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.31)

As a consequence, for any $s \ge s_0$,

$$\begin{aligned} \Vert A u \Vert _s^\mathrm{{Lip}(\gamma )}\leqq C(s_0) | A |_{0, s_0, 0}^\mathrm{{Lip}(\gamma )}\Vert u \Vert _{s}^\mathrm{{Lip}(\gamma )}+ C(s) | A |_{0, s, 0}^\mathrm{{Lip}(\gamma )}\Vert u \Vert _{s_0}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.32)

Proof

See [10, Lemma 2.21] for the proof of (2.31). The estimate (2.32) then follows from Lemma 2.17. $\quad \square$

In the KAM reducibility scheme of Section  7, we need to consider $\mathrm{{Lip}(\gamma )}$-tame operators A which satisfy a stronger condition, referred to $\mathrm{{Lip}(\gamma )}$-modulo-tame.

Definition 2.19

($\mathrm{{Lip}(\gamma )}$-modulo-tame) A linear operator $A \,{:=}\, A(\omega )$ as in (2.12) is $\mathrm{{Lip}(\gamma )}$-modulo-tame if there exist numbers $s_1$, S with Let $s_0 \le s_1 < S$ and a non-decreasing function $[s_1, S] \rightarrow [0, + \infty )$, $s \mapsto {{\mathfrak {M}}}_{A}^\sharp (s)$, so that for any $s_1 \leqq s \leqq S$ and $u \in H^{s}$,

$$\begin{aligned}&\sup _{\omega \in { \mathtt {\Omega }}} \Vert \, | A(\omega ) | \, u \Vert _s + \gamma \sup _{\begin{array}{c} \omega _1, \omega _2 \in { \mathtt {\Omega }} \\ \omega _1 \ne \omega _2 \end{array}} \Big \Vert \, \frac{| A(\omega _1) - A(\omega _2) |}{|\omega _1- \omega _2|} u \Big \Vert _s \nonumber \\&\quad \leqq {{\mathfrak {M}}}_{A}^\sharp (s_1) \Vert u \Vert _{s} + {{\mathfrak {M}}}_{A}^\sharp (s) \Vert u \Vert _{s_1} . \end{aligned}$$

(2.33)

The constant ${{\mathfrak {M}}}_A^\sharp (s)$ is called a modulo-tame constant of the operator A.

Similarly as mentioned in Remark 2.15, the domain of definition of ${{\mathfrak {M}}}_A^\sharp (s)$ will often not be explicitly recorded. By Definition 2.19, if B is a $\mathrm{{Lip}(\gamma )}$-modulo-tame operator and A is a linear operator satisfying $| A_j^{j'} (\ell ) | \leqq | B_j^{j'} (\ell ) |$, then A is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant ${{\mathfrak {M}}}^\sharp _{A}(s)$ satisfying ${{\mathfrak {M}}}^\sharp _{A}(s) \leqq {{\mathfrak {M}}}^\sharp _{B}(s)$. Moreover, by comparing Definitions 2.19 and 2.14 (for $\sigma = 0$) one deduces the following lemma (cf. [10, Lemma 2.24] for details):

Lemma 2.20

An operator A which is $\mathrm{{Lip}(\gamma )}$-modulo-tame with modulo-tame constant ${{\mathfrak {M}}}_A^\sharp (s)$ is also $\mathrm{{Lip}(\gamma )}$-tame and ${{\mathfrak {M}}}_A^\sharp (s)$ is a tame constant for A.

The class of $\mathrm{{Lip}(\gamma )}$-modulo-tame operators (Definition 2.19) is closed under the operations coming up in the KAM reduction procedure, namely: sum and composition (Lemma 2.21); projections (Lemma 2.23); solution of the homological equation (Lemma 7.5). Let us give the precise statement of the first property.

Lemma 2.21

(Sum and composition [10, Lemma 2.25]) Let A, B be $\mathrm{{Lip}(\gamma )}$-modulo-tame operators with modulo-tame constants ${{\mathfrak {M}}}_A^\sharp (s)$ and, respectively , ${{\mathfrak {M}}}_B^\sharp (s)$. Then $A+ B$ is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant ${{\mathfrak {M}}}_{A + B}^\sharp (s)$ satisfying

$$\begin{aligned} {{\mathfrak {M}}}_{A + B}^\sharp (s) \leqq {{\mathfrak {M}}}_A^\sharp (s) + {{\mathfrak {M}}}_B^\sharp (s) . \end{aligned}$$

(2.34)

The composed operator $A \circ B$ is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant satisfying, for some $C \ge 1$,

$$\begin{aligned} {{\mathfrak {M}}}_{A B}^\sharp (s) \leqq C \, \big ( {{\mathfrak {M}}}_{A}^\sharp (s) {{\mathfrak {M}}}_{B}^\sharp (s_1) + {{\mathfrak {M}}}_{A}^\sharp (s_1) {{\mathfrak {M}}}_{B}^\sharp (s) \big ) . \end{aligned}$$

(2.35)

Assume in addition that $\langle \partial _{\varphi } \rangle ^{\mathtt {b}} A$, $\langle \partial _{\varphi } \rangle ^{\mathtt {b}} B$ (see Definition 2.4) are $\mathrm{{Lip}(\gamma )}$-modulo-tame with modulo-tame constants ${{\mathfrak {M}}}_{\langle \partial _{\varphi } \rangle ^{\mathtt {b}} A}^\sharp (s)$ and, respectively, ${{\mathfrak {M}}}_{\langle \partial _{\varphi } \rangle ^{\mathtt {b}} B}^\sharp (s)$. Then $\langle \partial _{\varphi } \rangle ^{\mathtt {b}} (A B)$ is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant ${{\mathfrak {M}}}_{\langle \partial _{\varphi } \rangle ^{\mathtt {b}} (A B)}^\sharp (s)$, bounded for some $C( {\mathtt {b}} ) \geqq 1$ by

$$\begin{aligned} C({\mathtt {b}})\Big ( {{\mathfrak {M}}}_{\langle \partial _{\varphi } \rangle ^{\mathtt {b}} A}^\sharp (s) {{\mathfrak {M}}}_{B}^\sharp (s_1) + {{\mathfrak {M}}}_{\langle \partial _{\varphi } \rangle ^{\mathtt {b}} A }^\sharp (s_1) {{\mathfrak {M}}}_{B}^\sharp (s) + {{\mathfrak {M}}}_{A}^\sharp (s) {{\mathfrak {M}}}_{ \langle \partial _{\varphi } \rangle ^{\mathtt {b}} B}^\sharp (s_1) + {{\mathfrak {M}}}_{A}^\sharp (s_1) {{\mathfrak {M}}}_{ \langle \partial _{\varphi } \rangle ^{\mathtt {b}} B}^\sharp (s)\Big ) .\nonumber \\ \end{aligned}$$

(2.36)

Iterating the tame estimates (2.35) and (2.36) for the composition of operators we get that, for any $n \geqq 2$,

$$\begin{aligned} {{\mathfrak {M}}}_{A^n}^\sharp (s) \leqq \big ( 2 C{{\mathfrak {M}}}_{A}^\sharp (s_1) \big )^{n - 1} {{\mathfrak {M}}}_{A}^\sharp (s), \end{aligned}$$

(2.37)

and

$$\begin{aligned} {{\mathfrak {M}}}_{\langle \partial _\varphi \rangle ^{ \mathtt {b}} A^n}^\sharp (s)\leqq & {} (4C(\mathtt {b}) C)^{n - 1} \Big ( {{\mathfrak {M}}}^\sharp _{\langle \partial _\varphi \rangle ^{\mathtt {b}} A}(s) \big [ {{\mathfrak {M}}}^\sharp _A(s_1) \big ]^{n - 1}\\&\qquad \qquad \qquad \qquad + {{\mathfrak {M}}}^\sharp _{\langle \partial _\varphi \rangle ^{\mathtt {b}} A}(s_1) {{\mathfrak {M}}}_A^\sharp (s) \big [ {{\mathfrak {M}}}_A^\sharp (s_1) \big ]^{n - 2} \Big ).\nonumber \end{aligned}$$

(2.38)

As an application of (2.37)–(2.38) we obtain the following:

Lemma 2.22

(Exponential map) Let A and $\langle \partial _{\varphi } \rangle ^{\mathtt {b}} A$ be $\mathrm{{Lip}(\gamma )}$-modulo-tame operators and assume that ${{\mathfrak {M}}}_{A}^\sharp : [s_1, S] \rightarrow [0, + \infty )$ is a modulo-tame constant satisfying

$$\begin{aligned} {{\mathfrak {M}}}_{A}^\sharp (s_1) \leqq 1 . \end{aligned}$$

(2.39)

Then the operators $\Phi ^{\pm 1} \,{:=}\, \mathrm{exp}(\pm A)$, $\Phi ^{\pm 1} - \mathrm{Id}$, and $\langle \partial _\varphi \rangle ^{\mathtt {b}}(\Phi ^{\pm 1} - \mathrm{Id})$ are $\mathrm{{Lip}(\gamma )}$-modulo-tame with modulo-tame constants satisfying, for any $s_1 \leqq s \leqq S$,

$$\begin{aligned} {{\mathfrak {M}}}_{ \Phi ^{\pm 1} - \mathrm{Id}}^\sharp (s) \!\!\lesssim & {} \! {{\mathfrak {M}}}^\sharp _{A}(s), \nonumber \\ {{\mathfrak {M}}}_{\langle \partial _\varphi \rangle ^{\mathtt {b}} (\Phi ^{\pm 1} - \mathrm{Id})}^\sharp (s)&\lesssim _{\mathtt {b}}&{{\mathfrak {M}} }_{\langle \partial _\varphi \rangle ^{\mathtt {b} } A}^\sharp (s) + {{\mathfrak {M}}}_A^\sharp (s) {{\mathfrak {M}} }_{\langle \partial _\varphi \rangle ^{\mathtt {b}} A}^\sharp (s_1) . \end{aligned}$$

(2.40)

Proof

In view of the identity $\Phi ^{\pm 1} - \mathrm{Id} = \sum _{n \geqq 1} \frac{(\pm A)^n}{n !}$ and the assumption (2.39) the claimed estimates follow by (2.37)–(2.38). $\quad \square$

Along the KAM reducibility scheme of Section 7.1 we need the following estimates for the operator $\Pi _N^\bot A \,{:=}\, A - \Pi _N A$ where $\Pi _N A$ is the smoothed operator, defined in (2.15):

Lemma 2.23

(Smoothing, [10, Lemma 2.27]) Suppose that $\langle \partial _{\varphi } \rangle ^{\mathtt {b}} A$, ${\mathtt {b}} \geqq 0$, is $\mathrm{{Lip}(\gamma )}$-modulo-tame. Then the operator $\Pi _N^\bot A$ (cf. Definition 2.4) is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant satisfying

$$\begin{aligned} {{\mathfrak {M}}}_{\Pi _N^\bot A}^\sharp (s) \leqq N^{- {\mathtt {b}} }{{\mathfrak {M}}}_{ \langle \partial _{\varphi } \rangle ^{\mathtt {b}} A}^\sharp (s) , \quad {{\mathfrak {M}}}_{\Pi _N^\bot A}^\sharp (s) \leqq {{\mathfrak {M}}}_{ A}^\sharp (s) . \end{aligned}$$

(2.41)

We will also encounter linear operators of the form $h \mapsto ( a_2, \, h )_{L^2_x} \, a_1$ where $a_1, a_2$ are smooth functions. According to the next lemma, such operators are modulo-tame regularizing. Recall that $\langle D \rangle$ denotes the Fourier multiplier with symbol $\langle \xi \rangle$.

Lemma 2.24

Let $a_1(\cdot ; \omega )$, $a_2(\cdot ; \omega )$ be functions in ${{{\mathcal {C}}}}^\infty ({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1, {\mathbb {C}})$ and $\omega \in \mathtt {\Omega }$. Consider the linear operator ${{{\mathcal {R}}}} : L^2_x \rightarrow L^2_x, \, h \mapsto ( a_2, h )_{L^2_x} \, a_1$. Then for any $\lambda \in {\mathbb {N}}^{{\mathbb {S}}_+}$ and $n_1, n_2 \geqq 0$, the operator $\langle D \rangle ^{n_1} \partial _\varphi ^\lambda {{{\mathcal {R}}}} \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying, for some $\sigma \equiv \sigma (n_1, n_2, \lambda ) > 0$ and, for any $s \ge s_0$,

$$\begin{aligned} {{\mathfrak {M}}}_{\langle D \rangle ^{n_1} \partial _\varphi ^\lambda {{{\mathcal {R}}}} \langle D \rangle ^{n_2}}(s) \lesssim _{ s, n_1, n_2, \lambda } \, ( \mathrm{max}_{i = 1, 2} \Vert a_i \Vert _{s+ \sigma } ) \cdot ( \mathrm{max}_{i = 1, 2} \Vert a_i \Vert _{s_0+ \sigma } ) . \end{aligned}$$

Proof

For any $n_1, n_2 \geqq 0$, $\lambda \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $h \in L^2_x$, one has that, for some combinatorial constants $c_{\lambda _1, \lambda _2}$,

$$\begin{aligned} \begin{aligned} \langle D \rangle ^{n_1} \partial _\varphi ^\lambda {{{\mathcal {R}}}} \langle D \rangle ^{n_2} h&= \sum _{\lambda _1 + \lambda _2 = \lambda } c_{\lambda _1, \lambda _2} \langle D \rangle ^{n_1}[\partial _\varphi ^{\lambda _1} a_1] \, \big ( \langle D \rangle ^{n_2}[\partial _\varphi ^{\lambda _2} a_2],\, h \big )_{L^2_x}, \end{aligned} \end{aligned}$$

where we used that the operator $\langle D \rangle$ is self-adjoint. The lemma then follows by (2.5). $\quad \square$

2.4 Tame Estimates

In this section we record tame estimates for compositions of functions and operators with the lift $\breve{\iota }$ of a torus embedding ${\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow {\mathcal {E}}_s$ (cf. (1.25)) of the form

$$\begin{aligned} \breve{\iota } (\varphi ) = (\varphi ,0,0) + \iota (\varphi ) \quad \mathrm{where} \quad \iota (\varphi ) = ( \Theta (\varphi ), y(\varphi ), w(\varphi ) ) \ \ \mathrm{is} \ \ (2 \pi {\mathbb {Z}})^{{{\mathbb {S}}}_+} \text {-periodic} , \end{aligned}$$

endowed with the norm $\Vert \iota \Vert _s^\mathrm{{Lip}(\gamma )}\,{:=}\, \Vert \Theta \Vert _{H^s_\varphi }^\mathrm{{Lip}(\gamma )}+ \Vert y \Vert _{H^s_\varphi }^\mathrm{{Lip}(\gamma )}+ \Vert w \Vert _s^\mathrm{{Lip}(\gamma )}$. The main results are Lemmas 2.25 and 2.26, whose relevance is described in Remark 2.27.

Note that the norm $\Vert \cdot \Vert _s$ of the Sobolev space $H^s = H^s({\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {T}}_1)$, introduced in (1.15), is equivalent to

$$\begin{aligned} \Vert \ \Vert _s = \Vert \ \Vert _{H^s_{\varphi ,x}} \sim _s \Vert \ \Vert _{H^s_\varphi L^2_x} + \Vert \ \Vert _{L^2_\varphi H^s_x} \, \end{aligned}$$

(2.42)

and that by interpolation estimates (which are proved using Young’s inequality), one has

$$\begin{aligned} \Vert w \Vert _{H^s_\varphi H^\sigma _x} \leqq \Vert w \Vert _{H^{s+\sigma }_\varphi L^2_x } + \Vert w \Vert _{L^{2}_\varphi H^{s+\sigma }_x } \lesssim _{s,\sigma } \Vert w \Vert _{s + \sigma }. \end{aligned}$$

(2.43)

Given a Banach space X with norm $\Vert \ \Vert _X$ and $s \in {\mathbb {N}}$, we denote by ${{{\mathcal {C}}}}^s({\mathbb {T}}^{{{\mathbb {S}}}_+}, X)$ the Banach space of ${{{\mathcal {C}}}}^s$-smooth maps $f: {\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow X$, equipped with the norm

$$\begin{aligned} \Vert f \Vert _{{{{\mathcal {C}}}}^s_\varphi X} \,{:=}\, \sum _{0 \leqq |\alpha | \leqq s} \Vert \partial _\varphi ^\alpha f \Vert ^{\sup }_X, \quad \Vert \partial _\varphi ^\alpha f \Vert ^{\sup }_X \,{:=}\, \sup _{\varphi \in {\mathbb {T}}^{{{\mathbb {S}}}_+}} \Vert \partial _\varphi ^\alpha f(\varphi ) \Vert _X . \end{aligned}$$

(2.44)

If X is a Banach space and H a Hilbert space, the following Sobolev embedding results hold: for any $s_1 \in {\mathbb {N}}$,

$$\begin{aligned}&H^{s+s_1} ({\mathbb {T}}^{{{\mathbb {S}}}_+}, X) \hookrightarrow {{{\mathcal {C}}}}^{s_1} ({\mathbb {T}}^{{{\mathbb {S}}}_+}, X) , \quad \forall s> |{{\mathbb {S}}}_+| , \nonumber \\&H^{s + s_1} ({\mathbb {T}}^{{{\mathbb {S}}}_+}, H) \hookrightarrow {{{\mathcal {C}}}}^{s_1} ({\mathbb {T}}^{{{\mathbb {S}}}_+}, H) , \quad \forall s > |{{\mathbb {S}}}_+|/2 . \end{aligned}$$

(2.45)

For the convenience of the reader, let us prove the two cases for $s_1 = 0$. To see that $H^{s} ({\mathbb {T}}^{{{\mathbb {S}}}_+}, X) \hookrightarrow {{{\mathcal {C}}}}^0 ({\mathbb {T}}^{{{\mathbb {S}}}_+}, X)$ for $s > |{{\mathbb {S}}}_+|$, consider for any given $u \in H^s ({\mathbb {T}}^{{{\mathbb {S}}}_+}, X)$ its Fourier expansion $u(\varphi ) = \sum _{\ell \in {\mathbb {Z}}^{{\mathbb {S}}_+}} u_\ell e^{\mathrm{i} \ell \cdot \varphi }$, $u_\ell \in X$. Integrating by parts, one has

$$\begin{aligned} \Vert u_\ell \Vert _X = \Big \Vert \frac{1}{(2 \pi )^{|{\mathbb {S}}_+|}} \int _{{\mathbb {T}}^{{\mathbb {S}}_+}} u(\varphi ) e^{- \mathrm{i} \ell \cdot \varphi }\, \hbox {d} \varphi \Big \Vert _X \lesssim _s \langle \ell \rangle ^{- s} \Vert u \Vert _{H^s_\varphi X}, \quad \forall \, \ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}. \end{aligned}$$

The claimed embedding then follows, since $s > |{{\mathbb {S}}}_+|$:

$$\begin{aligned} \Vert u \Vert _{C^0_\varphi X} \leqq \sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} \Vert u_\ell \Vert _X \lesssim _s \Vert u \Vert _{H^s_\varphi X} \sum _{\ell \in {\mathbb {Z}}^{{\mathbb {S}}_+}} \langle \ell \rangle ^{- s} \lesssim _s \Vert u \Vert _{H^s_\varphi X} . \end{aligned}$$

To see that $H^{s} ({\mathbb {T}}^{{{\mathbb {S}}}_+}, H) \hookrightarrow {{{\mathcal {C}}}}^0 ({\mathbb {T}}^{{{\mathbb {S}}}_+}, H)$ for $s > |{{\mathbb {S}}}_+| /2$, use Plancherel’s identity $\Vert u \Vert _{H^s_\varphi X}^2 \sim _s \sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} \Vert u_\ell \Vert _X^2 \langle \ell \rangle ^{2s}$ to conclude that $\sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} \Vert u_\ell \Vert _X \leqq (\sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} \Vert u_\ell \Vert _X^2 \langle \ell \rangle ^{2s})^{1/2} (\sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} \langle \ell \rangle ^{-2s})^{1/2} \lesssim _s \Vert u \Vert _{H^s_\varphi X}$.

On the Banach spaces ${{{\mathcal {C}}}}^s({\mathbb {T}}^{{\mathbb {S}}_+}, X)$ the following interpolation inequalities hold: for any integer $0 \leqq k \leqq s$,

$$\begin{aligned} \Vert f \Vert _{{{{\mathcal {C}}}}^k_\varphi X} \lesssim _s \Vert f \Vert _{{{{\mathcal {C}}}}^0_\varphi X}^{1 - \frac{k}{s}} \Vert f \Vert _{{{{\mathcal {C}}}}^s_\varphi X}^{\frac{k}{s}}. \end{aligned}$$

(2.46)

In the next lemma, we assume the tame estimates (2.47) for the function ${{\mathfrak {x}}} = (\theta , y, w) \mapsto a({{\mathfrak {x}}} )$ in the x-variable only, and we deduce tame estimates for the composed function $a( \breve{\iota } (\varphi ) )$ in the variables $(\varphi , x)$. Recall that ${{{\mathcal {E}}}}_s, E_s$ are defined in (1.25) and ${{{\mathcal {V}}}}^s(\delta )$ in (1.27). Let $\Omega$ be an open bounded subset of ${\mathbb {R}}^{{\mathbb {S}}_+}$. In more detail, the following holds:

Lemma 2.25

(Tame estimates for functions) Let $\sigma > 0$ and assume that, for any $s \geqq 0$, the map $a : ({{{\mathcal {V}}}}^{ \sigma }(\delta ) \cap {{{\mathcal {E}}}}_{s+\sigma }) \times \Omega \rightarrow H^s ({\mathbb {T}}_1)$ is ${{{\mathcal {C}}}}^\infty$ with respect to ${{\mathfrak {x}}} = (\theta , y, w)$, ${{{\mathcal {C}}}}^1$ with respect to $\omega$, and satisfies for any ${\mathfrak {x}} \in {{{\mathcal {V}}}}^{\sigma }(\delta ) \cap {{\mathcal {E}}}_{s+\sigma }$, $\alpha \in {\mathbb {N}}^{{{\mathbb {S}}}_+}$ with $|\alpha | \leqq 1$, and $l \geqq 1$, the tame estimates

$$\begin{aligned}&\Vert \partial _\omega ^\alpha a ({\mathfrak {x}}; \omega )\Vert _{H^s_x} \lesssim _s 1 + \Vert w \Vert _{H^{s + \sigma }_x},\nonumber \\&\Vert d^l \partial _\omega ^\alpha a ( {\mathfrak {x}}; \omega ) [\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^s_x} \lesssim _{s, l, \alpha } \sum _{j = 1}^l \Big ( \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_{s + \sigma }} \prod _{n \ne j} \Vert \widehat{{\mathfrak {x}}}_n\Vert _{E_{\sigma }} \Big )\\&+ \Vert w \Vert _{H^{s + \sigma }_x} \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j \Vert _{E_{\sigma }}.\nonumber \end{aligned}$$

(2.47)

Then for any torus embedding $\breve{\iota }$ with $\Vert \iota \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\leqq \delta$ and any maps ${\widehat{\iota }}, {\widehat{\iota }}_1, {\widehat{\iota }}_2 : {\mathbb {T}}^{{\mathbb {S}}_+} \rightarrow E_{s + s_0 + \sigma }$, the following tame estimates hold for any $s \geqq 0$:

• (i)
  $$\begin{aligned} \begin{aligned}&\\&\Vert a(\breve{\iota })\Vert _s^{\mathrm{{Lip}(\gamma )}} \lesssim _s 1 + \Vert \iota \Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} , \\&\Vert \hbox {d} a (\breve{\iota })[{\widehat{\iota }}] \Vert _{s}^{\mathrm{{Lip}(\gamma )}} \lesssim _s \Vert {\widehat{\iota }}\Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} + \Vert \iota \Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert {\widehat{\iota }}\Vert _{s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} , \\&\Vert d^2 a (\breve{\iota })[\widehat{\iota }_1, \widehat{\iota }_2] \Vert _{s}^{\mathrm{{Lip}(\gamma )}} \lesssim _s \Vert \widehat{\iota }_1\Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert \widehat{\iota }_2\Vert _{s_0 + \sigma } \\&\quad \quad + \Vert \widehat{\iota }_1\Vert _{s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert \widehat{\iota }_2\Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} + \Vert \iota \Vert _{s+ s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert \widehat{\iota }_1\Vert _{s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert \widehat{\iota }_2\Vert _{s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} . \end{aligned} \end{aligned}$$

  (2.48)

• (ii) If in addition $a(\theta , 0, 0; \omega ) = 0$, then $\Vert a (\breve{\iota }) \Vert _{s}^{\mathrm{{Lip}(\gamma )}} \lesssim _s \Vert \iota \Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}}$.

• (iii) If in addition $a(\theta , 0, 0; \omega ) =0,$ $\partial _y a(\theta , 0, 0; \omega ) = 0$, and $\partial _w a(\theta , 0, 0; \omega ) = 0$, then

  $$\begin{aligned} \begin{aligned}&\Vert a(\breve{\iota })\Vert _s^{\mathrm{{Lip}(\gamma )}} \lesssim _s \Vert \iota \Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert \iota \Vert _{s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} ,\\&\Vert da(\breve{\iota })[{\widehat{\iota }}]\Vert _s^{\mathrm{{Lip}(\gamma )}} \lesssim _s \Vert \iota \Vert _{s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert {\widehat{\iota }}\Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} + \Vert \iota \Vert _{s + s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} \Vert {\widehat{\iota }}\Vert _{s_0 + \sigma }^{\mathrm{{Lip}(\gamma )}} . \end{aligned} \end{aligned}$$

Proof

(i) It suffices to prove the estimates for $\Vert d^2 a(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2]\Vert _s$ and $\Vert d^2 a(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2] \Vert _{s}^\mathrm{lip}$ in (2.48) since the ones for $a(\breve{\iota })$ and $d a(\breve{\iota })$ then follow by Taylor expansions. By the hypothesis (2.47) with $l = 2$, $\alpha = 0$, we have, for any $\varphi \in {\mathbb {T}}^{{\mathbb {S}}_+}$, $s \geqq 0$,

$$\begin{aligned} \Vert d^2 a(\breve{\iota }(\varphi ))[{\widehat{\iota }}_1(\varphi ), {\widehat{\iota }}_2(\varphi )]\Vert _{H^s_x}&\lesssim _s \Vert {\widehat{\iota }}_1(\varphi )\Vert _{E_{s + \sigma }} \Vert {\widehat{\iota }}_2(\varphi )\Vert _{E_{\sigma }} + \Vert {\widehat{\iota }}_1(\varphi )\Vert _{E_{\sigma }} \Vert {\widehat{\iota }}_2(\varphi )\Vert _{E_{s + \sigma }} \nonumber \\&\quad + \Vert \iota (\varphi ) \Vert _{E_{s + \sigma }}\Vert {\widehat{\iota }}_1(\varphi )\Vert _{E_{ \sigma }} \Vert {\widehat{\iota }}_2(\varphi )\Vert _{E_{\sigma }}. \end{aligned}$$

(2.49)

Squaring the expressions on the left and right hand side of (2.49) and then integrating them with respect to $\varphi$, one concludes, using (2.42), (2.43), and the Sobolev embedding (2.45), that

$$\begin{aligned}&\Vert d^2 a(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2]\Vert _{L^2_\varphi H^s_x} \lesssim _s \Vert {\widehat{\iota }}_1\Vert _{s + \sigma } \Vert {\widehat{\iota }}_2\Vert _{s_0 + \sigma }\nonumber \\&+ \Vert {\widehat{\iota }}_1\Vert _{s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{s + \sigma } + \Vert \iota \Vert _{s + \sigma }\Vert {\widehat{\iota }}_1\Vert _{s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{s_0 + \sigma }. \end{aligned}$$

(2.50)

In order to estimate $\Vert d^2 a(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2] \Vert _{H^s_\varphi L^2_x}$, we estimate $\Vert d^2 a(\breve{\iota }) [{\widehat{\iota }}_1, {\widehat{\iota }}_2] \Vert _{{{{\mathcal {C}}}}^s_\varphi L^2_x}$. We claim that

$$\begin{aligned} \begin{aligned} \Vert d^2 a(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2]\Vert _{{{{\mathcal {C}}}}^s_\varphi L^2_x}&\lesssim _s \Vert {\widehat{\iota }}_1\Vert _{s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{s + s_0 + \sigma } \\&\quad + \Vert {\widehat{\iota }}_1\Vert _{s + s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{ s_0 + \sigma } + \Vert \iota \Vert _{s + s_0 + \sigma } \Vert {\widehat{\iota }}_1 \Vert _{s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{s_0 + \sigma }, \end{aligned} \end{aligned}$$

(2.51)

so that the estimate for $\Vert d^2 a(\breve{\iota }) [{\widehat{\iota }}_1, {\widehat{\iota }}_2] \Vert _s$ stated in (2.48) follows by (2.50), (2.51), and (2.42). The bound for $\Vert d^2 a(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2] \Vert _{s}^\mathrm{lip}$ is obtained in the same fashion.

Proof of (2.51). By the Leibnitz rule, for any $\beta \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $0 \leqq |\beta | \leqq s$,

$$\begin{aligned}&\partial _\varphi ^\beta \Big ( d^2 a(\breve{\iota }(\varphi ))[{\widehat{\iota }}_1(\varphi ), {\widehat{\iota }}_2(\varphi )] \Big )\nonumber \\&= \sum _{\beta _1 + \beta _2 + \beta _3 = \beta } c_{\beta _1, \beta _2, \beta _3} \partial _\varphi ^{\beta _1}(d^2 a(\breve{\iota }(\varphi ))) \big [ \partial _\varphi ^{\beta _2} {\widehat{\iota }}_1(\varphi ), \partial _\varphi ^{\beta _3} {\widehat{\iota }}_2(\varphi ) \big ], \end{aligned}$$

(2.52)

where $c_{\beta _1, \beta _2, \beta _3}$ are combinatorial constants. Each term in the latter sum is estimated individually. For $1 \leqq |\beta _1| \leqq s$, we have

$$\begin{aligned} \begin{aligned}&\partial _\varphi ^{\beta _1} (d^2 a(\breve{\iota } (\varphi ) )) \big [ \partial _\varphi ^{\beta _2} {\widehat{\iota }}_1(\varphi ), \partial _\varphi ^{\beta _3} {\widehat{\iota }}_2(\varphi ) \big ] \\&\quad = \sum _{\begin{array}{c} 1 \leqq m \leqq |\beta _1| \\ \alpha _1 + \cdots + \alpha _m = \beta _1 \end{array}} c_{\alpha _1, \cdots , \alpha _m} d^{m + 2} a(\breve{\iota }(\varphi )) \big [ \partial _\varphi ^{\alpha _1} \breve{\iota }(\varphi ), \cdots , \partial _\varphi ^{\alpha _m} \breve{\iota }(\varphi ), \partial _\varphi ^{\beta _2} {\widehat{\iota }}_1(\varphi ), \partial _\varphi ^{\beta _3} {\widehat{\iota }}_2(\varphi ) \big ] \end{aligned} \end{aligned}$$

for suitable combinatorial constants $c_{\alpha _1, \cdots , \alpha _m}$. Then, by (2.47) with $l = m + 2$, $\alpha = 0$, we have the bound

$$\begin{aligned}&\Vert \partial _\varphi ^{\beta _1} (d^2 a(\breve{\iota }))[\partial _\varphi ^{\beta _2} {\widehat{\iota }}_1, \partial _\varphi ^{\beta _3} {\widehat{\iota }}_2] \Vert _{{{{\mathcal {C}}}}^0_\varphi L^2_x} \\&\quad \lesssim _\beta \sum _{\begin{array}{c} 1 \leqq m \leqq |\beta _1| \\ \alpha _1 + \cdots + \alpha _m = \beta _1 \end{array}} (1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\alpha _1|}_\varphi E_{ \sigma }}) \cdots (1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\alpha _m|}_\varphi E_{ \sigma }}) \Vert {\widehat{\iota }}_1\Vert _{{{{\mathcal {C}}}}^{|\beta _2|}_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2\Vert _{{{{\mathcal {C}}}}^{|\beta _3|}_\varphi E_{ \sigma }}. \nonumber \end{aligned}$$

(2.53)

Arguing as in the proof of the formula (75) in [9], for any $j = 1, \ldots , m$, we have

$$\begin{aligned} (1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\alpha _j|}_\varphi E_{ \sigma }}) \lesssim _\beta (1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma }})^{1 - \frac{|\alpha _j|}{|\beta |}}(1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\beta |}_\varphi E_{ \sigma }})^{\frac{|\alpha _j|}{|\beta |}} , \end{aligned}$$

and, using the interpolation estimate (2.46), we get

$$\begin{aligned}&(1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\alpha _1|}_\varphi E_{ \sigma }}) \cdots ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\alpha _m|}_\varphi E_{ \sigma }}) \Vert {\widehat{\iota }}_1\Vert _{{{{\mathcal {C}}}}^{|\beta _2|}_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2\Vert _{{{{\mathcal {C}}}}^{|\beta _3|}_\varphi E_{ \sigma }} \\&\quad \lesssim _s \Vert {\widehat{\iota }}_1 \Vert ^{1 - \frac{|\beta _2|}{|\beta |}}_{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_1\Vert ^{\frac{|\beta _2|}{|\beta |}}_{{{{\mathcal {C}}}}^{|\beta |}_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2 \Vert ^{1 - \frac{|\beta _3|}{|\beta |}}_{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2\Vert ^{\frac{|\beta _3|}{|\beta |}}_{{{{\mathcal {C}}}}^{|\beta |}_\varphi E_{ \sigma }} \prod _{j = 1}^m ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma } })^{1 - \frac{|\alpha _j|}{|\beta |}} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\beta |}_\varphi E_{ \sigma }})^{\frac{|\alpha _j|}{|\beta |}} \nonumber \\&\lesssim _s \Vert {\widehat{\iota }}_1 \Vert ^{1 - \frac{|\beta _2|}{|\beta |}}_{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_1\Vert ^{\frac{|\beta _2|}{|\beta |}}_{{{{\mathcal {C}}}}^{|\beta |}_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2 \Vert ^{1 - \frac{|\beta _3|}{|\beta |}}_{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2\Vert ^{\frac{|\beta _3|}{|\beta |}}_{{{{\mathcal {C}}}}^{|\beta |}_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma } })^{m - \sum _{j = 1}^m\frac{|\alpha _j|}{|\beta |}} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{|\beta |}_\varphi E_{ \sigma }})^{\sum _{j = 1}^m\frac{|\alpha _j|}{|\beta |}} . \nonumber \end{aligned}$$

(2.54)

By (2.45), (2.43), $(1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^0_\varphi E_\sigma })^{m - 1} {\mathop {\lesssim }\limits ^{}} (1 + \Vert \iota \Vert _{s_0 + \sigma })^{m - 1} \lesssim (1+ \delta )^{m-1}$ and $\frac{\sum _{j =1}^m |\alpha _j|}{|\beta |} = \frac{|\beta _1|}{|\beta |} = 1 - \frac{|\beta _2|}{|\beta |} - \frac{|\beta _3|}{|\beta |}$, so that

$$\begin{aligned}&(2.54) \lesssim _s \Vert {\widehat{\iota }}_1 \Vert ^{ \frac{|\beta _1| + |\beta _3|}{|\beta |}}_{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_1\Vert ^{\frac{|\beta _2|}{|\beta |}}_{{{{\mathcal {C}}}}^{s}_\varphi E_{\sigma }} \Vert {\widehat{\iota }}_2 \Vert ^{\frac{|\beta _1| + |\beta _2|}{|\beta |}}_{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2\Vert ^{\frac{|\beta _3|}{|\beta |}}_{{{{\mathcal {C}}}}^{s}_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma }})^{\frac{|\beta _2| + |\beta _3|}{|\beta |}} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{s}_\varphi E_{ \sigma }})^{\frac{|\beta _1|}{|\beta |}} \\&\lesssim _s \Big ( \Vert {\widehat{\iota }}_1 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{s}_\varphi E_{ \sigma }}) \Big )^{\frac{|\beta _1|}{|\beta |}} \Big ( \Vert {\widehat{\iota }}_1\Vert _{{{{\mathcal {C}}}}^{s}_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma }}) \Big )^{\frac{|\beta _2|}{|\beta |}} \\&\quad \times \Big ( \Vert {\widehat{\iota }}_1 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{\sigma }} \Vert {\widehat{\iota }}_2 \Vert _{{{{\mathcal {C}}}}^s_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma }}) \Big )^{\frac{|\beta _3|}{|\beta |}} \end{aligned}$$

and, by the iterated Young inequality with exponents $|\beta |/ |\beta _1|$, $|\beta |/ |\beta _2|$, $|\beta |/|\beta _3|$, we conclude that the expression on the line (2.54) is bounded by

$$\begin{aligned}&\Vert {\widehat{\iota }}_1 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{s}_\varphi E_{ \sigma }}) + \Vert {\widehat{\iota }}_1\Vert _{{{{\mathcal {C}}}}^{s}_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma }}) \\&\quad + \Vert {\widehat{\iota }}_1 \Vert _{{{{\mathcal {C}}}}^0_\varphi E_{ \sigma }} \Vert {\widehat{\iota }}_2 \Vert _{{{{\mathcal {C}}}}^s_\varphi E_{ \sigma }} ( 1 + \Vert \iota \Vert _{{{{\mathcal {C}}}}^{0}_\varphi E_{ \sigma }}) \\&{\mathop {\lesssim _s}\limits ^{(2.45), (2.43)}} \Vert \iota \Vert _{s + s_0 + \sigma } \Vert {\widehat{\iota }}_1 \Vert _{s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{s_0 + \sigma }\\&\quad + \Vert {\widehat{\iota }}_1\Vert _{s + s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{ s_0 + \sigma } + \Vert {\widehat{\iota }}_1\Vert _{s_0 + \sigma } \Vert {\widehat{\iota }}_2\Vert _{s + s_0 + \sigma } . \end{aligned}$$

Clearly, the term (2.53) satisfies the same type of bound as (2.54). The left hand side in (2.52) with $\beta _1 = 0$ is estimated in the same way and thus (2.51) is proved.

Proof (ii)–(iii). Let $\varphi \mapsto \breve{\iota } (\varphi ) = (\theta (\varphi ), y(\varphi ), w(\varphi ))$ be a torus embedding. If $a(\theta , 0, 0) = 0$, one has by the mean value theorem $a(\breve{\iota }) = \int _0^1 d a(\breve{\iota }_t)[{\widehat{\iota }}]\, \hbox {d} t$ with $\breve{\iota }_t \,{:=}\, (1 -t) (\theta (\varphi ), 0, 0) + t \breve{\iota }(\varphi )$ and ${\widehat{\iota }} \,{:=}\, (0, y(\varphi ), w(\varphi ))$. If $a(\theta , 0, 0), \partial _y a(\theta , 0, 0 ), \partial _w a(\theta , 0, 0)$ vanish, we write $a(\breve{\iota }) = \int _0^1 (1 - t) d^2 a(\breve{\iota }_t)[{\widehat{\iota }}, {\widehat{\iota }}]\, \hbox {d} t$.

Items (ii)–(iii) follow by (i), noting that $\Vert {\widehat{\iota }} \Vert _s^\mathrm{{Lip}(\gamma )}= \Vert (0, y(\cdot ), w(\cdot ))\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim \Vert \iota \Vert _s^\mathrm{{Lip}(\gamma )}$ for any $s \geqq 0$. $\quad \square$

Given $M \in {\mathbb {N}}$, we define the constant

$$\begin{aligned} {\mathfrak {s}}_M\,{:=}\, \mathrm{max}\{ s_0, M + 1 \}. \end{aligned}$$

(2.55)

Lemma 2.26

(Tame estimates for smoothing operators) Assume that, for any $M \geqq 0$, there is $\sigma _M \geqq 0$ so that the following holds:

• Assumption A. For any $s \ge 0$, the map

  $$\begin{aligned} {{{\mathcal {R}}} } : ({{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{\mathcal {E}}}_{s+\sigma _M}) \times \Omega \rightarrow {\mathcal {B}} (H^s ({\mathbb {T}}_1) , H^{s + M + 1} ({\mathbb {T}}_1) ) \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ with respect to ${\mathfrak {x}}$, ${{{\mathcal {C}}}}^1$ with respect to $\omega$ and, for any ${\mathfrak {x}} \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{\mathcal {E}}}_{s+\sigma _M}$, $\alpha \in {\mathbb {N}}^{{{\mathbb {S}}}_+}$ with $|\alpha | \leqq 1$,

  $$\begin{aligned} \Vert \partial _\omega ^{\alpha } {{{\mathcal {R}}}} ({\mathfrak {x}}; \omega )[{\widehat{w}}]\Vert _{H_x^{s + M + 1}} \lesssim _{s, M} \Vert {\widehat{w}} \Vert _{H^s_x} + \Vert w \Vert _{H^{s + \sigma _M}_x} \Vert {\widehat{w}}\Vert _{L^2_x} , \end{aligned}$$

  and, for any $l \geqq 1$,

  $$\begin{aligned}&\Vert d^l \partial _\omega ^{\alpha } {{{\mathcal {R}}}} ({\mathfrak {x}}; \omega )[{\widehat{w}}][\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H_x^{s + M + 1}} \lesssim _{s, M, l} \Vert {\widehat{w}} \Vert _{H^s_x} \prod _{j=1}^l \Vert \widehat{{\mathfrak {x}}}_j \Vert _{E_{\sigma _M}} \\&\qquad + \Vert {\widehat{w}} \Vert _{L^2_x} \Big ( \Vert w \Vert _{H_x^{s+ \sigma _M}} \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j \Vert _{E_{\sigma _M}} + \sum _{j = 1}^l \big ( \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_{s + \sigma _M}} \prod _{n \ne j} \Vert \widehat{{\mathfrak {x}}}_n\Vert _{E_{\sigma _M}}\big ) \Big ). \end{aligned}$$

• Assumption B. For any $- M - 1 \leqq s \leqq 0$, the map

  $$\begin{aligned} {{{\mathcal {R}}} } : {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Omega \rightarrow {\mathcal {B}} (H^s ({\mathbb {T}}_1) , H^{s + M + 1} ({\mathbb {T}}_1) ) \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ with respect to ${\mathfrak {x}}$, ${{{\mathcal {C}}}}^1$ with respect to $\omega$ and, for any ${\mathfrak {x}} \in {{{\mathcal {V}}}}^{\sigma _M}(\delta )$, $\alpha \in {\mathbb {N}}^{{{\mathbb {S}}}_+}$ with $|\alpha | \leqq 1$, and $l \geqq 1$,

  $$\begin{aligned}&\Vert \partial _\omega ^{\alpha } {{{\mathcal {R}}}} ({\mathfrak {x}}; \omega )[{\widehat{w}}]\Vert _{H_x^{s + M + 1}} \lesssim _{s, M} \Vert {\widehat{w}} \Vert _{H^s_x} , \\&\Vert d^l \partial _\omega ^{\alpha } {{{\mathcal {R}}}} ({\mathfrak {x}}; \omega )[{\widehat{w}}][\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H_x^{s + M + 1}} \lesssim _{s, M, l} \Vert {\widehat{w}} \Vert _{H^{s}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \end{aligned}$$

Then for any $S \ge {\mathfrak {s}}_M$ and $\lambda \in {\mathbb {N}}^{{\mathbb {S}}_+}$, there is a constant $\sigma _M(\lambda ) > 0$, so that for any $\breve{\iota }(\varphi ) = (\varphi , 0, 0) + \iota (\varphi )$ with $\Vert \iota \Vert _{s_0 + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}\leqq \delta$ and any $n_1, n_2 \in {\mathbb {N}}$ satisfying $n_1 + n_2 \leqq M + 1$, the following holds:

• (i) The operator $\langle D \rangle ^{n_1} \partial _\varphi ^\lambda ({{{\mathcal {R}}}} \circ \breve{\iota }) \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying, for any ${\mathfrak {s}}_M\leqq s \leqq S$,

  $$\begin{aligned} {{\mathfrak {M}}}_{\langle D \rangle ^{n_1} \partial _{{\varphi }}^\lambda ({{{\mathcal {R}}}} \circ \breve{\iota }) \langle D \rangle ^{n_2}}(s) \lesssim _{S, M, \lambda } 1 + \Vert \iota \Vert _{s+ \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

• (ii) The operator $\langle D \rangle ^{n_1} \partial _{\varphi }^\lambda (d {{{\mathcal {R}}}}( \breve{\iota })[{\widehat{\iota }}]) \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying, for any ${\mathfrak {s}}_M\leqq s \leqq S$,

  $$\begin{aligned} {{\mathfrak {M}}}_{\langle D \rangle ^{n_1} \partial _{\varphi }^\lambda (d {{{\mathcal {R}}}}( \breve{\iota })[{\widehat{\iota }}]) \langle D \rangle ^{n_2}}(s) \lesssim _{S, M, \lambda } \Vert {\widehat{\iota }} \Vert _{s + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s+ \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}\Vert _{s_0 + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

• (iii) If in addition ${{{\mathcal {R}}}}(\theta , 0, 0; \omega ) = 0$, then the operator $\langle D \rangle ^{n_1} \partial _{\varphi }^\lambda ({{{\mathcal {R}}}} \circ \breve{\iota }) \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying, for any ${\mathfrak {s}}_M\leqq s \leqq S$,

  $$\begin{aligned} {{\mathfrak {M}}}_{\langle D \rangle ^{n_1} \partial _{\varphi }^\lambda ({{{\mathcal {R}}}} \circ \breve{\iota }) \langle D \rangle ^{n_2}}(s) \lesssim _{S, M, \lambda } \Vert \iota \Vert _{s+ \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

Remark 2.27

Let us comment on Lemma 2.26 and its applications. Under the above assumptions A and B for the operator valued map ${{\mathfrak {x}}} = (\theta , y, w) \mapsto {{{\mathcal {R}}}}({{\mathfrak {x}}} )$, with ${{{\mathcal {R}}}}({{\mathfrak {x}}})$ acting on spaces of functions of the x-variable only, we obtained tame estimates for the composed operator ${{{\mathcal {R}}}} ( \breve{\iota } (\varphi ) )$, acting on spaces of functions in the variables $(\varphi , x)$. Assumption B regarding the action of ${{{\mathcal {R}}}}({{\mathfrak {x}}} )$ on negative Sobolev spaces is used to prove that also $\langle D \rangle ^{n_1} {{{\mathcal {R}}}} ( \breve{\iota } (\varphi ) ) \langle D \rangle ^{n_2}$ with $n_1 + n_2 \leqq M + 1$ is a modulo-tame operator. Lemma 2.26 will be used in the proof of Lemma 6.4 to show that the remainder ${{\mathcal {R}}_M^{(1)} }$ in the expansion (6.23) is a tame operator satisfying (6.31). The verification that ${{\mathcal {R}}_M^{(1)} }$ satisfies the assumptions A and B of Lemma 2.26 is proved by applying Lemmata 3.5 and  3.7.

Proof

Since items (i) and (ii) can be proved in a similar way, we only prove (ii). For any given $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 \leqq M + 1$, set ${{{\mathcal {Q}}}} \,{:=}\, \langle D \rangle ^{n_1} {{{\mathcal {R}}}} \langle D \rangle ^{n_2}$. Assumption A implies that for any $s \geqq M + 1$ and any ${\mathfrak {x}} \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{{\mathcal {E}}}}_{s + \sigma _M}$, the operator ${{{\mathcal {Q}}}}(\mathfrak {x})$ is in ${{{\mathcal {B}}}}(H^s_x)$ and for any $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{s + \sigma _M}$ with $l \geqq 1$, and ${\widehat{w}} \in H^s_x$,

$$\begin{aligned} \begin{aligned}&\Vert {{{\mathcal {Q}}}} ({\mathfrak {x}})[{\widehat{w}}]\Vert _{H_x^{s}} \lesssim _{s, M} \Vert {\widehat{w}} \Vert _{H^s_x} + \Vert w \Vert _{H^{s + \sigma _M}_x} \Vert {\widehat{w}}\Vert _{H^{ M + 1}_x} , \\&\Vert d^l \big ({{{\mathcal {Q}}}}(\mathfrak {x})[{\widehat{w}}] \big )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l]\Vert _{H^s_x} \lesssim _{s, M, l} \Vert {\widehat{w}}\Vert _{H^s_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}} \\&\quad + \Vert {\widehat{w}}\Vert _{H^{M + 1}_x} \Big (\Vert \mathfrak {x}\Vert _{E_{s + \sigma _M }}\prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}} + \sum _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{s + \sigma _M }} \prod _{n \ne j} \Vert \widehat{\mathfrak {x}}_n \Vert _{E_{\sigma _M}}\Big ) . \end{aligned} \end{aligned}$$

(2.56)

Furthermore, Assumption B implies that for any ${\mathfrak {x}} \in {{{\mathcal {V}}}}^{\sigma _M}(\delta )$, the operator ${{{\mathcal {Q}}}} (\mathfrak {x} )$ is in ${{{\mathcal {B}}}}(L^2_x)$ and for any $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{ \sigma _M}$, $l \geqq 1$,

$$\begin{aligned} \begin{aligned}&\Vert {{{\mathcal {Q}}}} ({\mathfrak {x}})\Vert _{{{{\mathcal {B}}}}(L^2_x)} \lesssim _{M} 1 , \quad \Vert d^l {{{\mathcal {Q}}}}(\mathfrak {x})[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l]\Vert _{{{{\mathcal {B}}}}(L^2_x)} \lesssim _{M, l} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \\ \end{aligned} \end{aligned}$$

(2.57)

One computes by Leibniz’s rule that

$$\begin{aligned} \partial _{\varphi }^\lambda \big ( \hbox {d} {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[{\widehat{\iota }}(\varphi )] \big ) = \sum _{\begin{array}{c} 0 \leqq k \leqq |\lambda | \\ \lambda _1 + \ldots + \lambda _{k + 1} = \lambda \end{array}} c_{\lambda _1, \ldots , \lambda _{k + 1}} d^{k + 1} {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[\partial _{\varphi }^{\lambda _1} \breve{\iota }(\varphi ), \ldots , \partial _{\varphi }^{\lambda _k} \breve{\iota }(\varphi ), \partial _{\varphi }^{\lambda _{k + 1}} {\widehat{\iota }}(\varphi )],\nonumber \\ \end{aligned}$$

(2.58)

where $c_{\lambda _1, \ldots , \lambda _{k + 1}}$ are combinatorial constants.

Estimate of $\Vert \partial _{\varphi }^\lambda \big ( d {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[{\widehat{\iota }}(\varphi )] \big )[{\widehat{w}}] \Vert _{L^2_\varphi H^s_x}$. By (2.56), we have, for $s \geqq M + 1$,

$$\begin{aligned}&\Vert d^{k + 1} {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[\partial _{\varphi }^{\lambda _1} \breve{\iota }(\varphi ), \ldots , \partial _{\varphi }^{\lambda _k} \breve{\iota }(\varphi ), \partial _{\varphi }^{\lambda _{k + 1}} {\widehat{\iota }}(\varphi )][{\widehat{w}}(\varphi )] \Vert _{H^s_x} \\&\quad \lesssim _{s, M, k} \Vert {\widehat{w}}(\varphi ) \Vert _{H^s_x} \Vert \partial _{\varphi }^{\lambda _{k + 1}} {\widehat{\iota }}(\varphi )\Vert _{E_{ \sigma _M}} \prod _{n = 1}^k \Vert \partial _{\varphi }^{\lambda _n} \breve{\iota }(\varphi )\Vert _{E_{\sigma _M}} \nonumber \\&\qquad + \Vert {\widehat{w}}(\varphi ) \Vert _{H^{M + 1}_x} \Big ( \Vert \iota (\varphi ) \Vert _{E_{s + \sigma _M}} \Vert \partial _{\varphi }^{\lambda _{k + 1}} {\widehat{\iota }}(\varphi )\Vert _{E_{ \sigma _M}} \prod _{n = 1}^k \Vert \partial _{\varphi }^{\lambda _n} \breve{\iota }(\varphi )\Vert _{E_{\sigma _M}} \nonumber \\&\qquad + \sum _{j = 1}^k \Vert \partial _{\varphi }^{\lambda _j} \breve{\iota }(\varphi )\Vert _{E_{s + \sigma _M}} \big (\prod _{n \ne j} \Vert \partial _{\varphi }^{\lambda _n} \breve{\iota }(\varphi )\Vert _{E_{\sigma _M}}\big ) \Vert \partial _{\varphi }^{\lambda _{k + 1}} {\widehat{\iota }}(\varphi ) \Vert _{E_{\sigma _M}} \nonumber \\&\qquad + \Vert \partial _{\varphi }^{\lambda _{k + 1}} {\widehat{\iota }}(\varphi )\Vert _{E_{s + \sigma _M}} \prod _{n = 1}^k \Vert \partial _{\varphi }^{\lambda _n} \breve{\iota }(\varphi )\Vert _{E_{\sigma _M}} \Big ) . \nonumber \end{aligned}$$

(2.59)

Note that by the Sobolev embedding and (2.43), for any $s \geqq 0$, $\mu \in {\mathbb {N}}^{{\mathbb {S}}_+}$,

$$\begin{aligned} \Vert \partial _{\varphi }^\mu \breve{\iota }(\varphi ) \Vert _{E_{s}} \lesssim 1 + \Vert \partial _{\varphi }^\mu \iota \Vert _{C^0_\varphi E_{s}} \lesssim 1 + \Vert \iota \Vert _{s + s_0 + |\mu |} . \end{aligned}$$

(2.60)

Hence (2.58)–(2.59) and $\Vert \cdot \Vert _{L^2_\varphi H^s_x} \lesssim \Vert \cdot \Vert _s$ imply that for any $\breve{\iota }$ with $\Vert \iota \Vert _{s_0 + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}\leqq \delta$ and any $s \geqq M + 1$,

$$\begin{aligned}&\Vert \partial _{\varphi }^\lambda \big ( \hbox {d} {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[{\widehat{\iota }}(\varphi )] \big )[{\widehat{w}}(\varphi )] \Vert _{L^2_\varphi H^s_x}\\&\quad \lesssim _{s, M, \lambda } \Vert {\widehat{w}}\Vert _{s } \Vert {\widehat{\iota }}\Vert _{s_0 + \sigma _M(\lambda )} + \Vert {\widehat{w}}\Vert _{M + 1} \big ( \Vert \iota \Vert _{s + \sigma _M(\lambda )} \Vert {\widehat{\iota }}\Vert _{s_0 + \sigma _M(\lambda )} + \Vert {\widehat{\iota }}\Vert _{s + \sigma _M(\lambda )} \big )\nonumber \end{aligned}$$

(2.61)

for some constant $\sigma _M(\lambda ) > 0$.

Estimate of $\Vert \partial _{\varphi }^\lambda \big ( d {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[{\widehat{\iota }}(\varphi )] \big )\Vert _{H^s_\varphi {{{\mathcal {B}}}}(L^2_x)}$. For any $s \in {\mathbb {N}}$, $\beta \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $|\beta | \leqq s$, we need to estimate $\Vert \partial _\varphi ^{\beta + \lambda } \big ( d {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[{\widehat{\iota }}(\varphi )] \big ) \Vert _{L^2_\varphi {{{\mathcal {B}}}}(L^2_x)}$. As in (2.58) we have

$$\begin{aligned} \begin{aligned}&\partial _{\varphi }^{\beta + \lambda } \big ( d {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[{\widehat{\iota }}(\varphi )] \big )\\&= \sum _{\begin{array}{c} 0 \leqq k \leqq |\beta | + |\lambda | \\ \alpha _1 + \ldots + \alpha _{k + 1} = \beta + \lambda \end{array}} c_{\alpha _1, \ldots , \alpha _{k + 1}} d^{k + 1} {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[\partial _{\varphi }^{\alpha _1} \breve{\iota }(\varphi ), \ldots , \partial _{\varphi }^{\alpha _k} \breve{\iota }(\varphi ), \partial _{\varphi }^{\alpha _{k + 1}} {\widehat{\iota }}(\varphi )] \end{aligned} \end{aligned}$$

(2.62)

where $c_{\alpha _1, \ldots , \alpha _{k + 1}}$ are combinatorial constants. By (2.57) and (2.60) one obtains that

$$\begin{aligned}&\Vert d^{k + 1} {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[\partial _{\varphi }^{\alpha _1} \breve{\iota }(\varphi ), \ldots , \partial _{\varphi }^{\alpha _k} \breve{\iota }(\varphi ), \partial _{\varphi }^{\alpha _{k + 1}} {\widehat{\iota }}(\varphi )] \Vert _{L^2_\varphi {{{\mathcal {B}}}}(L^2_x)} \nonumber \\&\quad \lesssim _{\beta ,\lambda } \prod _{j = 1}^{k } (1 + \Vert \iota \Vert _{|\alpha _j |+ \eta _M }) \Vert {\widehat{\iota }}\Vert _{|\alpha _{k + 1}| + \eta _M} \end{aligned}$$

(2.63)

for some $\eta _M > 0$. Using the interpolation inequality (2.4), and arguing as in the proof of the formula (75) in [9], we have, for any $\breve{\iota }$ with $\Vert \iota \Vert _{\eta _M} \leqq 1$ and any $j = 1, \ldots , k$,

$$\begin{aligned} \begin{aligned} 1 + \Vert \iota \Vert _{|\alpha _j| + \eta _M }&\lesssim (1 + \Vert \iota \Vert _{\eta _M})^{1 - \frac{|\alpha _j|}{|\beta +\lambda |}} (1 + \Vert \iota \Vert _{|\beta + \lambda | + \eta _M})^{\frac{|\alpha _j|}{|\beta + \lambda |}}\\&{\mathop {\lesssim }\limits ^{\Vert \iota \Vert _{\eta _M} \leqq 1}} (1 + \Vert \iota \Vert _{|\beta + \lambda | + \eta _M})^{\frac{|\alpha _j|}{|\beta + \lambda |}} , \\ \Vert {\widehat{\iota }}\Vert _{|\alpha _{k + 1}| + \eta _M}&\lesssim \Vert {\widehat{\iota }}\Vert _{\eta _M}^{1 - \frac{|\alpha _{k + 1}|}{|\beta + \lambda |}} \Vert {\widehat{\iota }} \Vert _{|\beta + \lambda | + \eta _M}^{\frac{|\alpha _{k + 1}|}{|\beta + \lambda |}} . \end{aligned} \end{aligned}$$

Then by (2.63) and since $\sum _{j = 1}^k |\alpha _j| + |\alpha _{k + 1}| = |\beta + \lambda |$, it follows that

$$\begin{aligned}&\Vert d^{k + 1} {{{\mathcal {Q}}}}(\breve{\iota }(\varphi ))[\partial _{\varphi }^{\alpha _1} \breve{\iota }(\varphi ), \ldots , \partial _{\varphi }^{\alpha _k} \breve{\iota }(\varphi ), \partial _{\varphi }^{\alpha _{k + 1}} {\widehat{\iota }}(\varphi )] \Vert _{L^2_\varphi {{{\mathcal {B}}}}(L^2_x)} \nonumber \\&\quad \lesssim _{s,\lambda } (1 + \Vert \iota \Vert _{|\beta + \lambda |+ \eta _M})^{\frac{\sum _{j = 1}^k |\alpha _j|}{|\beta + \lambda |}} \Vert {\widehat{\iota }}\Vert _{\eta _M}^{1 - \frac{|\alpha _{k + 1}|}{|\beta + \lambda |}} \Vert {\widehat{\iota }} \Vert _{|\beta + \lambda | + \eta _M}^{\frac{|\alpha _{k + 1}|}{|\beta + \lambda |}} \nonumber \\&\quad \lesssim _{s,\lambda } \Big ((1 + \Vert \iota \Vert _{|\beta + \lambda |+ \eta _M}) \Vert {\widehat{\iota }}\Vert _{\eta _M} \Big )^{\frac{\sum _{j = 1}^k |\alpha _j| }{|\beta + \lambda |}} \Vert {\widehat{\iota }} \Vert _{|\beta + \lambda | + \eta _M}^{\frac{|\alpha _{k + 1}|}{|\beta + \lambda |}} \nonumber \\&\quad \lesssim _{s,\lambda } \Vert {\widehat{\iota }} \Vert _{|\beta + \lambda | + \eta _M} + \Vert \iota \Vert _{|\beta + \lambda |+ \eta _M} \Vert {\widehat{\iota }}\Vert _{\eta _M} \end{aligned}$$

(2.64)

where for the latter inequality we used Young’s inequality with exponents $\frac{|\beta +\lambda |}{\sum _{j = 1}^k |\alpha _j| }, \frac{|\beta +\lambda |}{ |\alpha _{k + 1}|}$. Combining (2.62) and (2.64) we obtain

$$\begin{aligned} \begin{aligned}&\Vert \partial _\varphi ^\lambda (d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}]) \Vert _{H^s_\varphi {{{\mathcal {B}}}}(L^2_x)} \lesssim _{s, M, \lambda } \Vert {\widehat{\iota }} \Vert _{s + |\lambda | + \eta _M} + \Vert \iota \Vert _{s + |\lambda |+ \eta _M} \Vert {\widehat{\iota }}\Vert _{\eta _M}. \end{aligned} \end{aligned}$$

(2.65)

Estimate of $\Vert \partial _\varphi ^\lambda (d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}])[{\widehat{w}}] \Vert _{H^s_\varphi L^2_x}$. Using that

$$\begin{aligned} \Big ( \sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} \Vert {\widehat{A}} (\ell ) \Vert _{ {{{\mathcal {B}}}}(L^2_x) }^2 \langle \ell \rangle ^{2s} \Big )^{1/2} \lesssim _{s_0} \Vert A \Vert _{H^{s+s_0}_\varphi {{{\mathcal {B}}}}(L^2_x) } \end{aligned}$$

one deduces from [9, Lemma 2.12] that for any $\breve{\iota }$ with $\Vert \iota \Vert _{2 s_0 + |\lambda | + \eta _M} \leqq 1$ and any $s \geqq s_0$,

$$\begin{aligned} \Vert \partial _\varphi ^\lambda (d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}])[{\widehat{w}}] \Vert _{H^s_\varphi L^2_x}&\lesssim _s \Vert \partial _\varphi ^\lambda (d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}]) \Vert _{H^{2 s_0}_\varphi {{{\mathcal {B}}}}(L^2_x)} \Vert {\widehat{w}} \Vert _{H^{s}_\varphi L^2_x}\nonumber \\&\quad + \Vert \partial _\varphi ^\lambda (d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}]) \Vert _{H^{s + s_0}_\varphi {{{\mathcal {B}}}}(L^2_x)} \Vert {\widehat{w}}\Vert _{H^{s_0}_\varphi L^2_x} \\&{\mathop {\lesssim _{s, M}}\limits ^{(2.65)}} \Vert {\widehat{w}}\Vert _s \Vert {\widehat{\iota }}\Vert _{2 s_0 + |\lambda | + \eta _M} \nonumber \\&\quad + \Vert {\widehat{w}}\Vert _{s_0} \big ( \Vert {\widehat{\iota }} \Vert _{s + s_0 + |\lambda | + \eta _M} + \Vert \iota \Vert _{s + s_0 + |\lambda | + \eta _M} \Vert {\widehat{\iota }} \Vert _{2 s_0 + |\lambda | + \eta _M} \big ) . \nonumber \end{aligned}$$

(2.66)

Increasing the constant $\sigma _M(\lambda )$ in (2.61) if needed, one infers from the estimates (2.61), (2.66) that for any $s \geqq {\mathfrak {s}}_M= \mathrm{max}\{ s_0, M + 1\}$, $\partial _\varphi ^\lambda (d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}])$ satisfies

$$\begin{aligned} \Vert \partial _\varphi ^\lambda (d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}]) [{\widehat{w}}] \Vert _s&\lesssim _{s, M, \lambda } \Vert {\widehat{w}}\Vert _s \Vert {\widehat{\iota }}\Vert _{ s_0 + \sigma _M(\lambda )} \nonumber \\&\quad + \Vert {\widehat{w}}\Vert _{{\mathfrak {s}}_M} \big ( \Vert {\widehat{\iota }} \Vert _{s + \sigma _M(\lambda )} + \Vert \iota \Vert _{s + \sigma _M(\lambda )} \Vert {\widehat{\iota }} \Vert _{s_0 + \sigma _M(\lambda )} \big ) .\qquad \end{aligned}$$

(2.67)

Furthermore, arguing similarly, one can show that for any $\omega _1, \omega _2 \in \Omega$, $\omega _1 \ne \omega _2$, the operator $\partial _\varphi ^\lambda \Delta _{\omega }(d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}])$ satisfies the estimate, for any $s \geqq {\mathfrak {s}}_M$

$$\begin{aligned} \begin{aligned} \gamma \frac{\Vert \partial _\varphi ^\lambda \Delta _{\omega }(d {{{\mathcal {Q}}}}(\breve{\iota })[{\widehat{\iota }}]) [{\widehat{w}}] \Vert _s}{|\omega _1 - \omega _2|}&\lesssim _{s, M, \lambda } \Vert {\widehat{w}}\Vert _s \Vert {\widehat{\iota }}\Vert _{ s_0 + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}\\&\quad + \Vert {\widehat{w}}\Vert _{{\mathfrak {s}}_M} \big ( \Vert {\widehat{\iota }} \Vert _{s + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0 + \sigma _M(\lambda )}^\mathrm{{Lip}(\gamma )}\big ) . \end{aligned} \end{aligned}$$

(2.68)

It then follows from (2.67) and (2.68) that there exists a tame constant ${{\mathfrak {M}}}_{\partial _\varphi ^\lambda (d {{{\mathcal {Q}}}} (\breve{\iota })[{\widehat{\iota }}]) }(s)$ for $\partial _\varphi ^\lambda (d {{{\mathcal {Q}}}} (\breve{\iota })[{\widehat{\iota }}])$ satisfying the estimate stated in item (ii).

