# Hamiltonian Birkhoff normal form for
gravity-capillary water waves with constant vorticity:
almost global existence

Massimiliano Berti, Alberto Maspero, Federico Murgante 111 International
School for Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy.
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract. We prove an almost global in time existence result of small
amplitude space periodic solutions of the 1D gravity-capillary water waves
equations with constant vorticity. The result holds for any value of gravity,
vorticity and depth and any surface tension belonging to a full measure set.
The proof demands a Hamiltonian paradifferential Birkhoff normal form
reduction for quasi-linear PDEs in presence of resonant wave interactions: the
normal form may be not integrable but it preserves the Sobolev norms thanks to
its Hamiltonian nature. A major difficulty is that usual paradifferential
calculus used to prove local well posedness (as the celebrated Alinhac good
unknown) does not preserve the Hamiltonian structure. A major novelty of this
paper is to develop an algorithmic perturbative procedure à la Darboux to
correct usual paradifferential transformations to symplectic maps, up to an
arbitrary degree of homogeneity. The symplectic correctors turn out to be
smoothing perturbations of the identity, and therefore only slightly modify
the paradifferential structure of the equations. The Darboux procedure which
recovers the nonlinear Hamiltonian structure is written in an abstract
functional setting, in order to be applicable also in other contexts.

Keywords: water waves equations, vorticity, Hamiltonian Birkhoff normal form,
para-differential calculus.

###### Contents

  1. 1 Introduction and main results
  2. I Abstract setting and Darboux symplectic corrector
    1. 2 Functional Setting
      1. 2.1 Paradifferential calculus
      2. 2.2 Spectrally localized maps
      3. 2.3 Approximate inverse of non–linear maps and flows
      4. 2.4 Pluri-homogeneous differential geometry
    2. 3 Hamiltonian formalism
      1. 3.1 Hamiltonian and symplectic structures
      2. 3.2 Hamiltonian systems up to homogeneity $N$
      3. 3.3 Linear symplectic flows
      4. 3.4 Paradifferential Hamiltonian structure
    3. 4 Construction of a Darboux symplectic corrector
  3. II Almost global existence of water waves
    1. 5 Paralinearization of the water waves equations with constant vorticity and its Hamiltonian structure
    2. 6 Block-diagonalization and reduction to constant coefficients
      1. 6.1 A complex good unknown of Alinhac
      2. 6.2 Block-Diagonalization at highest order
      3. 6.3 Reduction to constant coefficients of the highest order
      4. 6.4 Block-Diagonalization up to smoothing operators
      5. 6.5 Reduction to constant coefficients up to smoothing operators
    3. 7 Hamiltonian Birkhoff normal form
      1. 7.1 Hamiltonian correction up to homogeneity $N$
      2. 7.2 Super action preserving symbols and Hamiltonians
      3. 7.3 Birkhoff normal form reduction
    4. 8 Energy estimate and proof of Theorem 1.1
    5. A Conjugation lemmata
    6. B Non–resonance conditions
      1. B.1 Deep-water case
      2. B.2 Finite depth case

###  1 Introduction and main results

The study of the time evolution of water waves is a classical question, and
one of the major problems in fluid dynamics. The very first attempt of a
theory of water waves trace its origin in the Principia of Newton in the
second half of the seventeenth century, but one more century is needed for the
foundational works by giants such as Euler, Laplace, Lagrange, Cauchy,
Poisson, Bernoulli, and then by the British school with Russel, Robinson,
Green, Airy, Stokes among others. We refer to the historical overview in [31].

In this paper we consider the Euler equations of hydrodynamics for a
2-dimensional perfect and incompressible fluid with constant vorticity
$\gamma$, under the action of gravity and capillary forces at the free
surface. The fluid fills the time dependent region

$\mathcal{D}_{\eta}:=\big{\\{}(x,y)\in\mathbb{T}\times\mathbb{R}\ :\
-\mathtt{h}<y<\eta(t,x)\big{\\}}\,,\quad\mathbb{T}:=\mathbb{T}_{x}:=\mathbb{R}/(2\pi\mathbb{Z})\,,$
(1.1)

with depth $\mathtt{h}>0$, possibly infinite, and space periodic boundary
conditions. The unknowns are the free surface $y=\eta(t,x)$ of
$\mathcal{D}_{\eta}$ and the divergence free velocity field
$\begin{pmatrix}u(t,x,y)\\\ v(t,x,y)\end{pmatrix}$. In case of a fluid with
constant vorticity $v_{x}-u_{y}=\gamma$ (a property which is preserved along
the time evolution), the velocity field is the sum of the Couette flow
$\begin{pmatrix}-\gamma y\\\ 0\end{pmatrix}$, which carries all the vorticity
$\gamma$, and an irrotational field, expressed as the gradient of a harmonic
function $\Phi$, called the generalized velocity potential.

We study the water waves problem in the Hamiltonian Zakharov-Craig-Sulem [72,
33] formulation, extended by Constantin, Ivanov, Prodanov [28] and Wahlén [69]
for constant vorticity fluids. Denoting by $\psi(t,x)$ the evaluation of the
generalized velocity potential at the free interface
$\psi(t,x):=\Phi(t,x,\eta(t,x))$, one recovers $\Phi$ as the unique harmonic
function $\Delta\Phi=0$ in $\mathcal{D}_{\eta}$ with Dirichlet boundary
condition $\Phi=\psi$ at $y=\eta(t,x)$ and Neumann boundary condition
$\Phi_{y}(t,x,y)\to 0$ as $y\to-\mathtt{h}$. Imposing that the fluid particles
at the free surface remain on it along the evolution (kinematic boundary
condition) and that the pressure of the fluid plus the capillary forces at the
free surface is equal to the constant atmospheric pressure (dynamic boundary
condition), the time evolution of the fluid is determined by the non-local
quasi-linear equations

$\left\\{\begin{aligned} &\partial_{t}\eta=G(\eta)\psi+\gamma\eta\eta_{x}\\\
&\partial_{t}\psi=-g\eta-\frac{1}{2}\psi_{x}^{2}+\frac{1}{2}\frac{(\eta_{x}\psi_{x}+G(\eta)\psi)^{2}}{1+\eta_{x}^{2}}+\kappa\partial_{x}\Big{[}\frac{\eta_{x}}{(1+\eta_{x}^{2})^{\frac{1}{2}}}\Big{]}+\gamma\eta\psi_{x}+\gamma\partial_{x}^{-1}G(\eta)\psi\end{aligned}\right.$
(1.2)

where $g>0$ is the gravity constant, $\kappa>0$ is the surface tension
coefficient,
$\partial_{x}\big{[}\frac{\eta_{x}}{(1+\eta_{x}^{2})^{1/2}}\big{]}$ is the
curvature of the surface and $G(\eta)$ is the Dirichlet-Neumann operator

$G(\eta)\psi:=(-\Phi_{x}\eta_{x}+\Phi_{y})|_{y=\eta(x)}\,.$ (1.3)

The quantity $\int_{\mathbb{T}}\eta(x)\,{\rm d}x$ is a prime integral of (1.2)
(indeed $\int_{\mathbb{T}}G(\eta)\psi{\rm d}x=0$) and then, with no loss of
generality, we restrict to interfaces with zero average
$\int_{\mathbb{T}}\eta(x)\,{\rm d}x=0$. The component $\eta$ of the solution
of (1.2) will lie in a Sobolev space $H_{0}^{s+\frac{1}{4}}(\mathbb{T})$ of
periodic functions with zero mean. Moreover the vector field on the right hand
side of (1.2) depends only on $\eta$ and
$\psi-\frac{1}{2\pi}\int_{\mathbb{T}}\psi\,{\rm d}x$ (indeed $G(\eta)[1]=0$)
and therefore $\psi$ will evolve in a homogeneous Sobolev space
$\dot{H}^{{s-\frac{1}{4}}}(\mathbb{T})$ of periodic functions modulo
constants.

By [72, 33, 28, 69] the equations (1.2) are the Hamiltonian system

$\partial_{t}\begin{pmatrix}\eta\\\
\psi\end{pmatrix}=J_{\gamma}\begin{pmatrix}\nabla_{\eta}H_{\gamma}(\eta,\psi)\\\
\nabla_{\psi}H_{\gamma}(\eta,\psi)\end{pmatrix}\qquad\text{where}\qquad
J_{\gamma}:=\begin{pmatrix}0&{\rm Id}\\\ -{\rm
Id}&\gamma\partial_{x}^{-1}\end{pmatrix}$ (1.4)

and

$H_{\gamma}(\eta,\psi):=\frac{1}{2}\int_{\mathbb{T}}\big{(}\psi\,G(\eta)\psi+g\eta^{2}\big{)}\,{\rm
d}x+\kappa\int_{\mathbb{T}}\sqrt{1+\eta_{x}^{2}}\,{\rm
d}x+\frac{\gamma}{2}\int_{\mathbb{T}}\big{(}-\psi_{x}\eta^{2}+\frac{\gamma}{3}\eta^{3}\big{)}\,{\rm
d}x\,.$ (1.5)

The $L^{2}$-gradients $(\nabla_{\eta}H_{\gamma},\nabla_{\psi}H_{\gamma})$ in
(1.4) belong to (a dense subspace of) $\dot{L}^{2}(\mathbb{T})\times
L^{2}_{0}(\mathbb{T})$.

Since the bottom of $\mathcal{D}_{\eta}$ in (1.1) is flat, the Hamiltonian
vector field $X_{\gamma}$, defined by the right hand side of (1.2), is
translation invariant, namely

$X_{\gamma}\circ\tau_{\varsigma}=\tau_{\varsigma}\circ
X_{\gamma}\,,\quad\forall\varsigma\in\mathbb{R}\,,\qquad\text{where}\qquad\tau_{\varsigma}\colon
f(x)\mapsto f(x+\varsigma)$ (1.6)

is the translation operator. Equivalently the Hamiltonian $H_{\gamma}$ in
(1.5) satisfies $H_{\gamma}\circ\tau_{\varsigma}=H_{\gamma}$ for any
$\varsigma\in\mathbb{R}$. The associated conservation law induced by Noether
theorem is the momentum $\int_{\mathbb{T}}\psi(x)\eta_{x}(x)\,{\rm d}x$.

The main result of this paper (Theorem 1.1) is that, for almost all surface
tension coefficients $\kappa$, for any integer $N$, the solutions of the water
waves equations (1.2) with initial data (smooth enough) of size $\varepsilon$
small enough, are defined over a time interval of length at least
$c\varepsilon^{-N-1}$. This is the most general almost global existence in
time result for the solutions of the water waves equations with periodic
boundary conditions known so far. We present below the mathematical literature
concerning the local and global well posedness theory of water waves, focusing
on the maximal time life span of the solutions.

In order to state precisely the main theorem we define, for any
$s\in\mathbb{R}$, the Sobolev spaces

$H^{s}_{0}(\mathbb{T},\mathbb{C})=\Big{\\{}u(x)\in
H^{s}(\mathbb{T},\mathbb{C})\,:\,\int_{\mathbb{T}}u(x)\,{\rm
d}x=0\Big{\\}}\,,\quad\dot{H}^{s}(\mathbb{T},\mathbb{C})=H^{s}(\mathbb{T},\mathbb{C})/\mathbb{C}\,,$

equipped with the same norm

$\|u\|_{H^{s}_{0}}=\|u\|_{\dot{H}^{s}}=\Big{(}\sum_{n\in\mathbb{N}}\|\Pi_{n}u\|_{L^{2}}^{2}n^{2s}\Big{)}^{\frac{1}{2}}=\Big{(}\sum_{j\in\mathbb{Z}\setminus\\{0\\}}|u_{j}|^{2}|j|^{2s}\Big{)}^{\frac{1}{2}}$

where $\Pi_{n}$ denote the orthogonal projectors from
$L^{2}(\mathbb{T},\mathbb{C})$ on the subspaces spanned by $\\{e^{-{\rm
i}nx},e^{{\rm i}nx}\\}$ and $u_{j}$ are the Fourier coefficients of $u(x)$.
The quotient map induces an isometry between $H^{s}_{0}$ and $\dot{H}^{s}$ and
we shall often identify $H^{s}_{0}$ with $\dot{H}^{s}$. Our main result is the
following.

######  Theorem 1.1.

(Almost global in time gravity-capillary water waves with constant vorticity)
For any value of the gravity $g>0$, depth $\mathtt{h}\in(0,+\infty]$ and
vorticity $\gamma\in\mathbb{R}$, there is a zero measure set
${\mathcal{K}}\subset(0,+\infty)$ such that, for any surface tension
coefficient ${\kappa}\in(0,+\infty)\setminus{\mathcal{K}}$, for any $N$ in
$\mathbb{N}_{0}$, there is $s_{0}>0$ and, for any $s\geq s_{0}$, there are
$\varepsilon_{0}>0,c>0,C>0$ such that, for any
$0<\varepsilon<\varepsilon_{0}$, any initial datum

$(\eta_{0},\psi_{0})\in
H_{0}^{s+\frac{1}{4}}(\mathbb{T},\mathbb{R})\times{\dot{H}}^{s-\frac{1}{4}}(\mathbb{T},\mathbb{R})\quad\mbox{
with}\quad\|\eta_{0}\|_{H^{s+\frac{1}{4}}_{0}}+\|\psi_{0}\|_{{\dot{H}}^{s-\frac{1}{4}}}<\varepsilon\,,$

system (1.2) has a unique classical solution $(\eta,\psi)$ in

$C^{0}\Bigl{(}[-T_{\varepsilon},T_{\varepsilon}],H_{0}^{s+\frac{1}{4}}(\mathbb{T},\mathbb{R})\times{\dot{H}}^{s-\frac{1}{4}}(\mathbb{T},\mathbb{R})\Bigr{)}\quad\mbox{with}\quad
T_{\varepsilon}\geq c\varepsilon^{-N-1}\,,$ (1.7)

satisfying the initial condition $\eta|_{t=0}=\eta_{0},\psi|_{t=0}=\psi_{0}$.
Moreover

$\sup_{t\in[-T_{\varepsilon},T_{\varepsilon}]}\big{(}\|\eta\|_{H_{0}^{s+\frac{1}{4}}}+\|\psi\|_{\dot{H}^{s-\frac{1}{4}}}\big{)}\leq
C\varepsilon\,.$ (1.8)

Let us make some comments on the result.
1\. Comparison with [16]. We first discuss the relation between Theorem 1.1
and the result in Berti-Delort [16]. Theorem 1.1 extends the one in [16] in
two ways: ($i$) the equations (1.2) may have a non zero vorticity, whereas the
water waves in [16] are irrotational, i.e. $\gamma=0$. ($ii$) Also in the
irrotational case Theorem 1.1 is new since the almost global existence result
in [16] holds for initial data $(\eta_{0},\psi_{0})$ even in $x$, whereas
Theorem 1.1 applies to any $(\eta_{0},\psi_{0})$. We remark that, in the
irrotational case, the subspace of functions even in $x$ -the so called
standing waves- is invariant under evolution, whereas for $\gamma\neq 0$ it is
not invariant under the flow of (1.2) and the approach of [16] can not be
applied.
2\. Periodic setting vs $\mathbb{R}^{d}$. Global (and almost global) in time
results [76, 47, 77, 48, 54, 5, 51, 55, 39] have been proved for irrotational
water waves equations on $\mathbb{R}^{d}$ for sufficiently small, localized
and smooth initial data, exploiting the dispersive effects of the linear flow.
So far no global existence is known for the solutions of (1.2) in
$\mathbb{R}^{2}$, not even for irrotational fluids ([39] applies in
$\mathbb{R}^{3}$). The periodic setting is deeply different, as the linear
waves oscillate without decaying in time, and the long time dynamics of the
equations strongly depends on the presence of $N$-wave resonant interactions
and the Hamiltonian and reversible nature of the equations.
3\. Dispersion relation and non-resonant parameters. The water waves equations
(1.2) may be regarded as a quasi-linear complex PDE of the form

$\partial_{t}u=-{\rm i}\Omega(D)u+\mathcal{N}(u,\overline{u}),\quad
u(x)=\tfrac{1}{\sqrt{2\pi}}{\mathop{\sum}}_{j\in\mathbb{Z}\setminus\\{0\\}}u_{j}\,e^{{\rm
i}jx}\,,$

where $\mathcal{N}$ is a quadratic nonlinearity (depending on derivatives of
$u$) and $\Omega_{j}(\kappa)$ is the dispersion relation

$\Omega_{j}(\kappa):=\omega_{j}(\kappa)+\frac{\gamma}{2}\frac{{\mathtt{G}}(j)}{j}\,,\qquad\omega_{j}(\kappa):=\sqrt{{\mathtt{G}}(j)\Big{(}g+\kappa
j^{2}+\frac{\gamma^{2}}{4}\frac{{\mathtt{G}}(j)}{j^{2}}\Big{)}}\,,$ (1.9)

where ${\mathtt{G}}(\xi)=|\xi|\tanh(\mathtt{h}|\xi|)$ ($=|\xi|$ in infinite
depth) is the symbol of the Dirichlet-Neumann operator $G(0)$. The linear
frequencies $\Omega_{j}(\kappa)$ actually depend on
$(\kappa,g,\mathtt{h},\gamma)$. The restriction on the parameters required in
Theorem 1.1 arises to ensure the absence of $N$-wave resonant interactions

$\Omega_{j_{1}}(\kappa)\pm\ldots\pm\Omega_{j_{N}}(\kappa)\neq 0$ (1.10)

(with quantitative lower bounds as (1.13) below) among integer indices
$j_{1},\ldots,j_{N}$ which are not super-action preserving, cfr. Definition
7.4. In Theorem 1.1 we fix arbitrary $(g,\mathtt{h},\gamma)$ and require
$\kappa\notin\mathcal{K}$, but other choices are possible.
4\. Energy estimates. The life span estimate (1.7) and the bound (1.8) for the
solutions of (1.2) follow by an energy estimate for
$\|(\eta,\psi)\|_{X^{s}}:=\|\eta\|_{H_{0}^{s+\frac{1}{4}}}+\|\psi\|_{\dot{H}^{s-\frac{1}{4}}}$
of the form

${\|(\eta,\psi)(t)\|}_{X^{s}}^{2}\lesssim_{s,N}{\|(\eta,\psi)(0)\|}^{2}_{X^{s}}+\int_{0}^{t}\|(\eta,\psi)(\tau)\|^{N+3}_{X^{s}}\,{\rm
d}\tau\,.$ (1.11)

The fact that the right hand side in (1.11) contains the same norm $\|\
\|_{X^{s}}$ of the left hand side is non trivial because the equations (1.2)
are quasi-linear. The presence of the exponent $N$ is not trivial at all
because the nonlinearity in (1.2) vanishes only quadratically for
$(\eta,\psi)=(0,0)$. Actually it will be a major consequence of our
Hamiltonian Birkhoff normal form reduction, as we explain below.
5\. Long time existence of water waves. We now describe the long time
existence results proved in literature for space periodic water waves, with or
without capillarity and vorticity.

  1. ($i$)

$T_{\varepsilon}\geq c\varepsilon^{-1}$. The local well posedness theory for
free boundary Euler equations has been developed along several years in
different scenarios in [63, 71, 32, 74, 75, 60, 2, 4, 23, 62, 65, 61, 29, 66,
67, 3, 53, 7]. As a whole they prove the existence, for sufficiently nice
initial data, of classical smooth solutions on a small time interval. When
specialized to initial data of size $\varepsilon$ in some Sobolev space, imply
a time of existence larger than $c\varepsilon^{-1}$ (the nonlinearity in (1.2)
vanishes quadratically at zero). We remark that other large initial data can
lead to breakdown in finite time, see for example the papers [27, 30] on
“splash” singularities.

  2. ($ii$)

$T_{\varepsilon}\geq c\varepsilon^{-2}$. Wu [76], Ionescu-Pusateri [54],
Alazard-Delort [5] for pure gravity waves, and Ifrim-Tataru [52], Ionescu-
Pusateri [55] for $\kappa>0$, $g=0$ and $\mathtt{h}=+\infty$, proved that
small data of size $\varepsilon$ (periodic or on the line) give rise to
irrotational solutions defined on a time interval at least
$c\varepsilon^{-2}$. We quote [53] for $\kappa=0$, $g>0$, infinite depth and
constant vorticity, [50] for irrotational fluids, and [49] in finite depth.
All the previous results hold in absence of three wave interactions.
Exploiting the Hamiltonian nature of the water waves equations, Berti-Feola-
Franzoi [17] proved, for any value of gravity, capillarity and depth, an
energy estimate as (1.11) with $N=1$, and so a $c\varepsilon^{-2}$ lower bound
for the time of existence. The interesting fact is that in these cases three
wave interactions may occur, giving rise to the well known Wilton ripples in
fluid mechanics literature. We finally mention the
$\varepsilon^{-\frac{5}{3}+}$ long time existence result [56] for periodic 2D
gravity-capillary water waves (see [41, 46] for NLS).

  3. ($iii$)

$T_{\varepsilon}\geq c\varepsilon^{-3}$. A time of existence larger than
$c\varepsilon^{-3}$ has been recently proved for the pure gravity water waves
equations in deep water in Berti-Feola-Pusateri [18]. In this case four wave
interactions may occur, but the Hamiltonian Birkhoff normal form turns out to
be completely integrable by the formal computation in Zakharov-Dyachenko
[73],implying an energy estimate as (1.11) with $N=2$. This result has been
recently extended by S. Wu [78] for a larger class of initial data, developing
a novel approach in configuration space, and, even more recently, by Deng-
Ionescu-Pusateri [40] for waves with large period.

