(2.83). Using (2.80), (2.81), (2.74), (2.31) (with $m=-\varrho$ and
$\mu\leadsto\mu^{\prime}$) we get

$\displaystyle\|\Pi_{n_{0}}$ $\displaystyle
S_{q_{1}}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q_{1}}}U_{q_{1}})R_{q_{2}}(\Pi_{n_{q_{1}+1}}U_{q_{1}+1},\dots,\Pi_{n_{q_{1}+q_{2}}}U_{q_{1}+q_{2}})\Pi_{n_{q_{1}+q_{2}+1}}U_{q_{1}+q_{2}+1}\|_{L^{2}}$
$\displaystyle\leq C{\rm
max}\\{n_{1},\dots,n_{q_{1}}\\}^{\mu}(n^{\prime})^{m}\frac{\max_{2}\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}+1}\\}^{\mu^{\prime}}}{\max\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}+1}\\}^{\varrho}}\prod_{a=1}^{q_{1}+q_{2}+1}\|\Pi_{n_{a}}U_{a}\|_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{claimo},\eqref{denaco},m\geq
0}}{{\leq}}C\frac{\max_{2}\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}^{\mu^{\prime}+\mu}}{\max\\{n_{1},\dots,n_{q_{1}+q_{2}+1}\\}^{\varrho-m}}\prod_{a=1}^{q_{1}+q_{2}+1}\|\Pi_{n_{a}}U_{a}\|_{L^{2}}.$

This proves that (2.78) is in
${\widetilde{\mathcal{R}}}^{-\varrho+m}_{q_{1}+q_{2}}$.

In order to prove that the operators in (2.79) satisfy (2.34) (with
$m\leadsto-\varrho+m$ and $p\leadsto N$) we recall that $S_{q_{1}}(U,\dots,U)$
defines a spectrally localized map in $\mathcal{S}^{m}_{K,0,q_{1}}[r]$ (as
remarked below Definition 2.16) and $R_{q_{2}}(U,\dots,U)$ defines a smoothing
operator in $\mathcal{R}^{-\varrho}_{K,0,q_{2}}[r]$ thanks to Lemma 2.8. Then
the thesis follows by estimates (2.72) and (2.34). For instance consider the
last term in (2.79). For any $k=0,\ldots,K-K^{\prime}$

$\displaystyle\|{\partial_{t}^{k}\left(S_{>N}(U;t)R_{>N}(U;t)V\right)}\|_{\dot{H}^{s-\frac{3}{2}k-m+\varrho}}\leq
C\sum_{k^{\prime}+k^{\prime\prime}=k}\|U\|_{k^{\prime}+K^{\prime},s_{0}}^{N+1}\|R_{>N}(U;t)V\|_{k^{\prime\prime},s+\varrho}$
$\displaystyle\leq C\sum_{k^{\prime}+k^{\prime\prime}=k}\sum_{0\leq j\leq
k^{\prime\prime}\atop
j^{\prime}+j^{\prime\prime}=j}\|U\|_{k^{\prime}+K^{\prime},s_{0}}^{N+1}\,\left(\|{V}\|_{j^{\prime\prime},s}\|U\|_{j^{\prime}+K^{\prime},s_{0}}^{N+1}+\|{V}\|_{j^{\prime\prime},s_{0}}\|U\|_{j^{\prime}+K^{\prime},s_{0}}^{N}\|U\|_{j^{\prime}+K^{\prime},s}\right)$
$\displaystyle\leq
C\sum_{j^{\prime\prime}=0}^{k}\|{V}\|_{j^{\prime\prime},s}\|U\|_{k-j^{\prime\prime}+K^{\prime},s_{0}}^{2N+2}+\|{V}\|_{j^{\prime\prime},s_{0}}\|U\|_{k-j^{\prime\prime}+K^{\prime},s_{0}}^{2N+1}\|U\|_{k-j^{\prime\prime}+K^{\prime},s}$

using that $k^{\prime}\leq k-j^{\prime\prime}$ and $j^{\prime}\leq
k-j^{\prime\prime}$. This proves that (2.78) is in
$\mathcal{R}^{-\varrho+m}_{K,K^{\prime},N+1}[r]$ (as $\|U\|_{K,s_{0}}<1$).

Proof of $(ii)$.  Decomposing $S=\sum_{q=p}^{N}S_{q}+S_{>N}$ and
$S^{\prime}=\sum_{q=p^{\prime}}^{N}S^{\prime}_{q}+S^{\prime}_{>N}$ as in
(2.73), we have to show, on the one hand that

$S_{q_{1}}(U_{1},\dots,U_{q_{1}})S^{\prime}_{q_{2}}(U_{q_{1}+1},\dots,U_{q_{1}+q_{2}})$
(2.85)

is a homogeneous spectrally localized map in
${\widetilde{\mathcal{S}}}^{m+m^{\prime}}_{q_{1}+q_{2}}$ if $q_{1}+q_{2}\leq
N$ and, on the other hand, that

$\displaystyle S_{q_{1}}(U,\dots,U)S^{\prime}_{q_{2}}(U,\dots,U),\
q_{1}+q_{2}\geq N+1,$ (2.86) $\displaystyle
S_{>N}(U;t)S^{\prime}_{q_{2}}(U,\dots,U),\ q_{2}=p^{\prime},\dots,N,$
$\displaystyle S_{q_{1}}(U,\dots,U)S^{\prime}_{>N}(U;t),\ q_{1}=p,\dots,N,$
$\displaystyle S_{>N}(U;t)S^{\prime}_{>N}(U;t)$

are non–homogeneous spectrally localized map in
$\mathcal{S}^{m+m^{\prime}}_{K,K^{\prime},N+1}[r]$. We first prove that (2.85)
is in ${\widetilde{\mathcal{S}}}^{m+m^{\prime}}_{q_{1}+q_{2}}$. First of all
one has

$\displaystyle\|\Pi_{n_{0}}S_{q_{1}}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q_{1}}}U_{q_{1}})S^{\prime}_{q_{2}}(\Pi_{n_{q_{1}+1}}U_{q_{1}+1},\dots,\Pi_{n_{q_{1}+q_{2}}}U_{q_{1}+q_{2}})\Pi_{n_{q_{1}+q_{2}+1}}U_{q_{1}+q_{2}+1}\|_{L^{2}}$
(2.87)
$\displaystyle\leq\sum_{n^{\prime}}\|\Pi_{n_{0}}S_{q_{1}}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q_{1}}}U_{q_{1}})\Pi_{n^{\prime}}S^{\prime}_{q_{2}}(\Pi_{n_{q_{1}+1}}U_{q_{1}+1},\dots,\Pi_{n_{q_{1}+q_{2}}}U_{q_{1}+q_{2}})\Pi_{n_{q_{1}+q_{2}+1}}U_{q_{1}+q_{2}+1}\|_{L^{2}}.$

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

$\displaystyle n^{\prime}=\sum_{a=0}^{q_{1}}\sigma_{a}n_{a},\quad
n^{\prime}=\sum_{a=q_{1}+1}^{q_{1}+q_{2}+1}\sigma_{a}n_{a},$ (2.88)
$\displaystyle\begin{cases}\max\\{n_{1},\dots,n_{q_{1}}\\}\leq\delta
n^{\prime}\\\ \frac{n_{0}}{C}\leq n^{\prime}\leq
Cn_{0}\,,\end{cases}\quad\begin{cases}\max\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}}\\}\leq\delta^{\prime}n_{q_{1}+q_{2}+1}\\\
\frac{n^{\prime}}{C^{\prime}}\leq n_{q_{1}+q_{2}+1}\leq
C^{\prime}n^{\prime}\,,\end{cases}$

for some $C,C^{\prime}>1$ and $\delta,\delta^{\prime}>0$. Therefore

$\begin{cases}\max\\{n_{1},\dots,n_{q_{1}+q_{2}}\\}\leq\max\\{\delta^{\prime},\delta
C^{\prime}\\}n_{q_{1}+q_{2}+1}\\\ \frac{n_{0}}{CC^{\prime}}\leq
n_{q_{1}+q_{2}+1}\leq CC^{\prime}n_{0}\,.\end{cases}$