(iii) Since ${{{\mathcal {R}}}}(\theta , 0, 0) = 0$, one has by the mean value theorem ${{{\mathcal {R}}}}(\breve{\iota }) = \int _0^1 \hbox {d} {{{\mathcal {R}}}}(\breve{\iota }_t)[{\widehat{\iota }}]\, \hbox {d} t$ with $\breve{\iota }_t = (1-t) (\theta (\varphi ), 0, 0) + t \breve{\iota }(\varphi )$ and ${\widehat{\iota }}(\varphi ) \,{:=}\, (0, y(\varphi ), w(\varphi ))$. Since $\Vert {\widehat{\iota }} \Vert _s \lesssim \Vert \iota \Vert _s$ for any $s \geqq 0$, item (iii) is thus a direct consequence of (ii). $\quad \square$

2.5 Egorov Type Theorems

In this section we investigate operators obtained by conjugating a pseudo-differential operator of the form $a(\varphi , x) \partial _x^m$, $m \in {\mathbb {Z}}$, by the flow map of a transport equation. The main result is an Egorov type theorem, stated in Proposition 2.31, saying that such a conjugated operator is again a pseudo-differential operator, up to a smoothing remainder; it is used in Section 6.3.

Let $\Phi (\tau _0, \tau , \varphi )$ denote the flow of the transport equation

$$\begin{aligned} \partial _\tau \Phi (\tau _0, \tau , \varphi ) = B(\tau , \varphi ) \Phi (\tau _0, \tau , \varphi ) , \quad \Phi (\tau _0, \tau _0, \varphi ) = \mathrm{Id} , \end{aligned}$$

(2.69)

where $B(\tau , \varphi )$ is the transport operator, given by

$$\begin{aligned} \begin{aligned} B(\tau , \varphi )&\,{:=}\, \Pi _{\bot } \big ( b (\tau , \varphi , x) \partial _x + b_x(\tau , \varphi , x) \big ) ,\\&\quad b\equiv b (\tau , \varphi , x) \,{:=}\, \frac{ \beta (\varphi , x) }{ 1 + \tau \beta _x (\varphi , x) } , \end{aligned} \end{aligned}$$

(2.70)

$\Pi _\bot$ is the $L^2_x$-orthogonal projector $L^2_x \rightarrow L^2_\bot ({\mathbb {T}}_1)$, and $\beta (\varphi , x) \equiv \beta (\varphi , x; \omega )$ is a real valued function, which is ${{{\mathcal {C}}}}^\infty$ with respect to the variables $(\varphi , x)$ and Lipschitz continuous with respect to the parameter $\omega \in { \Omega }$. For brevity we set $\Phi (\tau , \varphi ) \,{:=}\, \Phi (0, \tau , \varphi )$ and $\Phi (\varphi )\,{:=}\, \Phi (0, 1, \varphi )$. Note that $\Phi (\varphi )^{- 1} = \Phi (1, 0 , \varphi )$ and that

$$\begin{aligned} \Phi (\tau _0, \tau , \varphi ) = \Phi (\tau , \varphi ) \circ \Phi (\tau _0, \varphi )^{- 1}. \end{aligned}$$

(2.71)

By standard hyperbolic estimates, equation (2.69) is well-posed. The flow $\Phi (\tau _0, \tau , \varphi )$ has the following properties:

Lemma 2.28

(Transport flow) Let $\lambda _0 \in {\mathbb {N}}$, $S > s_0$. For any $\lambda \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$, any $n_1, n_2 \in {\mathbb {R}}$ with $n_1 + n_2 = - \lambda - 1$, and any $s \geqq s_0$, there exist constants $\sigma (\lambda _0, n_1, n_2) > 0$ and $\delta \equiv \delta (S, \lambda _0, n_1, n_2) \in (0, 1)$ so that the following holds: if $\beta (\varphi , x))$ satisfies

$$\begin{aligned} \Vert \beta \Vert _{s_0 + \sigma (\lambda _0, n_1, n_2)}^\mathrm{{Lip}(\gamma )}\leqq \delta , \end{aligned}$$

(2.72)

then for any $m \in {\mathbb {S}}_+$, $\langle D \rangle ^{n_1} \partial _{\varphi _m}^\lambda \Phi (\tau _0, \tau , \varphi ) \langle D \rangle ^{ n_2}$ is a $\mathrm{{Lip}(\gamma )}$-tame operator with a tame constant satisfying

$$\begin{aligned}&{\mathfrak {M}}_{\langle D \rangle ^{ n_1} \partial _{\varphi _m}^\lambda \Phi (\tau _0, \tau , \varphi ) \langle D \rangle ^{ n_2}}(s) \lesssim _{S, \lambda _0, n_1, n_2} 1 + \Vert \beta \Vert _{s + \sigma (\lambda _0, n_1, n_2)}^\mathrm{{Lip}(\gamma )},\nonumber \\&\quad \forall s_0 \leqq s \leqq S, \quad \forall \tau _0, \tau \in [0, 1]. \end{aligned}$$

(2.73)

In addition, if $n_1 + n_2 = - \lambda - 2$, then $\langle D \rangle ^{ n_1} \partial _{\varphi _m}^\lambda ( \Phi (\tau _0, \tau , \varphi ) - \mathrm{Id} ) \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

$$\begin{aligned}&{{\mathfrak {M}}}_{\langle D \rangle ^{ n_1} \partial _{\varphi _m}^\lambda (\Phi (\tau _0, \tau , \varphi ) - \mathrm{Id}) \langle D \rangle ^{n_2}}(s) \lesssim _{S, \lambda _0, n_1, n_2} \Vert \beta \Vert _{s + \sigma (\lambda _0, n_1, n_2)}^\mathrm{{Lip}(\gamma )},\nonumber \\&\quad \forall s_0 \leqq s \leqq S , \quad \forall \tau _0, \tau \in [0, 1]. \end{aligned}$$

(2.74)

Furthermore, let $s_0< s_1 < S$, $n_1, n_2 \in {\mathbb {R}}$, $\lambda _0 \in {\mathbb {N}}$, $\lambda \leqq \lambda _0$ with $n_1 + n_2 =- \lambda - 1$, $m \in {\mathbb {S}}_+$. If $\beta _1$ and $\beta _2$ satisfy $\Vert \beta _i \Vert _{s_1 + \sigma (n_1, n_2)} \leqq \delta$ for some $\sigma (n_1, n_2) > 0$, and $\delta \in (0, 1)$ small enough, then

$$\begin{aligned}&\Vert \langle D \rangle ^{n_1} \partial _{\varphi _m}^\lambda \Delta _{12} \Phi (\tau _0, \tau , \varphi ) \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})}\nonumber \\&\quad \lesssim _{s_1, \lambda _0, n_1, n_2} \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma (n_1, n_2)}, \quad \forall \, \tau _0, \tau \in [0, 1] , \end{aligned}$$

(2.75)

where $\Delta _{12} \beta \,{:=}\, \beta _2 - \beta _1$ and $\Delta _{12} \Phi (\tau _0, \tau , \varphi )\,{:=}\, \Phi (\tau _0, \tau , \varphi ; \beta _2) - \Phi (\tau _0, \tau , \varphi ; \beta _1)$.

Proof

The proof of (2.73) is similar to the one of Propositions A.7, A.10 and A.11 in [10] and hence we omit it. (Essentially the only difference is that the vector field (2.70) is of order 1, whereas the vector field considered in [10] is of order $\frac{1}{2}$.) Using (2.73) we now prove (2.74). By (2.69), one has that $\Phi (\tau _0, \tau , \varphi ) - \mathrm{Id} = \int _{\tau _0}^\tau B(t, \varphi ) \Phi (\tau _0, t, \varphi )\, \hbox {d} t$. Then, for any $\lambda \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$ and any $n_1, n_2 \in {\mathbb {R}}$ with $n_1 + n_2 =- \lambda - 2$, one has, by Leibniz’ rule that,

$$\begin{aligned} \begin{aligned}&\langle D \rangle ^{n_1} \partial _{\varphi _m}^\lambda (\Phi (\tau _0, \tau , \varphi ) - \mathrm{Id}) \langle D \rangle ^{n_2} \\&\quad = \sum _{\lambda _1 + \lambda _2 = \lambda } c_{\lambda _1, \lambda _2} \int _{\tau _0}^\tau \big ( \langle D \rangle ^{n_1}\partial _{\varphi _m}^{\lambda _1}B(t, \varphi ) \langle D \rangle ^{ n_2 + \lambda _2 + 1} \big ) \big ( \langle D \rangle ^{- n_2 - \lambda _2 - 1} \partial _{\varphi _m}^{\lambda _2}\Phi (\tau _0, t, \varphi ) \langle D \rangle ^{n_2} \big )\, \hbox {d} t \\&\quad = \sum _{\lambda _1 + \lambda _2 = \lambda } c_{\lambda _1, \lambda _2} \int _{\tau _0}^\tau \big ( \langle D \rangle ^{n_1}\partial _{\varphi _m}^{\lambda _1}B(t, \varphi ) \langle D \rangle ^{- 1 - n_1 - \lambda _1} \big ) \big ( \langle D \rangle ^{- n_2 - \lambda _2 - 1} \partial _{\varphi _m}^{\lambda _2}\Phi (\tau _0, t, \varphi ) \langle D \rangle ^{n_2} \big )\, \hbox {d} t, \end{aligned} \end{aligned}$$

where $c_{\lambda _1, \lambda _2}$ are combinatorial constants and we used that $n_2 + \lambda _2 + 1 = -1 - n_1 - \lambda _1$. Recalling the definition (2.70) of B, using Lemmata 2.10, 2.18, 2.30-(i), and (2.73), one has that for any $s \geqq s_0$,

$$\begin{aligned}&\qquad \quad {\mathfrak {M}}_{\langle D \rangle ^{n_1} \partial _{\varphi _m}^{\lambda _1}B \langle D \rangle ^{- 1 - n_1 - \lambda _1}}(s) \lesssim _s |\langle D \rangle ^{n_1} B \langle D \rangle ^{- 1 - n_1 - \lambda _1}|_{0, s + \lambda _1, 0}^\mathrm{{Lip}(\gamma )}\lesssim _{s, \lambda _1, n_1} \Vert \beta \Vert _{s + \sigma (\lambda _1, n_1)}^\mathrm{{Lip}(\gamma )}, \nonumber \\&{\mathfrak {M}}_{ \langle D \rangle ^{- 1- n_2 - \lambda _2} \partial _{\varphi _m}^{\lambda _2}\Phi (\tau _0, t, \varphi ) \langle D \rangle ^{n_2}}(s) \lesssim _{s, \lambda _2, n_1, n_2} 1 + \Vert \beta \Vert _{s + \sigma (\lambda _2, n_1, n_2)}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.76)

Then (2.74) follows by (2.76), Lemma 2.16 and (2.72). The estimate (2.75) follows by similar arguments. $\quad \square$

For what follows we need to study the solutions of the characteristic ODE $\partial _\tau x = - b(\tau ,\varphi , x)$ associated with the transport operator defined in (2.70).

Lemma 2.29

(Characteristic flow) The characteristic flow $\gamma ^{\tau _0, \tau }(\varphi , x)$ defined by

$$\begin{aligned} \partial _\tau \gamma ^{\tau _0, \tau }(\varphi , x) = - b\big (\tau , \varphi , \gamma ^{\tau _0, \tau }(\varphi , x) \big ) , \quad \gamma ^{\tau _0, \tau _0}(\varphi , x) = x, \end{aligned}$$

(2.77)

is given by

$$\begin{aligned} \gamma ^{\tau _0, \tau } (\varphi , x) = x + \tau _0 \beta (\varphi , x) + \breve{\beta } (\tau , \varphi , x + \tau _0 \beta (\varphi , x)) , \end{aligned}$$

(2.78)

where $y \mapsto y + \breve{\beta } (\tau , \varphi , y)$ is the inverse of the diffeomorphism $x \mapsto x + \tau \beta (\varphi , x)$.

Proof

A direct computation proves that $\gamma ^{0,\tau } (y) = y + \breve{\beta } (\tau , \varphi , y)$ and therefore $\gamma ^{\tau ,0} (x) = x + \tau \beta ( \varphi , x)$. By the composition rule of the flow $\gamma ^{\tau _0, \tau } = \gamma ^{0, \tau } \circ \gamma ^{\tau _0, 0}$ we deduce (2.78). $\quad \square$

Lemma 2.30

There are constants $\sigma$ and $\delta > 0$ so that the following holds: if $\Vert \beta \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\leqq \delta$, then

• (i) $\Vert b \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert \beta \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}$ for any $s \geqq s_0$;

• (ii) $\Vert \gamma ^{\tau _0, \tau }(\varphi , x) - x \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert \beta \Vert _{s +\sigma }^\mathrm{{Lip}(\gamma )}$ for any $\tau _0, \tau \in [0, 1]$ and $s \geqq s_0$;

• (iii) for any $\Vert \beta _j \Vert _{s_1 + \sigma } \le \delta ,$ $j= 1,2$ with $s_1 > s_0$, $\Delta _{12} b \,{:=}\, b(\cdot ; \beta _2) - b(\cdot ; \beta _1)$ and $\Delta _{12} \gamma ^{\tau _0, \tau }\,{:=}\, \gamma ^{\tau _0, \tau }(\cdot ; \beta _2) - \gamma ^{\tau _0, \tau }(\cdot ; \beta _1)$ can be estimated in terms of $\Delta _{12} \beta \,{:=}\, \beta _2 - \beta _1$ as

  $$\begin{aligned} \Vert \Delta _{12} b\Vert _{s_1} \lesssim _{s_1} \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma } , \quad \Vert \Delta _{12} \gamma ^{\tau _0, \tau } \Vert _{s_1} \lesssim _{s_1} \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma }. \end{aligned}$$

Proof

Item (i) follows from the definition of b in (2.70) and Lemma 2.3. Item (ii) can be deduced from Lemma 2.1 and (2.78) and item (iii) follows by similar arguments. $\quad \square$

The main result of this section is the Egorov type theorem below, saying that the operator obtained by conjugating $a(\varphi , x) \partial _x^m$, $m \in {\mathbb {Z}}$, with the time one flow $\Phi (\varphi ) = \Phi (0, 1, \varphi )$ of the transport equation (2.69), remains a pseudo-differential operator with a homogenous asymptotic expansion up to a regularizing remainder satisfying the quantitative tame estimate (2.83).

Proposition 2.31

(Egorov) Let N, $\lambda _0 \in {\mathbb {N}}$ and $S > s_0$ be given and assume that $\beta (\cdot , \cdot ; \omega )$ and $a(\cdot , \cdot ; \omega )$ are in ${{{\mathcal {C}}}}^\infty ({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1)$ and Lipschitz continuous with respect to $\omega \in \Omega$. Then there exist constants $\sigma _N(\lambda _0)$, $\sigma _N > 0$, $\delta (S, N, \lambda _0) \in (0, 1)$, and $C_0 > 0$ so that the following holds: if

$$\begin{aligned} \Vert \beta \Vert _{s_0 + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )}\leqq \delta , \quad \Vert a \Vert _{s_0 + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )}\leqq C_0 , \end{aligned}$$

(2.79)

then for any $m \in {\mathbb {Z}}$, the conjugated operator

$$\begin{aligned} {{{\mathcal {P}}}}(\varphi ) \,{:=}\, \Phi (\varphi ) {{{\mathcal {P}}}}_0 (\varphi )\Phi (\varphi )^{- 1} , \quad {{{\mathcal {P}}}}_0 \,{:=}\, a(\varphi , x; \omega ) \partial _x^m, \end{aligned}$$

is a pseudo-differential operator of order m with an expansion of the form

$$\begin{aligned} {{{\mathcal {P}}}}(\varphi ) = \sum _{i = 0}^N p_{m - i} (\varphi , x; \omega ) \partial _x^{m - i} + {{{\mathcal {R}}}}_N(\varphi ) \end{aligned}$$

(2.80)

with the following properties:

• 1. The principal symbol $p_m$ of ${{\mathcal {P}}}$ is given by

  $$\begin{aligned} p_m(\varphi , x; \omega ) = \Big ( (1 + \breve{\beta }_y (\varphi , y; \omega ))^m a(\varphi , y; \omega ) \Big )|_{y = x + \beta (\varphi , x; \omega )} \end{aligned}$$

  (2.81)

  where $y \mapsto y + \breve{\beta }(\varphi , y; \omega )$ denotes the inverse of the diffeomorphism $x \mapsto x + \beta (\varphi , x; \omega )$.

• 2. For any $s \geqq s_0$ and $i = 1, \ldots , N$,

  $$\begin{aligned} \Vert p_m - a\Vert _s^\mathrm{{Lip}(\gamma )}, \ \Vert p_{m - i}\Vert _s^\mathrm{{Lip}(\gamma )}\, \lesssim _{s, N} \, \Vert \beta \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}+ \Vert a \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}\Vert \beta \Vert _{s_0 + \sigma _N}^\mathrm{{Lip}(\gamma )}.\nonumber \\ \end{aligned}$$

  (2.82)

• 3. For any $k \in {\mathbb {S}}_+$, any $\lambda \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$, and any $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 + \lambda _0 \leqq N - 1 - m$, the pseudo-differential operator $\langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda {{{\mathcal {R}}}}_N(\varphi ) \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

  $$\begin{aligned}&{{\mathfrak {M}}}_{\langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda {{{\mathcal {R}}}}_N(\varphi ) \langle D \rangle ^{n_2}}(s) \lesssim _{S, N, \lambda _0} \Vert \beta \Vert _{s + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )}+ \Vert a \Vert _{s + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )}\Vert \beta \Vert _{s_0 + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )},\nonumber \\&\quad \forall s_0 \leqq s \leqq S. \end{aligned}$$

  (2.83)

• 4. Let $s_1 > s_0$ and assume that $\Vert \beta _j \Vert _{s_1 + \sigma _N(\lambda _0)} \leqq \delta$ and $\Vert a_j\Vert _{s_1 + \sigma _N(\lambda _0)} \leqq C_0$, $j = 1,2$. Then

  $$\begin{aligned} \begin{aligned} \Vert \Delta _{12} p_{m - i} \Vert _{s_1} \lesssim _{s_1, N} \Vert \Delta _{12} a\Vert _{s_1 + \sigma _N} + \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma _N}, \quad i = 0, \ldots , N , \end{aligned} \end{aligned}$$

  and for any $k \in {\mathbb {S}}_+$, any $\lambda \leqq \lambda _0$, and any $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 + \lambda _0 \leqq N - 1 - m$,

  $$\begin{aligned} \Vert \langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda \Delta _{12} {{{\mathcal {R}}}}_N(\varphi ) \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})} \lesssim _{s_1, N, n_1, n_2} \Vert \Delta _{12} a\Vert _{s_1 + \sigma _N(\lambda _0)} + \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma _N(\lambda _0)} \end{aligned}$$

  where $\Delta _{12}a = a_2 - a_1$, $\Delta _{12}\beta = \beta _2 - \beta _1$, and $\Delta _{12} {{{\mathcal {R}}}}_N = {{{\mathcal {R}}}}^{(2)}_N - {{{\mathcal {R}}}}^{(1)}_N$ with ${{{\mathcal {R}}}}^{(j)}_N$ denoting the remainder in (2.80), corresponding to $a_j$, $\beta _j$ for $j = 1, 2$.

Proof

The $L^2_x$-orthogonal projector $\Pi _\bot : L^2({\mathbb {T}}_1) \rightarrow L^2_\bot ({\mathbb {T}}_1)$ is a Fourier multiplier of order 0, $\Pi _\bot = \mathrm{Op} ( \chi _\bot (\xi ) )$, where $\chi _\bot$ is a ${{{\mathcal {C}}}}^\infty ({\mathbb {R}}, {\mathbb {R}})$ cut-off function which is equal to 1 on a neighborhood of ${{\mathbb {S}}}^\bot$ and vanishes in a neighborhood of ${{\mathbb {S}}} \cup \{0\}$. We then decompose the operator $B(\tau , \varphi ) = \Pi _\bot (b(\tau , \varphi , x) \partial _x + b_x (\tau , \varphi , x))$ as $B(\tau , \varphi ) = B_1(\tau , \varphi ) + B_\infty (\tau , \varphi )$ with

$$\begin{aligned} B_1(\tau , \varphi )&:= b(\tau , \varphi , x) \partial _x + b_x (\tau , \varphi , x),\nonumber \\ B_\infty (\tau , \varphi )&:= \mathrm{Op}(b_\infty (\tau , \varphi , x, \xi )) \in OPS^{- \infty } \end{aligned}$$

(2.84)

swhere for some $\sigma > 0$, $B_\infty$ satisfies for any $s, m \geqq 0$ and $\alpha \in {\mathbb {N}}$ the estimate

$$\begin{aligned} |B_\infty |_{- m, s, \alpha }^\mathrm{{Lip}(\gamma )}\lesssim _{m, s, \alpha } \Vert \beta \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.85)

The conjugated operator ${{{\mathcal {P}}}}(\tau , \varphi ) \,{:=}\, \Phi (\tau , \varphi ) {{{\mathcal {P}}}}_0(\varphi ) \Phi (\tau , \varphi )^{- 1}$ solves the Heisenberg equation

$$\begin{aligned} \partial _\tau {{{\mathcal {P}}}}(\tau , \varphi ) = [B(\tau , \varphi ), {{{\mathcal {P}}}}(\tau , \varphi )], \quad {{{\mathcal {P}}}}(0, \varphi ) = {{{\mathcal {P}}}}_0(\varphi ) = a(\varphi , x; \omega ) \partial _x^m . \end{aligned}$$

(2.86)

Indeed, one has

$$\begin{aligned} \partial _\tau {{{\mathcal {P}}}}(\tau , \varphi )&= \partial _\tau \Phi (\tau , \varphi ) {{{\mathcal {P}}}}_0(\varphi ) \Phi (\tau , \varphi )^{- 1} + \Phi (\tau , \varphi ) {{{\mathcal {P}}}}_0(\varphi ) \partial _\tau \Big ( \Phi (\tau , \varphi )^{- 1} \Big ) \nonumber \\&= \partial _\tau \Phi (\tau , \varphi ) {{{\mathcal {P}}}}_0(\varphi ) \Phi (\tau , \varphi )^{- 1} - \Phi (\tau , \varphi ) {{{\mathcal {P}}}}_0(\varphi ) \Phi (\tau , \varphi )^{- 1} \partial _\tau \Phi (\tau , \varphi ) \Phi (\tau , \varphi )^{- 1} \nonumber \\&{\mathop {=}\limits ^{(2.69)}} B(\tau , \varphi ) \Phi (\tau , \varphi ) {{{\mathcal {P}}}}_0(\varphi ) \Phi (\tau , \varphi )^{- 1} \nonumber \\&\quad - \Phi (\tau , \varphi ) {{{\mathcal {P}}}}_0(\varphi ) \Phi (\tau , \varphi )^{- 1} B(\tau , \varphi ) \Phi (\tau , \varphi ) \Phi (\tau , \varphi )^{- 1} \nonumber \\&= [B(\tau , \varphi ), {{{\mathcal {P}}}}(\tau , \varphi )]. \end{aligned}$$

(2.87)

We look for an approximate solution of (2.86) of the form

$$\begin{aligned} {{{\mathcal {P}}}}_N(\tau , \varphi ) \,{:=}\, \sum _{i = 0}^N p_{m - i}(\tau , \varphi , x) \partial _x^{m - i} \end{aligned}$$

(2.88)

for suitable functions $p_{m - i}(\tau , \varphi , x)$ to be determined. By (2.84)

$$\begin{aligned}{}[B(\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )] = [B_1(\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )] + [B_\infty (\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )], \end{aligned}$$

(2.89)

where $[ B_\infty (\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )]$ is in $OPS^{- \infty }$, and $[B_1(\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )] = \sum _{i = 0}^N \big [ b \partial _x + b_x ,\, p_{m - i} \partial _x^{m - i} \big ]$. By Lemma 2.12, one has for any $i = 0, \ldots , N$,

$$\begin{aligned} \big [ b \partial _x + b_x, p_{m - i} \partial _x^{m - i} \big ]&= \big ( b (p_{m - i})_x - (m - i) b_x p_{m - i} \big ) \partial _x^{m - i} \\&\quad + \sum _{j = 1}^{N - i} g_{j}(b, p_{m -i}) \partial _x^{m - i - j} + {{{\mathcal {R}}}}_N(b, p_{m - i}) , \end{aligned}$$

where the functions $g_j(b, p_{m - i})(\tau , \varphi , x)$, $j = 0, \ldots , N - i$, and the remainders ${{{\mathcal {R}}}}_N(b, p_{m - i})$ can be estimated as follows: there exists $\sigma _N \,{:=}\, \sigma _N(m) > 0$ so that for any $s \geqq s_0$, (cf. Lemma 2.30-(i))

$$\begin{aligned} \Vert g_j (b, p_{m - i}) \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _{m, N, s} \Vert \beta \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}\Vert p_{m - i}\Vert _{s_0 + \sigma _N}^\mathrm{{Lip}(\gamma )}\nonumber \\&\quad + \Vert \beta \Vert _{s_0 + \sigma _N}^\mathrm{{Lip}(\gamma )}\Vert p_{m - i}\Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

(2.90)

and for any $s \geqq s_0$ and $\alpha \in {\mathbb {N}}$ (cf. Lemma 2.12-(ii))

$$\begin{aligned} |{{{\mathcal {R}}}}_N(b, p_{m - i})|_{m - N - 1, s, \alpha }^\mathrm{{Lip}(\gamma )}&\lesssim _{m, N, s, \alpha } \Vert \beta \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}\Vert p_{m - i}\Vert _{s_0 + \sigma _N}^\mathrm{{Lip}(\gamma )}\nonumber \\&\quad + \Vert \beta \Vert _{s_0 + \sigma _N}^\mathrm{{Lip}(\gamma )}\Vert p_{m - i}\Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.91)

Adding up the expansions for $\big [ b \partial _x + b_x, p_{m - i} \partial _x^{m - i} \big ]$, $0 \le i \le N$, yields

$$\begin{aligned} \big [ B_1(\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi ) \big ]&= \sum _{i = 0}^N \big ( b (p_{m - i})_x - (m - i) b_x p_{m - i} \big )\partial _x^{m - i} \nonumber \\&\quad + \sum _{i = 0}^N \sum _{j = 1}^{N - i} g_{j}(b, p_{m -i}) \partial _x^{m - i - j} + \sum _{i = 0}^N {{{\mathcal {R}}}}_N(b, p_{m - i}) \nonumber \\&= \sum _{i = 0}^N \big ( b (p_{m - i})_x - (m - i) b_x p_{m - i} \big ) \partial _x^{m - i}\nonumber \\&\quad + \sum _{k = 1}^N \sum _{j = 1}^k g_j (b, p_{m - k + j}) \partial _x^{m - k } + \sum _{i = 0}^N {{{\mathcal {R}}}}_N(b, p_{m - i}) \nonumber \\&= \big ( b (p_m)_x - m b_x p_m \big ) \partial _x^m\nonumber \\&\quad + \sum _{i = 1}^N \big ( b (p_{m - i})_x - (m - i) b_x p_{m - i} + {{\widetilde{g}}}_{ i} \big ) \partial _x^{m - i} + {{{\mathcal {Q}}}}_N , \end{aligned}$$

(2.92)

where

$$\begin{aligned} {{{\mathcal {Q}}}}_N \,{:=}\, \sum _{i = 0}^N {{{\mathcal {R}}}}_N(b, p_{m - i}) \in OPS^{m - N - 1} , \quad {{\widetilde{g}}}_{ i} \,{:=}\, \sum _{j= 1}^i g_j(b, p_{m - i + j}), \quad \forall 1 \le i \le N. \quad \nonumber \\ \end{aligned}$$

(2.93)

Defining for any $s \geqq 0$ and $1 \le i \le N$,

$$\begin{aligned}&\mathtt {M}_{< i}(s) \,{:=}\, \mathrm{max}\{ \Vert p_{m - k}\Vert _s^\mathrm{{Lip}(\gamma )}, \, k = 0, \ldots , i - 1 \} ,\nonumber \\&\quad \quad \mathtt {M}(s) \,{:=}\, \mathrm{max} \{ \Vert p_{m - i}\Vert _s^\mathrm{{Lip}(\gamma )}, \, i = 0, \ldots , N \} , \end{aligned}$$

(2.94)

we deduce from (2.90) and (2.91) that for any $s \geqq s_0$, $\alpha \in {\mathbb {N}}$, $i = 0, \ldots , N$,

$$\begin{aligned} \begin{aligned}&\Vert {{\widetilde{g}}}_i\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, N} \mathtt {M}_{< i}(s + \sigma _N) \Vert \beta \Vert _{s_0 + \sigma _N}^\mathrm{{Lip}(\gamma )}+ \mathtt {M}_{< i}(s_0 + \sigma _N) \Vert \beta \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}\\&|{{{\mathcal {Q}}}}_N|_{m - N - 1, s, \alpha }^\mathrm{{Lip}(\gamma )}\lesssim _{s, N} \mathtt {M}(s + \sigma _N) \Vert \beta \Vert _{s_0 + \sigma _N}^\mathrm{{Lip}(\gamma )}+ \mathtt {M}(s_0 + \sigma _N) \Vert \beta \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}. \end{aligned} \end{aligned}$$

(2.95)

By (2.88), (2.89), and (2.92) the operator ${{{\mathcal {P}}}}_N(\tau , \varphi )$ solves the approximated Heisenberg equation

$$\begin{aligned} \partial _\tau {{{\mathcal {P}}}}_N(\tau , \varphi ) = [B(\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )] + OPS^{m - N - 1} , \end{aligned}$$

if the functions $p_{m - i}$ solve the transport equations

$$\begin{aligned} \begin{aligned}&\partial _\tau p_m = b (p_m)_x - m b_x p_m, \\&\partial _\tau p_{m - i} = b (p_{m - i})_x - (m - i) b_x p_{m - i} + {{\widetilde{g}}}_{ i} , \quad \forall \, i = 1, \ldots , N. \end{aligned} \end{aligned}$$

(2.96)

Note that, since ${{\widetilde{g}}}_i$ only depends on $p_{m - i + 1}, \ldots , p_{m }$, we can solve (2.96) inductively.

Determination of $p_m$. We solve the first equation in (2.96),

$$\begin{aligned} \partial _\tau p_m(\tau , \varphi , x)&= b(\tau , \varphi , x) \partial _x p_m (\tau , \varphi , x) - m b_x(\tau , \varphi , x) p_m(\tau , \varphi , x) , \\&\quad p_m(0, \varphi , x) = a(\varphi , x). \end{aligned}$$

By the method of characteristics we deduce that

$$\begin{aligned} p_m(\tau , \varphi , \gamma ^{0, \tau }(\varphi , x)) = \mathrm{exp}\Big ( - m \int _0^\tau b_x(t, \varphi , \gamma ^{0, t}(\varphi , x))\, \hbox {d} t \Big ) a(\varphi , x) \end{aligned}$$

(2.97)

where $\gamma ^{0, \tau }(\varphi , x)$ is given by (2.78). Differentiating the equation (2.77) with respect to the initial datum x, we get

$$\begin{aligned} \partial _\tau (\partial _x \gamma ^{\tau _0, \tau }(x)) = - b_x (\tau , \varphi , \gamma ^{\tau _0, \tau }(x)) \partial _x \gamma ^{\tau _0, \tau }(x), \quad \partial _x \gamma ^{\tau _0, \tau _0}(x) = 1 , \end{aligned}$$

implying that

$$\begin{aligned} \partial _x \gamma ^{\tau _0, \tau }(\varphi , x)= \mathrm{exp}\Big (- \int _{\tau _0}^\tau b_x(t, \varphi , \gamma ^{\tau _0, t}(\varphi , x))\, \hbox {d} t \Big ). \end{aligned}$$

(2.98)

From (2.97) and (2.98) we infer that

$$\begin{aligned} p_m(\tau ,\varphi , y) = \Big ( \big (\partial _x \gamma ^{0, \tau }(\varphi , x) \big )^m a(\varphi , x) \Big )|_{x = \gamma ^{\tau , 0}(\varphi , y)}. \end{aligned}$$

(2.99)

Evaluating the latter identity at $\tau = 1$ and using (2.78), we obtain (2.81).

Inductive determination of $p_{m - i}$. For $i = 1, \ldots , N$, we solve the inhomogeneous transport equation,

$$\begin{aligned} \partial _\tau p_{m - i} = b \partial _x p_{m - i} - (m - i) b_x p_{m - i} + {\widetilde{g}}_{ i} , \quad p_{m - i}(0, \varphi , x) = 0. \end{aligned}$$

By the method of characteristics one has

$$\begin{aligned} p_{m - i}(\tau , \varphi , y) = \int _0^\tau \mathrm{exp}\Big (- (m - i) \int _t^\tau b_x(s,\varphi , \gamma ^{\tau , s}(\varphi , y))\, \hbox {d} s \Big ) {\widetilde{g}}_{ i}(t ,\varphi , \gamma ^{\tau , t}(\varphi , y))\, \hbox {d} t.\nonumber \\ \end{aligned}$$

(2.100)

The functions $p_{m - i}(\varphi , y)$ in the expansion (2.80) are then given by $p_{m - i}(\varphi , y) \,{:=} p_{m -i}(1,\varphi , y)$. Next we prove the estimates for $p_{m-i}$ stated in (2.82). They follow from the following

Lemma 2.32

There exist $\sigma _N^{(N)}> \sigma _N^{(N - 1)}> \cdots> \sigma _N^{(0)} > 0$ so that for any $i \in \{ 1, \ldots , N \}$, $\tau \in [0,1]$, and $s \geqq s_0$,

$$\begin{aligned} \begin{aligned} \Vert p_m(\tau , \cdot ) - a\Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \Vert \beta \Vert _{s + \sigma _N^{(0)}}^\mathrm{{Lip}(\gamma )}+ \Vert a \Vert _{s + \sigma _N^{(0)}}^\mathrm{{Lip}(\gamma )}\Vert \beta \Vert _{s_0 + \sigma _N^{(0)}}^\mathrm{{Lip}(\gamma )}, \\ \Vert p_{m - i}(\tau , \cdot )\Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \Vert \beta \Vert _{s + \sigma _N^{(i)}}^\mathrm{{Lip}(\gamma )}+ \Vert a \Vert _{s + \sigma _N^{(i)}}^\mathrm{{Lip}(\gamma )}\Vert \beta \Vert _{s_0 + \sigma _N^{(i)}} . \end{aligned} \end{aligned}$$

(2.101)

Proof of Lemma 2.32

We argue by induction. First we prove the claimed estimate for $p_m - a$ with $p_m$ given by (2.99). Recall that $\gamma ^{0, \tau }(\varphi , x) = x + \breve{\beta } (\tau , \varphi , x)$ and $\gamma ^{\tau , 0}(\varphi , y) = y + \tau \beta ( \varphi , y)$ (cf. (2.78)). Since $a( \varphi , y + \tau \beta (\varphi , y) ) - a(\varphi , y) = \int _0^\tau a_x(\varphi , y + t \beta (\varphi , y)) \beta (\varphi , y) \hbox {d}t$, the claimed estimate for $p_m$ then follows by Lemmata 2.1, 2.30 and assumption (2.79). Now assume that for any $k \in \{ 1, \ldots , i - 1 \}$, $1 \le i \le N$, the function $p_{m - k}$, given by (2.100), satisfies the estimates (2.101). The ones for $p_{m - i}$ then follow by Lemmata 2.1, 2.3, 2.30, (2.95), (2.94), and (2.79). $\quad \square$

Continuing the proof of Proposition 2.31, note that in view of the definition (2.88) of ${{{\mathcal {P}}}}_N(\tau , \varphi )$, it follows from (2.101), Lemma 2.10, (2.22) and (2.21) that for any $\alpha \in {\mathbb {N}}$, $\tau \in [0, 1]$, and $s \geqq s_0$,

$$\begin{aligned} |{{{\mathcal {P}}}}_N(\tau , \varphi ) |_{m, s, \alpha }^\mathrm{{Lip}(\gamma )}\lesssim _{m, s, N, \alpha } \Vert a \Vert _{s }^\mathrm{{Lip}(\gamma )}+ \Vert \beta \Vert _{s + \sigma _N^{(N)}}^\mathrm{{Lip}(\gamma )}+ \Vert a \Vert _{s + \sigma _N^{(N)}}^\mathrm{{Lip}(\gamma )}\Vert \beta \Vert _{s_0 + \sigma _N^{(N)}}^\mathrm{{Lip}(\gamma )}.\nonumber \\ \end{aligned}$$

(2.102)

By (2.89), (2.92), and (2.96) we deduce that ${{{\mathcal {P}}}}_N(\tau , \varphi )$ solves

$$\begin{aligned} \begin{aligned}&\partial _\tau {{{\mathcal {P}}}}_N(\tau , \varphi ) = [B(\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )] - {{{\mathcal {Q}}}}_N^{(1)}(\tau , \varphi ), \quad {{{\mathcal {P}}}}_N(0, \varphi ) = a \partial _x^m , \\&{{{\mathcal {Q}}}}_N^{(1)}(\tau , \varphi ) \,{:=}\, {{{\mathcal {Q}}}}_N(\tau , \varphi ) + [B_\infty (\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )] \in OPS^{m - N - 1} , \end{aligned} \end{aligned}$$

(2.103)

where ${{{\mathcal {Q}}}}_N$ is defined in (2.93).

Next we estimate the difference between ${{{\mathcal {P}}}}_N(\tau )$ and ${{{\mathcal {P}}}}(\tau )$. First we establish the following formula:

Lemma 2.33

The operator ${{{\mathcal {R}}}}_N(\tau , \varphi ) \,{:=}\, {{{\mathcal {P}}}}(\tau , \varphi ) - {{{\mathcal {P}}}}_N(\tau , \varphi )$ is given by

$$\begin{aligned} {{{\mathcal {R}}}}_N(\tau , \varphi ) = \int _0^\tau \Phi (\eta , \tau , \varphi ) {{{\mathcal {Q}}}}_N^{(1)}(\eta , \varphi ) \Phi (\tau , \eta , \varphi )\, \hbox {d} \eta . \end{aligned}$$

(2.104)

Proof of Lemma 2.33

Writing ${{{\mathcal {P}}}}_N(\tau , \varphi ) - {{{\mathcal {P}}}}(\tau , \varphi )$ as

$$\begin{aligned}&{{{\mathcal {P}}}}_N(\tau , \varphi ) - {{{\mathcal {P}}}}(\tau , \varphi ) = {{{\mathcal {V}}}}_N(\tau , \varphi ) \Phi (\tau , \varphi )^{- 1}, \nonumber \\&\quad \quad {{{\mathcal {V}}}}_N(\tau , \varphi ) \,{:=}\, {{{\mathcal {P}}}}_N(\tau , \varphi ) \Phi (\tau , \varphi ) - \Phi (\tau , \varphi ){{{\mathcal {P}}}}_0(\varphi ) , \end{aligned}$$

(2.105)

one verifies by a straightforward calculation that ${{{\mathcal {V}}}}_N(\tau )$ solves

$$\begin{aligned} \partial _\tau {{{\mathcal {V}}}}_N (\tau , \varphi ) = B(\tau , \varphi ){{{\mathcal {V}}}}_N(\tau , \varphi ) - {{{\mathcal {Q}}}}_N^{(1)}(\tau , \varphi ) \Phi (\tau , \varphi ), \quad {{{\mathcal {V}}}}_N(0, \varphi ) = 0, \end{aligned}$$

where ${{{\mathcal {Q}}}}_N^{(1)}$ is given in (2.103). By the variation of the constants formula,

$$\begin{aligned} {{{\mathcal {V}}}}_N(\tau , \varphi ) = - \int _0^\tau \Phi (\tau , \varphi ) \Phi (\eta , \varphi )^{- 1} {{{\mathcal {Q}}}}_N^{(1)}(\eta , \varphi ) \Phi (\eta , \varphi )\, \hbox {d} \eta \end{aligned}$$

and, by (2.105) and (2.71), we deduce (2.104). $\quad \square$

Using formula (2.104) we now prove the estimate for ${{{\mathcal {R}}}}_N(\tau , \varphi )$ stated in (2.83) of Proposition 2.31. First we estimate ${{{\mathcal {Q}}}}_N^{(1)} = {{{\mathcal {Q}}}}_N(\tau , \varphi ) + [B_\infty (\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )] \in OPS^{m - N - 1}$ (cf. (2.103)). The estimate of ${{{\mathcal {Q}}}}_N$, obtained from (2.95), (2.94), (2.101), and the one of $[B_\infty (\tau , \varphi ), {{{\mathcal {P}}}}_N(\tau , \varphi )],$ obtained from (2.85), (2.102), Lemma 2.11, imply that there exists a constant $\aleph _N > 0$ so that for any $s \geqq s_0$, $\alpha \in {\mathbb {N}}$,

$$\begin{aligned} |{{{\mathcal {Q}}}}_N^{(1)}(\eta , \varphi ) |_{m - N - 1, s, \alpha }^\mathrm{{Lip}(\gamma )}\, \lesssim _{m, s, \alpha , N} \, \Vert \beta \Vert _{s + \aleph _N}^\mathrm{{Lip}(\gamma )}+ \Vert a \Vert _{s + \aleph _N}^\mathrm{{Lip}(\gamma )}\Vert \beta \Vert _{s_0 + \aleph _N}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(2.106)

Let $k \in {\mathbb {S}}_+$, $\lambda _0$ with $\lambda \leqq \lambda _0$, and $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 + \lambda _0 + m \leqq N - 1$. In view of the formula (2.104) of ${{{\mathcal {R}}}}_N(\tau , \varphi )$, the claimed estimate of $\langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda {{{\mathcal {R}}}}_N(\tau , \varphi )\langle D \rangle ^{n_2}$ follows from corresponding ones of

$$\begin{aligned} \langle D \rangle ^{n_1}\partial _{\varphi _k}^{\lambda _1}\Phi (\eta , \tau , \varphi ) \partial _{\varphi _k}^{\lambda _2}{{{\mathcal {Q}}}}_N^{(1)}(\eta , \varphi ) \partial _{\varphi _k}^{\lambda _3}\Phi (\tau , \eta , \varphi ) \langle D \rangle ^{n_2} \end{aligned}$$

($\tau , \eta \in [0, 1]$ and $\lambda _1 + \lambda _2 +\lambda _3 = \lambda$) which we write as

$$\begin{aligned}&\Big ( \langle D \rangle ^{n_1}\partial _{\varphi _k}^{\lambda _1}\Phi (\eta , \tau , \varphi ) \langle D \rangle ^{- n_1 - \lambda _1 - 1 } \Big ) \Big ( \langle D \rangle ^{n_1 + \lambda _1 + 1} \partial _{\varphi _k}^{\lambda _2}{{{\mathcal {Q}}}}_N^{(1)}(\eta , \varphi ) \langle D \rangle ^{n_2 + \lambda _3 + 1} \Big ) \\&\quad \Big ( \langle D \rangle ^{- n_2 - \lambda _3 - 1 } \partial _{\varphi _k}^{\lambda _3}\Phi (\tau , \eta , \varphi ) \langle D \rangle ^{n_2} \Big ). \end{aligned}$$

By Lemma 2.28, one obtains tame constants for the operators

$$\begin{aligned} \langle D \rangle ^{n_1}\partial _{\varphi _k}^{\lambda _1}\Phi (\eta , \tau , \varphi ) \langle D \rangle ^{- n_1 - \lambda _1 - 1 }, \quad \langle D \rangle ^{- n_2 - \lambda _3 - 1 } \partial _{\varphi _k}^{\lambda _3}\Phi (\tau , \eta , \varphi ) \langle D \rangle ^{n_2} , \end{aligned}$$

and by the estimates (2.106), (2.21), and Lemmata 2.10, 2.18 a tame constant for

$$\begin{aligned} \langle D \rangle ^{n_1 + \lambda _1 + 1} \partial _{\varphi _k}^{\lambda _2}{{{\mathcal {Q}}}}_N^{(1)}(\eta , \varphi ) \langle D \rangle ^{n_2 + \lambda _3 + 1} , \end{aligned}$$

allowing us to deduce that the composition of these three operators satisfies the bound (2.83) (using also Lemma 2.16 together with the assumption (2.79)). This proves the bound (2.83) for ${\mathcal {R}}_N$.

Item 4. of Proposition 2.31 can be shown by similar arguments. $\quad \square$

In the sequel we also need to study the operator obtained by conjugating $\omega \cdot \partial _\varphi$ with the time one flow $\Phi (\varphi ) = \Phi (0, 1, \varphi )$ of the transport equation (2.69). A straightforward calculation shows that

$$\begin{aligned} \Phi (\varphi ) \circ (\omega \cdot \partial _\varphi ) \circ \Phi (\varphi )^{- 1} = \omega \cdot \partial _\varphi + \Phi (\varphi ) \circ \omega \cdot \partial _\varphi (\Phi (\varphi )^{- 1}), \end{aligned}$$

where, according to Definition 2.4-4, for any $\varphi$-dependent family of linear operators $A(\varphi )$, the operator $\omega \cdot \partial _\varphi A(\varphi )$ is defined as

$$\begin{aligned} \omega \cdot \partial _\varphi A(\varphi ) = \sum _{m \in {\mathbb {S}}_+ } \omega _m \partial _{\varphi _m} A(\varphi ) = \sum _{m \in {\mathbb {S}}_+ } \omega _m [\partial _{\varphi _m}, A(\varphi )]. \end{aligned}$$

We now show that the operator $\Phi (\varphi )\circ \omega \cdot \partial _\varphi (\Phi (\varphi )^{- 1})$ is a pseudo-differential operator of order one, admitting an expansion in decreasing symbols. More precisely, the following holds:

Proposition 2.34

(Conjugation of $\omega \cdot \partial _\varphi$) Let N, $\lambda _0 \in {\mathbb {N}}$ and $S > s_0$ and assume that $\beta (\cdot , \cdot ; \omega )$ is in ${{{\mathcal {C}}}}^\infty ({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1)$ and Lipschitz continuous with respect to $\omega \in \Omega$. Then there exist constants $\sigma _N(\lambda _0)$, $\sigma _N > 0$, $\delta (S, N, \lambda _0) \in (0, 1)$, and $C_0 > 0$ so that the following holds: if

$$\begin{aligned} \Vert \beta \Vert _{s_0 + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )}\leqq \delta , \end{aligned}$$

(2.107)

then ${\Psi }(\varphi ) \,{:=}\, \Phi (\varphi ) \circ \omega \cdot \partial _\varphi ( \Phi (\varphi )^{- 1})$ is a pseudo-differential operator of order 1 with an expansion of the form

$$\begin{aligned} { \Psi }(\varphi ) = \sum _{i = 0}^N p_{1 - i} (\varphi , x ; \omega ) \partial _x^{1 - i} + {{{\mathcal {R}}}}_N(\varphi ) \end{aligned}$$

(2.108)

with the following properties:

• 1. For any $i = 0, \ldots , N$ and $s \geqq s_0$, $\Vert p_{1 - i}\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, N} \Vert \beta \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}$.

• 2. For any $k \in {\mathbb {S}}_+$, any $\lambda \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$, and any $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 + \lambda _0 \leqq N - 1 - m$, the pseudo-differential operator $\langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda {{{\mathcal {R}}}}_N(\varphi ) \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

  $$\begin{aligned} {{\mathfrak {M}}}_{\langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda {{{\mathcal {R}}}}_N(\varphi ) \langle D \rangle ^{n_2}}(s) \lesssim _{S, N, \lambda _0} \Vert \beta \Vert _{s + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )}, \quad \forall \, s_0 \leqq s \leqq S. \end{aligned}$$

• 3. Let $s_0< s_1 < S$ and assume that $\Vert \beta _j \Vert _{s_1 + \sigma _N(\lambda _0)} \leqq \delta$, $j = 1,2$. Then

  $$\begin{aligned} \begin{aligned} \Vert \Delta _{12} p_{1 - i} \Vert _{s_1} \lesssim _{s_1, N} \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma _N}, \quad i = 0, \ldots , N , \end{aligned} \end{aligned}$$

  and, for any $\lambda \leqq \lambda _0$, $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 + \lambda _0 \leqq N - 2$, and $k \in {\mathbb {S}}_+$

  $$\begin{aligned} \Vert \langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda \Delta _{12} {{{\mathcal {R}}}}_N(\varphi ) \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})} \lesssim _{s_1, N, n_1, n_2} \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma _N(\lambda _0)} \end{aligned}$$

  where $\Delta _{12}\beta = \beta _2 - \beta _1$, $\Delta _{12}p_{1-i} = p^{(2)}_{1-i} - p^{(1)}_{1-i}$, and $\Delta _{12} {{{\mathcal {R}}}}_N = {{{\mathcal {R}}}}^{(2)}_N - {{{\mathcal {R}}}}^{(1)}_N$. Here $p^{(j)}_{1-i}$ and ${{{\mathcal {R}}}}^{(j)}_N$ denote the coefficient $p_{m-i}$ and the remainder ${\mathcal {R}}_N$ in the expansion (2.108), corresponding to $\beta _j$ for $j = 1, 2$.

Proof

We have that $\Psi (\varphi ) = \Psi (1, \varphi )$ where $\Psi (\tau , \varphi ) \,{:=}\, \Phi (\tau , \varphi ) \circ \omega \cdot \partial _\varphi ( \Phi (\tau , \varphi )^{- 1})$. Arguing as in (2.87), one sees that the operator $\Psi (\tau , \varphi )$ solves the inhomogeneous Heisenberg equation

$$\begin{aligned} \partial _\tau \Psi (\tau , \varphi ) = [B(\tau , \varphi ), \Psi (\tau , \varphi ))] - \omega \cdot \partial _\varphi ( B(\tau , \varphi )) , \quad \Psi (0, \varphi ) = 0. \end{aligned}$$

The latter equation can be solved in a similar way as (2.86) by looking for approximate solutions of the form of a pseudo-differential operator of order 1, admitting an expansion of the form (2.108) (cf. (2.88)). The proof then proceeds in the same way as the one for Proposition 2.31 and hence is omitted. $\quad \square$

We finish this section by the following application of Proposition 2.31 to Fourier multipliers.

Lemma 2.35

Let N, $\lambda _0 \in {\mathbb {N}}$, $m \in {\mathbb {Z}}$, and $S > s_0$ and assume that ${{{\mathcal {Q}}}}$ is a Lipschitz family of Fourier multipliers with an expansion of the form

$$\begin{aligned} {{{\mathcal {Q}}}} = \sum _{i = 0}^N c_{m - i}(\omega ) \partial _x^{m - i} + {{{\mathcal {Q}}}}_N (\omega ) , \quad {{{\mathcal {Q}}}}_N (\omega ) \in {{{\mathcal {B}}}}(H^s, H^{s + N + 1 - m}), \quad \forall s \geqq 0.\nonumber \\ \end{aligned}$$

(2.109)

Then there exist $\sigma _N(\lambda _0)$, $\sigma _N > 0$, and $\delta = \delta (S, N, \lambda _0) \in (0, 1)$ so that the following holds: if

$$\begin{aligned} \Vert \beta \Vert _{s_0 + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )}\leqq \delta , \end{aligned}$$

(2.110)

then $\Phi (\varphi ) {{{\mathcal {Q}}}} \Phi (\varphi )^{- 1}$ is an operator of the form ${{{\mathcal {Q}}}} + {{{\mathcal {Q}}}}_\Phi (\varphi ) + {{{\mathcal {R}}}}_N(\varphi )$ with the following properties:

• 1. ${{{\mathcal {Q}}}}_\Phi (\varphi ) = \sum _{i = 0}^N \alpha _{m - i}(\varphi , x; \omega ) \partial _x^{m - i}$ where for any $s \geqq s_0$,

  $$\begin{aligned} \Vert \alpha _{m - i} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, N} \Vert \beta \Vert _{s + \sigma _N}^\mathrm{{Lip}(\gamma )}, \quad i = 0, \ldots , N . \end{aligned}$$

  (2.111)

• 2. For any $k \in {\mathbb {S}}_+$, $\lambda \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$, and $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 + \lambda _0 \leqq N - m - 2$, the operator $\langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda {{{\mathcal {R}}}}_N \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

  $$\begin{aligned} {{\mathfrak {M}}}_{\langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda {{{\mathcal {R}}}}_N \langle D \rangle ^{n_2}}(s) \lesssim _{S, N, \lambda _0} \Vert \beta \Vert _{s + \sigma _N(\lambda _0)}^\mathrm{{Lip}(\gamma )},\quad \forall s_0 \leqq s \leqq S. \end{aligned}$$

  (2.112)

• 3. Let $s_0< s_1 < S$ and assume that $\Vert \beta _j \Vert _{s_1 + \sigma _N(\lambda _0)} \leqq \delta$, $j = 1,2$. Then

  $$\begin{aligned} \begin{aligned} \Vert \Delta _{12} \alpha _{m - i} \Vert _{s_1} \lesssim _{s_1, N} \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma _N}, \quad i = 0, \ldots , N , \end{aligned} \end{aligned}$$

  and, for any $k \in {\mathbb {S}}_+$, $\lambda \leqq \lambda _0$, and $n_1, n_2 \in {\mathbb {N}}$ with $n_1 + n_2 + \lambda _0 \leqq N - m - 2$,

  $$\begin{aligned} \Vert \langle D \rangle ^{n_1}\partial _{\varphi _k}^\lambda \Delta _{12} {{{\mathcal {R}}}}_N(\varphi ) \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})} \lesssim _{s_1, N, n_1, n_2} \Vert \Delta _{12} \beta \Vert _{s_1 + \sigma _N(\lambda _0)} \end{aligned}$$

  where $\Delta _{12}\beta = \beta _2 - \beta _1$, $\Delta _{12}\alpha _{m-i} = \alpha ^{(2)}_{m-i} - \alpha ^{(1)}_{m-i}$, and $\Delta _{12} {{{\mathcal {R}}}}_N = {{{\mathcal {R}}}}^{(2)}_N - {{{\mathcal {R}}}}^{(1)}_N$. Here $\alpha ^{(j)}_{m-i}$ and ${{{\mathcal {R}}}}^{(j)}_N$ denote the coefficient $\alpha _{m-i}$ and, respectively, remainder ${\mathcal {R}}_N$, corresponding to $\beta _j$ for $j = 1, 2$.

Proof

Applying Proposition 2.31 to $\Phi (\varphi ) \partial _x^{m - i} \Phi (\varphi )^{- 1}$ for $i = 0, \ldots , N$, we get

$$\begin{aligned} \begin{aligned} \Phi (\varphi ) \Big ( \sum _{i = 0}^N c_{m -i} (\omega ) \partial _x^{m - i} \Big ) \Phi (\varphi )^{- 1}&= \sum _{i = 0}^N c_{m - i}(\omega ) \partial _x^{m - i} + {{{\mathcal {Q}}}}_\Phi (\varphi ) + {{{\mathcal {R}}}}_N^{(1)}(\varphi ) \end{aligned} \end{aligned}$$

where ${{{\mathcal {Q}}}}_\Phi (\varphi ) = \sum _{i = 0}^N \alpha _{m - i}(\varphi , x; \omega ) \partial _x^{m - i}$ with $\alpha _{m - i}$ satisfying (2.111) and the remainder ${{{\mathcal {R}}}}_N^{(1)}(\varphi )$ satisfying (2.112). Next we write $\Phi (\varphi ){{{\mathcal {Q}}}}_N \Phi (\varphi )^{- 1} = {{{\mathcal {Q}}}}_N + {{{\mathcal {R}}}}_N^{(2)}(\varphi )$ where

$$\begin{aligned} \begin{aligned} {{{\mathcal {R}}}}_N^{(2)}(\varphi )&: = \big ( \Phi (\varphi ) - \mathrm{Id} \big ) {{{\mathcal {Q}}}}_N \Phi (\varphi )^{- 1} + {{{\mathcal {Q}}}}_N \big (\Phi (\varphi )^{- 1} - \mathrm{Id} \big ). \end{aligned} \end{aligned}$$

We then argue as in the proof of the estimate of the remainder ${{{\mathcal {R}}}}_N(\tau , \varphi )$ in Proposition 2.31. Using Lemma 2.28 and the assumption that ${{{\mathcal {Q}}}}_N$ is a Fourier multiplier in ${{{\mathcal {B}}}}(H^s, H^{s + N + 1 - m})$ we get that ${{{\mathcal {R}}}}_N^{(2)}(\varphi )$ satisfies (2.112), and ${{{\mathcal {R}}}}_N(\varphi ) = {{{\mathcal {R}}}}_N^{(1)}(\varphi ) + {{{\mathcal {R}}}}_N^{(2)}(\varphi )$ satisfies (2.112) as well. Item 3. follows by similar arguments. $\quad \square$

3 Integrable Features of KdV

In this section we discuss the canonical coordinates which are used to prove the existence of quasi-periodic solutions of the perturbed KdV equation (1.6) close to the finite gap manifold ${\mathcal {M}}_{{{\mathbb {S}}}_+}$, cf. (1.10), via a Nash–Moser iterative scheme. These coordinates were constructed in [20] specifically for this purpose. Their main properties were described in broad terms in the Introduction (cf. (P1)–(P6)). We discuss their features in detail in Sections 3.2–3.3. In Section 3.4 we record properties of the KdV-frequencies, which will be needed in particular to derive the measure estimates of Section 8.2.

3.1 Birkhoff Coordinates

According to [21], the KdV equation (1.1) on the torus admits global canonical coordinates, called Birkhoff coordinates, so that equation (1.1) can be solved by quadrature, cf. Theorem 3.1. We use them to describe the ${{\mathbb {S}}}_+$-gap potentials. Unfortunately they are not suited for implementing a Nash–Moser iteration scheme for the search of quasi-periodic solutions of (1.6), since they do not seem to possess an expansion in terms of pseudo-differential operators.

The Birkhoff coordinates $z_k$, $k \ne 0$, take values in the sequence space $h^0_0$ (cf. (1.23)), which we endow with the standard Poisson bracket defined by $\{ z_n, z_{k} \} = \mathrm{i} 2 \pi n \, \delta _{k,-n}$ for any $n, k \in {\mathbb {Z}}\setminus \{0\}$.

Theorem 3.1

(Birkhoff coordinates, [21]) There exists a real analytic diffeomorphism $\Psi ^{kdv} : h^0_0 \rightarrow H^0_0 ({\mathbb {T}}_1)$ so that the following holds:

• (i) for any $s \in {\mathbb {Z}}_{\geqq 0}$, $\Psi ^{kdv}(h^s_0) \subseteq H^s_0 ({\mathbb {T}}_1)$ and $\Psi ^{kdv}: h^s_0 \rightarrow H^s_0 ({\mathbb {T}}_1)$ is a real analytic symplectic diffeomorphism.

• (ii) ${H}^{kdv} \circ \Psi ^{kdv}: h^1_0 \rightarrow {\mathbb {R}}$ is a real analytic function of the actions $I_k \,{:=}\, \frac{1}{2\pi k} z_k z_{- k}$, $k \ge 1$. The KdV Hamiltonian, viewed as a function of the actions $(I_k)_{k \ge 1}$, is denoted by ${\mathcal {H}}^{kdv}_o$.

• (iii) $\Psi ^{kdv}(0) = 0$ and the differential $d_0\Psi ^{kdv}$ of $\Psi ^{kdv}$ at 0 is the inverse Fourier transform ${{{\mathcal {F}}}}^{- 1}$.