  4. ($iv$)

$T_{\varepsilon}\geq c_{N}\varepsilon^{-N}$ for any $N$. Berti-Delort [16]
proved, for almost all the values of the surface tension
$\kappa\in(0,+\infty)$, an almost global existence result as in Theorem 1.1
for the solutions of (1.2) in the case of zero vorticity $\gamma=0$ and for
initial data $(\eta_{0},\psi_{0})$ even in $x$. The restriction on the
capillary parameter arises to imply the absence of $N$-wave interactions, for
any $N$. As already said, Theorem 1.1 extends this result for any $\gamma$ and
for any periodic initial data, see comment 1.

The results [16, 18, 17] are based on paradifferential calculus. We remark
that all the transformations performed to get energy estimates, as the
celebrated Alinhac good unknown [6, 2, 4, 5], are not symplectic. In [18, 17]
an a-posteriori identification argument allows to prove that the corresponding
quadratic and cubic Poincaré-Birkhoff normal forms are nevertheless
Hamiltonian. This argument does not work for any $N$. We note that also the
local well posedness approach of S.Wu [74, 75] introduces coordinates which
break the Hamiltonian nature of the equations.

The lack of preservation of the Hamiltonian structure is a substantial
difficulty in order to deduce long time existence results. A major novelty of
this paper is to provide an effective tool to recover, in the framework of
paradifferential calculus, the nonlinear Hamiltonian structure, at any degree
of homogeneity $N$. The present approach is in principle applicable to a wide
range of quasi-linear PDEs.
6\. Open problem: we do not know if the almost global solutions of the Cauchy
problem proved in Theorem 1.1 are global in time or not, being (1.2) a quasi-
linear system of equations with periodic boundary conditions (no dispersive
effects of the flow nor conservation laws at the regularity level of the local
well posedeness theory can be exploited). Nevertheless several families of
time periodic/quasi-periodic solutions of (1.2) have been constructed in the
last years in [70, 1, 19, 21] (other KAM results for pure gravity water waves
are proved in [64, 57, 8, 20, 42]). We point out that it could also happen
that small, smooth and localized initial data lead to solutions which blow-up
in finite time (as it happen for quasilinear wave equations [58] and for
compressible Euler equations [68]). The following natural question therefore
remains still open: what happens to the solutions of (1.2) which do not start
on a KAM invariant torus for times longer than the ones provided by Theorem
1.1?

We now illustrate some of the main ideas of our approach.
1\. Paradifferential Hamiltonian Birkhoff normal form. For PDEs on a compact
manifold –where dispersion is not available– a natural tool to extend the life
span of solutions is to implement normal form ideas. This approach has been
developed for Hamiltonian semilinear PDEs starting with the seminal works [9,
13, 11] (for very recent advances see e.g. [26, 24, 25, 12]), and for quasi-
linear ones by Delort [36, 37]. These methods do not work for the quasi-linear
equations (1.2), as we explain below. The long time existence result of
Theorem 1.1 relies on a novel paradifferential Hamiltonian Birkhoff normal
form reduction for quasi-linear PDEs in presence of resonant wave
interactions.

The situation is substantially more difficult than in [16] which exploits only
the reversible structure of the water waves, and it is preserved by usual
paradifferential calculus. On the subspace of functions even in $x$, it
implies that its normal form possesses the actions $|u_{n}|^{2}$ as prime
integrals (on the subspace of even functions the linear frequencies
$\omega_{j}(\kappa)$ in the dispersion relation are simple). On the other
hand, without this restriction, the $\omega_{j}(\kappa)$ in (1.9) are double
and the approach in [16] fails. We remark that, in view of the $8$-wave
resonant interactions (1.15) described below, also for $\gamma\neq 0$ the
approach in [16] fails. In order to prove Theorem 1.1, it is necessary to
change strategy and preserve the Hamiltonian nature of the normal form to show
that the super-actions

$\|\Pi_{n}u\|_{L^{2}}^{2}=|u_{-n}|^{2}+|u_{n}|^{2}\,,\quad\forall
n\in\mathbb{N}\,,$ (1.12)

are prime integrals. This is a major difficulty since usual paradifferential
calculus transformations performed to get energy estimates do not preserve the
Hamiltonian structure.
2\. Symplectic Darboux corrector. In order to preserve the Hamiltonian
structure along the normal form reduction -it is sufficient up to homogeneity
$N$\- we construct symplectic correctors of usual paradifferential
transformations. We remind that the first step to apply paradifferential
calculus to PDEs relies on a suitable para-linearization of the equations. For
Hamiltonian PDEs, the paradifferential part inherits a linear Hamiltonian
structure that is preserved by performing “linearly symplectic”
transformations. The aim of the abstract Theorem 4.1 is to correct
paradifferential (more generally spectrally localized) linearly symplectic
maps (up to homogeneity $N$) to nonlinear symplectic ones, up to an arbitrary
degree of homogeneity. Theorem 4.1 is proved via Darboux-type arguments. The
Darboux corrector turns out to be a smoothing perturbation of the identity. As
a consequence it only slightly modifies the paradifferential structure of the
PDE.

Symplectic corrections via Darboux-type arguments have been used in different
contexts by Kuksin-Perelman [59], Bambusi [10], Cuccagna [34, 35], Bambusi-
Maspero [14, 15]. The present case is much more delicate since the symplectic
form to be corrected might be an unbounded perturbation of the standard one
(in all the above works it is a smoothing perturbation). This requires a novel
analysis that we describe below. The present approach is quite efficient in
PDE applications, since it systematically allows to symplectically correct
usual paradifferential transformations which lead to energy estimates.

Paradifferential calculus has been also developed by Delort [36, 37] for
Hamiltonian quasi-linear Klein-Gordon equations on spheres, with a different
approach. Also in these works the Hamiltonian structure is preserved only up
to homogeneity $N$.
3\. Non-resonance conditions. A key ingredient to achieve the Hamiltonian
Birkhoff normal form reduction which possesses the super-actions (1.12) as
prime integrals, are the non-resonance conditions (B.1) for the linear
frequencies $\Omega_{j}(\kappa)$ in (1.9) proved in Theorem B.1, which
exclude, for almost all surface tension coefficients, $N$-wave interactions,

$|\Omega_{j_{1}}(\kappa)\pm\ldots\pm\Omega_{j_{N}}(\kappa)|\gtrsim\max(|j_{1}|,\ldots,|j_{N}|)^{-\tau}\,,$
(1.13)

for all integer indices $j_{1},\ldots,j_{N}$ which are not super-action
preserving. Their proof is based on the Delort-Szeftel Theorem 5.1 in [38]
about measure estimates for sublevels of subanalytic functions.
4\. $\mathtt{S}\mathtt{A}\mathtt{P}$-Hamiltonians. Thanks to the non-resonance
conditions (B.1) we eliminate the Hamiltonian monomials which do not Poisson
commute with the super-actions (1.12), cfr. Lemma 7.7. The remaining
monomials, which we call super-action-preserving
($\mathtt{S}\mathtt{A}\mathtt{P}$) (Definition 7.8), have either the
integrable form $|z_{j_{1}}|^{2}\ldots|z_{j_{m}}|^{2}$ or the form

$z_{j_{1}}\overline{z_{-j_{1}}}\ldots
z_{j_{m}}\overline{z_{-j_{m}}}\times\text{integrable monomial}\,$ (1.14)

(with not necessarily distinct indexes $j_{1},\ldots,j_{m}$). The not
integrable monomials (1.14) allow an exchange of energy between the Fourier
modes $\\{z_{j_{a}},$ $z_{-j_{a}}\\}$, $a=1,\ldots,m$, but, thanks to the
Hamiltonian structure, each super-action $|z_{j_{a}}|^{2}+|z_{-j_{a}}|^{2}$
remains constant in time.

We may not expect to get an integrable Hamiltonian Birkhoff normal form for
the water waves equations (1.2) starting from the degree of homogeneity $8$.
Actually, using the conservation of momentum, the fourth order Hamiltonian
Birkhoff normal form is integrable, see Remark 7.16. The same holds if
$\gamma\neq 0$ also at degree $6$. But there are $8$-wave resonant
interactions corresponding to $\mathtt{S}\mathtt{A}\mathtt{P}$ not integrable
monomials

$z_{n_{1}}\overline{z_{-n_{1}}}z_{n_{2}}\overline{z_{-n_{2}}}z_{-n_{3}}\overline{z_{n_{3}}}z_{-n_{4}}\overline{z_{n_{4}}}$

(which are momentum preserving if $n_{1}+n_{2}=n_{3}+n_{4}$) for any
$\kappa>0$ and any $\gamma\in\mathbb{R}$. Indeed, for any positive integer
$n_{1},n_{2},n_{3},n_{4}$ we have, if $\mathtt{h}=+\infty$,

$\displaystyle\Omega_{n_{1}}(\kappa)-\Omega_{-n_{1}}(\kappa)+\Omega_{n_{2}}(\kappa)-\Omega_{-n_{2}}(\kappa)+\Omega_{-n_{3}}(\kappa)-\Omega_{n_{3}}(\kappa)+\Omega_{-n_{4}}(\kappa)-\Omega_{n_{4}}(\kappa)$
(1.15)
$\displaystyle\stackrel{{\scriptstyle\eqref{omegonejin}}}{{=}}\gamma({\rm
sign}(n_{1})+{\rm sign}(n_{2})-{\rm sign}(n_{3})-{\rm sign}(n_{4}))\equiv
0\,.$

The analytical difficulties of the loss of derivatives caused by the quasi-
linearity of the equations and the small divisors in (1.10) along the Birkhoff
normal form reduction is overcome by preserving the paradifferential structure
of the equations. The final outcome is that the water waves system in
Hamiltonian Birkhoff normal form satisfies an energy estimate of the form

${\|z(t)\|}^{2}_{\dot{H}^{s}}\leq{\|z(0)\|}^{2}_{\dot{H}^{s}}+C(s)\int_{0}^{t}{\|z(\tau)\|}^{N+3}_{\dot{H}^{s}}\,{\rm
d}\tau\,.$ (1.16)

5\. Comparison with the approach in [36, 37] and [16]. The Hamiltonian
approach to paradifferential calculus in [36, 37] is developed for quasi-
linear Klein-Gordon equations and can not be applied to prove Theorem 1.1.
Indeed, since the Klein-Gordon dispersion relation is asymptotically linear,
it is not required a reduction to $x$-independent paradifferential operators
up to smoothing remainders: since the commutator between first order
paradifferential operators is still a first order paradifferential operator,
it is possible to implement a Hamiltonian Birkhoff normal form reduction in
degrees of homogeneity, in the same spirit of semilinear PDEs. This approach
can not be applied for (1.2) since the dispersion relation (1.9) is super-
linear. It is for this reason that we first reduce in Proposition 7.2 the
paralinearized water waves equations to $x$-independent symbols up to
smoothing remainders. This was done in [16] for $\gamma=0$ (in a different
way) but breaking the Hamiltonian structure (see [44] for NLS). Incidentally
we mention that the paradifferential normal form in [16] is not a Birkhoff
normal form: for standing waves it is not needed to reduce the $x$-independent
symbols to deduce that the actions $|u_{n}|^{2}$ are prime integrals.

Summarizing, the proof of Theorem 1.1 demands

  * •

a reduction of the water waves equations (1.2) to paradifferential
$x$-independent symbols up to smoothing remainders, done in [16] for
$\gamma=0$ (in a different way) losing the Hamiltonian structure, and,
additionally, reduce the $x$-independent symbols to super-action preserving
Birkhoff normal form;

  * •

preserve the Hamiltonian structure of the Birkhoff normal form, goal achieved
in [36, 37] but only for Klein-Gordon equations.

The resolution of these requirements is a main achievement of this paper.

Before presenting further ideas of the proof of Theorem 1.1 we state the
following byproduct of the Darboux-type Theorem 4.1 concerning a symplectic
version of the Alinhac good unknown. Such result may be of separate interest
and use for water waves results in other contexts.
Symplectic good unknown up to homogeneity $N$. The celebrated Alazard, Burq,
Zuily approach [2, 4, 6] to local well posedness extends Lannes [60]
introducing the nonlinear, not symplectic, Alinach good unknown

$\omega:=\psi-{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(B(\eta,\psi)\right)}\eta\qquad\text{where}\qquad
B(\eta,\psi):=(\Phi_{y})(x,y)_{|y=\eta(x)}$

and $\Phi$ is the generalized harmonic velocity potential in (1.3) (the
notation ${{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\cdot\right)}$
refers to a paradifferential operator in the Weyl quantization, according to
Definition 2.4). The nonlinear map

${{\mathcal{G}}_{A}\begin{pmatrix}\eta\\\ \psi\end{pmatrix}={{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}{\begin{pmatrix}1&0\\\
-B(\eta,\psi)&1\end{pmatrix}}\begin{pmatrix}\eta\\\ \psi\end{pmatrix}\,,}$
(1.17)

although not symplectic, is linearly symplectic, namely

${{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}{\begin{pmatrix}1&0\\\
-B(\eta,\psi)&1\end{pmatrix}}^{\top}\,E_{0}\,{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}{\begin{pmatrix}1&0\\\
-B(\eta,\psi)&1\end{pmatrix}}}=E_{0}\qquad\text{where}\qquad
E_{0}:=\begin{pmatrix}0&-{\rm Id}\\\ {\rm Id}&0\end{pmatrix}\,.$ (1.18)

A direct corollary of Theorem 4.1 is the following result, proved at the end
of Section 4. We refer to Definition 2.7 for the precise definition of
smoothing operators.

######  Theorem 1.2.

(Symplectic good unknown up to homogeneity $N$) Let $N\in\mathbb{N}$. There
exists a pluri-homogeneous matrix of real smoothing operators $R_{\leq
N}(\cdot)$ in
$\Sigma_{1}^{N}\widetilde{\mathcal{R}}_{q}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{\mathbb{C}})$
for any $\varrho\geq 0$ such that

$\big{(}{\rm Id}+R_{\leq N}(\cdot)\big{)}\circ{{\mathcal{G}}_{A}}(\eta,\psi)$
(1.19)

is symplectic up to homogeneity $N$, according to (3.40).

Let us make some comments about Theorem 1.2 and the more general Theorem 4.1.
1\. The pluri-homogeneous smoothing correcting operators $R_{\leq N}(\cdot)$
in (1.19) are constructed in Proposition 4.7 by a Darboux deformation argument
à la Moser. More precisely the $R_{\leq N}(\cdot)$ are defined as approximate
inverses of approximate flows, up to homogeneity $N$, generated by smoothing
vector fields, which are algorithmically determined by the Darboux mechanism
and depend only on the pluri-homogeneous components of $\mathcal{G}$ up to
degree $N$ (more in general of ${\bf B}_{\leq N}$ in (4.1)).
2\. The Alinhac good unknown map (1.17) is bounded, but Theorem 4.1 also holds
for a (spectrally localized) map ${\bf B}_{\leq N}(U)$ in (4.1) which is
unbounded. This is the case for example when ${\bf B}_{\leq N}(U)$ is the
Taylor expansion of a linear flow generated by an unbounded operator, as we
discuss later in (1.23).
3\. We do not expect to find in Theorem 4.1 a corrector which produces a
completely symplectic transformation of the phase space, but only up to an
arbitrary degree of homogeneity $N$. In the Darboux approach of Section 4 this
is because the equation (4.28) for the smoothing vector field $Y^{\tau}$,
whose flow defines the symplectic corrector, can be solved only in homogeneity
having the form $E_{c}Y^{\tau}=R(V,Y^{\tau},{\rm d}_{V}Y^{\tau})$ and so
losing derivatives, see Remark 4.6. We remark that also the transformations in
[36, 37] are symplectic at degree of homogeneity $\leq N$. A similar problem
appears in [45].
4\. Darboux perturbative methods for Hamiltonian PDEs have been developed in
different contexts in [10, 14, 15, 34, 35, 59]. In all these cases, the
perturbed symplectic form is a smoothing perturbation of the standard one and
thus Darboux correctors are symplectic maps. On the other hand, in this work
the perturbed symplectic tensor is a (possibly) unbounded perturbation of the
standard one,

$E_{{\leq N}}(V)=E_{c}+\text{(possibly) unbounded operator}\,.$ (1.20)

A key tool to overcome this difficulty is the structural Lemma 4.5.
5\. A symplectic map up to homogeneity $N$, transforms a Hamiltonian system up
to homogeneity $N$ into another Hamiltonian system up to homogeneity $N$, see
Lemma 3.15.

#### Further ideas of proof and plan of the paper

The paper is divided in

  1. 1.

Part I) containing the abstract functional setting used along the paper and
the Darboux result;

  2. 2.

Part II) with the proof of the almost global in time Theorem 1.1.

We first illustrate the way we proceed to preserve the Hamiltonian structure,
up to homogeneity $N$, in a generic transformation step along the proof of
Theorem 1.1.
Symplectic conjugation step up to homogeneity $N$. Consider a real-to-real
system in paradifferential form

$\partial_{t}U=X(U)={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U;t,x,\xi)\right)}[U]+R(U;t)[U]\
,\quad U=\begin{bmatrix}u\\\ \overline{u}\end{bmatrix}\ ,$ (1.21)

where $A(U;t,x,\xi)$ is a matrix of symbols and $R(U;t)$ are
$\varrho$-smoothing operators, which admit a homogeneous expansion up to
homogeneity $N$, whereas the terms with homogeneity $>N$ are dealt, as in
[16], as time dependent symbols and remainders, see Section 2.1. This is quite
convenient from a technical point of view because it does not demand much
information about the higher degree terms. Moreover this enables to directly
use the paralinearization of the Dirichlet-Neumann operator proved in [16].
System (1.21) is Hamiltonian up to homogeneity $N$, namely the homogeneous
components of the vector field $X(U)$ of degree $\leq N+1$ have the
Hamiltonian form

$J_{c}\nabla H(U)\qquad\text{where}\qquad J_{c}:=\begin{bmatrix}0&-{\rm i}\\\
{\rm i}&0\end{bmatrix}$ (1.22)

is the Poisson tensor and $H(U)$ is a real valued pluri-homogeneous
Hamiltonian of degree $\leq N+2$. Moreover the paradifferential operator
${{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}$ in (1.21)
is a linear Hamiltonian operator, up to homogeneity $N$, namely of the form
${{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}=J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(B(U)\right)}$ where $B(U)$ is a
symmetric operator up to homogeneity $N$, see Definition 3.6.

In order to prove energy estimates for (1.21) we transform it under several
changes of variables. Actually we do not really perform changes of variables
of the phase space, but we proceed in the time dependent setting due to the
high homogeneity terms. Let us discuss a typical transformation step. Let
${\mathcal{G}}(U;t):={\mathcal{G}}^{\tau}(U;t){|_{\tau=1}}$ be the time
$1$-flow

$\partial_{\tau}\mathcal{G}^{\tau}(U;t)=J_{c}\,{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\big{(}B(U;\tau,t,x,\xi)\big{)}\mathcal{G}^{\tau}(U;t)\,,\quad\mathcal{G}^{0}(U;t)={\rm
Id}\,,$ (1.23)

generated by a linearly Hamiltonian operator $J_{c}\,{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\big{(}B(U;\tau,t,x,\xi)\big{)}$ up to
homogeneity $N$. The transformation $\mathcal{G}(U;t)$ is invertible and
bounded on $\dot{H}^{s}(\mathbb{T})\times\dot{H}^{s}(\mathbb{T})$ for any
$s\in\mathbb{R}$ and it admits a pluri-homogeneous expansion
${\mathcal{G}}_{\leq N}(U)$, which is an unbounded operator if the generator
$J_{c}\,{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\big{(}B\big{)}$ is
unbounded, see Section 3.3. If $U$ solves (1.21) then the variable

$W:={\mathcal{G}}(U;t)U$ (1.24)

solves a new system in paradifferential form

$\partial_{t}W=X_{+}(W)={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A_{+}(W;t,x,\xi)\right)}[W]+R_{+}(W;t)[W]$
(1.25)

(actually the symbols and remainders of homogeneity $>N$ in (1.25) are still
expressed in terms of $U$, but for simplicity we skip to discuss this issue
here). In Section 6 we perform several transformations of this kind, choosing
suitable generators $J_{c}\,{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\big{(}B\big{)}$ (either bounded or
unbounded) in order to obtain a diagonal matrix $A_{+}$ with $x$-independent
symbols.

We remark that, with this procedure, since the time one flow map
${\mathcal{G}}(U;t)$ of the linear Hamiltonian system (1.23) is only linearly
symplectic up to homogeneity $N$, namely

${\mathcal{G}(U;t)}^{\top}E_{c}{\mathcal{G}}(U;t)=E_{c}+E_{>N}(U;t)\,,\quad
E_{>N}(U;t)=O(\|U\|^{N+1})\,,$

where $E_{c}:=J_{c}^{-1}$ is the standard symplectic tensor, the new system
(1.25) is not Hamiltonian anymore, not even its pluri-homogeneous components
of degree $\leq N+1$. The new system (1.25) is only linearly symplectic, up to
homogeneity $N$, see Lemma 3.9. In order to obtain a new Hamiltonian system up
to homogeneity $N$, we use the Darboux results of Section 4 to construct
perturbatively a “symplectic corrector” of the transformation (1.24).