This proves that $S_{q_{1}}\circ S^{\prime}_{q_{2}}$ fulfills the localization
property of Definition 2.16 $(i)$ (see (2.71)). In addition (2.85) satisfies
the corresponding condition (2.32). Using (2.87), (2.74) we get

$\displaystyle\|\Pi_{n_{0}}S_{q_{1}}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q_{1}}}U_{q_{1}})S^{\prime}_{q_{2}}(\Pi_{n_{q_{1}+1}}U_{q_{1}+1},\dots,\Pi_{n_{q_{1}+q_{2}}}U_{q_{1}+q_{2}})\Pi_{n_{q_{1}+q_{2}+1}}U_{q_{1}+q_{2}+1}\|_{L^{2}}$
$\displaystyle\leq C{\rm
max}\\{n_{1},\dots,n_{q_{1}}\\}^{\mu}(n^{\prime})^{m}{\rm
max}\\{n_{q_{1}+1},\dots,n_{q_{1}+q_{2}}\\}^{\mu^{\prime}}n_{q_{1}+q_{2}+1}^{m^{\prime}}\prod_{a=1}^{q_{1}+q_{2}+1}\|\Pi_{n_{a}}U_{a}\|_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{denaco3}}}{{\leq}}C\max\\{n_{1},\dots,n_{q_{1}+q_{2}}\\}^{\mu^{\prime}+\mu}(n_{q_{1}+q_{2}+1})^{m+m^{\prime}}\prod_{a=1}^{q_{1}+q_{2}+1}\|\Pi_{n_{a}}U_{a}\|_{L^{2}}$

which proves that $S_{q_{1}}\circ S^{\prime}_{q_{2}}$ satisfies (2.74). In
order to prove that the terms in (2.79) satisfy (2.72) we first note that,
thanks to $(i)$ of Lemma 2.18, we have that
$S_{q_{1}}(U,\dots,U)\in\mathcal{S}^{m}_{K,0,q_{1}}[r]$ and
$S^{\prime}_{q_{2}}(U,\dots,U)\in\mathcal{S}^{m^{\prime}}_{K,0,q_{2}}[r]$ and
then the thesis follows using (2.72). For instance consider the first term in
(2.86). Using twice (2.72) (with $K^{\prime}=0$) we get, for any
$k=0,\ldots,K$,

$\displaystyle\|{\partial_{t}^{k}\left(S_{q_{1}}(U;t)S^{\prime}_{q_{2}}(U;t)V\right)}\|_{\dot{H}^{s-\frac{3}{2}k-m_{1}-m_{2}}}$
$\displaystyle\leq
C\sum_{k^{\prime}+k^{\prime\prime}=k}\|U\|_{k^{\prime},s_{0}}^{q_{1}}\|S_{q_{2}}^{\prime}(U;t)V\|_{k^{\prime\prime},s-m_{2}}$
$\displaystyle\leq C\sum_{k^{\prime}+k^{\prime\prime}=k}\sum_{0\leq j\leq
k^{\prime\prime}\atop
j^{\prime}+j^{\prime\prime}=j}\|U\|_{k^{\prime},s_{0}}^{q_{1}}\,\|U\|_{j^{\prime},s_{0}}^{q_{2}}\,\|{V}\|_{j^{\prime\prime},s}$
$\displaystyle\leq
C\sum_{j^{\prime\prime}=0}^{k}\|U\|_{k-j^{\prime\prime},s_{0}}^{q_{1}+q_{2}}\|{V}\|_{j^{\prime\prime},s}$

where in the last step we used that $j^{\prime}\leq k-j^{\prime\prime}$ and
$k^{\prime}\leq k-j^{\prime\prime}$. The last line is $\eqref{piovespect}$
with $p$ replaced by $q_{1}+q_{2}\geq N+1$.

Proof of $(iii)$: We now consider the internal composition (2.77) in the
homogeneous case. For simplicity of notation we consider the case
$q_{2}=\dots=q_{p}=0$, $S^{(2)}=\dots=S^{(p)}={\rm Id}$ and
$q_{1}=\overline{q}=:q$, the general case follows in the same way. So we need
to show that $S(S^{(1)}(U)U,U,\ldots,U)$ is a spectrally localized map in
$\widetilde{\mathcal{S}}^{m}_{p+q}$. We first estimate

$\displaystyle\|\Pi_{n_{0}}S\big{(}S^{(1)}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q}}U_{q})\Pi_{n_{q+1}}U_{q+1},\Pi_{n_{q+2}}U_{q+2},\dots,\Pi_{n_{q+p}}U_{q+p}\big{)}\Pi_{n_{p+q+1}}U_{p+q+1}\|_{L^{2}}$
(2.89)
$\displaystyle\leq\sum_{n^{\prime}}\|\Pi_{n_{0}}S\big{(}\Pi_{n^{\prime}}S^{(1)}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q}}U_{q})\Pi_{n_{q+1}}U_{q+1},\Pi_{n_{q+2}}U_{q+2},\dots,\Pi_{n_{q+p}}U_{q+p}\big{)}\Pi_{n_{p+q+1}}U_{p+q+1}\|_{L^{2}}.$

Thanks to the conditions (2.32) for $S$ and $S^{(1)}$ the indices in the above
sum satisfy, for some choice of signs $\sigma_{a}$, $a=1,\dots,p+q+1$, the
restrictions $n^{\prime}=\sum_{a=1}^{q+1}\sigma_{a}n_{a}$ and
$n^{\prime}=\sigma_{0}n_{0}+\sum_{a=q+2}^{p+q+1}\sigma_{a}n_{a}$, proving that
$S(S^{(1)}(U)U,U,\ldots,U)$ fulfills (2.32). Moreover the condition (2.71) for
$S$ and $S^{(1)}$ imply the existence of $\delta,\delta_{1}>0$, $C,C_{1}>1$
such that

$\begin{cases}\max\\{n^{\prime},n_{q+2},\dots,n_{q+p}\\}\leq\delta
n_{q+p+1}\\\ \frac{n_{0}}{C}\leq n_{p+q+1}\leq
Cn_{0}\,,\end{cases}\quad\begin{cases}\max\\{n_{1},\dots,n_{q}\\}\leq\delta_{1}n_{q+1}\\\
\frac{n^{\prime}}{C_{1}}\leq n_{q+1}\leq C_{1}n^{\prime}\,,\end{cases}$ (2.90)

and therefore
$\max\\{n_{1},\dots,n_{p+q}\\}\leq\max(\delta\delta_{1}C_{1},\delta
C_{1})n_{q+p+1}$ and $\frac{n_{0}}{C}\leq n_{p+q+1}\leq Cn_{0}$, proving that
$S(S^{(1)}(U)U,U,\ldots,U)$ fulfills (2.71). We now prove it fulfills also
(2.74). We get

$\displaystyle\|\Pi_{n_{0}}S(S^{(1)}(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q}}U_{q})\Pi_{n_{q+1}}U_{q+1},\Pi_{n_{q+2}}U_{q+2},\dots,\Pi_{n_{q+p}}U_{q+p})\Pi_{n_{p+q+1}}U_{p+q+1}\|_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{triangolo4},\eqref{hom:sest}}}{{\lesssim}}\sum_{n^{\prime}}\max(n^{\prime},n_{q+2},\ldots,n_{q+p})^{\mu}\,n_{p+q+1}^{m}\,\max(n_{1},\ldots,n_{q})^{\mu^{\prime}}\,n_{q+1}^{\ell_{1}}\,\prod_{a=1}^{p+q+1}\|\Pi_{n_{a}}U_{a}\|_{L^{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{doppiacondizione},n_{q+1}\sim
n^{\prime}}}{{\lesssim}}\sum_{n^{\prime}}\max(n_{q+1},n_{q+2},\ldots,n_{q+p})^{\mu+\max(0,\ell_{1})}\,n_{p+q+1}^{m}\,\max(n_{1},\ldots,n_{q})^{\mu^{\prime}}\,\prod_{a=1}^{p+q+1}\|\Pi_{n_{a}}U_{a}\|_{L^{2}}$
$\displaystyle\lesssim\max(n_{1},\ldots,n_{p+q})^{\mu+\mu^{\prime}+\max(0,\ell_{1})}\,n_{p+q+1}^{m}\,\prod_{a=1}^{p+q+1}\|\Pi_{n_{a}}U_{a}\|_{L^{2}}\
,$

proving that $S(S^{(1)}(U)U,U,\ldots,U)$ is a spectrally localized map in
$\widetilde{\mathcal{S}}^{m}_{p+q}$. Finally $S(S^{(1)}(U)U,U,\ldots,U)$
satisfies also (2.33), concluding the proof that it is a spectrally localized
map in $\widetilde{\mathcal{S}}^{m}_{p+q}$.