By Theorem 3.1, the KdV equation, expressed in the Birkhoff coordinates $(z_n)_{n \ne 0}$, reads

$$\begin{aligned} \partial _t z_n = \mathrm{i} \omega _n^{kdv}((I_k)_{k \ge 1}) z_n , \,\, \forall n \in {\mathbb {Z}}\setminus \{0\} , \quad \omega _{\pm m}^{kdv}((I_k)_{k \ge 1}) : = \pm \partial _{I_m} {{\mathcal {H}}}_o^{kdv}((I_k)_{k \ge 1}) , \,\, \forall m \ge 1, \end{aligned}$$

and its solutions are given by $z(t) \,{:=}\, (z_n(t))_{n \ne 0}$ where

$$\begin{aligned} z_{n}(t) = z_{n}(0) \mathrm{exp}\big (\mathrm{i} \omega _{n}^{kdv}\big ((I_k^{(0)})_{k \ge 1}\big ) \, t \big ) , \,\,\, \forall n \in {\mathbb {Z}}\setminus \{0\} , \quad I_k^{(0)} : = \frac{1}{2\pi k} z_k(0) z_{- k}(0), \,\,\, \forall {k \ge 1}. \end{aligned}$$

Let us consider a finite set ${{\mathbb {S}}}_+\subset {\mathbb {N}}_+ = \{1, 2, \ldots \}$ and define

$$\begin{aligned} {{\mathbb {S}}}\,{:=}\, {{\mathbb {S}}}_+\cup (- {{\mathbb {S}}}_+) , \quad {{\mathbb {S}}}^\bot _+ \,{:=}\, {\mathbb {N}}_+ \setminus {{\mathbb {S}}}_+, \quad {{\mathbb {S}}}^\bot \,{:=}\, {{\mathbb {S}}}^{\bot }_+ \cup (- {{\mathbb {S}}}^\bot _+) \subset {\mathbb {Z}}\setminus \{0\} . \end{aligned}$$

In Birkhoff coordinates, a ${{\mathbb {S}}}_+-$gap solution of the KdV equation, also referred to as ${{\mathbb {S}}}_+-$gap solution, is a solution of the form

$$\begin{aligned} z_n(t) = \mathrm{exp}\big (\mathrm{i} \omega ^{kdv}_n( \nu , 0) t \big ) z_n(0) , \,\, z_n(0) \ne 0 , \,\, \forall n \in {{\mathbb {S}}}, \quad z_n(t) = 0 , \,\, \forall n \in {{\mathbb {S}}}^\bot ,\nonumber \\ \end{aligned}$$

(3.1)

where $\nu : = ( I_k^{(0)})_{k \in {{\mathbb {S}}}_+} \in {\mathbb {R}}_{> 0}^{{{\mathbb {S}}}_+}$ and, by a slight abuse of notation, we write

$$\begin{aligned} \omega ^{kdv}_n( I , (I_k)_{k \in {{\mathbb {S}}}_+^\bot }) = \omega ^{kdv}_n( (I_k)_{k \geqq 1} ) , \quad I \,{:=}\, (I_k)_{k \in {{\mathbb {S}}}_+} \in {\mathbb {R}}_{> 0}^{{{\mathbb {S}}}_+} . \end{aligned}$$

(3.2)

Such solutions are quasi-periodic in time with frequency vector (cf. (1.11))

$$\begin{aligned} \omega ^{kdv} (\nu ) = \big ( \omega _n^{kdv}(\nu , 0) \big )_{n \in {{\mathbb {S}}}_+} \in {\mathbb {R}}^{{{\mathbb {S}}}_+}, \end{aligned}$$

(3.3)

parametrized by $\nu \in {\mathbb {R}}^{{{\mathbb {S}}}_+}_{> 0}$. The map $\nu \mapsto \omega ^{kdv} (\nu )$ is a local analytic diffeomorphism, see Remark 3.10.

When written in action-angle coordinates $\theta = (\theta _n)_{n \in {{\mathbb {S}}}_+} \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$, $I = (I_n)_{n \in {{\mathbb {S}}}_+} \in {\mathbb {R}}_{> 0}^{{{\mathbb {S}}}_+}$, which are related to the complex Birkhoff coordinates $z_n= z_n(\theta , I)$, $n \ne 0$ by

$$\begin{aligned} z_{\pm n}(\theta , I) \,{:=}\, \sqrt{2\pi n I_n} e^{\mp \mathrm{i} \theta _n} , \quad \forall \, n \in {{\mathbb {S}}}_+, \quad z_n(\theta , I) = 0 , \quad \forall n \in {{\mathbb {S}}}^\bot , \end{aligned}$$

(3.4)

the ${{\mathbb {S}}}_+$-gap solution (3.1) reads

$$\begin{aligned} \theta (t) = \theta ^{(0)} - \omega ^{kdv} (\nu ) t , \quad I(t) = \nu , \quad z_n(t) = 0 , \quad \forall n \in {{\mathbb {S}}}^\bot . \end{aligned}$$

Furthermore, we introduce the map $\Psi _{{{\mathbb {S}}}_+} : {\mathbb {T}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \rightarrow {\mathcal {M}}_{{{\mathbb {S}}}_+} \subset \cap _{s \ge 0} H^s_0({\mathbb {T}}_1)$, which coordinatizes the manifold ${\mathcal {M}}_{{{\mathbb {S}}}_+}$ of ${{\mathbb {S}}}_+$-gap potentials (cf. (1.10)),

$$\begin{aligned} \Psi _{{{\mathbb {S}}}_+}(\theta , I) \,{:=}\, \Psi ^{kdv} ((z_n(\theta , I))_{n \in {\mathbb {Z}}\setminus \{ 0\}}) \end{aligned}$$

(3.5)

where $z_n(\theta , I)$, $n \ne 0$, are given by (3.4).

3.2 Normal Form Coordinates for the KdV Equation

Theorem 3.2 below rephrases Theorem 1.1 in [20], in a form taylored to our needs. A key property of the normal form coordinates is stated in Theorem 3.2-$(\mathbf{AE1})$, saying that they admit an expansion in terms of pseudo-differential operators. This property, together with the additional Corollaries 3.3 and 3.4 below, allow to prove, in Section 3.3, that the linearized Hamiltonian vector field $\partial _x d_\bot \nabla _w {\mathcal {H}}_\varepsilon$ admits an expansion in terms of classical pseudo-differential operators, up to smoothing remainders which satisfy tame estimates. These key results are needed for implementing our diagonalization procedure of the linearized operator carried out in Sections 6–7.

We consider an open bounded set $\Xi \subset {\mathbb {R}}^{{\mathbb {S}}_+}_{> 0}$ so that (1.14) holds for some $\delta > 0$. Recall that ${{{\mathcal {V}}}}^s(\delta ) \subset {\mathcal {E}}_s$, ${{{\mathcal {V}}}}(\delta ) = {{{\mathcal {V}}}}^0(\delta )$, are defined in (1.27) and the spaces ${\mathcal {E}}_s$ and $E_s$ are given by (1.25). The elements in ${\mathcal {E}}_s$ are denoted by ${\mathfrak {x}} = (\theta , y, w)$ whereas the ones in $E_s$ by $\widehat{ \mathfrak {x}} = ({\widehat{\theta }}, {\widehat{y}}, {\widehat{w}})$. The space ${{{\mathcal {V}}}}(\delta ) \cap {{{\mathcal {E}}}}_s$ is endowed with the symplectic form

$$\begin{aligned} {{{\mathcal {W}}}} \,{:=}\, \Big ( {\mathop \sum }_{j \in {{\mathbb {S}}}_+} \hbox {d} y_j \wedge \hbox {d} \theta _j \Big ) \oplus {{{\mathcal {W}}}}_\bot , \end{aligned}$$

(3.6)

where ${{{\mathcal {W}}}}_\bot$ is the restriction to $L^2_\bot ({\mathbb {T}}_1)$ of the symplectic form ${{{\mathcal {W}}}}_{L^2_0}$ defined in (1.9). The Poisson structure ${{{\mathcal {J}}}}$ corresponding to ${\mathcal {W}}$, defined by the identity $\{ F, G \} = {{{\mathcal {W}}}}(X_F, X_G) = \big \langle \nabla F ,\, {{{\mathcal {J}}}} \nabla G \big \rangle$, is the unbounded operator

$$\begin{aligned} {\mathcal {J}} : E_s \rightarrow E_s , \quad ({\widehat{\theta }}, {\widehat{y}}, {\widehat{w}}) \mapsto (- {\widehat{y}}, {\widehat{\theta }}, \partial _x {\widehat{w}}) , \end{aligned}$$

(3.7)

where $\langle , \ \rangle$ is the inner product (1.26).

Theorem 3.2

(Normal form KdV coordinates with pseudo-differential expansion, [20]) Let ${{\mathbb {S}}}_+\subseteq {\mathbb {N}}$ be finite, $\Xi$ an open bounded subset of ${\mathbb {R}}_{> 0}^{{{\mathbb {S}}}_+}$ so that (1.14) holds for some $\delta > 0$. Then, for $\delta > 0$ sufficiently small, there exists a canonical ${{{\mathcal {C}}}}^\infty$ family of diffeomorphisms $\Psi _\nu : {{{\mathcal {V}}}}(\delta ) \rightarrow \Psi _\nu ({{{\mathcal {V}}}}(\delta )) \subseteq L^2_0 ({\mathbb {T}}_1), \, (\theta , y, w)\mapsto q$, $\nu \in \Xi$, with the property that $\Psi _\nu$ extends $\Psi _{{{\mathbb {S}}}_+}$, introduced in (3.5), namely

$$\begin{aligned} \Psi _\nu (\theta , y , 0 ) = \Psi _{{{\mathbb {S}}}_+}(\theta , \nu + y) , \quad \forall (\theta , y, 0 ) \in {{{\mathcal {V}}}}(\delta ), \quad \forall \nu \in \Xi , \end{aligned}$$

(3.8)

and is compatible with the scale of Sobolev spaces $H^s_0({\mathbb {T}}_1), s \in {\mathbb {N}}$, in the sense that $\Psi _\nu \big ( {{{\mathcal {V}}}}(\delta )\cap {{\mathcal {E}}}_s \big ) \subseteq H^s_0({\mathbb {T}}_1)$ and that $\Psi _\nu : {{{\mathcal {V}}}}(\delta )\cap {{\mathcal {E}}}_s \rightarrow H^s_0({\mathbb {T}}_1)$ is a ${{{\mathcal {C}}}}^\infty -$diffeomorphism onto its image, so that the following holds:

• (AE1) (Asymptotic expansion of $\Psi _\nu$) For any integer $M \ge 1$, $\nu \in \Xi$, ${\mathfrak {x}} = (\theta , y , w) \in {\mathcal {V}}(\delta )$, $\Psi _\nu ({\mathfrak {x}})$ admits an asymptotic expansion of the form

  $$\begin{aligned} \Psi _\nu ( \theta , y , w ) = \Psi _{{{\mathbb {S}}}_+}(\theta , \nu + y) + w + \sum _{k = 1}^M a_{-k }^\Psi ({\mathfrak {x}}; \nu ) \, \partial _x^{- k} w + {{{\mathcal {R}}}}_{M }^\Psi ({\mathfrak {x}}; \nu )\nonumber \\ \end{aligned}$$

  (3.9)

  where ${{{\mathcal {R}}}}_{M}^\Psi (\theta , y, 0; \nu ) = 0$ and, for any $s \in {\mathbb {N}}$ and $1 \le k \le M$, the functions

  $$\begin{aligned}&{{{\mathcal {V}}}}(\delta ) \times \Xi \rightarrow H^s({\mathbb {T}}_1),\, ({\mathfrak {x}}, \nu ) \mapsto a_{-k }^\Psi ({\mathfrak {x}}; \nu ) , \\&\quad \quad ({{{\mathcal {V}}}}(\delta ) \cap {{\mathcal {E}}}_s) \times \Xi \rightarrow H^{s + M +1}({\mathbb {T}}_1),\, ({\mathfrak {x}},\nu ) \mapsto {\mathcal {R}}_{M }^\Psi ({\mathfrak {x}}; \nu ) , \end{aligned}$$

  are ${{{\mathcal {C}}}}^\infty$.

• (AE2) (Asymptotic expansion of $\hbox {d}\Psi _\nu ^\top$) For any ${\mathfrak {x}} \in {{{\mathcal {V}}}}^1(\delta )$ (cf. definition (1.27)), $\nu \in \Xi$, the transpose $d \Psi _\nu ({\mathfrak {x}})^\top$ of the differential $d \Psi _\nu ({\mathfrak {x}}) : E_1 \rightarrow H^1_0 ({\mathbb {T}}_1)$ is a bounded linear operator $d \Psi _\nu ({\mathfrak {x}})^\top : H^1_0({\mathbb {T}}_1) \rightarrow E_1$, and, for any ${\widehat{q}} \in H^1_0 ({\mathbb {T}}_1)$ and integer $M \ge 1$, $\hbox {d}\Psi _\nu ({\mathfrak {x}})^\top [{\widehat{q}}]$ admits an expansion of the form

  $$\begin{aligned} d \Psi _\nu ({\mathfrak {x}})^\top [{\widehat{q}}]&= \Big ( 0, 0, \, {\Pi }_\bot {\widehat{q}} + \Pi _\bot \sum _{k = 1}^M a_{-k}^{d \Psi ^\top }({\mathfrak {x}}; \nu ) \partial _x^{- k}{\widehat{q}} \, + \Pi _\bot \sum _{k = 1}^M (\partial _x^{- k} w) \, {\mathcal {A}}_{-k}^{\hbox {d}\Psi ^\top }({\mathfrak {x}}; \nu )[{\widehat{q}}] \Big ) \nonumber \\&\quad + {{{\mathcal {R}}}}_{M}^{d \Psi ^\top }({\mathfrak {x}}; \nu )[{\widehat{q}}] \end{aligned}$$

  (3.10)

  where, for any $s \ge 1$ and $1 \le k \le M$,

  $$\begin{aligned} \begin{aligned}&{{{\mathcal {V}}}}^1(\delta ) \times \Xi \rightarrow H^s ({\mathbb {T}}_1) , \,\, ({\mathfrak {x}}, \nu ) \mapsto a_{-k}^{d\Psi ^\top } ({\mathfrak {x}}; \nu ) , \\&{{{\mathcal {V}}}}^1(\delta ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H^1_0 ({\mathbb {T}}_1), H^s({\mathbb {T}}_1)), \, \, ({\mathfrak {x}}, \nu ) \mapsto {\mathcal {A}}_{-k}^{d\Psi ^\top } ({\mathfrak {x}}; \nu ), \\&({{{\mathcal {V}}}}^1 (\delta ) \cap {{\mathcal {E}}}_s) \times \Xi \rightarrow {{{\mathcal {B}}}}(H^s_0 ({\mathbb {T}}_1), E_{s +M +1}) , \, \, ({\mathfrak {x}}, \nu ) \mapsto {{{\mathcal {R}}}}_M^{d\Psi ^\top }({\mathfrak {x}}; \nu ) , \end{aligned} \end{aligned}$$

  are ${{\mathcal {C}}}^\infty$. Furthermore,

  $$\begin{aligned} a_{- 1}^{d \Psi ^\top }(\mathfrak {x}; \nu ) = - a_{- 1}^\Psi (\mathfrak {x}; \nu ). \end{aligned}$$

  (3.11)

• (AE3) (Normal form) For any $\nu \in \Xi$, the Hamiltonian ${{{\mathcal {H}}}}^{kdv} ( \cdot \, ; \nu ) \,{:=}\, H^{kdv} \circ \Psi _\nu : {{{\mathcal {V}}}}^1(\delta ) \rightarrow {\mathbb {R}}$ (cf. definition (1.27)) is in normal form up to order three, meaning that

  $$\begin{aligned} {{{\mathcal {H}}}}^{kdv} (\theta , y, w; \nu )&= \omega ^{kdv}(\nu ) \cdot y + \frac{1}{2} \big ( \Omega ^{kdv}(D ;\nu ) w, w \big )_{L^2_x} \nonumber \\&\quad + \frac{1}{2} \Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) [y] \cdot y+ {{{\mathcal {R}}}}^{kdv} (\theta , y , w; \nu ) \end{aligned}$$

  (3.12)

  where $\omega ^{kdv}(\nu ) = (\omega _n^{kdv}(\nu , 0))_{n \in {\mathbb {S}}_+},$

  $$\begin{aligned} \begin{aligned}&\Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) \,{:=}\, (\partial _{I_j} \omega ^{kdv}_k (\nu , 0))_{j, k \in {\mathbb {S}}_+}, \quad \quad \Omega ^{kdv}(D ; \nu ) w: = \sum _{n \in {{\mathbb {S}}}^\bot } \Omega _n^{kdv}(\nu ) w_n e^{\mathrm{i} 2\pi n x}, \\&\Omega _n^{kdv}(\nu ) \,{:=}\, \frac{1}{2\pi n} \omega _n^{kdv}(\nu , 0), \ \forall n \in {{\mathbb {S}}}^\bot , \quad w = \sum _{n \in {{\mathbb {S}}}^\bot } w_n e^{\mathrm{i} 2\pi n x}, \end{aligned} \end{aligned}$$

  (3.13)

  and ${{{\mathcal {R}}}}^{kdv} : {{{\mathcal {V}}}}^1(\delta ) \times \Xi \rightarrow {\mathbb {R}}$ is a ${{{\mathcal {C}}}}^\infty$ map satisfying

  $$\begin{aligned} {{{\mathcal {R}}}}^{kdv}(\theta , y, w; \nu ) = O\big ( \, ( \Vert y \Vert + \Vert w \Vert _{H^1_x})^3 \, \big ) , \end{aligned}$$

  (3.14)

  and has the property that, for any $s \ge 1$, its $L^2-$gradient

  $$\begin{aligned} ({{{\mathcal {V}}}}^1(\delta ) \cap {{\mathcal {E}}}_s) \times \Xi \rightarrow E_{s}, \, ({\mathfrak {x}}, \nu ) \mapsto \nabla {{{\mathcal {R}}}}^{kdv}({\mathfrak {x}}; \nu ) = \big ( \nabla _{\theta } {{{\mathcal {R}}}}^{kdv}({\mathfrak {x}}; \nu ), \nabla _{y} {{{\mathcal {R}}}}^{kdv}({\mathfrak {x}}; \nu ), \nabla _{w} {{{\mathcal {R}}}}^{kdv}({\mathfrak {x}}; \nu ) \big ) \end{aligned}$$

  is a ${{\mathcal {C}}}^\infty$ map as well. As a consequence

  $$\begin{aligned} \nabla {{{\mathcal {R}}}}^{kdv}(\theta , 0, 0; \nu ) = 0 , \quad d_\bot \nabla {{{\mathcal {R}}}}^{kdv}(\theta , 0, 0; \nu ) = 0 , \quad \partial _y \nabla {{{\mathcal {R}}}}^{kdv}(\theta , 0, 0; \nu ) = 0 .\nonumber \\ \end{aligned}$$

  (3.15)

• (Est1) For any $\nu \in \Xi$, $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, ${\mathfrak {x}} \in {{{\mathcal {V}}}} (\delta )$, $1 \le k \le M$, $\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l \in E_0$, $s \in {\mathbb {N}}$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha a_{- k}^\Psi ({\mathfrak {x}}; \nu ) \Vert _{H^s_x} \lesssim _{s, k, \alpha } \, 1 , \quad \Vert d^l \partial _\nu ^\alpha a_{- k}^\Psi ({\mathfrak {x}}; \nu )[\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^s_x}\lesssim _{s, k, l, \alpha } \, \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_0}. \end{aligned} \end{aligned}$$

  Similarly, for any $\nu \in \Xi$, $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, ${\mathfrak {x}} \in {{{\mathcal {V}}}} (\delta ) \cap {{\mathcal {E}}}_s$, $\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l \in E_s$, $s \in {\mathbb {N}}$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^\Psi ({\mathfrak {x}}; \nu )\Vert _{H^{s + M + 1}_x} \lesssim _{s, M, \alpha } \Vert w \Vert _{H^s_x }, \quad \\&\Vert d^l \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^\Psi ({\mathfrak {x}}; \nu )[\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^{s + M + 1}_x} \lesssim _{s, M, l, \alpha } \sum _{j = 1}^l \Big ( \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_s} \prod _{i \ne j} \Vert \widehat{{\mathfrak {x}}}_i\Vert _{E_0} \Big ) \\&\quad + \Vert w \Vert _{H^s_x} \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_0}. \end{aligned} \end{aligned}$$

• (Est2) For any $\nu \in \Xi$, $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, ${\mathfrak {x}} \in {{{\mathcal {V}}}}^1(\delta )$, $1 \le k \le M$, $\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l \in E_1$, $s \ge 1$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha a_{-k}^{{d \Psi }^\top }({\mathfrak {x}}; \nu )\Vert _{H^s_x} \lesssim _{s, k, \alpha } 1 , \quad \Vert d^l \partial _\nu ^\alpha a_{-k}^{{d \Psi }^\top }({\mathfrak {x}}; \nu )[\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^s_x} \lesssim _{s, k, l, \alpha } \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_1} , \\&\Vert \partial _\nu ^\alpha {\mathcal {A}}_{-k}^{{d \Psi }^\top }({\mathfrak {x}}; \nu )\Vert _{{{{\mathcal {B}}}}(H_0^1, H^s_x)} \lesssim _{s, k, \alpha } \, 1 , \\&\Vert d^l \partial _\nu ^\alpha {\mathcal {A}}_{-k}^{{d \Psi }^\top }({\mathfrak {x}} ; \nu )[\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{{{{\mathcal {B}}}}(H_0^1, H^s_x)} \lesssim _{s, k, l, \alpha } \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_1}. \end{aligned} \end{aligned}$$

  Similarly, for any $\nu \in \Xi$, $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, ${\mathfrak {x}} \in {{{\mathcal {V}}}}^1(\delta ) \cap {{\mathcal {E}}}_s$, $\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l \in E_s$, ${\widehat{q}} \in H^s_0$, $s \ge 1$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{d\Psi ^\top } ({\mathfrak {x}}; \nu ) [{\widehat{q}}]\Vert _{E_{s + M + 1}} \lesssim _{s, M, \alpha } \Vert {\widehat{q}}\Vert _{H^s_x}+ \Vert w \Vert _{H^s_x} \Vert {\widehat{q}}\Vert _{H^1_x}, \\&\Vert d^l \big ( \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{d \Psi ^\top } ({\mathfrak {x}}; \nu ) [{\widehat{q}}] \big )[\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{E_{s + M + 1}} \lesssim _{s, M, l, \alpha } \Vert {\widehat{q}}\Vert _{H^s_x} \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_1} \\&\quad + \Vert {\widehat{q}}\Vert _{H^1_x} \sum _{j = 1}^l \Big ( \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_s} \prod _{i \ne j} \Vert \widehat{{\mathfrak {x}}}_i\Vert _{E_1} \Big ) + \Vert {\widehat{q}}\Vert _{H^1_x} \Vert w\Vert _{H^s_x} \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j \Vert _{E_1}. \end{aligned} \end{aligned}$$

We now apply Theorem 3.2 to obtain in the two corollaries below novel results concerning the extensions of $d \Psi _\nu (\mathfrak {x})^\top$ and $\hbox {d}\Psi _\nu (\mathfrak {x})$ to Sobolev spaces of negative order. These results will be used to deduce Lemmata  3.5 and 3.7, which allow to verify the assumptions A and B of Lemma 2.26 for the remainders, as explained in Remark 2.27. The spaces ${\mathcal {E}}_s$, $E_s$ for negative s are defined as in (1.25).

Corollary 3.3

(Extension of $d \Psi _\nu (\mathfrak {x})^\top$ and its asymptotic expansion) Let $M \geqq 1$. There exists $\sigma _M > 0$ so that for any $\mathfrak {x} \in {{{\mathcal {V}}}}^{\sigma _M}(\delta )$ and $\nu \in \Xi$, the operator $d \Psi _\nu (\mathfrak {x})^\top$ extends to a bounded linear operator $d \Psi _\nu (\mathfrak {x})^\top : H^{- M - 1}_0({\mathbb {T}}_1) \rightarrow E_{- M - 1}$ and for any ${\widehat{q}} \in H^{- M - 1}_0({\mathbb {T}}_1)$, $d \Psi _\nu (\mathfrak {x})^\top [{\widehat{q}} ]$ admits an expansion of the form

$$\begin{aligned} d \Psi _\nu (\mathfrak {x})^\top [{\widehat{q}} ]&= \Big ( 0,0, \Pi _\bot {\widehat{q}} + \Pi _\bot \sum _{k = 1}^M a_{- k}^{ext}(\mathfrak {x}; \nu ; d \Psi ^\top ) \partial _x^{- k} {\widehat{q}}\Big )\nonumber \\&\quad + {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[{\widehat{q}}] \end{aligned}$$

(3.16)

with the following properties:

• (i) For any $s \geqq 0$, the maps

  $$\begin{aligned} \begin{aligned}&{{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi \rightarrow H^s({\mathbb {T}}_1) , \quad (\mathfrak {x}, \nu ) \mapsto a_{- k}^{ext}(\mathfrak {x}; \nu ; d \Psi ^\top ), \quad 1 \le k \le M , \end{aligned} \end{aligned}$$

  are ${{{\mathcal {C}}}}^\infty$. They satisfy $a_{- 1}^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top ) = a_{-1}^{d \Psi ^\top }({\mathfrak {x}}; \nu )$ (cf. Theorem 3.2-$\mathbf{(AE2)}$) and for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, and $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha a_{- k}^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top ) \Vert _{H^s_x} \lesssim _{s, M, \alpha } 1, \\&\Vert \partial _\nu ^\alpha d^l a_{- k}^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l] \Vert _{H^s_x} \lesssim _{s, M, l, \alpha } \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \end{aligned} \end{aligned}$$

  (3.17)

• (ii) For any $- 1 \leqq s \leqq M+ 1$, the map

  $$\begin{aligned} {{{\mathcal {R}}}}_M^{ext}(\cdot ; \cdot ;\hbox {d}\Psi ^\top ) : {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H_0^{- s}({\mathbb {T}}_1), E_{M + 1 - s}) \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, ${\widehat{q}} \in H^{- s}_0({\mathbb {T}}_1)$, and $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[{\widehat{q}}] \Vert _{E_{M + 1- s}} \lesssim _{ M, \alpha } \Vert {\widehat{q}}\Vert _{H_x^{- s}}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l][{\widehat{q}}] \Vert _{E_{M + 1 - s}} \lesssim _{s, M, l, \alpha } \Vert {\widehat{q}}\Vert _{H^{- s}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \\ \end{aligned}\nonumber \\ \end{aligned}$$

  (3.18)

• (iii) For any $s \geqq 1$, the map

  $$\begin{aligned} {{{\mathcal {R}}}}_M^{ext}(\cdot ; \cdot ;\hbox {d}\Psi ^\top ) : \big ({{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{{\mathcal {E}}}}_{s + \sigma _M}\big ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H_0^{ s}({\mathbb {T}}_1), E_{s + M + 1} ) \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{s + \sigma _M}$, ${\widehat{q}} \in H^{s}_0({\mathbb {T}}_1)$, and $(\mathfrak {x}, \nu ) \in \big ({{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{{\mathcal {E}}}}_{s + \sigma _M}\big ) \times \Xi$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[{\widehat{q}}] \Vert _{E_{M + 1+ s}} \lesssim _{ s, M, \alpha } \Vert {\widehat{q}}\Vert _{H_x^{s}} + \Vert \mathfrak {x}\Vert _{s + \sigma _M} \Vert {\widehat{q}}\Vert _{H^1_x} , \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top ) [\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l] [{\widehat{q}}] \Vert _{E_{M + 1+ s}} \lesssim _{ s, M, l, \alpha } \Vert {\widehat{q}}\Vert _{H_x^s} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{\sigma _M}} \\&\quad \quad + \Vert {\widehat{q}}\Vert _{H^1_x} \Big ( \sum _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{s + \sigma _M}} \prod _{i \ne j} \Vert \widehat{\mathfrak {x}}_i\Vert _{E_{\sigma _M}} + \Vert \mathfrak {x} \Vert _{E_{s + \sigma _M}} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}} \Big ). \end{aligned}\nonumber \\ \end{aligned}$$

  (3.19)

Proof

By Theorem 3.2, for any $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}(\delta ) \times \Xi$, the differential $d \Psi _\nu (\mathfrak {x}) : E_0 \rightarrow L^2_0({\mathbb {T}}_1)$ is bounded and, for any $M \geqq 1$, differentiating (3.9), $d \Psi _\nu (\mathfrak {x}) [\widehat{\mathfrak {x}}]$ admits the expansion for any $\widehat{\mathfrak {x}}= ({\widehat{\theta }}, {\widehat{y}}, {\widehat{w}}) \in E_0$ of the form

$$\begin{aligned}&\hbox {d}\Psi _\nu (\mathfrak {x}) [\widehat{\mathfrak {x}}] = {\widehat{w}} + \sum _{k = 1}^M a_{- k}^\Psi (\mathfrak {x}; \nu ) \partial _x^{- k} {\widehat{w}} + {{{\mathcal {R}}}}_M^{(1)}(\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}], \\&{{{\mathcal {R}}}}^{(1)}_M (\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}] \,{:=}\, \sum _{k = 1}^M (\partial _x^{- k} w) \hbox {d}a_{- k}^\Psi (\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}] +\hbox {d}{{{\mathcal {R}}}}_M^\Psi (\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}] + d_{\theta ,y} \Psi _{{{\mathbb {S}}}_+}(\theta , \nu + y) [{\widehat{\theta }}, {\widehat{y}}] . \nonumber \end{aligned}$$

(3.20)

For $\sigma _M \geqq M$, the map ${{{\mathcal {R}}}}_M^{(1)} : {{{\mathcal {V}}}}^{\sigma _M} (\delta ) \times \Xi \rightarrow {{{\mathcal {B}}}}(E_0, H^{M + 1}({\mathbb {T}}_1))$ is ${{{\mathcal {C}}}}^\infty$ and satisfies, by Theorem 3.2-$\mathbf{(Est1)}$, for any $\alpha \in {\mathbb {N}}^{{{\mathbb {S}}}_+}$, $l \geqq 1$,

$$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{(1)}(\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}] \Vert _{H^{M + 1}_x} \lesssim _{M, \alpha } \Vert \widehat{\mathfrak {x}}\Vert _{E_0}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{(1)}(\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l] [\widehat{\mathfrak {x}}] \Vert _{H^{M + 1}_x} \lesssim _{M, l, \alpha } \Vert \widehat{\mathfrak {x}}\Vert _{E_0} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{\sigma _M}}. \end{aligned} \end{aligned}$$

(3.21)

Now consider the transpose operator $d \Psi _\nu (\mathfrak {x})^\top : L^2_0({\mathbb {T}}_1) \rightarrow E_0$. By (3.20), for any ${\widehat{q}} \in L^2_0({\mathbb {T}}_1)$, one has

$$\begin{aligned} d \Psi _\nu (\mathfrak {x})^\top [{\widehat{q}} ] = \Big (0,0, \Pi _\bot {\widehat{q}} + \Pi _\bot \sum _{k = 1}^M (- 1)^k \partial _x^{- k}\big ( a_{-k}^\Psi (\mathfrak {x}; \nu ) \ {\widehat{q}} \big ) \Big ) + {{{\mathcal {R}}}}_M^{(1)}(\mathfrak {x}; \nu )^\top [{\widehat{q}}].\nonumber \\ \end{aligned}$$

(3.22)

Since each function $a_{-k}^\Psi (\mathfrak {x}; \nu )$ is ${{{\mathcal {C}}}}^\infty$ and ${{{\mathcal {R}}}}^{(1)}_M (\mathfrak {x}; \nu )^\top : H^{- M - 1} ({\mathbb {T}}_1) \rightarrow E_0$ is bounded, the right hand side of (3.22) defines a linear operator in ${{{\mathcal {B}}}} ( H^{- M - 1}_0({\mathbb {T}}_1), E_{- M - 1})$, which we also denote by $d \Psi _\nu (\mathfrak {x})^\top$. By (2.12), the expansion (3.22) yields one of the form (3.16) where by (3.21) and Theorem 3.2-$\mathbf{(Est1)}$, the remainder ${{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )$ satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, and ${\widehat{q}} \in H^{-M - 1}_0({\mathbb {T}}_1)$

$$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[{\widehat{q}}] \Vert _{E_{0}} \lesssim _{ M, \alpha } \Vert {\widehat{q}}\Vert _{H_x^{- M - 1}}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d \Psi ^\top )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l][{\widehat{q}}] \Vert _{E_{0}} \lesssim _{M, l, \alpha } \Vert {\widehat{q}}\Vert _{H^{- M - 1}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \end{aligned} \end{aligned}$$

(3.23)

The restriction of the operator $d \Psi _\nu (\mathfrak {x})^\top : H^{- M - 1}_0({\mathbb {T}}_1) \rightarrow E_{- M - 1}$ to $H_0^1({\mathbb {T}}_1)$ coincides with (3.10) and, by the uniqueness of an expansion of this form,

$$\begin{aligned} \begin{aligned}&a_{- k}^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top ) = a_{-k}^{d \Psi ^\top }({\mathfrak {x}}; \nu ) , \quad k = 1, \ldots , M, \\&{{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; \hbox {d} \Psi ^\top )[{\widehat{q}}]= \sum _{k = 1}^M (\partial _x^{- k} w ) {\mathcal {A}}_{-k}^{\hbox {d}\Psi ^\top }({\mathfrak {x}}; \nu )[{\widehat{q}}] \, + {{{\mathcal {R}}}}_{M}^{\hbox {d} \Psi ^\top }({\mathfrak {x}}; \nu )[{\widehat{q}}] , \quad \forall {\widehat{q}} \in H_0^1({\mathbb {T}}_1) . \end{aligned} \end{aligned}$$

The claimed estimates (3.17) and (3.19) then follow by Theorem 3.2-$\mathbf{(Est2)}$. In particular we have, for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, ${\widehat{q}} \in H^{1}_0({\mathbb {T}}_1)$,

$$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[{\widehat{q}}] \Vert _{E_{M + 2}} \lesssim _{ M, \alpha } \Vert {\widehat{q}}\Vert _{H_x^{1}}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ;\hbox {d}\Psi ^\top )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l][{\widehat{q}}] \Vert _{E_{M + 2}} \lesssim _{M, l, \alpha } \Vert {\widehat{q}}\Vert _{H^{1}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \\ \end{aligned} \end{aligned}$$

(3.24)

Finally the estimates (3.18) follow by interpolation between (3.23) and (3.24). $\quad \square$

Corollary 3.4

(Extension of $d_\bot \Psi _\nu (\mathfrak {x})$ and its asymptotic expansion) Let $M \geqq 1$. There exists $\sigma _M > 0$ so that for any $\mathfrak {x} \in {{{\mathcal {V}}}}^{\sigma _M}(\delta )$ and $\nu \in \Xi$, the operator $d_\bot \Psi _\nu (\mathfrak {x})$ extends to a bounded linear operator, $d_\bot \Psi _\nu (\mathfrak {x}): H_\bot ^{- M - 2}({\mathbb {T}}_1) \rightarrow H^{- M - 2}_0({\mathbb {T}}_1)$, and for any ${\widehat{w}} \in H_\bot ^{- M - 2}({\mathbb {T}}_1)$, $d_\bot \Psi _\nu (\mathfrak {x})[{\widehat{w}}]$ admits an expansion

$$\begin{aligned} d_\bot \Psi _\nu (\mathfrak {x})[{\widehat{w}}] = {\widehat{w}} + \sum _{k = 1}^M a_{- k}^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi ) \partial _x^{- k} {\widehat{w}} + {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[{\widehat{w}}] \end{aligned}$$

(3.25)

with the following properties:

• (i) For any $s \geqq 0$, the maps

  $$\begin{aligned} \begin{aligned}&{{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi \rightarrow H^s({\mathbb {T}}_1) , \quad (\mathfrak {x}, \nu ) \mapsto a_{- k}^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi ), \quad 1 \le k \le M , \end{aligned} \end{aligned}$$

  are ${{{\mathcal {C}}}}^\infty$. They satisfy $a_{- 1}^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi ) = a_{-1}^\Psi ({\mathfrak {x}}; \nu )$ (cf. Theorem 3.2-$\mathbf{(AE1)}$) and for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, and $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha a_{- k}^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi ) \Vert _{H^s_x} \lesssim _{s, M, \alpha } 1, \\&\Vert \partial _\nu ^\alpha d^l a_{- k}^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l] \Vert _{H^s_x} \lesssim _{s, M, l, \alpha } \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \end{aligned} \end{aligned}$$

  (3.26)

• (ii) For any $0 \leqq s \leqq M + 2$, the map

  $$\begin{aligned} {{{\mathcal {R}}}}_M^{ext}(\cdot , \cdot ; d_\bot \Psi ) : {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H_\bot ^{- s}({\mathbb {T}}_1), H^{M + 1 - s}({\mathbb {T}}_1)) \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies, for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, ${\widehat{w}} \in H^{- s}_\bot ({\mathbb {T}}_1)$, and $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[{\widehat{w}}] \Vert _{H^{M + 1- s}_x} \lesssim _{ M, \alpha } \Vert {\widehat{w}}\Vert _{H_x^{- s}}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l][{\widehat{w}}] \Vert _{H^{M + 1 - s}_x} \lesssim _{s, M, l, \alpha } \Vert {\widehat{w}}\Vert _{H^{- s}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \\ \end{aligned}\nonumber \\ \end{aligned}$$

  (3.27)

• (iii) For any $s \geqq 0$, the map

  $$\begin{aligned} {{{\mathcal {R}}}}_M^{ext}(\cdot , \cdot ; d_\bot \Psi ) : \big ( {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{{\mathcal {E}}}}_{s + \sigma _M} \big ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H_\bot ^{ s}({\mathbb {T}}_1), H^{M + 1 + s}({\mathbb {T}}_1)) \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{s + \sigma _M}$, ${\widehat{w}} \in H^{s}_\bot ({\mathbb {T}}_1)$, and $(\mathfrak {x}, \nu ) \in \big ( {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{{\mathcal {E}}}}_{s + \sigma _M} \big ) \times \Xi$,

  $$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[{\widehat{w}}] \Vert _{H^{M + 1+ s}_x} \lesssim _{ s, M, \alpha } \Vert {\widehat{w}}\Vert _{H_x^s} + \Vert \mathfrak {x}\Vert _{E_{s + \sigma _M}} \Vert {\widehat{w}}\Vert _{L^2_x} , \\&\Vert \partial _\nu ^\alpha d^l \big ( {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[{\widehat{w}}] \big )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l] \Vert _{H^{M + 1+ s}_x} \lesssim _{ s, M, l, \alpha } \Vert {\widehat{w}}\Vert _{H_x^s} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{\sigma _M}} \\&\quad + \Vert {\widehat{w}}\Vert _{L^2_x} \Big ( \sum _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{s + \sigma _M}} \prod _{i \ne j} \Vert \widehat{\mathfrak {x}}_i\Vert _{E_{\sigma _M}} + \Vert \mathfrak {x} \Vert _{E_{s + \sigma _M}} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}} \Big ). \end{aligned}\nonumber \\ \end{aligned}$$

  (3.28)

Proof

By Theorem 3.2-$\mathbf{(AE2)}$, for any $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}^1 (\delta ) \times \Xi$, the operator $d_\bot \Psi _\nu (\mathfrak {x})^\top : H_0^1({\mathbb {T}}_1) \rightarrow H^1_\bot ({\mathbb {T}}_1)$ is bounded and for any $M \geqq 1$ and ${\widehat{q}} \in H_0^1({\mathbb {T}}_1)$, $d_\bot \Psi _\nu (\mathfrak {x})^\top [\widehat{q} \, ]$ admits the expansion of the form

$$\begin{aligned} \begin{aligned}&d_\bot \Psi _\nu (\mathfrak {x})^\top [\widehat{q} \, ] = \Pi _\bot {\widehat{q}} + \Pi _\bot \sum _{k = 1}^M a_{- k}^{d \Psi ^\top }(\mathfrak {x}; \nu ) \partial _x^{- k} {\widehat{q}} + {{{\mathcal {R}}}}_M^{(2)}(\mathfrak {x}; \nu )[{\widehat{q}} \, ] , \\&{{{\mathcal {R}}}}^{(2)}_M (\mathfrak {x}; \nu )[ {\widehat{q}} \, ] \,{:=}\, \Pi _\bot \sum _{k = 1}^M (\partial _x^{- k} w) \, {{{\mathcal {A}}}}_{- k}^{d \Psi ^\top }(\mathfrak {x}; \nu )[{\widehat{q}}] + {{{\mathcal {R}}}}_M^{d \Psi ^\top }(\mathfrak {x}; \nu )[{\widehat{q}} \, ] . \end{aligned} \end{aligned}$$

(3.29)

For $\sigma _M \geqq M + 1$, the map ${{{\mathcal {R}}}}_M^{(2)} : {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H_0^1({\mathbb {T}}_1), H_\bot ^{M + 2}({\mathbb {T}}_1))$ is ${{{\mathcal {C}}}}^\infty$ and by Theorem 3.2-$\mathbf{(Est2)}$, satisfies for any $\alpha \in {\mathbb {N}}^{{{\mathbb {S}}}_+}$ and $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$

$$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{(2)}(\mathfrak {x}; \nu )[{\widehat{q}}] \Vert _{H^{M + 2}_x} \lesssim _{M, \alpha } \Vert {\widehat{q}}\Vert _{H^1_x}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{(2)}(\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l][{\widehat{q}}] \Vert _{H^{M + 2}_x} \lesssim _{M, l, \alpha } \Vert {\widehat{q}}\Vert _{H^1_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{\sigma _M}}. \end{aligned} \end{aligned}$$

(3.30)

Now consider the transpose operator $\big (d_\bot \Psi _\nu (\mathfrak {x})^\top \big )^\top : H^{- 1}_\bot ({\mathbb {T}}_1) \rightarrow H_0^{- 1}({\mathbb {T}}_1)$. It defines an extension of $d_\bot \Psi _\nu (\mathfrak {x})$ to $H^{- 1}_\bot ({\mathbb {T}}_1)$, which we denote again by $d_\bot \Psi _\nu (\mathfrak {x})$. By (3.29), for any ${\widehat{w}} \in H_\bot ^{- 1}({\mathbb {T}}_1)$, one has

$$\begin{aligned} d_\bot \Psi _\nu (\mathfrak {x})[{\widehat{w}}] = {\widehat{w}} + \sum _{k = 1}^M (- 1)^k \partial _x^{- k}\big ( a_{-k}^{d \Psi ^\top }(\mathfrak {x}; \nu ) {\widehat{w}} \big ) + {{{\mathcal {R}}}}_M^{(2)}(\mathfrak {x}; \nu )^\top [{\widehat{w}}]. \end{aligned}$$

(3.31)

Since each function $a_{-k}^{d \Psi ^\top }(\mathfrak {x}; \nu )$ is ${{{\mathcal {C}}}}^\infty$ and the operator ${{{\mathcal {R}}}}_M^{(2)}(\mathfrak {x}; \nu )^\top : H^{- M - 2}_\bot ({\mathbb {T}}_1) \rightarrow H_0^{- 1}({\mathbb {T}}_1)$ is bounded, the right hand side of (3.31) defines a linear operator in ${{{\mathcal {B}}}} ( H^{- M - 2}_0({\mathbb {T}}_1), E_{- M - 2})$, which we also denote by $d \Psi _\nu (\mathfrak {x})$. By (2.12), the expansion (3.31) yields one of the form (3.25) where by (3.30) and Theorem 3.2-$\mathbf{(Est2)}$, the remainder ${{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d \Psi ^\top )$ satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, and ${\widehat{w}} \in H^{-M - 2}_0({\mathbb {T}}_1)$

$$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[{\widehat{w}}] \Vert _{H^{- 1}_x} \lesssim _{ M, \alpha } \Vert {\widehat{w}}\Vert _{H_x^{- M - 2}}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l][{\widehat{w}}] \Vert _{H^{- 1}_x} \lesssim _{M, l, \alpha } \Vert {\widehat{w}}\Vert _{H_x^{- M - 2}} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \\ \end{aligned} \end{aligned}$$

(3.32)

The restriction of the expansion (3.31) to $L^2_\bot ({\mathbb {T}}_1)$ coincides with the one of $d_\bot \Psi _\nu (\mathfrak {x})[{\widehat{w}}]$, obtained by differentiating (3.9) (see (3.20)). It then follows from the uniqueness of an expansion of this form that

$$\begin{aligned} \begin{aligned}&a_{- k}^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi ) = a_{-k }^\Psi ({\mathfrak {x}}; \nu ) , \quad k = 1, \ldots , M, \\&{{{\mathcal {R}}}}_M^{ext}({\mathfrak {x}}; \nu ; d_\bot \Psi )[{\widehat{w}}]= \sum _{k = 1}^M (\partial _x^{- k} w) d_\bot a_{-k }^\Psi ({\mathfrak {x}}; \nu )[{\widehat{w}}] + d_\bot {{{\mathcal {R}}}}_{M }^\Psi ({\mathfrak {x}}; \nu ) [{\widehat{w}}] , \quad \forall {\widehat{w}} \in L^2_\bot ({\mathbb {T}}_1) . \end{aligned} \end{aligned}$$

The claimed estimates (3.26) and (3.28) thus follow by Theorem 3.2-$\mathbf{(Est1)}$. In particular, for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{\sigma _M}$, and ${\widehat{w}} \in L^2_\bot ({\mathbb {T}}_1)$,

$$\begin{aligned} \begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[{\widehat{w}}] \Vert _{H^{M + 1}_x} \lesssim _{ M, \alpha } \Vert {\widehat{w}}\Vert _{L^2_x}, \\&\Vert \partial _\nu ^\alpha d^l {{{\mathcal {R}}}}_M^{ext}(\mathfrak {x}; \nu ; d_\bot \Psi )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l][{\widehat{w}}] \Vert _{H^{M + 1}_x} \lesssim _{M, l, \alpha } \Vert {\widehat{w}}\Vert _{L^2_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}. \end{aligned} \end{aligned}$$

(3.33)

The claimed estimates (3.27) are then obtained by interpolating between (3.32) and (3.33). $\quad \square$

3.3 Expansions of Linearized Hamiltonian Vector Fields

In this section, we apply Theorem 3.2 and Corollaries 3.3–3.4, to obtain asymptotic expansions of the linearized Hamiltonian vector fields $\partial _x d_\bot \nabla _w {{{\mathcal {P}}}}$ and $\partial _x d_\bot \nabla _w {{{\mathcal {H}}}}^{kdv}$. These expansions are key for implementing the KAM reduction procedure of Sections 6–7 to obtain an approximate right inverse of $\omega \cdot \partial _\varphi - d X_{{\mathcal {H}}_\varepsilon }$.

For any Hamiltonian of the form $P(u) = \int _{{\mathbb {T}}_1} f(x, u(x), u_x(x))\, \hbox {d}x$ with a ${{{\mathcal {C}}}}^\infty$-smooth density

$$\begin{aligned} f : {\mathbb {T}}_1 \times {\mathbb {R}}\times {\mathbb {R}}\mapsto {\mathbb {R}}, \ (x, \zeta _0, \zeta _1 ) \mapsto f (x, \zeta _0, \zeta _1 ) , \end{aligned}$$

(3.34)

define

$$\begin{aligned} {{{\mathcal {P}}}} \,{:=}\, P \circ \Psi _\nu , \quad {{{\mathcal {P}}}} (\mathfrak {x}; \nu )\,{:=}\, P (\Psi _\nu (\mathfrak {x})), \quad \mathfrak {x} = (\theta , y, w) , \end{aligned}$$

(3.35)

where $\Psi _\nu$ is the coordinate transformation of Theorem 3.2. As a first result, we provide an expansion of the linearized Hamiltonian vector field $\partial _x d_\bot \nabla _w {\mathcal {P}}$.

Lemma 3.5

(Expansion of $\partial _x d_\bot \nabla _w {{{\mathcal {P}}}}$) Let $P(u) = \int _{{\mathbb {T}}_1} f(x, u, u_x)\, \hbox {d}x$ with $f \in {{{\mathcal {C}}}}^{\infty }({\mathbb {T}}_1 \times {\mathbb {R}}\times {\mathbb {R}})$. For any $M \in {\mathbb {N}}$ there is $\sigma _M \ge M+1$ so that for any ${\mathfrak {x}} \in {{{\mathcal {V}}}}^{\sigma _M}(\delta )$ and $\nu \in \Xi$, the operator $\partial _x d_\bot \nabla _w {{{\mathcal {P}}}}({\mathfrak {x}}; \nu )$ admits an expansion of the form

$$\begin{aligned} \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}({\mathfrak {x}}; \nu )[\cdot ]&= {\Pi }_\bot \sum _{k = 0}^{M+3} a_{3 -k}({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}) \, \partial _x^{3 - k}[\cdot ] \, \nonumber \\&\quad + {{{\mathcal {R}}}}_M({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}})[\cdot ] \end{aligned}$$

(3.36)

with the following properties:

• 1. For any $s \geqq 0$, the maps

  $$\begin{aligned}&( {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{\mathcal {E}}}_{s+\sigma _M} ) \times \Xi \rightarrow H^s({\mathbb {T}}_1) , \quad ({\mathfrak {x}} ; \nu ) \mapsto a_{3 - k}( {\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}} ) , \\&\quad \quad \quad 0 \le k \le M + 3 , \end{aligned}$$

  are ${{{\mathcal {C}}}}^\infty$, and satisfy for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{ {\mathfrak {x}}}_l \in E_{s + \sigma _M}$, and $({\mathfrak {x}}, \nu ) \in \big ({{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{\mathcal {E}}}_{s+\sigma _M}\big ) \times \Xi$,

  $$\begin{aligned}&\Vert \partial _\nu ^\alpha a_{3 - k}( {\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}) \Vert _{H^s_x} \lesssim _{s, M, \alpha } 1 + \Vert w \Vert _{H^{s + \sigma _M}_x}, \\&\Vert \partial _\nu ^\alpha d^l a_{3 - k}( {\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}})[\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^s_x} \lesssim _{s, M, l, \alpha } \sum _{j = 1}^l \big ( \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_{s + \sigma _M}} \prod _{n \ne j} \Vert \widehat{{\mathfrak {x}}}_n\Vert _{E_{\sigma _M}}\big ) \nonumber \\&\quad + \Vert w \Vert _{H^{s + \sigma _M}_x} \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j \Vert _{E_{\sigma _M}}. \nonumber \end{aligned}$$

  (3.37)

• 2. For any $0 \leqq s \leqq M + 1$, the map

  $$\begin{aligned} {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H^{-s}({\mathbb {T}}_1), H_\bot ^{M + 1 - s}({\mathbb {T}}_1)) , \quad (\mathfrak {x}, \nu ) \mapsto {{{\mathcal {R}}}}_M({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}) , \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{ \sigma _M}$, $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi$, and ${\widehat{w}} \in H^{-s}_\bot ({\mathbb {T}}_1)$,

  $$\begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}})[{\widehat{w}}]\Vert _{H^{ M + 1 - s}_x} \lesssim _{s, M, \alpha } \Vert {\widehat{w}}\Vert _{H^{-s}_x} , \\&\Vert \partial _\nu ^\alpha d^l \big ( {{{\mathcal {R}}}}_M({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}})[{\widehat{w}}] \big )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l]\Vert _{H^{M + 1 -s}_x} \lesssim _{s, M, l, \alpha } \Vert {\widehat{w}}\Vert _{H^{-s}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}} . \nonumber \end{aligned}$$

  (3.38)

• 3. For any $s \geqq 0$, the map

  $$\begin{aligned} ( {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{{\mathcal {E}}}}_{ s + \sigma _M} ) \times \Xi \rightarrow {{{\mathcal {B}}}}(H^s({\mathbb {T}}_1), H_\bot ^{s + M + 1}({\mathbb {T}}_1)) , \quad (\mathfrak {x}, \nu ) \mapsto {{{\mathcal {R}}}}_M({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}) , \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{s + \sigma _M}$, $(\mathfrak {x}, \nu ) \in ({{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{{\mathcal {E}}}}_{ s + \sigma _M}) \times \Xi$, and ${\widehat{w}} \in H^s_\bot ({\mathbb {T}}_1)$,

  $$\begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}})[{\widehat{w}}]\Vert _{H^{s + M + 1}_x} \lesssim _{s, M, \alpha } \Vert {\widehat{w}}\Vert _{H^s_x} + \Vert w\Vert _{H^{s + \sigma _M}_x} \Vert {\widehat{w}}\Vert _{L^2_x}, \nonumber \\&\Vert \partial _\nu ^\alpha d^l \big ( {{{\mathcal {R}}}}_M({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}})[{\widehat{w}}] \big )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l]\Vert _{H^{s + M + 1}_x} \lesssim _{s, M, l, \alpha } \Vert {\widehat{w}}\Vert _{H^s_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}}\nonumber \\&\quad + \Vert {\widehat{w}}\Vert _{L^2_x} \big ( \Vert w\Vert _{H^{s + \sigma _M}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{\sigma _M}} + \sum _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{s + \sigma _M}} \prod _{i \ne j} \Vert \widehat{\mathfrak {x}}_i \Vert _{E_{\sigma _M}} \big ) . \end{aligned}$$

  (3.39)

Remark 3.6

The coefficient $a_3$ in (3.36) can be computed as $a_3 ({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}} ) = - (\partial _{\zeta _1}^2 f)(x, u, u_x) \big |_{u = \Psi _\nu ({\mathfrak {x}})}$.

Proof

Differentiating (3.35) we have that

$$\begin{aligned} \nabla {{\mathcal {P}}} ( {\mathfrak {x}} ; \nu ) = (d \Psi _\nu ( {\mathfrak {x}}))^\top \big [ \nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big ] , \end{aligned}$$

(3.40)

where by (3.34),

$$\begin{aligned} \nabla P ( u ) = \Pi _0^\bot \big [ (\partial _{\zeta _0} f) (x, u, u_x )- \big ( (\partial _{\zeta _1} f ) (x, u, u_x) \big )_x \big ] \end{aligned}$$

(3.41)

and $\Pi _0^\bot$ is the $L^2$-orthogonal projector of $L^2({\mathbb {T}}_1)$ onto $L^2_0({\mathbb {T}}_1)$. By (3.40), the $w-$component $\nabla _w {{\mathcal {P}}}( {\mathfrak {x}}; \nu )$ of $\nabla {{\mathcal {P}}}( {\mathfrak {x}}; \nu )$ equals $(d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top \big [ \nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big ]$. Differentiating it with respect to w in direction ${\widehat{w}}$ then yields

$$\begin{aligned} d_\bot \nabla _w {{\mathcal {P}}} ( {\mathfrak {x}}; \nu ) [{\widehat{w}}]&= (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top \big [ d \nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big [ d_\bot \Psi _\nu ({\mathfrak {x}}) [ {\widehat{w}} ]\big ] \big ] \nonumber \\&\quad + \big ( d_\bot (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top [{\widehat{w}}]\big ) \big [ \nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big ]. \end{aligned}$$

(3.42)

Analysis of the first term on the right hand side of (3.42): Evaluating the differential $d \nabla P(u)$ of (3.41) at $u = \Psi _\nu ({\mathfrak {x}})$, one gets

$$\begin{aligned} \begin{aligned}&\hbox {d}(\nabla P) (\Psi _\nu ({\mathfrak {x}}))[h] = \Pi _0^\bot \big ( b_2({\mathfrak {x}}; \nu ) \partial _x^2 h +b_1({\mathfrak {x}}; \nu ) \partial _x h + b_0({\mathfrak {x}} ; \nu ) h \big ), \\&b_2({\mathfrak {x}}; \nu ) \,{:=}\, - \partial _{\zeta _1}^2 f (x, u, u_x)\Big |_{u = \Psi _\nu ({\mathfrak {x}})}, \quad b_1({\mathfrak {x}}; \nu ) \,{:=}\, (b_2({\mathfrak {x}} ; \nu ))_x , \\&b_0({\mathfrak {x}}; \nu ) \,{:=}\, \big ( (\partial _{\zeta _0}^2 f)(x, u, u_x) - \big ( (\partial ^2_{\zeta _0\zeta _1} f) (x, u, u_x) \big )_x \big ) \big |_{u = \Psi _\nu ({\mathfrak {x}})}. \end{aligned} \end{aligned}$$

(3.43)

By Lemma 2.3 and Theorem 3.2 one infers that for any $s \geqq 0$, the maps

$$\begin{aligned} ({{{\mathcal {V}}}}^{3}(\delta ) \cap {{\mathcal {E}}}_{s+3}) \times \Xi \rightarrow H^s_x , \quad ({\mathfrak {x}}; \nu ) \mapsto b_i({\mathfrak {x}}; \nu ) , \quad i = 0,1,2 , \end{aligned}$$

are ${{{\mathcal {C}}}}^\infty$ and satisfy for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{s + 3}$, and $({\mathfrak {x}}, \nu ) \in \big ({{{\mathcal {V}}}}^{3} (\delta ) \cap {{\mathcal {E}}}_{s+3}\big ) \times \Xi$,

$$\begin{aligned}&\Vert \partial _\nu ^\alpha b_i({\mathfrak {x}}; \nu )\Vert _{H^s_x} \lesssim _{s, \alpha } 1 + \Vert w \Vert _{H^{s + 3}_x}, \\&\Vert \partial _\nu ^\alpha d^l b_i(\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l] \Vert _{H^s_x} \lesssim _{s, l, \alpha } \sum _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{s + 3}} \prod _{i \ne j} \Vert \widehat{\mathfrak {x}}_i\Vert _{E_{3}} + \Vert w \Vert _{H^{s + 3}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_3}.\nonumber \end{aligned}$$

(3.44)

By Corollary 3.3 (expansion of $(d_\bot \Psi _\nu )^\top$), Corollary 3.4 (expansion of $d_\bot \Psi _\nu$), (3.44) (estimates of $b_i$), (3.43) (formula for $d (\nabla P) (\Psi _\nu ({\mathfrak {x}}))$), and Lemma 2.12 (composition), one obtains the expansion

$$\begin{aligned}&\partial _x (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top \big [\hbox {d}\nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big [ d_\bot \Psi _\nu ({\mathfrak {x}}) [ \cdot ] \big ] \big ]\nonumber \\&\quad = \Pi _\bot \sum _{k = 0}^{M + 3} a_{3 - k}^{(1)}({\mathfrak {x}}; \nu ) \partial _x^{3 - k} + R_1({\mathfrak {x}}; \nu ), \end{aligned}$$

(3.45)

where $a_{3}^{(1)}(\mathfrak {x}; \nu ) = b_2({\mathfrak {x}}; \nu )$, the functions $a_{3 - k}^{(1)}(\mathfrak {x}; \nu )$, $k = 0, \ldots , M+ 3$, and the remainder $R_1({\mathfrak {x}}; \nu )$ satisfy the claimed properties 1–3 of the lemma, in particular (3.37)–(3.39).

Analysis of the second term on the right hand side of (3.42): Since $d \Psi _\nu (\mathfrak {x})$ is symplectic, $d \Psi _\nu (\mathfrak {x})^\top = {{{\mathcal {J}}}}^{- 1}\hbox {d}\Psi _\nu (\mathfrak {x})^{- 1} \partial _x$ where ${{{\mathcal {J}}}}$ is the Poisson operator defined in (3.7), implying that for any ${\widehat{w}} \in H^1_\bot ({\mathbb {T}}_1)$,

$$\begin{aligned} \begin{aligned} d_\bot \big (d \Psi _\nu (\mathfrak {x})^\top \big ) [{\widehat{w}}]&= - {{{\mathcal {J}}}}^{- 1}\hbox {d}\Psi _\nu (\mathfrak {x})^{- 1} \big (d_\bot \hbox {d}\Psi _\nu (\mathfrak {x})[{\widehat{w}}] \big )\hbox {d}\Psi _\nu (\mathfrak {x})^{- 1} \partial _x \\&= -\hbox {d}\Psi _\nu (\mathfrak {x})^\top \partial _x^{- 1}\hbox {d}\big (d_\bot \Psi _\nu (\mathfrak {x})[{\widehat{w}}] \big )\big [ {{{\mathcal {J}}}}\hbox {d}\Psi _\nu (\mathfrak {x})^\top \cdot \big ]. \end{aligned} \end{aligned}$$

By this identity we get

$$\begin{aligned}&\partial _x \big ( d_\bot (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top [\cdot ]\big ) \big [ \nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big ]\\&\quad = - \partial _x d \Psi _\nu (\mathfrak {x})^\top \partial _x^{- 1}\hbox {d}\big (d_\bot \Psi _\nu (\mathfrak {x})[\cdot ] \big )\big [ {{{\mathcal {J}}}}\hbox {d}\Psi _\nu (\mathfrak {x})^\top \nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big ].\nonumber \end{aligned}$$

(3.46)

Arguing as for the first term on the right hand side of (3.42) (cf. (3.45)) one gets an expansion of the form

$$\begin{aligned} \partial _x \big ( d_\bot (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top [\cdot ]\big ) \big [ \nabla P (\Psi _\nu ({\mathfrak {x}} ) ) \big ] = \Pi _\bot \sum _{k = 3}^{M + 3} a_{3 - k}^{(2)}({\mathfrak {x}}; \nu ) \partial _x^{3 - k} + R_2({\mathfrak {x}}; \nu ),\nonumber \\ \end{aligned}$$

(3.47)

where the functions $a_{3 - k}^{(2)}(\mathfrak {x}; \nu )$, $k = 3, \ldots , M+ 3$, and the remainder $R_2({\mathfrak {x}}; \nu )$ satisfy the claimed properties 1-3 of the lemma, in particular (3.37)–(3.39).

Conclusion: By (3.42) and the above analysis of the expansions (3.45) and (3.47), the lemma and Remark 3.6 follow. $\quad \square$

As a second result of this section we derive an expansion for the linearized Hamiltonian vector field $\partial _x d_\bot \nabla _w {{\mathcal {H}}}^{kdv}$ where ${{\mathcal {H}}}^{kdv}(\mathfrak {x}; \nu )= H^{kdv}( \Psi _\nu (\mathfrak {x}))$ (cf. Theorem 3.2-(AE3)). We recall that the family of Fourier multipliers $\Omega ^{kdv}( D; \nu )$, $\nu \in \Xi$, is defined in (3.13).