Let us say some words about the construction of the symplectic corrector. We
remark that the perturbed symplectic tensor $E_{{\leq N}}(V)$ induced by the
non-symplectic transformation ${\mathcal{G}}_{\leq N}(U)$ is not a smoothing
perturbation of the standard Poisson tensor $E_{c}$, cfr. (1.20). However,
Lemmata 4.4 and 4.5 prove that, for any pluri-homogeneous vector field $X(V)$,
we have

$E_{{\leq N}}(V)[X(V)]=E_{c}X(V)+\nabla\mathcal{W}(V)+\text{ smoothing vector
fields}+\text{high homogeneity terms}$

where $\mathcal{W}(V)$ is a scalar function. This algebraic structural
property enables to prove the Darboux Proposition 4.7, thus Theorem 4.1, via a
deformation argument à la Moser. We also remark that the operators $R_{\leq
N}(\cdot)$ of Theorem 4.1 are smoothing for arbitrary $\varrho\geq 0$, since
they have $2$ equivalent frequencies, namely
${\rm\max}_{2}(n_{1},\ldots,n_{p+1})\sim\max(n_{1},\ldots,n_{p+1})$ in (2.38),
arising by applications of Lemma 2.21. This property compensates the presence
of unbounded operators in ${\mathcal{G}}_{\leq N}(U)$.

In conclusion, Theorem 4.1 provides a nonlinear map $W+R_{\leq N}(W)W$, where
$R_{\leq N}(W)$ are pluri-homogeneous $\varrho$-smoothing operators for
arbitrary $\varrho>0$, such that the pluri-homogeneous map

$\mathcal{D}_{N}(U):=({\rm Id}+R_{\leq N}(\cdot))\circ\mathcal{G}_{\leq
N}(U)U$

is symplectic up to homogeneity $N$, i.e.

$\big{[}{\rm d}_{U}\mathcal{D}_{N}(U)\big{]}^{\top}E_{c}\,\big{[}{\rm
d}_{U}\mathcal{D}_{N}(U)\big{]}=E_{c}+E_{>N}(U)$ (1.26)

where $E_{>N}(U)$ is an operator of homogeneity degree $\geq N+1$. As a
consequence, since (1.21) is Hamiltonian up to homogeneity $N$, the variable

$Z(t):=\mathcal{D}(U(t);t):=W(t)+R_{\leq N}(W(t))=({\rm Id}+R_{\leq
N}(\cdot))\circ\mathcal{G}(U(t);t)U(t)$

satisfies a system which is Hamiltonian up to homogeneity $N$ as well, and
which has, since $R_{\leq N}(\cdot)$ are smoothing operators, the same
paradifferential form as in (1.25),

$\partial_{t}Z=X_{++}(Z)={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A_{++}(Z;t,x,\xi)\right)}[Z]+R_{++}(Z;t)[Z]\,.$
(1.27)

This is the content of Theorem 7.1. Note that the matrix of symbols
$A_{++}(Z;t,x,\xi)$ in (1.27) is obtained by substituting in
$A_{+}(W;t,x,\xi)$ the relation $W=Z-R_{\leq N}(Z)+\ldots$ obtained inverting
$Z=W+R_{\leq N}(W)$ approximately up to homogeneity $N$. This procedure is
rigorously justified in Lemmata A.4 and A.5.

Scheme of proof of Theorem 1.1. In part II we apply the abstract formalism
developed in part I to prove Theorem 1.1. We proceed as follows.
Section 5: paralinearization of the water waves equations. In Section 5 we
first paralinearize the water waves equations (1.2), we introduce the Wahlén
variables $(\eta,\zeta)$ in (5.2) and the complex variable $U$ in (5.6) which
diagonalizes the linearized equations at zero. The resulting paralinearized
equations (5.37) are a Hamiltonian system of the form

$\partial_{t}U=J_{c}\nabla H_{\gamma}(U)$ (1.28)

where $H_{\gamma}$ is the Hamiltonian in (1.5) written in the variable $U$.
Our goal is to perform several changes of variable to prove energy estimates
for (5.37), i.e. (1.28), valid up to times of order $\varepsilon^{-N-1}$. We
split the proof in two major steps.
Section 6: Hamiltonian paradifferential normal form. In Section 6.1 we
introduce the good unknown of Alinhac ${\mathcal{G}(U)}$ (written in complex
coordinates), obtaining a system which has energy estimates for times of order
$\varepsilon^{-1}$. The Alinhac good unknown is not symplectic and therefore
the transformed system (6.4) is not Hamiltonian anymore. Next we transform
(6.4) into a diagonal matrix of $x$-independent symbols up to smoothing
remainders, in order to compensate along the Birkhoff normal form reduction
process the loss of derivatives due to the small divisors and the quasi-linear
nature of the water waves equations, see Proposition 6.1. The resulting system

$\partial_{t}W={{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left({\rm
i}\mathtt{m}_{\frac{3}{2}}(U;t,\xi)\right)}W+R(U;t)W$ (1.29)

is no longer Hamiltonian. In (1.29) the imaginary part of the symbol
$\mathtt{m}_{\frac{3}{2}}$ has order zero and homogeneity larger than $N$,
whereas $R(U;t)$ is a smoothing remainder vanishing linearly in $U$.
Section 7: Hamiltonian Birkhoff normal form. In order to recover the
Hamiltonian structure we apply the symplectic corrector given by Theorem 4.1:
using Theorem 7.1 and Lemmata A.4 and A.5, we obtain in Proposition 7.2 system
(7.4) which is Hamiltonian up to homogeneity $N$. We perform the Hamiltonian
Birkhoff normal form reduction for any value of the surface tension $\kappa$
outside the set ${\mathcal{K}}$ defined in Theorem B.1. Iteratively we first
reduce the $p$–homogeneous $x$-independent paradifferential symbol to its
super-action-preserving component, via the linear flow generated by an
unbounded Fourier multiplier, see (7.42). Since such transformation is only
linearly symplectic, we apply again Theorem 7.1 to recover a Hamiltonian
system up to homogeneity $N$, see system (7.56). Finally we reduce the
$(p+1)$-homogeneous component of the Hamiltonian smoothing vector field to its
super-action preserving part, see (7.70). The key property is that a super-
action preserving Hamiltonian Poisson commutes with the super-actions defined
in (1.12). After $N$ iterations, the final outcome is the Hamiltonian Birkhoff
normal form system (7.21), which has the form

$\partial_{t}Z=J_{c}\nabla H^{(\mathtt{S}\mathtt{A}\mathtt{P})}(Z)+{{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(-{\rm
i}(\mathtt{m}_{\frac{3}{2}})_{>N}(U;t,\xi)\right)}Z+R_{>N}(U;t)U$ (1.30)

where $H^{(\mathtt{S}\mathtt{A}\mathtt{P})}(Z)$ is a super-action preserving
Hamiltonian (Definition 7.8) and the higher order homogeneity paradifferential
and smoothing terms admit energy estimates in Sobolev spaces (the imaginary
part of the symbol $(\mathtt{m}_{\frac{3}{2}})_{>N}$ has order zero).
Section 8: energy estimates. The Hamiltonian Birkhoff normal form equation
$\partial_{t}Z=J_{c}\nabla H^{(\mathtt{S}\mathtt{A}\mathtt{P})}(Z)$ obtained
neglecting the terms of homogeneity larger than $N$ in (1.30) possesses the
super–actions $|z_{-n}|^{2}+|z_{n}|^{2}$, for any $n\in\mathbb{N}$, as prime
integrals. Thus it preserves the Sobolev norms and the solutions of (1.30)
with initial data of size $\varepsilon$ have energy estimates up to times of
order $\varepsilon^{-N-1}$. In conclusion, since the Sobolev norms of $U$ in
(1.28) and $Z$ in (1.30) are equivalent, we deduce energy estimates for
(1.28),

${\|U(t)\|}_{\dot{H}^{s}}^{2}\lesssim_{s,K}{\|U(0)\|}^{2}_{\dot{H}^{s}}+\int_{0}^{t}\|U(\tau)\|^{N+3}_{\dot{H}^{s}}\,{\rm
d}\tau$

valid up to times of order $\varepsilon^{-N-1}$. A standard bootstrap argument
concludes the proof of Theorem 1.1.

Notation: The notation $A\lesssim B$ means that there exists a constant $C\geq
0$ such that $A\leq CB$. We denote $\mathbb{N}=\\{1,2,\ldots\\}$ and
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$.

##  Part I Abstract setting and Darboux symplectic corrector

###  2 Functional Setting

This section contains the abstract functional setting used along the paper. In
Section 2.1 we present definitions and results about para-differential
calculus following Berti-Delort [16], but defining the different notion of
$m$-operators. Using the same classes of symbols and smoothing operators of
[16] has the advantage to directly rely on the result in [16] concerning the
paralinearization of the Dirichlet-Neumann operator with multilinear
expansions. In Section 2.2 we introduce the notion of spectrally localized
maps which includes, as a particular case, paradifferential operators of any
order. Then we prove several properties of spectrally localized maps among
which that the transpose of the differential of a homogeneous spectrally
localized map is smoothing, see Lemma 2.21. This result generalizes a lemma
which has been proved in Feola-Iandoli [45] for paradifferential operators,
and it is relevant for producing the Hamiltonian corrections to the
homogeneous components of the vector field in Section 4, by means of a Darboux
approximate procedure. In Section 2.3 we construct approximate inverses of
non–linear maps and approximate flows up to an arbitrary degree of
homogeneity. In Section 2.4 we introduce the formalism of pluri-homogeneous
$k$-forms, Lie derivatives and Cartan’s magic formula. Let us first fix some
notation used along the paper.

##### Function spaces.

Along the paper we deal with real parameters

$s\geq s_{0}\gg K\gg\varrho\gg N$ (2.1)

where $N\in\mathbb{N}_{0}$ is the constant in Theorem 1.1.

Given an interval $I\subset\mathbb{R}$ symmetric with respect to $t=0$ and
$s\in\mathbb{R}$, we define the space

$C_{*}^{K}(I,\dot{H}^{s}\left(\mathbb{T},\mathbb{C}^{2})\right):=\bigcap_{k=0}^{K}C^{k}\big{(}I,\dot{H}^{s-\frac{3}{2}k}(\mathbb{T},\mathbb{C}^{2})\big{)}$

endowed with the norm

$\sup_{t\in I}\|U(t,\cdot)\|_{K,s}\qquad{\rm
where}\qquad\|U(t,\cdot)\|_{K,s}:=\sum_{k=0}^{K}\|\partial_{t}^{k}U(t,\cdot)\|_{\dot{H}^{s-\frac{3}{2}k}}\,,$
(2.2)

and we also consider its subspace

$C_{*\mathbb{R}}^{K}(I,\dot{H}^{s}\left(\mathbb{T},\mathbb{C}^{2})\right):=\Big{\\{}U\in
C_{*}^{K}(I,\dot{H}^{s}\left(\mathbb{T},\mathbb{C}^{2})\right)\ :\
U=\begin{pmatrix}u\\\ \overline{u}\end{pmatrix}\Big{\\}}\,.$

Given $r>0$ we set $B^{K}_{s}(I;r)$ the ball of radius $r$ in
$C_{*}^{K}(I,\dot{H}^{s}\left(\mathbb{T},\mathbb{C}^{2})\right)$ and by
$B^{K}_{s,\mathbb{R}}(I;r)$ the ball of radius $r$ in
$C_{*\mathbb{R}}^{K}(I,\dot{H}^{s}\left(\mathbb{T},\mathbb{C}^{2})\right)$.

The parameter $s$ in (2.2) denotes the spatial Sobolev regularity of the
solution $U(t,\cdot)$ and $K$ its regularity in the time variable. The
gravity-capillary water waves vector field loses $3/2$-derivatives, and
therefore, differentiating the solution $U(t)$ for $k$-times in the time
variable, there is a loss of $\tfrac{3}{2}k$-spatial derivatives. The
parameter $\varrho$ in (2.1) denotes the order where we decide to stop our
regularization of the system and depends on the number $N$ of steps of
Birkhoff normal form that we will perform and the smallness of the small
divisors due to the resonances.

We denote
$\dot{L}^{2}(\mathbb{T},\mathbb{C}):=\dot{H}^{0}(\mathbb{T},\mathbb{C})$ and
$\dot{L}^{2}_{r}:=\dot{L}^{2}(\mathbb{T},\mathbb{R})=\dot{H}^{0}(\mathbb{T},\mathbb{R})$
the subspace of $\dot{L}^{2}(\mathbb{T},\mathbb{C})$ made by real valued
functions. Given $u,v\in\dot{L}^{2}(\mathbb{T},\mathbb{C})$ we define

$\langle
u,v\rangle_{\dot{L}^{2}_{r}}:=\int_{\mathbb{T}}\Pi_{0}^{\bot}u(x)\,\Pi_{0}^{\perp}v(x)\,{\rm
d}x\,,\quad\text{respectively}\quad\langle
u,v\rangle_{\dot{L}^{2}}:=\int_{\mathbb{T}}\Pi_{0}^{\bot}u(x)\,\overline{\Pi_{0}^{\bot}v(x)}\,{\rm
d}x\,,$ (2.3)

where $\Pi_{0}^{\perp}u:=u-\frac{1}{2\pi}\int u(x)\,{\rm d}x$ is the projector
onto the zero mean functions.

We also consider the non-degenerate bilinear form on
$\dot{L}^{2}(\mathbb{T};\mathbb{C}^{2})$

$\Big{\langle}\begin{pmatrix}v_{1}^{+}\\\
v_{1}^{-}\end{pmatrix},\begin{pmatrix}v_{2}^{+}\\\
v_{2}^{-}\end{pmatrix}\Big{\rangle}_{r}:=\langle
v_{1}^{+},v_{2}^{+}\rangle_{\dot{L}^{2}_{r}}+\langle
v_{1}^{-},v_{2}^{-}\rangle_{\dot{L}^{2}_{r}}\,.$ (2.4)

Fourier expansions. Given a $2\pi$-periodic function $u(x)$ in the homogeneous
space $\dot{L}^{2}(\mathbb{T},\mathbb{C})$, we identify $u(x)$ with its zero
average representative and we expand it in Fourier series as

$u(x)=\sum_{j\in\mathbb{Z}\setminus\\{0\\}}\widehat{u}(j)\frac{e^{{\rm
i}jx}}{\sqrt{2\pi}},\quad\widehat{u}(j):=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{T}}u(x)e^{-{\rm
i}jx}\,{\rm d}x\,.$ (2.5)

We shall expand a function $\begin{pmatrix}u^{+}\\\ u^{-}\end{pmatrix}$ as

$\begin{pmatrix}u^{+}\\\
u^{-}\end{pmatrix}=\sum_{\sigma\in\pm}\sum_{j\in\mathbb{Z}\setminus\\{0\\}}{\mathtt{q}}^{\sigma}u_{j}^{\sigma}\frac{e^{{\rm
i}\sigma jx}}{\sqrt{2\pi}},\quad u^{\sigma}_{j}:=\widehat{u^{\sigma}}(\sigma
j)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{T}}u^{\sigma}(x)e^{-{\rm i}\sigma
jx}\,{\rm d}x$ (2.6)

where

${\mathtt{q}}^{+}:=\begin{pmatrix}1\\\
0\end{pmatrix},\quad{\mathtt{q}}^{-}:=\begin{pmatrix}0\\\ 1\end{pmatrix}\,.$
(2.7)

For $n\in\mathbb{N}$ we denote by $\Pi_{n}$ the orthogonal projector from
$L^{2}(\mathbb{T},\mathbb{C})$ to the linear subspace spanned by $\\{e^{{\rm
i}nx},e^{-{\rm i}nx}\\}$,

$(\Pi_{n}u)(x):=\widehat{u}(n)\frac{e^{{\rm
i}nx}}{\sqrt{2\pi}}+\widehat{u}(-n)\frac{e^{-{\rm i}nx}}{\sqrt{2\pi}}\,,$
(2.8)

and we denote by $\Pi_{n}$ also the corresponding projector in
$L^{2}(\mathbb{T},\mathbb{C}^{2})$.

If $\mathcal{U}=(U_{1},\dots,U_{p})$ is a $p$-tuple of functions and
$\vec{n}=(n_{1},\dots,n_{p})\in\mathbb{N}^{p}$, we set

$\Pi_{\vec{n}}\mathcal{U}:=\big{(}\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{p}}U_{p}\big{)}\,,\qquad\tau_{\varsigma}\mathcal{U}:=\big{(}\tau_{\varsigma}U_{1},\dots,\tau_{\varsigma}U_{p}\big{)}\,.$

For $\vec{\jmath}_{p}=(j_{1},\dots,j_{p})\in(\mathbb{Z}\setminus\\{0\\})^{p}$
and $\vec{\sigma}_{p}=(\sigma_{1},\dots,\sigma_{p})\in\\{\pm\\}^{p}$ we denote
$|\vec{\jmath}_{p}|:=\max(|j_{1}|,\ldots,|j_{p}|)$ and

$u_{\vec{\jmath}_{p}}^{\vec{\sigma}_{p}}:=u_{j_{1}}^{\sigma_{1}}\dots
u_{j_{p}}^{\sigma_{p}}\,,\qquad\vec{\sigma}_{p}\cdot\vec{\jmath}_{p}:=\sigma_{1}j_{1}+\ldots+\sigma_{p}j_{p}\,.$
(2.9)

Note that, under the translation operator $\tau_{\varsigma}$ defined in (1.6),
the Fourier coefficients of $\tau_{\varsigma}u$ transform as

$(\tau_{\varsigma}u)^{\sigma}_{j}=e^{{\rm i}\sigma
j\varsigma}u^{\sigma}_{j}\,.$

We finally denote

${\mathfrak{T}}_{p}:=\Big{\\{}(\vec{\jmath}_{p},\vec{\sigma}_{p})\in(\mathbb{Z}\setminus\\{0\\})^{p}\times\\{\pm\\}^{p}\colon\
\ \vec{\sigma}_{p}\cdot\vec{\jmath}_{p}=0\Big{\\}}\,.$ (2.10)

Real-to-real operators and vector fields. Given a linear operator
$R(U)[\cdot]$ acting on $\dot{L}^{2}(\mathbb{T};\mathbb{C})$ we associate the
linear operator defined by the relation

$\overline{R}(U)[v]:=\overline{R(U)[\overline{v}]}\,,\quad\forall
v:\mathbb{T}\rightarrow\mathbb{C}\,.$ (2.11)

An operator $R(U)$ is real  if $R(U)=\overline{R}(U)$. We say that a matrix of
operators acting on $\dot{L}^{2}(\mathbb{T};\mathbb{C}^{2})$ is _real-to-real_
, if it has the form

$R(U)=\left(\begin{matrix}R_{1}(U)&R_{2}(U)\\\
\overline{R_{2}}(U)&\overline{R_{1}}(U)\end{matrix}\right)\,,$ (2.12)

for any $U$ in

$\dot{L}^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2}):=\Big{\\{}V\in\dot{L}^{2}(\mathbb{T},\mathbb{C}^{2})\,:\,V=\begin{pmatrix}v\\\
\overline{v}\end{pmatrix}\Big{\\}}\,.$ (2.13)

We define similarly $\dot{H}^{s}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})$. A
real-to-real matrix of operators $R(U)$ acts in the subspace
$\dot{L}^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})$.

If $R_{1}(U)$ and $R_{2}(U)$ are real-to-real operators then also
$R_{1}(U)\circ R_{2}(U)$ is real-to-real.

Similarly we will say that a vector field

$X(U):=\begin{pmatrix}X(U)^{+}\\\ X(U)^{-}\end{pmatrix}\quad\text{is real-to-
real if}\quad\overline{X(U)^{+}}=X(U)^{-}\,,\quad\forall
U\in\dot{L}^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})\,.$ (2.14)

####  2.1 Paradifferential calculus

We first introduce the paradifferential operators (Definition 2.4) following
[16]. Then we define the new class of $m$-Operators (Definition 2.5) that, for
$m\leq 0$, are the smoothing ones (Definition 2.7), and we prove properties of
$m$-operators under transposition and composition.

##### Classes of symbols.

We give the definition of the classes of symbols that we use. Roughly speaking
the class $\widetilde{\Gamma}_{p}^{m}$ contains symbols of order $m$ and
homogeneity $p$ in $U$, whereas the class $\Gamma_{K,K^{\prime},p}^{m}$
contains non-homogeneous symbols of order $m$ that vanishes at degree at least
$p$ in $U$ and that are $(K-K^{\prime})$-times differentiable in $t$; we can
think the parameter $K^{\prime}$ like the number of time derivatives of $U$
that are contained in the symbols. In the following we denote
$\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}):=\bigcap_{s\in\mathbb{R}}\dot{H}^{s}(\mathbb{T};\mathbb{C}^{2})$.

######  Definition 2.1.

Let $m\in\mathbb{R}$, $p,N\in\mathbb{N}_{0}$, $K,K^{\prime}\in\mathbb{N}_{0}$
with $K^{\prime}\leq K$, and $r>0$.