Proof of $(iv)$: By the estimate below (2.34) for ${\bf M}_{0}(U;t)$, for any
$U\in B_{s_{0}}^{K}(I;r)$ and any $k=0,\dots,K-K^{\prime}$, $\|{\bf
M}_{0}(U;t)U\|_{k,s_{0}}\lesssim\|U\|_{k,s_{0}}$. Then estimate (2.72) for
$S({\bf M}_{0}(U;t)U;t)$ for any $0\leq k\leq K-K^{\prime}$ follows from the
ones for $S(U;t)$ arguing as in $(iii)$ of Proposition 2.15.

Proof of $(v)$: It follows computing explicitly the differential ${\rm
d}_{U}(S(U)U)[V]$, evaluating it at $U\leadsto S_{1}(U;t)U$ and $V\leadsto
S_{2}(U;t)U$ and using item $(ii)$ and $(iii)$ of the proposition. ∎

The following lemma proves that the internal composition of a spectrally
localized map with a map, is a spectrally localized map plus a smoothing
operator whose transpose is another smoothing operator.

######  Lemma 2.20.

Let $p\in\mathbb{N}$, $q\in\mathbb{N}_{0}$, $m\in\mathbb{R}$ and
$m^{\prime}\geq 0$. Let $S(U)$ be a matrix of spectrally localized homogeneous
maps in $\widetilde{\mathcal{S}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$
and $M(U)$ be a matrix of homogeneous $m^{\prime}$-operators in
$\widetilde{\mathcal{M}}_{q}^{m^{\prime}}\otimes\mathcal{M}_{2}(\mathbb{C})$.
Then

$S(M(U)U,U,\dots,U)=S^{\prime}(U)+R(U)$ (2.91)

where

  * •

$S^{\prime}(U)$ is a matrix of spectrally localized homogeneous maps in
$\widetilde{\mathcal{S}}_{p+q}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$;

  * •

$R(U)$ is a matrix of homogeneous smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p+q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$, and $R(U)^{\top}$ is a matrix of homogeneous
smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p+q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$ as well.

###### Proof.

By multilinearity we expand

$\displaystyle
S\big{(}M(U_{1},\dots,U_{q})U_{q+1},U_{q+2},\dots,U_{p+q}\big{)}$ (2.92)
$\displaystyle=\\!\\!\\!\\!\\!\\!\sum_{(n^{\prime},\,n_{0},\,\dots,n_{p+q+1})\in\mathcal{N}}\\!\\!\\!\\!\\!\\!\Pi_{n_{0}}S\big{(}\Pi_{n^{\prime}}M(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q}}U_{q})\Pi_{n_{q+1}}U_{q+1},\Pi_{n_{q+2}}U_{q+2},\dots,\Pi_{n_{p+q}}U_{p+q}\big{)}\Pi_{n_{p+q+1}}$

where $\mathcal{N}$ is a subset of $\mathbb{N}^{p+q+3}$ made by indexes
fulfilling, by restrictions (2.71) and (2.32), the following conditions: there
exist signs $\\{\sigma_{j},\epsilon_{j},\epsilon^{\prime}\\}\subset\\{\pm\\}$
and constants $\delta>0$, $C>1$ such that

$\begin{cases}\max(n^{\prime},n_{q+2},\dots,n_{p+q})\leq\delta
n_{p+q+1}&(a)\\\ \quad C^{-1}n_{0}\leq n_{p+q+1}\leq Cn_{0}&(b)\\\
n^{\prime}=\sum_{j=1}^{q+1}\sigma_{j}n_{j},\quad
n_{0}=\epsilon^{\prime}n^{\prime}+\sum_{j=q+2}^{p+q+1}\epsilon_{j}n_{j}\,&(c)\
.\end{cases}$ (2.93)

We fix $\delta^{\prime}>\delta$ and we denote by $\mathcal{N}^{\prime}$ the
subset of $\mathcal{N}$ made by indices which satisfy the additional
restriction

$\max(n_{1},\dots,n_{q+1})\leq\delta^{\prime}n_{p+q+1}\,.$ (2.94)

Then we define

$\displaystyle S^{\prime}(U_{1},\dots,U_{p+q})$ (2.95)
$\displaystyle:=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(n^{\prime},\,n_{0},\,\dots,n_{p+q+1})\in\mathcal{N}^{\prime}}\\!\\!\\!\\!\\!\\!\\!\\!\Pi_{n_{0}}S\big{(}\Pi_{n^{\prime}}M(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q}}U_{q})\Pi_{n_{q+1}}U_{q+1},\Pi_{n_{q+2}}U_{q+2},\dots,\Pi_{n_{p+q}}U_{p+q}\big{)}\Pi_{n_{p+q+1}}.$

By (2.93) and (2.94) one has that $S^{\prime}(U)$ fulfills the spectral
condition $n_{0}\sim n_{p+q+1}$,
$\max(n_{1},\ldots,n_{p+q})\leq\delta^{\prime}n_{p+q+1}$ which is the
condition (2.71). Moreover using (2.74) and (2.31) we bound

$\displaystyle\Big{\|}\Pi_{n_{0}}S\big{(}\Pi_{n^{\prime}}M(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q}}U_{q})\Pi_{n_{q+1}}U_{q+1},\Pi_{n_{q+2}}U_{q+2},\dots,\Pi_{n_{p+q}}U_{p+q}\big{)}\Pi_{n_{p+q+1}}U_{p+q+q}\Big{\|}_{L^{2}}$
$\displaystyle\lesssim\max(n_{1},\ldots,n_{p+q})^{\mu+m^{\prime}}\,n_{p+q+1}^{m}\,\prod_{a=1}^{p+q+1}{\|\Pi_{n_{a}}U_{a}\|}_{L^{2}}$
(2.96)

for some $\mu>0$. Finally by (2.93)$(c)$ one has also that (2.32) holds. One
checks that also (2.33) holds true. We have proved that $S^{\prime}(U)$ is a
matrix of spectrally localized maps in
$\widetilde{\mathcal{S}}^{m}_{p+q}\otimes\mathcal{M}_{2}(\mathbb{C})$.