Lemma 3.7

(Expansion of $\partial _x d_\bot \nabla _w {{{\mathcal {H}}}}^{kdv}$) For any $M \in {\mathbb {N}}$ there is $\sigma _M \ge M+1$ so that, for any $({\mathfrak {x}}, \nu ) \in {\mathcal {V}}^{\sigma _M } (\delta )\times \Xi$, the operator $\partial _x d_\bot \nabla _w {{{\mathcal {H}}}}^{kdv}({\mathfrak {x}}; \nu )$ admits an expansion of the form

$$\begin{aligned} \begin{aligned}&\partial _x d_\bot \nabla _w {{{\mathcal {H}}}}^{kdv}({\mathfrak {x}}; \nu )[\cdot ] = \partial _x \Omega ^{kdv}(D ;\nu ) [\cdot ] + \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}({\mathfrak {x}}; \nu )[\cdot ], \\&\partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}({\mathfrak {x}}; \nu )[ \cdot ] = \, {\Pi }_\bot \sum _{k = 0}^{M + 1} a_{1 -k}({\mathfrak {x}} ; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) \, \partial _x^{1 - k} [\cdot ]\\&\quad + {{{\mathcal {R}}}}_M ({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv})[\cdot ] , \end{aligned} \end{aligned}$$

(3.48)

with the following properties:

• 1. For any $s \geqq 0$, the maps

  $$\begin{aligned} \begin{aligned}&( {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{\mathcal {E}}}_{s + \sigma _M} ) \times \Xi \rightarrow H^s ({\mathbb {T}}_1), \,\, ({\mathfrak {x}} , \nu ) \mapsto a_{1 -k}({\mathfrak {x}} ; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) ,\\&\quad \quad 0 \le k \le M + 1 , \end{aligned} \end{aligned}$$

  are ${{{\mathcal {C}}}}^\infty$ and satisfy for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l \in E_{s + \sigma _M}$, and $({\mathfrak {x}}, \nu ) \in ({{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{\mathcal {E}}}_{s + \sigma _M}) \times \Xi$,

  $$\begin{aligned}&\Vert \partial _\nu ^\alpha a_{1 - k}({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv})\Vert _{H^s_x} \lesssim _{s, k, \alpha } \Vert y \Vert + \Vert w \Vert _{H_x^{s + \sigma _M}}, \quad \nonumber \\&\Vert d^l \partial _\nu ^\alpha a_{1 - k}({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv} )[\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^s_x} \lesssim _{s, k, l, \alpha } \sum _{j = 1}^l \big ( \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_{s + \sigma _M}} \prod _{n \ne j} \Vert \widehat{{\mathfrak {x}}}_n\Vert _{E_{\sigma _M}}\big )\nonumber \\&+ ( \Vert y \Vert + \Vert w \Vert _{H_x^{s + \sigma _M}}) \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_{\sigma _M}}. \end{aligned}$$

  (3.49)

• 2. For any $0 \leqq s \leqq M+1$, the map

  $$\begin{aligned} {{{\mathcal {R}}}}_M(\cdot ; \cdot ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) : {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi \rightarrow {\mathcal {B}} (H_\bot ^{-s }({\mathbb {T}}_1) ,H_\bot ^{M + 1 - s}({\mathbb {T}}_1)) \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{ \sigma _M}$, $(\mathfrak {x}, \nu ) \in {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \times \Xi$, and ${\widehat{w}} \in H^{-s}_\bot ({\mathbb {T}}_1)$,

  $$\begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M ({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla {{{\mathcal {R}}}}^{kdv})[{\widehat{w}}]\Vert _{H^{M + 1 -s}_x} \lesssim _{s, M, \alpha } ( \Vert y \Vert + \Vert w \Vert _{H^{ \sigma _M}_x}) \Vert {\widehat{w}}\Vert _{H^{-s}_x} , \end{aligned}$$

  (3.50)
  $$\begin{aligned}&\Vert d^l \partial _\nu ^\alpha {{{\mathcal {R}}}}_M ({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla {{{\mathcal {R}}}}^{kdv}) [{\widehat{w}}][\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^{ M + 1 - s}_x} \lesssim _{s, M, l,\alpha } \Vert {\widehat{w}} \Vert _{H^{-s}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_{\sigma _M}} . \end{aligned}$$

  (3.51)

• 3. For any $s \geqq 0$, the map

  $$\begin{aligned} \begin{aligned}&{{{\mathcal {R}}}}_M( \cdot ; \cdot ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) : ( {{{\mathcal {V}}}}^{\sigma _M}(\delta ) \cap {{\mathcal {E}}}_{s + \sigma _M} ) \times \Xi \rightarrow {\mathcal {B}} (H_\bot ^{s }({\mathbb {T}}_1) ,H_\bot ^{s + M + 1}({\mathbb {T}}_1)) , \end{aligned} \end{aligned}$$

  is ${{{\mathcal {C}}}}^\infty$ and satisfies for any $\alpha \in {\mathbb {N}}^{{\mathbb {S}}_+}$, $\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l \in E_{s + \sigma _M}$, $(\mathfrak {x}, \nu ) \in ({{{\mathcal {E}}}}_{s + \sigma _M} \cap {{{\mathcal {V}}}}^{\sigma _M}(\delta )) \times \Xi$, and ${\widehat{w}} \in H^s_\bot ({\mathbb {T}}_1)$,

  $$\begin{aligned}&\Vert \partial _\nu ^\alpha {{{\mathcal {R}}}}_M ({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla {{{\mathcal {R}}}}^{kdv})[{\widehat{w}}]\Vert _{H^{s + M + 1}_x} \nonumber \\&\lesssim _{s, M, \alpha } ( \Vert y \Vert + \Vert w \Vert _{H^{s + \sigma _M}_x}) \Vert {\widehat{w}}\Vert _{L^2_x} + ( \Vert y \Vert + \Vert w\Vert _{H^{\sigma _M}_x}) \Vert {\widehat{w}} \Vert _{H^{s}_x} , \end{aligned}$$

  (3.52)
  $$\begin{aligned}&\Vert d^l \partial _\nu ^\alpha {{{\mathcal {R}}}}_M ({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla {{{\mathcal {R}}}}^{kdv}) [{\widehat{w}}][\widehat{{\mathfrak {x}}}_1, \ldots , \widehat{{\mathfrak {x}}}_l]\Vert _{H^{s + M + 1}_x} \lesssim _{s, M, l,\alpha } \Vert {\widehat{w}} \Vert _{H^s_x} \prod _{j=1}^l \Vert \widehat{{\mathfrak {x}}}_j \Vert _{E_{\sigma _M}} \nonumber \\&\quad + \Vert {\widehat{w}} \Vert _{L^2_x} \sum _{j = 1}^l \Big ( \Vert \widehat{{\mathfrak {x}}}_j\Vert _{E_{s + \sigma _M}} \prod _{n \ne j} \Vert \widehat{{\mathfrak {x}}}_n\Vert _{E_{\sigma _M}} \Big ) + \Vert {\widehat{w}} \Vert _{L^2_x} \Vert w \Vert _{H^{s+ \sigma _M}_x} \prod _{j = 1}^l \Vert \widehat{{\mathfrak {x}}}_j \Vert _{E_{\sigma _M}} . \nonumber \\ \end{aligned}$$

  (3.53)

Proof

Differentiating ${{\mathcal {H}}}^{kdv}(\mathfrak {x}; \nu ) = H^{kdv}( \Psi _\nu (\mathfrak {x}))$, we get

$$\begin{aligned} \nabla _w {{\mathcal {H}}}^{kdv} ( \mathfrak {x}; \nu ) = (d_\bot \Psi _\nu ( \mathfrak {x}))^\top \big [ \nabla H^{kdv} (\Psi _\nu (\mathfrak {x} ) ) \big ] \end{aligned}$$

(3.54)

where by (1.2),

$$\begin{aligned} \nabla H^{kdv} ( u ) = \Pi _0^\bot ( 3 u^2 - u_{xx} \big ) \end{aligned}$$

(3.55)

and where $\Pi _0^\bot$ is the $L^2$-orthogonal projector onto $L^2_0({\mathbb {T}}_1)$. Differentiating (3.54) with respect to w in direction ${\widehat{w}}$ we get

$$\begin{aligned} d_\bot \nabla _w {{\mathcal {H}}}^{kdv} ( \mathfrak {x}; \nu ) [{\widehat{w}} ]&= (d_\bot \Psi _\nu ( \mathfrak {x}))^\top \big [\hbox {d}\nabla H^{kdv} (\Psi _\nu (\mathfrak {x} ) ) [d_\bot \Psi _\nu (\mathfrak {x})[{\widehat{w}} ]]\big ] \nonumber \\&\quad + \big ( d_\bot (d_\bot \Psi _\nu ( \mathfrak {x}))^\top [{\widehat{w}} ]\big ) \big [ \nabla H^{kdv} (\Psi _\nu (\mathfrak {x} ) ) \big ] . \end{aligned}$$

(3.56)

On the other hand, by (3.12)

$$\begin{aligned} d_\bot \nabla _w {{\mathcal {H}}}^{kdv} ( \mathfrak {x} ; \nu ) = \Omega ^{kdv} (D ; \nu ) + d_\bot \nabla _w {{\mathcal {R}}}^{kdv} ( \mathfrak {x} ; \nu ) \end{aligned}$$

and by (3.15) $d_\bot \nabla _w {{\mathcal {R}}}^{kdv} ( \theta , 0, 0 ; \nu ) = 0$, implying that

$$\begin{aligned} \begin{aligned}&d_\bot \nabla _w {{\mathcal {H}}}^{kdv} ( \theta , 0, 0; \nu ) = \Omega ^{kdv}(D; \nu ), \\&d_\bot \nabla _w {{\mathcal {R}}}^{kdv} ( \mathfrak {x}; \nu ) = d_\bot \nabla _w {{\mathcal {H}}}^{kdv} ( \theta , y, w; \nu ) - d_\bot \nabla _w {{\mathcal {H}}}^{kdv} ( \theta , 0, 0; \nu ) . \end{aligned} \end{aligned}$$

(3.57)

In order to obtain the expansion (3.48) it thus suffices to expand $d_\bot \nabla _w {{\mathcal {H}}}^{kdv} ( \theta , y, w; \nu ) ) [{\widehat{w}} ]$ and then subtract from it the expansion of $d_\bot \nabla _w {{\mathcal {H}}}^{kdv} ( \theta , 0, 0; \nu ) ) [{\widehat{w}} ]$. We analyze separately the two terms in (3.56).

Analysis of the first term on the right hand side of (3.56): Evaluating the differential $d \nabla H^{kdv}(u)$ at $u = \Psi _\nu ({\mathfrak {x}})$, one gets

$$\begin{aligned} \begin{aligned}&\hbox {d}(\nabla H^{kdv}) (\Psi _\nu ({\mathfrak {x}}))[h] = \Pi _0^\bot \big (- \partial _x^2 h + b_0(\mathfrak {x}; \nu ) h \big ) , \quad b_0({\mathfrak {x}}; \nu ) \,{:=}\, 6 \Psi _\nu (\mathfrak {x}) . \end{aligned} \end{aligned}$$

(3.58)

By Theorem 3.2-(AE1) and the estimates (Est1), the function $b_0({\mathfrak {x}}; \nu )$ satisfies, for any $s \geqq 0$,

$$\begin{aligned}&\Vert \partial _\nu ^\alpha b_0 ({\mathfrak {x}}; \nu )\Vert _{H^s_x} \lesssim _{s, \alpha } 1 + \Vert w \Vert _{H^{s + 1}_x}, \\&\quad \Vert \partial _\nu ^\alpha d^l b_0(\mathfrak {x}; \nu )[\widehat{\mathfrak {x}}_1, \ldots , \widehat{\mathfrak {x}}_l] \Vert _{H^s_x} \lesssim _{s, l, \alpha } \sum _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j\Vert _{E_{s + 1}} \prod _{i \ne j} \Vert \widehat{\mathfrak {x}}_i\Vert _{E_{1}} + \Vert w \Vert _{H^{s + 1}_x} \prod _{j = 1}^l \Vert \widehat{\mathfrak {x}}_j \Vert _{E_1}. \nonumber \end{aligned}$$

(3.59)

By Corollary 3.3 (expansion of $(d_\bot \Psi _\nu )^\top$), Corollary 3.4 (expansion of $d_\bot \Psi _\nu$), (3.59) (estimates of $b_0$), (3.58) $\big ($formula for $d (\nabla H^{kdv}) (\Psi _\nu ({\mathfrak {x}}))\big )$, and Lemma 2.12 (composition), one obtains the expansion

$$\begin{aligned}&\partial _x (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top \big [ d \nabla H^{kdv} (\Psi _\nu ({\mathfrak {x}} ) ) \big [ d_\bot \Psi _\nu ({\mathfrak {x}}) [ \cdot ] \big ] \big ] \nonumber \\&\quad = \Pi _\bot \big ( - \partial _{x}^3 - (a_{- 1}^\Psi (\mathfrak {x}; \nu ) + a_{- 1}^{d \Psi ^\top }(\mathfrak {x}; \nu )) \partial _x^2 + \sum _{k = 0}^{M + 1} a_{1 - k}^{(1)}({\mathfrak {x}}; \nu ) \partial _x^{1 - k} \, \big ) + R_1({\mathfrak {x}}; \nu ) \nonumber \\&{\mathop {=}\limits ^{(3.11)}} \Pi _\bot \big ( - \partial _{x}^3 + \sum _{k = 0}^{M + 1} a_{1 - k}^{(1)}({\mathfrak {x}}; \nu ) \partial _x^{1 - k} \big ) + R_1({\mathfrak {x}}; \nu ), \end{aligned}$$

(3.60)

where the functions $a_{1 - k}^{(1)}(\mathfrak {x}; \nu )$, $k = 0, \ldots , M+ 1$ and the remainder $R_1({\mathfrak {x}}; \nu )$ satisfy the properties 1–3 stated in Lemma 3.5, in particular (3.37)–(3.39).

Analysis of the second term on the right hand side of (3.56): By (3.46) one has

$$\begin{aligned} \partial _x \big ( d_\bot (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top [\cdot ]\big ) \big [ \nabla H^{kdv} (\Psi _\nu ({\mathfrak {x}} ) ) \big ]&= - \partial _x\hbox {d}\Psi _\nu (\mathfrak {x})^\top \partial _x^{- 1}\hbox {d}\big (d_\bot \Psi _\nu (\mathfrak {x})[\cdot ] \big )\\&\quad \big [ {{{\mathcal {J}}}} d \Psi _\nu (\mathfrak {x})^\top \nabla H^{kdv} (\Psi _\nu (\mathfrak {x} ) ) \big ]. \end{aligned}$$

Arguing as for the first term on the right hand side of (3.56) one obtains an expansion of the form

$$\begin{aligned} \partial _x \big ( d_\bot (d_\bot \Psi _\nu ( {\mathfrak {x}}))^\top [\cdot ]\big ) \big [ \nabla H^{kdv} (\Psi _\nu ({\mathfrak {x}} ) ) \big ] = \Pi _\bot \sum _{k = 0}^{M + 1} a_{1 - k}^{(2)}({\mathfrak {x}}; \nu ) \partial _x^{1 - k} + R_2({\mathfrak {x}}; \nu ),\nonumber \\ \end{aligned}$$

(3.61)

where $a_{1}^{(2)}(\mathfrak {x}; \nu ) = 0$ (cf. (3.16)) and where the functions $a_{1 - k}^{(2)}(\mathfrak {x}; \nu )$, $k = 1, \ldots , M+ 1$ and the remainder $R_2({\mathfrak {x}}; \nu )$ satisfy the properties 1-3 of Lemma 3.5, in particular (3.37)–(3.39).

Conclusion: Combining (3.56), (3.57), (3.60), and (3.61) one obtains the claimed expansion (3.48) with

$$\begin{aligned} \begin{aligned}&a_{1 - k}({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) \,{:=}\, a_{1 - k}^{(1)}(\mathfrak {x}; \nu ) - a_{1 - k}^{(1)}(\theta , 0, 0; \nu ) + a_{1 - k}^{(2)}(\mathfrak {x}; \nu ) - a_{1 - k}^{(2)}(\theta , 0, 0; \nu ) \\&{{{\mathcal {R}}}}_M ({\mathfrak {x}}; \nu ; \partial _x d_\bot \nabla {{{\mathcal {R}}}}^{kdv}) \,{:=}\, R_1(\mathfrak {x} ; \nu ) - R_1(\theta , 0, 0; \nu ) + R_2(\mathfrak {x}; \nu ) - R_2(\theta , 0, 0; \nu ) . \end{aligned} \end{aligned}$$

Since $a_{1 - k}^{(1)}(\mathfrak {x}; \nu )$, $R_1({\mathfrak {x}}; \nu )$, and $a_{1 - k}^{(2)}(\mathfrak {x}; \nu )$, $R_2 ({\mathfrak {x}}; \nu )$ satisfy properties 1-3 of Lemma 3.5, in particular (3.37)–(3.39), the claimed estimates (3.49)–(3.53) then follow by the mean value theorem. $\quad \square$

3.4 Frequencies of KdV

In this section we record properties of the KdV frequencies $\omega ^{kdv}_n$, used in particular for the measure estimates in Section 8.2, and discuss an expansion of $\partial _x \Omega ^{kdv}( D; \nu )$ (cf. (3.13)) needed in Section 6.

Recall that the family of operators $\Omega ^{kdv}( D; \nu )$, introduced in (3.13), is defined for $\nu \in \Xi \subset {\mathbb {R}}^{{\mathbb {S}}_+}_{> 0}$. Actually, it is defined on all of ${\mathbb {R}}^{{\mathbb {S}}_+}_{> 0}$ (cf. (3.2)) and according to [20, Lemma 4.1], $\partial _x \Omega ^{kdv}(D; I)$ can be written as

$$\begin{aligned} \partial _x \Omega ^{kdv}(D; I) = - \partial _x^3 + Q_{-1}^{kdv} (D; I ), \end{aligned}$$

(3.62)

where $Q_{-1}^{kdv} (D; I)$ is a family of Fourier multiplier operators of order $- 1$ with an expansion in homogeneous components up to any order.

Lemma 3.8

For any $M \in {\mathbb {N}}$ and $I \in {\mathbb {R}}_{>0}^{{{\mathbb {S}}}_+}$, $\, Q_{-1}^{kdv} (D; I )$ admits an expansion of the form

$$\begin{aligned}&Q_{-1}^{kdv} (D; I ) = \Omega _{-1}^{kdv} (D; I) + {{{\mathcal {R}}}}_M (D; I; Q_{-1}^{kdv}),\nonumber \\&\quad \quad \Omega _{-1}^{kdv} (\xi ; I) = \sum _{k= 1}^{ M} a_{-k} (I; \Omega _{-1}^{kdv}) \chi _0(\xi ) ( \mathrm{i} 2 \pi \xi )^{-k} , \end{aligned}$$

(3.63)

where the functions $a_{-k} (I; \Omega _{-1}^{kdv})$ are real analytic and bounded on compact subsets of ${\mathbb {R}}_{>0}^{{{\mathbb {S}}}_+}$ and vanish identially for k even, and where ${{{\mathcal {R}}}}_M (D; I ; Q_{-1}^{kdv})$ is a Fourier multiplier operator with multipliers

$$\begin{aligned} {{{\mathcal {R}}}}_M (n; I; Q_{-1}^{kdv}) = \frac{{{{\mathcal {R}}}}_{M}^{\omega _n}(I)}{(2\pi n)^{M+1}}, \quad {{{\mathcal {R}}}}_M (- n; I; Q_{-1}^{kdv}) = - {{{\mathcal {R}}}}_M (n; I; Q_{-1}^{kdv}), \quad \forall n \in {{\mathbb {S}}}_+^\bot ,\nonumber \\ \end{aligned}$$

(3.64)

where the functions $I \mapsto {{{\mathcal {R}}}}_{M}^{\omega _n}(I)$ are real analytic and satisfy, for any $j \in {{\mathbb {S}}}_+,$ $\beta \in {\mathbb {N}}$,

$$\begin{aligned} \sup _{n \in {{\mathbb {S}}}^\bot } |{{{\mathcal {R}}}}_{M}^{\omega _n}(I)| \leqq C_M , \quad \, \sup _{n \in {{\mathbb {S}}}^\bot } | \partial _{I_j}^{\beta } {{{\mathcal {R}}}}_{ M}^{\omega _n}(I)| \leqq C_{M,\beta } , \end{aligned}$$

(3.65)

uniformly on compact subsets of ${\mathbb {R}}_{>0}^{{{\mathbb {S}}}_+}$.

Proof

The result follows from [20, Lemma C.7]. $\quad \square$

Lemma 3.9

(Non-degeneracy of KdV frequencies, [21, Proposition 15.5]) For any finite subset ${{\mathbb {S}}}_+\subset {{\mathbb {N}}}$ the following holds on ${\mathbb {R}}^{{{\mathbb {S}}}_+}_{>0}$:

• (i) The map $I \mapsto \mathrm{det}\big ( (\partial _{I_k} \omega _n^{kdv}(I, 0))_{k,n \in {{\mathbb {S}}}_+} \big )$ is real analytic and does not vanish identically.

• (ii) For any $\ell \in {{\mathbb {Z}}}^{{{\mathbb {S}}}_+} and \ j, k \in {{\mathbb {S}}}^\bot$ with $(\ell , j,k) \ne (0,j,j)$, the following functions are real analytic and do not vanish identically:

  $$\begin{aligned} \sum _{n\in {{{\mathbb {S}}}_+}} \ell _n \omega ^{kdv}_n + \omega ^{kdv}_j \not = 0, \quad \sum _{n \in {{{\mathbb {S}}}_+}} \ell _n \omega ^{kdv}_n + \omega ^{kdv}_j - \omega ^{kdv}_k \not = 0. \end{aligned}$$

  (3.66)

Remark 3.10

It was shown in [12] that for any $I \in {\mathbb {R}}^{{{\mathbb {S}}}_+}_{>0}$, $\mathrm{det} \big ( (\partial _{I_k} \omega _n^{kdv}(I, 0)) _{k,n \in {{\mathbb {S}}}_+} \big ) \ne 0.$

Finally, we record the following asymptotics of the KdV frequencies, used in Section 7,

$$\begin{aligned} \omega ^{kdv}_n (I,0) - (2 \pi n)^3 = O( n^{-1} ) , \quad n \, \partial _{I} \omega ^{kdv}_n (I,0) = O( 1 ) , \end{aligned}$$

(3.67)

uniformly on compact sets of actions $I \in {\mathbb {R}}_{>0}^{{{\mathbb {S}}}_+}$ (cf. for example [22, Proposition 8.1]).

4 Nash–Moser Theorem

The purpose of this short section is to state Theorem 4.1 which reformulates Theorem 1.1 in the canonical coordinates described in the previous section. In Section 8.3, we derive Theorem 1.1 from Theorem 4.1.

In the canonical coordinates $\mathfrak {x} = (\theta , y, w) \in {{{\mathcal {V}}}}(\delta ) \cap {{\mathcal {E}}}_s$, defined by Theorem 3.2, with symplectic 2-form given by (3.6), the Hamiltonian equation (1.6) reads as

$$\begin{aligned} \partial _t \theta = - \nabla _y {{\mathcal {H}}}_\varepsilon , \quad \partial _t y = \nabla _{\theta } {{\mathcal {H}}}_\varepsilon , \quad \partial _t w = \partial _x \nabla _w {{\mathcal {H}}}_\varepsilon , \end{aligned}$$

(4.1)

where ${{\mathcal {H}}}_\varepsilon \,{:=}\, H_\varepsilon \circ \Psi _\nu$ and $H_\varepsilon$ is given by (1.7). More explicitly,

$$\begin{aligned}&{{\mathcal {H}}}_\varepsilon (\mathfrak {x}; \nu ) = {{\mathcal {H}}}^{kdv} (\mathfrak {x}; \nu ) + \varepsilon {{\mathcal {P}}}(\mathfrak {x}; \nu ) ,\nonumber \\&{{\mathcal {H}}}^{kdv} = H^{kdv} \circ \Psi _\nu , \quad {{\mathcal {P}}} = P \circ \Psi _\nu , \quad \nu \in \Xi , \end{aligned}$$

(4.2)

where ${{\mathcal {H}}}^{kdv} (\mathfrak {x}; \nu )$ has the normal form expansion (3.12). We denote by $X_{ {{\mathcal {H}}}_\varepsilon }$ the Hamiltonian vector field associated to ${{{\mathcal {H}}}}_\varepsilon$. For $\varepsilon = 0$, the Hamiltonian system (4.1) possesses, for any value of the parameter $\nu \in \Xi$, the invariant torus ${\mathbb {T}}^{{{\mathbb {S}}}_+} \times \{ 0 \} \times \{0 \}$, filled by quasi-periodic finite gap solutions of the KdV equation with frequency vector $\omega ^{kdv} (\nu ) \,{:=}\, (\omega ^{kdv}_n (\nu ,0))_{n \in {{\mathbb {S}}}_+}$ introduced in (1.11).

By our choice of $\Xi ,$ the map $- \omega ^{kdv} : \Xi \rightarrow \Omega \,{:=}\, - \omega ^{kdv} (\Xi )$ is a real analytic diffeomorphism. In the sequel, we consider $\nu$ as a function of the parameter $\omega \in \Omega$, namely

$$\begin{aligned} \nu \equiv \nu (\omega ) : = (\omega ^{kdv})^{-1} ( - \omega )\ . \end{aligned}$$

(4.3)

To keep the notation simpler, we often will not record the dependence of the Hamiltonian ${{{\mathcal {H}}}}_\varepsilon$ on $\nu = (\omega ^{kdv})^{-1} ( - \omega )$. Consider the set of diophantine frequencies in $\Omega$,

$$\begin{aligned} \mathtt {DC}(\gamma , \tau ) \,{:=}\, \Big \{ \omega \in \Omega : |\omega \cdot \ell | \geqq \frac{\gamma }{\langle \ell \rangle ^\tau } , \quad \forall \ell \in {\mathbb {Z}}^{{\mathbb {S}}_+} \setminus \{ 0 \} \Big \}. \end{aligned}$$

(4.4)

For any torus embedding ${\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow {{{\mathcal {V}}}}(\delta ) \cap {{\mathcal {E}}}_s$, $\varphi \mapsto ( \theta (\varphi ), y(\varphi ), w(\varphi ) )$, close to the identity, consider its lift

$$\begin{aligned} \breve{\iota } : {\mathbb {R}}^{{{\mathbb {S}}}_+} \rightarrow {\mathbb {R}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times H^s_\bot ({\mathbb {T}}_1) , \quad \breve{\iota }(\varphi ) = (\varphi , 0, 0) + \iota (\varphi ) , \end{aligned}$$

(4.5)

where $\iota (\varphi ) = ( {\Theta } (\varphi ), y(\varphi ), w(\varphi ))$ and where $\Theta (\varphi ) \,{:=}\, \theta (\varphi ) - \varphi$ is $(2 \pi {\mathbb {Z}})^{{{\mathbb {S}}}_+}$ periodic. Often we will refer to the torus embedding $\breve{\iota }$ simply as torus. We look for a torus embedding $\breve{\iota }$ such that ${\mathcal {F}}_\omega ( \iota , \zeta ) = 0$ where

$$\begin{aligned} {\mathcal {F}}_\omega ( \iota , \zeta ) \,{:=}\, \left( \begin{aligned}&\omega \cdot \partial _{\varphi } \theta (\varphi ) + (\nabla _y {{\mathcal {H}}}_\varepsilon ) (\breve{\iota } (\varphi ) ) \\&\omega \cdot \partial _{\varphi } y (\varphi ) - (\nabla _\theta {{\mathcal {H}}}_\varepsilon ) (\breve{\iota } (\varphi ) ) - \zeta \\&\omega \cdot \partial _{\varphi } w (\varphi ) - \partial _x (\nabla _w {{\mathcal {H}}}_\varepsilon ) (\breve{\iota } (\varphi ) ) \\ \end{aligned} \right) . \end{aligned}$$

(4.6)

The additional variable $\zeta \in {\mathbb {R}}^{{{\mathbb {S}}}_+}$ is introduced in order to control the average of the y-component of the linearized Hamiltonian equations – see Section 5, in particular (5.35). Actually any invariant torus for $X_{{{{\mathcal {H}}}}_{\varepsilon , \zeta }} = X_{{{{\mathcal {H}}}}_\varepsilon } + (0, \zeta , 0 )$ with modified Hamiltonian

$$\begin{aligned} {{{\mathcal {H}}}}_{\varepsilon , \zeta } (\theta , y, w) \,{:=}\, {{{\mathcal {H}}}}_\varepsilon (\theta , y, w) + \zeta \cdot \theta ,\quad \zeta \in {\mathbb {R}}^{{{\mathbb {S}}}_+} \, , \end{aligned}$$

(4.7)

is invariant for $X_{{{{\mathcal {H}}}}_\varepsilon }$ due to (5.5). Note that ${{{\mathcal {H}}}}_{\varepsilon , \zeta }$ is not periodic in $\theta$, but that its Hamiltonian vector field is. The Lipschitz Sobolev norm of the periodic part $\iota (\varphi ) = (\Theta (\varphi ), y(\varphi ), w(\varphi ))$ of the embedded torus (4.5) is defined by

$$\begin{aligned} \Vert \iota \Vert _s^\mathrm{{Lip}(\gamma )}\,{:=}\, \Vert \Theta \Vert _s^\mathrm{{Lip}(\gamma )}+ \Vert y \Vert _s^\mathrm{{Lip}(\gamma )}+ \Vert w \Vert _s^\mathrm{{Lip}(\gamma )}\end{aligned}$$

(4.8)

where $\Vert w \Vert _s^\mathrm{{Lip}(\gamma )}$ is the Lipschitz Sobolev norm introduced in (2.1) and

$$\begin{aligned} \Vert \Theta \Vert _s^\mathrm{{Lip}(\gamma )}\equiv \Vert \Theta \Vert _{H^s_\varphi }^\mathrm{{Lip}(\gamma )}\,{:=}\, \Vert \Theta \Vert _{H^s ({\mathbb {T}}^{{{\mathbb {S}}}_+}, {\mathbb {R}}^{{{\mathbb {S}}}_+})}^\mathrm{{Lip}(\gamma )}, \quad \Vert y \Vert _s^\mathrm{{Lip}(\gamma )}\equiv \Vert y \Vert _{H^s_\varphi }^\mathrm{{Lip}(\gamma )}\,{:=}\, \Vert y \Vert _{H^s ({\mathbb {T}}^{{{\mathbb {S}}}_+}, {\mathbb {R}}^{{{\mathbb {S}}}_+})}^\mathrm{{Lip}(\gamma )}.\nonumber \\ \end{aligned}$$

(4.9)

Theorem 4.1

There exist ${\bar{s}} > (|{{\mathbb {S}}}_+| + 1) / 2$ and $\varepsilon _0 > 0$ so that for any $0 <\varepsilon \leqq \varepsilon _0$, there is a measurable subset $\Omega _\varepsilon \subseteq \Omega$ satisfying

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \, \frac{\mathrm{meas}( \Omega _\varepsilon )}{\mathrm{meas}( \Omega )} = 1 \end{aligned}$$

(4.10)

and for any $\omega \in \Omega _\varepsilon$, there exists a torus embedding with lift $\breve{\iota }_\omega : {\mathbb {R}}^{{{\mathbb {S}}}_+} \rightarrow {\mathbb {R}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times H^{{\bar{s}}}_\bot ({\mathbb {T}}_1)$ (cf. (4.5)) which satisfies the estimate

$$\begin{aligned} \Vert \breve{\iota }_\omega - (\varphi , 0,0) \Vert _{{\bar{s}}}^\mathrm{{Lip}(\gamma )}= O( \varepsilon \gamma ^{-2}), \quad \gamma = \varepsilon ^{{\mathfrak {a}}} , \ 0 < {{\mathfrak {a}}} \ll 1 , \end{aligned}$$

(4.11)

and solves

$$\begin{aligned} \omega \cdot \partial _\varphi \breve{\iota }_\omega (\varphi ) - X_{{{{\mathcal {H}}}}_\varepsilon (\cdot ; \nu )}( \breve{\iota }_\omega (\varphi )) = 0 , \quad \quad \nu = (\omega ^{kdv})^{-1}( - \omega ) . \end{aligned}$$

(4.12)

As a consequence, the embedded torus $\breve{\iota }_\omega ({\mathbb {T}}^{{{\mathbb {S}}}_+})$ is invariant under the flow of the Hamiltonian vector field $X_{{{\mathcal {H}}}_\varepsilon (\cdot ; \nu )}$ and is filled by quasi-periodic solutions of (4.1) with frequency vector $\omega \in \Omega _\varepsilon$. Furthermore, the quasi-periodic solution $\breve{\iota }_\omega (\omega t) = (\omega t, 0, 0) + \iota _\omega (\omega t)$ is linearly stable.

Remark 4.2

Up to the end of Section 7, $\gamma \in (0, 1)$ is assumed to be a constant independent of $\varepsilon$ with $\varepsilon \gamma ^{-3} \ll 1$. Only in Section 8, $\gamma$ and $\varepsilon$ are required to be related by $\gamma = \varepsilon ^{{\mathfrak {a}}}$ for some $0 < {{\mathfrak {a}}} \ll 1$.

Theorem 4.1 is proved in Section 8 and is applied to deduce Theorem 1.1 (cf. Section 8.3 for details). At the core of the proof of Theorem 4.1 is the construction of an approximate right inverse of the linearized operator $d_{\iota , \zeta } {\mathcal {F}}_{\omega } ( \iota , \zeta )$ at an approximate solution. This construction is carried out in Sections 5–7.

Along the proof we shall use the following tame estimates of the Hamiltonian vector field $X_{{{{\mathcal {H}}}}_\varepsilon }$ with respect to the norm $\Vert \cdot \Vert _s^\mathrm{{Lip}(\gamma )}$ in (4.8). Using the expansion (3.12) provided in Theorem 3.2, and the definition of ${{{\mathcal {P}}}}$ in (3.35), we decompose the Hamiltonian ${{{\mathcal {H}}}}_\varepsilon$, defined in (4.2), as

$$\begin{aligned} {{{\mathcal {H}}}}_\varepsilon= & {} {{{\mathcal {N}}}} + {{{\mathcal {P}}}}_\varepsilon \quad \mathrm{where}\nonumber \\ {{{\mathcal {N}}}}(y, w;\nu ):= & {} \omega ^{kdv}(\nu ) \cdot y + \frac{1}{2} \Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) [y] \cdot y + \frac{1}{2} \big ( \Omega ^{kdv}(D ;\nu )w,\,w \big )_{L^2_x},\nonumber \\ {{{\mathcal {P}}}}_\varepsilon:= & {} {{{\mathcal {R}}}}^{kdv} + \varepsilon {{{\mathcal {P}}}}. \end{aligned}$$

(4.13)

The Hamiltonian vector field of ${{{\mathcal {P}}}}_\varepsilon$ and ${{{\mathcal {H}}}}_\varepsilon$ satisfy the following tame estimates:

Lemma 4.3

There exists $\sigma _1 = \sigma _1({\mathbb {S}}_+) > 0$ so that for any $s \ge 0,$ any torus embedding $\breve{\iota }$ of the form (4.5) with $\Vert \iota \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}\leqq \delta$, and any maps ${\widehat{\iota }}, {\widehat{\iota }}_1, {\widehat{\iota }}_2 : {\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow E_s$, the following tame estimates hold:

$$\begin{aligned} \begin{aligned} \Vert X_{{{{\mathcal {P}}}}_\varepsilon } (\breve{\iota }) \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \varepsilon (1 + \Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}) + \Vert \iota \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}, \\ \Vert \hbox {d}X_{{{{\mathcal {P}}}}_\varepsilon }(\breve{\iota })[{\widehat{\iota }}] \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s (\varepsilon + \Vert \iota \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}) \Vert {\widehat{\iota }}\Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}, \\ \Vert d^2 X_{{{{\mathcal {H}}}}_\varepsilon }(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2] \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \Vert {\widehat{\iota }}_1\Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}_2\Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}+ \Vert {\widehat{\iota }}_1\Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}_2\Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}\\&\quad + \Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}(\Vert {\widehat{\iota }}_1\Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}_2\Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}. \end{aligned} \end{aligned}$$

Proof

By (4.13), one has $X_{{{{\mathcal {P}}}}_\varepsilon } = \varepsilon X_{{{\mathcal {P}}}} + X_{{{{\mathcal {R}}}}^{kdv}}$ and $d^2 X_{{{{\mathcal {H}}}}_\varepsilon } = d^2 X_{{\mathcal {N}}} + d^2 X_{{{{\mathcal {P}}}}_\varepsilon }$. The claimed estimate for $d X_{{{{\mathcal {P}}}}_\varepsilon }(\breve{\iota })[{\widehat{\iota }}]$ follows noting that, by Lemmata 3.5, 2.25, 2.26, $\Vert \hbox {d}X_{{{\mathcal {P}}}}(\breve{\iota })[{\widehat{\iota }}] \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert {\widehat{\iota }}\Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}$ and, by Lemmata 3.7, 2.25, 2.26,

$$\begin{aligned} \Vert \hbox {d}X_{{{{\mathcal {R}}}}^{kdv}}(\breve{\iota })[{\widehat{\iota }}] \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert \iota \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}\Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

The one for $\Vert X_{{{{\mathcal {P}}}}_\varepsilon } (\breve{\iota }) \Vert _s^\mathrm{{Lip}(\gamma )}$ is obtained by the mean value theorem. Indeed, one has $\Vert X_{{{{\mathcal {P}}}}} (\breve{\iota }) \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s 1 + ( \Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )})$, and taking into account (3.15), $\Vert X_{{{{\mathcal {R}}}}^{kdv}} (\breve{\iota }) \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert \iota \Vert _{s_0 + \sigma _1}^\mathrm{{Lip}(\gamma )}\Vert \iota \Vert _{s + \sigma _1}^\mathrm{{Lip}(\gamma )}$. Finally, the estimate for $d^2 X_{{{{\mathcal {H}}}}_\varepsilon }(\breve{\iota })[{\widehat{\iota }}_1, {\widehat{\iota }}_2]$ is verified using again Lemmata 3.5, 3.7, 2.25, 2.26. $\quad \square$

5 Approximate Inverse

In order to prove Theorem 4.1 we implement a Nash–Moser iteration scheme that leads to a solution of ${\mathcal {F}}_\omega (\iota , \zeta ) = 0$ (cf. (4.6)). For this purpose we construct an almost-approximate right inverse of the linearized operator

$$\begin{aligned} d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota , \zeta )[{\widehat{\iota }} , {\widehat{\zeta }} ] = \omega \cdot \partial _\varphi {\widehat{\iota }} - d_\iota X_{{{{\mathcal {H}}}}_{\varepsilon }} ( \breve{\iota } ) [{\widehat{\iota }} ] - (0, {\widehat{\zeta }}, 0 ), \end{aligned}$$

(5.1)

where ${{{\mathcal {H}}}}_\varepsilon = {{{\mathcal {N}}}} + {{{\mathcal {P}}}}_\varepsilon$ is the Hamiltonian in (4.13). Note that the perturbation ${{{\mathcal {P}}}}_\varepsilon$ and the differential $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota , \zeta )$ are independent of $\zeta$. Thus, in the sequel, we often write $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )$ instead of $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota , \zeta )$. The construction of an almost-approximate right inverse of $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )$ is the main result of this section, stated in Theorem 5.7. It is proved under the assumption A-I, introduced below (cf. (5.29)–(5.32)). In Theorem 7.11, these assumptions are stated as a theorem, and its proof is given in Section 7.2.

Throughout this section we assume that the following ansatz holds:

• Ansatz. The maps $\omega \mapsto \iota (\omega ) \,{:=}\, \breve{\iota }(\varphi ; \omega ) - (\varphi ,0,0)$, and $\omega \mapsto \zeta (\omega )$ are Lipschitz continuous with respect to $\omega \in \Omega$, and for $0< \gamma <1$

  $$\begin{aligned} \Vert \iota \Vert _{\mu _0}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2} , \quad \Vert Z\Vert _{s_0}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon , \end{aligned}$$

  (5.2)

  where Z, referred to as error function, is defined by

  $$\begin{aligned} Z(\varphi ) \,{:=}\, {{{\mathcal {F}}}}_\omega (\iota , \zeta ) (\varphi ) = \omega \cdot \partial _\varphi \breve{\iota } (\varphi ) - X_{{{{\mathcal {H}}}}_\varepsilon }(\breve{\iota } (\varphi )) - (0, \zeta , 0) . \end{aligned}$$

  (5.3)

We already mention that at each step of the Nash–Moser scheme of Theorem 8.1, the above ’Ansatz’ is shown to hold, with the constant $\mu _0$ specified in Theorem 5.7, depending on $|{\mathbb {S}}_+|$ and $\tau$ (given in Section 8).

Let us first give an outline of the proof of Theorem 5.7. Since the $\theta$-, y-, and w-components of the linear operator $d_\iota X_{{{{\mathcal {H}}}}_{\varepsilon }} ( \breve{\iota } )$ form a coupled system, it turns out to be difficult to invert the operator $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )$ in (5.1). To overcome this difficulty, we use the approach developed in [3, 8,9,10], consisting in transforming $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )$ into approximately triangular form, see (5.33). Let us describe in broad terms how to achieve this: If the error function Z, defined in (5.3), vanishes, then the torus $\breve{\iota }$ is invariant for the Hamiltonian ${{{\mathcal {H}}}}_{\varepsilon , \zeta }$ defined in (4.7). Furthermore, by (5.5) below, also $\zeta$ vanishes in this case, implying that $\breve{\iota }$ is invariant for the Hamiltonian ${{{\mathcal {H}}}}_\varepsilon$. Hence the invariant torus $\breve{\iota }$ is isotropic (cf. [8]) (where $\breve{\iota }$ being isotropic means that the pullback of the symplectic form by $\breve{\iota }$ vanishes) and there exist symplectic coordinates in a neighborhood of this torus (cf. (5.16)) so that when expressed in these coordinates, the linearized equations form a triangular system as described in (5.33). In general, the torus $\breve{\iota }$ is only “approximately” invariant up to order O(Z) and the linearized equations can only be approximately conjugated to a triangular system as in (5.33). Taking all this into account, we proceed as follows: given an approximately invariant torus $\breve{\iota }(\varphi ) = (\theta (\varphi ), y(\varphi ), w(\varphi ))$ satisfying (5.2), we first construct an isotropic torus $\breve{\iota }_\delta (\varphi ) = (\theta (\varphi ), y_\delta (\varphi ), w(\varphi ))$ which is close to $\breve{\iota }$ (cf. Lemma 5.4). Note that by (5.13), ${{{\mathcal {F}}}}(\breve{\iota }_\delta , \zeta )$ is also of the order O(Z) . Since $\breve{\iota }_\delta$ is isotropic, the diffeomorphism $(\phi , \eta , v) \mapsto (\theta , y, w) = G_\delta (\phi , \eta , v)$ defined in (5.16) is symplectic. In these coordinates, the torus $\breve{\iota }_\delta$ reads $\varphi \mapsto (\phi , \eta , v) = ( \varphi , 0, 0)$, and the transformed Hamiltonian ${{{\mathcal {K}}}} \,{:=}\, {{{\mathcal {H}}}}_{\varepsilon , \zeta } \circ G_\delta$ takes the form (5.18), where the terms $\partial _\phi {{{\mathcal {K}}}}_{00}$, ${{{\mathcal {K}}}}_{10} + \omega$, and ${{{\mathcal {K}}}}_{01}$ are of the order O(Z) (cf. Lemma 5.5). Neglecting terms of the order O(Z), the problem of finding an approximate right inverse of the operator $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )$ is reduced to the task of inverting the operator ${{\mathbb {D}}}$ in (5.33). The system (5.34) is solved in a triangular fashion as follows. First we solve the second equation in the system (5.34), cf. (5.35)–(5.36). Then we solve the third equation in (5.34) using the assumption A-I, cf. (5.37). Finally, to determine ${\widehat{\phi }}$, we solve the equation (5.38), cf. (5.41)–(5.42). In conclusion, we prove that the operator (5.44) is an almost-approximate right inverse of the operator $d_{\iota , \zeta }{{{\mathcal {F}}}}_\omega (\iota )$ which satisfies tame estimates – see Theorem 5.7 for details.

We start our construction by noting that the 2-form ${{{\mathcal {W}}}}$ given by (3.6) is exact:

$$\begin{aligned} {{{\mathcal {W}}}} = \Big ( {\mathop \sum }_{j \in {{\mathbb {S}}}_+}\hbox {d}y_j \wedge d \theta _j \Big ) \oplus {{{\mathcal {W}}}}_\bot =\hbox {d}\Lambda , \end{aligned}$$

where $\Lambda$ is the Liouville 1-form

$$\begin{aligned} \Lambda _{(\theta , y, w)}[{\widehat{\theta }}, {\widehat{y}}, {\widehat{w}}] \,{:=}\, {\mathop \sum }_{j \in {{\mathbb {S}}}_+} y_j {\widehat{\theta }}_j + \frac{1}{2} \big ( \partial _x^{- 1} w,\,{\widehat{w}} \big )_{L^2_x} . \end{aligned}$$

(5.4)

The pullbacks $\breve{\iota }^* \Lambda$ and $\breve{\iota }^* {{{\mathcal {W}}}}$ of $\Lambda$ and respectively ${\mathcal {W}}$ by a torus embedding $\breve{\iota }$ are related by $\breve{\iota }^* {{{\mathcal {W}}}} =\hbox {d}\, \breve{\iota }^* \Lambda$. Recall that the embedding $\breve{\iota }$ is said to be isotropic if $\breve{\iota }^* {{{\mathcal {W}}}} = 0$.

First, we provide an estimate of $\zeta$ in terms of the error function $Z(\varphi )$ defined in (5.3). We recall that by the ansatz (5.2), $Z(\varphi )$ and $\zeta$ are Lipschitz continuous with respect to the parameter $\omega \in \Omega$.

Lemma 5.1

One has

$$\begin{aligned} |\zeta |^\mathrm{{Lip}(\gamma )}\lesssim \Vert Z \Vert _{s_0}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(5.5)

Proof

We follow the arguments in [8, Lemma 3] and [3, Lemma 6.1]. Since the Hamiltonian ${{{\mathcal {H}}}}_\varepsilon$ is autonomous, the “restricted” action functional

$$\begin{aligned} G : {\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow {\mathbb {R}}, \quad \psi \mapsto {G}(\psi ) \,{:=}\, \int _{{\mathbb {T}}^{{{\mathbb {S}}}_+}} \Big ( - \Lambda _{\breve{\iota }^{(\psi )}}(\omega \cdot \partial _\varphi \breve{\iota }^{(\psi )}(\varphi )) - {{{\mathcal {H}}}}_{\varepsilon }(\breve{\iota }^{(\psi )}(\varphi ))\Big )\, \hbox {d} \varphi \end{aligned}$$

is constant, where $\breve{\iota }^{(\psi )} (\varphi ) \,{:=}\, \breve{\iota }(\psi + \varphi )$ and $\Lambda _{\breve{\iota }(\psi +\varphi )}$ is the canonical one form $\Lambda$ defined in (5.4), evaluated at $\breve{\iota }(\psi +\varphi )$. Using that $\partial _\psi { G}(0) = 0$, a direct calculation shows that $\zeta$ can be expressed in terms of $Z(\varphi ) = (Z_\theta (\varphi ), Z_y(\varphi ), Z_w(\varphi ))$ as

$$\begin{aligned} \zeta = \frac{1}{(2 \pi )^{|{{\mathbb {S}}}_+|}}\int _{{\mathbb {T}}^{{{\mathbb {S}}}_+}} \Big ( [\partial _\varphi y(\varphi )]^\top Z_\theta (\varphi )- [\partial _\varphi \theta (\varphi )]^\top Z_y(\varphi ) - [\partial _\varphi w(\varphi )]^\top \partial _x^{- 1} Z_w(\varphi ) \Big ) \, \hbox {d} \varphi . \end{aligned}$$

The latter formula together with the tame estimates (2.5) and the ansatz (5.2) imply (5.5). $\quad \square$

For an approximately invariant torus embedding $\breve{\iota } = (\theta (\varphi ), y(\varphi ), w (\varphi ) )$, the 1-form

$$\begin{aligned} \breve{\iota }^* \Lambda = {\mathop \sum }_{k \in {\mathbb {S}}_+} a_k (\varphi )\hbox {d}\varphi _k ,\quad a_k(\varphi ) \,{:=}\, [\partial _\varphi \theta (\varphi )]^\top y (\varphi ) \cdot {{\underline{e}}}_k + \frac{1}{2} ( \partial _x^{- 1} w (\varphi ), \, \partial _{\varphi _k} w(\varphi ) )_{L^2_x},\nonumber \\ \end{aligned}$$

(5.6)

is only “approximately closed”, in the sense that

$$\begin{aligned} \breve{\iota }^* {{{\mathcal {W}}}} =\hbox {d}\, \breve{\iota }^* \Lambda = {\mathop \sum }_{\begin{array}{c} k, j \in {\mathbb {S}}_+ \\ k < j \end{array}} A_{k j}(\varphi )\hbox {d}\varphi _k \wedge \hbox {d}\varphi _j,\quad A_{k j} (\varphi ) \,{:=}\, \partial _{\varphi _k} a_j(\varphi ) - \partial _{\varphi _j} a_k(\varphi ) ,\nonumber \\ \end{aligned}$$

(5.7)

is of the order O(Z). Here ${\underline{e}}_k$, $k \in {{\mathbb {S}}}_+$, denotes the standard basis of ${\mathbb {R}}^{{\mathbb {S}}_+}.$ More precisely, the following lemma holds:

Lemma 5.2

Let $\omega \in {\mathtt {DC}} (\gamma , \tau )$ (cf. (4.4)). Then for any $k, j \in {{\mathbb {S}}}_+$, the coefficient $A_{kj}$ in (5.7) satisfies, for some $\sigma = \sigma (\tau , {\mathbb {S}}_+) > 0$, $\forall \, s\ge s_0$,

$$\begin{aligned} \Vert A_{k j} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \gamma ^{-1} \big (\Vert Z \Vert _{s+ \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert Z \Vert _{s_0+ \sigma }^\mathrm{{Lip}(\gamma )}\Vert \iota \Vert _{s+\sigma }^\mathrm{{Lip}(\gamma )}\big ) . \end{aligned}$$

(5.8)

Remark 5.3

In the sequel the constant $\sigma = \sigma ( \tau , {\mathbb {S}}_+ )$, referred to as loss of derivatives, will be tacitly increased in the course of our arguments if needed.

Proof

For any $j, k \in {{\mathbb {S}}}_+$, the coefficient $A_{kj}$ satisfies the identity $\omega \cdot \partial _\varphi A_{k j} =$ ${{{\mathcal {W}}}}\big ( \partial _\varphi Z(\varphi ) {\underline{e}}_k , \partial _\varphi \breve{\iota }(\varphi ) {\underline{e}}_j \big ) +$ ${{{\mathcal {W}}}} \big (\partial _\varphi \breve{\iota } (\varphi ) {\underline{e}}_k , \partial _\varphi Z(\varphi ) {\underline{e}}_j \big )$ (cf. [8, Lemma 5]). The estimate (5.8) then follows by (5.2) and (2.10). $\quad \square$

As in [3, 8] we first modify the approximate torus $\breve{\iota }$ to obtain an isotropic torus $\breve{\iota }_\delta$ which is still approximately invariant. Let $\Delta _\varphi \,{:=}\, \sum _{k\in {\mathbb {S}}_+} \partial _{\varphi _k}^2$.

Lemma 5.4

(Isotropic torus) Let $\omega \in {\mathtt {DC}} (\gamma , \tau )$. The torus $\breve{\iota }_\delta (\varphi ) \,{:=} (\theta (\varphi ), y_\delta (\varphi ), w (\varphi ) )$ defined by

$$\begin{aligned} y_\delta (\varphi ) \,{:=}\, y(\varphi ) - [\partial _\varphi \theta (\varphi )]^{- \top } \rho (\varphi ) , \ \ \rho (\varphi ) = (\rho _j(\varphi ))_{j \in {\mathbb {S}}_+} , \ \ \rho _j(\varphi ) \,{:=}\, \Delta _\varphi ^{-1} {\mathop \sum }_{ k \in {\mathbb {S}}_+} \partial _{\varphi _k} A_{k j}(\varphi ) ,\nonumber \\ \end{aligned}$$

(5.9)

is isotropic and there exists $\sigma = \sigma (\tau , {\mathbb {S}}_+) > 0$ so that for any $s \geqq s_0$,

$$\begin{aligned} \Vert y_\delta - y \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

(5.10)

$$\begin{aligned} \Vert y_\delta - y \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \gamma ^{-1} \big (\Vert Z \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert Z \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\big ) , \end{aligned}$$

(5.11)

$$\begin{aligned} \Vert d_\iota \iota _\delta [ {\widehat{\iota }} ] \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \Vert {\widehat{\iota }} \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0}^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

(5.12)

$$\begin{aligned} \Vert {{{\mathcal {F}}}}_\omega (\iota _\delta , \zeta ) \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _s \gamma ^{- 1}\big ( \Vert Z \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert Z \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\big ) . \end{aligned}$$

(5.13)

Proof

The isotropy of the modified torus $\breve{\iota }_\delta$ is proved in [8]. By a standard Neumann series argument and using the ansatz (5.2), it follows that for any $s \geqq s_0$, there is a constant $C(s) > 0$ so that

$$\begin{aligned} \Vert [\partial _\varphi \theta ]^{- \top } - \mathrm{Id} \Vert _s^\mathrm{{Lip}(\gamma )}\leqq C(s) \Vert \iota \Vert _{s + 1}^\mathrm{{Lip}(\gamma )}, \quad \Vert [\partial _\varphi \theta ]^{- \top }\Vert _s^\mathrm{{Lip}(\gamma )}\leqq 1 + C(s) \Vert \iota \Vert _{s + 1}^\mathrm{{Lip}(\gamma )}.\nonumber \\ \end{aligned}$$

(5.14)

Furthermore, by the estimate (5.8) for the coefficients $A_{kj}$ and the definition (5.9) of $\rho$, one gets

$$\begin{aligned} \Vert \rho \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \gamma ^{-1} \big (\Vert Z \Vert _{s+ \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert Z \Vert _{s_0+ \sigma }^\mathrm{{Lip}(\gamma )}\Vert \iota \Vert _{s+\sigma }^\mathrm{{Lip}(\gamma )}\big ). \end{aligned}$$

(5.15)

The estimates (5.10), (5.11) then follow by using (5.14), (5.15), the interpolation estimate (2.5), the ansatz (5.2), $\varepsilon \gamma ^{- 2} \ll 1$, and the definition of $a_k$, $k \in {{\mathbb {S}}}_+$, in (5.6). The estimate (5.12) follows by similar arguments. To prove (5.13), it suffices to estimate ${{{\mathcal {F}}}}_\omega (\iota _\delta , \zeta ) - {{{\mathcal {F}}}}_\omega (\iota , \zeta )$. One computes

$$\begin{aligned} {{{\mathcal {F}}}}_\omega (\iota _\delta , \zeta ) - {{{\mathcal {F}}}}_\omega (\iota , \zeta ) = \begin{pmatrix} \Omega _{{\mathbb {S}}_+}(\nu )[y_\delta - y] \\ \omega \cdot \partial _\varphi (y_\delta - y_0 ) \\ 0 \end{pmatrix}\, + X_{{{{\mathcal {P}}}}_\varepsilon }(\iota _\delta ) - X_{{{{\mathcal {P}}}}_\varepsilon }(\iota ) . \end{aligned}$$

The estimate (5.13) then follows by using the mean value theorem to bound $X_{{{{\mathcal {P}}}}_\varepsilon }(\iota _\delta ) - X_{{{{\mathcal {P}}}}_\varepsilon }(\iota )$, together with Lemma 4.3 and by applying the estimate (5.11) on $y_\delta - y$ (using also the ansatz (5.2)). $\quad \square$

In order to find an approximate inverse of the linearized operator $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota _\delta )$, we introduce the symplectic diffeomorphism $G_\delta : (\phi , \eta , v) \mapsto (\theta , y , w)$ of the phase space ${\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {R}}^{{\mathbb {S}}_+} \times L^2_\bot ({\mathbb {T}}_1)$, defined by

$$\begin{aligned} \begin{pmatrix} \theta \\ y \\ w \end{pmatrix} \,{:=}\, G_\delta \begin{pmatrix} \phi \\ \eta \\ v \end{pmatrix} \,{:=}\, \begin{array}{l} \theta (\phi ) \\ y_\delta (\phi ) + [\partial _\phi \theta (\phi )]^{-\top } \eta - \big [ (\partial _\theta {\tilde{w}}) (\theta (\phi )) \big ]^\top (\partial _x^{- 1} v) \\ w(\phi ) + v \end{array}\qquad \qquad \end{aligned}$$

(5.16)

where ${\tilde{w}} \,{:=}\, w \circ \theta ^{-1}$. Since $\breve{\iota }_\delta$ is an isotropic torus embedding (cf. Lemma 5.4), $G_\delta$ is symplectic by [8, Lemma 2]. In the new coordinates, $\breve{\iota }_\delta$ is the trivial embedded torus $\varphi \mapsto (\phi , \eta , v ) = (\varphi , 0, 0 )$ and the Hamiltonian vector field $X_{{{{\mathcal {H}}}}_{\varepsilon , \zeta }}$ (with ${{{\mathcal {H}}}}_{\varepsilon , \zeta }$ defined in (4.7)) is given by

$$\begin{aligned} X_{{{{\mathcal {K}}}}} = (d G_\delta )^{-1} X_{{{{\mathcal {H}}}}_{\varepsilon , \zeta }} \circ G_\delta , \quad {{{\mathcal {K}}}} \equiv {{{\mathcal {K}}}}_{\varepsilon , \zeta } \,{:=}\, {{{\mathcal {H}}}}_{\varepsilon , \zeta } \circ G_\delta . \end{aligned}$$

(5.17)

The Taylor expansion of ${{{\mathcal {K}}}}$ in $\eta , v$ at the trivial torus $\varphi \mapsto (\phi , \eta , v ) = (\varphi , 0, 0)$ is of the form

$$\begin{aligned} {{{\mathcal {K}}}}(\phi , \eta , v, \zeta )&= \theta (\phi ) \cdot \zeta + {{{\mathcal {K}}}}_{00}(\phi ) + {{{\mathcal {K}}}}_{10}(\phi ) \cdot \eta + ({{{\mathcal {K}}}}_{0 1}(\phi ), v)_{L^2_x} + \frac{1}{2} {{{\mathcal {K}}}}_{2 0}(\phi ) \eta \cdot \eta \nonumber \\&\quad + \big ( {{{\mathcal {K}}}}_{11}(\phi ) \eta , v \big )_{L^2_x } + \frac{1}{2} \big ({{{\mathcal {K}}}}_{02}(\phi ) v , v \big )_{L^2_x} + {{{\mathcal {K}}}}_{\geqq 3}(\phi , \eta , v) \end{aligned}$$

(5.18)

where ${{{\mathcal {K}}}}_{\geqq 3}$ comprises all the terms which are at least cubic in the variables $(\eta , v )$, and where ${{{\mathcal {K}}}}_{00}(\phi ) \in {\mathbb {R}}$, ${{{\mathcal {K}}}}_{10}(\phi ) \in {\mathbb {R}}^{{\mathbb {S}}_+}$, ${{{\mathcal {K}}}}_{01}(\phi ) \in L^2_\bot ({\mathbb {T}}_1)$, ${{{\mathcal {K}}}}_{20}(\phi )$ is a $|{\mathbb {S}}_+|\times |{\mathbb {S}}_+|$ real matrix, ${{{\mathcal {K}}}}_{02}(\phi ) : L^2_\bot ({\mathbb {T}}_1) \rightarrow L^2_\bot ({\mathbb {T}}_1)$ is a linear self-adjoint operator, and ${{{\mathcal {K}}}}_{11}(\phi ) : {\mathbb {R}}^{{\mathbb {S}}_+} \rightarrow L^2_\bot ({\mathbb {T}}_1)$ is a linear operator of finite rank. At an exact solution of ${\mathcal {F}}_\omega (\iota , \zeta ) = 0$ one has $Z = 0$ and ${{{\mathcal {K}}}}_{00} = \mathrm{const}$, ${{{\mathcal {K}}}}_{10} = - \omega$, ${{{\mathcal {K}}}}_{01} = 0$.

Denote by $\mathrm{Id}_\bot$ the identity operator on $L^2_\bot ({\mathbb {T}}_1)$. The linear transformation $d G_\delta |_{(\varphi , 0, 0)} \equiv d G_\delta (\varphi , 0, 0)$ then reads

$$\begin{aligned} d G_\delta |_{(\varphi , 0, 0)} \begin{pmatrix} {\widehat{\phi }} \, \\ {\widehat{\eta }} \\ {\widehat{v}} \end{pmatrix} \,{:=}\, \begin{pmatrix} \partial _\phi \theta (\varphi ) &{}\quad 0 &{}\quad 0 \\ \partial _\phi y_\delta (\varphi ) &{}\quad [\partial _\phi \theta (\varphi )]^{-\top } &{}\quad - [(\partial _\theta {\tilde{w}})(\theta (\varphi ))]^\top \partial _x^{- 1} \\ \partial _\phi w(\varphi ) &{}\quad 0 &{}\quad \mathrm{Id}_\bot \end{pmatrix} \begin{pmatrix} {\widehat{\phi }} \, \\ {\widehat{\eta }} \\ {\widehat{v}} \end{pmatrix} .\nonumber \\ \end{aligned}$$

(5.19)

It approximately transforms the linearized operator $d_{\iota , \zeta }{{{\mathcal {F}}}}_\omega (\iota _\delta )$ (see the proof of Theorem 5.7) into the one obtained when the Hamiltonian system with Hamiltonian ${\mathcal {K}}_{\varepsilon , \zeta }$ (cf. (5.17)) is linearized at $(\phi , \eta , v ) = (\varphi , 0, 0 )$, differentiated also with respect to $\zeta$, and $\partial _t$ is replaced by $\omega \cdot \partial _\varphi$,

$$\begin{aligned} \begin{pmatrix} {\widehat{\phi }} \\ {\widehat{\eta }} \\ {\widehat{v}} \\ {\widehat{\zeta }} \end{pmatrix} \mapsto \begin{pmatrix} \omega \cdot \partial _\varphi {\widehat{\phi }} + \partial _\phi {{{\mathcal {K}}}}_{10}(\varphi )[{\widehat{\phi }} \, ] + {{{\mathcal {K}}}}_{2 0}(\varphi ){\widehat{\eta }} + {{{\mathcal {K}}}}_{11}^\top (\varphi ) {\widehat{v}} \\ \omega \cdot \partial _\varphi {\widehat{\eta }} - \big (\partial _\phi \theta (\varphi ) \big )^\top [{\widehat{\zeta }}] - \partial _\phi \big (\partial _\phi \theta (\varphi )^\top [\zeta ]\big )[{\widehat{\phi }}] - \partial _{\phi \phi } {{{\mathcal {K}}}}_{00}(\varphi )[{\widehat{\phi }}] - [\partial _\phi {{{\mathcal {K}}}}_{10}(\varphi )]^\top {\widehat{\eta }} - [\partial _\phi {{{\mathcal {K}}}}_{01}(\varphi )]^\top {\widehat{v}} \\ \omega \cdot \partial _\varphi {\widehat{v}} - \partial _x \{ \partial _\phi {{{\mathcal {K}}}}_{01}(\varphi )[{\widehat{\phi }}] + {{{\mathcal {K}}}}_{11}(\varphi ) {\widehat{\eta }} + {{{\mathcal {K}}}}_{02}(\varphi ) {\widehat{v}} \} \end{pmatrix} .\nonumber \\ \end{aligned}$$

(5.20)

Using (5.2) and (5.10), one shows as in [3] that the operator ${\widehat{\iota }} = ({\widehat{\phi }}, {\widehat{\eta }}, {\widehat{v}}) \mapsto dG_\delta [{\widehat{\iota }}]$ satisfies for any $s \ge s_0$

$$\begin{aligned}&\Vert dG_\delta (\varphi ,0,0) [{\widehat{\iota }}] \Vert _s^\mathrm{{Lip}(\gamma )}, \, \Vert dG_\delta (\varphi ,0,0)^{-1} [{\widehat{\iota }}] \Vert _s^\mathrm{{Lip}(\gamma )}\nonumber \\&\quad \lesssim _s \, \Vert {\widehat{\iota }} \Vert _{s}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0 }^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

(5.21)

$$\begin{aligned}&\Vert d^2 G_\delta (\varphi ,0,0)[{\widehat{\iota }}_1, {\widehat{\iota }}_2] \Vert _s^\mathrm{{Lip}(\gamma )}\\&\quad \lesssim _s \Vert {\widehat{\iota }}_1\Vert _s^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}_2 \Vert _{s_0}^\mathrm{{Lip}(\gamma )}+ \Vert {\widehat{\iota }}_1\Vert _{s_0}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}_2 \Vert _{s}^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}_1 \Vert _{s_0}^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }}_2\Vert _{s_0}^\mathrm{{Lip}(\gamma )}.\nonumber \end{aligned}$$

(5.22)

The next lemma provides estimates for the coefficients of the Taylor expansion (5.18) of the Hamiltonian ${{{\mathcal {K}}}}$.