($i$) $p$-homogeneous symbols. We denote by $\widetilde{\Gamma}^{m}_{p}$ the
space of symmetric $p$-linear maps from
$(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{p}$ to the space of
$C^{\infty}$ functions from $\mathbb{T}\times\mathbb{R}$ to $\mathbb{C}$,
$(x,\xi)\mapsto a(\mathcal{U};x,\xi)$, satisfying the following: there exist
$\mu>0$ and, for any $\alpha,\beta\in\mathbb{N}_{0}$, there is a constant
$C>0$ such that

$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(\Pi_{\vec{n}}\mathcal{U};x,\xi)|\leq
C|\vec{n}|^{\mu+\alpha}\langle\xi\rangle^{m-\beta}\prod_{j=1}^{p}\|\Pi_{n_{j}}U\|_{L^{2}}$
(2.15)

for any
$\mathcal{U}=(U_{1},\dots,U_{p})\in(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{p}$
and $\vec{n}=(n_{1},\dots,n_{p})\in\mathbb{N}^{p}$. Moreover we assume that,
if for some $(n_{0},\dots,n_{p})\in\mathbb{N}_{0}\times\mathbb{N}^{p}$,
$\Pi_{n_{0}}a\left(\Pi_{n_{1}}U_{1},\dots\Pi_{n_{p}}U_{p};\cdot\right)\not=0$,
then there is a choice of signs $\sigma_{0},\dots,\sigma_{p}\in\\{-1,1\\}$
such that $\sum_{j=0}^{p}\sigma_{j}n_{j}=0$. In addition we require the
translation invariance property

$a\left(\tau_{\varsigma}\mathcal{U};x,\xi\right)=a\left(\mathcal{U};x+\varsigma,\xi\right),\quad\forall\varsigma\in\mathbb{R}\,,$
(2.16)

where $\tau_{\varsigma}$ is the translation operator in (1.6).

For $p=0$ we denote by $\widetilde{\Gamma}^{m}_{0}$ the space of constant
coefficients symbols $\xi\mapsto a(\xi)$ which satisfy (2.15) with $\alpha=0$
and the right hand side replaced by $C\langle\xi\rangle^{m-\beta}$.

We denote by $\Sigma_{p}^{N}\widetilde{\Gamma}^{m}_{q}$ the class of pluri-
homogeneous symbols $\sum_{q=p}^{N}a_{q}$ with
$a_{q}\in\widetilde{\Gamma}_{q}^{m}$. For $p\geq N+1$ we mean that the sum is
empty.

($ii$) Non-homogeneous symbols.  We denote by $\Gamma_{K,K^{\prime},p}^{m}[r]$
the space of functions $(U;t,x,\xi)\mapsto a(U;t,x,\xi)$, defined for $U\in
B_{s_{0}}^{K^{\prime}}(I;r)$ for some $s_{0}$ large enough, with complex
values, such that for any $0\leq k\leq K-K^{\prime}$, any $\sigma\geq s_{0}$,
there are $C>0$, $0<r(\sigma)<r$ and for any $U\in
B_{s_{0}}^{K}(I;r(\sigma))\cap
C_{*}^{k+K^{\prime}}(I,\dot{H}^{\sigma}(\mathbb{T};\mathbb{C}^{2}))$ and any
$\alpha,\beta\in\mathbb{N}_{0}$, with $\alpha\leq\sigma-s_{0}$ one has the
estimate

$|\partial_{t}^{k}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(U;t,x,\xi)|\leq
C\langle\xi\rangle^{m-\beta}\|U\|_{k+K^{\prime},s_{0}}^{p-1}\|U\|_{k+K^{\prime},\sigma}\,.$
(2.17)

If $p=0$ the right hand side has to be replaced by
$C\langle\xi\rangle^{m-\beta}$.

($iii$) Symbols. We denote by $\Sigma\Gamma_{K,K^{\prime},p}^{m}[r,N]$ the
space of functions $(U;t,x,\xi)\mapsto a(U;t,x,\xi),$ with complex values such
that there are homogeneous symbols $a_{q}\in\widetilde{\Gamma}_{q}^{m}$,
$q=p,\dots,N$ and a non-homogeneous symbol
$a_{>N}\in\Gamma_{K,K^{\prime},N+1}^{m}$ such that

$a(U;t,x,\xi)=\sum_{q=p}^{N}a_{q}(U,\dots,U;x,\xi)+a_{>N}(U;t,x,\xi)\,.$
(2.18)

We denote by
$\Sigma\Gamma_{K,K^{\prime},p}^{m}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$ the
space of $2\times 2$ matrices with entries in
$\Sigma\Gamma_{K,K^{\prime},p}^{m}[r,N]$.

We say that a symbol $a(U;t,x,\xi)$ is _real_ if it is real valued for any
$U\in B^{K^{\prime}}_{s_{0},\mathbb{R}}(I;r)$.

$\bullet$ If $a(\mathcal{U};\cdot)$ is a homogeneous symbol in
$\widetilde{\Gamma}_{p}^{m}$ then $a(U,\ldots,U;\cdot)$ belongs to
$\Gamma^{m}_{K,0,p}[r]$, for any $r>0$.

$\bullet$ If $a$ is a symbol in $\Sigma\Gamma^{m}_{K,K^{\prime},p}[r,N]$ then
$\partial_{x}a\in\Sigma\Gamma^{m}_{K,K^{\prime},p}[r,N]$ and
$\partial_{\xi}a\in\Sigma\Gamma^{m-1}_{K,K^{\prime},p}[r,N]$. If in addition
$b$ is a symbol in $\Sigma\Gamma^{m^{\prime}}_{K,K^{\prime},p^{\prime}}[r,N]$
then $ab\in\Sigma\Gamma^{m+m^{\prime}}_{K,K^{\prime},p+p^{\prime}}[r,N]$.

######  Remark 2.2.

(Fourier representation of symbols) The translation invariance property (2.16)
means that the dependence with respect to the variable $x$ of a symbol
$a(\mathcal{U};x,\xi)$ enters only through the functions $\mathcal{U}(x)$,
implying that a symbol $a_{q}(U;x,\xi)$ in $\widetilde{\Gamma}_{q}^{m}$,
$m\in\mathbb{R}$, has the form (recall notation (2.9))

$a_{q}(U;x,\xi)=\\!\\!\\!\\!\sum_{\vec{\jmath}\in{(\mathbb{Z}\setminus\\{0\\}})^{q},\vec{\sigma}\in\\{\pm
1\\}^{{}^{q}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(a_{q}\right)_{\vec{\jmath}}^{\vec{\sigma}}(\xi)u_{\vec{\jmath}}^{\vec{\sigma}}e^{{\rm
i}\vec{\sigma}\cdot\vec{\jmath}x}$ (2.19)

where $(a_{q})_{\vec{\jmath}}^{\vec{\sigma}}(\xi)\in\mathbb{C}$ are Fourier
multipliers of order $m$ satisfying: there exists $\mu>0$, and for any
$\beta\in\mathbb{N}_{0}$, there is $C_{\beta}>0$ such that

$|\partial_{\xi}^{\beta}\left(a_{q}\right)_{\vec{\jmath}}^{\vec{\sigma}}(\xi)|\leq
C_{\beta}|\vec{\jmath}\,|^{\mu}\langle\xi\rangle^{m-\beta},\quad\forall(\vec{\jmath},\vec{\sigma})\in(\mathbb{Z}\setminus\\{0\\})^{q}\times\\{\pm\\}^{q}\,.$
(2.20)

A symbol $a_{q}(U;x,\xi)$ as in (2.19) is real if

$\overline{\left(a_{q}\right)_{\vec{\jmath}}^{\vec{\sigma}}(\xi)}=\left(a_{q}\right)_{\vec{\jmath}}^{-\vec{\sigma}}(\xi)\,.$
(2.21)

By (2.19) a symbol $a_{1}$ in $\widetilde{\Gamma}_{1}^{m}$ can be written as
$a_{1}(U;x,\xi)=\sum_{\begin{subarray}{c}j\in\mathbb{Z}\setminus\\{0\\},\sigma=\pm\end{subarray}}(a_{1})^{\sigma}_{j}(\xi)u_{j}^{\sigma}e^{{\rm
i}\sigma jx}$, and therefore, if $a_{1}$ is independent of $x$, it is actually
$a_{1}\equiv 0$.

We also define classes of functions in analogy with our classes of symbols.

######  Definition 2.3.

(Functions) Let $p,N\in\mathbb{N}_{0}$, $K,K^{\prime}\in\mathbb{N}_{0}$ with
$K^{\prime}\leq K$, $r>0$. We denote by $\widetilde{\mathcal{F}}_{p}$, resp.
$\mathcal{F}_{K,K^{\prime},p}[r]$, $\Sigma\mathcal{F}_{K,K^{\prime},p}[r,N]$,
the subspace of $\widetilde{\Gamma}^{0}_{p}$, resp.
$\Gamma^{0}_{K,K^{\prime},p}[r]$, resp.
$\Sigma\Gamma^{0}_{K,K^{\prime},p}[r,N]$, made of those symbols which are
independent of $\xi$. We write $\widetilde{\mathcal{F}}^{\mathbb{R}}_{p}$,
resp. $\mathcal{F}_{K,K^{\prime},p}^{\mathbb{R}}[r]$,
$\Sigma\mathcal{F}_{K,K^{\prime},p}^{\mathbb{R}}[r,N]$, to denote functions in
$\widetilde{\mathcal{F}}_{p}$, resp. $\mathcal{F}_{K,K^{\prime},p}[r]$,
$\Sigma\mathcal{F}_{K,K^{\prime},p}[r,N]$, which are real valued for any $U\in
B^{K^{\prime}}_{s_{0},\mathbb{R}}(I;r)$.

##### Paradifferential quantization.

Given $p\in\mathbb{N}_{0}$ we consider functions $\chi_{p}\in
C^{\infty}(\mathbb{R}^{p}\times\mathbb{R};\mathbb{R})$ and $\chi\in
C^{\infty}(\mathbb{R}\times\mathbb{R};\mathbb{R})$, even with respect to each
of their arguments, satisfying, for $0<\delta\ll 1$,

$\displaystyle{\rm{supp}}\,\chi_{p}\subset\\{(\xi^{\prime},\xi)\in\mathbb{R}^{p}\times\mathbb{R};|\xi^{\prime}|\leq\delta\langle\xi\rangle\\}\,,\qquad\chi_{p}(\xi^{\prime},\xi)\equiv
1\,\,\,\rm{for}\,\,\,|\xi^{\prime}|\leq\delta\langle\xi\rangle/2\,,$
$\displaystyle\rm{supp}\,\chi\subset\\{(\xi^{\prime},\xi)\in\mathbb{R}\times\mathbb{R};|\xi^{\prime}|\leq\delta\langle\xi\rangle\\}\,,\qquad\quad\chi(\xi^{\prime},\xi)\equiv
1\,\,\,\rm{for}\,\,\,|\xi^{\prime}|\leq\delta\langle\xi\rangle/2\,.$

For $p=0$ we set $\chi_{0}\equiv 1$. We assume moreover that

$|\partial_{\xi}^{\alpha}\partial_{\xi^{\prime}}^{\beta}\chi_{p}(\xi^{\prime},\xi)|\leq
C_{\alpha,\beta}\langle\xi\rangle^{-\alpha-|\beta|}\,,\
\forall\alpha\in\mathbb{N}_{0},\,\beta\in\mathbb{N}_{0}^{p}\,,\ \
|\partial_{\xi}^{\alpha}\partial_{\xi^{\prime}}^{\beta}\chi(\xi^{\prime},\xi)|\leq
C_{\alpha,\beta}\langle\xi\rangle^{-\alpha-\beta},\
\forall\alpha,\,\beta\in\mathbb{N}_{0}\,.$

If $a(x,\xi)$ is a smooth symbol we define its Weyl quantization as the
operator acting on a $2\pi$-periodic function $u(x)$ (written as in (2.5)) as

${\rm
Op}^{W}(a)u=\frac{1}{\sqrt{2\pi}}\sum_{k\in\mathbb{Z}}\Big{(}\sum_{j\in\mathbb{Z}}\widehat{a}\big{(}k-j,\frac{k+j}{2}\big{)}\widehat{u}(j)\Big{)}\frac{e^{{\rm
i}kx}}{\sqrt{2\pi}}$

where $\widehat{a}(k,\xi)$ is the $k^{th}-$Fourier coefficient of the
$2\pi-$periodic function $x\mapsto a(x,\xi)$.

######  Definition 2.4.

(Bony-Weyl quantization) If a is a symbol in $\widetilde{\Gamma}^{m}_{p}$,
respectively in $\Gamma^{m}_{K,K^{\prime},p}[r]$, we set

$\displaystyle
a_{\chi_{p}}(\mathcal{U};x,\xi):=\sum_{\vec{n}\in\mathbb{N}^{p}}\chi_{p}\left(\vec{n},\xi\right)a(\Pi_{\vec{n}}\mathcal{U};x,\xi)\,,$
(2.22) $\displaystyle
a_{\chi}(U;t,x,\xi):=\frac{1}{2\pi}\int_{\mathbb{R}}\chi(\xi^{\prime},\xi)\widehat{a}(U;t,\xi^{\prime},\xi)e^{{\rm
i}\xi^{\prime}x}{\rm d}\xi^{\prime}\,,$

where in the last equality $\widehat{a}$ stands for the Fourier transform with
respect to the $x$ variable, and we define the _Bony-Weyl_ quantization of $a$
as

${{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a(\mathcal{U};\cdot))={\rm
Op}^{W}(a_{\chi_{p}}(\mathcal{U};\cdot))\,,\qquad{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a(U;t,\cdot))={\rm
Op}^{W}(a_{\chi}(U;t,\cdot))\,.$ (2.23)

If $a$ is a symbol in $\Sigma\Gamma^{m}_{K,K^{\prime},p}[r,N]$, we define its
_Bony-Weyl_ quantization

${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a(U;t,\cdot))=\sum_{q=p}^{N}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a_{q}(U,\ldots,U;\cdot))+{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a_{>N}(U;t,\cdot))\,.$

We will use also the notation

${{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(a(U;t,x,\xi)\right)}:={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\begin{bmatrix}a(U;t,x,\xi)&0\\\
0&\overline{a^{\vee}(U;t,x,\xi)}\end{bmatrix}\right)}\,,\quad
a^{\vee}(x,\xi):=a(x,-\xi)\,.$ (2.24)

$\bullet$ The operator ${{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a)$
acts on homogeneous spaces of functions, see Proposition 3.8 of [16].

$\bullet$ If $a$ is a homogeneous symbol, the two definitions of quantization
in (2.23) differ by a smoothing operator according to Definition 2.7 below.
With the first regularization in (2.22) we guarantee the important property
that ${{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a)$ is a spectrally
localized map according to Definition 2.16 below.

$\bullet$ The action of ${{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a)$ on
homogeneous spaces only depends on the values of the symbol $a=a(U;t,x,\xi)$
(or $a(\mathcal{U};t,x,\xi)$) for $|\xi|\geq 1$. Therefore, we may identify
two symbols $a(U;t,x,\xi)$ and $b(U;t,x,\xi)$ if they agree for $|\xi|\geq
1/2$. In particular, whenever we encounter a symbol that is not smooth at
$\xi=0$, such as, for example, $a=g(x)|\xi|^{m}$ for
$m\in\mathbb{R}\setminus\\{0\\}$, or ${\rm sign}(\xi)$, we will consider its
smoothed out version $\chi(\xi)a$, where $\chi\in
C^{\infty}(\mathbb{R};\mathbb{R})$ is an even and positive cut-off function
satisfying

$\chi(\xi)=0\;\;{\rm if}\;\;|\xi|\leq\tfrac{1}{8}\,,\quad\chi(x)=1\;\;{\rm
if}\;\;|\xi|>\tfrac{1}{4},\quad\partial_{\xi}\chi(\xi)>0\quad\forall\xi\in\big{(}\tfrac{1}{8},\tfrac{1}{4}\big{)}\,.$

$\bullet$ Definition 2.4 is independent of the cut-off functions $\chi_{p}$,
$\chi$, up to smoothing operators that we define below (see Definition 2.7),
see the remark at pag. 50 of [16].

$\bullet$ If for some $(n_{0},\dots,n_{p+1})\in\mathbb{N}^{p+2}$,
$\Pi_{n_{0}}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a(\Pi_{\vec{n}}\mathcal{U};\cdot))\Pi_{n_{p+1}}U_{p+1}\not=0\,,$
then there exist signs $\epsilon_{j}\in\\{\pm\\}$, $j=0,\dots,p+1$, such that
$\sum_{0}^{p+1}\epsilon_{j}n_{j}=0$ and the indices satisfy (see Proposition
3.8 in [16])

$n_{0}\sim n_{p+1},\quad n_{j}\leq C\delta n_{p+1}\,,\quad n_{j}\leq C\delta
n_{0}\,,\quad j=1,\dots,p\,.$ (2.25)

$\bullet$ Given a paradifferential operator $A={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(a(x,\xi)\right)}$ it results

$\overline{A}={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\overline{a(x,-\xi)}\right)}\,,\quad
A^{\top}={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(a(x,-\xi)\right)}\,,\quad
A^{*}={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\overline{a(x,\xi)}\right)}\,,$
(2.26)

where $A^{\top}$ and $A^{*}$ denote respectively the transposed and adjoint
operator with respect to the complex, respectively real, scalar product of
$\dot{L}^{2}$ in (2.3). It results $A^{*}=\overline{A}^{\top}$.

$\bullet$ A paradifferential operator $A={{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a(x,\xi))$ is real (i.e.
$A=\overline{A}$) if

$\overline{a(x,\xi)}=a^{\vee}(x,\xi)\quad\text{ where}\quad
a^{\vee}(x,\xi):=a(x,-\xi)\,.$ (2.27)

$\bullet$ A matrix of paradifferential operators ${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U;t,x,\xi))$ is real-to-real, i.e.
(2.12) holds, if and only if the matrix of symbols $A(U;t,x,\xi)$ has the form

$A(U;x,\xi)=\left(\begin{matrix}{a}(U;t,x,\xi)&{b}(U;t,x,\xi)\\\
{\overline{b^{\vee}(U;t,x,\xi)}}&{\overline{a^{\vee}(U;t,x,\xi)}}\end{matrix}\right)\,\,.$
(2.28)

##### Classes of $m$-Operators and smoothing Operators.

Given integers $(n_{1},\ldots,n_{p+1})\in\mathbb{N}^{p+1}$, we denote by
$\max_{2}(n_{1},\ldots,n_{p+1})$ the second largest among
$n_{1},\ldots,n_{p+1}$. We shall often use that $\max_{2}$ is monotone in each
component, i.e. if $n^{\prime}_{j}\geq n_{j}$ for some $j$, then

${\rm max}_{2}(n_{1},\ldots,n_{j},\ldots,n_{p})\leq{\rm
max}_{2}(n_{1},\ldots,n^{\prime}_{j},\ldots,n_{p})\ .$ (2.29)

In addition $\max_{2}$ is non decreasing by adding elements, namely

${\rm max}_{2}(n_{1},\ldots,n_{p})\leq{\rm
max}_{2}(n_{1},\ldots,n_{p},n_{p+1})\,.$ (2.30)

We now define the $m$-operators. The class $\widetilde{\mathcal{M}}^{m}_{p}$
denotes multilinear operators that lose $m$ derivatives and are
$p$-homogeneous in $U$, while the class $\mathcal{M}_{K,K^{\prime},p}^{m}$
contains non-homogeneous operators which lose $m$ derivatives, vanish at
degree at least $p$ in $U$, satisfy tame estimates and are
$(K-K^{\prime})$-times differentiable in $t$. The constant $\mu$ in (2.31)
takes into account possible loss of derivatives in the “low” frequencies.

######  Definition 2.5.

(Classes of $m$-operators) Let $m\in\mathbb{R}$, $p,N\in\mathbb{N}_{0}$
$K,K^{\prime}\in\mathbb{N}_{0}$ with $K^{\prime}\leq K$, and $r>0$.

(i) $p$-homogeneous $m$-operators. We denote by
$\widetilde{\mathcal{M}}^{m}_{p}$ the space of $(p+1)$-linear operators $M$
from
$(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{p}\times\dot{H}^{\infty}(\mathbb{T};\mathbb{C})$
to $\dot{H}^{\infty}(\mathbb{T};\mathbb{C})$ which are symmetric in
$(U_{1},\ldots,U_{p})$, of the form

$(U_{1},\ldots,U_{p+1})\to M(U_{1},\ldots,U_{p})U_{p+1}$

that satisfy the following. There are $\mu\geq 0$, $C>0$ such that

$\|\Pi_{n_{0}}M(\Pi_{\vec{n}}\mathcal{U})\Pi_{n_{p+1}}U_{p+1}\|_{L^{2}}\leq
C{\rm\max}_{2}(n_{1},\ldots,n_{p+1})^{\mu}\max(n_{1},\ldots,n_{p+1})^{m}\prod_{j=1}^{p+1}\|\Pi_{n_{j}}U_{j}\|_{L^{2}}$
(2.31)

for any
$\mathcal{U}=(U_{1},\ldots,U_{p})\in(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{p}$,
any $U_{p+1}\in\dot{H}^{\infty}(\mathbb{T};\mathbb{C})$,
$\vec{n}=(n_{1},\ldots,n_{p})$ in $\mathbb{N}^{p}$, any
$n_{0},n_{p+1}\in\mathbb{N}$. Moreover, if

$\Pi_{n_{0}}M(\Pi_{n_{1}}U_{1},\ldots,\Pi_{n_{p}}U_{p})\Pi_{n_{p+1}}U_{p+1}\neq
0\,,$ (2.32)

then there is a choice of signs $\sigma_{0},\ldots,\sigma_{p+1}\in\\{\pm 1\\}$
such that $\sum_{j=0}^{p+1}\sigma_{j}n_{j}=0$. In addition we require the
translation invariance property

$M(\tau_{\varsigma}{\mathcal{U}})[\tau_{\varsigma}U_{p+1}]=\tau_{\varsigma}\big{(}M({\mathcal{U}})U_{p+1}\big{)}\,,\quad\forall\varsigma\in\mathbb{R}\,.$
(2.33)

We denote $\widetilde{\mathcal{M}}_{p}:=\cup_{m\geq
0}\widetilde{\mathcal{M}}_{p}^{m}$ and
$\Sigma_{p}^{N}\widetilde{\mathcal{M}}^{m}_{q}$ the class of pluri-homogeneous
operators $\sum_{q=p}^{N}M_{q}$ with $M_{q}$ in
$\widetilde{\mathcal{M}}^{m}_{q}$. For $p\geq N+1$ we mean that the sum is
empty. We set
$\Sigma_{p}\widetilde{\mathcal{M}}^{m}_{q}:=\bigcup_{N\in\mathbb{N}}\Sigma_{p}^{N}\widetilde{\mathcal{M}}^{m}_{q}$.