Then, recalling (2.92) and (2.95), we define

$\displaystyle R(U_{1},\dots,U_{p+q})$ (2.97)
$\displaystyle:=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(n^{\prime},\,n_{0},\,\dots,n_{p+q+1})\in\mathcal{N}\setminus\mathcal{N}^{\prime}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Pi_{n_{0}}S\big{(}\Pi_{n^{\prime}}M(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{q}}U_{q})\Pi_{n_{q+1}}U_{q+1},\Pi_{n_{q+2}}U_{q+2},\dots,\Pi_{n_{p+q}}U_{p+q}\big{)}\Pi_{n_{p+q+1}}.$

We claim that there is $C^{\prime}>0$ such that if
$(n^{\prime},\,n_{0},\,\dots,n_{p+q+1})\in\mathcal{N}\setminus\mathcal{N}^{\prime}$
then

$\begin{cases}\max(n_{q+2},\dots,n_{p+q})\leq\delta^{\prime}n_{p+q+1}&(a)\\\
C^{-1}n_{0}\leq n_{p+q+1}\leq Cn_{0}&(b)\\\ n_{p+q+1}\leq
C^{\prime}\max_{2}(n_{1},\dots,n_{q+1})&(c)\\\ \max(n_{1},\dots,n_{q+1})\leq
C^{\prime}\max_{2}(n_{1},\dots,n_{q+1})&(d)\,.\end{cases}$ (2.98)

Before proving (2.98) we note that it implies

$\displaystyle\max(n_{1},\dots,n_{p+q+1})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}(1+\delta^{\prime})\max(n_{1},\dots,n_{q+1},n_{p+q+1})$
$\displaystyle\stackrel{{\scriptstyle(c)+(d)}}{{\leq}}(1+\delta^{\prime})C^{\prime}{\rm
max}_{2}(n_{1},\dots,n_{q+1})\leq(1+\delta^{\prime})C^{\prime}{\rm
max}_{2}(n_{1},\dots,n_{p+q+1})$

proving that $\max(n_{1},\dots,n_{p+q+1})\sim{\rm
max}_{2}(n_{1},\dots,n_{p+q+1})$. Then by (2.97) and (2.96) we obtain

$\displaystyle{\|\Pi_{n_{0}}R(\Pi_{n_{1}}U_{1},\ldots,\Pi_{n_{p+q}}U_{p+q})\Pi_{n_{p+q+1}}U_{p+q+1}\|}_{L^{2}}\lesssim\max(n_{1},\ldots,n_{p+q+1})^{\widetilde{\mu}}\,\prod_{a=1}^{p+q+1}{\|\Pi_{n_{a}}U_{a}\|}_{L^{2}}$
$\displaystyle\lesssim\frac{\max_{2}(n_{1},\ldots,n_{p+q+1})^{\widetilde{\mu}+\varrho}}{\max(n_{1},\ldots,n_{p+q+1})^{\varrho}}\prod_{a=1}^{p+q+1}{\|\Pi_{n_{a}}U_{a}\|}_{L^{2}}\qquad\text{with}\qquad\widetilde{\mu}=\mu+m^{\prime}+\max(m,0)\,,$

showing that $R(U)$ is a $(p+q)$–homogeneous smoothing operator in
${\widetilde{\mathcal{R}}}^{-\varrho}_{p+q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$.

We now prove (2.98). Note that $(a)$ and $(b)$ of (2.98) follow by $(a)$ and
$(b)$ of (2.93) and $\delta^{\prime}>\delta$. Then note that if
$(n^{\prime},\,n_{0},\,\dots,n_{p+q+1})\in\mathcal{N}\setminus\mathcal{N}^{\prime}$
then

$\max(n_{1},\dots,n_{q+1})>\delta^{\prime}n_{p+q+1}\,.$ (2.99)

Then, by $(c)$ of (2.93), one has

$\max(n_{1},\ldots,n_{q+1})=\epsilon^{\prime}n^{\prime}+\sum_{n_{j}\leq{\rm\max}_{2}(n_{1},\ldots,n_{q+1})}\epsilon_{j}n_{j}\leq
n^{\prime}+q{\rm\max}_{2}(n_{1},\ldots,n_{q+1})$ (2.100)

so that ${\rm
max}_{2}(n_{1},\ldots,n_{q+1})\geq\frac{1}{q}\big{(}\max(n_{1},\ldots,n_{q+1})-n^{\prime}\big{)}$.
We deduce using (2.99) and (2.93)$(a)$ that

${\rm\max}_{2}(n_{1},\ldots,n_{q+1})\geq\frac{\delta^{\prime}-\delta}{q}\
n_{p+q+1}$ (2.101)

thus proving (2.98) $(c)$. Finally using (2.100), (2.93) (a) and (2.101) we
get

$\max(n_{1},\dots,n_{q+1})\leq
q\big{(}\frac{\delta}{\delta^{\prime}-\delta}+1\big{)}{\rm
max}_{2}(n_{1},\dots,n_{q+1})$

which proves $(d)$ of (2.98).

We finally prove that $R(U)^{\top}$ is a smoothing operator. First note that
if $\Pi_{n_{0}}R(\Pi_{\vec{n}}\mathcal{U})^{\top}\Pi_{n_{p+q+1}}\neq 0$ then
$\Pi_{n_{p+q+1}}R(\Pi_{\vec{n}}\mathcal{U})\Pi_{n_{0}}\neq 0$, which implies
that $C^{-1}n_{p+q+1}\leq n_{0}\leq Cn_{p+q+1}$ by (2.98) $(b)$. Then
$R(U)^{\top}$ is a smoothing operator in
${\widetilde{\mathcal{R}}}^{-\varrho}_{p+q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$ by Lemma 2.10 and Remark 2.11. ∎

We finally prove the following lemma which generalizes a result in [45] for
paradifferential operators: the transpose of the internal differential of a
spectrally localized map is a smoothing operator (with two equivalent
frequencies).

######  Lemma 2.21.

Let $p\in\mathbb{N}$ and $m\in\mathbb{R}$. Given a matrix of spectrally
localized $p$–homogeneous maps
$S(U)\in{\widetilde{\mathcal{S}}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$,
consider

$V\mapsto L(U)V:=\frac{1}{p}{\rm d}_{U}S(U)[V]U=S(V,U,\dots,U)U\,\,.$ (2.102)

Then the transposed map $L(U)^{\top}$ is a matrix of smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ for
any $\varrho\geq 0$.

###### Proof.

We first note that the matrix entries $L(U)^{\top}$ defined as in (2.43) are

$\displaystyle[L^{\top}]^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}_{\vec{\jmath}_{p},j,k}\stackrel{{\scriptstyle\eqref{emmetop}}}{{=}}$
$\displaystyle
L^{\vec{\sigma}_{p},\sigma,\sigma^{\prime}}_{\vec{\jmath}_{p},-k,-j}$
$\displaystyle\stackrel{{\scriptstyle\eqref{M.coeff}}}{{=}}$
$\displaystyle\int_{\mathbb{T}}L\Big{(}\mathtt{q}^{\sigma_{1}}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}},\dots,\mathtt{q}^{\sigma_{p}}\frac{e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\frac{\mathtt{q}^{\sigma}e^{-{\rm
i}\sigma
kx}}{\sqrt{2\pi}}\Big{]}\cdot\mathtt{q}^{\sigma^{\prime}}\frac{e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\,{\rm d}x$
$\displaystyle\stackrel{{\scriptstyle\eqref{effonetra}}}{{=}}$
$\displaystyle\int_{\mathbb{T}}S\Big{(}\mathtt{q}^{\sigma}\frac{e^{-{\rm
i}\sigma kx}}{\sqrt{2\pi}},\mathtt{q}^{\sigma_{1}}\frac{e^{{\rm
i}\sigma_{1}j_{1}x}}{\sqrt{2\pi}}\dots,\mathtt{q}^{\sigma_{p-1}}\frac{e^{{\rm
i}\sigma_{p-1}j_{p-1}x}}{\sqrt{2\pi}}\Big{)}\Big{[}\frac{\mathtt{q}^{\sigma_{p}}e^{{\rm
i}\sigma_{p}j_{p}x}}{\sqrt{2\pi}}\Big{]}\cdot\mathtt{q}^{\sigma^{\prime}}\frac{e^{{\rm
i}\sigma^{\prime}jx}}{\sqrt{2\pi}}\,{\rm d}x$
$\displaystyle\stackrel{{\scriptstyle\eqref{M.coeff}}}{{=}}$ $\displaystyle
S^{\sigma,\sigma_{1},\dots,\sigma_{p-1},\sigma_{p},\sigma^{\prime}}_{-k,j_{1},\dots,j_{p-1},j_{p},-j}\,$
(2.103)

which, by (2.76), are different from zero only if

${\rm max}\\{|k|,|j_{1}|,\dots,|j_{p-1}|\\}\leq\delta|j_{p}|\,,\quad
C^{-1}|j|\leq|j_{p}|\leq C|j|\,.$ (2.104)