Lemma 5.5

There exists $\sigma \,{:=}\, \sigma ( \tau , {\mathbb {S}}_+ ) > 0$ so that for any $s \ge s_0$

$$\begin{aligned} \begin{aligned}&\Vert \partial _\phi {{{\mathcal {K}}}}_{00} \Vert _s^\mathrm{{Lip}(\gamma )}+ \Vert {{{\mathcal {K}}}}_{10} + \omega \Vert _s^\mathrm{{Lip}(\gamma )}+ \Vert {{{\mathcal {K}}}}_{0 1} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \gamma ^{- 1}\big ( \Vert Z \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert Z \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\big ). \\&\Vert {{{\mathcal {K}}}}_{20} - \Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \varepsilon + \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}, \\&\Vert {{{\mathcal {K}}}}_{11} \eta \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \varepsilon \gamma ^{- 2} \Vert \eta \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert \eta \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}, \\&\Vert {{{\mathcal {K}}}}_{11}^\top v \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \varepsilon \gamma ^{- 2} \Vert v \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert v \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}. \end{aligned} \end{aligned}$$

(5.23)

Proof

First we prove the claimed estimates for ${{{\mathcal {K}}}}_{00}, {{{\mathcal {K}}}}_{10}, {{{\mathcal {K}}}}_{0 1}$ and then the ones for ${{{\mathcal {K}}}}_{20}, {{{\mathcal {K}}}}_{11}, {{{\mathcal {K}}}}_{11}^\top$.

Estimates of ${{{\mathcal {K}}}}_{00}, {{{\mathcal {K}}}}_{10}, {{{\mathcal {K}}}}_{0 1}$: The Hamiltonian vector field associated to the Hamiltonian ${{{\mathcal {K}}}}$ in (5.17) is given by $X_{{{{\mathcal {K}}}}} \,{:=}\, (- \nabla _\eta {{{\mathcal {K}}}} ,\,\, \nabla _\phi {{{\mathcal {K}}}} + (\partial _\phi \theta )^\top \zeta , \,\, \partial _x \nabla _{v} {{{\mathcal {K}}}})$. Furthermore, since $\breve{\iota }_\delta (\varphi ) = G_\delta ( \varphi , 0, 0)$, the directional derivative $\omega \cdot \partial _\varphi \breve{\iota }_\delta (\varphi )$ equals $d G_\delta (\varphi , 0, 0)[(\omega , 0, 0)]$. Using the transformation law of vector fields we get that

$$\begin{aligned} \begin{aligned} {{{\mathcal {F}}}}_\omega ( \iota _\delta , \zeta )(\varphi )&= \omega \cdot \partial _\varphi \breve{\iota }_{\delta }(\varphi ) - X_{{{{\mathcal {H}}}}_{\varepsilon , \zeta }} (\breve{\iota }_\delta (\varphi )) \\&\quad = \hbox {d} G_\delta (\varphi , 0, 0) [(\omega , 0, 0)] -\hbox {d}G_\delta (\varphi , 0, 0)X_{{{{\mathcal {K}}}}} ( \varphi , 0, 0), \end{aligned} \end{aligned}$$

or

$$\begin{aligned} X_{{{{\mathcal {K}}}}} ( \varphi , 0, 0) = (\omega , 0, 0) - (d G_\delta (\varphi , 0, 0))^{- 1} {{{\mathcal {F}}}}_\omega (\iota _\delta , \zeta ). \end{aligned}$$

(5.24)

Furthermore, by (5.18), one computes

$$\begin{aligned} X_{{{\mathcal {K}}}}(\varphi , 0, 0) = \big (- {{{\mathcal {K}}}}_{1 0}(\varphi ), \, \partial _\varphi \theta (\varphi )[\zeta ] + \nabla _\varphi {{{\mathcal {K}}}}_{00}(\varphi ), \, \partial _x {{{\mathcal {K}}}}_{0 1}(\varphi ) \big ). \end{aligned}$$

(5.25)

By comparing the two expressions (5.24) and (5.25), and by using the estimate (5.5) of $\zeta$, the estimate (5.13) of ${{{\mathcal {F}}}}_\omega (\iota _\delta , \zeta )$, the estimate (5.195.21) of $d G_\delta$, the ansatz (5.2) and that $\Vert [\partial _\varphi \theta ]^\top \Vert _s^\mathrm{{Lip}(\gamma )}\leqq 1 + C(s) \Vert \iota \Vert _s^\mathrm{{Lip}(\gamma )}$, one gets the first estimate in (5.23).

Estimates of ${{{\mathcal {K}}}}_{20}, {{{\mathcal {K}}}}_{11}, {{{\mathcal {K}}}}_{11}^\top$: We prove the claimed bound for ${{{\mathcal {K}}}}_{2 0}$ and ${{{\mathcal {K}}}}_{11}$. The estimate for ${{{\mathcal {K}}}}_{11}^\top$ can be proved arguing similarly. By (5.18), one has ${{{\mathcal {K}}}}_{2 0}(\varphi ) = \partial _\eta \nabla _\eta {{{\mathcal {K}}}}(\varphi , 0, 0)$ and ${{{\mathcal {K}}}}_{1 1}(\varphi ) = \partial _\eta \nabla _v {{{\mathcal {K}}}}(\varphi , 0, 0)$. Furthermore, taking into account the formulae (4.7), (5.16), (5.17), one then infers that

$$\begin{aligned} \begin{aligned} {{{\mathcal {K}}}}_{2 0}(\varphi )&= [\partial _\varphi \theta (\varphi )]^{- 1} \partial _y \nabla _y {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }(\varphi )) [\partial _\varphi \theta (\varphi )]^{- \top } \\&{\mathop {=}\limits ^{(4.13)}} [\partial _\varphi \theta (\varphi )]^{- 1}\Omega _{{\mathbb {S}}_+}^{kdv}(\nu )[\partial _\varphi \theta (\varphi )]^{- \top } + [\partial _\varphi \theta (\varphi )]^{- 1}\partial _y \nabla _y {{{\mathcal {P}}}}_\varepsilon (\breve{\iota }(\varphi ))[\partial _\varphi \theta (\varphi )]^{- \top }, \\ {{{\mathcal {K}}}}_{1 1}(\varphi )&= - [(\partial _\theta {\tilde{w}})(\theta (\varphi ))] \partial _y \nabla _y {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )) [\partial _\varphi \theta (\varphi )]^{-\top } + \partial _y \nabla _w {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )) [\partial _\varphi \theta (\varphi )]^{-\top } \\&{\mathop {=}\limits ^{(4.13)}} - [(\partial _\theta {\tilde{w}})(\theta (\varphi ))] \Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) [\partial _\varphi \theta (\varphi )]^{-\top } - [(\partial _\theta {\tilde{w}})(\theta (\varphi ))] \partial _y \nabla _y {{{\mathcal {P}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )) [\partial _\varphi \theta (\varphi )]^{-\top } \\&\quad + \partial _y \nabla _w {{{\mathcal {P}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )) [\partial _\varphi \theta (\varphi )]^{-\top } . \end{aligned} \end{aligned}$$

By $\Vert [\partial _\varphi \theta (\varphi )]^{- 1}- \mathrm{Id} \Vert _s^\mathrm{{Lip}(\gamma )}+ \Vert [\partial _\varphi \theta (\varphi )]^{- \top } - \mathrm{Id} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert \iota \Vert _{s + 1}^\mathrm{{Lip}(\gamma )}$ (cf. (5.14)), the estimates for $\partial _y \nabla _y {{{\mathcal {P}}}}_\varepsilon (\breve{\iota }_\delta )$, $\partial _y \nabla _w {{{\mathcal {P}}}}_\varepsilon (\breve{\iota }_\delta )$ of Lemma 4.3, the interpolation estimate (2.5), and (5.2), one obtains the claimed bounds. $\quad \square$

In order to construct an almost-approximate right inverse of (5.20), we need that

$$\begin{aligned} {{{\mathcal {L}}}}_\omega \,{:=}\, \Pi _\bot \big (\omega \cdot \partial _\varphi - \partial _x {{{\mathcal {K}}}}_{02}(\varphi ) \big )_{|L^2_\bot } \end{aligned}$$

(5.26)

is “almost-invertible”, that is invertible up to a remainder of order $O(N_{n - 1}^{- \mathtt {a}})$, where

$$\begin{aligned} N_n \,{:=}\, K_n^p , \quad \forall n \geqq 0, \end{aligned}$$

(5.27)

and

$$\begin{aligned} K_n \,{:=}\, K_0^{\chi ^{n}} , \quad \chi \,{:=}\, 3/ 2 , \end{aligned}$$

(5.28)

are the scales used in the nonlinear Nash–Moser iteration in Section 8.1. The constants ${\mathtt {a}}$, $K_0$ are given in the almost-invertibility assumption A-I below.

Based on results obtained in Sections 6–7, the almost invertibility of ${{{\mathcal {L}}}}_\omega$ is proved in Theorem 7.11, but here it is stated as an assumption to avoid the involved definition of the subset $\Omega _o$ of the set $\Omega$ of frequency vectors $\omega$, for which ${{{\mathcal {L}}}}_\omega$ can be shown to admit an almost-approximate right inverse. Recall that $\mathtt {DC}(\gamma , \tau )$ is the set of diophantine frequencies in $\Omega$, defined in (4.4).

• A-I Almost-invertibility of ${{{\mathcal {L}}}}_\omega$. There exists a subset $\Omega _o \subset \mathtt {DC}(\gamma , \tau )$ such that, for all $\omega \in \Omega _o$, the operator ${{{\mathcal {L}}}}_\omega$ in (5.26) admits a decomposition

  $$\begin{aligned} {{{\mathcal {L}}}}_\omega = {{{\mathcal {L}}}}_\omega ^< + {{{\mathcal {R}}}}_\omega + {{{\mathcal {R}}}}_\omega ^\bot \end{aligned}$$

  (5.29)

  with the following properties: there exist constants $K_0$, $N_0$, $\sigma$, $\tau _1$, $\mu (\mathtt {b})$, $\mathtt {a}$, p, ${\mathfrak {s}}_M> 0$, so that for any $S > {\mathfrak {s}}_M$, ${\mathfrak {s}}_M\leqq s \leqq S$ and $\omega \in \Omega _o$ the following holds:

  • (i) The operators ${{{\mathcal {R}}}}_\omega$, ${{{\mathcal {R}}}}_\omega ^\bot$ satisfy the estimates

    $$\begin{aligned} \Vert {{{\mathcal {R}}}}_\omega h \Vert _s^\mathrm{{Lip}(\gamma )}&\lesssim _S \varepsilon \gamma ^{- 2} N_{n - 1}^{- {\mathtt {a}}}\big ( \Vert h \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ N_0^{\tau _1} \gamma ^{- 1}\Vert \iota \Vert _ {s + \mu (\mathtt {b}) + \sigma }^\mathrm{{Lip}(\gamma )}\Vert h \Vert _{{\mathfrak {s}}_M+ \sigma }^\mathrm{{Lip}(\gamma )}\big ), \end{aligned}$$

    (5.30)
    $$\begin{aligned} \Vert {{{\mathcal {R}}}}_\omega ^\bot h \Vert _{{\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}&\lesssim _{S,b} K_n^{- b} \big ( \Vert h \Vert _{{\mathfrak {s}}_M+ b + \sigma }^\mathrm{{Lip}(\gamma )}+ N_0^{\tau _1} \gamma ^{- 1}\Vert \iota \Vert _ {{\mathfrak {s}}_M+ \mu (\mathtt {b}) + \sigma + b }^\mathrm{{Lip}(\gamma )}\Vert h \Vert _{{\mathfrak {s}}_M+ \sigma }^\mathrm{{Lip}(\gamma )}\big ), \quad \forall b > 0. \nonumber \\ \end{aligned}$$

    (5.31)

  • (ii) The operator ${{{\mathcal {L}}}}_\omega ^<$ admits a right inverse. More precisely, for any $g \in H^{s+\sigma }_{\bot }({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1)$, there is a solution $h \in H^{s}_{\bot }({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1)$ of the linear equation ${{{\mathcal {L}}}}_\omega ^< h = g$, denoted by $({{{\mathcal {L}}}}_\omega ^<)^{- 1} g$, satisfying the tame estimates

    $$\begin{aligned} \Vert ({{{\mathcal {L}}}}_\omega ^<)^{- 1} g \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _S \gamma ^{-1} \big ( \Vert g \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ N_0^{\tau _1} \gamma ^{- 1} \Vert \iota \Vert _{s + \mu ({\mathtt {b}}) + \sigma }^\mathrm{{Lip}(\gamma )}\Vert g \Vert _{{\mathfrak {s}}_M+ \sigma }^\mathrm{{Lip}(\gamma )}\big ) . \end{aligned}$$

    (5.32)

In order to find an almost-approximate inverse of the linear operator (5.20) and hence of $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota _\delta )$, note that the remainder ${{{\mathcal {L}}}}_\omega - {{{\mathcal {L}}}}_\omega ^< = {{{\mathcal {R}}}}_\omega + {{{\mathcal {R}}}}_\omega ^\bot$ is small (cf. (5.30)–(5.31) in A-I) and that by Lemma 5.5 and by the estimate (5.5) of $\zeta$, the terms $\partial _\phi {{{\mathcal {K}}}}_{10}$, $\partial _{\phi \phi } {{{\mathcal {K}}}}_{00}$, $\partial _\phi {{{\mathcal {K}}}}_{01}$ and $\partial _\phi \big (\partial _\phi \theta (\varphi )^\top [\zeta ]\big )$ in (5.20) are of the order O(Z). Therefore, it suffices to invert the operator

$$\begin{aligned} {{\mathbb {D}}} [{\widehat{\phi }}, {\widehat{\eta }}, {\widehat{v}}, {\widehat{\zeta }} ] \,{:=}\, \begin{pmatrix} \omega \cdot \partial _\varphi {\widehat{\phi }} + {{{\mathcal {K}}}}_{20}(\varphi ) {\widehat{\eta }} + {{{\mathcal {K}}}}_{11}(\varphi )^\top {\widehat{v}}\\ \omega \cdot \partial _\varphi {\widehat{\eta }} - \partial _\phi \theta (\varphi )^\top {\widehat{\zeta }} \\ {{{\mathcal {L}}}}_\omega ^{<} {\widehat{v}} - \partial _x {{{\mathcal {K}}}}_{11}(\varphi ){\widehat{\eta }} \end{pmatrix} \end{aligned}$$

(5.33)

obtained by neglecting in (5.20) the terms $\partial _\phi {{{\mathcal {K}}}}_{10}$, $\partial _{\phi \phi } {{{\mathcal {K}}}}_{00}$, $\partial _\phi {{{\mathcal {K}}}}_{01}$, $\partial _\phi \big (\partial _\phi \theta (\varphi )^\top [\zeta ]\big )$ and by replacing ${{{\mathcal {L}}}}_\omega$ by ${{{\mathcal {L}}}}_\omega ^<$ (cf. (5.29)). We look for a right inverse of ${{\mathbb {D}}}$ by solving the system

$$\begin{aligned} {{\mathbb {D}}} [{\widehat{\phi }}, {\widehat{\eta }}, {\widehat{v}}, {\widehat{\zeta }}] = \begin{pmatrix} g_1 \\ g_2 \\ g_3 \end{pmatrix}. \end{aligned}$$

(5.34)

We first consider the second equation in (5.34), $\omega \cdot \partial _\varphi {\widehat{\eta }} = g_2 + \partial _\phi \theta (\varphi )^\top {\widehat{\zeta }}$. Since $\partial _\varphi \theta (\varphi ) = \mathrm{Id} + \partial _\varphi \Theta (\varphi )$, the average $\langle \partial _\varphi \theta ^\top \rangle _\varphi = \frac{1}{(2\pi )^{|{{\mathbb {S}}}_+|}} \int _{{\mathbb {T}}^{{{\mathbb {S}}}_+}} \partial _\varphi \theta ^\top (\varphi )\hbox {d}\varphi$ equals the identity matrix $\mathrm{Id}$ of ${\mathbb {R}}^{{{\mathbb {S}}}_+}$. We then define

$$\begin{aligned} {\widehat{\zeta }} \,{:=}\, - \langle g_2 \rangle _\varphi \ \end{aligned}$$

(5.35)

so that $\langle g_2 + \partial _\phi \theta (\varphi )^\top {\widehat{\zeta }} \rangle _\varphi$ vanishes and define

$$\begin{aligned} {\widehat{\eta }} \,{:=}\, {\widehat{\eta }}_0 + {\widehat{\eta }}_1 , \quad {\widehat{\eta }}_1 \,{:=}\, ( \omega \cdot \partial _\varphi )^{-1} \big ( g_2 + \partial _\phi \theta (\varphi )^\top {\widehat{\zeta }} \, \big ), \end{aligned}$$

(5.36)

where the constant vector ${\widehat{\eta }}_0 \in {\mathbb {R}}^{{{\mathbb {S}}}_+}$ will be determined in order to control the average of the first equation in (5.34). Next we consider the third equation in (5.34), $({{{\mathcal {L}}}}_\omega ^<) {\widehat{v}} = g_3 + \partial _x {{{\mathcal {K}}}}_{11}(\varphi ) {\widehat{\eta }}$, which, by assumption (5.32) on the invertibility of ${{{\mathcal {L}}}}_\omega ^<$, has the solution

$$\begin{aligned} {\widehat{v}} = ({{{\mathcal {L}}}}_\omega ^<)^{-1} \big ( g_3 + \partial _x {{{\mathcal {K}}}}_{11}(\varphi ) {\widehat{\eta }}_1 \big ) + ({{{\mathcal {L}}}}_\omega ^<)^{-1} \partial _x {{{\mathcal {K}}}}_{11}(\varphi ) {\widehat{\eta }}_0 . \end{aligned}$$

(5.37)

Finally, we solve the first equation in (5.34). After substituting the solutions ${\widehat{\zeta }}$, ${\widehat{\eta }}$ (cf. (5.35), (5.36)), and ${\widehat{v}}$ (cf. (5.37)), this equation becomes

$$\begin{aligned} \omega \cdot \partial _\varphi {\widehat{\phi }} = g_1 + M_1{\widehat{\eta }}_0 + M_2 g_2 + M_3 g_3 - M_4 \langle g_2 \rangle _\varphi , \end{aligned}$$

(5.38)

where ${\widehat{\phi }}$ and ${\widehat{\eta }}_0$ are the unknowns and where $M_j: \varphi \mapsto M_j(\varphi )$, $1 \le j \le 4$, are defined as

$$\begin{aligned}&M_1(\varphi ) \,{:=}\, - {{{\mathcal {K}}}}_{2 0}(\varphi ) - {{{\mathcal {K}}}}_{11}(\varphi )^\top ({{{\mathcal {L}}}}_\omega ^{<})^{- 1} \partial _x {{{\mathcal {K}}}}_{11}(\varphi ) , \end{aligned}$$

(5.39)

$$\begin{aligned}&M_2(\varphi ) \,{:=}\, M_1(\varphi )[\omega \cdot \partial _\varphi ]^{-1} , \quad M_3(\varphi ) \,{:=}\, - {{{\mathcal {K}}}}_{11} (\varphi )^\top ({{{\mathcal {L}}}}_\omega ^<)^{-1} , \quad M_4(\varphi ) \,{:=}\, M_2(\varphi ) \partial _\phi \theta (\varphi )^\top . \end{aligned}$$

(5.40)

In order to solve equation (5.38) we have to choose ${\widehat{\eta }}_0$ so that the average of the right hand side vanishes. By Lemma 5.5, by the ansatz (5.2) and by the tame estimates (5.32), the $\varphi$-averaged matrix is $\langle M_1 \rangle _\varphi = - \Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) + O( \varepsilon \gamma ^{-2 })$. Since the matrix $\Omega _{{\mathbb {S}}_+}^{kdv}(\nu ) = (\partial _{I_k} \omega _n^{kdv}(\nu , 0)) _{k,n \in {{\mathbb {S}}}_+}$ is invertible (cf. Lemma 3.9-(i), Remark 3.10), $\langle M_1 \rangle _\varphi$ is invertible for $\varepsilon \gamma ^{- 2}$ small enough and $\langle M_1 \rangle _\varphi ^{-1} = - \Omega _{{\mathbb {S}}_+}^{kdv}(\nu )^{- 1} + O(\varepsilon \gamma ^{- 2})$. We then define

$$\begin{aligned} {\widehat{\eta }}_0 \,{:=}\, - \langle M_1 \rangle _\varphi ^{-1} \Big ( \langle g_1 \rangle _\varphi + \langle M_2 g_2 \rangle _\varphi + \langle M_3 g_3 \rangle _\varphi - \langle M_4 \rangle _\varphi \langle g_2 \rangle _\varphi \Big ) . \end{aligned}$$

(5.41)

With this choice of ${\widehat{\eta }}_0$, the equation (5.38) has the solution

$$\begin{aligned} {\widehat{\phi }} = (\omega \cdot \partial _\varphi )^{-1} \big ( g_1 + M_1{\widehat{\eta }}_0 + M_2 g_2 + M_3 g_3 - M_4 \langle g_2 \rangle _\varphi \big ) . \end{aligned}$$

(5.42)

Altogether, we have obtained a solution $({\widehat{\phi }}, {\widehat{\eta }}, {\widehat{v}}, {\widehat{\zeta }})$ of the linear system (5.34).

Proposition 5.6

Assume (5.2) (Ansatz) with $\mu _0 = \mu (\mathtt {b}) + \sigma$ and that the estimates (5.32) (item (ii) of A-I) hold. Then, for any $\omega \in \Omega _o$ and any $g \,{:=}\, (g_1, g_2, g_3)$ with $g_1, g_2 \in H^{s+\sigma }({\mathbb {T}}^{{\mathbb {S}}_+}, {\mathbb {R}}^{{{\mathbb {S}}}_+})$, $g_3 \in H^{s+\sigma }_{\bot }({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1)$, and ${\mathfrak {s}}_M\leqq s \leqq S$, the system (5.34) has a solution $({\widehat{\phi }}, {\widehat{\eta }}, {\widehat{v}}, {\widehat{\zeta }} )$, where ${\widehat{\phi }}$, ${\widehat{\eta }}$, ${\widehat{v}}$, ${\widehat{\zeta }}$ are defined in (5.35)–(5.37), (5.41)–(5.42). We denote $({\widehat{\phi }}, {\widehat{\eta }}, {\widehat{v}}, {\widehat{\zeta }} )$ by ${{\mathbb {D}}}^{-1} g$. It satisfies the tame estimates

$$\begin{aligned} \Vert {{\mathbb {D}}}^{-1} g \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _S \gamma ^{-2} \big ( \Vert g \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ N_0^{\tau _1} \gamma ^{- 1} \Vert \iota \Vert _{s + \mu (\mathtt {b}) + \sigma }^\mathrm{{Lip}(\gamma )}\Vert g \Vert _{{\mathfrak {s}}_M+ \sigma }^\mathrm{{Lip}(\gamma )}\big ). \end{aligned}$$

(5.43)

Proof

The proposition follows by the definitions of ${\widehat{\zeta }}$ (cf. (5.35)), ${\widehat{\eta }}_1$ (cf. (5.36)), ${\widehat{v}}$ (cf. (5.37)), ${\widehat{\eta }}_0$ (cf. (5.41)), ${\widehat{\phi }}$ (cf. (5.42)), the definitions of $M_j$, $1 \le j \le 4$, in (5.39)–(5.40), by the estimates of Lemma 5.5, and the ansatz (5.2) as well as the estimates (5.32) for $({\mathcal {L}}_\omega ^<)^{-1}$ (item (ii) in A-I). $\quad \square$

Let $\widetilde{G}_\delta : (\phi , \eta , v, \zeta ) \mapsto \big ( G_\delta (\phi , \eta , v), \, \zeta \big )$ and note that its differential $\hbox {d} \widetilde{G}_\delta (\phi , \eta , v, \zeta )$ is independent of $\zeta$. In the sequel, we denote it by $d \widetilde{G}_\delta (\phi , \eta , v)$ or $d \widetilde{G}_\delta |_{ (\phi , \eta , v )}$. Finally we prove that the operator

$$\begin{aligned} \mathbf{T}_0 = \mathbf{T}_0(\iota ) \,{:=}\, \hbox {d} { {\widetilde{G}}}_\delta |_{(\varphi ,0,0)} \circ {{\mathbb {D}}}^{-1} \circ \big (d G_\delta |_{(\varphi ,0,0)} \big )^{-1} \end{aligned}$$

(5.44)

is an almost-approximate right inverse for $d_{\iota ,\zeta } {{{\mathcal {F}}}}_\omega (\iota )$ meaning that $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota ) \circ \mathbf{T}_0(\iota ) - \mathrm{Id}$ can be estimated in terms of the error function $Z = {{{\mathcal {F}}}}_\omega (\iota )$ (’approximate’, cf. (5.47)) and of terms which are small (’almost’, cf. (5.48), (5.49)). Let $\Vert (\phi , \eta , v, \zeta ) \Vert _s^\mathrm{{Lip}(\gamma )}\,{:=}\, \max \{ \Vert (\phi , \eta , v) \Vert _s^\mathrm{{Lip}(\gamma )},$ $| \zeta |^\mathrm{{Lip}(\gamma )}\}$.

Theorem 5.7

(Almost-approximate inverse) Assume the almost-invertibility assumption A-I of ${{{\mathcal {L}}}}_\omega$. Then there exists $\sigma _2 = \sigma _2(\tau , {\mathbb {S}}_+ ) > 0$ so that, if (5.2) (Ansatz) holds with $\mu _0 \geqq {\mathfrak {s}}_M+ \mu (\mathtt {b}) + \sigma _2$, then for any $\omega \in \Omega _o$ and any $g = (g_1, g_2, g_3)$ with $g_1, g_2 \in H^{s+\sigma }({\mathbb {T}}^{{\mathbb {S}}_+}, {\mathbb {R}}^{{{\mathbb {S}}}_+})$, $g_3 \in H^{s+\sigma }_{\bot }({\mathbb {T}}^{{\mathbb {S}}_+} \times {\mathbb {T}}_1)$, and ${\mathfrak {s}}_M\leqq s \leqq S$, $\mathbf{T}_0(\iota ) g$, defined by (5.44), satisfies

$$\begin{aligned} \Vert \mathbf{T}_0(\iota ) g \Vert _{s}^\mathrm{{Lip}(\gamma )}\lesssim _S \gamma ^{-2} \Big (\Vert g \Vert _{s + \sigma _2}^\mathrm{{Lip}(\gamma )}+ N_0^{\tau _1} \gamma ^{- 1} \Vert \iota \Vert _{s + \mu ({\mathtt {b}}) + \sigma _2 }^\mathrm{{Lip}(\gamma )}\Vert g \Vert _{{\mathfrak {s}}_M+ \sigma _2}^\mathrm{{Lip}(\gamma )}\Big ). \end{aligned}$$

(5.45)

Moreover $\mathbf{T}_0(\iota )$ is an almost-approximate right inverse of $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )$. More precisely,

$$\begin{aligned} d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota ) \circ \mathbf{T}_0(\iota ) - \mathrm{Id} = {{{\mathcal {P}}}} + {{{\mathcal {P}}}}_\omega + {{{\mathcal {P}}}}_\omega ^\bot , \end{aligned}$$

(5.46)

where the operators ${{{\mathcal {P}}}}$, ${{{\mathcal {P}}}}_\omega$, ${{{\mathcal {P}}}}_\omega ^\bot$ are defined in the course of the proof and satisfy the following estimates:

$$\begin{aligned} \Vert {{{\mathcal {P}}}} g \Vert _{{\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}&\lesssim _S \gamma ^{-3 } \Vert {{{\mathcal {F}}}}_\omega (\iota , \zeta ) \Vert _{{\mathfrak {s}}_M+ \sigma _2}^\mathrm{{Lip}(\gamma )}\Big (1 + N_0^{\tau _1} \gamma ^{- 1} \Vert \iota \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + \sigma _2}^\mathrm{{Lip}(\gamma )}\Big )\Vert g \Vert _{{\mathfrak {s}}_M+ \sigma _2}^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

(5.47)

$$\begin{aligned} \Vert {{{\mathcal {P}}}}_\omega g \Vert _{{\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}&\lesssim _S \varepsilon \gamma ^{- 4} N_{n - 1}^{- {\mathtt {a}}} \big ( 1 + N_0^{\tau _1} \gamma ^{- 1}\Vert \iota \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + \sigma _2}^\mathrm{{Lip}(\gamma )}\big ) \Vert g \Vert _{{\mathfrak {s}}_M+ \sigma _2}^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

(5.48)

$$\begin{aligned} \Vert {{{\mathcal {P}}}}_\omega ^\bot g\Vert _{{\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}&\lesssim _{S, b} \gamma ^{- 2} K_n^{- b } \big ( \Vert g \Vert _{{\mathfrak {s}}_M+ \sigma _2 + b }^\mathrm{{Lip}(\gamma )}+ N_0^{\tau _1} \gamma ^{- 1}\Vert \iota \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + \sigma _2 +b }^\mathrm{{Lip}(\gamma )}\big \Vert g \Vert _{{\mathfrak {s}}_M+\sigma _2}^\mathrm{{Lip}(\gamma )}\big ),\,\, \quad \forall b > 0 . \end{aligned}$$

(5.49)

Proof

The bound (5.45) follows from the definition of $\mathbf{T}_0(\iota )$ in (5.44), the estimates (5.43) of $\Vert {{\mathbb {D}}}^{-1} g \Vert _s^\mathrm{{Lip}(\gamma )}$, and the ones of $dG_\delta (\varphi ,0,0)$ and of its inverse in (5.195.21). It remains to estimate $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota ) \circ \mathbf{T}_0(\iota ) - \mathrm{Id}$. The operators ${{{\mathcal {P}}}}$, ${{{\mathcal {P}}}}_\omega$, ${{{\mathcal {P}}}}_\omega ^\bot$ in (5.46) are defined as follows: by the formula (5.1) for $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )$ and since only the $y-$components of $\breve{\iota }_\delta$ and $\breve{\iota }$ differ from each other (cf. (5.9)), one has $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota ) = d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota _\delta ) + {{{\mathcal {E}}}}_0$ where, by the mean value theorem, ${{{\mathcal {E}}}}_0$ can be written as

$$\begin{aligned} {{{\mathcal {E}}}}_0 [{\widehat{\iota }}, {\widehat{\zeta }}] = \int _0^1 \partial _y \big ( d_\iota X_{{{{\mathcal {H}}}}_\varepsilon } (\theta , y_\delta + s (y - y_\delta ), w) [ {\widehat{\iota }}]\big )\hbox {d}s \, [y - y_\delta ] . \end{aligned}$$

(5.50)

Denote by $\kappa \,{:=}\, (\phi , \eta , v)$ the symplectic coordinates defined by $G_\delta$. Let also $\breve{\kappa } (\varphi ) = (\varphi , 0, 0) + \kappa (\varphi )$ the torus embedding defined by $\breve{\iota }(\varphi ) = G_\delta ({\breve{\kappa }(\varphi )})$ where $G_\delta$ is the symplectic transformation given by (5.16). The nonlinear operator ${{{\mathcal {F}}}}_\omega$ (cf. (4.6)) is transformed under $G_\delta$ into

$$\begin{aligned} {{{\mathcal {F}}}}_\omega ( \iota , \zeta ) (\varphi ) =\hbox {d}G_\delta ( {\breve{\kappa }} (\varphi ) ) [ \omega \cdot \partial _\varphi \breve{\kappa }(\varphi ) - X_{{{{\mathcal {K}}}}} ( \breve{\kappa }(\varphi )) ] \end{aligned}$$

(5.51)

where ${{{\mathcal {K}}}}= {{{\mathcal {H}}}}_{\varepsilon , \zeta } \circ G_\delta$ (cf. (5.17)). Differentiating (5.51) at the trivial torus $\breve{\kappa }_{0} (\varphi ) = G_\delta ^{-1}(\breve{\iota }_\delta ) (\varphi ) = (\varphi , 0 , 0 )$, we get

$$\begin{aligned} d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota _\delta ) =&\, d G_\delta ( \breve{\kappa }_0) \big ( \omega \cdot \partial _\varphi - d_{\kappa , \zeta } X_{{{{\mathcal {K}}}}}( \breve{\kappa }_0) \big )\hbox {d}{\widetilde{G}}_\delta ( \breve{\kappa }_0)^{-1} + {{{\mathcal {E}}}}_1 , \end{aligned}$$

(5.52)

$$\begin{aligned} {{{\mathcal {E}}}}_1 \,{:=}\, \,&d^2 G_\delta ( \breve{\kappa }_0) \big [ d G_\delta (\breve{\kappa }_0)^{-1} {{{\mathcal {F}}}}_\omega (\iota _\delta , \zeta ), \, \hbox {d}G_\delta (\breve{\kappa }_0)^{-1} \Pi [ \, \cdot \, ] \, \big ] , \end{aligned}$$

(5.53)

where $\Pi$ denotes the projection $\Pi : ({\widehat{\iota }}, {\widehat{\zeta }} ) \mapsto {\widehat{\iota }}$. Let us consider the operator $\omega \cdot \partial _\varphi - d_{\kappa , \zeta } X_{{{{\mathcal {K}}}}}( \breve{\kappa }_0)$ in more detail. By the definition (5.33) of ${\mathbb {D}}$ and the discussion following it, we decompose $\omega \cdot \partial _\varphi - d_{\kappa , \zeta } X_{{{\mathcal {K}}}}( \breve{\kappa }_0)$ as

$$\begin{aligned} \omega \cdot \partial _\varphi - d_{\kappa , \zeta } X_{{{\mathcal {K}}}}( \breve{\kappa }_0) = {\mathbb {D}} + R_Z + {{\mathbb {R}}}_\omega + {{\mathbb {R}}}_\omega ^\bot \end{aligned}$$

(5.54)

where in view of (5.20),

$$\begin{aligned}&R_Z [ {\widehat{\phi }}, {\widehat{\eta }}, {\widehat{v}}, {\widehat{\zeta }}] \,{:=}\, \begin{pmatrix} \partial _\phi {{{\mathcal {K}}}}_{10}(\varphi ) [{\widehat{\phi }} ] \\ - \partial _{\phi \phi } {{{\mathcal {K}}}}_{00} (\varphi ) [ {\widehat{\phi }} ] - \partial _\phi \big (\partial _\phi \theta (\varphi )^\top [\zeta ]\big )[{\widehat{\phi }}] - [\partial _\phi {{{\mathcal {K}}}}_{10}(\varphi )]^\top {\widehat{\eta }} - [\partial _\phi {{{\mathcal {K}}}}_{01}(\varphi )]^\top {\widehat{v}} \\ - \partial _x \big ( \partial _{\phi } {{{\mathcal {K}}}}_{01}(\varphi )[ {\widehat{\phi }} ] \big ) \end{pmatrix} , \\&{{\mathbb {R}}}_\omega [{\widehat{\phi }}, {\widehat{y}}, {\widehat{w}}, {\widehat{\zeta }}] \,{:=}\, \begin{pmatrix} 0 \\ 0 \\ {{{\mathcal {R}}}}_\omega [{\widehat{w}}] \end{pmatrix},\quad {{\mathbb {R}}}_\omega ^\bot [{\widehat{\phi }}, {\widehat{y}} , {\widehat{w}}, {\widehat{\zeta }}] \,{:=}\, \begin{pmatrix} 0 \\ 0 \\ {{{\mathcal {R}}}}_\omega ^\bot [{\widehat{w}}] \end{pmatrix}\, \end{aligned}$$

with ${{{\mathcal {R}}}}_\omega$ and ${{{\mathcal {R}}}}_\omega ^\bot$ given by (5.29). By (5.50) and (5.52)–(5.54) we get the decomposition

$$\begin{aligned} d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota )&= d G_\delta (\breve{\kappa }_0) \circ {{\mathbb {D}}} \circ \big ( d {{\widetilde{G}}}_\delta (\breve{\kappa }_0)\big )^{-1} + {{{\mathcal {E}}}} + {{{\mathcal {E}}}}_\omega + {{{\mathcal {E}}}}_\omega ^\bot \end{aligned}$$

(5.55)

where

$$\begin{aligned} {{{\mathcal {E}}}} \,{:=}\, {{{\mathcal {E}}}}_0 + {{{\mathcal {E}}}}_1 + d G_\delta ( \breve{\kappa }_0) R_Z \big (d {{\widetilde{G}}}_\delta (\breve{\kappa }_0)\big )^{-1}, \end{aligned}$$

(5.56)

$$\begin{aligned} {{{\mathcal {E}}}}_\omega \,{:=}\, \hbox {d} G_\delta ( \breve{\kappa }_0) {{\mathbb {R}}}_\omega \big (d {{\widetilde{G}}}_\delta ( \breve{\kappa }_0)\big )^{-1}, \quad {{{\mathcal {E}}}}_\omega ^\bot \,{:=}\, \hbox {d}G_\delta ( \breve{\kappa }_0) {{\mathbb {R}}}_\omega ^\bot \big (\hbox {d}{{\widetilde{G}}}_\delta ( \breve{\kappa }_0)\big )^{-1} . \end{aligned}$$

(5.57)

Letting the operator $\mathbf{T}_0 = \mathbf{T}_0(\iota )$ (cf. (5.44)) act from the right to both sides of the identity (5.55) and taking into accout that $\breve{\kappa }_0 (\varphi ) = (\varphi , 0, 0 )$, one obtains

$$\begin{aligned}&\quad \quad d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota ) \circ \mathbf{T}_0 - \mathrm{Id} ={{{\mathcal {P}}}} + {{{\mathcal {P}}}}_\omega + {{{\mathcal {P}}}}_\omega ^\bot , \quad {{{\mathcal {P}}}} \,{:=}\, {{{\mathcal {E}}}} \circ \mathbf{T}_0 , \\&\quad \quad {{{\mathcal {P}}}}_\omega \,{:=}\, {\mathcal {E}}_\omega \circ \mathbf{T}_0 , \quad {{{\mathcal {P}}}}_\omega ^\bot \,{:=}\, {\mathcal {E}}_\omega ^\bot \circ \mathbf{T}_0 \, . \end{aligned}$$

To obtain the claimed estimate for ${{\mathcal {P}}}$ we first need to estimate ${\mathcal {E}}$. By (5.2) (Ansatz), (5.5) (estimate for $\zeta$), Lemma 5.5 (estimates of the components of $R_Z$), (5.10)–(5.13) (estimates related to $\iota _\delta$), and (5.195.21)–(5.22) (estimates of $dG_\delta (\breve{\kappa }_0)$ and its inverse) one infers that

$$\begin{aligned} \Vert {{{\mathcal {E}}}} [\, {\widehat{\iota }}, {\widehat{\zeta }} \, ] \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \gamma ^{- 1} \Big ( \Vert Z \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert Z \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}+ \Vert Z \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\Vert \iota \Vert _{s+\sigma }^\mathrm{{Lip}(\gamma )}\Vert {\widehat{\iota }} \Vert _{s_0 + \sigma }^\mathrm{{Lip}(\gamma )}\Big ),\nonumber \\ \end{aligned}$$

(5.58)

for some $\sigma > 0$, where Z is the error function, $Z = {\mathcal {F}}_\omega (\iota , \zeta )$ (cf. (5.3)). The claimed estimate (5.47) for ${\mathcal {P}}$ then follows from (5.58), the estimate (5.45) of $\mathbf{T}_0$, and the ansatz (5.2). The claimed estimates (5.48), (5.49) for ${{{\mathcal {P}}}}_\omega$ and, respectively, ${{{\mathcal {P}}}}_\omega ^\bot$ follow by the estimates (5.30)–(5.31) of ${{{\mathcal {R}}}}_\omega$ and ${{{\mathcal {R}}}}_\omega ^\bot$ (cf. A-I), the estimate (5.45) of $\mathbf{T}_0$, the estimate (5.195.21) of $dG_\delta (\breve{\kappa }_0)$ and its inverse, and the ansatz (5.2). $\quad \square$

The goal of Sections 6 and  7 below is to prove that the Hamiltonian operator ${{\mathcal {L}}}_\omega$, defined in (5.26), satisfies the almost-invertibility A-I, including the tame estimates (5.30)–(5.32).

6 Reduction of ${{\mathcal {L}}}_\omega$ Up to Order Zero

The goal of this section is to reduce the Hamiltonian operator ${{\mathcal {L}}}_\omega$, defined in (5.26), to a differential operator of order three with constant coefficients, up to an operator of order zero – see ${{\mathcal {L}}}^{(4)}_\omega$ defined in (6.69). It is the starting point for the KAM reduction scheme, implemented in Section 7, which will reduce ${{\mathcal {L}}}^{(4)}_\omega$ to a diagonal operator with constant coefficients. The main result of this section is Proposition 6.7.

In the sequel, we consider torus embeddings $\breve{\iota }(\varphi ) = (\varphi , 0, 0) + \iota (\varphi )$ with $\iota (\cdot ) \equiv \iota (\cdot \ ; \omega )$, $\omega \in \mathtt {DC}(\gamma ,\tau )$ (cf. (4.4)), satisfying

$$\begin{aligned} \Vert \iota \Vert _{ \mu _0}^{\mathrm {Lip}(\gamma )} \lesssim \varepsilon \gamma ^{-2} , \quad \varepsilon \gamma ^{- 2} \leqq \delta (S) , \end{aligned}$$

(6.1)

where $\mu _0 \,{:=}\, \mu _0(\tau , {{\mathbb {S}}}_+) > s_0$ and $S > \mu _0$ are sufficiently large, $0< \delta (S) < 1$ is sufficiently small, and $0< \gamma < 1$. The index S of the Sobolev space $H^S_\bot$ will be fixed in (8.4), along the Nash Moser iteration scheme of Section 8.1. In the course of the Nash–Moser iteration we will verify that (6.1) is satisfied by each approximate solution—see the bounds (8.8).

Notation. For a quantity $g(\iota ) \equiv g(\breve{\iota })$ such as an operator, a map, or a scalar function, depending on $\breve{\iota }(\varphi ) = (\varphi , 0, 0) + \iota (\varphi )$, we denote for any two such tori embeddings $\breve{\iota }_1$, $\breve{\iota }_2$ by $\Delta _{12} g$ the difference

$$\begin{aligned} \Delta _{12} g \,{:=}\, g (\iota _2) - g (\iota _1) . \end{aligned}$$

6.1 Expansion of ${{\mathcal {L}}}_\omega$

As a first step, we derive an expansion of the operator ${{\mathcal {L}}}_\omega = \Pi _\bot \big (\omega \cdot \partial _\varphi - \partial _x {{{\mathcal {K}}}}_{02}(\varphi ) \big )_{|L^2_\bot }$, defined in (5.26).

Lemma 6.1

The Hamiltonian operator $\partial _x {{\mathcal {K}}}_{02}(\varphi )$ acting on $L^2_\bot ({\mathbb {T}}_1)$ is of the form

$$\begin{aligned} \partial _x {{\mathcal {K}}}_{02}(\varphi ) = \Pi _\bot \partial _x ( d_\bot \nabla _w {{\mathcal {H}}}_\varepsilon ) ( \breve{\iota }_\delta (\varphi )) + R(\varphi ) \end{aligned}$$

(6.2)

where ${{\mathcal {H}}}_\varepsilon$ is the Hamiltonian defined in (4.2) and the remainder $R(\varphi )$ is given by

$$\begin{aligned} R(\varphi )[h] = {\mathop \sum }_{j \in {{\mathbb {S}}}_+} \big (h,\,g_j \big )_{L^2_x} \chi _j, \quad \forall h \in L^2_\bot ({\mathbb {T}}_1) , \end{aligned}$$

(6.3)

with functions $g_j, \chi _j \in H^s_\bot$, $j \in {{\mathbb {S}}}_+$, satisfying, for some $\sigma \,{:=}\, \sigma (\tau , {{\mathbb {S}}}_+) > 0$ and any $s \geqq s_0$

$$\begin{aligned} \Vert g_j\Vert _s^{\mathrm{{Lip}(\gamma )}} +\Vert \chi _j\Vert _s^{\mathrm{{Lip}(\gamma )}} \lesssim _s \varepsilon + \Vert \iota \Vert _{s + \sigma }^{\mathrm{{Lip}(\gamma )}}. \end{aligned}$$

(6.4)

Let $s_1 \geqq s_0$ and let $\breve{\iota }_1, \breve{\iota }_2$ be torus embeddings satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma$. Then, for any $j \in {{\mathbb {S}}}_+$,

$$\begin{aligned} \Vert \Delta _{12} g_j \Vert _{s_1} +\Vert \Delta _{12} \chi _j \Vert _{s_1} \lesssim _{s_1} \Vert \iota _2 - \iota _1 \Vert _{s_1 + \sigma } . \end{aligned}$$

(6.5)

Proof

The operator ${{{\mathcal {K}}}}_{02}(\varphi )$ is defined by ${{{\mathcal {K}}}}_{02}(\varphi ) = d_\bot \nabla _v {{{\mathcal {K}}}}(\varphi , 0, 0) = d_\bot \nabla _v ({{{\mathcal {H}}}}_\varepsilon \circ G_\delta )(\varphi , 0, 0)$. Differentiating the Hamiltonian $({{{\mathcal {H}}}}_\varepsilon \circ G_\delta )(\varphi , \eta , v) = {{{\mathcal {H}}}}_\varepsilon \big (\theta (\varphi ), y_\delta (\varphi ) + L_1(\varphi ) \eta + L_2(\varphi ) w, w(\varphi ) + v \big )$ with respect to v, we get $\nabla _v ({{{\mathcal {H}}}}_\varepsilon \circ G_\delta )(\varphi , \eta , v) =$ $L_2(\varphi )^\top \partial _y {{{\mathcal {H}}}}_\varepsilon (G_\delta (\varphi , \eta , v)) + \nabla _w {{{\mathcal {H}}}}_\varepsilon (G_\delta (\varphi , \eta , v))$, where we used that by (5.16) $L_1(\varphi ) \,{:=}\, [\partial _\phi \theta _0(\varphi )]^{- \top }$ and $L_2(\varphi ) \,{:=}\, - [\partial _\theta {{\tilde{w}}}(\theta (\varphi ))]^{\top } \partial _x^{- 1}$. Since $G_\delta (\varphi , 0, 0) = \breve{\iota }_\delta (\varphi )$, it then follows that

$$\begin{aligned} \partial _x d_\bot \nabla _v ({{{\mathcal {H}}}}_\varepsilon \circ G_\delta )(\varphi , 0, 0) = \partial _x d_\bot \nabla _w {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )) + R(\varphi ) \end{aligned}$$

where $R(\varphi ) \,{:=}\, R_1(\varphi ) + R_2(\varphi ) + R_3(\varphi )$ with $R_1(\varphi ) \,{:=}\, \partial _x L_2(\varphi )^\top \partial _{yy} {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )) L_2(\phi )$ and

$$\begin{aligned} R_2(\varphi )\,{:=}\, \partial _x L_2(\varphi )^\top d_\bot \partial _{y} {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )) , \quad R_3(\varphi ) \,{:=}\, \partial _x \partial _y \nabla _w {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }_\delta (\varphi ))L_2(\varphi ) . \end{aligned}$$

Each of the linear operators $R_1, R_2, R_3$ is of the form (6.3) since it is the composition of linear operators, at least one of which has finite rank. For example, expressing the linear operator $L_2(\phi ) : L^2_\bot ({\mathbb {T}}_1) \rightarrow {\mathbb {R}}^{{\mathbb {S}}_+}$ in terms of the canonical basis ${\underline{e}}_j$, $j \in {{\mathbb {S}}}_+$, $L_2(\phi )[h] = \sum _{j \in {{\mathbb {S}}}_+} \big (h,\, L_2(\phi )^\top [ {{\underline{e}}}_j] \big )_{L^2_x} \, {{\underline{e}}}_j$, $\forall \ h \in L^2_\bot$, one obtains

$$\begin{aligned} R_1(\varphi )[h] = {\mathop \sum }_{j \in {{\mathbb {S}}}_+} \big (h,\, L_2(\varphi )^\top [{{\underline{e}}}_j] \big )_{L^2_x} \, A_1(\varphi ) [{{\underline{e}}}_j], \quad A_1(\varphi ) \,{:=}\, \partial _x L_2(\varphi )^\top \partial _{yy} {{{\mathcal {H}}}}_\varepsilon (\breve{\iota }_\delta (\varphi )), \end{aligned}$$

showing that it has the form (6.3). By similar arguments one concludes the same for $R_2$ and $R_3$. Let us prove that $R_1$ satisfies the estimates (6.4). By the explicit form of $L_2(\varphi )$, Lemma 2.1-(ii) and (5.2), one gets

$$\begin{aligned} \Vert L_2(\varphi )[{\underline{e}}_j] \Vert _s^\mathrm{{Lip}(\gamma )}, \ \Vert L_2(\varphi )^\top [{\underline{e}}_j] \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}, \quad \text {for some } \quad \sigma > 0.\nonumber \\ \end{aligned}$$

(6.6)

Furthermore, since ${{{\mathcal {H}}}}_\varepsilon = {{{\mathcal {N}}}} + {{{\mathcal {P}}}}_\varepsilon$ with ${\mathcal {N}}$ and ${{{\mathcal {P}}}}_\varepsilon$ given by (4.13), one has $\partial _{yy} {{{\mathcal {H}}}}_\varepsilon = \Omega _{{\mathbb {S}}_+}^{kdv} + \partial _{yy} {{{\mathcal {P}}}}_\varepsilon$, $\partial _y \nabla _w {{{\mathcal {H}}}}_\varepsilon = \partial _y \nabla _w {{{\mathcal {P}}}}_\varepsilon$ and $d_\bot \partial _y {{{\mathcal {H}}}}_\varepsilon = d_\bot \partial _y {{{\mathcal {P}}}}_\varepsilon$. Using the estimates in Lemma 4.3 and (5.2), we then infer the bound $\Vert A_1 [{\underline{e}}_j] \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _s \Vert \iota \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}$ for some $\sigma > 0$, implying together with (6.6) the claimed estimate in (6.4). By similar arguments, one obtains the ones for $R_2$ and $R_3$, as well as the estimate (6.5). $\quad \square$

By Lemma 6.1 the linear Hamiltonian operator ${{{\mathcal {L}}}}_\omega$ has the form

$$\begin{aligned} {{{\mathcal {L}}}}_\omega = {{{\mathcal {L}}}}_\omega ^{(0)} - R , \quad {{{\mathcal {L}}}}_\omega ^{(0)} \,{:=}\, \omega \cdot \partial _{\varphi } - \Pi _{\bot } \partial _x ( d_\bot \nabla _w {{\mathcal {H}}}_\varepsilon ) (\breve{\iota }_\delta (\varphi ) ) , \end{aligned}$$

(6.7)

where here and in the sequel, we write $\omega \cdot \partial _\varphi$ instead of $\Pi _\bot \ \omega \cdot \partial _\varphi |_{L^2_\bot }$ in order to simplify notation. The operator ${{{\mathcal {L}}}}_\omega$ is defined for any $\omega \in \Omega$. In a next step we prove in Lemmata 6.2 and 6.3 below that the Hamiltonian operator ${{{\mathcal {L}}}}^{(0)}_\omega$, acting on $L^2_\bot ({\mathbb {T}}_1)$, is the sum of a pseudo-differential operator of order three, a Fourier multiplier with $\varphi -$independent coefficients and a small smoothing remainder.

First note that, since ${{\mathcal {H}}}_\varepsilon = {{\mathcal {H}}}^{kdv} + \varepsilon {{\mathcal {P}}}$ (cf. (4.2)) and $\partial _x d_\bot \nabla _w {{{\mathcal {H}}}}^{kdv} = \partial _x \Omega ^{kdv} + \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}$ (cf. (3.12)), we have

$$\begin{aligned} {{{\mathcal {L}}}}^{(0)}_\omega&= \omega \cdot \partial _\varphi + \partial _x^3 - \Pi _{\bot } Q_{-1}^{kdv} (D; \omega ) - \Pi _{\bot } \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}(\breve{\iota }_\delta ) - \varepsilon \Pi _{\bot } \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}(\breve{\iota }_\delta ) \end{aligned}$$

(6.8)

where we write $\partial _x^3$ instead of $\partial _x^3|_{L^2_\bot }$ and where $Q_{-1}^{kdv} (D; \omega )$ is given by (cf. (3.62))

$$\begin{aligned} Q_{-1}^{kdv} (D; \omega ) \equiv Q_{-1}^{kdv} ( D; \nu (\omega )) = \partial _x \Omega ^{kdv}(D; \nu (\omega ) ) + \partial _x^3 \end{aligned}$$

(6.9)

with $\nu (\omega )$ defined in (4.3).

Lemma 6.2

(Asymptotic expansion of ${{\mathcal {L}}}^{(0)}_\omega$) For any $M \in {\mathbb {N}}$, the Hamiltonian operator ${{\mathcal {L}}}^{(0)}_\omega$, $\omega \in \Omega$, acting on $L^2_\bot ({\mathbb {T}}_1)$, defined in (6.7), admits an expansion of the form

$$\begin{aligned} {{{\mathcal {L}}}}^{(0)}_\omega&\,{:=}\, \omega \cdot \partial _\varphi - \Pi _{\bot } \Big ( a_3^{(0)} \partial _x^3 + 2 (a_3^{(0)})_x \partial _x^2 + a_1^{(0)} \partial _x + \mathrm{Op}( r_0^{(0)}) + Q_{-1}^{kdv} (D ; \omega ) \Big )\nonumber \\&\quad + {{\mathcal {R}}_M^{(0)}}( \breve{\iota }_\delta (\varphi ); \omega ), \end{aligned}$$

(6.10)

where $a_3^{(0)} \,{:=}\, a_3^{(0)} (\varphi , x; \omega )$, $a_1^{(0)} \,{:=}\, a_1^{(0)} (\varphi , x; \omega )$ are real valued functions satisfying for any $s \ge s_0$

$$\begin{aligned} \Vert a_3^{(0)} + 1 \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon ( 1 + \Vert \iota \Vert _{s+ \sigma _M }^\mathrm{{Lip}(\gamma )}) , \quad \Vert a_1^{(0)} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}\nonumber \\ \end{aligned}$$

(6.11)

for some $\sigma _M > 0$ and where the pseudo-differential symbol $r_0^{(0)} \,{:=}\, r_0^{(0)} (\varphi , x, \xi ; \omega )$ has an expansion in homogeneous components

$$\begin{aligned} r_0^{(0)}(\varphi , x, \xi ; \omega ) = \sum _{k=0}^{M} a_{-k}^{(0)}(\varphi , x; \omega ) (\mathrm{i} 2 \pi \xi )^{-k} \chi _0(\xi ) \end{aligned}$$

(6.12)

(with $\chi _0$ defined in (2.18)) where the coefficients $a_{-k}^{(0)} \,{:=}\, a_{-k}^{(0)}(\varphi , x; \omega )$ satisfy

$$\begin{aligned} \sup _{k=0, \ldots , M} \Vert a_{-k}^{(0)} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}, \quad \forall s \geqq s_0. \end{aligned}$$

(6.13)

Furthermore, the remainder is defined by

$$\begin{aligned} {{\mathcal {R}}_M^{(0)}}( \breve{\iota }_\delta (\varphi ); \omega )&\,{:=}\, - {{{\mathcal {R}}}}_M (\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) \nonumber \\&\quad - \varepsilon {{{\mathcal {R}}}}_M (\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}) \end{aligned}$$

(6.14)

where the latter two remainders are given by (3.48) and (3.36) with $\nu (\omega ) = (\omega ^{kdv})^{- 1}(- \omega )$.

Finally, for any $s_1 \geqq s_0$ and any torus embeddings $\breve{\iota }_1, \breve{\iota }_2$ satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M$ it follows that for any $0 \le k \le M +1$,

$$\begin{aligned} \Vert \Delta _{12} a_3^{(0)}\Vert _{s_1} \lesssim _{s_1, M} \varepsilon \Vert \iota _1 - \iota _2 \Vert _{s_1 + \sigma _M}, \quad \Vert \Delta _{12} a_{1 - k}^{(0)}\Vert _{s_1} \lesssim _{s_1, M} \Vert \iota _1 - \iota _2 \Vert _{s_1 + \sigma _M}.\nonumber \\ \end{aligned}$$

(6.15)

Proof

By the definition (6.8) of ${{{\mathcal {L}}}}^{(0)}_\omega$, the expansion (3.48) of $\partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}$, the expansion (3.36) of $\partial _x d_\bot \nabla _w {{{\mathcal {P}}}}$, and the formula for the coefficient of $\partial _x^2$, described in Lemma 2.7, one obtains (6.10) with

$$\begin{aligned} \begin{aligned} a_3^{(0)}(\varphi , x; \omega )&\,{:=}\, - 1 + \varepsilon a_3(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}), \\ a_1^{(0)}(\varphi , x; \omega )&\,{:=}\, a_1(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) + \varepsilon a_1(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}), \\ a_{-k}^{(0)}(\varphi , x; \omega )&\,{:=}\, a_{- k}(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv}) + \varepsilon a_{- k}(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}) , \quad k = 0, \ldots , M, \end{aligned} \end{aligned}$$

and $\nu (\omega ) = (\omega ^{kdv})^{- 1}(- \omega )$. By Lemma 3.7-1, the functions $a_{1 - k}({\mathfrak {x}}; \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv})$, $0 \le k \le M + 1$, satisfy the hypothesis of Lemma 2.25-(ii). In view of (5.10) one then infers that for any $s \geqq s_0$

$$\begin{aligned} \Vert a_{1 - k}(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {R}}}}^{kdv})\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}\end{aligned}$$

for some $\sigma _M > 0$. Similarly, by the first item of Lemma 3.5, the functions $a_{3 - k}(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}})$, $0 \le k \le M + 3$, satisfy the hypothesis of Lemma 2.25-(i), implying that for any $s \geqq s_0$,

$$\begin{aligned} \Vert a_{3 - k}(\breve{\iota }_\delta (\varphi ); \nu (\omega ) ; \partial _x d_\bot \nabla _w {{{\mathcal {P}}}}) \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} 1 + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}\end{aligned}$$

for some $\sigma _M > 0$, proving (6.11), (6.13). The estimates (6.15) follow by similar arguments. $\quad \square$

We remark that in the finitely many steps of our reduction procedure, described in the subsequent sections, the loss of derivatives $\sigma _M = \sigma _M(\tau , {{\mathbb {S}}}_+) > 0$ might have to be increased, but the notation will not be changed.

We finish this section by showing that the operator $Q_{-1}^{kdv} (D; \omega )$, which is a Fourier multiplier with $\varphi -$independent coefficients, admits an expansion as described in the following lemma:

Lemma 6.3

For any $M \in {\mathbb {N}}$,

$$\begin{aligned} Q_{-1}^{kdv} (D; \omega ) = \sum _{k = 1}^M c_{ - k}^{kdv}(\omega ) \partial _x^{- k} + {{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) \end{aligned}$$

(6.16)

where for any $1 \le k \le M$, the function $\Omega \rightarrow {\mathbb {R}}, \, \omega \mapsto c_{ - k}^{kdv}(\omega )$ is Lipschitz and where ${{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ): L^2_\bot ({\mathbb {T}}_1) \rightarrow L^2_\bot ({\mathbb {T}}_1)$ is a Lipschitz family of diagonal operators of order $- M - 1$. Furthermore, for any $n_1, n_2 \in {\mathbb {N}}$, $n_1 + n_2 \leqq M + 1$, the operator $\langle D \rangle ^{n_1} {{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying ${{\mathfrak {M}}}_{\langle D \rangle ^{n_1} {{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) \langle D \rangle ^{n_2}}(s) \leqq C(s, M)$ for any $s \geqq s_0$ and $C(s, M) > 0$.