(ii) Non-homogeneous $m$-operators. We denote by
$\mathcal{M}^{m}_{K,K^{\prime},p}[r]$ the space of operators $(U,t,V)\mapsto
M(U;t)V$ defined on $B^{K^{\prime}}_{s_{0}}(I;r)\times I\times
C^{0}_{*}(I,\dot{H}^{s_{0}}(\mathbb{T},\mathbb{C}))$ for some $s_{0}>0$, which
are linear in the variable $V$ and such that the following holds true. For any
$s\geq s_{0}$ there are $C>0$ and $r(s)\in]0,r[$ such that for any $U\in
B^{K}_{s_{0}}(I;r(s))\cap
C^{K}_{*}(I,\dot{H}^{s}(\mathbb{T},\mathbb{C}^{2}))$, any $V\in
C^{K-K^{\prime}}_{*}(I,\dot{H}^{s}(\mathbb{T},\mathbb{C}))$, any $0\leq k\leq
K-K^{\prime}$, $t\in I$, we have that

$\|{\partial_{t}^{k}\left(M(U;t)V\right)}\|_{\dot{H}^{s-\frac{3}{2}k-m}}\leq
C\\!\\!\\!\\!\sum_{k^{\prime}+k^{\prime\prime}=k}\\!\\!\\!\\!\|{V}\|_{k^{\prime\prime},s}\|{U}\|_{k^{\prime}+K^{\prime},s_{0}}^{p}+\|{V}\|_{k^{\prime\prime},s_{0}}\|U\|_{k^{\prime}+K^{\prime},s_{0}}^{p-1}\|{U}\|_{k^{\prime}+K^{\prime},s}\,.$
(2.34)

In case $p=0$ we require the estimate
$\|{\partial_{t}^{k}\left(M(U;t)V\right)}\|_{\dot{H}^{s-\frac{3}{2}k-m}}\leq
C\|{V}\|_{k,s}$.

(iii) $m$-Operators. We denote by
$\Sigma\mathcal{M}^{m}_{K,K^{\prime},p}[r,N]$, the space of operators
$(U,t,V)\to M(U;t)V$ such that there are homogeneous $m$-operators $M_{q}$ in
$\widetilde{\mathcal{M}}^{m}_{q}$, $q=p,\ldots,N$ and a non–homogeneous
$m$-operator $M_{>N}$ in $\mathcal{M}^{m}_{K,K^{\prime},N+1}[r]$ such that

$M(U;t)V=\sum_{q=p}^{N}M_{q}(U,\ldots,U)V+M_{>N}(U;t)V\,.$ (2.35)

We denote

$\widetilde{\mathcal{M}}_{p}:=\bigcup_{m\geq
0}\widetilde{\mathcal{M}}_{p}^{m}\,,\ \
\mathcal{M}_{K,K^{\prime},p}[r]:=\bigcup_{m\geq
0}\mathcal{M}_{K,K^{\prime},p}^{m}[r]\,,\ \
\Sigma\mathcal{M}_{K,K^{\prime},p}[r,N]:=\bigcup_{m\geq
0}\Sigma\mathcal{M}^{m}_{K,K^{\prime},p}[r,N]\,,$

and
$\Sigma\mathcal{M}_{K,K^{\prime},p}^{m}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$
the space of $2\times 2$ matrices whose entries are operators in
$\Sigma\mathcal{M}^{m}_{K,K^{\prime},p}[r,N]$.

$\bullet$ If $M(U,\dots,U)$ is a $p$–homogeneous $m$-operator in
${\widetilde{\mathcal{M}}}_{p}^{m}$ then the differential of the non–linear
map $M(U,\dots,U)U$, ${\rm
d}_{U}\big{(}M(U,\dots,U)U\big{)}V=pM(V,U,\dots,U)U+M(U,\dots,U)V$ is a
$p$–homogeneous $m$-operator in ${\widetilde{\mathcal{M}}}_{p}^{m}$. This
follows because the right hand side of (2.31) is symmetric in
$(n_{1},\ldots,n_{p+1})$.

$\bullet$ If $m_{1}\leq m_{2}$ then
$\Sigma\mathcal{M}^{m_{1}}_{K,K^{\prime},p}[r,N]\subseteq\Sigma\mathcal{M}^{m_{2}}_{K,K^{\prime},p}[r,N]$.

$\bullet$ Notation for $p$-homogeneous $m$-operators: if
$M(U_{1},\ldots,U_{p})$ is a $p$-homogeneous $m$-operator, we shall often
denote by $M(U):=M(U,\ldots,U)$ the corresponding polynomial and say that
$M(U)$ is in ${\widetilde{\mathcal{M}}}_{p}^{m}$. Viceversa, a polynomial can
be represented by a $(p+1)$-linear form $M(U_{1},\ldots,U_{p})U_{p+1}$ not
necessarily symmetric in the internal variables. If it fulfills the symmetric
estimate (2.31), the polynomial is generated by the $m$-operator in
${\widetilde{\mathcal{M}}}_{p}^{m}$ obtained by symmetrization of the internal
variables. We will do this consistently without mentioning it further.

$\bullet$ Notation for projection on homogeneous components: given an operator
$M(U;t)$ in $\Sigma\mathcal{M}_{K,K^{\prime},p}^{m}[r,N]$ of the form (2.35)
we denote by

$\mathcal{P}_{\leq
N}[M(U;t)]:=\sum_{q=p}^{N}M_{q}(U)\,,\quad\text{resp.}\quad\mathcal{P}_{q}[M(U;t)]:=M_{q}(U)\,,$
(2.36)

the projections on the pluri-homogeneous, resp. homogeneous, operators in
$\Sigma_{p}^{N}\widetilde{\mathcal{M}}^{m}_{q}$ , resp. in
$\widetilde{\mathcal{M}}_{q}^{m}$. Given an integer $p\leq p^{\prime}\leq N$
we also denote

$\mathcal{P}_{\geq
p^{\prime}}[M(U;t)]:=\sum_{q=p^{\prime}}^{N}M_{q}(U),\quad\mathcal{P}_{\leq
p^{\prime}}[M(U;t)]:=\sum_{q=p}^{p^{\prime}}M_{q}(U)\,.$

The same notation will be also used to denote pluri-homogeneous/homogeneous
components of symbols.

######  Remark 2.6.

Definition 2.5 of homogeneous $m$-operators is different than the one in
Definition 3.9 in [16], due to the different bound (2.31). However for $m\geq
0$ the class of homogeneous $m$-operators contains the class of homogeneous
maps of order $m$ in Definition 3.9 of [16], and in view of (2.31) is
contained in the class of maps of order $m+\mu$ of [16]. On the other hand the
class of non-homogeneous $m$-operators coincides with the class of non-
homogeneous maps in Definition 3.9 of [16].

If $m\leq 0$ the operators in $\Sigma\mathcal{M}^{m}_{K,K^{\prime},p}[r,N]$
are referred to as smoothing operators.

######  Definition 2.7.

(Smoothing operators) Let $\varrho\geq 0$. A $(-\varrho)$-operator $R(U)$
belonging to $\Sigma\mathcal{M}^{-\varrho}_{K,K^{\prime},p}[r,N]$ is called a
smoothing operator. Along the paper will use also the notation

$\displaystyle\widetilde{\mathcal{R}}^{-\varrho}_{p}:=\widetilde{\mathcal{M}}^{-\varrho}_{p}\,,\quad\Sigma_{p}^{N}\widetilde{\mathcal{R}}^{-\varrho}_{q}:=\Sigma_{p}^{N}\widetilde{\mathcal{M}}^{-\varrho}_{q}\,,$
(2.37)
$\displaystyle\mathcal{R}^{-\varrho}_{K,K^{\prime},p}[r]:=\mathcal{M}^{-\varrho}_{K,K^{\prime},p}[r]\,,\quad\Sigma\mathcal{R}^{-\varrho}_{K,K^{\prime},p}[r,N]:=\Sigma\mathcal{M}^{-\varrho}_{K,K^{\prime},p}[r,N]\,.$

$\bullet$ Given $\varrho\geq 0$, an operator $R(U)$ belongs to
$\widetilde{\mathcal{R}}^{-\varrho}_{p}$ if and only if there is
$\mu=\mu(\varrho)\geq 0$ and $C>0$ such that

$\|\Pi_{n_{0}}R(\Pi_{\vec{n}}\mathcal{U})\Pi_{n_{p+1}}U_{p+1}\|_{L^{2}}\leq
C\frac{{\rm\max}_{2}(n_{1},\ldots,n_{p+1})^{\mu}}{\max(n_{1},\ldots,n_{p+1})^{\varrho}}\prod_{j=1}^{p+1}\|\Pi_{n_{j}}U_{j}\|_{L^{2}}\
.$ (2.38)

We remark that Definition 2.7 of smoothing operators coincides with Definition
$3.7$ in [16].

$\bullet$ In view of (2.31) and (2.38) a homogeneous $m$-operator in
$\widetilde{\mathcal{M}}^{m}_{p}$ with the property that, on its support,
$\max_{2}(n_{1},\ldots,n_{p+1})\sim\max(n_{1},\ldots,n_{p+1})$ is actually a
smoothing operator in $\widetilde{\mathcal{R}}^{-\varrho}_{p}$ for any
$\varrho\geq 0$.

$\bullet$ The Definition 2.7 of smoothing operators is modeled to gather
remainders which satisfy either the property
$\max_{2}(n_{1},\ldots,n_{p+1})\sim\max(n_{1},\ldots,n_{p+1})$ or arise as
remainders of compositions of paradifferential operators, see Proposition 2.14
below, and thus have a fixed order $\varrho$ of regularization.

######  Lemma 2.8.

If $M(U)$ is a $p$–homogeneous $m$-operator in
$\widetilde{\mathcal{M}}_{p}^{m}$ then for any $K\in\mathbb{N}_{0}$ and $0\leq
k\leq K$ there exists $s_{0}>0$ such that for any $s\geq s_{0}$, for any $U\in
C^{K}_{*}(I,\dot{H}^{s}(\mathbb{T},\mathbb{C}^{2}))$, any $v\in
C^{K}_{*}(I,\dot{H}^{s}(\mathbb{T},\mathbb{C}))$, one has

$\displaystyle\|\partial_{t}^{k}(M(U_{1},\dots,U_{p})v)\|_{\dot{H}^{s-m-\frac{3}{2}k}}\lesssim_{K}\sum_{k_{1}+\dots+k_{p+1}=k}\Big{(}$
$\displaystyle{\|v\|}_{k_{p+1},s}\prod_{a=1}^{p}{\|U_{a}\|}_{k_{a},s_{0}}$
(2.39)
$\displaystyle+{\|v\|}_{k_{p+1},s_{0}}\sum_{\overline{a}=1}^{p}{\|U_{\overline{a}}\|}_{k_{\overline{a}},s}\prod_{a=1\atop
a\neq\overline{a}}^{p}{\|U_{a}\|}_{k_{a},s_{0}}\Big{)}\,.$

In particular $M(U)$ is a non-homogeneous $m$-operator in
${\mathcal{M}}^{m}_{K,0,p}[r]$ for any $r>0$ and $K\in\mathbb{N}_{0}$.

###### Proof.

For any $0\leq k\leq K$ we estimate

$\|\partial_{t}^{k}(M(U_{1},\dots,U_{p})v)\|_{\dot{H}^{s-m-\frac{3}{2}k}}\lesssim_{K}\sum_{k_{1}+\dots+k_{p+1}=k}\|M(\partial_{t}^{k_{1}}U_{1},\dots,\partial_{t}^{k_{p}}U_{p})\partial_{t}^{k_{p+1}}v\|_{\dot{H}^{s-m-\frac{3}{2}k}}.$

We now estimate each term in the above sum. We denote
$\vec{n}_{p+1}:=(n_{1},\ldots,n_{p+1})\in\mathbb{N}^{p+1}$,
$\vec{\sigma}_{p+1}:=(\sigma_{1},\ldots,\sigma_{p+1})\in\\{\pm\\}^{p+1}$ and
$\mathcal{I}(n_{0},\vec{\sigma}_{p+1}):=\left\\{\vec{n}_{p+1}\in\mathbb{N}^{p+1}\colon
n_{0}=\sigma_{1}n_{1}+\ldots+\sigma_{p+1}n_{p+1}\right\\}$. We get

$\displaystyle\|M(\partial_{t}^{k_{1}}U_{1},\dots,\partial_{t}^{k_{p}}U_{p})\partial_{t}^{k_{p+1}}v\|_{\dot{H}^{s-m-\frac{3}{2}k}}=\big{\|}n_{0}^{s-m-\frac{3}{2}k}\|\Pi_{n_{0}}(M(\partial_{t}^{k_{1}}U_{1},\dots,\partial_{t}^{k_{p}}U_{p})\partial_{t}^{k_{p+1}}v)\|_{L^{2}}\big{\|}_{\ell^{2}_{n_{0}}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{omoresti2}}}{{\leq}}\Big{\|}n_{0}^{s-m-\frac{3}{2}k}\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\vec{\sigma}_{p+1}\in\\{\pm\\}^{p+1}\atop\vec{n}_{p+1}\in\mathcal{I}(n_{0},\vec{\sigma}_{p+1})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\big{\|}\Pi_{n_{0}}(M(\Pi_{n_{1}}\partial_{t}^{k_{1}}U_{1},\dots,\Pi_{n_{p}}\partial_{t}^{k_{p}}U_{p})\Pi_{n_{p+1}}\partial_{t}^{k_{p+1}}v)\|_{L^{2}}\Big{\|}_{\ell^{2}_{n_{0}}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{omomap}}}{{\lesssim}}\sum_{\vec{\sigma}_{p+1}}\Big{\|}\sum_{\vec{n}_{p+1}\in\mathcal{I}(n_{0},\vec{\sigma}_{p+1})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\rm\max}_{2}(n_{1},\ldots,n_{p+1})^{\mu}{\rm\max}(n_{1},\ldots,n_{p+1})^{s-\frac{3}{2}k}\prod_{a=1}^{p}\|\Pi_{n_{a}}\partial_{t}^{k_{a}}U_{a}\|_{L^{2}}\|\Pi_{n_{p+1}}\partial_{t}^{k_{p+1}}v\|_{L^{2}}\Big{\|}_{\ell^{2}_{n_{0}}}$

where in the last inequality we also used that
$n_{0}\lesssim\max\\{n_{1},\dots,n_{p+1}\\}$ and $s-m-\frac{3}{2}k\geq 0$ to
bound $n_{0}^{s-m-\frac{3}{2}k}$. For any choice of
$\vec{\sigma}_{p+1}\in\\{\pm\\}^{p+1}$, we split the internal sum in $p+1$
components

$\sum_{\overline{a}=1}^{p+1}\Sigma_{\vec{\sigma}_{p+1}}^{(\overline{a})},\qquad\Sigma_{\vec{\sigma}_{p+1}}^{(\overline{a})}:=\sum_{\begin{subarray}{c}\vec{n}_{p+1}\in\mathcal{I}(n_{0},\vec{\sigma}_{p+1})\\\
\max(n_{1},\ldots,n_{p+1})=n_{\overline{a}}\end{subarray}}\\!\\!\\!\\!.$

We first deal with the term $\Sigma_{\vec{\sigma}_{p+1}}^{(p+1)}$. In this
case we bound

$\displaystyle\|\Pi_{n_{p+1}}\partial_{t}^{k_{p+1}}v\|_{L^{2}}\leq\frac{\widetilde{c}_{n_{p+1}}}{n_{p+1}^{s-\frac{3}{2}k_{p+1}}}\,\|\partial_{t}^{k_{p+1}}v\|_{\dot{H}^{s-\frac{3}{2}k_{p+1}}}\leq\frac{\widetilde{c}_{n_{p+1}}}{n_{p+1}^{s-\frac{3}{2}k_{p+1}}}\,\|v\|_{k_{p+1},s}$
$\displaystyle\|\Pi_{n_{a}}\partial_{t}^{k_{a}}U_{a}\|_{L^{2}}\leq\frac{c_{n_{a}}^{(a)}}{n_{a}^{\mu+1-{k_{a}}}}\,\|\partial_{t}^{k_{a}}U_{a}\|_{\dot{H}^{\mu+1-k_{a}}}\leq\frac{c_{n_{a}}^{(a)}}{n_{a}^{\mu+1-{k_{a}}}}\,\|U_{a}\|_{k_{a},\mu+1},\quad
a=1,\ldots,p$

for some sequences $(\widetilde{c}_{n})_{n\in\mathbb{N}}$,
$(c_{n}^{(a)})_{n\in\mathbb{N}}$ in $\ell^{2}(\mathbb{N})$. With these bounds,
and using
$\max(n_{1},\ldots,n_{p+1})^{s-\frac{3}{2}k}=n_{p+1}^{s-\frac{3}{2}k}\leq
n_{p+1}^{s-\frac{3}{2}k_{p+1}}\,n_{1}^{-\frac{3}{2}k_{1}}\ldots
n_{p}^{-\frac{3}{2}k_{p}}$, we get

$\displaystyle\Big{\|}\Sigma_{\vec{\sigma}_{p+1}}^{(p+1)}\Big{\|}_{\ell^{2}_{n_{0}}}\lesssim\Big{\|}\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}(\vec{n}_{p+1})\in\mathbb{N}^{p+1}\\\
n_{0}=\sigma_{1}n_{1}+\dots+\sigma_{p+1}n_{p+1}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\
\widetilde{c}_{n_{p+1}}\,\frac{c_{n_{1}}^{(1)}}{n_{1}}\times\dots\times\frac{c_{n_{p}}^{(p)}}{n_{p}}\Big{\|}_{\ell^{2}_{n_{0}}}\,\|v\|_{k_{p+1},s}\prod_{a=1}^{p}\|U_{a}\|_{k_{a},\mu+1}\
.$

Applying Young inequality for convolution of sequences and using that
$\big{(}c_{n}^{(a)}n^{-1}\big{)}_{n\in\mathbb{N}}\in\ell^{1}(\mathbb{N})$, we
finally arrive at

$\Big{\|}\Sigma_{\vec{\sigma}_{p+1}}^{(p+1)}\Big{\|}_{\ell^{2}_{n_{0}}}\lesssim{\|v\|}_{k_{p+1},s}\,\prod_{a=1}^{p}{\|U_{a}\|}_{k_{a},s_{0}}\
,\ \ \ \ \ k_{1}+\ldots+k_{p+1}=k\,,$

which is the first term of inequality (2.39) with
$s_{0}=\max(m+\frac{3}{2}K,\mu+1)$. Proceeding similarly we obtain, for any
$\overline{a}=1,\ldots,p$,

$\|\Sigma_{\vec{\sigma}_{p+1}}^{(\overline{a})}\|_{\ell^{2}_{n_{0}}}\lesssim\|v\|_{k_{p+1},\mu+1}\,{\|U_{\overline{a}}\|}_{k_{\overline{a}},s}\prod_{a\neq\overline{a}}{\|U_{a}\|}_{k_{a},\mu+1}\,$

which are terms in the sum in the second line of (2.39). If $U_{a}=U$ for any
$a$, we deduce by (2.39) and the estimate
${\|U\|}_{k_{a},\sigma}\leq{\|U\|}_{k-k_{p+1},\sigma}$ that $M(U)$ fulfills
(2.34) with $K^{\prime}=0$, $k^{\prime\prime}:=k_{p+1}$,
$k^{\prime}=k-k_{p+1}$. Hence $M(U)$ belongs to $\mathcal{M}^{m}_{K,0,p}[r]$
for any $r>0$ and $K\in\mathbb{N}_{0}$. ∎

$\bullet$ A pluri-homogeneous nonlinear map $Z+R_{\leq N}(Z)Z$ where $R_{\leq
N}(Z)$ is in
$\Sigma_{1}^{N}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
satisfies the following bound: for any $K\in\mathbb{N}_{0}$ there is $s_{0}>0$
such that for any $s\geq s_{0}$, $0<r<r_{0}(s,K)$ small enough and any $Z\in
B_{s_{0}}^{K}(I;r)\cap
C^{K}_{*\mathbb{R}}(I;\dot{H}^{s}(\mathbb{T},\mathbb{C}^{2}))$ one has

$2^{-1}\|Z\|_{k,s}\leq\|Z+R_{\leq N}(Z)Z\|_{k,s}\leq
2\|Z\|_{k,s}\,,\quad\forall 0\leq k\leq K\,.$ (2.40)

Fourier representation of $m$-operators. We may also represent a matrix of
operators

$M(U)=\begin{pmatrix}M^{+}_{+}(U)&M^{-}_{+}(U)\\\
M^{+}_{-}(U)&M^{-}_{-}(U)\end{pmatrix}\in\widetilde{\mathcal{M}}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$
(2.41)

through their Fourier matrix elements, see (2.6), writing

$\displaystyle M(U)V=\begin{pmatrix}(M(U)V)^{+}\\\
(M(U)V)^{-}\end{pmatrix},\quad(M(U)V)^{\sigma}=\sum_{\begin{subarray}{c}(\vec{\jmath}_{p},j,k)\in(\mathbb{Z}\setminus\\{0\\})^{p+2}\\\
(\vec{\sigma}_{p},\sigma^{\prime})\in\\{\pm\\}^{p+1}\\\ \sigma
k=\vec{\sigma}_{p}\cdot\vec{\jmath}_{p}+\sigma^{\prime}j\end{subarray}}M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}u_{\vec{\jmath}_{p}}^{\vec{\sigma}_{p}}v^{\sigma^{\prime}}_{j}\frac{e^{{\rm
i}\sigma kx}}{\sqrt{2\pi}}\,,$ (2.42)

where 222 Given $u=\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix},\,v=\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix}\in\mathbb{C}^{2}$ we set $u\cdot v:=u_{1}v_{1}+u_{2}v_{2}$.