The restriction (2.104) implies that

$\max(|j_{1}|,\dots,|j_{p}|,|j|)\sim{\rm max}_{2}(|j_{1}|,\dots,|j_{p}|,|j|)$
(2.105)

because

$\displaystyle\max(|j_{1}|,\dots,|j_{p}|,|j|)$
$\displaystyle\leq(1+\delta)\max(|j_{p}|,|j|)$ $\displaystyle\leq
C(1+\delta){\rm max}_{2}(|j_{p}|,|j|)\leq C(1+\delta){\rm
max}_{2}(|j_{1}|,\dots,|j_{p}|,|j|)\,.$

Finally by (2.103), we estimate

$\displaystyle\big{|}[L^{\top}]^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}_{\vec{\jmath}_{p},j,k}\big{|}=\big{|}S^{\sigma,\sigma_{1},\dots,\sigma_{p-1},\sigma_{p},\sigma^{\prime}}_{-k,j_{1},\dots,j_{p-1},j_{p},-j}\big{|}$
$\displaystyle\stackrel{{\scriptstyle\eqref{smoocaraspec}}}{{\leq}}C{\rm
max}\\{|k|,|j_{1}|,\dots,|j_{p-1}|\\}^{\mu}|j_{p}|^{m}$
$\displaystyle\stackrel{{\scriptstyle\eqref{altrospec}}}{{\leq}}C^{\prime}{\rm
max}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu+\max(m,0)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{segunda}}}{{\leq}}C^{\prime\prime}\frac{{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu+\max(m,0)+\varrho}}{{\rm
max}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\varrho}}$

implying, in view of Lemma 2.9 (with $m\leadsto-\varrho$ and
$\mu\leadsto\mu+\max(m,0)+\varrho$), that $L(U)^{\top}$ is a matrix of
smoothing operators in
${\widetilde{\mathcal{R}}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$. ∎

We conclude this section with a lemma which shall be used in Section 3.4.

######  Lemma 2.22.

Let $p\in\mathbb{N}$ and $\varrho\geq 0$. Let $S(U)$ be a matrix of
$p$-homogeneous spectrally localized maps in
${\widetilde{\mathcal{S}}}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ of the form

$S(U)=L(U)+R(U)\,,$ (2.106)

where $L(U)^{\top}$ and $R(U)$ are matrices of $p$-homogeneous smoothing
operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$.
Then $S(U)$ is a matrix of $p$–homogeneous smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$.

###### Proof.

In view of (2.47) and (2.48) (for
$L(U)^{\top}\in\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$),

$|L_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|=|(L^{\top})_{\vec{\jmath}_{p},-k,-j}^{\vec{\sigma}_{p},\sigma,\sigma^{\prime}}|\leq
C{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{\mu}\max\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{-\varrho}\
.$ (2.107)

Since $S(U)$ is spectrally localized, its non zero Fourier coefficients
$S_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$ (defined
as in (2.43)) satisfy $|j|\sim|k|$ (see (2.76)) and, in view of (2.106),

$\displaystyle|S_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|\leq|L_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|+|R_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|$
$\displaystyle\stackrel{{\scriptstyle\eqref{L.t.s},\eqref{smoocara}}}{{\lesssim}}\frac{{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{\mu}}{\max\\{|j_{1}|,\dots,|j_{p}|,|k|\\}^{\varrho}}+\frac{{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu}}{\max\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\varrho}}$
$\displaystyle\stackrel{{\scriptstyle|j|\sim|k|}}{{\lesssim}}\frac{{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\mu}}{\max\\{|j_{1}|,\dots,|j_{p}|,|j|\\}^{\varrho}}$

proving that $S(U)$ is a smoothing operator in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$. ∎

####  2.3 Approximate inverse of non–linear maps and flows

In this section we construct an approximate version of two fundamental
non–linear operations that we will need: the inverse of a non–linear map and
the flow generated by a non–linear vector field. We first provide the
definition of an approximate inverse of a map, up to homogeneity $N$.

######  Definition 2.23.

(Approximate inverse up to homogeneity $N$) Let $p,N\in\mathbb{N}$ with $p\leq
N$. Consider

$\Psi_{\leq N}(U)=U+M_{\leq N}(U)U\quad\text{where}\quad M_{\leq
N}(U)\in\Sigma_{p}^{N}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
(2.108)

is a matrix of pluri–homogeneous operators. We say that

$\Phi_{\leq N}(V)=V+\breve{M}_{\leq
N}(V)V\quad\text{where}\quad\breve{M}_{\leq
N}(V)\in\Sigma_{p}^{N}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,,$
(2.109)

is an approximate inverse of $\Psi_{\leq N}(U)$ up to homogeneity $N$ if

$\displaystyle\big{(}{\rm Id}+M_{\leq N}(\Phi_{{\leq N}}(V))\big{)}\big{(}{\rm
Id}+\breve{M}_{\leq N}(V)\big{)}={\rm Id}+M^{\prime}_{>N}(V)$ (2.110)
$\displaystyle\big{(}{\rm Id}+\breve{M}_{\leq N}(\Psi_{{\leq
N}}(U))\big{)}\big{(}{\rm Id}+M_{\leq N}(U)\big{)}={\rm
Id}+M^{\prime\prime}_{>N}(U)$

where $M^{\prime}_{>N}(V)$ and $M^{\prime\prime}_{>N}(U)$ are
pluri–homogeneous matrices of operators in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.

Note that, if $\Phi_{\leq N}(V)$ is an approximate inverse up to homogeneity
$N$ of $\Psi_{\leq N}(U)$ then

$\Psi_{\leq N}\circ\Phi_{{\leq N}}(V)=V+M^{\prime}_{>N}(V)V\,,\quad\Phi_{\leq
N}\circ\Psi_{{\leq N}}(U)=U+M^{\prime\prime}_{>N}(U)U\,,$ (2.111)

and, by differentiation and taking the transpose

$\displaystyle{\rm d}_{U}\Psi_{\leq N}(\Phi_{{\leq N}}(V)){\rm
d}_{V}\Phi_{{\leq N}}(V)={\rm Id}+M^{1}_{>N}(V)$ (2.112) $\displaystyle{\rm
d}_{V}\Phi_{{\leq N}}(\Psi_{\leq N}(U)){\rm d}_{U}\Psi_{\leq N}(U)={\rm
Id}+M^{2}_{>N}(U)\,$ $\displaystyle{\rm d}_{V}\Phi_{\leq N}(V)^{\top}\,{\rm
d}_{U}\Psi_{\leq N}(\Phi_{\leq N}(V))^{\top}={\rm Id}+M^{3}_{>N}(V)$
$\displaystyle{\rm d}_{U}\Psi_{\leq N}(\Phi_{\leq N}(V))^{\top}\,{\rm
d}_{V}\Phi_{\leq N}(V)^{\top}={\rm Id}+M^{4}_{>N}(V)$

where $M^{a}_{>N}(V)$, $a=1,\ldots,4$ are other pluri-homogeneous operators in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
(the differential of a homogeneous $m$-operator is a homogeneous $m$-operator
by the first remark after Definition 2.5, so is its transpose by Lemma 2.10).

The following lemma ensures the existence of an approximate inverse.

######  Lemma 2.24.

(Approximate inverse) Let $p,N\in\mathbb{N}$ with $p\leq N$. Consider
$\Psi_{\leq N}(U)=U+M_{\leq N}(U)U$ as in (2.108). Then there exists an
approximate inverse of $\Psi_{\leq N}(U)$ up to homogeneity $N$ (according to
Definition 2.23) of the form (2.109) with

$\mathcal{P}_{p}[\breve{M}_{\leq N}(V)]=-\mathcal{P}_{p}[M_{\leq N}(V)]\,.$
(2.113)

Moreover, if $M_{\leq N}(U)$ in (2.108) is a matrix of pluri–homogeneous

  * (i)

spectrally localized maps in
$\Sigma_{p}^{N}{\widetilde{\mathcal{S}}}_{q}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$,
$m\geq 0$, then $\breve{M}_{\leq N}$ in (2.109) is a matrix of pluri-
homogeneous spectrally localized maps in
$\Sigma_{p}^{N}{\widetilde{\mathcal{S}}}_{q}^{m(N-p+1)}\otimes\mathcal{M}_{2}(\mathbb{C})$
and $M^{\prime}_{>N}$, $M^{\prime\prime}_{>N}$ in (2.110) belong to
$\Sigma_{N+1}{\widetilde{\mathcal{S}}}_{q}^{m(N-p+2)}\otimes\mathcal{M}_{2}(\mathbb{C})$;

  * (ii)

smoothing operators in
$\Sigma_{p}^{N}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for some $\varrho\geq 0$, then $\breve{M}_{\leq N}$ in (2.109) is a matrix of
pluri-homogeneous smoothing operators in
$\Sigma_{p}^{N}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
and $M^{\prime}_{>N}$, $M^{\prime\prime}_{>N}$ in (2.110) belong to
$\Sigma_{N+1}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.