Proof

By Lemma 3.8, $Q_{-1}^{kdv} (D; \omega ) = \Omega _{-1}^{kdv} (D; \omega ) + {{{\mathcal {R}}}}_M (D; \omega ; Q_{-1}^{kdv})$ where

$$\begin{aligned} \begin{aligned}&\Omega _{-1}^{kdv} (D; \omega ) h = \sum _{k= 1}^{ M} \sum _{\xi \in {\mathbb {S}}^\bot } a_{-k} (\omega ; \Omega _{-1}^{kdv}) \chi _0(\xi ) ( \mathrm{i} 2 \pi \xi )^{-k} h_\xi (\varphi ) e^{\mathrm{i} 2 \pi x \xi }, \\&{{{\mathcal {R}}}}_M (D; \omega ; Q_{-1}^{kdv})[h] = \sum _{\xi \in {\mathbb {S}}^\bot } \frac{{{{\mathcal {R}}}}_{M}^{\omega _\xi }(\omega )}{(2\pi \xi )^{M+1}} h_\xi (\varphi ) e^{\mathrm{i} 2 \pi x \xi }. \end{aligned} \end{aligned}$$

Hence (6.16) holds with $c_{- k}^{kdv}(\omega ) \,{:=}\, a_{-k} (\omega ; \Omega _{-1}^{kdv})$, $k = 1,\ldots , M$ and ${{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) = {{{\mathcal {R}}}}_M (D; \omega ; Q_{-1}^{kdv})$. For any integer $n_1, n_2$ such that $n_1 + n_2 \leqq M + 1$, one has that

$$\begin{aligned} \langle D \rangle ^{n_1} {{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) \langle D \rangle ^{n_2} h = \sum _{\xi \in {\mathbb {S}}^\bot } \frac{{{{\mathcal {R}}}}_{M}^{\omega _\xi }(\omega ) \langle \xi \rangle ^{n_1 + n_2}}{(2\pi \xi )^{M+1}} h_\xi (\varphi ) e^{\mathrm{i} 2 \pi x \xi }, \end{aligned}$$

where, by (3.65), $\Big | \frac{{{{\mathcal {R}}}}_{M}^{\omega _\xi }(\omega ) \langle \xi \rangle ^{n_1 + n_2}}{(2\pi \xi )^{M+1}} \Big | \leqq C_M$. Therefore $\Vert \langle D \rangle ^{n_1} {{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) \langle D \rangle ^{n_2} h \Vert _s \lesssim _M \Vert h \Vert _s$ for any $s \geqq 0$. The correpsonding Lipschitz estimate is proved in a similar way. $\quad \square$

6.2 Quasi-periodic Reparametrization of Time

The goal of this section is to conjugate the operator ${{\mathcal {L}}}_\omega = {{{\mathcal {L}}}}_\omega ^{(0)} - R$ in (6.7) to the operator ${{\mathcal {L}}}_\omega ^{(1)}$, defined in (6.22), which admits an expansion of the form (6.23) with the property that its highest order coefficient $a_3^{(1)}$ satisfies (6.25). This property will allow us in Section 6.3 to conjugate ${{\mathcal {L}}}_\omega ^{(1)}$ to an operator with constant highest order coefficient (cf. (6.42)).

The operator $\Phi ^{(1)}$, by which ${{\mathcal {L}}}_\omega$ is conjugated, is induced by the change of variable $\varphi$, defined by the quasi-periodic reparametrization of time,

$$\begin{aligned} \vartheta = \varphi + \alpha ^{(1)} (\varphi ) \omega \quad \text {or equivalently} \quad \varphi = \vartheta + \breve{\alpha }^{(1)} (\vartheta ) \omega \end{aligned}$$

where $\alpha ^{(1)}: {\mathbb {T}}^{{{\mathbb {S}}}_+} \rightarrow {\mathbb {R}}$, is a small, real valued function chosen below (cf. (6.19)). In more detail, $\Phi ^{(1)}$ and its inverse $(\Phi ^{(1)})^{-1}$ are given by

$$\begin{aligned} (\Phi ^{(1)} h) (\varphi , x) \,{:=}\, h( \varphi + \alpha ^{(1)} (\varphi ) \omega , x ) , \quad ((\Phi ^{(1)})^{-1} h) (\vartheta , x) \,{:=}\, h( \vartheta + \breve{\alpha }^{(1)} (\vartheta ) \omega , x ) .\nonumber \\ \end{aligned}$$

(6.17)

First recall that the coefficient $a_3^{(0)}$ in the expansion (6.10) satisfies $a_3^{(0)} = -1 + O(\varepsilon )$ (cf. (6.11)). Hence the the cube root $(a_3^{(0)}(\varphi ,x))^{\frac{1}{3}}$ is smooth.

Lemma 6.4

Let $m_3$ be the constant

$$\begin{aligned} m_3(\omega ) \,{:=}\, \frac{1}{(2 \pi )^{|{{\mathbb {S}}}_+|}} \int _{{\mathbb {T}}^{{{\mathbb {S}}}_+}} \Big ( \int _{{\mathbb {T}}_1} \frac{\hbox {d}x}{ (a_3^{(0)}(\vartheta ,x; \omega ))^{\frac{1}{3}}} \Big )^{-3}\, \mathrm{d}\vartheta , \quad \forall \omega \in \Omega , \end{aligned}$$

(6.18)

and define, for $\omega \in \mathtt {DC}(\gamma ,\tau )$, the function

$$\begin{aligned} \breve{\alpha }^{(1)} (\vartheta ; \omega ) \,{:=}\, (\omega \cdot \partial _{\varphi } )^{-1} \Big [ \frac{1}{m_3} \Big ( \int _{{\mathbb {T}}_1} \frac{\hbox {d}x}{ (a_3^{(0)}(\vartheta ,x; \omega ))^{\frac{1}{3}}} \Big )^{-3} - 1 \Big ] . \end{aligned}$$

(6.19)

Then for any $M \in {\mathbb {N}}$, there exists a constant $\sigma _M > 0$ so that the following holds:

• (i) The constant $m_3$ satisfies

  $$\begin{aligned} |m_3 + 1|^\mathrm{{Lip}(\gamma )}\lesssim _M \varepsilon \, \end{aligned}$$

  (6.20)

  and for any $s \geqq s_0$, $\alpha ^{(1)}, \breve{\alpha }^{(1)}$ satisfy

  $$\begin{aligned} \Vert \alpha ^{(1)} \Vert _s^\mathrm{{Lip}(\gamma )}, \Vert \breve{\alpha }^{(1) }\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon \gamma ^{- 1}(1 + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}). \end{aligned}$$

  (6.21)

• (ii) The Hamiltonian operator

  $$\begin{aligned} {{\mathcal {L}}}_\omega ^{(1)}:= & {} \frac{1}{ \rho } \Phi ^{(1)} {{\mathcal {L}}}_\omega \, (\Phi ^{(1)})^{- 1} ,\nonumber \\ \rho (\vartheta ):= & {} , \Phi ^{(1)} ( 1 + \omega \cdot \partial _{\vartheta } \breve{\alpha }^{(1)} ) = 1 + \Phi ^{(1)} ( \omega \cdot \partial _{\vartheta } \breve{\alpha }^{(1)} ) , \end{aligned}$$

  (6.22)

  admits an expansion of the form

  $$\begin{aligned} {{\mathcal {L}}}_\omega ^{(1)} = \omega \cdot \partial _\vartheta - \Big ( a_3^{(1)} \partial _x^3 + 2 (a_3^{(1)})_x \partial _x^2 + a_1^{(1)} \partial _x + \mathrm{Op}( r_0^{(1)}) + Q_{-1}^{kdv} (D ;\omega )\Big ) + {{\mathcal {R}}_M^{(1)}}\nonumber \\ \end{aligned}$$

  (6.23)

  where the coefficients $a_3^{(1)} = a_3^{(1)}(\vartheta , x; \omega )$ and $a_1^{(1)} = a_1^{(1)} (\vartheta , x; \omega )$ are real valued and satisfy

  $$\begin{aligned} \Vert a_3^{(1)} +1 \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon (1 + \Vert \iota \Vert _{s + \sigma _M}) , \quad \Vert a_1^{(1)} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}, \quad \forall s \geqq s_0 ,\nonumber \\ \end{aligned}$$

  (6.24)

  and

  $$\begin{aligned} \int _{{\mathbb {T}}_1} \frac{\hbox {d}x}{ (a_3^{(1)}( \vartheta , x; \omega ))^{\frac{1}{3}} } = m_3^{- \frac{1}{3}} , \quad \forall \vartheta \in {\mathbb {T}}^{{{\mathbb {S}}}_+} . \end{aligned}$$

  (6.25)

  The function $r_0^{(1)} \equiv r_0^{(1)} (\vartheta , x, \xi ; \omega )$ in (6.23) is a pseudo-differential symbol in the symbol class $S^0$ (cf. (2.8)) and admits an expansion of the form

  $$\begin{aligned} r_0^{(1)} (\vartheta , x, \xi ; \omega ) = \sum _{k = 0}^M a_{- k}^{(1)}(\vartheta , x; \omega ) (\mathrm{i} 2 \pi \xi )^{ - k} \chi _0(\xi ), \end{aligned}$$

  (6.26)

  where $\chi _0$ is defined in (2.18) and where for any $0 \le k \le M$, $s \geqq s_0$,

  $$\begin{aligned} \Vert a_{- k}^{(1)} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (6.27)

  Furthermore, the function $\rho$ appearing in (6.22) satisfies

  $$\begin{aligned} \Vert \rho - 1\Vert _s^\mathrm{{Lip}(\gamma )}, \ \Vert \rho ^{- 1} - 1\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (6.28)

  Let $s_1 \geqq s_0$ and let $\breve{\iota }_1$, $\breve{\iota }_2$ be torus embeddings, satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M$. Then

  $$\begin{aligned} \begin{aligned}&| \Delta _{12} m_3 |, \quad \Vert \Delta _{12} \alpha ^{(1)} \Vert _{s_1}, \ \Vert \Delta _{12}\breve{\alpha }^{(1)} \Vert _{s_1}, \ \Vert \Delta _{12} a_1^{(1)} \Vert _{ s_1}, \\&\quad \ \Vert \Delta _{12} \rho ^{\pm 1}\Vert _{s_1} \lesssim _{s_1, M} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M} , \\&\Vert \Delta _{12} a_{- k}^{(1)} \Vert _{s_1} \, \lesssim _{s_1, M} \, \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M}, \quad \forall k= 0, \ldots , M. \end{aligned} \end{aligned}$$

  (6.29)

• (iii) Let $S > {\mathfrak {s}}_M$ where ${\mathfrak {s}}_M$ is defined in (2.55). Then the maps $(\Phi ^{(1)})^{\pm 1}$ are $\mathrm{{Lip}(\gamma )}$-1-tame operators with tame constants satisfying

  $$\begin{aligned} {\mathfrak {M}}_{(\Phi ^{(1)})^{\pm 1}}(s) \lesssim _{S, M} 1 + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}, \quad \forall s_0 + 1 \leqq s \leqq S. \end{aligned}$$

  (6.30)

  For any given $\lambda _0 \in {\mathbb {N}}$, there exists a constant $\sigma _M(\lambda _0) > 0$ so that for any $m \in {{\mathbb {S}}}_+$, $\lambda , n_1, n_2 \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$ and $n_1 + n_2 + \lambda _0 \leqq M + 1$, the operator $\partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1}{{\mathcal {R}}_M^{(1)}} \langle D \rangle ^{n_2}$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

  $$\begin{aligned} {{\mathfrak {M}}}_{\partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1}{{\mathcal {R}}_M^{(1)}} \langle D \rangle ^{n_2}} (s) \, \lesssim _{S, M} \, \varepsilon + \Vert \iota \Vert _{s+\sigma _M(\lambda _0)}^\mathrm{{Lip}(\gamma )}, \quad \forall {\mathfrak {s}}_M\leqq s \leqq S . \end{aligned}$$

  (6.31)

  If in addition, $s_1 \geqq {\mathfrak {s}}_M$ and $\breve{\iota }_1, \breve{\iota }_2$ are torus embeddings, satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M(\lambda _0)$, then

  $$\begin{aligned} \Vert \partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1} \Delta _{12} {{\mathcal {R}}_M^{(1)}} \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})} \lesssim _{s_1, M, \lambda _0} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M(\lambda _0)}. \end{aligned}$$

  (6.32)

Proof

Writing $\Pi _\bot$ as $\mathrm{Id} + (\Pi _\bot - \mathrm{Id} )$, the operator ${{{\mathcal {L}}}}_\omega ^{(0)}$ (cf. (6.10) ) becomes

$$\begin{aligned}&{{{\mathcal {L}}}}_\omega ^{(0)} = \omega \cdot \partial _\varphi - \Big ( a_3^{(0)} \partial _x^3 + 2 (a_3^{(0)})_x \partial _x^2 + a_1^{(0)} \partial _x + \mathrm{Op}( r_0^{(0)}) + Q_{-1}^{kdv} (D ; \omega ) \Big ) \\&\quad + {{{\mathcal {R}}}}_M^{(I)}( \breve{\iota }_\delta (\varphi ); \omega ) + {{\mathcal {R}}}_M^{(0)}( \breve{\iota }_\delta (\varphi ); \omega ) , \end{aligned}$$

where ${{{\mathcal {R}}}}_M^{(I)}( \breve{\iota }_\delta (\varphi ); \omega ) \,{:=}\, (\mathrm{Id} - \Pi _\bot ) \big ( a_3^{(0)} \partial _x^3 + 2 (a_3^{(0)})_x \partial _x^2 + a_1^{(0)} \partial _x + \mathrm{Op}( r_0^{(0)}) \big )$. Since $(\mathrm{Id} - \Pi _\bot )\partial _x^3 h = 0$ for any $h \in H^s_\bot$, the operator ${{{\mathcal {R}}}}_M^{(I)} = {{{\mathcal {R}}}}_M^{(I)}( \breve{\iota }_\delta (\varphi ); \omega )$ can be written as

$$\begin{aligned} {{{\mathcal {R}}}}_M^{(I)} = (\mathrm{Id} - \Pi _\bot )\Big ( (a_3^{(0)} + 1) \partial _x^3 + 2 (a_3^{(0)})_x \partial _x^2 + a_1^{(0)} \partial _x + \mathrm{Op}( r_0^{(0)}) \Big ) , \end{aligned}$$

(6.33)

and is a finite rank operator of the form (6.3) with functions $g_j, \chi _j \in H^s_\bot$ satisfying (6.4) (use (6.11), (6.13)).

The estimate (6.30) follows by Lemma 2.1-(iii) and (6.21). Note that

$$\begin{aligned} \Phi ^{(1)} \circ \omega \cdot \partial _\varphi \circ (\Phi ^{(1)})^{-1} = \rho (\vartheta ) \omega \cdot \partial _\vartheta , \quad \rho \,{:=}\, \Phi ^{(1)} ( 1 + \omega \cdot \partial _{\varphi } \breve{\alpha }^{(1)} ) , \end{aligned}$$

and that any Fourier multiplier g(D) is left unchanged under conjugation, that is, $\Phi ^{(1)} g(D) (\Phi ^{(1)})^{-1} = g(D)$. Using (6.7) and (6.10), we obtain (6.23) where

$$\begin{aligned} a_3^{(1)} \,{:=}\, \Phi ^{(1)} \Big ( \frac{a_3^{(0)}}{1 + \omega \cdot \partial _\varphi \breve{\alpha }^{(1)}} \Big ) , \end{aligned}$$

(6.34)

$a_1^{(1)} \,{:=}\, \frac{1}{\rho } \Phi ^{(1)} (a_1^{(0)})$, $r_0^{(1)}$ is of the form (6.26) with $a_{- k}^{(1)} \,{:=}\, \frac{1}{\rho } \Phi ^{(1)}(a_{-k}^{(0)})$, and the remainder ${{{\mathcal {R}}}}_M^{(1)}$ is given by

$$\begin{aligned} {{{\mathcal {R}}}}_M^{(1)}&= \frac{1}{\rho } \Phi ^{(1)} {{{\mathcal {R}}}}_M^{(I)} (\Phi ^{(1)})^{- 1} + \frac{1}{\rho } \Phi ^{(1)} {{{\mathcal {R}}}}_M^{(0)}(\breve{\iota }_\delta (\varphi )) (\Phi ^{(1)})^{- 1} \nonumber \\&\quad - \frac{1}{\rho } \Phi ^{(1)} R (\varphi ) (\Phi ^{(1)})^{- 1} . \end{aligned}$$

(6.35)

We choose $\breve{\alpha }^{(1)}$ such that (6.25) holds, yielding (6.18), (6.19). We now verify the estimates, stated in items (i) and (ii). Recall that we assume throughout that (6.1) holds. The estimates (6.20)–(6.21) follow by (6.18), (6.19), (6.11), and by using Lemma 2.1-(iii), Lemma 2.3. The estimate (6.28) on $\rho$ follows by the definition (6.22), (6.19), and by applying Lemma 2.1-(iii), Lemma 2.3. Hence, by Lemma 2.1 and the estimates (6.11), (6.13), and (6.28), we deduce (6.27). The estimates (6.29) are obtained by similar arguments. Let us now prove item (iii). The estimate (6.30) follows from Lemma 2.1-(iii). Since $(\Phi ^{(1)})^{\pm 1}$ commutes with every Fourier multiplier, we get

$$\begin{aligned} \frac{1}{\rho } \langle D \rangle ^{n_1} \Phi ^{(1)} {{{\mathcal {R}}}}_M^{(0)}(\breve{\iota }_\delta (\varphi )) (\Phi ^{(1)})^{- 1} \langle D \rangle ^{n_2} = \frac{1}{\rho } \langle D \rangle ^{n_1} {{{\mathcal {R}}}}_M^{(0)} ( \breve{\iota }_{\delta ,\alpha } ( \varphi )) \langle D \rangle ^{n_2} \end{aligned}$$

(6.36)

where $\breve{\iota }_{\delta ,\alpha } ( \varphi ) \,{:=}\, \breve{\iota }_\delta ( \varphi + \alpha ^{(1)} (\varphi ) \omega )$. By Lemma 2.1, (5.10), and (6.21) one has $\Vert \iota _{\delta ,\alpha } \Vert ^\mathrm{{Lip}(\gamma )}_s \lesssim _s \Vert \iota \Vert ^\mathrm{{Lip}(\gamma )}_{s+\sigma _M}$. Moreover, by (6.3), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{\rho } \Phi ^{(1)} R (\varphi ) (\Phi ^{(1)})^{- 1} h = {\mathop \sum }_{j \in {{\mathbb {S}}}_+} \big ( h ,\, (\Phi ^{(1)} g_j) \big )_{L^2_x} \frac{1}{\rho } (\Phi ^{(1)} \chi _j) , \forall h \in L^2_\bot , \end{aligned} \end{aligned}$$

(6.37)

and by (6.33), the conjugated operator $\frac{1}{\rho } \Phi ^{(1)} {{{\mathcal {R}}}}_M^{(I)} (\Phi ^{(1)})^{- 1} h$ has the same form. The estimates (6.31) are then obtained by using (6.36), (6.14), and Lemmata 3.5, 3.7, 2.26 to estimate the first term on the right hand side of (6.35) and by (6.37), (6.30), (6.4) and Lemma 2.24, to estimate the second and third term in (6.35). The estimates (6.32) are proved by similar arguments. $\quad \square$

6.3 Elimination of the $(\varphi , x)$-Dependence of the Highest Order Coefficient

The goal of this section is to remove the $(\varphi , x)$-dependence of the coefficient $a_3^{(1)}(\varphi , x)$ of the Hamiltonian operator ${{\mathcal {L}}}^{(1)}_\omega$, given by (6.22)–(6.23), where we rename $\vartheta$ to $\varphi$. Actually this step will at the same time also remove the coefficient of $\partial _x^2$ thanks to the Hamiltonian nature of the subprincipal operator of order 2, described in Lemma 2.7. We achieve these goals by conjugating the operator ${{\mathcal {L}}}^{(1)}_\omega$ by the flow $\Phi ^{(2)} (\tau , \varphi )$, acting on $L^2_\bot ({\mathbb {T}}_1)$, defined by the transport equation

$$\begin{aligned} \partial _\tau \Phi ^{(2)} (\tau , \varphi ) = \Pi _{\bot } \partial _x \big ( b^{(2)} (\tau , \varphi , x) \Phi ^{(2)} (\tau , \varphi ) \big ) , \quad \Phi ^{(2)} (0, \varphi ) = \mathrm{Id}_{\bot } , \end{aligned}$$

(6.38)

where

$$\begin{aligned} b^{(2)} \equiv b^{(2)} (\tau , \varphi , x) \,{:=}\, \frac{ \beta ^{(2)} (\varphi , x) }{ 1 + \tau \beta ^{(2)}_x (\varphi , x) } , \end{aligned}$$

and $\beta ^{(2)} (\varphi , x)$ is a small, real valued periodic function chosen in (6.40) below. The flow $\Phi ^{(2)} (\tau , \varphi )$ is well defined for $0 \le \tau \le 1$ and satisfies the tame estimates provided in Lemma 2.28. Since the vector field $\Pi _{\bot } \partial _x \big ( b^{(2)} h \big )$, $h \in H^s_\bot ({\mathbb {T}}_1)$, is Hamiltonian (it is generated by the Hamiltonian $\frac{1}{2} \int _{{\mathbb {T}}_1} b^{(2)} h^2 \, \hbox {d}x$), each $\Phi ^{(2)} (\tau , \varphi )$, $0 \le \tau \le 1$, $\varphi \in {\mathbb {T}}^{{\mathbb {S}}_+}$ is a symplectic linear isomorphism of $H^s_\bot ({\mathbb {T}}_1)$. Therefore the time one conjugated operator,

$$\begin{aligned} {{\mathcal {L}}}_\omega ^{(2)} \,{:=}\, \Phi ^{(2)} {{\mathcal {L}}}_\omega ^{(1)} \big (\Phi ^{(2)} \big )^{-1} , \quad \Phi ^{(2)} \,{:=}\, \Phi ^{(2)}(1, \varphi ) , \end{aligned}$$

(6.39)

is a Hamiltonian operator acting on $H^s_\bot ({\mathbb {T}}_1)$.

Given the $(\tau , \varphi )$-dependent family of diffeomorphisms of the torus ${\mathbb {T}}_1$, $x \mapsto y = x + \tau \beta ^{(2)} (\varphi , x)$, we denote the family of its inverses by $y \mapsto x = y + \breve{\beta }^{(2)} (\tau , \varphi , y)$.

Lemma 6.5

(Reduction to constant coefficients of the third order term) Let $\breve{\beta }^{(2)} ( \varphi , y; \omega ) \equiv \breve{\beta }^{(2)}(1, \varphi , y; \omega )$ be the real valued, periodic function

$$\begin{aligned} \breve{\beta }^{(2)}( \varphi , y; \omega ) \,{:=}\, \partial _y^{-1} \Big ( \frac{ m_3^{1/3} }{ ( a_3^{(1)} (\varphi , y; \omega ) )^{1/3} } - 1 \Big ) \end{aligned}$$

(6.40)

(which is well defined by (6.25)) and let $M \in {\mathbb {N}}$. Then there exists $\sigma _M > 0$ so that the following holds:

• (i) For any $s \geqq s_0$

  $$\begin{aligned} \Vert \beta ^{(2)}\Vert _s^\mathrm{{Lip}(\gamma )}, \ \Vert \breve{\beta }^{(2)}\Vert _s^\mathrm{{Lip}(\gamma )}\, \lesssim _{s, M} \, \varepsilon \big ( 1 + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}\big ) . \end{aligned}$$

  (6.41)

• (ii) The Hamiltonian operator ${{\mathcal {L}}}_\omega ^{(2)}$ in (6.39) admits an expansion of the form

  $$\begin{aligned} {{\mathcal {L}}}_\omega ^{(2)} = \omega \cdot \partial _\varphi - \big ( m_3 \partial _x^3 + a_1^{(2)} \partial _x + \mathrm{Op}( r_0^{(2)}) + Q_{-1}^{kdv} (D; \omega ) \big ) + {{\mathcal {R}}_M^{(2)}}\nonumber \\ \end{aligned}$$

  (6.42)

  where $a_1^{(2)} \,{:=}\, a_1^{(2)} (\varphi , x; \omega )$ is a real valued, periodic function, satisfying

  $$\begin{aligned} \Vert a_1^{(2)} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (6.43)

  The pseudo-differential symbol $r_0^{(2)} \equiv r_0^{(2)} (\varphi , x, \xi ; \omega )$ is in $S^0$ and satisfies, for any $s \geqq s_0$, the estimate

  $$\begin{aligned} | \mathrm{Op}( r_0^{(2)}) |_{0, s, 0}^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}.\nonumber \\ \end{aligned}$$

  (6.44)

  Let $s_1 \geqq s_0$ and let $\breve{\iota }_1, \breve{\iota }_2$ be torus embeddings satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M$. Then

  $$\begin{aligned}&\Vert \Delta _{12} \beta ^{(2)}\Vert _{s_1}, \ \Vert \Delta _{12} \breve{\beta }^{(2)}\Vert _{s_1}, \ \Vert \Delta _{12} a_1^{(2)} \Vert _{s_1}, \ |\Delta _{12} \mathrm{Op}( r_0^{(2)}) |_{0, s_1, 0} \nonumber \\&\quad \quad \, \lesssim _{s_1, M} \, \Vert \iota _1 - \iota _2 \Vert _{s_1 + \sigma _M}. \end{aligned}$$

  (6.45)

• (iii) Let $S> {\mathfrak {s}}_M$. Then the symplectic maps $(\Phi ^{(2)})^{\pm 1}$ are $\mathrm{{Lip}(\gamma )}$-1 tame operators with tame constants satisfying

  $$\begin{aligned} {\mathfrak {M}}_{(\Phi ^{(2)})^{\pm 1}}(s) \lesssim _{S, M} 1 + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}, \quad \forall s_0 + 1 \leqq s \leqq S. \end{aligned}$$

  (6.46)

  Let $\lambda _0 \in {\mathbb {N}}$. Then there exists a constant $\sigma _M(\lambda _0) > 0$ so that for any $\lambda , n_1, n_2 \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$ and $n_1 + n_2 + \lambda _0 \leqq M - 1$, the operator $\partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1}{{\mathcal {R}}_M^{(2)}} \langle D \rangle ^{n_2}$, $m \in {{\mathbb {S}}}_+$, is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

  $$\begin{aligned} {{\mathfrak {M}}}_{\partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1} {{\mathcal {R}}_M^{(2)}} \langle D \rangle ^{n_2} } (s) \, \lesssim _{S, M, \lambda _0} \, \varepsilon + \Vert \iota \Vert _{s+\sigma _M(\lambda _0)}^\mathrm{{Lip}(\gamma )}, \quad \forall {\mathfrak {s}}_M\leqq s \leqq S . \end{aligned}$$

  (6.47)

  Let $s_1 \geqq {\mathfrak {s}}_M$ and $\breve{\iota }_1$, $\breve{\iota }_2$ be torus embeddings satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M(\lambda _0)$. Then

  $$\begin{aligned} \Vert \partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1} \Delta _{12}{{\mathcal {R}}_M^{(2)}} \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})} \, \lesssim _{s_1, M, \lambda _0 } \, \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M(\lambda _0)}. \end{aligned}$$

  (6.48)

Proof

We use the Egorov type results proved in Section 2.5. According to (6.23), (6.26), the conjugated operator is given by

$$\begin{aligned}&{{{\mathcal {L}}}}_\omega ^{(2)} = \Phi ^{(2)} {{{\mathcal {L}}}}_\omega ^{(1)} (\Phi ^{(2)})^{- 1} \\&= \omega \cdot \partial _\varphi - \Phi ^{(2)} a_3^{(1)} \partial _x^3 (\Phi ^{(2)})^{- 1} - 2 \Phi ^{(2)} (a_3^{(1)})_x \partial _x^2 (\Phi ^{(2)})^{- 1} - \Phi ^{(2)} a_1^{(1)} \partial _x (\Phi ^{(2)})^{- 1} \nonumber \\&\ \ - \sum _{k=0}^{M} \Phi ^{(2)} a_{-k}^{(1)} \partial _x^{-k} (\Phi ^{(2)})^{- 1} - \Phi ^{(2)} Q_{- 1}^{kdv}(D ; \omega ) (\Phi ^{(2)})^{- 1}\nonumber \\&\quad + \Phi ^{(2)} {{{\mathcal {R}}}}_M^{(1)} (\Phi ^{(2)})^{- 1} + \Phi ^{(2)} \big ( \omega \cdot \partial _{\varphi } \, \, ( \Phi ^{(2)} )^{-1} \big ). \nonumber \end{aligned}$$

(6.49)

By (6.40), (6.20), (6.24) and Lemmata 2.1, 2.3, the estimate (6.41) follows. Using (6.1) with $\mu _0 > 0$ large enough, the estimate (6.41) implies that $\Vert \beta ^{(2)}\Vert _{s_0 + \sigma _M(\lambda _0)}^\mathrm{{Lip}(\gamma )}\lesssim _{M, \lambda _0} \varepsilon \gamma ^{- 2}$, where the constant $\sigma _M(\lambda _0)$ is the constant appearing in the smallness conditions (2.79), (2.107), (2.110). Now we apply Proposition 2.31 to expand the terms

$$\begin{aligned}&\Phi ^{(2)} a_3^{(1)} \partial _x^3 (\Phi ^{(2)})^{- 1} , \quad 2 \Phi ^{(2)} (a_3^{(1)})_x \partial _x^2 (\Phi ^{(2)})^{- 1} , \\&\quad \quad \Phi ^{(2)} a_{1-k}^{(1)} \partial _x^{1-k} (\Phi ^{(2)})^{- 1} , \ \ 0 \le k \le M+1, \end{aligned}$$

Lemma 2.35 to expand the term $\Phi ^{(2)} Q_{- 1}^{kdv}(D; \omega ) (\Phi ^{(2)})^{- 1}$, and Proposition 2.34 to expand $\Phi ^{(2)} \big ( \omega \cdot \partial _{\varphi } \, \, ( \Phi ^{(2)} )^{-1} \big )$. Using also the bounds (6.11), (6.13), (6.41) one deduces (6.43), (6.44). By the choice of $\breve{\beta }^{(2)}$ in (6.40) and Proposition 2.31, the coefficient of the highest order term of $\Phi ^{(2)} a_3^{(1)} \partial _x^3 (\Phi ^{(2)})^{- 1}$ (and hence of ${{{\mathcal {L}}}}_\omega ^{(2)}$) is given by

$$\begin{aligned} \big ( [1 + \breve{\beta }^{(2)}_y (\varphi , y)]^3 a_3^{(1)}(\varphi , y) \big )|_{y = x + \beta ^{(2)}(\varphi , x)} = m_3 \end{aligned}$$

which is constant in $(\varphi , x)$ by (6.25). Since $\Phi ^{(2)}$ is symplectic, the operator ${{{\mathcal {L}}}}_\omega ^{(2)}$ is Hamiltonian and hence by Lemma 2.7, the second order term equals $2 (m_3)_x \partial _{x}^2$, which vanishes since $m_3$ is constant. The remainder $\Phi ^{(2)} {{{\mathcal {R}}}}_M^{(1)} (\Phi ^{(2)})^{- 1}$ can be estimated by arguing as at the end of the proof of Proposition 2.31 (estimate of ${{{\mathcal {R}}}}_N(\tau , \varphi )$), using Lemma 2.28 to estimate $\Phi ^{(2)}$, $(\Phi ^{(2)})^{- 1}$, the estimate (6.31) for ${{{\mathcal {R}}}}_M^{(1)}$, the estimate (6.41) of $\beta ^{(2)}$, $\breve{\beta }^{(2)}$, and (6.1) with $\mu _0$ large enough. The estimates (6.46) follow from (2.73) and (6.41). The bounds (6.45), (6.48) are derived by similar arguments. $\quad \square$

6.4 Elimination of the x-Dependence of the First Order Coefficient

The goal of this section is to remove the x-dependence of the coefficient $a_1^{(2)} (\varphi ,x)$ of the Hamiltonian operator ${{\mathcal {L}}}^{(2)}_\omega$ in (6.39), (6.42). To this end, we conjugate the operator ${{\mathcal {L}}}^{(2)}_\omega$ by the variable transformation induced by the flow $\Phi ^{(3)}(\tau , \varphi )$, acting on $L^2_\bot ({\mathbb {T}}_1 )$, defined by

$$\begin{aligned} \partial _\tau \Phi ^{(3)} (\tau , \varphi ) = \Pi _{\bot } \big ( b^{(3)} (\varphi , x) \partial _{x}^{-1} \Phi ^{(3)} (\tau , \varphi ) \big ) , \quad \Phi ^{(3)} (0) = \mathrm{Id}_{\bot } , \end{aligned}$$

(6.50)

where $b^{(3)} (\varphi , x)$ is a small, real valued, periodic function, chosen in (6.52) below. Since the vector field $\Pi _{\bot } \big ( b^{(3)} \partial _x^{-1} h \big )$, $h \in H^s_\bot ({\mathbb {T}}_1)$, is Hamiltonian (it is generated by the Hamiltonian $\frac{1}{2} \int _{{\mathbb {T}}_1} b^{(3)} (\partial _x^{-1} h)^2 \, \hbox {d}x$), each $\Phi ^{(3)} (\tau , \varphi )$ is a symplectic linear isomorphism of $H^s_\bot$ for any $0 \le \tau \le 1$ and $\varphi \in {\mathbb {T}}^{{{\mathbb {S}}}_+}$, and the time one conjugated operator,

$$\begin{aligned} {{\mathcal {L}}}_\omega ^{(3)} \,{:=}\, \Phi ^{(3)} {{\mathcal {L}}}_\omega ^{(2)} \big (\Phi ^{(3)} \big )^{-1} , \quad \Phi ^{(3)} \,{:=}\, \Phi ^{(3)}(1) , \end{aligned}$$

(6.51)

is Hamiltonian.

Lemma 6.6

Let $b^{(3)} (\varphi , x; \omega )$ be the real valued periodic function

$$\begin{aligned}&b^{(3)}(\varphi , x; \omega ) \,{:=}\, \frac{1}{3 m_3} \partial _x^{-1} \Big ( a_1^{(2)}(\varphi , x; \omega ) - \langle a_1^{(2)} \rangle _x (\varphi ; \omega ) \Big ) , \nonumber \\&\quad \quad \langle a_1^{(2)} \rangle _x (\varphi ; \omega ) \,{:=}\, \int _{{\mathbb {T}}_1} a_1^{(2)} (\varphi , x; \omega )\,d x \end{aligned}$$

(6.52)

and let $M \in {\mathbb {N}}$. Then there exists $\sigma _M > 0$ with the following properties:

• (i) For any $s \geqq s_0$,

  $$\begin{aligned} \Vert b^{(3)}\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}\end{aligned}$$

  (6.53)

  and the symplectic maps $(\Phi ^{(3)})^{\pm 1}$ are $\mathrm{{Lip}(\gamma )}$-tame and with tame constants satisfying

  $$\begin{aligned} {\mathfrak {M}}_{(\Phi ^{(3)})^{\pm 1}}(s) \lesssim _{s, M} 1 + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (6.54)

• (ii) The Hamiltonian operator in (6.51) admits an expansion of the form

  $$\begin{aligned} {{\mathcal {L}}}_\omega ^{(3)} = \omega \cdot \partial _\varphi - \big ( m_3 \partial _x^3 + a_1^{(3)}(\varphi ) \partial _x + \mathrm{Op}( r_0^{(3)}) + Q_{-1}^{kdv} (D; \omega ) \big ) + {{\mathcal {R}}_M^{(3)}} \end{aligned}$$

  (6.55)

  where the real valued, periodic function $a_1^{(3)} (\varphi ; \omega ) \,{:=}\, \langle a_1^{(2)} \rangle _x(\varphi ; \omega )$ satisfies

  $$\begin{aligned} \Vert a_1^{(3)} \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

  (6.56)

  and $r_0^{(3)} \,{:=}\, r_0^{(3)} (\varphi , x, \xi ; \omega )$ is a pseudo-differential symbol in $S^0$ satisfying for any $s \geqq s_0$,

  $$\begin{aligned} |\mathrm{Op}(r_0^{(3)})|_{0, s, 0}^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (6.57)

  Let $s_1 \geqq s_0$ and let $\breve{\iota }_1, \breve{\iota }_2$ be torus embeddings satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M$. Then

  $$\begin{aligned} \begin{aligned}&\Vert \Delta _{12} b^{(3)} \Vert _{s_1} , \Vert \Delta _{12} a_1^{(3)}\Vert _{s_1} \lesssim _{s_1, M} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M} , \\&\quad \quad |\Delta _{12} \mathrm{Op}( r_0^{(3)}) |_{0, s_1, 0} \lesssim _{s_1, M} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M}. \end{aligned} \end{aligned}$$

  (6.58)

• (iii) Let $S > {\mathfrak {s}}_M$, $\lambda _0 \in {\mathbb {N}}$. Then there exists a constant $\sigma _M(\lambda _0) > 0$ so that for any $m \in {{\mathbb {S}}}_+$ and $\lambda , n_1, n_2 \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$ and $n_1 + n_2 + \lambda _0\leqq M - 1$, the operator $\langle D \rangle ^{n_1}\partial _{\varphi _m}^\lambda {{\mathcal {R}}_M}^{(3)} \langle D \rangle ^{n_2}$, is $\mathrm{{Lip}(\gamma )}$-tame with tame constant satisfying

  $$\begin{aligned} {{\mathfrak {M}}}_{\partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1}{{\mathcal {R}}_M^{(3)}} \langle D \rangle ^{n_2}} (s) \lesssim _{S, M, \lambda _0} \varepsilon + \Vert \iota \Vert _{s+\sigma _M(\lambda _0)}^\mathrm{{Lip}(\gamma )}, \quad \forall {\mathfrak {s}}_M\leqq s \leqq S . \end{aligned}$$

  (6.59)

  Let $s_1 \geqq {\mathfrak {s}}_M$ and let $\breve{\iota }_1, \breve{\iota }_2$ be torus embeddings satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M(\lambda _0)$. Then

  $$\begin{aligned} \Vert \partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1} \Delta _{12} {{\mathcal {R}}_M^{(3)}} \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})} \lesssim _{s_1, M, \lambda _0} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M(\lambda _0)}. \end{aligned}$$

  (6.60)

Proof

The estimate (6.53) follows by the definition (6.52) and (6.43), (6.20). We now provide estimates for the flow

$$\begin{aligned} \Phi ^{(3)}(\tau ) = \mathrm{exp}\big (\tau \Pi _\bot b^{(3)} (\varphi , x; \omega ) \partial _{x}^{-1} \big ) , \quad \forall \tau \in [- 1 , 1] . \end{aligned}$$

By (2.20), Lemma 2.10, and (6.53), one infers that for any $s \geqq s_0$, $|\Pi _\bot b^{(3)} \partial _x^{- 1}|_{- 1, s, 0}^\mathrm{{Lip}(\gamma )} \lesssim _{s, M} \, \varepsilon + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}.$ Therefore, by Lemma 2.13, there exists $\sigma _M > 0$ such that, if (6.1) holds with $\mu _0 \geqq \sigma _M$, then, for any $s \geqq s_0$,

$$\begin{aligned} \sup _{\tau \in [- 1, 1]} |\Phi ^{(3)}(\tau ) - \mathrm{Id}|_{0, s, 0}^\mathrm{{Lip}(\gamma )}\lesssim _{s} \varepsilon + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(6.61)

The latter estimate, together with Lemma 2.18, imply (6.54).

By (6.42) and using Lemma 6.3 for the operator $Q_{- 1}^{kdv}(D; \omega )$, one has that

$$\begin{aligned} \Phi ^{(3)} {{{\mathcal {L}}}}_\omega ^{(2)} (\Phi ^{(3)})^{- 1} = \omega \cdot \partial _\varphi - \Phi ^{(3)} \big ( m_3 \partial _x^3 + a_1^{(2)} \partial _x \big ) (\Phi ^{(3)})^{- 1} - Q^{kdv}_{- 1}(D ; \ \omega ) + {{{\mathcal {R}}}}_0^{(I)} + {{{\mathcal {R}}}}_M^{(3)} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {{{\mathcal {R}}}}_0^{(I)}&\,{:=}\, - \Phi ^{(3)} \mathrm{Op}( r_0^{(2)}) (\Phi ^{(3)})^{- 1} + \Phi ^{(3)} \big ( \omega \cdot \partial _\varphi (\Phi ^{(3)})^{- 1} \big ) \\&\quad - (\Phi ^{(3)} - \mathrm{Id}_\bot )\Pi _\bot \Big ( \sum _{k = 1}^M c_{ - k}^{kdv}(\omega ) \partial _x^{- k} \Big ) (\Phi ^{(3)})^{- 1} \\&\quad - \Pi _\bot \Big ( \sum _{k = 1}^{M } c_{ - k}^{kdv}(\omega ) \partial _x^{- k} \Big ) \Big ( (\Phi ^{(3)})^{- 1} - \mathrm{Id}_\bot \Big ) , \\ {{{\mathcal {R}}}}_M^{(3)}&\,{:=}\, \Phi ^{(3)} {{{\mathcal {R}}}}_M^{(2)} (\Phi ^{(3)})^{- 1} - (\Phi ^{(3)} - \mathrm{Id}_\bot ) {{{\mathcal {R}}}}_{M }( \omega , Q_{-1}^{kdv}) (\Phi ^{(3)})^{- 1} \\&\quad - {{{\mathcal {R}}}}_{M }( \omega , Q_{-1}^{kdv}) \big ( (\Phi ^{(3)})^{- 1} - \mathrm{Id}_\bot \big ). \end{aligned} \end{aligned}$$

(6.62)

Note that ${{{\mathcal {R}}}}_0^{(I)}$ is a pseudo-differential operator in $OPS^0$ (cf. Lemma 2.13). Moreover, by a Lie expansion and using (6.50), one has

$$\begin{aligned} \begin{aligned}&\Phi ^{(3)} \big ( m_3 \partial _x^3 + a_1^{(2)} \partial _x \big ) (\Phi ^{(3)})^{- 1} = m_3 \partial _x^3 + a_1^{(2)} \partial _x + [ \Pi _\bot b^{(3)} \partial _x^{- 1}, \, m_3 \partial _x^3 + a_1^{(2)} \partial _x] \\&\quad \ \ + \int _0^1 (1 - \tau ) \Phi ^{(3)}(\tau ) \Big [\Pi _\bot b^{(3)} \partial _x^{- 1}, \Big [\Pi _\bot b^{(3)} \partial _x^{- 1}, m_3 \partial _x^3 + a_1^{(2)} \partial _x\Big ] \Big ] \Phi ^{(3)}(\tau )^{- 1}\, \hbox {d} \tau \\&\quad = m_3 \partial _x^3 + \big ( a_1^{(2)} - 3 m_3 b^{(3)}_x\big ) \partial _x + {{{\mathcal {R}}}}_0^{(II)},\\ {\mathcal {R}}_0^{(II)}&:= - 3 m_3 b^{(3)}_{xx} - m_3 b^{(3)}_{xxx} \partial _x^{- 1} + [ \Pi _\bot b^{(3)} \partial _x^{- 1} , a_1^{(2)} \partial _x] + [ (\Pi _\bot - \mathrm{Id}) b^{(3)} \partial _x^{- 1}, m_3 \partial _x^3] \\&\quad + \int _0^1 (1 - \tau ) \Phi ^{(3)}(\tau ) \Big [\Pi _\bot b^{(3)} \partial _x^{- 1},\, \Big [\Pi _\bot b^{(3)} \partial _x^{- 1}, m_3 \partial _x^3 + a_1^{(2)} \partial _x\Big ] \Big ] \Phi ^{(3)}(\tau )^{- 1}\, \hbox {d} \tau \in OPS^0. \end{aligned}\nonumber \\ \end{aligned}$$

(6.63)

Note that ${{{\mathcal {R}}}}_0^{(II)}$ is also a pseudo-differential operator in $OPS^0$ (cf. Lemma 2.13). Hence, (6.62)–(6.63) and the choice of $b^{(3)}$ in (6.52) lead to the expansion (6.55) with ${{{\mathcal {R}}}}_M^{(3)}$ given by (6.62) and

$$\begin{aligned} \mathrm{Op}( r_0^{(3)}) \,{:=}\, - {{{\mathcal {R}}}}_0^{(I)} + {{{\mathcal {R}}}}_0^{(II)}. \end{aligned}$$

(6.64)

The estimate (6.56) follows by (6.24).

The estimate (6.57) on the operator $\mathrm{Op}(r_0^{(3)})$ follows by the definitions (6.62), (6.63), (6.64), by applying the estimates (6.20), (6.43), (6.44), (6.53), (6.61), (2.20), (2.21), (2.22), (2.24), (2.26) (using the ansatz (6.1) with $\mu _0$ large enough). Next we estimate the remainder ${{{\mathcal {R}}}}_M^{(3)}$, defined in (6.62). We only consider the second term in the definition of ${{{\mathcal {R}}}}_M^{(3)}$, since the estimates of the first and third terms can be obtained similarly. We recall that the operator ${{{\mathcal {R}}}}_M(Q_{-1}^{kdv}; \omega )$ is $\varphi$-independent. For $m \in {{\mathbb {S}}}_+$ and $\lambda , n_1, n_2 \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$ and $n_1 + n_2 + \lambda _0\leqq M - 2$, one has

$$\begin{aligned}&\langle D \rangle ^{n_1} \partial _{\varphi _m}^\lambda \Big ( (\Phi ^{(3)} - \mathrm{Id}_\bot ) {{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) (\Phi ^{(3)})^{- 1} \Big ) \langle D \rangle ^{n_2} \\&\quad = \sum _{\lambda _1 + \lambda _2 = \lambda } C_{\lambda _1, \lambda _2} \langle D \rangle ^{n_1} \partial _{\varphi _m}^{\lambda _1} (\Phi ^{(3)} - \mathrm{Id}_\bot ) {{{\mathcal {R}}}}_M(Q_{-1}^{kdv}; \omega ) \partial _{\varphi _m}^{\lambda _2} (\Phi ^{(3)})^{- 1} \langle D \rangle ^{n_2} \nonumber \\&\quad = \sum _{\lambda _1 + \lambda _2 = \lambda } C_{\lambda _1, \lambda _2} \Big ( \langle D \rangle ^{n_1} \partial _{\varphi _m}^{\lambda _1} (\Phi ^{(3)} - \mathrm{Id}_\bot ) \langle D \rangle ^{- n_1} \Big ) \nonumber \\&\qquad \Big ( \langle D \rangle ^{n_1} {{{\mathcal {R}}}}_M( Q_{-1}^{kdv}; \omega ) \langle D \rangle ^{n_2} \Big ) \Big ( \langle D \rangle ^{- n_2} \partial _{\varphi _m}^{\lambda _2} (\Phi ^{(3)})^{- 1} \langle D \rangle ^{n_2} \Big ) . \nonumber \end{aligned}$$

(6.65)

By the estimates (2.21), (2.24), (6.61) and Lemma 2.18, one has

$$\begin{aligned} \begin{aligned}&{\mathfrak {M}}_{\langle D \rangle ^{n_1} \partial _{\varphi _m}^{\lambda _1} (\Phi ^{(3)} - \mathrm{Id}_\bot ) \langle D \rangle ^{- n_1}}(s) \lesssim _s |\langle D \rangle ^{n_1} \partial _{\varphi _m}^{\lambda _1} (\Phi ^{(3)} - \mathrm{Id}_\bot ) \langle D \rangle ^{- n_1}|_{0, s, 0}^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s + \sigma _M(\lambda _0)}^\mathrm{{Lip}(\gamma )}, \\&{\mathfrak {M}}_{\langle D \rangle ^{- n_2} \partial _{\varphi _m}^{\lambda _2} (\Phi ^{(3)})^{- 1} \langle D \rangle ^{n_2} }(s) \lesssim _s |\langle D \rangle ^{- n_2} \partial _{\varphi _m}^{\lambda _2} (\Phi ^{(3)})^{- 1} \langle D \rangle ^{n_2} |_{0, s, 0}^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} 1 + \Vert \iota \Vert _{s + \sigma _M(\lambda _0)}^\mathrm{{Lip}(\gamma )}, \end{aligned} \end{aligned}$$

and therefore, by Lemmata 2.16, 6.16 and using (6.1), the operator (6.65) satisfies (6.59). The estimates (6.58), (6.60) follow by similar arguments. $\quad \square$

6.5 Elimination of the $\varphi$-Dependence of the First Order Term

The goal of this section is to remove the $\varphi$-dependence of the coefficient $a_1^{(3)} (\varphi )$ of the Hamiltonian operator ${{\mathcal {L}}}^{(3)}_\omega$ in (6.51), (6.55). We conjugate the operator ${{\mathcal {L}}}^{(3)}_\omega$ by the variable transformation $\Phi ^{(4)} \equiv \Phi ^{(4)} (\varphi )$,

$$\begin{aligned} (\Phi ^{(4)} w)(\varphi , x) = w(\varphi , x + b^{(4)} (\varphi )) , \quad ((\Phi ^{(4)})^{-1} h)(\varphi , x) = h(\varphi , x - b^{(4)} (\varphi )) \, , \end{aligned}$$

where $b^{(4)} (\varphi )$ is a small, real valued, periodic function, chosen in (6.67) below. Note that $\Phi ^{(4)}$ is the time-one flow of the transport equation $\partial _\tau w = b^{(4)}(\varphi ) \partial _x w$. Each $\Phi ^{(4)} (\varphi )$ is a symplectic linear isomorphism of $H^s_\bot ({\mathbb {T}}_1)$, and hence the conjugated operator

$$\begin{aligned} {{\mathcal {L}}}_\omega ^{(4)} \,{:=}\, \Phi ^{(4)}{{\mathcal {L}}}_\omega ^{(3)} \big (\Phi ^{(4)} \big )^{- 1} \end{aligned}$$

(6.66)

is Hamiltonian.

Proposition 6.7

(Reduction of ${{{\mathcal {L}}}}_\omega$ to constant coefficients up to order zero) Assume that $\omega \in \mathtt {DC}(\gamma ,\tau )$ (cf. (4.4)). Let $b^{(4)} (\varphi )$ be the real valued, periodic function

$$\begin{aligned} b^{(4)}(\varphi ; \omega ) \,{:=}\, - (\omega \cdot \partial _\varphi )^{-1} \big ( a_1^{(3)}(\varphi ; \omega ) - m_1 \big ) , \quad m_1 \,{:=}\, \frac{1}{(2 \pi )^{|{{\mathbb {S}}}_+|}} \int _{{\mathbb {T}}^{{{\mathbb {S}}}_+}} a_1^{(3)} (\varphi ; \omega ) \, \hbox {d} \varphi \end{aligned}$$

(6.67)

and let $M \in {\mathbb {N}}$. Then there exists $\sigma _M > 0$ with the following properties:

• (i) The constant $m_1$ and the function $b^{(4)}$ satisfy

  $$\begin{aligned} |m_1|^\mathrm{{Lip}(\gamma )}\lesssim _M \varepsilon \gamma ^{- 2}, \quad \Vert b^{(4)}\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \gamma ^{- 1}\big ( \varepsilon + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}) , \quad \forall s \geqq s_0 . \end{aligned}$$

  (6.68)

• (ii) The Hamiltonian operator in (6.66) admits an expansion of the form

  $$\begin{aligned} {{\mathcal {L}}}_\omega ^{(4)} = \omega \cdot \partial _\varphi - \big ( m_3 \partial _x^3 + m_1 \partial _x + \mathrm{Op}( r_0^{(4)}) + Q_{-1}^{kdv} ( D; \omega )\big ) + {{\mathcal {R}}_M^{(4)}} \end{aligned}$$

  (6.69)

  where $m_3$, given by (6.18), satisfies $|m_3 + 1|^\mathrm{{Lip}(\gamma )}\lesssim _M \varepsilon$ (cf. (6.20)), and $r_0^{(4)} \,{:=}\, r_0^{(4)} (\varphi , x, \xi ; \omega )$ is a pseudo-differential symbol in $S^0$ satisfying

  $$\begin{aligned} | \mathrm{Op}( r_0^{(4)})|_{0, s, 0}^\mathrm{{Lip}(\gamma )}\lesssim _{s, M} \varepsilon + \Vert \iota \Vert _{s+\sigma _M}^\mathrm{{Lip}(\gamma )}, \quad \forall s \geqq s_0. \end{aligned}$$

  (6.70)

  Let $s_1 \geqq s_0$ and let $\breve{\iota }_1, \breve{\iota }_2$ be torus embeddings satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M$. Then

  $$\begin{aligned} \begin{aligned}&|\Delta _{12} m_1|,\, \Vert \Delta _{12} b^{(4)} \Vert _{s_1} \, \lesssim _{s_1, M} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M}, \\&\quad \quad |\Delta _{12} \mathrm{Op}( r_0^{(4)}) |_{0, s_1, 0} \, \lesssim _{s_1, M} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M}. \end{aligned} \end{aligned}$$

  (6.71)

• (iii) Let $S> {\mathfrak {s}}_M$. Then the maps $(\Phi ^{(4)})^{\pm 1}$ are $\mathrm{{Lip}(\gamma )}$-tame operators with tame constants satisfying

  $$\begin{aligned} {\mathfrak {M}}_{(\Phi ^{(4)})^{\pm 1}}(s) \lesssim _{S, M} 1 + \Vert \iota \Vert _{s + \sigma _M}^\mathrm{{Lip}(\gamma )}, \quad \forall s_0 \leqq s \leqq S. \end{aligned}$$

  (6.72)

  Let $\lambda _0 \in {\mathbb {N}}$. Then there exists a constant $\sigma _M(\lambda _0) > 0$ so that for any $\lambda , n_1, n_2 \in {\mathbb {N}}$ with $\lambda \leqq \lambda _0$ and $n_1 + n_2 + 2 \lambda _0 \leqq M - 3$, the operator $\partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1}{{\mathcal {R}}_M^{(4)}} \langle D \rangle ^{n_2}$, $m \in {{\mathbb {S}}}_+$, is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying

  $$\begin{aligned} {{\mathfrak {M}}}_{\partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1}{{\mathcal {R}}_M^{(4)}} \langle D \rangle ^{n_2}} (s) \lesssim _{S, M, \lambda _0} \varepsilon + \Vert \iota \Vert _{s+\sigma _M(\lambda _0)}^\mathrm{{Lip}(\gamma )}, \quad \forall {\mathfrak {s}}_M\leqq s \leqq S . \end{aligned}$$

  (6.73)

  Let $s_1 \geqq {\mathfrak {s}}_M$ and let $\breve{\iota }_1, \breve{\iota }_2$ be two tori satisfying (6.1) with $\mu _0 \geqq s_1 + \sigma _M(\lambda _0)$. Then

  $$\begin{aligned} \Vert \partial _{\varphi _m}^\lambda \langle D \rangle ^{n_1} \Delta _{12} {{\mathcal {R}}_M^{(4)}} \langle D \rangle ^{n_2} \Vert _{{{{\mathcal {B}}}}(H^{s_1})} \lesssim _{s_1, M, \lambda _0} \Vert \iota _1 - \iota _2\Vert _{s_1 + \sigma _M(\lambda _0)}. \end{aligned}$$

  (6.74)

Proof

The estimates (6.68) are direct consequences of (6.56) and of (6.1). Note that

$$\begin{aligned} \Phi ^{(4)} \circ \omega \cdot \partial _\varphi \circ (\Phi ^{(4)})^{- 1} = \omega \cdot \partial _\varphi - \big (\omega \cdot \partial _\varphi b^{(4)} \big ) \partial _x . \end{aligned}$$

A straightforward calculation then shows that for any pseudo-differential operator $\mathrm{Op}(a(\varphi , x, \xi ))$,

$$\begin{aligned} \Phi ^{(4)} \mathrm{Op}(a (\varphi , x, \xi )) \big (\Phi ^{(4)} \big )^{- 1} = \mathrm{Op}( a (\varphi , x + b^{(4)}(\varphi ) , \xi )) . \end{aligned}$$

Hence, by (6.55) and the definition (6.67), one obtains (6.69) with

$$\begin{aligned} \mathrm{Op}\big (r_0^{(4)}(\varphi , x, \xi ) \big ) = \mathrm{Op}\big (r_0^{(3)}(\varphi , x + b^{(4)}(\varphi ), \xi ) \big ) , \quad {{{\mathcal {R}}}}_M^{(4)} \,{:=}\, \Phi ^{(4)} {{{\mathcal {R}}}}_M^{(3)} (\Phi ^{(4)})^{- 1}.\nonumber \\ \end{aligned}$$

(6.75)

The estimates (6.70) for $\mathrm{Op}( r_0^{(4)})$ follow from Lemma 2.1, using (6.68), (6.57), and (6.1). The estimates (6.73) for the operator ${{{\mathcal {R}}}}_M^{(4)}$ ared obtained from (6.59), (6.68) arguing as in the proof of the estimates of the remainder ${{{\mathcal {R}}}}_N(\tau , \varphi )$ (with $\beta$ given by $b^{(4)}$) at the end of the proof of Proposition 2.31. The estimates (6.72) follow by Lemma 2.1 and (6.68). Finally, the estimates (6.71), (6.74) are obtained by similar arguments. $\quad \square$

7 KAM Reduction of the Linearized Operator

The main result of this section is Theorem 7.11, stating that the assumptions A-I concerning the almost-invertibility of ${{{\mathcal {L}}}}_\omega$ in Section 5 are satisfied. The key ingredient for its proof is Theorem 7.3, which affirms that the Hamiltonian operator ${{\mathcal {L}}}_\omega ^{(4)}$ in (6.69), renamed ${\mathtt {L}}_0$ in (7.1), can be brought in almost diagonal form. This completes the diagonalization of the Hamiltonian operator ${{\mathcal {L}}}_\omega$, defined in (6.7), which was started in Section 6. The Hamiltonian operator ${\mathtt {L}}_0$ is diagonalized by applying a KAM-reducibility iterative scheme, developed in [10], for perturbations of diagonal operators which are modulo-tame (cf. Definition 2.19). In Lemma 7.1 we prove that the initial remainder ${\mathtt {R}}_0$ is indeed modulo-tame. We recall that the class of modulo-tame operators is closed under the operations coming up in the KAM reduction procedure, namely: sum and composition (Lemma 2.21); projections (Lemma 2.23); solution of the homological equation (Lemma 7.5).

As in Section 6, we again consider in the sequel torus embeddings $\breve{\iota }(\varphi ) = (\varphi , 0, 0) + \iota (\varphi )$ with $\iota (\cdot ) \equiv \iota (\cdot \ ; \omega )$, $\omega \in \mathtt {DC}(\gamma ,\tau )$ (cf. (4.4)), satisfying (6.1), $\Vert \iota \Vert _{ \mu _0}^{\mathrm {Lip}(\gamma )} \lesssim \varepsilon \gamma ^{-2}$, $\varepsilon \gamma ^{- 2} \leqq \delta (S)$, where $\mu _0 \,{:=}\, \mu _0(\tau , {{\mathbb {S}}}_+) > s_0$ and $S > \mu _0$ are sufficiently large, $0< \delta (S) < 1$ is sufficiently small, and $0< \gamma < 1$.

Recall that by (6.69), the operator ${{\mathcal {L}}}_\omega ^{(4)}$ is given by $\omega \cdot \partial _\varphi - \big ( m_3 \partial _x^3 + m_1 \partial _x + \mathrm{Op}( r_0^{(4)}) + Q_{-1}^{kdv} ( D; \omega )\big ) + {{\mathcal {R}}_M^{(4)}}$ and acts on $H^s_\bot$. In view of the reduction scheme, implemented in this section, it is useful to rename it to

$$\begin{aligned} {\mathtt {L}}_0 \,{:=}\, \omega \cdot \partial _\varphi + \mathrm{i} {\mathtt {D}}_0 + {\mathtt {R}}_0 , \end{aligned}$$

(7.1)

where $\omega \in \mathtt {DC}(\gamma ,\tau )$ (cf. (4.4)) and, in view of (6.9), (3.13), (4.3),

$$\begin{aligned}&{\mathtt {D}}_0 \,{:=}\, \mathrm{diag}_{j \in {{\mathbb {S}}}^\bot } ( \mu _j^{0}) , \quad \mu _j^{0} \,{:=}\, m_3 (2 \pi j)^3 - m_1 2 \pi j - q_j(\omega ) , \nonumber \\&q_j(\omega ) \,{:=}\, \omega _j^{kdv} \big ( \nu (\omega ), 0 \big ) - (2 \pi j)^3 , \end{aligned}$$

(7.2)

$$\begin{aligned}&{\mathtt {R}}_0 \,{:=}\, - \mathrm{Op}( r_0^{(4)}) + {{\mathcal {R}}_M^{(4)}} . \end{aligned}$$

(7.3)

We recall that $m_3: \Omega \rightarrow {\mathbb {R}}$ is given by (6.18) and $m_1: \mathtt {DC}(\gamma ,\tau ) \rightarrow {\mathbb {R}}$ by (6.67). Furthermore, note that $\mu _{-j}^{0} = - \mu _{j}^{0}$ for any $j \in {{\mathbb {S}}}^\bot$. By (3.67) we have

$$\begin{aligned} \sup _{j \in {{\mathbb {S}}}^\bot } |j| |q_j|^\mathrm{sup}, \ \sup _{j \in {{\mathbb {S}}}^\bot } |j| |q_j|^\mathrm{lip} \, \lesssim \, 1, \end{aligned}$$

(7.4)

and, by (6.20), (6.68) and $\varepsilon \gamma ^{-3} \leqq 1$,

$$\begin{aligned} |\mu _j^{0} - \mu _{j'}^{0}|^\mathrm{lip} \lesssim _M |j^3 - j'^3| , \quad \forall j, j' \in {{\mathbb {S}}}^\bot . \end{aligned}$$

(7.5)

The operator ${\mathtt {R}}_0$ satisfies the tame estimates of Lemma 7.1 below. We first fix the constants

$$\begin{aligned} \begin{aligned}&{\mathtt {b}} \,{:=}\, [{\mathtt {a}}] + 2 \in {\mathbb {N}}, \quad {\mathtt {a}} \,{:=}\, 3 \tau _1 + 1 , \quad \tau _1 \,{:=}\, 2 \tau + 1 ,\\&\mu (\mathtt {b}) \,{:=}\, s_0 + \mathtt {b} + \sigma _M + \sigma _M(\mathtt {b}) + 1 , \quad M \,{:=}\, 2(s_0 + \mathtt {b}) + 4 , \end{aligned} \end{aligned}$$

(7.6)

where $[\mathtt {a}]$ denotes the integer part of $\mathtt {a}$ and the constants $\sigma _M$, $\sigma _M(\mathtt {b})$ are the ones introduced in Lemma 6.7. The integer M is related to the order of smoothing of the remainder ${{\mathcal {R}}_M^{(4)}}$ in (6.69) (cf. (6.73)). Note that M only depends on the number of frequencies $|{{\mathbb {S}}}_+|$ and the diophantine constant $\tau$.

Lemma 7.1

Let $\mathtt {b}$ and M be given as in (7.6) and $S > {\mathfrak {s}}_M$ with ${\mathfrak {s}}_M$ given by (2.55).