$\displaystyle
M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}:=$
$\displaystyle\int_{\mathbb{T}}M\Big{(}\mathtt{q}^{\sigma_{1}}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}},\dots,\mathtt{q}^{\sigma_{p}}\frac{e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\frac{\mathtt{q}^{\sigma^{\prime}}e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\Big{]}\cdot\mathtt{q}^{\sigma}\frac{e^{-{\rm
i}\sigma kx}}{\sqrt{2\pi}}\,{\rm d}x$ (2.43) $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}M^{\sigma^{\prime}}_{\sigma}\Big{(}\mathtt{q}^{\sigma_{1}}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}},\dots,\mathtt{q}^{\sigma_{p}}\frac{e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\frac{e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\Big{]}\,\frac{e^{-{\rm i}\sigma
kx}}{\sqrt{2\pi}}\,{\rm d}x\,\in\mathbb{C}\ ,$

and ${\mathtt{q}}^{\pm}$ are defined in (2.7). In (2.42) we have exploited the
translation invariance property (2.33) which implies that if
$M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}\not=0$
then

$\sigma k=\vec{\sigma}_{p}\cdot\vec{\jmath}_{p}+\sigma^{\prime}j\,.$ (2.44)

Note also that since $M$ is symmetric in the internal entries, the
coefficients
$M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$ in (2.43)
satisfy the following symmetric property: for any permutation $\pi$ of
$\\{1,\ldots,p\\}$, it results

$M_{j_{\pi(1)},\ldots,j_{\pi(p)},j,k}^{\sigma_{\pi(1)},\ldots,\sigma_{\pi(p)},\sigma^{\prime},\sigma}=M_{j_{1},\ldots,j_{p},j,k}^{\sigma_{1},\ldots,\sigma_{p},\sigma^{\prime},\sigma}\,.$
(2.45)

The operator $M(U)$ is real-to-real, according to definition (2.12), if and
only if its coefficients fulfill

$\overline{M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}}=M_{\vec{\jmath}_{p},j,k}^{-\vec{\sigma}_{p},-\sigma^{\prime},-\sigma}\
.$ (2.46)

The matrix entries of the transpose operator $M(U)^{\top}$ with respect to the
non-degenerate bilinear form (2.4) are

$\displaystyle(M^{\top})_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}=$
$\displaystyle\int_{\mathbb{T}}M^{\top}\Big{(}\mathtt{q}^{\sigma_{1}}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}},\dots,\mathtt{q}^{\sigma_{p}}\frac{e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\frac{\mathtt{q}^{\sigma^{\prime}}e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\Big{]}\cdot\frac{\mathtt{q}^{\sigma}e^{-{\rm
i}\sigma kx}}{\sqrt{2\pi}}\,{\rm d}x$ (2.47)
$\displaystyle=\int_{\mathbb{T}}M\Big{(}\mathtt{q}^{\sigma_{1}}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}},\dots,\mathtt{q}^{\sigma_{p}}\frac{e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\frac{\mathtt{q}^{\sigma}e^{-{\rm
i}\sigma
kx}}{\sqrt{2\pi}}\Big{]}\cdot\frac{\mathtt{q}^{\sigma^{\prime}}e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\,{\rm
d}x=M_{\vec{\jmath}_{p},-k,-j}^{\vec{\sigma}_{p},\sigma,\sigma^{\prime}}\,.$

One can directly verify that
$(M^{\top})_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$
fulfill (2.44), (2.45). If $M(U)$ is real-to-real (i.e. (2.46) holds) then
$M(U)^{\top}$ is real-to-real as well.

######  Lemma 2.9.

(Characterization of $m$-operators in Fourier basis) Let $m\in\mathbb{R}$. A
real-to-real linear operator $M(U)$ as in (2.42)-(2.43) is a matrix of
$p$-homogeneous $m$-operators in
$\widetilde{\mathcal{M}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$ if and
only if its coefficients
$M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$ satisfy
(2.44), (2.45), (2.46) and there exist $\mu>0$ and $C>0$ such that for any
$(\vec{\jmath}_{p},j)\in(\mathbb{Z}\setminus\\{0\\})^{p+1}$,
$(\vec{\sigma}_{p},\sigma)\in\\{\pm\\}^{p+1}$,

$|M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|\leq
C{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu}\max\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{m}\,.$
(2.48)

###### Proof.

Let $M(U)$ be a matrix of $m$-operators in
$\widetilde{\mathcal{M}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$. Then by
(2.43), applying Cauchy-Schwartz inequality and recalling (2.8) we get

$\displaystyle|M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|$
$\displaystyle\leq\Big{\|}{M^{\sigma^{\prime}}_{\sigma}}\Big{(}\mathtt{q}^{\sigma_{1}}\Pi_{|j_{1}|}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}},\dots,\mathtt{q}^{\sigma_{p}}\Pi_{|j_{p}|}\frac{e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\Pi_{|j|}\frac{e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\Big{]}\Big{\|}_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{omoresti2}}}{{\leq}}$
$\displaystyle\sum_{\begin{subarray}{c}\epsilon_{1},\dots,\epsilon_{p},\epsilon\in\\{\pm\\}\\\
n_{0}=\epsilon_{1}|j_{1}|+\dots+\epsilon_{p}|j_{p}|+\epsilon|j|\end{subarray}}\Big{\|}\Pi_{n_{0}}{M^{\sigma^{\prime}}_{\sigma}}\Big{(}\mathtt{q}^{\sigma_{1}}\Pi_{|j_{1}|}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}},\dots,\mathtt{q}^{\sigma_{p}}\Pi_{|j_{p}|}\frac{e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\Pi_{|j|}\frac{e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\Big{]}\Big{\|}_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{omomap}}}{{\leq}}$ $\displaystyle
C2^{p+1}{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu}\max\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{m}$

proving (2.48). Viceversa suppose that $M(U)$ is an operator as in
(2.41)-(2.43) with coefficients satisfying (2.48). Then, for any
$\sigma,\sigma^{\prime}\in\\{\pm\\}$,

$\displaystyle\|\Pi_{n_{0}}{M^{\sigma^{\prime}}_{\sigma}}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{p}}U_{p})\Pi_{n_{p+1}}v^{\sigma^{\prime}}\|_{L^{2}}=\Big{\|}\\!\\!\sum_{\begin{subarray}{c}j_{1}=\pm
n_{1},\dots,j_{p}=\pm n_{p}\\\ j=\pm n_{p+1},\,k=\pm
n_{0}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}(u_{1})_{j_{1}}^{\sigma_{1}}\cdots(u_{p})_{j_{p}}^{\sigma_{p}}v^{\sigma^{\prime}}_{j}\frac{e^{{\rm
i}\sigma kx}}{\sqrt{2\pi}}\Big{\|}_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{smoocara}}}{{\leq}}C\\!\\!\\!\\!\sum_{\begin{subarray}{c}j_{1}=\pm
n_{1},\dots,j_{p}=\pm n_{p}\\\ j=\pm n_{p+1},\,k=\pm
n_{0}\end{subarray}}\\!\\!\\!\\!{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu}\max\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{m}|(u_{1})_{j_{1}}^{\sigma_{1}}|\cdots|(u_{p})_{j_{p}}^{\sigma_{p}}||v^{\sigma^{\prime}}_{j}|$
$\displaystyle\leq C2^{p+2}{\rm
max}_{2}\\{n_{1},\dots,n_{p},n_{p+1}\\}^{\mu}\max\\{n_{1},\dots,n_{p},n_{p+1}\\}^{m}\prod_{\ell=1}^{p}\|\Pi_{n_{\ell}}U_{\ell}\|_{L^{2}}\|\Pi_{n_{p+1}}v\|_{L^{2}}$

proving (2.31). ∎

The transpose of a matrix of $m$-operators is a $m^{\prime}$-operator.

######  Lemma 2.10.

(Transpose of $m$-Operators) Let $p\in\mathbb{N}_{0}$, $m\in\mathbb{R}$. If
$M(U)$ is a matrix of $p$–homogeneous $m$-operators in
$\widetilde{\mathcal{M}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$ then
$M(U)^{\top}$ (where the transpose is computed with respect to the non-
degenerate bilinear form (2.4)) is a matrix of $p$-homogeneous operators in
${\widetilde{\mathcal{M}}}_{p}^{m^{\prime}}\otimes\mathcal{M}_{2}(\mathbb{C})$
for some $m^{\prime}\geq\max(m,0)$.

If in addition there exists $C>1$ such that

$M_{\vec{\jmath}_{p},k,j}^{\vec{\sigma}_{p},\sigma,\sigma^{\prime}}\neq
0\quad\Rightarrow\quad C^{-1}|k|\leq|j|\leq C|k|$ (2.49)

then
$M(U)^{\top}\in{\widetilde{\mathcal{M}}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$.

###### Proof.

By (2.47), (2.48) (applied to
$M_{\vec{\jmath}_{p},-k,-j}^{\vec{\sigma}_{p},\sigma,\sigma^{\prime}}$) and
since, by (2.44), $|k|\leq(p+1)\max\\{|j|,|j_{1}|,\dots,|j_{p}|\\}$, we deduce
that

$\displaystyle|(M^{\top})_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|$
$\displaystyle\lesssim{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{\mu}\max\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{m}$
(2.50)
$\displaystyle\lesssim\max\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\max(m,0)+\mu}$

proving (2.48) for $M(U)^{\top}$ with $m\leadsto m^{\prime}:=\max(m,0)+\mu$.

If in addition (2.49) holds true then
$\max\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{m}\sim\max\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{m}$
for any $m\in\mathbb{R}$ and similarly
$\max_{2}\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{\mu}\sim\max_{2}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu}$
for any $\mu\geq 0$. We deduce by (2.50) that $M(U)^{\top}$ is in
$\widetilde{\mathcal{M}}^{m}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$. ∎

######  Remark 2.11.

(Transpose of Smoothing operators) If $R(U)$ is a matrix of smoothing
operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ and
the spectral condition (2.49) holds true, then $R(U)^{\top}$ is a smoothing
operator in the same class. Without the spectral condition (2.49) this might
fail: for example consider $R(U)$ such that its transpose is

$R(U_{1},\ldots,U_{p})^{\top}V:={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(V,U_{1},\ldots,U_{p-1})\right)}U_{p}\
,\ \
A(\cdot)\in\widetilde{\Gamma}^{\mu}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})\ .$
(2.51)

As a consequence of Lemma 2.21 below, we have that $R(U)=[R(U)^{\top}]^{\top}$
is in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ for
any $\varrho\geq 0$, but $R(U)^{\top}$ in (2.51) is a $\mu$-operator.

We conclude this subsection with the paralinearization of the product (see
e.g. Lemma 7.2 in [16]).

######  Lemma 2.12.

(Bony paraproduct decomposition) Let $u_{1},u_{2}$ be functions in
$H^{\sigma}(\mathbb{T};\mathbb{C})$ with $\sigma>\frac{1}{2}$. Then

$u_{1}u_{2}={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(u_{1}\right)}u_{2}+{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(u_{2}\right)}u_{1}+R_{1}(u_{1})u_{2}+R_{2}(u_{2})u_{1}$
(2.52)

where for $j=1,2$, $R_{j}$ is a homogeneous smoothing operator in
$\widetilde{\mathcal{R}}^{-\varrho}_{1}$ for any $\varrho\geq 0$.

Composition theorems. Let
$\sigma(D_{x},D_{\xi},D_{y},D_{\eta}):=D_{\xi}D_{y}-D_{x}D_{\eta}$ where
$D_{x}:=\frac{1}{{\rm i}}\partial_{x}$ and $D_{\xi},D_{y},D_{\eta}$ are
similarly defined. The following is Definition 3.11 in [16].

######  Definition 2.13.

(Asymptotic expansion of composition symbol) Let $p$, $p^{\prime}$ in
$\mathbb{N}_{0}$, $K,K^{\prime}\in\mathbb{N}_{0}$ with $K^{\prime}\leq K$,
$\varrho\geq 0$, $m,m^{\prime}\in\mathbb{R}$, $r>0$. Consider symbols
$a\in\Sigma\Gamma_{K,K^{\prime},p}^{m}[r,N]$ and
$b\in\Sigma\Gamma^{m^{\prime}}_{K,K^{\prime},p^{\prime}}[r,N]$. For $U$ in
$B_{\sigma}^{K}(I;r)$ we define, for $\varrho<\sigma-s_{0}$, the symbol

$(a\\#_{\varrho}b)(U;t,x,\xi):=\sum_{k=0}^{\varrho}\frac{1}{k!}\left(\frac{{\rm
i}}{2}\sigma(D_{x},D_{\xi},D_{y},D_{\eta})\right)^{k}\Big{[}a(U;t,x,\xi)b(U;t,y,\eta)\Big{]}_{|_{\begin{subarray}{c}x=y,\xi=\eta\end{subarray}}}$
(2.53)

modulo symbols in
$\Sigma\Gamma^{m+m^{\prime}-\varrho}_{K,K^{\prime},p+p^{\prime}}[r,N]$.

$\bullet$ The symbol $a\\#_{\varrho}b$ belongs to
$\Sigma\Gamma^{m+m^{\prime}}_{K,K^{\prime},p+p^{\prime}}[r,N]$.

$\bullet$ We have that $a\\#_{\varrho}b=ab+\frac{1}{2{\rm i}}\\{a,b\\}$ up to
a symbol in $\Sigma\Gamma^{m+m^{\prime}-2}_{K,K^{\prime},p+p^{\prime}}[r,N]$,
where

$\\{a,b\\}:=\partial_{\xi}a\partial_{x}b-\partial_{x}a\partial_{\xi}b$

denotes the Poisson bracket.

$\bullet$ If $c$ is a symbol in
$\Sigma\Gamma^{m^{\prime\prime}}_{K,K^{\prime},p^{\prime\prime}}[r,N]$ then
$a\\#_{\varrho}b\\#_{\varrho}c+c\\#_{\varrho}b\\#_{\varrho}a-2abc$ is a symbol
in
$\Sigma\Gamma^{m+m^{\prime}+m^{\prime\prime}-2}_{K,K^{\prime},p+p^{\prime}+p^{\prime\prime}}[r,N]$.

$\bullet$
$\overline{a^{\vee}}\\#_{\varrho}\overline{b^{\vee}}=\overline{a\\#_{\varrho}b}^{\vee}$
where $a^{\vee}$ is defined in (2.27).

The following result is proved in Proposition $3.12$ in [16].

######  Proposition 2.14.

(Composition of Bony-Weyl operators) Let $p,q,N,K,K^{\prime}\in\mathbb{N}_{0}$
with $K^{\prime}\leq K$, $\varrho\geq 0$, $m,m^{\prime}\in\mathbb{R}$, $r>0$.
Consider symbols $a\in\Sigma{\Gamma}^{m}_{K,K^{\prime},p}[r,N]$ and
$b\in\Sigma{\Gamma}^{m^{\prime}}_{K,K^{\prime},q}[r,N]$. Then

${{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(a(U;t,x,\xi)\right)}\circ{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(b(U;t,x,\xi)\right)}-{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left((a\\#_{\varrho}b)(U;t,x,\xi)\right)}$
(2.54)

is a smoothing operator in
$\Sigma{\mathcal{R}}^{-\varrho+m+m^{\prime}}_{K,K^{\prime},p+q}[r,N]$.

We now prove other composition results concerning $m$-operators.

######  Proposition 2.15.

(Compositions of $m$-operators) Let
$p,p^{\prime},N,K,K^{\prime}\in\mathbb{N}_{0}$ with $K^{\prime}\leq K$ and
$r>0$. Let $m,m^{\prime}\in\mathbb{R}$. Then

  * $(i)$

If $M(U;t)$ is in $\Sigma\mathcal{M}^{m}_{K,K^{\prime},p}[r,N]$ and
$M^{\prime}(U;t)$ is in
$\Sigma\mathcal{M}^{m^{\prime}}_{K,K^{\prime},p^{\prime}}[r,N]$ then the
composition $M(U;t)\circ M^{\prime}(U;t)$ is in
$\Sigma\mathcal{M}^{m+\max(m^{\prime},0)}_{K,K^{\prime},p+p^{\prime}}[r,N]$.

  * $(ii)$

If $M(U)$ is a homogeneous $m$-operator in $\widetilde{\mathcal{M}}_{p}^{m}$
and $M^{(\ell)}(U;t)$, $\ell=1,\dots,p+1$, are matrices of
$m_{\ell}$-operators in
$\Sigma\mathcal{M}^{m_{\ell}}_{K,K^{\prime},q_{\ell}}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$
with $m_{\ell}\in\mathbb{R}$, $q_{\ell}\in\mathbb{N}_{0}$, then

$M(M^{(1)}(U;t)U,\ldots,M^{(p)}(U;t)U)M^{(p+1)}(U;t)$

belongs to
$\Sigma\mathcal{M}_{K,K^{\prime},p+\overline{q}}^{m+\overline{m}}[r,N]$ with
$\overline{m}:=\sum_{\ell=1}^{p+1}\max(m_{\ell},0)$ and
$\overline{q}:=\sum_{\ell=1}^{p+1}q_{\ell}$.

  * $(iii)$

If $M(U;t)$ is in ${\mathcal{M}}_{K,0,p}^{m}[\breve{r}]$ for any
$\breve{r}\in\mathbb{R}^{+}$ and ${\bf M}_{0}(U;t)$ belongs to
$\mathcal{M}^{0}_{K,K^{\prime},0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$, then
$M({\bf M}_{0}(U;t)U;t)$ is in $\mathcal{M}^{m}_{K,K^{\prime},p}[r]$.

  * (iv)

Let $c$ be a homogeneous symbol in $\widetilde{\Gamma}_{p}^{m}$ and
$M^{(\ell)}(U;t)$, $\ell=1,\dots,p$, be operators in
$\Sigma\mathcal{M}_{K,K^{\prime},q_{\ell}}[r,N]$ with
$q_{\ell}\in\mathbb{N}_{0}$. Then

$U\rightarrow b(U;t,x,\xi):=c(M^{(1)}(U;t)U,\ldots,M^{(p)}(U;t)U;t,x,\xi)$

is a symbol in $\Sigma\Gamma^{m}_{K,K^{\prime},p+\overline{q}}[r,N]$ with
$\overline{q}:=q_{1}+\dots+q_{p}$ and

${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(c(W_{1},\ldots,W_{p};t,x,\xi))_{|W_{\ell}=M^{(\ell)}(U;t)U}={{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(b(U;t,x,\xi))+R(U;t)$

where $R(U;t)$ is a smoothing operator in
$\Sigma\mathcal{R}^{-\varrho}_{K,K^{\prime},p+\overline{q}}[r,N]$ for any
$\varrho\geq 0$.

###### Proof.

Proof of $(ii)$: It is sufficient to prove the thesis for $m_{\ell}\geq 0$
otherwise we regard $M^{(\ell)}(U;t)$ as a $0$-operator in
$\Sigma\mathcal{M}^{0}_{K,K^{\prime},p}[r,N]$. We decompose, for any
$\ell=1,\dots,p+1$,
$M^{(\ell)}=\sum_{a=q_{\ell}}^{N}M_{a}^{(\ell)}+M^{(\ell)}_{>N}$ as in (2.35).
Given integers $a_{\ell}\in[q_{\ell},N]$, $\ell=1,\dots p+1$, we use the
notation $\overline{a}_{\ell}:=a_{1}+\dots+a_{\ell}+\ell$ and
$\overline{a}_{0}:=0$. Note that
$\overline{a}_{p+1}-1=a_{1}+\ldots+a_{p+1}+p\geq\overline{q}+p$. We also
denote the vector with $a_{\ell}$ elements

$\mathcal{U}_{\ell}:=(U_{\overline{a}_{\ell-1}+1},\ldots,U_{\overline{a}_{\ell}-1})\
,\quad\ell=1,\ldots,p+1\,.$

By multi-linearity we have to show on the one hand that, if
$\overline{a}_{p+1}-1\leq N$ then

$M\Big{(}M^{(1)}_{a_{1}}(\mathcal{U}_{1})U_{a_{1}+1},\dots,M^{(p)}_{a_{p}}(\mathcal{U}_{p})U_{\overline{a}_{p}}\Big{)}M^{(p+1)}_{a_{p+1}}(\mathcal{U}_{p+1})$
(2.55)

is a homogeneous operator in
${\widetilde{\mathcal{M}}}^{m+\overline{m}}_{\overline{a}_{p+1}-1}$. On the
other hand, if $\overline{a}_{p+1}-1\geq N+1$ then

$M\Big{(}M^{(1)}_{a_{1}}(U,\dots,U)U,\dots,M^{(p)}_{a_{p}}(U,\dots,U)U\Big{)}M^{(p+1)}_{a_{p+1}}(U,\dots,U)\quad$
(2.56)

is a non–homogeneous operator in
$\mathcal{M}^{m+\overline{m}}_{K,K^{\prime},N+1}[r]$, having included for
notational convenience the inhomogeneous term as
$M^{(\ell)}_{N+1}(U,\dots,U):=M^{(\ell)}_{>N}(U;t)$ which belongs to
$\mathcal{M}^{m_{\ell}}_{K,K^{\prime},N+1}[r]$.