###### Proof.

We expand in homogeneous components

$M(U):={\rm Id}+M_{\leq N}(U)={\rm
Id}+\sum_{q=p}^{N}M_{q}(U)\quad\text{with}\quad
M_{q}(U)\in{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,.$
(2.114)

In order to solve the first equation in (2.110) we look for a
pluri–homogeneous operator

$\breve{M}(V)={\rm Id}+\breve{M}_{\leq
N}(V)=\breve{M}_{0}(V)+\sum_{a=p}^{N}\breve{M}_{a}(V)\quad\text{with}\quad\breve{M}_{0}(V):={\rm
Id}\,,\
\breve{M}_{a}(V)\in{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,,$
(2.115)

such that

$M\big{(}\breve{M}(V)V\big{)}\breve{M}(V)={\rm
Id}+M_{>N}(V)\quad\text{with}\quad
M_{>N}(V)\in\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,.$
(2.116)

By (2.114), (2.115) we get

$\displaystyle M\big{(}\breve{M}(V)V\big{)}\breve{M}(V)$ $\displaystyle={\rm
Id}+\sum_{\ell=p}^{N}\breve{M}_{\ell}(V)+\sum_{q=p}^{N}\sum_{0\leq a_{1},\dots
a_{q+1}\leq
N}M_{q}(\breve{M}_{a_{1}}(V)V,\dots,\breve{M}_{a_{q}}(V)V)\breve{M}_{a_{q+1}}(V)$
$\displaystyle={\rm
Id}+\sum_{\ell=p}^{N}\Big{[}\breve{M}_{\ell}(V)+\sum_{\begin{subarray}{c}p\leq
q\leq\ell\\\
q+a_{1}+\dots+a_{q+1}=\ell\end{subarray}}M_{q}(\breve{M}_{a_{1}}(V)V,\dots,\breve{M}_{a_{q}}(V)V)\breve{M}_{a_{q+1}}(V)\Big{]}+M_{>N}(V)$

and therefore equation (2.116) is recursively solved by defining, for any
$\ell=p,\dots,N,$

$\breve{M}_{\ell}(V,\dots,V):=-\sum_{q+a_{1}+\dots+a_{q+1}=\ell}M_{q}(\breve{M}_{a_{1}}(V)V,\dots,\breve{M}_{a_{q}}(V)V)\breve{M}_{a_{q+1}}(V)\,.$
(2.117)

Note that each $\breve{M}_{\ell}(V,\dots,V)$ is a matrix of homogeneous
operators by $(ii)$ of Proposition 2.15. We proved the first identity in
(2.110).

We now prove the second identity in (2.110). Using the same recursive
procedure we find a matrix of pluri–homogeneous operators of the form

$M^{\prime\prime}(U)={\rm
Id}+\sum_{n=p}^{N}M^{\prime\prime}_{n}(U)\quad\text{with}\quad
M^{\prime\prime}_{n}(U)\in{\widetilde{\mathcal{M}}}_{n}\otimes\mathcal{M}_{2}(\mathbb{C})$
(2.118)

such that $M^{\prime\prime}(U)$ is an approximate right inverse of
$\breve{M}(V)$, i.e.

$\breve{M}\big{(}M^{\prime\prime}(U)U\big{)}M^{\prime\prime}(U)={\rm
Id}+M_{>N}(U)\quad\text{with}\quad
M_{>N}(U)\in\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,.$
(2.119)

Applying (2.116) with $V=M^{\prime\prime}(U)U$ and right–composing it with
$M^{\prime\prime}(U)$ defined in (2.118), we obtain by (2.119) and Proposition
2.15 that

$M\big{(}U+M_{>N}(U)U\big{)}=M^{\prime\prime}(U)+M^{\prime}_{>N}(U)$ (2.120)

where $M_{>N}(U)$ and $M^{\prime}_{>N}(U)$ are operators in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.
Then, expanding the left hand side of (2.120) by multilinearity, we get

$M\big{(}U\big{)}-M^{\prime\prime}(U)=\sum_{q=p}^{N}\sum_{\ell=1}^{q}\binom{q}{\ell}M_{q}(\underbrace{M_{>N}(U)U,\dots,M_{>N}(U)U}_{\ell-
times},U,\dots,U)+M^{\prime}_{>N}(U)=\tilde{M}_{>N}(U)$

where $\tilde{M}_{>N}(U)$ is in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
thanks to $(ii)$ of Proposition 2.15. This implies, since both $M(U)$ and
$M^{\prime\prime}(U)$ are pluri–homogeneous operators up to homogeneity $N$
(cfr. (2.114), (2.118)), that $M\big{(}U\big{)}=M^{\prime\prime}(U)$ and, by
(2.119), we conclude that

$\breve{M}\big{(}M(U)U\big{)}M(U)={\rm Id}+M_{>N}(U)\,.$ (2.121)

This proves, recalling the notation (2.114), (2.115), (2.108), the second
identity in (2.110). Moreover for $\ell=p$ the sum in (2.117) reduces to the
unique element with $q=p$, $a_{1}=\dots=a_{q+1}=0$ and
$\breve{M}_{p}(V)=-M_{p}(V)$, proving (2.113).

$(i)$ If $M_{\leq N}(U)$ is a spectrally localized map in
$\Sigma_{p}^{N}{\widetilde{\mathcal{S}}}^{m}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
we claim that $\breve{M}_{\ell}(V)$ is a spectrally localized map in
${\widetilde{\mathcal{S}}}^{m+m(\ell-p)}_{\ell}\otimes\mathcal{M}_{2}(\mathbb{C})$
for $\ell=p,\dots,N$. For $\ell=p$ by (2.113) we have
$\breve{M}_{p}(V)=-M_{p}(V)$ which is in
${\widetilde{\mathcal{S}}}^{m}_{\ell}\otimes\mathcal{M}_{2}(\mathbb{C})$. Then
supposing inductively that $\breve{M}_{a}(V)$ is in
${\widetilde{\mathcal{S}}}^{m+m(a-p)}_{a}\otimes\mathcal{M}_{2}(\mathbb{C})$
for $a=p,\dots\ell-1$, we deduce by $(ii)$ and $(iii)$ of Proposition 2.19,
that each term in the sum in (2.117) is a spectrally localized map in
${\widetilde{\mathcal{S}}}^{m+(m+m(a_{q+1}-p))}_{\ell}\otimes\mathcal{M}_{2}(\mathbb{C})$
which is included in
${\widetilde{\mathcal{S}}}^{m+m(\ell-p)}_{\ell}\otimes\mathcal{M}_{2}(\mathbb{C})$
using that $a_{q+1}\leq\ell-1$.

$(ii)$ In the same way, if $M_{\leq N}(U)$ is a smoothing operator in
$\Sigma_{p}^{N}{\widetilde{\mathcal{R}}}_{q}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$,
thanks to $(ii)$ of Proposition 2.15 one proves recursively that
$\breve{M}_{\ell}(V)$ are smoothing operators in
${\widetilde{\mathcal{R}}}_{\ell}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$.
∎

We now define the approximate flow of a smoothing $\tau$–dependent vector
field.