• (i) The operators ${\mathtt {R}}_0$, $[ {\mathtt {R}}_0 , \partial _x ]$, $\partial _{\varphi _m}^{s_0} [ {\mathtt {R}}_0 , \partial _x ]$, $\partial _{\varphi _m}^{s_0 + {\mathtt {b}}} {\mathtt {R}}_0$, $\partial _{\varphi _m}^{s_0 + {\mathtt {b}}} [ {\mathtt {R}}_0, \partial _x ]$, $m \in {{\mathbb {S}}}_+$, are $\mathrm{{Lip}(\gamma )}$-tame with tame constants

  $$\begin{aligned}&{{\mathbb {M}}}_0 (s) \,{:=}\, \max _{m \in {{\mathbb {S}}}_+} \big \{ {{\mathfrak {M}}}_{ {\mathtt {R}}_0 }(s), {{\mathfrak {M}}}_{[ {\mathtt {R}}_0 , \partial _x ] }(s), {{\mathfrak {M}}}_{ \partial _{\varphi _m}^{s_0} {\mathtt {R}}_0 }(s), {{\mathfrak {M}}}_{ \partial _{\varphi _m}^{s_0} [ {\mathtt {R}}_0 , \partial _x ] }(s) \big \} , \end{aligned}$$

  (7.7)
  $$\begin{aligned}&{{\mathbb {M}}}_0 (s, {\mathtt {b}}) \,{:=}\, \max _{m \in {{\mathbb {S}}}_+} \big \{ {{\mathfrak {M}}}_{ \partial _{\varphi _m}^{s_0 + {\mathtt {b}}} {\mathtt {R}}_0 }(s), {{\mathfrak {M}}}_{ \partial _{\varphi _m}^{s_0 + {\mathtt {b}}} [ {\mathtt {R}}_0, \partial _x ] }(s) \big \} , \end{aligned}$$

  (7.8)

  satisfying, for any ${\mathfrak {s}}_M\leqq s \leqq S$,

  $$\begin{aligned} {{\mathfrak {M}}}_0(s, \mathtt {b}) \,{:=}\, \mathrm{max}\{ {{\mathbb {M}}}_0 (s), {{\mathbb {M}}}_0 (s, {\mathtt {b}}) \} \lesssim _S \varepsilon + \Vert \iota \Vert _{s + \mu (\mathtt {b})}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (7.9)

  Assuming that (6.1) (ansatz for $\breve{\iota }$) holds with $\mu _0 \geqq {\mathfrak {s}}_M+ \mu (\mathtt {b})$, the latter estimate yields ${{\mathfrak {M}}}_0({\mathfrak {s}}_M, \mathtt {b}) \lesssim _S \varepsilon \gamma ^{- 2}$.

• (ii) For any torus embeddings $\breve{\iota }_1$, $\breve{\iota }_2$ satisfying (6.1), one has for any $m \in {\mathbb {S}}_+$ and any $\lambda \in {\mathbb {N}}$ with $\lambda \leqq s_0 + \mathtt {b}$

  $$\begin{aligned} \Vert \partial _{\varphi _m}^\lambda \Delta _{12} \mathtt {R}_0 \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})}, \ \Vert \partial _{\varphi _m}^\lambda [\Delta _{12} \mathtt {R}_0, \partial _x] \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} \, \lesssim \Vert \iota _1 - \iota _2\Vert _{{\mathfrak {s}}_M+ \mu (\mathtt {b})} .\nonumber \\ \end{aligned}$$

  (7.10)

Proof

• (i) Since the assertions for the various operators are proved in the same way, we restrict ourselves to show that there are tame constants ${{\mathfrak {M}}}_{\partial _{\varphi _m}^{s_0 + \mathtt {b}} [\mathtt {R}_0, \partial _x]}(s)$, $m \in {{\mathbb {S}}}_+$, satisfying the bound in (7.9). The two operators $\mathrm{Op}( r_0^{(4)})$ and ${{\mathcal {R}}_M^{(4)}}$ in the definition (7.3) of ${\mathtt {R}}_0$ are treated separately. By Lemma 2.18, for any $m \in {{\mathbb {S}}}_+$, the operator $\partial _{\varphi _m}^{s_0 + \mathtt {b}}[\mathrm{Op}( r_0^{(4)}), \partial _x ] = - \mathrm{Op} \big ( \partial _{\varphi _m}^{s_0 + \mathtt {b}} \partial _x r_0^{(4)} \big )$ is $\mathrm{{Lip}(\gamma )}$-tame with a tame constant satisfying for any $s_0 \leqq s \leqq S$,

  $$\begin{aligned} \begin{aligned} {{\mathfrak {M}}}_{\partial _{\varphi _m}^{s_0 + \mathtt {b}}[\mathrm{Op}( r_0^{(4)}), \partial _x ]}(s)&{\mathop {\lesssim _s}\limits ^{(2.31)}} \Big | \mathrm{Op}\Big ( \partial _{\varphi _m}^{s_0 + \mathtt {b}} \partial _x r_0^{(4)} \Big ) \Big |_{0, s, 0}^\mathrm{{Lip}(\gamma )}\\&\lesssim _s \Big | \mathrm{Op}( r_0^{(4)} ) \Big |_{0, s +s_0 + \mathtt {b} + 1, 0}^\mathrm{{Lip}(\gamma )}\\&{\mathop {\lesssim _s}\limits ^{(6.70)}} \varepsilon + \Vert \iota \Vert _{s + s_0 + \mathtt {b} + 1 + \sigma _M}^\mathrm{{Lip}(\gamma )}. \end{aligned} \end{aligned}$$

  (7.11)

  Next consider, for any given $m \in {{\mathbb {S}}}_+$, the operator $\partial _{\varphi _m}^{s_0 + \mathtt {b}} [{{{\mathcal {R}}}}_M^{(4)}, \partial _x]$. Recalling that $\langle D \rangle$ denotes the Fourier multiplier with symbol $\langle \xi \rangle$, one has

  $$\begin{aligned} \begin{aligned} \partial _{\varphi _m}^{s_0 + \mathtt {b}} [{{{\mathcal {R}}}}_M^{(4)}, \partial _x]&= \partial _{\varphi _m}^{s_0 + \mathtt {b}}{{{\mathcal {R}}}}_M^{(4)} \langle D \rangle \langle D \rangle ^{- 1}\partial _x - \langle D \rangle ^{- 1} \partial _x \langle D \rangle \partial _{\varphi _m}^{s_0 + \mathtt {b}}{{{\mathcal {R}}}}_M^{(4)} . \end{aligned} \end{aligned}$$

  Since $\langle D \rangle ^{- 1}\partial _x$ admits a tame constant ${{\mathfrak {M}}}_{\langle D \rangle ^{- 1}\partial _x}(s)$ bounded by 1, it follows by (6.73) that, for any ${\mathfrak {s}}_M\leqq s \leqq S$,

  $$\begin{aligned} {{\mathfrak {M}}}_{\partial _{\varphi _m}^{s_0 + \mathtt {b}} [{{{\mathcal {R}}}}_M^{(4)}, \partial _x]}(s) \lesssim _S \, \varepsilon + \Vert \iota \Vert _{s + \sigma _M(\mathtt {b})}^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

  (7.12)

  Combining (7.11), (7.12) and recalling the definition of $\mu (\mathtt {b})$ in (7.6) one infers that $\partial _{\varphi _m}^{s_0 + \mathtt {b}} [\mathtt {R}_0, \partial _x]$ admits a tame constant ${{\mathfrak {M}}}_{\partial _{\varphi _m}^{s_0 + \mathtt {b}} [\mathtt {R}_0, \partial _x]}(s)$, satisfying the claimed bound.

• (ii) The estimate (7.10) follows by similar arguments using (6.71) and (6.74) with $s_1 = {\mathfrak {s}}_M$. $\quad \square$

We perform the almost reducibility scheme for ${\mathtt {L}}_0$ along the scale

$$\begin{aligned} N_{- 1} \,{:=}\, 1,\quad N_\nu \,{:=}\, N_0^{\chi ^\nu }, \ \nu \geqq 0,\quad \chi \,{:=}\, 3/2 \end{aligned}$$

(7.13)

(with $N_0$ specified in Theorem 7.2 below), assuming that at each induction step the second order Melnikov non-resonance conditions (7.18) hold.

Theorem 7.2

(Almost reducibility) There exists ${\overline{\tau }} \,{:=}\, {\overline{\tau }} (\tau , {{\mathbb {S}}}_+) > 0$ so that for any $S > {\mathfrak {s}}_M$, there is $N_0 \,{:=}\, N_0 (S, {\mathtt {b}}) \in {\mathbb {N}}$, satisfying

$$\begin{aligned} N_0^{{\overline{\tau }}} {{\mathfrak {M}}}_0({\mathfrak {s}}_M, {\mathtt {b}}) \gamma ^{- 1} \leqq 1, \end{aligned}$$

(7.14)

so that the following holds: for any $\nu \in {\mathbb {N}}$,

• $\mathbf{(S1)_{\nu }}$ There exists a Hamiltonian operator ${\mathtt {L}}_\nu$, acting on $H^s_\bot$ and defined for $\omega \in {\mathtt {\Omega }}_\nu ^\gamma$, of the form

  $$\begin{aligned} {\mathtt {L}}_\nu \,{:=}\, \omega \cdot \partial _\varphi + \mathrm{i} {\mathtt {D}}_\nu + {\mathtt {R}}_\nu , \quad {\mathtt {D}}_\nu \,{:=}\, \mathrm{diag}_{j \in {{\mathbb {S}}}^\bot } \mu _j^\nu ,\quad \mu _j^\nu \in {\mathbb {R}}, \end{aligned}$$

  (7.15)

  where for any $j \in {{\mathbb {S}}}^\bot$, $\mu _j^\nu$, also denoted by $\mu _j^\nu (\omega )$ or $\mu _j^\nu (\omega ; \iota )$, is a $\mathrm{{Lip}(\gamma )}$-function of the form

  $$\begin{aligned} \mu _j^\nu (\omega ) \,{:=}\, \mu _j^0(\omega ) + r_j^\nu (\omega ), \end{aligned}$$

  (7.16)

  with $\mu _j^{(0)}$ defined by (7.2) and

  $$\begin{aligned} \mu _{-j}^{\nu } = - \mu _j^{\nu } , \quad |r_j^\nu |^\mathrm{{Lip}(\gamma )}\leqq C(S) \varepsilon \gamma ^{- 2}. \end{aligned}$$

  (7.17)

  If $\nu =0$, the set of frequency vectors $\mathtt {\Omega }_\nu ^\gamma$ is defined to be the set ${\mathtt {\Omega }}_0^\gamma \,{:=}\, {\mathtt {D} \mathtt {C}}(\gamma , \tau )$, and if $\nu \geqq 1$,

  $$\begin{aligned} {\mathtt {\Omega }}_\nu ^\gamma&= {\mathtt {\Omega }}_\nu ^\gamma (\iota ) \,{:=}\, \Big \{ \omega \in {\mathtt {\Omega }}_{\nu - 1}^\gamma \, : |\omega \cdot \ell + \mu _j^{\nu - 1} - \mu _{j'}^{\nu - 1}| \geqq \gamma \frac{|j^{3} - j'^{3}|}{ \langle \ell \rangle ^{\tau }}, \ \forall |\ell | \leqq N_{\nu - 1}, j, j' \in {{\mathbb {S}}}^\bot \Big \} . \nonumber \\ \end{aligned}$$

  (7.18)

  Note that ${\mathtt {\Omega }}_{\nu +1}^\gamma \subseteq {\mathtt {\Omega }}_\nu ^\gamma$ for any $\nu \ge 0$. The operators ${\mathtt {R}}_\nu$ and $\langle \partial _\varphi \rangle ^{\mathtt {b} } {\mathtt {R}}_\nu$ are $\mathrm{{Lip}(\gamma )}$-modulo-tame with modulo-tame constants

  $$\begin{aligned} {{\mathfrak {M}}}_\nu ^\sharp (s) \,{:=}\, {{\mathfrak {M}}}_{{\mathtt {R}}_\nu }^\sharp (s) , \quad {{\mathfrak {M}}}_\nu ^\sharp (s, {\mathtt {b}}) \,{:=}\, {{\mathfrak {M}}}_{ \langle \partial _\varphi \rangle ^{\mathtt {b}} {\mathtt {R}}_\nu }^\sharp (s) \, , \end{aligned}$$

  (7.19)

  satisfying, for some $C_* ({\mathfrak {s}}_M, \mathtt {b}) > 0$ and any $s \in [{\mathfrak {s}}_M, S]$,

  $$\begin{aligned} {{\mathfrak {M}}}_\nu ^\sharp (s) \leqq C_* ({\mathfrak {s}}_M, \mathtt {b}) {{\mathfrak {M}}}_0 (s, {\mathtt {b}}) N_{\nu - 1}^{- {\mathtt {a}}},\quad {{\mathfrak {M}}}_\nu ^\sharp ( s, \mathtt {b}) \leqq C_* ({\mathfrak {s}}_M, \mathtt {b}) {{\mathfrak {M}}}_0 (s, {\mathtt {b}}) N_{\nu - 1}\,\nonumber \\ \end{aligned}$$

  (7.20)

  with $N_{\nu -1}$ given by (7.13). Moreover, if $\nu \geqq 1$ and $\omega \in {\mathtt {\Omega }}_\nu ^\gamma$, there exists a Hamiltonian operator ${ \Psi }_{\nu - 1}$ acting on $H^s_\bot$, so that the corresponding symplectic time one flow

  $$\begin{aligned} { \Phi }_{\nu - 1} \,{:=}\, \exp ( { \Psi }_{\nu - 1}) \end{aligned}$$

  (7.21)

  conjugates ${\mathtt {L}}_{\nu - 1}$ to

  $$\begin{aligned} {\mathtt {L}}_\nu = { \Phi }_{\nu - 1} {\mathtt {L}}_{\nu - 1} { \Phi }_{\nu - 1}^{- 1}. \end{aligned}$$

  (7.22)

  The operators $\Psi _{\nu - 1}$ and $\langle \partial _\varphi \rangle ^{\mathtt {b}} \Psi _{\nu - 1}$ are $\mathrm{{Lip}(\gamma )}$-modulo-tame with modulo-tame constants satisfying, for any $s \in [{\mathfrak {s}}_M, S]$, (with $\tau _1$, $\mathtt {a}$ defined in (7.6))

  $$\begin{aligned}&{{\mathfrak {M}}}_{\Psi _{\nu - 1}}^\sharp (s) \leqq \frac{C( {\mathfrak {s}}_M, \mathtt {b})}{\gamma } N_{\nu - 1}^{\tau _1} N_{\nu - 2}^{- \mathtt {a}} {{\mathfrak {M}}}_0 (s, {\mathtt {b}}) , \nonumber \\&\quad \quad {{\mathfrak {M}}}_{\langle \partial _\varphi \rangle ^{\mathtt {b}} \Psi _{\nu - 1}}^\sharp (s) \leqq \frac{C( {\mathfrak {s}}_M, \mathtt {b})}{\gamma } N_{\nu - 1}^{\tau _1} N_{\nu - 2} {{\mathfrak {M}}}_0 (s, {\mathtt {b}}) . \end{aligned}$$

  (7.23)

• $\mathbf{(S2)_{\nu }}$ For any $j \in {\mathbb {S}}^\bot$, there exists a Lipschitz extension ${{\widetilde{\mu }}}_j^\nu : \Omega \rightarrow {\mathbb {R}}$ of $\mu _j^\nu : \mathtt {\Omega }_\nu ^\gamma \rightarrow {\mathbb {R}}$, where ${\widetilde{\mu }}_j^0 = m_3 (2 \pi j)^3 - {\widetilde{m}}_1 2 \pi j - q_j(\omega )$ (cf. (7.2)) and ${\widetilde{m}}_1 : \Omega \rightarrow {\mathbb {R}}$ is a Lipschitz extension of $m_1$, satisfying $|{\widetilde{m}}_1|^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2}$ (recall that by (6.18) $m_3$ is defined on $\Omega$); for any $\nu \geqq 1$,

  $$\begin{aligned} |{\widetilde{\mu }}_j^\nu - {\widetilde{\mu }}_j^{\nu - 1}|^\mathrm{{Lip}(\gamma )}\lesssim {\mathfrak {M}}_{\nu - 1}^\sharp ({\mathfrak {s}}_M) \lesssim {{\mathfrak {M}}}_0 ({\mathfrak {s}}_M, {\mathtt {b}}) N_{\nu - 1}^{- {\mathtt {a}}}. \end{aligned}$$

  If needed, we indicate the dependence of ${\widetilde{\mu }}_j^\nu$ on the torus embedding by writing ${\widetilde{\mu }}_j^\nu (\omega ; \iota )$ or ${\widetilde{\mu }}_j^\nu ( \iota )$.

• $\mathbf{(S3)_{\nu }}$ Let $\breve{\iota }_1$, $\breve{\iota }_2$ be torus embeddings satisfying (6.1) with $\mu _0 \geqq {\mathfrak {s}}_M+ \mu (\mathtt {b})$. Then, for all $\omega \in {\mathtt {\Omega }}_\nu ^{\gamma _1}( \iota _1) \cap {\mathtt {\Omega }}_\nu ^{\gamma _2}( \iota _2)$ with $\gamma _1, \gamma _2 \in [\gamma /2, 2 \gamma ]$ and $0< \gamma < 1/2$, we have

  $$\begin{aligned}&\Vert |{\mathtt {R}}_\nu (\iota _1) - {\mathtt {R}}_\nu (\iota _2)| \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} \lesssim _{S} N_{\nu - 1}^{- \mathtt {a}} \Vert \iota _1 - \iota _2\Vert _{{\mathfrak {s}}_M+ \mu (\mathtt {b}) }, \end{aligned}$$

  (7.24)
  $$\begin{aligned}&\Vert |\langle \partial _\varphi \rangle ^{\mathtt {b}}({\mathtt {R}}_\nu (\iota _1) - {\mathtt {R}}_\nu (\iota _2)) | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} \lesssim _{S} N_{\nu - 1} \Vert \iota _1 - \iota _2\Vert _{ {\mathfrak {s}}_M+ \mu (\mathtt {b}) }. \end{aligned}$$

  (7.25)

  Moreover, if $\nu \geqq 1$, then for any $j \in {{\mathbb {S}}}^\bot$, $r_j^{\nu }$, given by (7.16), satisfies

  $$\begin{aligned}&\big |(r_j^\nu (\iota _1) - r_j^\nu (\iota _2)) - (r_j^{\nu - 1}(\iota _1) - r_j^{\nu - 1}(\iota _2)) \big | \lesssim \Vert |{\mathtt {R}}_\nu (\iota _1) - {\mathtt {R}}_\nu (\iota _2)| \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} , \end{aligned}$$

  (7.26)
  $$\begin{aligned}&\ |r_j^{\nu }(\iota _1) - r_j^{\nu }(\iota _2)| \lesssim _S \Vert \iota _1 - \iota _2 \Vert _{ {\mathfrak {s}}_M+ \mu (\mathtt {b}) }. \end{aligned}$$

  (7.27)

• $\mathbf{(S4)_{\nu }}$ Let $\breve{\iota }_1$, $\breve{\iota }_2$ be torus embeddings as in $\mathbf{(S3)_{\nu }}$ and $0< \rho < \gamma /2$. Then

  $$\begin{aligned} C(S) N_{\nu - 1}^\tau \Vert \iota _1 - \iota _2 \Vert _{ {\mathfrak {s}}_M+ \mu (\mathtt {b}) } \leqq \rho \quad \Longrightarrow \quad {\mathtt {\Omega }}_\nu ^\gamma (\iota _1) \subseteq {\mathtt {\Omega }}_\nu ^{\gamma - \rho }(\iota _2) . \end{aligned}$$

  (7.28)

Before proving Theorem 7.2 in the subsequent section, we discuss the following application. Theorem 7.2 implies that for any $n \ge 1$, the symplectic invertible operator

$$\begin{aligned} U_n \,{:=}\, \Phi _{n - 1} \circ \ldots \circ \Phi _0 , \quad \omega \in \bigcap _{\nu = 0}^{n } {\mathtt {\Omega }}_\nu ^\gamma , \end{aligned}$$

(7.29)

almost diagonalizes ${\mathtt {L}}_0$, meaning that (7.32) below holds. Note that since ${\mathtt {\Omega }}_{\nu }^\gamma \subset {\mathtt {\Omega }}_{\nu -1}^\gamma$ (cf. (7.18)), one has

$$\begin{aligned} \bigcap _{\nu = 0}^{n } {\mathtt {\Omega }}_\nu ^\gamma = {\mathtt {\Omega }}_n^{ \gamma } . \end{aligned}$$

(7.30)

By the same arguments as in [10], one infers the following theorem from Theorem 7.2.

Theorem 7.3

(KAM almost-reducibility) Assume the ansatz (6.1) with $\mu _0 \geqq {\mathfrak {s}}_M+ \mu (\mathtt {b})$. Then for any $S > {\mathfrak {s}}_M$ there exist $N_0 \,{:=}\, N_0(S, \mathtt {b}) > 0$, $0< \delta _0 \,{:=}\, \delta _0(S) < 1$, so that if

$$\begin{aligned} N_0^{{\overline{\tau }}} \varepsilon \gamma ^{- 3} \leqq \delta _0 , \end{aligned}$$

(7.31)

with ${\overline{\tau }} = {\overline{\tau }} (\tau , {{\mathbb {S}}}_+)$ given by Theorem 7.2, the following holds: For any $n \in {\mathbb {N}}$ and $\omega$ in ${\mathtt {\Omega }}_{n}^{ \gamma }$, the operator $U_n$, introduced in (7.29), is well defined and ${\mathtt {L}}_{n } \,{:=}\, {U}_n {\mathtt {L}}_0 {U}_n^{- 1}$ satisfies

$$\begin{aligned} {\mathtt {L}}_{n } = \omega \cdot \partial _\varphi + \mathrm{i} {\mathtt {D}}_n + {\mathtt {R}}_n \end{aligned}$$

(7.32)

where ${\mathtt {D}}_n$ and ${\mathtt {R}}_n$ are defined in (7.15) (with $\nu = n$). The operator ${\mathtt {R}}_n$ is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant

$$\begin{aligned} {{\mathfrak {M}}}^\sharp _{{\mathtt {R}}_n}(s) \lesssim _S N_{n - 1}^{- {\mathtt {a}}} (\varepsilon + \Vert \iota \Vert _{s + \mu (\mathtt {b}) }^\mathrm{{Lip}(\gamma )}) , \quad \forall {\mathfrak {s}}_M\leqq s \leqq S. \end{aligned}$$

(7.33)

Moreover, the operator ${\mathtt {L}}_n$ is Hamiltonian, ${U}_n$, ${U}_n^{- 1}$ are symplectic, and $U_{n}^{\pm 1} - \mathrm{Id}_\bot$ are $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant satisfying

$$\begin{aligned} {{\mathfrak {M}}}^\sharp _{U_{n}^{\pm 1} - \mathrm{Id}_\bot } (s) \lesssim _S \gamma ^{-1} N_0^{\tau _1} (\varepsilon + \Vert \iota \Vert _{s + \mu (\mathtt {b}) }^\mathrm{{Lip}(\gamma )}), \quad \forall {\mathfrak {s}}_M\leqq s \leqq S . \end{aligned}$$

(7.34)

Here $\mathrm{Id}_\bot$ denotes the identity operator on $L^2_\bot ({\mathbb {T}}_1)$ and $\tau _1$ is defined in (7.6).

7.1 Proof of Theorem 7.2

Theorem 7.2 is proved by induction. We first prove the base case $\nu = 0$ and then the induction step.

Base case $\nu = 0$. The items $\mathbf{(S1)}_{0}$, ..., $\mathbf{(S4)}_{0}$ are proved separately.

Proof of $\mathbf{(S1)}_{0}$. Properties (7.15)–(7.17) for $\nu = 0$ follow by (7.1)–(7.2) with $r_j^0 = 0$. Furthermore (7.20) holds for $\nu = 0$ in view of the following lemma, which can be proved by the same arguments used in the proof of Lemma 7.6 in [10].

Lemma 7.4

${{\mathfrak {M}}}_0^\sharp (s)$, ${{\mathfrak {M}}}_0^\sharp ( s, \mathtt {b}) \lesssim _{ \mathtt {b}} {{\mathfrak {M}}}_0 (s, {\mathtt {b}})$ where ${{\mathfrak {M}}}_0 (s, {\mathtt {b}})$ is defined in (7.9).

Proof of $\mathbf{(S2)}_0$. For any $j \in {\mathbb {S}}^\bot$, $\mu _j^{0}$ is defined in (7.2). Note that $m_3(\omega )$ and $q_j(\omega )$ are already defined on the whole parameter space $\Omega$. By the Kirszbraun extension Theorem for Lipschitz functions (which is a particular case of the Whitney extension Theorem as recorded in [1, Appendix B], see also Theorem 3, page 174 in [31]) and (6.68) there is an extension ${\widetilde{m}}_1$ on $\Omega$ of $m_1$ satisfying the estimate $|{\widetilde{m}}_1|^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2}$. This proves $\mathbf{(S2)}_0$.

Proof of $\mathbf{(S3)}_0$. The estimates (7.24), (7.25) at $\nu = 0$ follow by arguing as in the proof of $\mathbf{(S3)}_0$ in [10].

Proof of $\mathbf{(S4)}_{0}$. By the definition of $\mathtt {\Omega }_0^\gamma$ one has $\mathtt {\Omega }_0^\gamma (\iota _1) = {\mathtt {D} \mathtt {C}}(\gamma , \tau ) \subseteq {\mathtt {D} \mathtt {C}}(\gamma - \rho , \tau ) = \mathtt {\Omega }_0^{\gamma - \rho }(\iota _2)$.

Induction step. In this paragraph, we describe how to define $\Psi _\nu$, $\Phi _\nu$, ${\mathtt {L}}_{\nu + 1}$ etc., at the iterative step. To simplify notation we drop the index $\nu$ and write $+$ instead of $\nu + 1$. So, for example we write ${\mathtt {L}}$ for ${\mathtt {L}}_{\nu }$, ${\mathtt {L}}_+$ for ${\mathtt {L}}_{\nu +1}$, $\Psi$ for $\Psi _{\nu }$, N for $N_\nu$, ${{\mathfrak {M}}}^\sharp (s)$ for ${{\mathfrak {M}}}_\nu ^\sharp (s)$, etc. We conjugate ${\mathtt {L}}$ by the symplectic time one flow map

$$\begin{aligned} \Phi \,{:=}\, \exp ( \Psi ) \end{aligned}$$

(7.35)

generated by a Hamiltonian vector field $\Psi$ acting in $H^s_\bot$. By a Lie expansion we get

$$\begin{aligned} \begin{aligned}&\Phi \mathtt {L} \Phi ^{- 1} = \Phi ( \omega \cdot \partial _\varphi + \mathrm{i} \mathtt {D} ) \Phi ^{- 1} + \Phi \mathtt {R} \Phi ^{- 1} \\&\quad = \omega \cdot \partial _\varphi + \mathrm{i} \mathtt {D} - \omega \cdot \partial _\varphi \Psi - \mathrm{i} [\mathtt {D}, \Psi ] + \Pi _N \mathtt {R} + \Pi _N^\bot \mathtt {R} - \int _0^1 \mathrm{exp}( \tau \Psi )[\mathtt {R}, \Psi ] \mathrm{exp}(- \tau \Psi )\, \hbox {d} \tau \\&\quad \quad + \int _0^1 (1 - \tau ) \mathrm{exp}( \tau \Psi ) \Big [\omega \cdot \partial _\varphi \Psi + \mathrm{i} [\mathtt {D}, \Psi ] , \Psi \Big ] \mathrm{exp}(- \tau \Psi )\, \hbox {d} \tau \end{aligned}\nonumber \\ \end{aligned}$$

(7.36)

where the projector $\Pi _N$ is defined in (2.15) and $\Pi _N^\bot = \mathrm{Id }_\bot - \Pi _N$. We want to solve the homological equation

$$\begin{aligned} - \omega \cdot \partial _\varphi \Psi - \mathrm{i} [{\mathtt {D}}, \Psi ] + \Pi _{N} {\mathtt {R}} = [{\mathtt {R}}] \quad \mathrm{where} \quad [{\mathtt {R}}] \,{:=}\, \mathrm{diag}_{j \in {{\mathbb {S}}}^\bot } {\mathtt {R}}_j^j(0) . \end{aligned}$$

(7.37)

The solution of (7.37) is

$$\begin{aligned}&\Psi _j^{j'}(\ell ) \,{:=}\, {\left\{ \begin{array}{ll} \dfrac{{\mathtt {R}}_j^{j'}(\ell )}{\mathrm{i} (\omega \cdot \ell + \mu _j - \mu _{j'})} \quad \forall (\ell , j, j') \ne (0, j, j) , \, \, | \ell | \leqq N , \, j, j' \in {\mathbb {S}}^\bot \\ 0 \quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

(7.38)

Note that the denominators in (7.38) do not vanish for $\omega \in {\mathtt {\Omega }}_{\nu +1}^\gamma$ (cf. (7.18)).

Lemma 7.5

(Homological equations) (i) The solution $\Psi$ of the homological equation (7.37), given by (7.38) for $\omega \in {\mathtt {\Omega }}_{\nu +1}^{\gamma }$, is a $\mathrm{{Lip}(\gamma )}$-modulo-tame operator with a modulo-tame constant satisfying

$$\begin{aligned} {{\mathfrak {M}}}_{\Psi }^\sharp (s) \lesssim N^{\tau _1} \gamma ^{-1} {{\mathfrak {M}}}^\sharp (s) , \quad {{\mathfrak {M}}}_{\langle \partial _\varphi \rangle ^{\mathtt {b}} \Psi }^\sharp (s) \lesssim N^{\tau _1} \gamma ^{-1} {{\mathfrak {M}}}^\sharp (s, {\mathtt {b}}), \end{aligned}$$

(7.39)

where $\tau _1 \,{:=}\, 2 \tau + 1$. Moreover $\Psi$ is Hamiltonian.

(ii) Let $\breve{\iota }_1$, $\breve{\iota }_2$ be torus embeddings and define $\Delta _{12} \Psi \,{:=}\, \Psi (\iota _2) - \Psi (\iota _1)$. If $\gamma /2 \leqq \gamma _1, \gamma _2 \leqq 2 \gamma < 1$ then, for any $\omega \in {\mathtt {\Omega }}_{\nu + 1}^{\gamma _1}(\iota _1) \cap {\mathtt {\Omega }}_{\nu + 1}^{\gamma _2}(\iota _2)$,

$$\begin{aligned}&\Vert |\Delta _{12} \Psi | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})}\leqq C N^{2 \tau } \gamma ^{- 2} \big ( \Vert |{\mathtt {R}}(\iota _2)| \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} \Vert \iota _1 - \iota _2 \Vert _{{\mathfrak {s}}_M+ \mu (\mathtt {b})} + \Vert | \Delta _{1 2} {\mathtt {R}} | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} \big ) , \end{aligned}$$

(7.40)

$$\begin{aligned}&\Vert |\langle \partial _\varphi \rangle ^{\mathtt {b}}\Delta _{12} \Psi | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} \lesssim _{\mathtt {b}} N^{2 \tau } \gamma ^{- 2} \big ( \Vert |\langle \partial _\varphi \rangle ^{\mathtt {b}}{\mathtt {R}}(\iota _2) | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})} \Vert \iota _1 - \iota _2 \Vert _{ {\mathfrak {s}}_M+ \mu (\mathtt {b})} + \Vert | \langle \partial _\varphi \rangle ^{\mathtt {b}}\Delta _{12} {\mathtt {R}} | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})}\big ) . \nonumber \\ \end{aligned}$$

(7.41)

Proof

Since ${\mathtt {R}}$ is Hamiltonian, one infers from Definition 2.5 and Lemma 2.6-(iii) that the operator $\Psi$ defined in (7.38) is Hamiltonian as well. We now prove (7.39). Let $\omega \in {\mathtt {\Omega }}_{\nu +1}^{\gamma }$. By (7.18), and the definition of $\Psi$ in (7.38), it follows that for any $(\ell , j, j') \in {\mathbb {Z}}^{{{\mathbb {S}}}_+} \times {{\mathbb {S}}}^\bot \times {{\mathbb {S}}}^\bot$, with $|\ell | \leqq N$, $(\ell , j, j') \ne (0, j, j)$,

$$\begin{aligned} |\Psi _j^{j'}(\ell )| \lesssim \langle \ell \rangle ^\tau \gamma ^{-1} |{\mathtt {R}}_j^{j'}(\ell )| \end{aligned}$$

(7.42)

and

$$\begin{aligned}&\Delta _\omega \Psi _j^{j'}(\ell ) = \dfrac{\Delta _{\omega } {\mathtt {R}}_j^{j'}(\ell )}{\delta _{\ell j j'}(\omega _1)} - {\mathtt {R}}_j^{j'}(\ell ;\omega _2) \dfrac{\Delta _{\omega } \delta _{\ell j j'}}{\delta _{\ell j j'}(\omega _1) \delta _{\ell j j'}(\omega _2)},\nonumber \\&\delta _{\ell j j'}(\omega ) \,{:=}\, \mathrm{i} (\omega \cdot \ell + \mu _j - \mu _{j'}). \end{aligned}$$

By (7.5), (7.16), (7.17) one gets $|\Delta _\omega \delta _{\ell j j'}| \lesssim (\langle \ell \rangle + |j^3 - j'^3| ) |\omega _1 - \omega _2|,$ and therefore, using also (7.18), we deduce that

$$\begin{aligned} |\Delta _\omega \Psi _j^{j'}(\ell )| \lesssim \langle \ell \rangle ^\tau \gamma ^{- 1} |\Delta _{\omega } {\mathtt {R}}_j^{j'}(\ell )| + \langle \ell \rangle ^{2 \tau + 1} \gamma ^{- 2} |{\mathtt {R}}_j^{j'}(\ell ;\omega _2) | |\omega _1 - \omega _2| .\nonumber \\ \end{aligned}$$

(7.43)

Recalling the definitions (2.33), (7.19), using (7.42), (7.43), and arguing as in the proof of the estimates (7.61) in [10, Lemma 7.7], one then deduces (7.39). The estimates (7.40)–(7.41) can be obtained similarly. $\quad \square$

By (7.36)–(7.37) one has

$$\begin{aligned} \begin{aligned} {\mathtt {L}}_+&= \Phi {\mathtt {L}} \Phi ^{- 1} = \omega \cdot \partial _\varphi + \mathrm{i} {\mathtt {D}}_+ + {\mathtt {R}}_+ \end{aligned} \end{aligned}$$

which proves (7.22) and (7.15) at the step $\nu + 1$, with

$$\begin{aligned} \begin{aligned}&\mathrm{i} {\mathtt {D}}_+ \,{:=}\, \mathrm{i} {\mathtt {D}} + [{\mathtt {R}}] , \quad \\&{\mathtt {R}}_+ = \Pi _N^\bot \mathtt {R} - \int _0^1 \mathrm{exp}( \tau \Psi )[\mathtt {R}, \Psi ] \mathrm{exp}(- \tau \Psi )\, \hbox {d} \tau \\&\quad + \int _0^1 (1 - \tau ) \mathrm{exp}( \tau \Psi ) \big [\Pi _N \mathtt {R} - [\mathtt {R}] , \Psi \big ] \mathrm{exp}(- \tau \Psi )\, \hbox {d} \tau . \end{aligned} \end{aligned}$$

(7.44)

The operator ${\mathtt {L}}_+$ has the same form as $\mathtt {L}$. More precisely, ${\mathtt {D}}_+$ is diagonal and ${\mathtt {R}}_+$ is the sum of an operator supported on high frequencies and one which is quadratic in $\Psi$ and ${ \mathtt {R}}$. The new normal form ${\mathtt {D}}_+$ has the following properties:

Lemma 7.6

(New diagonal part) (i) The new normal form is

$$\begin{aligned}&{\mathtt {D}}_+ = {\mathtt {D}} - \mathrm{i} [{\mathtt {R}}] , \quad {\mathtt {D}}_+ \,{:=}\, \mathrm{diag}_{j \in {{\mathbb {S}}}^\bot } \mu _j^+,\nonumber \\&\mu _j^+ \,{:=}\, \mu _j + {\mathtt {r}}_j\in {\mathbb {R}}, \quad {\mathtt {r}}_j\,{:=}\, -\mathrm{i} {\mathtt {R}}_j^j(0), \ \ \forall j \in {\mathbb {S}}^\bot , \end{aligned}$$

(7.45)

with

$$\begin{aligned} \mu _{-j}^+ = - \mu _j^+ , \quad |\mu _j^+ - \mu _j|^\mathrm{{Lip}(\gamma )}= |\mathtt {r}_j|^\mathrm{{Lip}(\gamma )}\lesssim {{\mathfrak {M}}}^\sharp ({\mathfrak {s}}_M) . \end{aligned}$$

(ii) For any tori $\breve{\iota }_1(\omega )$, $\breve{\iota }_2(\omega )$ and any $\omega \in {\mathtt {\Omega }}_{\nu }^{\gamma _1}(\iota _1) \cap {\mathtt {\Omega }}_\nu ^{\gamma _2}(\iota _2)$, one has

$$\begin{aligned} | {\mathtt {r}}_j (\iota _1) -{\mathtt {r}}_j (\iota _2) | \lesssim \Vert | \Delta _{1 2} {\mathtt {R}} | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})}. \end{aligned}$$

(7.46)

Proof

By the definition (7.19) of ${{\mathfrak {M}}}^\sharp ({\mathfrak {s}}_M)$ and using (2.30) (with ${\mathfrak {s}}_M= s_1$) we have that $|\mu _j^+ - \mu _j|^\mathrm{{Lip}(\gamma )}\leqq |{\mathtt {R}}_j^j(0) |^\mathrm{{Lip}(\gamma )}\lesssim {{\mathfrak {M}}}^\sharp ({\mathfrak {s}}_M)$. Since ${\mathtt {R}} (\varphi )$ is Hamiltonian, Lemma 2.6 implies that ${\mathtt {r}}_j = - \mathrm{i} {\mathtt {R}}_j^j(0)$, $j \in {\mathbb {S}}^\bot$, are odd in j and real. The estimate (7.46) is proved in the same way by using $|\Delta _{1 2} {\mathtt {R}}_j^j(0) | \leqq C \Vert | \Delta _{1 2} {\mathtt {R}} | \Vert _{{{{\mathcal {B}}}}(H^{{\mathfrak {s}}_M})}$. $\quad \square$

Induction step. Assuming that $(\mathbf{S1})_{\nu }$–$(\mathbf{S4})_{\nu }$ are true for a given $\nu \geqq 0$, we show in this paragraph that each of the statements $(\mathbf{S1})_{\nu + 1}$, $(\mathbf{S2})_{\nu + 1}$, $(\mathbf{S3})_{\nu + 1}$, and $(\mathbf{S4})_{\nu + 1}$ hold.

Proof of $(\mathbf{S1})_{\nu + 1}$. By Lemma 7.5, for any $\omega \in {\mathtt {\Omega }}_{\nu + 1}^\gamma$, the solution $\Psi _{\nu }$ of the homological equation (7.37), defined in (7.38), is well defined and that by (7.39), (7.20), it satisfies the estimates (7.23) at $\nu + 1$. In particular, the estimate (7.23) for $\nu + 1$, $s = {\mathfrak {s}}_M$ together with (7.6), (7.14) imply that

$$\begin{aligned} {{\mathfrak {M}}}^\sharp _{\Psi _{\nu }}({\mathfrak {s}}_M) \lesssim _{\mathtt {b}} N_{\nu }^{\tau _1} N_{\nu - 1}^{- \mathtt {a}} \gamma ^{- 1} {{\mathfrak {M}}}_0({\mathfrak {s}}_M, {\mathtt {b}}) \leqq 1 . \end{aligned}$$

(7.47)

By Lemma 2.22 and using again Lemma 7.5 one infers that

$$\begin{aligned} \begin{aligned} {{\mathfrak {M}}}^\sharp _{\Phi _\nu ^{\pm 1}}({\mathfrak {s}}_M)&\lesssim 1, \\ {{\mathfrak {M}}}^\sharp _{\langle \partial _\varphi \rangle ^{\mathtt {b}} \Phi _\nu ^{\pm 1}}({\mathfrak {s}}_M)&\lesssim 1 + {{\mathfrak {M}}}_{\langle \partial _\varphi \rangle ^\mathtt {b} \Psi _\nu }^\sharp ({\mathfrak {s}}_M) \lesssim 1 + N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}^\sharp _\nu ({\mathfrak {s}}_M, \mathtt {b}), \\ {{\mathfrak {M}}}^\sharp _{\Phi _\nu ^{\pm 1}}(s)&\lesssim 1 + {{\mathfrak {M}}}^\sharp _{\Psi _\nu }(s) \lesssim _s 1 + N_\nu ^{\tau _1} \gamma ^{-1} {{\mathfrak {M}}}_\nu ^\sharp (s) , \\ {{\mathfrak {M}}}^\sharp _{\langle \partial _\varphi \rangle ^{\mathtt {b}} \Phi _\nu ^{\pm 1}}(s)&\lesssim 1 + {{\mathfrak {M}} }_{\langle \partial _\varphi \rangle ^{\mathtt {b}} \Psi _\nu }^\sharp (s) + {{\mathfrak {M}}}_{\Psi _\nu }^\sharp (s) {{\mathfrak {M}} }_{\langle \partial _\varphi \rangle ^{\mathtt {b} } \Psi _\nu }^\sharp ({\mathfrak {s}}_M) ,\\&{\mathop {\lesssim }\limits ^{(7.14), (7.20), (7.39)}} 1 + N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}^\sharp _\nu (s, \mathtt {b}) + N_\nu ^{2 \tau _1} N_{\nu - 1} \gamma ^{- 1} {{\mathfrak {M}}}^\sharp _\nu (s). \end{aligned} \end{aligned}$$

(7.48)

By Lemma 7.6, by the estimate (7.20) and Lemma 7.1, the operator ${\mathtt {D}}_{\nu +1}$ is diagonal and its eigenvalues $\mu _j^{\nu +1} : {\mathtt {\Omega }}_{\nu +1}^\gamma \rightarrow {\mathbb {R}}$ satisfy (7.17) at $\nu + 1$.

Next we estimate the remainder ${\mathtt {R}}_{\nu + 1}$ defined in (7.44).

Lemma 7.7

(Nash–Moser iterative scheme) The operator ${\mathtt {R}}_{\nu + 1}$ is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant satisfying

$$\begin{aligned} {{\mathfrak {M}}}_{\nu + 1}^\sharp (s ) \lesssim N_\nu ^{- {\mathtt {b}}} {{\mathfrak {M}}}_\nu ^\sharp (s, {\mathtt {b}}) + N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}_\nu ^\sharp (s) {{\mathfrak {M}}}_\nu ^\sharp ({\mathfrak {s}}_M). \end{aligned}$$

(7.49)

The operator $\langle \partial _\varphi \rangle ^{ {\mathtt {b}}} {\mathtt {R}}_{\nu + 1}$ is $\mathrm{{Lip}(\gamma )}$-modulo-tame with a modulo-tame constant satisfying

$$\begin{aligned}&{{\mathfrak {M}}}_{\nu + 1}^\sharp (s, {\mathtt {b}}) \lesssim _{\mathtt {b}} {{\mathfrak {M}}}_\nu ^\sharp (s, \mathtt {b}) + N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}_\nu ^\sharp (s, \mathtt {b}) {{\mathfrak {M}}}_\nu ^\sharp ({\mathfrak {s}}_M) \nonumber \\&\quad + N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}_\nu ^\sharp ({\mathfrak {s}}_M, \mathtt {b}) {{\mathfrak {M}}}_\nu ^\sharp (s) . \end{aligned}$$

(7.50)

Proof

We treat each of the three terms in the definition (7.44) of ${\mathtt {R}}_{\nu + 1}$ separately. By Lemma 2.23 and the definition (7.19) of ${{\mathfrak {M}}}_{ \nu }^\sharp (s, \mathtt {b})$, we have

$$\begin{aligned} \begin{aligned}&{{\mathfrak {M}}}_{\Pi _{N_\nu }^\bot \mathtt {R}_\nu }^\sharp (s) \leqq N_\nu ^{- {\mathtt {b}} }{{\mathfrak {M}}}_{ \langle \partial _{\varphi } \rangle ^{\mathtt {b}} \mathtt {R}_\nu }^\sharp (s) = N_\nu ^{- {\mathtt {b}} } {{\mathfrak {M}}}_{ \nu }^\sharp (s, \mathtt {b}) , \\&\quad {{\mathfrak {M}}}_{ \langle \partial _\varphi \rangle ^{\mathtt {b}} \Pi _{N_\nu }^\bot {\mathtt {R}}_\nu }^\sharp (s) \leqq {{\mathfrak {M}}}_{ \langle \partial _\varphi \rangle ^{\mathtt {b}} {\mathtt {R}}_\nu }^\sharp (s) = {{\mathfrak {M}}}_\nu ^{\sharp } (s, \mathtt {b}). \end{aligned} \end{aligned}$$

(7.51)

We now estimate the second term $\mathtt {G}_\nu \,{:=}\, - \int _0^1 \mathrm{exp}( \tau \Psi _\nu )[\mathtt {R}_\nu , \Psi _\nu ] \mathrm{exp}(- \tau \Psi _\nu )\, \hbox {d} \tau$ in (7.44). By applying (7.39), together with the composition Lemma 2.21, one obtains that

$$\begin{aligned} \begin{aligned}&{{\mathfrak {M}}}_{[\mathtt {R}_\nu , \Psi _\nu ]}^\sharp (s) \lesssim N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}^\sharp _\nu (s) {{\mathfrak {M}}}^\sharp _\nu ({\mathfrak {s}}_M), \\&{{\mathfrak {M}}}_{\langle \partial _\varphi \rangle ^{\mathtt {b}}[\mathtt {R}_\nu , \Psi _\nu ]}^\sharp (s) \lesssim N_\nu ^{\tau _1} \gamma ^{- 1} \big ({{\mathfrak {M}}}^\sharp _\nu (s, \mathtt {b}) {{\mathfrak {M}}}^\sharp _\nu ({\mathfrak {s}}_M) + {{\mathfrak {M}}}^\sharp _\nu ({\mathfrak {s}}_M, \mathtt {b}) {{\mathfrak {M}}}^\sharp _\nu (s) \big ). \end{aligned} \end{aligned}$$

(7.52)

Note that the estimates (7.48) also hold for the maps $\mathrm{exp}(\pm \tau \Psi _\nu )$, uniformly in $\tau \in [- 1 , 1]$. Therefore, taking into account (7.52) and (7.48), Lemma 2.21, the induction hypothesis for the estimate (7.20), the smallness condition (7.14) (with ${{\bar{\tau }}}$ large enough) and (7.6), $N_\nu ^{\tau _1}{{\mathfrak {M}}}_\nu ^\sharp ({\mathfrak {s}}_M) \gamma ^{- 1}\leqq 1$, one concludes that

$$\begin{aligned} \begin{aligned}&{{\mathfrak {M}}}_{\mathtt {G}_n}^\sharp (s) \lesssim N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}^\sharp _\nu (s) {{\mathfrak {M}}}^\sharp _\nu ({\mathfrak {s}}_M), \\&{{\mathfrak {M}}}_{\langle \partial _\varphi \rangle ^{\mathtt {b}}\mathtt {G}_n}^\sharp (s) \lesssim N_\nu ^{\tau _1} \gamma ^{- 1} \big ({{\mathfrak {M}}}^\sharp _\nu (s, \mathtt {b}) {{\mathfrak {M}}}^\sharp _\nu ({\mathfrak {s}}_M) + {{\mathfrak {M}}}^\sharp _\nu ({\mathfrak {s}}_M, \mathtt {b}) {{\mathfrak {M}}}^\sharp _\nu (s) \big ). \end{aligned} \end{aligned}$$

(7.53)

The estimate of the third term in (7.44) is obtained in a similar way. Together with (7.51) and (7.53) it yields (7.49)–(7.50). $\quad \square$

The estimates (7.49), (7.50), and (7.6), now allow us to prove that (7.20) holds at the step $\nu + 1$.

Lemma 7.8

${{\mathfrak {M}}}_{\nu + 1}^ \sharp (s) \leqq C_*({\mathfrak {s}}_M, \mathtt {b}) {{\mathfrak {M}}}_0(s, \mathtt {b}) N_\nu ^{- \mathtt {a}}$ and ${{\mathfrak {M}}}_{\nu + 1}^\sharp (s,\mathtt {b}) \leqq C_*({\mathfrak {s}}_M, \mathtt {b}) {{\mathfrak {M}}}_0 (s, \mathtt {b}) N_\nu$.

Proof

By (7.49), the induction hypothesis for the estimate (7.20) we get

$$\begin{aligned} {{\mathfrak {M}}}_{\nu + 1}^\sharp (s)&\leqq C_*({\mathfrak {s}}_M, \mathtt {b})N_{\nu }^{- \mathtt {b}} N_{\nu - 1} {{\mathfrak {M}}}_0(s, \mathtt {b}) + C C_*({\mathfrak {s}}_M, \mathtt {b})^2N_\nu ^{\tau _1} \gamma ^{- 1} {{\mathfrak {M}}}_0(s, \mathtt {b}) {{\mathfrak {M}}}_0({\mathfrak {s}}_M, \mathtt {b}) N_{\nu - 1}^{- 2 \mathtt {a}} \\&\leqq C_*(s_0, \mathtt {b}) N_\nu ^{- \mathtt {a}} {{\mathfrak {M}}}_0(s, \mathtt {b}) \end{aligned}$$

where for the latter inequality we used the definition (7.6) of the constants and the bound (7.14) with ${\bar{\tau }}$ and $N_0 \,{:=}\, N_0(S, \mathtt {b}) > 0$ large enough. Similarly, by (7.50), (7.20),

$$\begin{aligned} {{\mathfrak {M}}}_{\nu + 1}^\sharp (s, \mathtt {b})&\leqq C_*(s_0, \mathtt {b})N_{\nu - 1} {{\mathfrak {M}}}_0(s, \mathtt {b}) + C C_*(s_0, \mathtt {b})^2 N_\nu ^{\tau _1} N_{\nu - 1}^{1 - \mathtt {a}} \gamma ^{- 1} {{\mathfrak {M}}}_0(s, \mathtt {b}) {{\mathfrak {M}}}_0({\mathfrak {s}}_M, \mathtt {b}) \nonumber \\&\leqq C_*(s_0, \mathtt {b}) N_\nu {{\mathfrak {M}}}_0(s, \mathtt {b}) \nonumber \end{aligned}$$

where for the latter inequality, we again used (7.6) and (7.14) with $N_0 \,{:=}\, N_0(S, \mathtt {b}) > 0$ large enough. $\quad \square$

Proof of $(\mathbf{S2})_{\nu + 1}$. By Lemma 7.6, for any $j \in {\mathbb {S}}^\bot$, $\mu _j^{\nu + 1} = \mu _j^\nu + \mathtt {r}_j^\nu$ where $|\mathtt {r}_j^\nu |^\mathrm{{Lip}(\gamma )}\lesssim {\mathfrak {M}}_0({\mathfrak {s}}_M, \mathtt {b}) N_\nu ^{- \mathtt {a}}$. Then $(\mathbf{S2})_{\nu + 1}$ follows by defining ${\widetilde{\mu }}_j^{\nu + 1} \,{:=}\, {\widetilde{\mu }}_j^\nu + \widetilde{\mathtt {r}}_j^\nu$ where $\widetilde{\mathtt {r}}_j^\nu : \Omega \rightarrow {\mathbb {R}}$ is a Lipschitz extension of ${\mathtt {r}}_j^\nu$ (using again the Kirszbraun extension theorem).

Proof of $(\mathbf{S3})_{\nu + 1}$. The proof follows by induction arguing as in the proof of $(\mathbf{S2})_{\nu + 1}$.

Proof of $(\mathbf{S4})_{\nu + 1}$. The proof is the same as that of $(\mathbf{S3})_{\nu + 1}$ in [2, Theorem 4.2]. $\quad \square$

7.2 Almost-Invertibility of ${{\mathcal {L}}}_\omega$

By (7.32), for any $\omega \in {\mathtt {\Omega }}^\gamma _n$, we have that $\mathtt {L}_0 = U_n^{-1} \mathtt {L}_n U_n$ where $U_n$ is defined in (7.29) and thus

$$\begin{aligned} {{\mathcal {L}}}_\omega = {{{\mathcal {V}}}}_n^{-1} {\mathtt {L}}_n {{{\mathcal {V}}}}_n , \quad {{{\mathcal {V}}}}_n \,{:=}\, U_n \Phi ^{(4)} \cdots \Phi ^{(1)} , \end{aligned}$$

(7.54)

where ${{{\mathcal {L}}}}_\omega$ is the operator introduced in (5.26) and $\Phi ^{(1)}$, ..., $\Phi ^{(4)}$ are the transformations constructed in Lemmas 6.4, 6.5, 6.6, and respectively, Proposition 6.7.

Lemma 7.9

There exists $\sigma = \sigma (\tau , {\mathbb {S}}_+) > 0$ such that, if (7.31) and (6.1) with $\mu _0 \geqq {\mathfrak {s}}_M+ \mu (\mathtt {b}) + \sigma$ hold, then the operators ${{{\mathcal {V}}}}_n^{\pm 1}$ satisfy for any ${\mathfrak {s}}_M\leqq s \leqq S$ the estimate

$$\begin{aligned} \Vert {{{\mathcal {V}}}}_n^{\pm 1} h \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim _S \Vert h \Vert _{s + \sigma }^\mathrm{{Lip}(\gamma )}+ N_0^{\tau _1} \gamma ^{- 1}\Vert \iota \Vert _{s + \mu (\mathtt {b}) + \sigma }^\mathrm{{Lip}(\gamma )}\Vert h \Vert _{{\mathfrak {s}}_M+ \sigma }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(7.55)

Proof

The claimed estimates follow from the estimates (6.30), (6.46), (6.54), (6.72), and (7.34) together with the Lemmata 2.16, 2.17, 2.20. $\quad \square$

We now decompose the operator ${\mathtt {L}}_n = \omega \cdot \partial _\varphi + \mathrm{i} {\mathtt {D}}_n + {\mathtt {R}}_n$ in (7.32) as

$$\begin{aligned} {\mathtt {L}}_n = {{\mathfrak {L}}}_n^{<} + {\mathtt {R}}_n + {\mathtt {R}}_n^\bot \end{aligned}$$

(7.56)

with

$$\begin{aligned} {{\mathfrak {L}}}_n^{<} \,{:=}\, \Pi _{K_n} \big ( \omega \cdot \partial _\varphi + \mathrm{i} {\mathtt {D}}_n \big ) \Pi _{K_n} + \Pi _{K_n}^\bot , \quad {\mathtt {R}}_n^\bot \,{:=}\, \Pi _{K_n}^\bot \big ( \omega \cdot \partial _\varphi + \mathrm{i} {\mathtt {D}}_n \big ) \Pi _{K_n}^\bot - \Pi _{K_n}^\bot ,\nonumber \\ \end{aligned}$$

(7.57)

where the diagonal operator ${\mathtt {D}}_n$ is defined in (7.15) (with $\nu = n$), $K_n = K_0^{\chi ^n}$ is the scale of the nonlinear Nash–Moser iterative scheme introduced in (5.28), and $\Pi _{K_n}^\bot = \mathrm{Id}_\bot - \Pi _{K_n}$ with $\Pi _{K_n}$ denoting the projector in (2.2). The diagonal constant coefficient operator ${{\mathfrak {L}}}_n^{<}$ can be inverted assuming the following standard non-resonance conditions:

Lemma 7.10

(First order Melnikov non-resonance conditions) Let $n \ge 0$. Then for any $\omega$ in

$$\begin{aligned} {\mathtt {\Lambda }}_{n + 1}^{\gamma } \,{:=}\, {\mathtt {\Lambda }}_{n + 1}^{\gamma } ( \iota ) \,{:=}\, \big \{ \omega \in \mathtt {\Omega } : |\omega \cdot \ell + {\widetilde{\mu }}_j^n| \geqq 2\gamma |j|^{3} \langle \ell \rangle ^{- \tau } , \quad \forall | \ell | \leqq K_n,\ j \in {\mathbb {S}}^\bot \big \} ,\nonumber \\ \end{aligned}$$

(7.58)

the operator ${{\mathfrak {L}}}_n^<$ in (7.57) is invertible and

$$\begin{aligned} \Vert ({{\mathfrak {L}}}_n^<)^{- 1} g \Vert _s^\mathrm{{Lip}(\gamma )}\lesssim \gamma ^{- 1} \Vert g \Vert _{s + 2 \tau + 1 }^\mathrm{{Lip}(\gamma )}. \end{aligned}$$

(7.59)

By (7.54), (7.56), Theorem 7.3, estimates (7.59), (7.60), (7.55), and using that, for all $b > 0$,

$$\begin{aligned} \Vert {\mathtt {R}}_n^\bot h \Vert _{{\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}\lesssim K_n^{- b} \Vert h \Vert _{{\mathfrak {s}}_M+ b + 3}^\mathrm{{Lip}(\gamma )}, \quad \Vert {\mathtt {R}}_n^\bot h\Vert _s^\mathrm{{Lip}(\gamma )}\lesssim \Vert h \Vert _{s + 3}^\mathrm{{Lip}(\gamma )}, \end{aligned}$$

(7.60)

we deduce the following theorem, stating the assumption A-I on the almost-invertibility of ${{{\mathcal {L}}}}_\omega$ in Section 5:

Theorem 7.11

(Almost-invertibility of ${{{\mathcal {L}}}}_\omega$) Assume the ansatz (6.1) with $\mu _0 \geqq {\mathfrak {s}}_M+ \mu (\mathtt {b})$. Let ${\mathtt {a}}, {\mathtt {b}}, M$ as in (7.6), and $S > {\mathfrak {s}}_M$. There exists $\sigma = \sigma (\tau , {\mathbb {S}}_+) > 0$ so that, if (7.31) and (6.1) hold with $\mu _0 \geqq {\mathfrak {s}}_M+ \mu (\mathtt {b}) + \sigma$, then, for any $n \ge 0$ and any

$$\begin{aligned} \omega \in {\varvec{\Omega }}_{n + 1}^{\gamma } = {\varvec{\Omega }}_{n + 1}^{\gamma } (\iota ) \,{:=}\, {\mathtt {\Omega }}_{n + 1}^\gamma (\iota ) \cap {\mathtt {\Lambda }}_{n + 1}^{\gamma } (\iota ) \end{aligned}$$

(7.61)

(see (7.18), (7.58)), the operator ${{\mathcal {L}}}_\omega$, defined in (5.26), can be decomposed as

$$\begin{aligned} {{\mathcal {L}}}_\omega = {{\mathcal {L}}}_\omega ^{<} + {{{\mathcal {R}}}}_\omega + {{{\mathcal {R}}}}_\omega ^\bot , \quad {{\mathcal {L}}}_\omega ^{<} \,{:=}\, {{{\mathcal {V}}}}_n^{-1} {{\mathfrak {L}}}_n^{<} {{{\mathcal {V}}}}_n ,\quad {{{\mathcal {R}}}}_\omega \,{:=}\, {{{\mathcal {V}}}}_n^{- 1} {\mathtt {R}}_n {{{\mathcal {V}}}}_n ,\quad {{{\mathcal {R}}}}_\omega ^\bot \,{:=}\, {{{\mathcal {V}}}}_n^{- 1} {\mathtt {R}}_n^\bot {{{\mathcal {V}}}}_n \, , \end{aligned}$$

(7.62)

where ${{\mathcal {L}}}_\omega ^{<}$ is invertible and satisfies (5.32) and the operators ${{{\mathcal {R}}}}_\omega$ and ${{{\mathcal {R}}}}_\omega ^\bot$ satisfy (5.30)–(5.31).

8 Proof of Theorem 1.1

The goal of this section is to prove Theorem 1.1. In Section 8.3 we deduce Theorem 1.1 from Theorem 4.1. The proof of the latter theorem is also given in Section 8.3, relying on the Nash–Moser Theorem 8.1 and its Corollary 8.4 in Section 8.1, and the measure estimate of Proposition 8.7 in Section 8.2.

8.1 The Nash–Moser Iteration

The main result of this section is Theorem 8.1 which implements a Nash–Moser iteration scheme by providing a sequence of better and better approximate solutions of the equation ${{{\mathcal {F}}}}_\omega ( \iota , \zeta ) = 0$ (cf. (4.6)) under the assumption that $\omega$ satisfies nonresonance conditions, that is $\omega$ belongs to the sets ${{{\mathcal {G}}}}_n$ defined in (8.10). A key ingredient into its proof is Theorem 5.7, concerning the existence of an approximate right inverse of $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega ( \iota , \zeta )$.

To describe the iteration scheme, first recall that $L^2_\varphi = L^2_\varphi ({\mathbb {T}}^{{{\mathbb {S}}}_+}, {\mathbb {R}}^{{{\mathbb {S}}}_+})$ (cf. (4.9)) and $L^2_\bot = L^2({\mathbb {T}}^{{{\mathbb {S}}}_+}, L^2_\bot ({\mathbb {T}}_1))$ (cf. (1.24)). We then introduce finite-dimensional subspaces of $L^2_\varphi \times L^2_\varphi \times L^2_\bot$, defined for any $n \in {\mathbb {N}}$ as

$$\begin{aligned} \mathtt {E}_n \,{:=}\, \big \{ \iota (\varphi ) = (\Theta , y , w) (\varphi ) , \ \ \Theta = \Pi _n \Theta , \ y = \Pi _n y , \ w = \Pi _n w \big \} \end{aligned}$$

where, by a slight abuse of notation, $\Pi _n : L^2_\bot \rightarrow \cap _{s \ge 0}H^s_\bot$ denotes the projector $\Pi _{K_n}$, introduced in (2.2),

$$\begin{aligned} \Pi _n : \, w = \sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}, j \in {{\mathbb {S}}}^\bot } w_{\ell , j} e^{\mathrm{i} (\ell \cdot \varphi + 2 \pi jx)} \quad \mapsto \quad \Pi _n w \,{:=}\, \sum _{|(\ell ,j)| \leqq K_n} w_{\ell , j} e^{\mathrm{i} (\ell \cdot \varphi + 2 \pi jx)},\nonumber \\ \end{aligned}$$

(8.1)

with $K_n = K_0^{\chi ^n}$, $n \ge 1$, (cf. (5.28)) and $K_0 > 1$ a constant chosen in Theorem 8.1 below. We also denote by $\Pi _n$ the $L^2-$ orthogonal projector on $L^2_\varphi ,$ $L^2_\varphi \rightarrow \cap _{s \ge 0} H^s_{\varphi }$, $v = \sum _{\ell \in {\mathbb {Z}}^{{{\mathbb {S}}}_+}} v_{\ell } e^{\mathrm{i} \ell \cdot \varphi } \mapsto \Pi _n(v) = \sum _{|\ell | \leqq K_n} v_{\ell } e^{\mathrm{i} \ell \cdot \varphi }$. The projectors $\Pi _n$, $n \ge 0$, are smoothing operators on the Sobolev spaces $H^s_\bot$ (and $H^s_{\varphi }$), meaning that $\Pi _n$ and $\Pi _n^\bot \,{:=}\, \mathrm{Id} - \Pi _n$ satisfy the smoothing properties (2.3).