We first study (2.55). First of all, using the notation
$\vec{n}_{\ell}:=(n_{\overline{a}_{\ell-1}+1},\ldots,n_{\overline{a}_{\ell}-1})$
one has

$\displaystyle\|\Pi_{n_{0}}M\Big{(}M^{(1)}_{a_{1}}(\Pi_{\vec{n}_{1}}\mathcal{U}_{1})\Pi_{n_{\overline{a}_{1}}}U_{\overline{a}_{1}},\dots,M^{(p)}_{a_{p}}(\Pi_{\vec{n}_{p}}\mathcal{U}_{p})\Pi_{n_{\overline{a}_{p}}}U_{\overline{a}_{p}}\Big{)}M^{(p+1)}_{a_{p+1}}(\Pi_{\vec{n}_{p+1}}\mathcal{U}_{p+1})\Pi_{n_{\overline{a}_{p+1}}}U_{\overline{a}_{p+1}}\|_{L^{2}}$
$\displaystyle\leq\\!\\!\\!\\!\\!\sum_{n^{\prime}_{1},\dots,n_{p+1}^{\prime}}\\!\\!\\!\\!\\!\|\Pi_{n_{0}}M\Big{(}\Pi_{n^{\prime}_{1}}M^{(1)}_{a_{1}}(\Pi_{\vec{n}_{1}}\mathcal{U}_{1})\Pi_{n_{\overline{a}_{1}}}U_{\overline{a}_{1}},\dots,\Pi_{n^{\prime}_{p}}M^{(p)}_{a_{p}}(\Pi_{\vec{n}_{p}}\mathcal{U}_{p})\Pi_{n_{\overline{a}_{p}}}U_{\overline{a}_{p}}\Big{)}$
$\displaystyle\qquad\qquad\circ\Pi_{n^{\prime}_{p+1}}M^{(p+1)}_{a_{p+1}}(\Pi_{\vec{n}_{p+1}}\mathcal{U}_{p+1})\Pi_{n_{\overline{a}_{p+1}}}U_{\overline{a}_{p+1}}\|_{L^{2}}\,.$
(2.57)

Thanks to the conditions (2.32) for $M$ and $M^{(\ell)}$ the indices in the
above sum satisfy, for some choice of signs $\sigma_{b},\epsilon_{j}$,
$b=1,\dots,\overline{a}_{p+1}$, $j=1,\dots,p+1$, the restrictions

$n_{0}=\sum_{j=1}^{p+1}\epsilon_{j}n^{\prime}_{j}\,,\quad
n^{\prime}_{\ell}=\sum_{b=\overline{a}_{\ell-1}+1}^{\overline{a}_{\ell}}\sigma_{b}n_{b}\,,\quad\ell=1,\dots,p+1\,.$
(2.58)

As a consequence (2.55) satisfies the corresponding condition (2.32) and

$n^{\prime}_{\ell}\lesssim\max\\{\vec{n}_{\ell},n_{\overline{a}_{\ell}}\\}\
,\qquad\forall\ell=1,\dots,p+1\ ,$ (2.59)

then we have

$\displaystyle{\rm max}\\{n^{\prime}_{1},\dots,n^{\prime}_{p+1}\\}\lesssim{\rm
max}\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}$ (2.60) $\displaystyle{\rm
max}_{2}\\{n^{\prime}_{1},\dots,n^{\prime}_{p+1}\\}\stackrel{{\scriptstyle\eqref{denaco34}}}{{\lesssim}}{\rm
max}_{2}\left\\{\max\\{\vec{n}_{1},n_{\overline{a}_{1}}\\},\ldots,\max\\{\vec{n}_{p+1},n_{\overline{a}_{p+1}}\\}\right\\}\lesssim{\rm
max}_{2}\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}$

where in the last inequality we used that
$\\{n_{1},\ldots,n_{\overline{a}_{p+1}}\\}$ is the disjoint union of the sets
$\\{\vec{n}_{\ell},n_{\overline{a}_{\ell}}\\}_{\ell=1,\ldots,p+1}$.
Using (2.58), (2.31) for $M$ and $M^{(\ell)}$ we get, with
$\vec{n}^{\prime}:=(n_{1}^{\prime},\dots,n^{\prime}_{p+1})$

$\displaystyle\eqref{inininni35}\lesssim{\rm
max}_{2}\\{\vec{n}^{\prime}\\}^{\mu}{\rm
max}\\{\vec{n}^{\prime}\\}^{m}\prod_{\ell=1}^{p+1}{\rm
max}_{2}\\{\vec{n}_{\ell},n_{\overline{a}_{\ell}}\\}^{\mu_{\ell}}{\rm
max}\\{\vec{n}_{\ell},n_{\overline{a}_{\ell}}\\}^{m_{\ell}}\prod_{b=1}^{\overline{a}_{p+1}}\|\Pi_{n_{b}}U_{b}\|_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{denaco35},\eqref{mono:max2},m_{\ell}\geq
0}}{{\lesssim}}{\rm
max}_{2}\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}^{\overline{\mu}}\max\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}^{\overline{m}}{\rm
max}\\{\vec{n}^{\prime}\\}^{m}\prod_{b=1}^{\overline{a}_{p+1}}\|\Pi_{n_{b}}U_{b}\|_{L^{2}},$
(2.61)

where $\overline{\mu}:=\mu+\mu_{1}+\dots+\mu_{p+1}$ and recall
$\overline{m}=m_{1}+\dots+m_{p+1}$. We claim that

${\rm max}\\{\vec{n}^{\prime}\\}^{m}\lesssim{\rm
max}_{2}\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}^{\mu^{\prime\prime}}\max\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}^{m}$
(2.62)

for some $\mu^{\prime\prime}\geq 0$. Then (2.31), (2.61) and (2.62) imply that
the operator in (2.55) belongs to
${\widetilde{\mathcal{M}}}_{\overline{a}_{p+1}-1}^{m+\overline{m}}$.

We now prove (2.62). If $m\geq 0$ it follows by (2.60) with
$\mu^{\prime\prime}=0$. So from now on we consider $m<0$. We fix
$\overline{\ell}$ such that

$\max\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}={\rm
max}\\{\vec{n}_{\overline{\ell}},\ n_{\overline{a}_{\overline{\ell}}}\\}$
(2.63)

and we distinguish two cases:

Case $1$: $n^{\prime}_{\overline{\ell}}\geq\frac{1}{2}{\rm
max}\\{\vec{n}_{\overline{\ell}},\ n_{\overline{a}_{\overline{\ell}}}\\}$. In
this case

${\rm max}\\{\vec{n}^{\prime}\\}^{m}\stackrel{{\scriptstyle
m<0}}{{\leq}}(n^{\prime}_{\overline{\ell}})^{m}\stackrel{{\scriptstyle{\rm
Case}\,1}}{{\leq}}2^{|m|}\,{\rm
max}\\{\vec{n}_{\overline{\ell}},n_{\overline{a}_{\overline{\ell}}}\\}^{m}\stackrel{{\scriptstyle\eqref{scelgouguali}}}{{=}}2^{|m|}\max\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}^{m}$

which proves (2.62) with $\mu^{\prime\prime}=0$.

Case $2$: $n^{\prime}_{\overline{\ell}}<\frac{1}{2}{\rm
max}\\{\vec{n}_{\overline{\ell}},\ n_{\overline{a}_{\overline{\ell}}}\\}$. In
view of the momentum condition (2.58),

$\displaystyle\max\\{\vec{n}_{\overline{\ell}},\
n_{\overline{a}_{\overline{\ell}}}\\}$ $\displaystyle\leq
a_{\overline{\ell}}{\rm max}_{2}\\{\vec{n}_{\overline{\ell}},\
n_{\overline{a}_{\overline{\ell}}}\\}+n^{\prime}_{\overline{\ell}}$
$\displaystyle\stackrel{{\scriptstyle{\rm
Case}\,2}}{{\leq}}a_{\overline{\ell}}{\rm
max}_{2}\\{\vec{n}_{\overline{\ell}},\
n_{\overline{a}_{\overline{\ell}}}\\}+\frac{1}{2}{\rm
max}\\{\vec{n}_{\overline{\ell}},\ n_{\overline{a}_{\overline{\ell}}}\\}$

and consequently

$\max\\{\vec{n}_{\overline{\ell}},\ n_{\overline{a}_{\overline{\ell}}}\\}\leq
2a_{\overline{\ell}}{\rm max}_{2}\\{\vec{n}_{\overline{\ell}},\
n_{\overline{a}_{\overline{\ell}}}\\}\,.$ (2.64)

Then, since $m<0$,

$\displaystyle\max\\{\vec{n}^{\prime}\\}^{m}$ $\displaystyle\leq
1\stackrel{{\scriptstyle\eqref{smoopercaso35}}}{{\leq}}(2a_{\overline{\ell}})^{|m|}\
{\rm max}_{2}\\{\vec{n}_{\overline{\ell}},\
n_{\overline{a}_{\overline{\ell}}}\\}^{|m|}\
\max\\{\vec{n}_{\overline{\ell}},\ n_{\overline{a}_{\overline{\ell}}}\\}^{m}$
$\displaystyle\stackrel{{\scriptstyle\eqref{scelgouguali},\eqref{mono:max2}}}{{\leq}}$
$\displaystyle(2a_{\overline{\ell}})^{|m|}{\rm
max}_{2}\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}^{|m|}\max\\{n_{1},\dots,n_{\overline{a}_{p+1}}\\}^{m}$

which proves (2.62) with $\mu^{\prime\prime}=|m|$. This concludes the proof of
(2.62) and then that the operator in (2.55) is in
${\widetilde{\mathcal{M}}}_{\overline{a}_{p+1}-1}^{m+\overline{m}}$.

Now we prove that the operator in (2.56) is in
$\mathcal{M}^{m+\overline{m}}_{K,K^{\prime},N+1}[r]$. We have to verify (2.34)
with $m\leadsto m+\overline{m}$ and $p\leadsto N+1$. For simplicity we denote
$M^{(\ell)}_{a_{\ell}}(U)=M^{(\ell)}_{a_{\ell}}(U,\ldots,U)$. First we apply
(2.39) to $M(U)\in\widetilde{\mathcal{M}}^{m}_{p}$ (with $U_{\ell}\leadsto
M^{(\ell)}_{a_{\ell}}(U)U$, $v\leadsto M^{(p+1)}_{a_{p+1}}(U)v$ and $s\leadsto
s-\overline{m}$) getting, for any $k=0,\ldots,K-K^{\prime}$,

$\displaystyle\|\partial_{t}^{k}\big{(}\eqref{nonomo35}v\big{)}\|_{\dot{H}^{s-m-\overline{m}-\frac{3}{2}k}}$
$\displaystyle\lesssim_{K}\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k_{1}+\dots+k_{p+1}=k}\\!\\!\\!\\!\\!\Big{(}{\|M^{(p+1)}_{a_{p+1}}(U)v\|}_{k_{p+1},s-\overline{m}}\prod_{\ell=1}^{p}\
{\|M^{(\ell)}_{a_{\ell}}(U)U\|}_{k_{\ell},s_{0}}$ (2.65)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!+{\|M^{(p+1)}_{a_{p+1}}(U)v\|}_{k_{p+1},s_{0}}\sum_{\overline{\ell}=1}^{p}{\|M^{(\overline{\ell})}_{a_{\overline{\ell}}}(U)U\|}_{k_{\overline{\ell}},s-\overline{m}}\prod_{\ell=1\atop\ell\neq\overline{\ell}}^{p}{\|M^{(\ell)}_{a_{\ell}}(U)U\|}_{k_{\ell},s_{0}}\Big{)}\,.$
(2.66)

Then we estimate line (2.65) where, by Lemma 2.8, each
$M^{(\ell)}_{a_{\ell}}(U)$ is in $\mathcal{M}^{m_{\ell}}_{K,0,a_{\ell}}[r]$
and $M^{(\ell)}_{N+1}(U)$ belongs to
$\mathcal{M}^{m_{\ell}}_{K,K^{\prime},N+1}[r]$. For any $\ell=1,\ldots,p$,
using (2.34) (with $m\leadsto m_{\ell}$ and $p\leadsto a_{\ell}$) we bound
(since $k_{\ell}\leq k-k_{p+1}$ and $0\leq k\leq K-K^{\prime}$)

$\displaystyle{\|M^{(\ell)}_{a_{\ell}}(U)U\|}_{k_{\ell},s_{0}}\leq
C{\|U\|}_{k_{\ell}+K^{\prime},s_{0}+\overline{m}}^{a_{\ell}+1}\leq
C{\|U\|}_{k-k_{p+1}+K^{\prime},s_{0}+\overline{m}}^{a_{\ell}+1}\,,$ (2.67)
$\displaystyle{\|M^{(p+1)}_{a_{p+1}}(U)v\|}_{k_{p+1},s-\overline{m}}\leq
C\sum_{k^{\prime}+k^{\prime\prime}\leq
k_{p+1}}{\|v\|}_{k^{\prime\prime},s}{\|U\|}_{k^{\prime}+K^{\prime},s_{0}}^{a_{p+1}}+{\|v\|}_{k^{\prime\prime},s_{0}}{\|U\|}_{k^{\prime}+K^{\prime},s_{0}}^{a_{p+1}-1}{\|U\|}_{k^{\prime}+K^{\prime},s}$
$\displaystyle\qquad\qquad\qquad\qquad\leq C\sum_{k^{\prime\prime}\leq
k_{p+1}}{\|v\|}_{k^{\prime\prime},s}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}}^{a_{p+1}}+{\|v\|}_{k^{\prime\prime},s_{0}}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}}^{a_{p+1}-1}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s}$

since $k^{\prime}\leq k-k^{\prime\prime}$ (being $k_{p+1}\leq k$). By (2.67)
and since $k-k_{p+1}+K^{\prime}\leq k-k^{\prime\prime}+K^{\prime}$, we get

$\displaystyle\eqref{due1}$
$\displaystyle\lesssim_{K,s}\sum_{k^{\prime\prime}\leq
k}{\|v\|}_{k^{\prime\prime},s}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}+\overline{m}}^{a_{1}+\dots+a_{p+1}+p}+{\|v\|}_{k^{\prime\prime},s_{0}}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}+\overline{m}}^{a_{1}+\dots+a_{p+1}+p-1}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s}$
$\displaystyle\lesssim_{K,s}\sum_{k^{\prime\prime}\leq
k}{\|v\|}_{k^{\prime\prime},s}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}+\overline{m}}^{N+1}+{\|v\|}_{k^{\prime\prime},s_{0}}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}+\overline{m}}^{N}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s}$
(2.68)

because $a_{1}+\dots+a_{p+1}+p=\overline{a}_{p+1}-1\geq N+1$ (for
$\|U\|_{K,s_{0}+\overline{m}}<1$). Regarding (2.66), we first bound by (2.34)
(and (2.67) with $s=s_{0}+\overline{m}$)

$\displaystyle{\|M^{(\overline{\ell})}_{a_{\overline{\ell}}}(U)U\|}_{k_{\overline{\ell}},s-\overline{m}}\leq
C{\|U\|}_{k_{\overline{\ell}}+K^{\prime},s_{0}}^{a_{\overline{\ell}}}{\|U\|}_{k_{\overline{\ell}}+K^{\prime},s}\leq
C{\|U\|}_{k-k_{p+1}+K^{\prime},s_{0}}^{a_{\overline{\ell}}}{\|U\|}_{k-k_{p+1}+K^{\prime},s}$
$\displaystyle{\|M^{(p+1)}_{a_{p+1}}(U)v\|}_{k_{p+1},s_{0}}\leq
C\sum_{k^{\prime\prime}\leq
k_{p+1}}{\|v\|}_{k^{\prime\prime},s_{0}+\overline{m}}\
{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}+\overline{m}}^{a_{p+1}}$

and, proceeding similarly to the previous computation, we deduce

$\eqref{due2}\lesssim_{K,s}\sum_{k^{\prime\prime}\leq
k}{\|v\|}_{k^{\prime\prime},s_{0}+\overline{m}}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s_{0}+\overline{m}}^{N}{\|U\|}_{k-k^{\prime\prime}+K^{\prime},s}\,.$
(2.69)

Hence (2.68), (2.69) imply that for any $k=0,\ldots,K-K^{\prime}$

$\|\partial_{t}^{k}\big{(}\eqref{nonomo35}v\big{)}\|_{\dot{H}^{s-m-\overline{m}-\frac{3}{2}k}}\lesssim_{K,s}\\!\\!\\!\\!\\!\\!\sum_{k^{\prime}+k^{\prime\prime}=k}\\!\\!\\!\\!{\|v\|}_{k^{\prime\prime},s}{\|U\|}_{k^{\prime}+K^{\prime},s_{0}+\overline{m}}^{N+1}+{\|v\|}_{k^{\prime\prime},s_{0}+\overline{m}}{\|U\|}_{k^{\prime}+K^{\prime},s_{0}+\overline{m}}^{N}{\|U\|}_{k^{\prime}+K^{\prime},s}$
(2.70)

proving that the operator (2.56) satisfies (2.34) with $m\leadsto
m+\overline{m}$, $p\leadsto N+1$ and $s_{0}\leadsto s_{0}+\overline{m}$.
Proof of $(i)$: Decomposing $M=\sum_{q=p}^{N}M_{q}+M_{>N}$ and
$M^{\prime}=\sum_{q=p^{\prime}}^{N}M^{\prime}_{q}+M^{\prime}_{>N}$ as in
(2.35), by item $(ii)$ we deduce that $M_{q_{1}}(U)M^{\prime}_{q_{2}}(U)$ is
in ${\widetilde{\mathcal{M}}}^{m+\max{\\{m^{\prime},0\\}}}_{q_{1}+q_{2}}$ if
$q_{1}+q_{2}\leq N$, and $M_{q_{1}}(U)M^{\prime}_{q_{2}}(U)$, $q_{1}+q_{2}\geq
N+1$ and $M_{q_{1}}(U)M^{\prime}_{>N}(U;t)$, $q_{1}=p,\dots,N,$ are in
$\mathcal{M}^{m+\max\\{m^{\prime},0\\}}_{K,K^{\prime},N+1}[r]$. Furthermore,
proceeding as in the proof of (2.70), one shows that
$M_{>N}(U;t)M^{\prime}_{q_{2}}(U)$, $q_{2}=p^{\prime},\dots,N$, and
$M_{>N}(U;t)M^{\prime}_{>N}(U;t)$ are operators in
$\mathcal{M}^{m+\max\\{m^{\prime},0\\}}_{K,K^{\prime},N+1}[r]$.

Proof of $(iii)$: As ${\bf
M}_{0}(U;t)\in\mathcal{M}^{0}_{K,K^{\prime},0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$,
for any $U\in B^{K^{\prime}}_{s_{0}}(I;r)$, the function ${\bf M}_{0}(U;t)U\in
B^{0}_{s_{0}}(I;Cr)$, for some $C\geq 1$. Hence the composition $M({\bf
M}_{0}(U;t)U)$ is a well defined operator. Moreover, for any $U\in
B_{s_{0}}^{K}(I;r(\sigma))$, we have the quantitative estimate ${\|{\bf
M}_{0}(U;t)U\|}_{k,\sigma}\lesssim{\|U\|}_{k,\sigma}$, for
$k=0,\ldots,K-K^{\prime}$ and $\sigma\geq s_{0}$. To bound the time
derivatives of $M({\bf M}_{0}(U;t)U)$ we use (2.34) and the previous estimate
to get, for any $k=0,\ldots,K-K^{\prime}$

$\displaystyle{\|\partial_{t}^{k}\left(M({\bf
M}_{0}(U;t)U)V\right)\|}_{s-\frac{3}{2}k-m}$
$\displaystyle\quad\lesssim\sum_{k^{\prime}+k^{\prime\prime}=k}{\|{\bf
M}_{0}(U;t)U\|}_{k^{\prime},s_{0}}^{p}{\|V\|}_{k^{\prime\prime},s}+{\|{\bf
M}_{0}(U;t)U\|}_{k^{\prime},s}{\|{\bf
M}_{0}(U;t)U\|}_{k^{\prime},s_{0}}^{p-1}{\|V\|}_{k^{\prime\prime},s_{0}}$
$\displaystyle\quad\lesssim\sum_{k^{\prime}+k^{\prime\prime}=k}\\!\\!\\!\\!\|V\|_{k^{\prime\prime},s}\|{U}\|_{k^{\prime}+K^{\prime},s_{0}}^{p}+\|{V}\|_{k^{\prime\prime},s_{0}}\|U\|_{k^{\prime}+K^{\prime},s_{0}}^{p-1}\|{U}\|_{k^{\prime}+K^{\prime},s}$

proving (2.34).

Proof of $(iv)$: It follows, in view of Remark 2.6, by Proposition 3.17-$(i)$
and Proposition 3.18 in [16]. ∎

We shall use the following facts which follow by Proposition 2.15.