######  Definition 2.25.

(Approximate flow of a smoothing vector field up to homogeneity $N$) Let
$p,N\in\mathbb{N}$ with $p\leq N$. An _approximate flow up to homogeneity $N$_
of a $\tau$–dependent pluri-homogeneous smoothing vector field $X^{\tau}(Z)$
in $\Sigma_{p+1}\widetilde{\mathfrak{X}}_{q}^{-\varrho}$, defined for
$\tau\in[0,1]$ and some $\varrho\geq 0$, is a non–linear map,

$\mathcal{F}_{\leq N}^{\tau}(Z)=Z+F_{\leq
N}^{\tau}(Z)Z\,,\qquad\tau\in[0,1]\,,$ (2.122)

where $F_{\leq N}^{\tau}(Z)$ is a matrix of pluri–homogeneous,
$\tau$–dependent, smoothing operators in
$\Sigma_{p}^{N}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
with estimates uniform in $\tau\in[0,1]$, solving

$\partial_{\tau}\mathcal{F}^{\tau}_{\leq
N}(Z)=X^{\tau}(\mathcal{F}^{\tau}_{\leq
N}(Z))+R^{\tau}_{>N}(Z)Z\,,\quad\mathcal{F}^{0}_{\leq N}(Z)=Z\,,$ (2.123)

where $R^{\tau}_{>N}(Z)$ is a matrix of $\tau$–dependent, smoothing operators
in
$\Sigma_{N+1}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
with estimates uniform in $\tau\in[0,1]$.

The following lemma ensures the existence of an approximate flow.

######  Lemma 2.26.

(Approximate flow) Let $p,N\in\mathbb{N}$ with $p\leq N$. Consider a
pluri–homogeneous $\tau$–dependent smoothing vector field $X^{\tau}(Z)$ in
$\Sigma_{p+1}\widetilde{\mathfrak{X}}^{-\varrho}_{q}$, defined for
$\tau\in[0,1]$ and some $\varrho\geq 0$. Then

  * •

there exists an approximate flow $\mathcal{F}^{\tau}_{\leq N}$ according to
Definition 2.25;

  * •

denoting by $G_{p}^{\tau}(Z)Z$ with
$G^{\tau}_{p}(Z)\in{\widetilde{\mathcal{R}}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$
the (p+1)-homogeneous component of $X^{\tau}(Z)$, then the $p$-homogeneous
component of the smoothing operator $F_{\leq N}^{\tau}(Z)$ in (2.122) is

$\mathcal{P}_{p}(F_{\leq
N}^{\tau}(Z))=\int_{0}^{\tau}G_{p}^{\tau^{\prime}}(Z){\rm d}\tau^{\prime}\,.$
(2.124)

###### Proof.

We write $X^{\tau}(Z):=G^{\tau}(Z)Z$ with
$G^{\tau}(Z)=\sum_{q=p}^{M}G^{\tau}_{q}(Z)$ and
$G^{\tau}_{q}(Z)\in{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
and we look for an approximate flow solution of (2.123) of the form

$\mathcal{F}_{\leq
N}^{\tau}(Z)=Z+\sum_{\ell=p}^{N}F^{\tau}_{\ell}(Z,\dots,Z)Z\qquad\text{with}\qquad
F_{\ell}^{\tau}(Z)\in\widetilde{\mathcal{R}}^{-\varrho}_{\ell}\otimes\mathcal{M}_{2}(\mathbb{C})\,.$

Since $G^{\tau}(Z)=\sum_{q=p}^{M}G^{\tau}_{q}(Z)$ then, using the notation
$F_{0}^{\tau}(Z):={\rm Id}$, we expand by multilinearity

$X^{\tau}(\mathcal{F}_{\leq
N}^{\tau}(Z))=\sum_{q=p}^{M}G_{q}^{\tau}(\mathcal{F}_{\leq
N}^{\tau}(Z),\dots,\mathcal{F}_{\leq N}^{\tau}(Z))\mathcal{F}_{\leq
N}^{\tau}(Z)=\sum_{\mathtt{a}=p}^{N+(N+1)M}\breve{X}_{\mathtt{a}}^{\tau}(Z)Z$

where

$\breve{X}_{\mathtt{a}}^{\tau}(Z):=\sum_{\begin{subarray}{c}q=p,\ldots,M\\\
\ell_{1},\dots,\ell_{q+1}\in\\{0,p,\dots,N\\}\\\
\ell_{1}+\dots+\ell_{q+1}+q=\mathtt{a}\end{subarray}}G_{q}^{\tau}\big{(}F_{\ell_{1}}^{\tau}(Z)Z,\dots,F_{\ell_{q}}^{\tau}(Z)Z\big{)}F_{\ell_{q+1}}^{\tau}(Z)\
.$ (2.125)

Then we solve (2.123) defining recursively for $\mathtt{a}=p,\dots,N$,

$F_{\mathtt{a}}^{\tau}(Z):=\int_{0}^{\tau}\breve{X}^{\tau^{\prime}}_{\mathtt{a}}(Z)\,{\rm
d}\tau^{\prime},\qquad
R_{>N}^{\tau}(Z):=\sum_{\mathtt{a}=N+1}^{N+(N+1)M}X^{\tau}_{\mathtt{a}}(Z)\ .$

Using recursively formula (2.125) and Proposition 2.15 one verifies that each
$\breve{X}_{\mathtt{a}}^{\tau}(Z)$ is a $\mathtt{a}$-homogeneous smoothing
operator in
$\widetilde{\mathcal{R}}_{\mathtt{a}}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$,
so is $F_{\mathtt{a}}^{\tau}(Z)$, and $R_{>N}^{\tau}(Z)$ is a pluri-
homogeneous smoothing operator in
$\Sigma_{N+1}\widetilde{\mathcal{R}}_{\mathtt{a}}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$.
Note that for $\mathtt{a}=p$ the sum in (2.125) reduces to the indices $q=p$,
$\ell_{1}=\dots=\ell_{q+1}=0$. As a consequence
$F_{p}^{\tau}(Z)=\int_{0}^{\tau}G^{\tau^{\prime}}_{p}(Z)\,{\rm
d}\tau^{\prime}$ proving (2.124). ∎

####  2.4 Pluri-homogeneous differential geometry

In this section we introduce pluri-homogeneous $k$-forms. We revisit the
classical identities of differential geometry (${\rm d}^{2}=0$, Cartan’s magic
formula) for $k=0,1,2$ which are the only cases needed in this paper.

######  Definition 2.27.

($r$-homogeneous $k$-form) Let $p\in\mathbb{N}_{0}$, $k=0,1,2$ and set
$r:=p+2-k$. A $r$-homogeneous $k$-form is a $(r+k)$-linear map from
$(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{r}\times(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{k}$
to $\mathbb{C}$ of the form $\Lambda(U_{1},\dots,U_{r})[V_{1},\dots,V_{k}]$,
symmetric in the variables ${\mathcal{U}}:=(U_{1},\dots,U_{r})$ and
antisymmetric in the entries ${\mathcal{V}}:=(V_{1},\dots,V_{k})$, satisfying
the following: there are constants $C>0$ and $m\geq 0$ such that

$\displaystyle|\Lambda$
$\displaystyle(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{r}}U_{r})[\Pi_{n_{r+1}}V_{1},\dots,\Pi_{n_{r+k}}V_{k}]|$
$\displaystyle\leq C{\rm
max}\\{n_{1},\dots,n_{r+k}\\}^{m}\prod_{j=1}^{r}\|\Pi_{n_{j}}U_{j}\|_{L^{2}}\prod_{\ell=1}^{k}\|\Pi_{n_{r+\ell}}V_{\ell}\|_{L^{2}}$
(2.126)

for any $\mathcal{U}\in(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{r}$, any
$\mathcal{V}\in(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{k}$, and any
$(n_{1},\ldots,n_{r+k})$ in $\mathbb{N}^{r+k}$. Moreover, if

$\Lambda(\Pi_{n_{1}}U_{1},\dots,\Pi_{n_{r}}U_{r})[\Pi_{n_{r+1}}V_{1},\dots,\Pi_{n_{r+k}}V_{k}]\neq
0\,,$

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

$\Lambda(\tau_{\varsigma}\mathcal{U})[\tau_{\varsigma}{\mathcal{V}}]=\Lambda(\mathcal{U})[{\mathcal{V}}]\,,\quad\forall\varsigma\in\mathbb{R}\,.$
(2.127)

We also require that $\Lambda({\mathcal{U}})[{\mathcal{V}}]$ is real valued
for any ${\mathcal{U}}\in(L^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2}))^{r}$
and ${\mathcal{V}}\in(L^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2}))^{k}$,
cfr. (2.13). We denote by $\widetilde{\Uplambda}^{k}_{r}$ the space of
$r$-homogeneous $k$-forms and by $\Sigma_{r}^{N}\widetilde{\Uplambda}^{k}_{q}$
the space of pluri–homogeneous $k$-forms. We set
$\Sigma_{r}\widetilde{\Uplambda}^{k}_{q}:=\cup_{N\geq
r}\Sigma_{r}^{N}\widetilde{\Uplambda}^{k}_{q}$.