For the Nash–Moser Theorem 8.1, stated below, we introduce the constants

$$\begin{aligned}&{{\overline{\sigma }}} \,{:=}\, \max \{ \sigma _1, \sigma _2 \}\, , \quad {\mathtt {b}} \,{:=}\, [{\mathtt {a}}] + 2, \quad {\mathtt {a}} = 3 \tau _1 + 1, \quad \tau _1 = 2 \tau + 1 , \quad \chi = 3/ 2 , \end{aligned}$$

(8.2)

$$\begin{aligned}&{\mathtt {a}}_1 \,{:=}\, \mathrm{max}\{12 {{\overline{\sigma }}} + 13, \, p \tau + 3 + \chi (\mu (\mathtt {b}) + 2 {{\overline{\sigma }}}) \}, \quad \mathtt {a}_2 \,{:=}\, \chi ^{- 1} \mathtt {a}_1 - \mu (\mathtt {b}) - 2 {{\overline{\sigma }}} , \end{aligned}$$

(8.3)

$$\begin{aligned}&{\mathtt {b}}_1 \,{:=}\, {\mathtt {a}}_1 + \mu ({\mathtt {b}}) + 3 {{\overline{\sigma }}} + 4 + \frac{2}{3} \mu _1 , \quad \mu _1 \,{:=}\, 3( \mu ({\mathtt {b}}) + 2{{\overline{\sigma }}} + 2 ) + 1 , \quad S \,{:=}\, {\mathfrak {s}}_M+ {\mathtt {b}}_1 , \end{aligned}$$

(8.4)

where $\sigma _1$ is defined in Lemma 4.3, $\sigma _2$ in Theorem 5.7, ${\mathtt {a}}$, $\mu (\mathtt {b})$ in (7.6) , and ${\mathfrak {s}}_M$ in (2.55). The number p is the exponent in (5.27) and is requested to satisfy

$$\begin{aligned} p {\mathtt {a}} > (\chi - 1 ) \cdot {\mathtt {a}}_1 + \chi \cdot ({{\overline{\sigma }}} + 4) \ {\mathop {=}\limits ^{\chi = 3/2}} \ \frac{1}{2} {\mathtt {a}}_1 + \frac{3}{2} ({{\overline{\sigma }}} + 4) . \end{aligned}$$

(8.5)

In view of the definition (8.3) of ${\mathtt {a}}_1$, we can define $p \,{:=}\, p(\tau , {{\mathbb {S}}}_+)$ as

$$\begin{aligned} p \,{:=}\, \frac{12 {{\overline{\sigma }}} + 17 + \chi \cdot (\mu (\mathtt {b}) + 2 {{\overline{\sigma }}})}{ \mathtt {a}} . \end{aligned}$$

(8.6)

We denote by $\Vert W \Vert _{s}^\mathrm{{Lip}(\gamma )}\,{:=}\, \max \{ \Vert \iota \Vert _{s}^\mathrm{{Lip}(\gamma )}, | \zeta |^\mathrm{{Lip}(\gamma )}\}$ the norm of a map

$$\begin{aligned} W \,{:=}\, ( \iota , \zeta ): {\mathtt {\Omega }} \rightarrow \big ( H^{s}_\varphi \times H^{s}_\varphi \times H^{s}_\bot \big ) \times {\mathbb {R}}^{{{\mathbb {S}}}_+} , \ \ \omega \mapsto W (\omega ) = ( \iota (\omega ), \zeta (\omega )) . \end{aligned}$$

Theorem 8.1

(Nash–Moser) There exist $0< \delta _0 < 1$ (small) and $C_* > 0$ (large) so that if

$$\begin{aligned}&\tau _2 \,{:=}\, \mathrm{max}\{ p {\overline{\tau }} + 3, \ 4 {{\overline{\sigma }}} + 4 + {\mathtt {a}}_1 \} , \quad \gamma \,{:=}\, \varepsilon ^{\mathfrak {a}},\quad 0< {\mathfrak {a}}< \frac{1}{ \tau _2}, \nonumber \\&\quad \quad \quad K_0 \,{:=}\, \gamma ^{- 1}, \quad \varepsilon K_0^{\tau _2} = \varepsilon ^{1 - {\mathfrak {a}}\tau _2}< \delta _0 , \end{aligned}$$

(8.7)

where ${\overline{\tau }} \,{:=}\, {\overline{\tau }}(\tau , {{\mathbb {S}}}_+)$ is defined as in Theorem 7.2, then the following holds: for any $n \in {\mathbb {N}}$

• $({{{\mathcal {P}}}}1)_{n}$ (Estimates in low norms) Let ${{\tilde{W}}}_0 \,{:=}\, (0, 0)$. If $n \ge 1,$ then there exists a $\mathrm{{Lip}(\gamma )}$-function

  $$\begin{aligned} {{\tilde{W}}}_n : {\mathbb {R}}^{{{\mathbb {S}}}_+} \rightarrow \mathtt {E}_{n -1} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} , \ \omega \mapsto {{\tilde{W}}}_n (\omega ) \,{:=}\, ({{\tilde{\iota }}}_n, {{\tilde{\zeta }}}_n ) , \end{aligned}$$

  satisfying

  $$\begin{aligned} \Vert {{\tilde{W}}}_n \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {{\overline{\sigma }}}}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2}. \end{aligned}$$

  (8.8)

  Let ${{\tilde{U}}}_n \,{:=}\, U_0 + {{\tilde{W}}}_n$ where $U_0 \,{:=}\, (\varphi ,0,0, 0)$. The difference ${{\tilde{H}}}_n \,{:=}\, {{\tilde{U}}}_{n} - {{\tilde{U}}}_{n-1}$ satisfies

  $$\begin{aligned} \Vert {{\tilde{H}}}_1 \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {{\overline{\sigma }}}}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2} \quad (n =1), \quad \Vert {{\tilde{H}}}_{n} \Vert _{ {\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {{\overline{\sigma }}}}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2} K_{n - 1}^{- \mathtt {a}_2} \quad (n \ge 2)\, ,\nonumber \\ \end{aligned}$$

  (8.9)

  where $K_n = K_0^{\chi ^n}$ (cf. (5.28)).

• $({{{\mathcal {P}}}}2)_{n}$ Define

  $$\begin{aligned} {{{\mathcal {G}}}}_{0} \,{:=}\, \Omega , \quad {{{\mathcal {G}}}}_{n} \,{:=}\, {{{\mathcal {G}}}}_{n-1} \cap \ {\varvec{\Omega }}_{n }^{ \gamma }({{\tilde{\iota }}}_{n-1}) \quad \forall n \ge 1 , \end{aligned}$$

  (8.10)

  where ${\varvec{\Omega }}_{n}^{ \gamma }({{\tilde{\iota }}}_{n-1})$ is defined in (7.61). Then for any $\omega \in {{{\mathcal {G}}}}_{n}$

  $$\begin{aligned} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{ {\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}\leqq C_* \varepsilon K_{n - 1}^{- {\mathtt {a}}_1} , \quad K_{-1} \,{:=}\, 1 . \end{aligned}$$

  (8.11)

• $({{{\mathcal {P}}}}3)_{n}$ (Estimates in high norms)   $\Vert {{\tilde{W}}}_n \Vert _{ {\mathfrak {s}}_M+ {\mathtt {b}}_1}^\mathrm{{Lip}(\gamma )}\leqq C_* \varepsilon K_{n - 1}^{\mu _1},$    $\forall \omega \in {{{\mathcal {G}}}}_{n}$.

Proof

The theorem can be proved in a by now standard way (cf. [1, 10]). We argue by induction. To simplify notation, we write within this proof $\Vert \cdot \Vert$ instead of $\Vert \cdot \Vert ^{\mathrm{{Lip}(\gamma )}}$.

Step 1: Proof of $({{{\mathcal {P}}}}1)_0$, $({\mathcal {P}}2)_0$, $({\mathcal {P}}3)_0$. Note that $({{{\mathcal {P}}}}1)_0$ and $({{{\mathcal {P}}}}3)_0$ are trivially satisfied and hence it remains to verify (8.11) for $n = 0$. By (4.6), (4.13), (4.3), and Lemma 4.3, there exists $C_* > 0$ large enough so that $\Vert {{{\mathcal {F}}}}_\omega (U_0) \Vert _{ {\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}\le \varepsilon C_*$.

Step 2: Proof of induction step. Assuming that $({{{\mathcal {P}}}}1)_n$, $({\mathcal {P}}2)_n$, $({\mathcal {P}}3)_n$ hold for some $n \geqq 0$, it is to prove that $({{{\mathcal {P}}}}1)_{n+1}$, $({\mathcal {P}}2)_{n+1}$, $({\mathcal {P}}3)_{n+1}$ hold. We are going to define the approximation ${{\tilde{U}}}_{n+1}$ by a modified Nash–Moser scheme. To this end, we prove the almost-approximate invertibility of the linearized operator

$$\begin{aligned} L_n = L_n(\omega ) \,{:=}\, \hbox {d}_{\iota ,\zeta } {{{\mathcal {F}}}}_\omega ( {\tilde{\iota }}_n(\omega )) \end{aligned}$$

(8.12)

by applying Theorem 5.7 to ${ L}_n(\omega )$. To prove that the assumptions (5.29)–(5.32) in Theorem 5.7 hold, we apply Theorem 7.11 with $\iota = {\tilde{\iota }}_n$. By choosing $\varepsilon$ small enough it follows from (8.7) that $N_0 = K_0^p = \gamma ^{-p}= \varepsilon ^{-p {\mathfrak {a}}}$ and the smallness condition (7.31) required in Theorem 7.11 holds. In addition, (6.1) holds by (8.9). Therefore Theorem 7.11 applies, and we deduce that (5.29)–(5.32) hold for all $\omega$ in the set ${\varvec{\Omega }}_{n + 1}^{\gamma }({\tilde{\iota }}_n)$, defined in (7.61). Now we apply Theorem 5.7 to the linearized operator $L_n(\omega )$ with ${\Omega }_o = {\varvec{\Omega }}_{n + 1}^{\gamma }({\tilde{\iota }}_n)$ and $S = {\mathfrak {s}}_M+ \mathtt {b}_1$ (cf. (8.4)). It implies that there exists an almost-approximate inverse $\mathbf{T}_n \,{:=}\, { \mathbf{T}}_n (\omega , {{\tilde{\iota }}}_n(\omega ))$ satisfying

$$\begin{aligned} \Vert \mathbf{T}_n g \Vert _s&\lesssim _{{\mathfrak {s}}_M+ \mathtt {b}_1} \gamma ^{-2 } \big ( \Vert g \Vert _{s + {{\overline{\sigma }}}} + K_0^{\tau _1 p} \gamma ^{- 1} \Vert {\tilde{\iota }}_n \Vert _{s + \mu ({\mathtt {b}}) + {{\overline{\sigma }}} } \Vert g \Vert _{{\mathfrak {s}}_M+ {{\overline{\sigma }}} }\big ) , \quad \forall {\mathfrak {s}}_M\leqq s \leqq {\mathfrak {s}}_M+ \mathtt {b}_1 , \end{aligned}$$

(8.13)

where we used that ${{\overline{\sigma }}} \geqq \sigma _2$ (cf. (8.2)), $\sigma _2$ is the loss of regularity constant appearing in the estimate (5.45), and $N_0 = K_0^p$. Furthermore, by (8.7)–(8.8), one obtains

$$\begin{aligned} K_0^{\tau _1 p} \gamma ^{- 1} \Vert {{\tilde{W}}}_n \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {{\overline{\sigma }}} } \leqq 1 , \end{aligned}$$

(8.14)

and hence, for the special value $s = {\mathfrak {s}}_M$, (8.13) becomes

$$\begin{aligned} \Vert \mathbf{T}_n g \Vert _{{\mathfrak {s}}_M}&\lesssim _{ \mathtt {b}_1} \gamma ^{-2 } \Vert g \Vert _{{\mathfrak {s}}_M+ {{\overline{\sigma }}}}. \end{aligned}$$

(8.15)

For all $\omega \in {{{\mathcal {G}}}}_{n + 1} = {{{\mathcal {G}}}}_n \cap {\varvec{\Lambda }}_{n + 1}^{\gamma }({\tilde{\iota }}_n)$ (cf. (8.10)), we define

$$\begin{aligned} U_{n + 1} \,{:=}\, {{\tilde{U}}}_n + H_{n + 1}, \quad H_{n + 1} \,{:=}\, ( {\widehat{\iota }}_{n+1}, {\widehat{\zeta }}_{n+1}) \,{:=}\, - {\varvec{\Pi }}_{n } \mathbf{T}_n \Pi _{n } {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \in \mathtt {E}_n \times {\mathbb {R}}^{{{\mathbb {S}}}_+},\nonumber \\ \end{aligned}$$

(8.16)

where ${\varvec{\Pi }}_n$ is defined by (cf. (8.1))

$$\begin{aligned} {\varvec{\Pi }}_n ({\iota }, \zeta ) \,{:=}\, (\Pi _n \iota , \zeta ), \quad {\varvec{\Pi }}_n^\bot (\iota , \zeta ) \,{:=}\, (\Pi _n^\bot \iota , 0), \quad \forall (\iota , \zeta ). \end{aligned}$$

(8.17)

To show that the iterative scheme in (8.16) is rapidly converging, we write

$$\begin{aligned} {{{\mathcal {F}}}}_\omega (U_{n + 1}) = {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) + L_n H_{n + 1} + Q_n, \end{aligned}$$

(8.18)

where $L_n \,{:=}\, \hbox {d}_{\iota ,\zeta } {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)$ and $Q_n$ is defined by (8.18). Then, by the definition of $H_{n+1}$ in (8.16), and by taking into account (8.17), one has

$$\begin{aligned} {{{\mathcal {F}}}}_\omega (U_{n + 1})&= {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) - L_n {\varvec{\Pi }}_{n } \mathbf{T}_n \Pi _{n } {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) + Q_n \nonumber \\&= {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) - L_n \mathbf{T}_n \Pi _{n } {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) + L_n {\varvec{\Pi }}_n^\bot \mathbf{T}_n \Pi _{n } {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) + Q_n \nonumber \\&= \Pi _{n }^\bot {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) + R_n + Q_n + P_n \end{aligned}$$

(8.19)

where

$$\begin{aligned} R_n \,{:=}\, L_n {\varvec{\Pi }}_n^\bot \mathbf{T}_n \Pi _{n }{{{\mathcal {F}}}}_\omega ( {{\tilde{U}}}_n) , \quad P_n \,{:=}\, - ( L_n \mathbf{T}_n - \mathrm{Id}) \Pi _{n } {{{\mathcal {F}}}}_\omega ( {{\tilde{U}}}_n). \end{aligned}$$

(8.20)

We first note that for any $\omega \in \Omega$ and any $s \geqq {\mathfrak {s}}_M$ one has, by (4.6), Lemma 4.3, and (8.2) and (8.8) and using the triangle inequality,

$$\begin{aligned} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _s \lesssim _s \Vert {{{\mathcal {F}}}}_\omega (U_0)\Vert _s + \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) - {{{\mathcal {F}}}}_\omega (U_0)\Vert _s \lesssim _s \varepsilon + \Vert {{\tilde{W}}}_n\Vert _{s + {{\overline{\sigma }}}}, \end{aligned}$$

(8.21)

and, by (8.8), (8.7), (8.11),

$$\begin{aligned} K_0^{\tau _1 p} \gamma ^{- 1} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} \leqq 1. \end{aligned}$$

(8.22)

To conclude, we first need to prove the following inductive estimates of Nash–Moser type:

Lemma 8.2

Let $\mu _2 \,{:=}\, \mu ({\mathtt {b}}) + 3 {{\overline{\sigma }}} + 3$. Then for any $\omega \in {{{\mathcal {G}}}}_{n + 1}$,

$$\begin{aligned}&\Vert {{{\mathcal {F}}}}_\omega (U_{n + 1})\Vert _{{\mathfrak {s}}_M} \lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} K_n^{\mu _2 - {\mathtt {b}}_1} (\varepsilon + \Vert {{\tilde{W}}}_n \Vert _{{\mathfrak {s}}_M+ \mathtt {b}_1}) + K_n^{4 {{\overline{\sigma }}} + 4} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M}^2 \nonumber \\&\quad + \varepsilon K_{n - 1}^{- p {\mathtt {a}} } K_n^{{{\overline{\sigma }}} + 4} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M} , \end{aligned}$$

(8.23)

$$\begin{aligned}&\Vert W_1 \Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} \, \lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} \, K_0^2 \varepsilon , \nonumber \\&| W_{n + 1}\Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} \, \lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} \, K_n^{\mu ({\mathtt {b}}) + 2{{\overline{\sigma }}} + 2} (\varepsilon + \Vert {{\tilde{W}}}_n\Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} ), \quad n \geqq 1 . \end{aligned}$$

(8.24)

Proof of Lemma 8.2

We first estimate $H_{n +1}$, defined in (8.16).

Estimates of $H_{n+1}$. By (8.16) and (2.3), (8.13), (8.8), we get

$$\begin{aligned} \Vert H_{n + 1} \Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1}&\lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} \gamma ^{- 2} \big ( K_n^{{{\overline{\sigma }}} } \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} + K_n^{\mu ({\mathtt {b}}) + 2 {{\overline{\sigma }}} } K_0^{\tau _1 p} \gamma ^{- 1} \Vert {\tilde{\iota }}_n \Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1}\Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} \big ) \nonumber \\&{\mathop {\lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1}}\limits ^{(8.21), 8.22}} K_n^{\mu (\mathtt {b}) + 2 {{\overline{\sigma }}} } \gamma ^{-2} \big ( \varepsilon + \Vert {{\tilde{W}}}_n \Vert _{{\mathfrak {s}}_M+ \mathtt {b}_1} \big ) \nonumber \\&\quad {\mathop {\lesssim _{{\mathfrak {s}}_M+ \mathtt {b}_1}}\limits ^{\gamma ^{- 1} = K_0 \leqq K_n}} K_n^{\mu (\mathtt {b}) + 2 {{\overline{\sigma }}} + 2} \big ( \varepsilon + \Vert {{\tilde{W}}}_n \Vert _{{\mathfrak {s}}_M+ \mathtt {b}_1} \big ) , \end{aligned}$$

(8.25)

$$\begin{aligned}&| H_{n + 1}\Vert _{{\mathfrak {s}}_M} {\mathop {\lesssim _{{\mathfrak {s}}_M+ \mathtt {b}_1}}\limits ^{(8.15)}} \gamma ^{-2}K_{n}^{{{\overline{\sigma }}}} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} . \end{aligned}$$

(8.26)

Next we estimate the terms $Q_n$ in (8.18) and $P_n$, $R_n$ in (8.20) in $\Vert \ \Vert _{{\mathfrak {s}}_M}$ norm.

Estimate of $Q_n$. By (8.8), (8.16), (2.3), (8.26), (8.11), and since $\chi 2 {{\overline{\sigma }}} - \mathtt {a}_1 \leqq 0$ (see (8.3)), we infer that $\Vert {{\tilde{W}}}_n + t H_{n+1} \Vert _{{\mathfrak {s}}_M+ {{\overline{\sigma }}}} \lesssim \varepsilon \gamma ^{- 2} K_0^{2 \overline{\sigma } }$ for all $t \in [0,1]$. Since $\gamma ^{- 1} = K_0$, by (8.7) we can apply Lemma 4.3 and by Taylor’s formula, using (8.18), (4.6), (8.26), (2.3), and $\gamma ^{- 1} = K_0 \leqq K_n$, we get

$$\begin{aligned} \Vert Q_n \Vert _{{\mathfrak {s}}_M} \lesssim _{{\mathfrak {s}}_M+ \mathtt {b}_1} \Vert H_{n+1} \Vert _{{\mathfrak {s}}_M+{{\overline{\sigma }}}}^2 \lesssim _{{\mathfrak {s}}_M+ \mathtt {b}_1} K_n^{4 {{\overline{\sigma }}} + 4} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M}^2. \end{aligned}$$

(8.27)

Estimate of $P_n$. By (5.46), $L_n \mathbf{T}_n - \mathrm{Id} = {{{\mathcal {P}}}}({{\tilde{\iota }}}_n ) + {{{\mathcal {P}}}}_\omega ( {{\tilde{\iota }}}_n ) + {{{\mathcal {P}}}}_\omega ^\bot ( {{\tilde{\iota }}}_n )$. Accordingly, we decompose $P_n$ in (8.20) as $P_n = - P_n^{(1)} - P_{n , \omega } - P_{n, \omega }^\bot$, where

$$\begin{aligned}&P_n^{(1)} \,{:=}\, \Pi _n {{{\mathcal {P}}}}({{\tilde{\iota }}}_n ) \Pi _n {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) , \quad P_{n, \omega } \,{:=}\, \Pi _n {{{\mathcal {P}}}}_\omega ( {{\tilde{\iota }}}_n ) \Pi _n {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) , \\&\quad P_{n, \omega }^\bot \,{:=}\, \Pi _n {{{\mathcal {P}}}}_\omega ^\bot ({{\tilde{\iota }}}_n ) \Pi _n {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) . \end{aligned}$$

By (2.3),

$$\begin{aligned} \begin{aligned} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M+ {{\overline{\sigma }}}}&\leqq \Vert \Pi _n {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M+ {{\overline{\sigma }}}} + \Vert \Pi _n^\bot {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M+ {{\overline{\sigma }}}} \\&\leqq K_n^{{{\overline{\sigma }}}} ( \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} + K_n^{- \mathtt {b}_1} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M+ \mathtt {b}_1} ). \end{aligned} \end{aligned}$$

(8.28)

By (5.47), (8.14), (8.28), and using (8.21), (8.22), and $\gamma ^{- 1} = K_0 \leqq K_n$ one obtains

$$\begin{aligned} \Vert P_n^{(1)} \Vert _{{\mathfrak {s}}_M}&\lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} \gamma ^{-3} K_n^{2 {{\overline{\sigma }}} } \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} ( \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} + K_n^{- {{\mathtt {b}}}_1} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1} ) \nonumber \\&\lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} K_n^{2 {{\overline{\sigma }}} + 3 } \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} ( \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M} + K_n^{{{\overline{\sigma }}} - {{\mathtt {b}}}_1} (\varepsilon + \Vert {{\tilde{W}}}_n\Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} ) ) \nonumber \\&\lesssim _{s_0 + {\mathtt {b}}_1} K_n^{2 {{\overline{\sigma }}} + 3 } \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n)\Vert _{{\mathfrak {s}}_M}^2 + K_n^{3 {{\overline{\sigma }}} + 3 - {\mathtt {b}}_1 } (\varepsilon + \Vert {{\tilde{W}}}_n\Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} ). \end{aligned}$$

(8.29)

By (5.48), (8.14), (8.8), (2.3), we have

$$\begin{aligned} \Vert P_{n, \omega } \Vert _{{\mathfrak {s}}_M} \lesssim _{{\mathfrak {s}}_M+ \mathtt {b}_1} \varepsilon \gamma ^{- 4 } N_{n - 1}^{- {{\mathtt {a}}}} K_n^{{{\overline{\sigma }}} } \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M} {\mathop {\lesssim _{s_0 + {\mathtt {b}}_1}}\limits ^{\gamma ^{- 1} = K_0 \leqq K_n }} \varepsilon N_{n - 1}^{- {{\mathtt {a}}}} K_n^{{{\overline{\sigma }}} + 4} \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M},\nonumber \\ \end{aligned}$$

(8.30)

with $\mathtt {a}$ given as in (8.2). By (5.49), (2.3), (8.4), (8.11), (8.22) and by using (8.21), $\gamma ^{- 1} = K_0 \leqq K_n$, we get

$$\begin{aligned} \Vert P_{n, \omega }^\bot \Vert _{{\mathfrak {s}}_M}&\lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} K_{n}^{\mu ({{\mathtt {b}}}) + 2 {{\overline{\sigma }}} - {{\mathtt {b}}}_1} \gamma ^{-2 } ( \Vert {{{\mathcal {F}}}}_\omega ({{\tilde{U}}}_n) \Vert _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1} + \varepsilon \Vert {{\tilde{W}}}_n \Vert _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1}) \nonumber \\&\lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} K_{n}^{\mu ({{\mathtt {b}}}) + 3 {{\overline{\sigma }}} + 2 - {{\mathtt {b}}}_1} (\varepsilon + \Vert {{\tilde{W}}}_n \Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1 }). \end{aligned}$$

(8.31)

Estimate of $R_n$. Since $L_n = d_{\iota ,\zeta } {{{\mathcal {F}}}}_\omega ( {\tilde{\iota }}_n(\omega ))$ (cf. (8.12)), $d_{\iota , \zeta } {{{\mathcal {F}}}}_\omega (\iota , \zeta )[{\widehat{\iota }} , {\widehat{\zeta }} ] = \omega \cdot \partial _\varphi {\widehat{\iota }} - d_\iota X_{{{{\mathcal {H}}}}_{\varepsilon }} ( \breve{\iota } ) [{\widehat{\iota }} ] - (0, {\widehat{\zeta }}, 0 )$ (cf. (5.1)), and ${{{\mathcal {H}}}}_\varepsilon = {{{\mathcal {N}}}} + {{{\mathcal {P}}}}_\varepsilon$, one has that for any ${\widehat{U}} = ({\widehat{\iota }}, {\widehat{\zeta }})$,

$$\begin{aligned} L_n {\widehat{U}}&= \omega \cdot \partial _\varphi {\widehat{\iota }} - d_\iota X_{{{{\mathcal {H}}}}_\varepsilon }\big ( (\varphi , 0, 0) + {\tilde{\iota }}_n \big )[{\widehat{\iota }}] - (0, {\widehat{\zeta }}, 0) \nonumber \\&{\mathop {=}\limits ^{(4.13)}}\omega \cdot \partial _\varphi {\widehat{\iota }} - d_\iota X_{{{{\mathcal {N}}}}}\big ( (\varphi , 0, 0) + {\tilde{\iota }}_n \big )[{\widehat{\iota }}] - d_\iota X_{{{{\mathcal {P}}}}_\varepsilon }\big ( (\varphi , 0, 0) + {\tilde{\iota }}_n \big )[{\widehat{\iota }}] - (0, {\widehat{\zeta }}, 0) \end{aligned}$$

(8.32)

where by (4.13), $d_\iota X_{{{{\mathcal {N}}}}}\big ( (\varphi , 0, 0) + {\tilde{\iota }}_n \big )[{\widehat{\iota }}] = \big ( -\Omega _{{\mathbb {S}}_+}^{kdv}(\nu )[{\widehat{y}}], \, 0, \, \partial _x \Omega ^{kdv}(D; \nu )[{\widehat{w}}] \big ).$ By the estimate of $d_\iota X_{{{{\mathcal {P}}}}_\varepsilon }$ of Lemma 4.3, one then obtains $\Vert L_n {\widehat{U}} \Vert _{{\mathfrak {s}}_M} \lesssim \Vert {\widehat{U}}\Vert _{{\mathfrak {s}}_M+ {\overline{\sigma }}}$. Using (8.20), (8.13), (8.8), (2.3) and then (8.14), (8.21), (8.22), $\gamma ^{- 1} = K_0 \leqq K_n$, we get

$$\begin{aligned} \Vert R_n\Vert _{{\mathfrak {s}}_M}&\lesssim _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} K_n^{ \mu ({{\mathtt {b}}}) + 3 {\overline{\sigma }} + 2 - {{\mathtt {b}}}_1} (\varepsilon + \Vert {{\tilde{W}}}_n\Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1} ). \end{aligned}$$

(8.33)

Estimate of ${{{\mathcal {F}}}}_\omega (U_{n + 1})$. By (8.19), (2.3), (8.21), (8.27), (8.29)–(8.31), (8.33), (8.8), we get (8.23). Estimate of $W_1 = H_1$. By (8.16) and (8.13) one has

$$\begin{aligned} \Vert W_1 \Vert _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1} = \Vert H_1 \Vert _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1} \lesssim _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1} \gamma ^{-2} \Vert {{{\mathcal {F}}}}_\omega (U_0)\Vert _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1 + {\overline{\sigma }}} {\lesssim _{{\mathfrak {s}}_M+ {{\mathtt {b}}}_1} } \,\,\, \varepsilon \gamma ^{- 2} \, \,\, {\mathop {\lesssim }\limits ^{\gamma ^{- 1} = K_0}} K_0^2 \varepsilon \, \end{aligned}$$

implying the first estimate in (8.24).

Estimate of $W_{n+1} = {{\tilde{W}}}_n + H_{n+1}$, $n \geqq 1$. The claimed estimates for $W_{n+1}$ in (8.24) follows by (8.25). $\quad \square$

By Lemma 8.2 we get the following lemma, where for clarity we write $\Vert \cdot \Vert ^\mathrm{{Lip}(\gamma )}_s$ instead of $\Vert \cdot \Vert _s$ as above:

Lemma 8.3

For any $\omega \in {{{\mathcal {G}}}}_{n + 1}$, $n \ge 0$,

$$\begin{aligned}&\Vert {{{\mathcal {F}}}}_\omega (U_{n + 1}) \Vert _{{\mathfrak {s}}_M}^\mathrm{{Lip}(\gamma )}\leqq C_* \varepsilon K_n^{- {\mathtt {a}}_1} , \quad \quad \Vert W_{n + 1} \Vert _{{\mathfrak {s}}_M+ {\mathtt {b}}_1}^\mathrm{{Lip}(\gamma )}\leqq C_* K_n^{\mu _1} \varepsilon , \end{aligned}$$

(8.34)

$$\begin{aligned}&\Vert H_1 \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {\overline{\sigma }}}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{-2} , { \quad \quad \quad \ \ } \Vert H_{n + 1}\Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {\overline{\sigma }}}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2} K_n^{\mu ({\mathtt {b}}) + 2 {\overline{\sigma }} } K_{n - 1}^{- {\mathtt {a}}_1} , \ (n \geqq 1) . \end{aligned}$$

(8.35)

Proof of Lemma 8.3

First note that, by (8.10), ${{{\mathcal {G}}}}_{n + 1} \subset {{{\mathcal {G}}}}_n$ and so for any $\omega \in {{{\mathcal {G}}}}_{n + 1}$ (8.11) and the inequality in $({{{\mathcal {P}}}}3)_{n}$ holds. Then the first inequality in (8.34) follows by (8.23), $({{{\mathcal {P}}}}2)_{n}$, $({{{\mathcal {P}}}}3)_{n}$, $\gamma ^{- 1} = K_0 \leqq K_n$, and by (8.3)–(8.6). For $n = 0$ we use also (8.7). Concerning the second inequality in (8.34), note that for $n=0$, the inequality follows directly from the bound for $W_1$ in (8.24) since $\mu _1 \geqq 2$ (cf. (8.4)) and $C_* > 0$ is chosen large enough; the second inequality in (8.34) for $n \geqq 1$ is proved inductively by taking (8.24), $({{{\mathcal {P}}}}3)_{n}$, and the choice of $\mu _1$ in (8.4) into account and by choosing $K_0= \varepsilon ^{-\mathfrak {a}}$ large enough (that is, $\varepsilon$ small enough). Since $H_1 = W_1$, the first inequality in (8.35) follows since $\Vert H_1 \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {\overline{\sigma }}} \lesssim \gamma ^{- 2} \Vert {{{\mathcal {F}}}}_\omega (U_0) \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + 2 {\overline{\sigma }}} \lesssim \varepsilon \gamma ^{- 2}$. If $n \geqq 1$, estimate (8.35) follows by (2.3), (8.26) and (8.11). $\quad \square$

We are now in a position to finish the proof of Theorem 8.1. Denote by ${{\tilde{H}}}_{n + 1}$ the $\mathrm{{Lip}(\gamma )}$-extension of $(H_{n + 1})_{|{{{\mathcal {G}}}}_{n+1}}$ to the whole set $\Omega$ of parameters, provided by the Kirszbraun theorem. Then ${{\tilde{H}}}_{n + 1}$ satisfies the same bound as $H_{n+1}$ in (8.35) and therefore, by the definition of ${{\mathtt {a}}}_2$ in (8.3), the estimate (8.9) holds at $n + 1$.

Finally we define ${{\tilde{W}}}_{n+1} \,{:=}\, {{\tilde{W}}}_{n} + {{\tilde{H}}}_{n + 1}$ and ${{\tilde{U}}}_{n + 1} \,{:=}\, {{\tilde{U}}}_n + {{\tilde{H}}}_{n + 1}$, which both are defined for all $\omega \in \Omega$. Note that

$$\begin{aligned} {{\tilde{U}}}_{n + 1} = U_0 + {{\tilde{W}}}_n + {{\tilde{H}}}_{n + 1} = U_0 + {{\tilde{W}}}_{n + 1} \end{aligned}$$

and that for any $\omega \in {{{\mathcal {G}}}}_{n + 1}$, ${{\tilde{W}}}_{n + 1} = W_{n + 1}$, ${{\tilde{U}}}_{n + 1} = U_{n + 1}$. Hence $({{{\mathcal {P}}}}2)_{n + 1}$, $({{{\mathcal {P}}}}3)_{n + 1}$ follow from Lemma 8.3. Moreover by (8.9), which at this point has been proved up to the step $n + 1$, we have

$$\begin{aligned} \Vert {{\tilde{W}}}_{n + 1} \Vert _{{\mathfrak {s}}_M+ \mu ({{\mathtt {b}}}) + {\overline{\sigma }}}^\mathrm{{Lip}(\gamma )}\leqq {\mathop \sum }_{k = 1}^{n + 1} \Vert {{\tilde{H}}}_k \Vert _{{\mathfrak {s}}_M+ \mu ({{\mathtt {b}}}) + {\overline{\sigma }}}^\mathrm{{Lip}(\gamma )}\leqq C_* \varepsilon \gamma ^{-2} \end{aligned}$$

and thus also (8.8) holds at the step $n + 1$. This completes the proof of Theorem 8.1. $\quad \square$

Corollary 8.4

Let $\gamma = \varepsilon ^{\mathfrak {a}}$ with $\mathfrak {a} \in (0, {\mathfrak {a}}_0)$, ${\mathfrak {a}}_0 \,{:=}\, 1 / \tau _2$ with $\tau _2$ defined as in (8.7), and $K_0 = 1/ \gamma$. Then there is $\varepsilon _0 > 0$ so that for any $0 < \varepsilon \le \varepsilon _0$ the following holds:

• (i) there exists a function $U_\infty (\omega ) = (\breve{\iota }_\infty (\omega ), \zeta _\infty (\omega ))$, $\omega \in \Omega$, satisfying

  $$\begin{aligned} \Vert U_\infty - U_0 \Vert _{{\bar{s}}}^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2} , \quad \Vert U_\infty - {{{\tilde{U}}}}_n \Vert _{{\bar{s}}}^\mathrm{{Lip}(\gamma )}\, \lesssim \, \varepsilon \gamma ^{-2} K_{n }^{- {\mathtt {a}}_2}, \ n \geqq 1,\nonumber \\ \end{aligned}$$

  (8.36)

  where ${\bar{s}} \,{:=}\, {\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {\overline{\sigma }}$ (with ${\mathfrak {s}}_M$, $\mu ({\mathtt {b}})$, ${\overline{\sigma }}$ fixed in (2.55), (7.6), and respectively, (8.2));

• (ii) for any $\omega$ in the set

  $$\begin{aligned} \Omega _\varepsilon \,{:=}\, \bigcap _{n \geqq 0} {{{\mathcal {G}}}}_n = {\mathcal {G}}_0 \cap \bigcap _{n \geqq 1} {\varvec{\Omega }}_{n + 1}^{\gamma }({\tilde{\iota }}_{n-1}) {\mathop {=}\limits ^{(7.61)}} {\mathcal {G}}_0 \cap \Big [ \bigcap _{n \geqq 1} {\mathtt {\Lambda }} _{n}^{\gamma }({\tilde{\iota }}_{n-1}) \Big ] \cap \Big [ \bigcap _{n \geqq 1} {\mathtt {\Omega }}_{n}^{\gamma }({\tilde{\iota }}_{n-1}) \Big ],\nonumber \\ \end{aligned}$$

  (8.37)

  the torus embedding $\breve{\iota }_\omega (\varphi )$ solves (4.12).

Proof

For any $0 < \varepsilon \le \varepsilon _0$ with $\varepsilon _0$ small enough, the smallness condition $\varepsilon K_0^{\tau _2} < \delta _0$ in (8.7) holds and Theorem 8.1 applies. By Theorem 8.1-$({{{\mathcal {P}}}}1)_n$ the sequence $({{{\tilde{U}}}}_n)_{n \ge 1}$ converges as $n \rightarrow \infty$ to a function $U_\infty (\omega )$, satisfying (8.36). By Theorem 8.1-$({{{\mathcal {P}}}}2)_n$, for any $\omega \in \Omega _\varepsilon$, we have that ${{{\mathcal {F}}}}_\omega (U_\infty (\omega )) = 0$. Formula (5.5) implies that $\zeta _\infty (\omega ) = 0$ for any $\omega \in \Omega _\varepsilon$ and item (ii) is proved. $\quad \square$

In order to complete the proof of Theorem 4.1 it only remains to establish the measure estimate (4.10).

8.2 Measure Estimates

The measure estimate (4.10) of Theorem 4.1 for the subset $\Omega _\varepsilon = \cap _{n \geqq 0} {{{\mathcal {G}}}}_n$ defined in (8.37) of nonresonant frequency vectors will be deduced from the measure estimate of Proposition 8.7, using that by Lemmas 8.5 and 8.6, the set $\Omega \setminus \Omega _\varepsilon$ is included in $\Omega \setminus \Omega _\infty ^\gamma$, where $\Omega _\infty ^\gamma$ is introduced in Lemma 8.6. The main result of this section is Proposition 8.7. In all of this section, the assumptions of Theorem 8.1hold with the constants given as in (8.2)–(8.7) .

In order to prove Lemmata 8.5 and  8.6, we first recall that by (8.10), ${\mathcal {G}}_0 = \Omega$ and for $n \ge 1$, ${{{\mathcal {G}}}}_{n} \,{:=}\, {{{\mathcal {G}}}}_{n-1} \cap \ {\varvec{\Omega }}_{n}^{ \gamma }({{\tilde{\iota }}}_{n-1})$ where ${\varvec{\Omega }}_{n}^{\gamma } (\iota ) = {{\mathtt {\Omega }}}_{n }^\gamma (\iota ) \cap {{\mathtt {\Lambda }}}_{n}^{\gamma }(\iota )$ (cf. (7.61), (7.18), (7.58)).

Lemma 8.5

For any $n \ge 0$, the set

$$\begin{aligned} {{{\mathcal {G}}}}_\infty \,{:=}\, {\mathcal {G}}_0 \cap \Big [ \bigcap _{n \geqq 1} {\mathtt {\Omega }}_n^{2 \gamma }(\iota _\infty ) \Big ] \cap \Big [ \bigcap _{n \geqq 1} {\mathtt {\Lambda }} _n^{2 \gamma }( \iota _\infty ) \Big ] \end{aligned}$$

(8.38)

is contained in ${{{\mathcal {G}}}}_n$ and hence ${{{\mathcal {G}}}}_\infty \subseteq \bigcap _{n \geqq 0 } {{{\mathcal {G}}}}_n$.

Proof

We apply the inclusion property (7.28). By (8.36), (5.27), we have, for any $n \geqq 2$,

$$\begin{aligned} C(S) N_{n-1}^{ \tau } \Vert \iota _\infty - {\tilde{\iota }}_{n-1} \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {\overline{\sigma }}}&\leqq C(S) C K_{n-1}^{p \tau - {\mathtt {a}}_2} \varepsilon \gamma ^{- 2} \leqq \gamma \end{aligned}$$

taking $\varepsilon$ small enough, by (8.7) and using ${\mathtt {a}}_2 \geqq p \tau$ (see (8.3)). For $n=1$, one also has $C(S) N_{0}^{ \tau } \Vert \iota _\infty - {{\tilde{\iota }}}_{0} \Vert _{{\mathfrak {s}}_M+ \mu ({\mathtt {b}}) + {\overline{\sigma }}} \leqq \gamma$ using the first inequality in (8.36) and recalling that $K_0 = \gamma ^{-1}$, $\gamma = \varepsilon ^{\mathfrak {a}}$ and $1 - {\mathfrak {a}}(3 + \tau p) > 0$ (indeed $\mathfrak {a} = 1/ \tau _2$ where $\tau _2$ is defined in (8.7) and we take ${\bar{\tau }} > \tau$ large enough, see Theorem 7.2). Recall also that $S = {\mathfrak {s}}_M+ {\mathtt {b}}_1$ has been fixed in (8.4). Therefore (7.28) in Theorem 7.2-$(\mathbf{S3})_\nu$ gives ${\mathtt {\Omega }}_n^{2 \gamma }( \iota _\infty ) \subseteq {{\mathtt {\Omega }}}_n^{ \gamma }( {\tilde{\iota }}_{n-1} )$, $\forall n \geqq 1$. By similar arguments we deduce that ${\mathtt {\Lambda }}_n^{2 \gamma }( \iota _\infty ) \subseteq \mathtt \Lambda _n^{\gamma }( {\tilde{\iota }}_{n-1} )$, and the lemma is proved. $\quad \square$

By Theorem 7.2-$\mathbf{(S2)}_n$, it follows that for any $j \in {\mathbb {S}}^\bot$, ${\widetilde{\mu }}_j^n : \Omega \rightarrow {\mathbb {R}}$, $n \ge 0$, is a Cauchy sequence with respect to the norm $| \cdot |^\mathrm{{Lip}(\gamma )}$. We denote its limit by $\mu _j^\infty$,

$$\begin{aligned} \mu _j^\infty \,{:=}\, \lim _{n \rightarrow \infty } {\widetilde{\mu }}_j^n(\iota _\infty ) , \quad {\widetilde{\mu }}_j^n(\iota _\infty ) \equiv {\widetilde{\mu }}_j^n(\omega ; \iota _\infty ) , \quad j \in {\mathbb {S}}^\bot . \end{aligned}$$

(8.39)

By Theorem 7.2 one has for any $j \in {\mathbb {S}}^\bot$,

$$\begin{aligned} \mu _{-j}^\infty = - \mu _j^\infty , \quad | \mu _j^\infty - {\widetilde{\mu }}_j^n(\iota _\infty )|^\mathrm{{Lip}(\gamma )}\lesssim \varepsilon \gamma ^{- 2} N_{n - 1}^{- {{\mathtt {a}}}} , \ n \geqq 0 . \end{aligned}$$

(8.40)

Lemma 8.6

The set $\Omega _\infty ^\gamma$ is contained in ${{{\mathcal {G}}}}_\infty$ where ${\mathcal {G}}_\infty$ is defined in (8.38) and

$$\begin{aligned} \Omega _\infty ^\gamma&\,{:=}\, \Big \{ \omega \in \mathtt {DC}(4 \gamma , \tau ) : |\omega \cdot \ell + \mu _j^\infty - \mu _{j'}^\infty | \geqq \frac{4 \gamma |j^3 - j'^3|}{\langle \ell \rangle ^\tau }, \ \forall (\ell , j, j') \in {\mathbb {Z}}^{{\mathbb {S}}_+} \times {{\mathbb {S}}}^\bot \times {{\mathbb {S}}}^\bot ,\, \nonumber \\&\quad \quad \quad \quad \quad |\omega \cdot \ell + \mu _j^\infty | \geqq \frac{4 \gamma |j|^3}{\langle \ell \rangle ^\tau }, \ \forall (\ell , j) \in {\mathbb {Z}}^{{\mathbb {S}}_+} \times {\mathbb {S}}^\bot \Big \} . \end{aligned}$$

(8.41)

Proof

We have to verify that $\Omega _\infty ^\gamma$ is contained in each subset on the right hand side of (8.38). Since by (4.4), $\mathtt {DC}(4 \gamma , \tau ) \subseteq \Omega$ one has that $\Omega _\infty ^\gamma \subseteq \Omega {\mathop {=}\limits ^{(8.10)}} {{{\mathcal {G}}}}_0$. Next we prove that $\Omega _\infty ^\gamma \subseteq {{\mathtt {\Omega }}}_n^{2 \gamma }(\iota _\infty )$, $\forall n \geqq 1$. We argue by induction. Assume that $\Omega _\infty ^\gamma \subseteq {{\mathtt {\Omega }}}_n^{2 \gamma }(\iota _\infty )$ for some $n \geqq 1$. For all $\omega \in \Omega _\infty ^\gamma \subseteq {{\mathtt {\Omega }}}_n^{2 \gamma }(\iota _\infty )$, by (7.16), (8.39), (8.40), we get $|({\widetilde{\mu }}_j^n- {\widetilde{\mu }}_{j'}^n)(\iota _\infty ) - (\mu _j^\infty - \mu _{j'}^\infty ) | \leqq C \varepsilon \gamma ^{-2} N_{n-1}^{- {\mathtt {a}}}$. Therefore, let $(\ell , j, j') \in {\mathbb {Z}}^{{\mathbb {S}}_+} \times {{\mathbb {S}}}^\bot \times {{\mathbb {S}}}^\bot$, $|\ell | \leqq N_n$ with $(\ell , j, j') \ne (0, j, j)$ (recall (8.41)). If $j = j'$, then $\ell \ne 0$ and since $\Omega _\infty ^{ \gamma } \subseteq \mathtt {DC}(4 \gamma , \tau )$ we have

$$\begin{aligned} |\omega \cdot \ell + {\widetilde{\mu }}_j^n(\iota _\infty ) - {\widetilde{\mu }}_{j'}^n(\iota _\infty ) | = |\omega \cdot \ell | \geqq 4 \gamma \langle \ell \rangle ^{-\tau }. \end{aligned}$$

In case $j \ne j'$, one has

$$\begin{aligned} |\omega \cdot \ell + {\widetilde{\mu }}_j^n(\iota _\infty ) - {\widetilde{\mu }}_{j'}^n(\iota _\infty )|&\geqq |\omega \cdot \ell + \mu _j^\infty - \mu _{j'}^\infty | - C \varepsilon \gamma ^{- 2} N_{n-1}^{- {\mathtt {a}}} \\&\geqq \frac{4 \gamma |j^3 - j'^3|}{\langle \ell \rangle ^\tau } - C \varepsilon \gamma ^{- 2} N_{n-1}^{- {\mathtt {a}}} \geqq \frac{2 \gamma |j^3 - j'^3|}{\langle \ell \rangle ^\tau } , \end{aligned}$$

provided $\frac{1}{2} C \varepsilon \gamma ^{- 3} N_{n - 1}^{- {\mathtt {a}}} N_n^\tau \leqq 1$ (note that since $j \ne j'$, $|j^3 - j'^3 | \geqq 1$). The latter condition is fullfilled by (7.6), (8.7), by taking ${\bar{\tau }} > \tau$ large enough. In conclusion we have proved that $\Omega _\infty ^\gamma \subseteq {{\mathtt {\Omega }}}_{n + 1}^{2 \gamma }(\iota _\infty )$. Similarly we prove that $\Omega _\infty ^\gamma \subseteq {\mathtt {\Lambda }}_{n}^{2 \gamma }(\iota _\infty )$ for all $n \geqq 1$. $\quad \square$

In view of Lemmata  8.5 and 8.6, it suffices to estimate the Lebesgue measure $|\Omega \setminus \Omega _\infty ^\gamma |$ of $\Omega \setminus \Omega _\infty ^\gamma$ instead of the one of $\Omega \setminus \bigcap _{n \geqq 0 } {{{\mathcal {G}}}}_n$.

Proposition 8.7

(Measure estimates) Let $\tau > |{\mathbb {S}}_+| + 2$. Then there is ${{\mathfrak {a}}} \in (0,1)$ so that for $\varepsilon \gamma ^{-3}$ sufficiently small, one has $| \Omega \setminus \Omega _\infty ^\gamma | \lesssim \gamma ^{{\mathfrak {a}}}$.

Proof

By (8.41), we have

$$\begin{aligned} \Omega \setminus \Omega _\infty ^\gamma = \Omega \setminus \mathtt {DC}(4 \gamma , \tau ) \,\, \cup \, \bigcup _{(\ell , j, j') \in {\mathbb {Z}}^{{\mathbb {S}}_+} \times {{\mathbb {S}}}^\bot \times {{\mathbb {S}}}^\bot , (\ell , j, j') \ne (0, j, j) } {{{\mathcal {R}}}}_{\ell , j, j'} \,\, \cup \, \bigcup _{(\ell , j) \in {\mathbb {Z}}^{{\mathbb {S}}_+} \times {{\mathbb {S}}}^\bot } {{{\mathcal {Q}}}}_{\ell , j}\nonumber \\ \end{aligned}$$

(8.42)

where ${{{\mathcal {R}}}}_{\ell ,j,j'}$, ${{{\mathcal {Q}}}}_{\ell , j}$ denote the ’resonant’ sets

$$\begin{aligned} {{{\mathcal {R}}}}_{\ell ,j,j'}&\,{:=}\, \Big \{ \omega \in \mathtt {DC}(4 \gamma , \tau ) \ : \ | \omega \cdot \ell + \mu _j^\infty - \mu _{j'}^\infty |< \frac{4\gamma |j^3 - j'^3|}{ \langle \ell \rangle ^\tau } \Big \} ,\\ {{{\mathcal {Q}}}}_{\ell , j}&\,{:=}\, \Big \{ \omega \in \mathtt {DC}(4 \gamma , \tau ) : |\omega \cdot \ell + \mu _j^\infty | < \frac{4 \gamma |j|^3}{\langle \ell \rangle ^\tau } \Big \}. \end{aligned}$$

Note that ${{{\mathcal {R}}}}_{\ell , j, j } = \emptyset$. Furthermore, it is well known that $|\Omega \setminus \mathtt {DC}(4 \gamma , \tau )| \lesssim \gamma$. In order to prove Proposition 8.7 we shall use the following asymptotic properties of $\mu _j^\infty (\omega )$. For any $\omega$ in $\mathtt {DC}(4\gamma , \tau )$, we have ${\widetilde{\mu }}_j^{0} (\iota _\infty ) = \mu _j^{0} (\iota _\infty )$ (for simplicity $\mu _j^0 (\iota _\infty ) \equiv \mu _j^0 (\omega ; \iota _\infty )$) and we write $\mu _j^\infty (\omega ) = \mu _j^0 (\iota _\infty ) + r_j^\infty (\omega )$, where by (7.2)

$$\begin{aligned} \mu _j^0(\iota _\infty ) = m_3^\infty (\omega ) (2 \pi j)^3 - m_1^\infty (\omega ) 2 \pi j - q_j (\omega ) , \quad m_3^\infty \,{:=}\, m_3 ( \iota _\infty ), \quad m_1^\infty \,{:=}\, m_1 ( \iota _\infty ) . \end{aligned}$$

On $\mathtt {DC}(4 \gamma , \tau )$, the following estimates hold:

$$\begin{aligned} \begin{aligned}&|m_3^\infty + 1|^\mathrm{{Lip}(\gamma )}{\mathop {\lesssim }\limits ^{(6.20)}} \varepsilon , \quad \ |m_1^\infty |^\mathrm{{Lip}(\gamma )}{\mathop {\lesssim }\limits ^{(6.68)}} \varepsilon \gamma ^{- 2}, \\&\sup _{j \in {{\mathbb {S}}}^\bot } |j| |q_j|^\mathrm{sup}, \ \sup _{j \in {{\mathbb {S}}}^\bot } |j| |q_j|^\mathrm{lip} {\mathop {\lesssim }\limits ^{(7.4)}} 1, \quad |r_j^\infty |^\mathrm{{Lip}(\gamma )}{\mathop {\lesssim }\limits ^{(8.40)}} \varepsilon \gamma ^{- 2} . \end{aligned} \end{aligned}$$

(8.43)

From the latter estimates one infers the following standard lemma see [2, Lemma 5.3]). $\quad \square$

Lemma 8.8

• (i) If ${{{\mathcal {R}}}}_{\ell ,j,j'} \ne \emptyset$, then $|j^3 - j'^3 | \leqq C \langle \ell \rangle$ for some $C > 0$. In particular one has $j^2 + j'^2 \leqq C \langle \ell \rangle$.

• (ii) If ${{{\mathcal {Q}}}}_{\ell , j} \ne \emptyset$, then $|j|^3 \leqq C \langle \ell \rangle$ for some $C > 0$.

Lemma 8.8 can be used to estimate $|{{{\mathcal {R}}}}_{\ell ,j,j'} |$ and $|{{{\mathcal {Q}}}}_{\ell ,j}|$ for $|\ell |$ sufficiently large.

Lemma 8.9

• (i) If ${{{\mathcal {R}}}}_{\ell ,j,j'} \ne \emptyset$, then there exists $C_1 > 0$ with the following property: if $|\ell | \geqq C_1$, then $|{{{\mathcal {R}}}}_{\ell ,j,j'} | \lesssim \gamma |j^3 - j'^3| \langle \ell \rangle ^{- (\tau + 1)}$.

• (ii) If ${{{\mathcal {Q}}}}_{\ell ,j} \ne \emptyset$, then there exists $C_1 > 0$ with the following property: if $|\ell | \geqq C_1$, then $|{{{\mathcal {Q}}}}_{\ell ,j} | \lesssim \gamma |j|^3 \langle \ell \rangle ^{- (\tau +1)}$.

Proof of Lemma 8.9

We only prove item (i) since item (ii) can be proved in a similar way. Assume that ${{{\mathcal {R}}}}_{\ell , j, j'} \ne \emptyset$. Let ${\bar{\omega }}$ such that ${\bar{\omega }} \cdot \ell = 0$ and introduce the real valued function $s \mapsto \phi _{\ell ,j,k} (s)$,

$$\begin{aligned} \phi _{\ell ,j,j'} (s) \,{:=}\, f_{\ell ,j,j'} \big ( {\bar{\omega }} + s \frac{\ell }{|\ell |} \big ) , \quad f_{\ell ,j,j'} (\omega ) \,{:=}\, \omega \cdot \ell + \mu _j^\infty (\omega ) - \mu _{j'}^\infty (\omega ) . \end{aligned}$$

Using that by Lemma 8.8, $|j^3 - j'^3| \leqq C \langle \ell \rangle$, one infers from (8.43) that for $\varepsilon \gamma ^{- 2}$ small enough and $|\ell | \geqq C_1$ with $C_1$ large enough, $|\phi _{\ell ,j,j'} (s_2) - \phi _{\ell ,j,j'} (s_1)| \geqq \frac{|\ell |}{2} |s_2 - s_1|$. Since $\mathtt {DC}(4 \gamma , \tau ) \subseteq \Omega$ is bounded one sees by standard arguments that $\big | \big \{ s \in {\mathbb {R}}\, : \, {\bar{\omega }} + s \frac{\ell }{|\ell |} \in {{{\mathcal {R}}}}_{\ell , j, j'} \big \} \big | \lesssim \gamma |j^3 - j'^3| \langle \ell \rangle ^{- (\tau + 1)}$. The claimed estimate then follows by applying Fubini’s theorem. $\quad \square$

It remains to estimate the Lebesgue measure of the resonant sets ${{{\mathcal {R}}}}_{\ell , j, j'}$ and ${{{\mathcal {Q}}}}_{\ell , j}$ for $|\ell | \leqq C_1$.

Lemma 8.10

Assume that $|\ell | \leqq C_1$ and that $\varepsilon \gamma ^{- 3}$ is small enough. Then the following holds:

• (i) If ${{{\mathcal {R}}}}_{\ell , j, j'} \ne \emptyset$, then there are constants ${\mathfrak {a}} \in (0,1)$ and $C_2 > 0$ so that $|j|, |j'| \leqq C_2$ and $| {{{\mathcal {R}}}}_{\ell ,j,j'} | \lesssim \gamma ^{{\mathfrak {a}}}$.

• (ii) If ${{{\mathcal {Q}}}}_{\ell , j } \ne \emptyset$ then there are constants ${\mathfrak {a}} \in (0,1)$ and $C_2 > 0$ so that $|j| \leqq C_2$ and $| {{{\mathcal {Q}}}}_{\ell ,j} | \lesssim \gamma ^{{\mathfrak {a}}}$.

Proof of Lemma 8.10

We only prove item (i) since item (ii) can be proved in a similar way. If $|\ell | \leqq C_1$ and ${{{\mathcal {R}}}}_{\ell , j, j'} \ne \emptyset$, Lemma 8.8-(i) implies that there is a constant $C_2$ such that $|j|, |j'| \leqq j^2 + j'^2 \leqq C_2$. For $\varepsilon \gamma ^{-3}$ small enough one sees, using (8.43), the definition (7.2) of $\mu _j^0$, and the bounds $|\ell | \leqq C_1, |j|, |j'| \leqq C_2$, that $|\mu _j^\infty - \omega ^{kdv}_j| \lesssim \varepsilon \gamma ^{- 2} \lesssim \gamma$, implying that for some constant $C_3 > 0,$

$$\begin{aligned} {{{\mathcal {R}}}}_{\ell ,j,j'} \subset \big \{ \omega \in \Omega \ : \ |\omega \cdot \ell + \omega ^{kdv}_j ( \nu (\omega ),0 ) - \omega ^{kdv}_{j'} (\nu (\omega ),0) | \leqq C_3 \gamma \big \} . \end{aligned}$$

(8.44)

By Lemma 3.9, the function $\omega \mapsto \omega \cdot \ell + \omega ^{kdv}_j ( \nu (\omega ),0 ) - \omega ^{kdv}_{j'} (\nu (\omega ),0)$ is real analytic and not identically zero. Hence by the Weierstrass preparation theorem (cf. the proof of Proposition 3.1 in [11]), we deduce that the measure of the set on the right hand side of (8.44) is smaller than $\gamma ^{{\mathfrak {a}}}$ for some ${\mathfrak {a}} \in (0, 1)$ and $\gamma$ small enough. $\quad \square$

We are now in position to finish the proof of Proposition 8.7. By (8.42) and Lemmata 8.9–8.10 we have

$$\begin{aligned} |\Omega \setminus \Omega _\infty ^\gamma | \lesssim \gamma ^{{\mathfrak {a}}} + \gamma \sum _{ |\ell | \geqq C_1, |j|, |j'| \leqq C \langle \ell \rangle } \langle \ell \rangle ^{- \tau } \lesssim \gamma ^{{\mathfrak {a}}}, \end{aligned}$$

where we used the assumption that $\tau - 2>|{\mathbb {S}}_+|$. $\quad \square$

8.3 Proofs of Theorems 4.1 and 1.1

In this section, we complete the proof of Theorem 4.1 and then derive from it Theorem 1.1.

Proof of Theorem 4.1

In view of Corollary 8.4-(ii) for any $\omega \in \Omega _\varepsilon$, the embedded torus $\breve{\iota }_\omega ( {\mathbb {T}}^{{{\mathbb {S}}}_+} )$ is invariant under the flow of the Hamiltonian vector field $X_{{{{\mathcal {H}}}}_\varepsilon (\cdot , \nu )}$ and is filled by quasi-periodic solutions with frequency $\omega = - \omega ^{kdv} (\nu )$. The bound (4.11) follows by (8.36). The linear stability of the quasi-periodic solution $\breve{\iota }_\omega ( \omega t )$ follows by standard arguments as in [1, 3, 10] since for any $\omega \in \cap _{n \geqq 0} {{{\mathtt {\Omega }}}}_{n}^{ \gamma } (\iota _\infty )$ (cf. (7.30)) we obtain by Theorem 7.3 the complete diagonalization of ${\mathtt {L}_0 } ( \iota _\infty )$. It thus remains to prove the measure estimate (4.10) of Theorem 4.1. Since by Lemma 8.5 one has ${{{\mathcal {G}}}}_\infty \subset \Omega _\varepsilon$ and by Lemma 8.6, $\Omega _\infty ^\gamma \subseteq {{{\mathcal {G}}}}_\infty$ the claimed estimates of $|\Omega \setminus \Omega _\varepsilon |$ in (4.10) follow from the estimates of $|\Omega \setminus \Omega _\infty ^\gamma |$ established in Proposition 8.7. $\quad \square$

Proof of Theorem 1.1

Theorem 1.1 is in fact a reformulation of Theorem 4.1. Choose ${\bar{s}}$, $\varepsilon _0$, and $\Omega _\varepsilon$, $0 < \varepsilon \le \varepsilon _0$, as in Theorem 4.1. Using that $- \omega ^{kdv} : \Xi \rightarrow \Omega ,$ $\nu \mapsto - \omega ^{kdv}(\nu )$, is a diffeomorphism (cf. (1.13)), we define for any $0 < \varepsilon \le \varepsilon _0$ the set

$$\begin{aligned} \Xi _\varepsilon \,{:=}\, \Big \{ \nu \in \Xi : \nu = (\omega ^{kdv})^{- 1}(- \omega ) , \quad \omega \in \Omega _\varepsilon \Big \}. \end{aligned}$$

By Theorem 4.1, for any $\nu \in \Xi _\varepsilon$, there exists a torus embedding with lift $\breve{\iota }_\omega : {\mathbb {R}}^{{{\mathbb {S}}}_+} \rightarrow {\mathbb {R}}^{{{\mathbb {S}}}_+} \times {\mathbb {R}}^{{{\mathbb {S}}}_+} \times H^{{\bar{s}}}_\bot ({\mathbb {T}}_1)$ of the form $\breve{\iota }_\omega (\varphi ) = (\varphi , 0, 0) + \iota _\omega (\varphi )$ and $\omega \equiv \omega _\varepsilon (\nu ) = - \omega ^{kdv}(\nu )$, satisfying (cf. (4.11))

$$\begin{aligned} \quad \Vert \iota _\omega \Vert _{{\bar{s}}} = O( \varepsilon \gamma ^{-2}), \quad \gamma = \varepsilon ^{{\mathfrak {a}}} , \ 0 < {{\mathfrak {a}}} \ll 1 , \end{aligned}$$

(8.45)

so that $\breve{\iota }_\omega ( \omega t )$ is a linearly stable, quasi-periodic solution of (4.1). By the measure estimate (4.10) one has $\lim _{\varepsilon \rightarrow 0} |\Xi \setminus \Xi _\varepsilon | = 0$, which proves (1.16). By the definition of the symplectic diffeomorphism $\Psi _\nu$ (cf. Theorem 3.2) and the one of ${{\mathcal {H}}}_\varepsilon$ (cf. (4.2)), it then follows that

$$\begin{aligned} u_\varepsilon (\omega t, x; \nu ) \,{:=}\, \Psi _\nu \big ( (\omega t, 0, 0) + \iota _\omega (\omega t) \big ) \end{aligned}$$

(8.46)

is a quasi-periodic solution of the perturbed KdV equation (1.6). Furthermore, by (3.8) in Theorem 3.2 and (3.5), the finite gap solution $t \mapsto q(\omega t, x; \nu )$ of the KdV equation (1.1) (cf. (1.10)) satisfies

$$\begin{aligned} q(\omega t, x; \nu ) = \Psi _\nu (\omega t, 0, 0). \end{aligned}$$

(8.47)

It then follows from (8.45)–(8.47) that $\Vert u_\varepsilon (\cdot , \cdot \, ; \nu ) - q ( \cdot , \cdot \, ; \nu ) \Vert _{{\bar{s}}} \lesssim \varepsilon ^{1 - {{\mathfrak {b}}}}$ with ${\mathfrak {b}} = 2 {\mathfrak {a}}$, proving (1.17). $\quad \square$