$\bullet$ If $a(U;t,x,\xi)$ is a symbol in
$\Sigma\Gamma_{K,K^{\prime},p}^{m}[r,N]$ and $U$ is a solution of
$\partial_{t}U=M(U;t)U$ for some $M(U;t)$ in
$\Sigma\mathcal{M}_{K,0,0}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$, then
$\partial_{t}a(U;t,x,\xi)$ is a symbol in
$\Sigma\Gamma_{K,K^{\prime}+1,p}^{m}[r,N]$.

$\bullet$ If $R(U;t)$ is a smoothing operator in
$\Sigma\mathcal{R}^{-\varrho}_{K,K^{\prime},p}[r,N]$ and $U$ is a solution of
$\partial_{t}U=M(U;t)U$ for some $M(U;t)$ in
$\Sigma\mathcal{M}_{K,0,0}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$, then
$\partial_{t}R(U;t)$ is a smoothing operator
$\Sigma\mathcal{R}^{-\varrho}_{K,K^{\prime}+1,p}[r,N]$.

$\bullet$ If $M(U;t)$ is in $\Sigma\mathcal{M}^{m}_{K,K^{\prime},p}[r,N]$ and
$R(U;t)$ is in $\Sigma\mathcal{R}^{-\varrho}_{K,K^{\prime},p^{\prime}}[r,N]$
then the composition $M(U;t)\circ R(U;t)$ is in
$\Sigma\mathcal{M}^{m}_{K,K^{\prime},p+p^{\prime}}[r,N]$, and so it is not a
smoothing map.

####  2.2 Spectrally localized maps

We introduce the notion of a “spectrally localized” map. The class
$\widetilde{\mathcal{S}}^{m}_{p}$ denotes $p$-linear $m$-operators with the
spectral support similar to a para–differential operator (compare (2.25) and
(2.71)), i.e. the “ internal frequencies” are controlled by the “external”
ones which are equivalent. On the other hand the class
$\mathcal{S}_{K,K^{\prime},p}^{m}$ contains non-homogeneous $m$-operators
which vanish at degree at least $p$ in $U$, are $(K-K^{\prime})$-times
differentiable in $t$, and satisfy estimates similar to para–differential
operators, see (2.72). These maps include para–differential operators,
smoothing remainders which come from compositions of para–differential
operators (see (2.54)) and also linear flows generated by para–differential
operators. The class of spectrally localized maps do not enjoy a symbolic
calculus and it is reminiscent of the maps introduced in Definition 1.2.1 in
[37].

The class of spectrally localized maps is closed under transposition (Lemma
2.18) and under “external” and “internal” compositions, see Proposition 2.19.
A key property is that the transpose of the internal differential of a
spectrally localized map is a smoothing operator, see Lemma 2.21.

######  Definition 2.16.

(Spectrally localized maps) Let $m\in\mathbb{R}$, $p,N\in\mathbb{N}_{0}$,
$K,K^{\prime}\in\mathbb{N}_{0}$ with $K^{\prime}\leq K$ and $r>0$.

(i) Spectrally localized $p$-homogeneous maps. We denote by
$\widetilde{\mathcal{S}}^{m}_{p}$ the subspace of $m$-operators
$S(\mathcal{U})$ in $\widetilde{\mathcal{M}}^{m}_{p}$ satisfying the following
spectral condition: there exist $\delta>0$, $C>1$ such that for any
$(U_{1},\ldots,U_{p})\in(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{p}$,
for any $U_{p+1}\in\dot{H}^{\infty}(\mathbb{T};\mathbb{C})$ and for any
$n_{0},\ldots,n_{p+1}\in\mathbb{N}$ such that

$\Pi_{n_{0}}S(\Pi_{n_{1}}U_{1},\ldots,\Pi_{n_{p}}U_{p})\Pi_{n_{p+1}}U_{p+1}\neq
0\,,$

it results

$\max\\{n_{1},\dots,n_{p}\\}\leq\delta n_{p+1}\,,\qquad C^{-1}\,n_{0}\leq
n_{p+1}\leq C\,n_{0}\,.$ (2.71)

We denote
$\widetilde{\mathcal{S}}_{p}:=\bigcup_{m}\widetilde{\mathcal{S}}^{m}_{p}$ and
by $\Sigma_{p}^{N}\widetilde{\mathcal{S}}^{m}_{q}$ the class of pluri-
homogeneous spectrally localized maps of the form $\sum_{q=p}^{N}S_{q}$ with
$S_{q}\in\widetilde{\mathcal{S}}^{m}_{q}$ and
$\Sigma_{p}\widetilde{\mathcal{S}}^{m}_{q}:=\bigcup_{N\in\mathbb{N}}\Sigma_{p}^{N}\widetilde{\mathcal{S}}^{m}_{q}$.
For $p\geq N+1$ we mean that the sum is empty.

(ii) Non-homogeneous spectrally localized maps. We denote
$\mathcal{S}^{m}_{K,K^{\prime},p}[r]$ the space of maps $(U,t,V)\mapsto
S(U;t)V$ defined on $B^{K^{\prime}}_{s_{0}}(I;r)\times I\times
C^{0}_{*}(I,\dot{H}^{s_{0}}(\mathbb{T},\mathbb{C}))$ for some $s_{0}>0$, which
are linear in the variable $V$ and such that the following holds true. For any
$s\in\mathbb{R}$ there are $C>0$ and $r(s)\in]0,r[$ such that for any $U\in
B^{K}_{s_{0}}(I;r(s))\cap
C^{K}_{*}(I,\dot{H}^{s}(\mathbb{T},\mathbb{C}^{2}))$, any $V\in
C^{K-K^{\prime}}_{*}(I,\dot{H}^{s}(\mathbb{T},\mathbb{C}))$, any $0\leq k\leq
K-K^{\prime}$, $t\in I$, we have that

$\|{\partial_{t}^{k}\left(S(U;t)V\right)(t,\cdot)}\|_{\dot{H}^{s-\frac{3}{2}k-m}}\leq
C\sum_{k^{\prime}+k^{\prime\prime}=k}\|{U}\|_{k^{\prime}+K^{\prime},s_{0}}^{p}\|{V}\|_{k^{\prime\prime},s}\,.$
(2.72)

In case $p=0$ we require the estimate
$\|{\partial_{t}^{k}\left(S(U;t)V\right)}\|_{\dot{H}^{s-\frac{3}{2}k-m}}\leq
C\|{V}\|_{k,s}$.

We denote
$\mathcal{S}_{K,K^{\prime},N}[r]=\bigcup_{m}\mathcal{S}^{m}_{K,K^{\prime},N}[r]$.

(iii) Spectrally localized Maps. We denote by
$\Sigma\mathcal{S}^{m}_{K,K^{\prime},p}[r,N]$, the space of maps $(U,t,V)\to
S(U;t)V$ of the form

$S(U;t)V=\sum_{q=p}^{N}S_{q}(U,\ldots,U)V+S_{>N}(U;t)V$ (2.73)

where $S_{q}$ are spectrally localized homogeneous maps in
$\widetilde{\mathcal{S}}^{m}_{q}$, $q=p,\ldots,N$ and $S_{>N}$ is a
non–homogeneous spectrally localized map in
$\mathcal{S}^{m}_{K,K^{\prime},N+1}[r]$. We denote by
$\Sigma\mathcal{S}_{K,K^{\prime},p}^{m}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$
the space of $2\times 2$ matrices whose entries are spectrally localized maps
in $\Sigma\mathcal{S}^{m}_{K,K^{\prime},p}[r,N]$. We will use also the
notation $\Sigma\mathcal{S}_{K,K^{\prime},p}[r,N]:=\bigcup_{m\geq
0}\Sigma\mathcal{S}^{m}_{K,K^{\prime},p}[r,N]$.

$\bullet$ Note that (2.71) implies that $\max\\{n_{1},\ldots,n_{p+1}\\}\sim
n_{p+1}$ and
$\max_{2}\\{n_{1},\ldots,n_{p+1}\\}\sim\max\\{n_{1},\ldots,n_{p}\\}$ and
therefore, by (2.31), $S\in\widetilde{\mathcal{S}}^{m}_{p}$ if and only if
there are $\mu\geq 0$, $C>0$ such that

$\|\Pi_{n_{0}}S(\Pi_{n_{1}}U_{1},\ldots,\Pi_{n_{p}}U_{p})\Pi_{n_{p+1}}U_{p+1}\|_{L^{2}}\leq
C{\rm\max}(n_{1},\ldots,n_{p})^{\mu}\,n_{p+1}^{m}\prod_{j=1}^{p+1}\|\Pi_{n_{j}}U_{j}\|_{L^{2}}$
(2.74)

for any $n_{0},\ldots,n_{p+1}\in\mathbb{N}$.

$\bullet$ In view of Lemma 2.9 a matrix of spectrally localized maps $S(U)$ is
characterized in terms of its Fourier coefficients as follows: $S(U)$ is in
$\widetilde{\mathcal{S}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$ if and
only if $S_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$
(defined as in (2.43)) satisfy, for some $\mu\geq 0$, $C>0$,

$|S_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|\leq
C{\rm
max}\\{|j_{1}|,\dots,|j_{p}|\\}^{\mu}|j|^{m}\,,\quad\forall(\vec{\jmath}_{p},j,k)\in(\mathbb{Z}\setminus\\{0\\})^{p+2}\,,\
(\vec{\sigma}_{p},\sigma^{\prime},\sigma)\in\\{\pm\\}^{p+2}\,,$ (2.75)

and if
$S_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}\not=0$
then (2.44), (2.45) hold and, for some $\delta>0$,

$\max\\{|j_{1}|,\dots,|j_{p}|\\}\leq\delta|j|\,,\quad C^{-1}\,|k|\leq|j|\leq
C\,|k|\,.$ (2.76)

$\bullet$ If $S$ is a $p$–homogeneous spectrally localized map in
$\widetilde{\mathcal{S}}^{m}_{p}$ then $S(U)[V]$ defines a non–homogeneous
spectrally localized map in $\mathcal{S}_{K,0,p}^{m}[r]$ for any $r>0$. The
proof is similar to the one of Lemma 2.8 noting that the estimate (2.72) holds
for any $s\in\mathbb{R}$ because of the equivalence $n_{0}\sim
n_{p+1}\sim\max\\{n_{1},\dots,n_{p+1}\\}$ in (2.71).

$\bullet$ (Paradifferential operators as spectrally localized maps) If
$a(U;t,x,\xi)$ is a symbol in $\Sigma\Gamma^{m}_{K,K^{\prime},p}[r,N]$ then
the paradifferential operator ${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(a(U;t,x,\xi))$ is a spectrally
localized map in $\Sigma\mathcal{S}_{K,K^{\prime},p}^{m}[r,N]$. This is a
consequence of Proposition 3.8 in [16]. We remark that for a homogeneous
symbol this is a consequence of the choice of the first quantization in
(2.23).

$\bullet$ If $S(U,\dots,U)$ is a $p$–homogeneous spectrally localized map in
${\widetilde{\mathcal{S}}}_{p}$, then the differential of the non–linear map
$S(U,\dots,U)U$, ${\rm
d}_{U}\big{(}S(U,\dots,U)U\big{)}V=pS(V,U,\dots,U)U+S(U,\dots,U)V$ is a
$p$–homogeneous operator in ${\widetilde{\mathcal{M}}}_{p}$, _not_ necessarily
spectrally localized. Indeed the operator $S(V,U,\dots,U)U$ is _not_ , in
general, spectrally localized.

$\bullet$ If $m_{1}\leq m_{2}$ then
$\Sigma\mathcal{S}^{m_{1}}_{K,K^{\prime},p}[r,N]\subseteq\Sigma\mathcal{S}^{m_{2}}_{K,K^{\prime},p}[r,N]$.
If $K_{1}^{\prime}\leq K^{\prime}_{2}$ then
$\Sigma\mathcal{S}^{m}_{K,K^{\prime}_{1},p}[r,N]\subseteq\Sigma\mathcal{S}^{m}_{K,K^{\prime}_{2},p}[r,N]$.

######  Remark 2.17.

The constant $\delta>0$ in the spectral condition (2.71) is not assumed to be
small (unlike the one in (2.25) for paradifferential operators).

The class of matrices of spectrally localized homogeneous maps is closed under
transposition.

######  Lemma 2.18.

Let $p\in\mathbb{N}_{0}$, $m\in\mathbb{R}$. If $S(U)$ is a matrix of
$p$–homogeneous spectrally localized maps in
$\widetilde{\mathcal{S}}^{m}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ then
$S(U)^{\top}$ is in
$\widetilde{\mathcal{S}}^{m}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$, where the
transpose is computed with respect to the non-degenerate real bilinear form
(2.4).

###### Proof.

It results

$|(S^{\top})_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|\stackrel{{\scriptstyle\eqref{emmetop}}}{{=}}|S_{\vec{\jmath}_{p},-k,-j}^{\vec{\sigma}_{p},\sigma,\sigma^{\prime}}|\stackrel{{\scriptstyle\eqref{smoocaraspec}}}{{\leq}}C{\rm
max}\\{|j_{1}|,\dots,|j_{p}|\\}^{\mu}|k|^{m}\stackrel{{\scriptstyle\eqref{smoocaraspecres}}}{{\leq}}C^{\prime}{\rm
max}\\{|j_{1}|,\dots,|j_{p}|\\}^{\mu}|j|^{m}$

which means that
$(S^{\top})_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$
satisfies (2.75). Since $|j|\sim|k|$ then
$\max\\{|j_{1}|,\dots,|j_{p}|\\}\leq\delta C|k|$ hence (2.76) holds for
$(S^{\top})_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$.
∎

We now prove some further composition results for spectrally localized maps.

######  Proposition 2.19.

(Compositions of spectrally localized maps) Let
$p,p^{\prime},p^{\prime\prime},N,K,K^{\prime}\in\mathbb{N}_{0}$ with
$K^{\prime}\leq K$ and $r>0$. Let $m,m^{\prime}\in\mathbb{R}$ and $S(U;t)$ be
a spectrally localized map in $\Sigma\mathcal{S}^{m}_{K,K^{\prime},p}[r,N]$.
Then

  * $(i)$

if $R(U;t)$ is a smoothing operator in
$\Sigma\mathcal{R}^{-\varrho}_{K,K^{\prime},p^{\prime\prime}}[r,N]$ for some
$\varrho\geq 0$, then $S(U;t)\circ R(U;t)$ and $R(U;t)\circ S(U;t)$ are
smoothing operators in
$\Sigma\mathcal{R}^{-\varrho+\max(m,0)}_{K,K^{\prime},p+p^{\prime\prime}}[r,N]$.

  * $(ii)$

If $S^{\prime}(U;t)$ is in
$\Sigma\mathcal{S}^{m^{\prime}}_{K,K^{\prime},p^{\prime}}[r,N]$ then the
composition $S(U;t)\circ S^{\prime}(U;t)$ is in
$\Sigma\mathcal{S}^{m+m^{\prime}}_{K,K^{\prime},p+p^{\prime}}[r,N]$.

  * $(iii)$

If $S$ is in $\widetilde{\mathcal{S}}^{m}_{p}$ and $S^{(a)}(U;t)$ are
spectrally localized maps in
$\Sigma\mathcal{S}_{K,K^{\prime},q_{a}}^{\ell_{a}}[r,N]$ for some
$\ell_{a}\in\mathbb{R}$ and $q_{a}\in\mathbb{N}$, $a=1,\dots,p$, then also the
internal composition

$S(S^{(1)}(U;t)U,\dots,S^{(p)}(U;t)U)$ (2.77)

is a spectrally localized map in
$\Sigma\mathcal{S}^{m}_{K,K^{\prime},p+\overline{q}}[r,N]$ with
$\overline{q}:=\sum_{a=1}^{p}q_{a}$.

  * $(iv)$

If $S(U;t)$ is in ${\mathcal{S}}_{K,0,p}^{m}[\breve{r}]$ for any
$\breve{r}\in\mathbb{R}^{+}$ and ${\bf M}_{0}(U;t)$ belongs to
$\mathcal{M}^{0}_{K,K^{\prime},0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$, then
$S({\bf M}_{0}(U;t)U;t)$ is in $\mathcal{S}^{m}_{K,K^{\prime},p}[r]$.

  * $(v)$

If $S(U)$ is a matrix of $p$-homogeneous spectrally localized maps in
$\widetilde{\mathcal{S}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$ and
$S_{a}(U;t)$ are in
$\mathcal{S}^{m_{a}}_{K,K^{\prime},q_{a}}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$,
$a=1,2$, $m_{a}\in\mathbb{R}$, then

${\rm d}_{U}\left.(S(U)U)\right|_{S_{1}(U;t)U}\,[S_{2}(U;t)U]$

is in
$\mathcal{S}^{m+\max(m_{1},m_{2})}_{K,K^{\prime},p+pq_{1}+q_{2}}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$.

###### Proof.

Proof of $(i)$. The operator $R(U;t)\circ S(U;t)$ is in
$\Sigma\mathcal{R}^{-\varrho+\max(m,0)}_{K,K^{\prime},p+p^{\prime\prime}}[r,N]$
by Proposition 2.15-($i$) since $S(U;t)$ is in
$\Sigma\mathcal{M}^{m}_{K,K^{\prime},p}[r,N]$. Then we prove that $S(U;t)\circ
R(U;t)$ is in
$\Sigma\mathcal{R}^{-\varrho+\max(m,0)}_{K,K^{\prime},p+p^{\prime\prime}}[r,N]$.
It is sufficient to consider the case $m\geq 0$ since, if $m<0$, we regard
$S(U;t)$ as a spectrally localized map in
$\Sigma\mathcal{S}^{0}_{K,K^{\prime},p}[r,N]$. Decomposing
$S=\sum_{q=p}^{N}S_{q}+S_{>N}$ as in (2.73) and
$R=\sum_{q=p^{\prime\prime}}^{N}R_{q}+R_{>N}$ as in (2.35), we have to show,
on the one hand that

$S_{q_{1}}(U_{1},\dots,U_{q_{1}})R_{q_{2}}(U_{q_{1}+1},\dots,U_{q_{1}+q_{2}})$
(2.78)

is a homogeneous smoothing operator in
${\widetilde{\mathcal{R}}}^{-\varrho+m}_{q_{1}+q_{2}}$ if $q_{1}+q_{2}\leq N$
and, on the other hand, that

$\displaystyle S_{q_{1}}(U,\dots,U)R_{q_{2}}(U,\dots,U),\ q_{1}+q_{2}\geq
N+1,$ (2.79) $\displaystyle S_{>N}(U;t)R_{q_{2}}(U,\dots,U),\
q_{2}=p^{\prime\prime},\dots,N,$ $\displaystyle
S_{q_{1}}(U,\dots,U)R_{>N}(U;t),\ q_{1}=p,\dots,N,$ $\displaystyle
S_{>N}(U;t)R_{>N}(U;t)$

are non–homogeneous smoothing operators in
$\mathcal{R}^{-\varrho+m}_{K,K^{\prime},N+1}[r]$. We first study (2.78). First
of all one has

$\displaystyle\|\Pi_{n_{0}}S_{q_{1}}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q_{1}}}U_{q_{1}})R_{q_{2}}(\Pi_{n_{q_{1}+1}}U_{q_{1}+1},\dots,\Pi_{n_{q_{1}+q_{2}}}U_{q_{1}+q_{2}})\Pi_{n_{q_{1}+q_{2}+1}}U_{q_{1}+q_{2}+1}\|_{L^{2}}$
(2.80)
$\displaystyle\leq\sum_{n^{\prime}}\|\Pi_{n_{0}}S_{q_{1}}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q_{1}}}U_{q_{1}})\Pi_{n^{\prime}}R_{q_{2}}(\Pi_{n_{q_{1}+1}}U_{q_{1}+1},\dots,\Pi_{n_{q_{1}+q_{2}}}U_{q_{1}+q_{2}})\Pi_{n_{q_{1}+q_{2}+1}}U_{q_{1}+q_{2}+1}\|_{L^{2}}.$

Thanks to the conditions (2.32), (2.71) for $S$ and (2.32) for $R$ the indices
in the above sum satisfy, for some choice of signs $\sigma_{a}$,
$a=1,\dots,q_{1}+q_{2}+1$, the restrictions

$n^{\prime}=\sum_{a=0}^{q_{1}}\sigma_{a}n_{a},\quad
n^{\prime}=\sum_{a=q_{1}+1}^{q_{1}+q_{2}+1}\sigma_{a}n_{a},\quad\max\\{n_{1},\dots,n_{q_{1}}\\}\lesssim
n^{\prime},\quad n_{0}\sim n^{\prime}\,.$ (2.81)

As a consequence (2.78) satisfies the corresponding condition (2.32) and

$n^{\prime}\lesssim\max\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}+1}\\},\quad\max\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}\sim\max\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}+1}\\}.$
(2.82)

We also claim that

$\max\\{n_{1},\dots,n_{q_{1}}\\}\lesssim{\rm
max}_{2}\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}.$ (2.83)

Indeed, by (2.81) and (2.82), we have

$\max\\{n_{1},\dots,n_{q_{1}}\\}\lesssim\max\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}+1}\\},$
(2.84)

and we distinguish two cases. If
$\max\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}=\max\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}+1}\\}$
then we directly obtain $\max\\{n_{1},\dots,n_{q_{1}}\\}\leq{\rm
max}_{2}\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}$ which gives (2.83). If
$\max\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}=\max\\{n_{1},\dots,n_{q_{1}}\\}$ then
$\max\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}+1}\\}\leq{\rm
max}_{2}\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}$ which together with (2.84) proves