$\bullet$ A $r$-homogeneous $0$-form is also called a homogeneous Hamiltonian.

$\bullet$ Fourier representation of $(p+2)$-homogeneous $0$-forms: Let
$p\in\mathbb{N}_{0}$. A $(p+2)$-homogeneous $0$-form can be expressed in
Fourier as (recall (2.10))

$H(U)=\frac{1}{p+2}\sum_{(\vec{\jmath}_{p+2},\vec{\sigma}_{p+2})\in{\mathfrak{T}}_{p+2}}H_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}u_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}\,.$
(2.128)

The reality condition $H(U)\in\mathbb{R}$ for any $U\in
L^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})$ amounts to

$\overline{H_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}}=H_{\vec{\jmath}_{p+2}}^{-\vec{\sigma}_{p+2}}\,.$
(2.129)

Moreover the scalar coefficients
$H_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}\equiv H_{\
j_{1},\dots,j_{p+2}}^{\sigma_{1},\dots,\sigma_{p+2}}\in\mathbb{C}$ satisfy the
symmetric condition: for any permutation $\pi$ of $\\{1,\ldots,p+2\\}$

$H_{\
j_{\pi(1)},\dots,j_{\pi(p+2)}}^{\sigma_{\pi(1)},\dots,\sigma_{\pi(p+2)}}=H_{\
j_{1},\dots,j_{p+2}}^{\sigma_{1},\dots,\sigma_{p+2}}\,$ (2.130)

and, for some $m\geq 0$, the bound

$|H_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}|\lesssim\max(|j_{1}|,\ldots,|j_{p+2}|)^{m}\,.$
(2.131)

$\bullet$ Fourier representation of $(p+1)$-homogeneous $1$-forms: A
$(p+1)$-homogeneous $1$-form can be expressed in Fourier as

$\theta(U)[V]=\sum_{(\vec{\jmath}_{p+1},k,\vec{\sigma}_{p+1},\sigma)\in{\mathfrak{T}}_{p+2}}\Theta_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}u_{\vec{\jmath}_{p+1}}^{\vec{\sigma}_{p+1}}v_{k}^{\sigma}\,.$
(2.132)

The reality condition $\theta(U)[V]\in\mathbb{R}$ for any $U,V\in
L^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})$ amounts to

$\overline{\Theta_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}}=\Theta_{\vec{\jmath}_{p+1},k}^{-\vec{\sigma}_{p+1},-\sigma}\,.$
(2.133)

Moreover the coefficients satisfy, for some $m\geq 0$,

$|\Theta_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}|\lesssim\max(|j_{1}|,\ldots,|j_{p+1}|)^{m}\,.$
(2.134)

$\bullet$ Fourier representation of $p$-homogeneous $2$-forms: A
$p$-homogeneous $2$-form can be expressed in Fourier as

$\Lambda(U)[V_{1},V_{2}]=\sum_{(\vec{\jmath}_{p},j,k,\vec{\sigma}_{p},\sigma^{\prime},\sigma)\in{\mathfrak{T}}_{p+2}}\Lambda_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}u_{\vec{\jmath}_{p}}^{\vec{\sigma}_{p}}(v_{1})_{j}^{\sigma^{\prime}}(v_{2})_{k}^{\sigma}\,.$
(2.135)

The antisymmetry condition $\Lambda(U)[V_{1},V_{2}]=-\Lambda(U)[V_{2},V_{1}]$
amounts to

$\Lambda_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}=-\Lambda_{\vec{\jmath}_{p},k,j}^{\vec{\sigma}_{p},\sigma,\sigma^{\prime}}\,.$
(2.136)

The reality condition $\Lambda(U)[V_{1},V_{2}]\in\mathbb{R}$ for any
$U,V_{1},V_{2}\in L^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})$ amounts to

$\overline{\Lambda_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}}=\Lambda_{\vec{\jmath}_{p},j,k}^{-\vec{\sigma}_{p},-\sigma^{\prime},-\sigma}\,.$
(2.137)

Moreover the coefficients satisfy, for some $m\geq 0$,

$|\Lambda_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}|\lesssim\max(|\vec{\jmath}_{p}|,|j|)^{m}\,.$
(2.138)

######  Definition 2.28 (Homogeneous vector fields).

Let $m\in\mathbb{R}$ and $p,N\in\mathbb{N}_{0}$. We denote by
$\widetilde{\mathfrak{X}}_{p+1}^{m}$ the space of $(p+1)$-homogeneous vector
fields of the form $X(U)=M(U)U$ where $M(U)$ is a matrix of $p$-homogeneous
$m$-operators in
${\widetilde{\mathcal{M}}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$. We
denote $\widetilde{\mathfrak{X}}_{p+1}:=\cup_{m\geq
0}\widetilde{\mathfrak{X}}_{p+1}^{m}$ and
$\Sigma_{p+1}^{N+1}\widetilde{\mathfrak{X}}^{m}_{q}$ the class of pluri-
homogeneous vector fields. We also set
$\Sigma_{p+1}\widetilde{\mathfrak{X}}_{q}:=\bigcup_{N\in\mathbb{N}}\Sigma_{p+1}^{N+1}\widetilde{\mathfrak{X}}_{q}$.
The vector fields in $\widetilde{\mathfrak{X}}_{p+1}^{-\varrho}$, $\varrho\geq
0$, are called smoothing.

Note that $X(U)=M(U)U$ is real-to-real in the sense of (2.14) if and only if
the operator $M(U)$ is real-to-real in the sense of (2.12).

$\bullet$ Fourier representation of $(p+1)$-homogeneous vector fields: A
$(p+1)$-homogeneous vector field can be expressed in Fourier as: for any
$\sigma=\pm$,

$X(U)^{\sigma}=\sum_{k\in\mathbb{Z}\setminus\\{0\\}}X(U)^{\sigma}_{k}\,\frac{e^{{\rm
i}\sigma kx}}{\sqrt{2\pi}},\qquad
X(U)^{\sigma}_{k}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(\vec{\jmath}_{p+1},k,\vec{\sigma}_{p+1},-\sigma)\in{\mathfrak{T}}_{p+2}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!X_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}u^{\vec{\sigma}_{p+1}}_{\vec{\jmath}_{p+1}}\,,$
(2.139)

the last sum being in $(\vec{\jmath}_{p+1},\vec{\sigma}_{p+1})$, with
coefficients
$X_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}\in\mathbb{C}$ given by

$X_{\
j_{1},\ldots,j_{p},j_{p+1},k}^{\sigma_{1},\ldots,\sigma_{p},\sigma_{p+1},\sigma}=\frac{1}{p+1}\left(M_{\
j_{1},\ldots,j_{p},j_{p+1},k}^{\sigma_{1},\ldots,\sigma_{p},\sigma_{p+1},\sigma}+M_{\
j_{p+1},\ldots,j_{p},j_{1},k}^{\sigma_{p+1},\ldots,\sigma_{p},\sigma_{1},\sigma}+\cdots+M_{\
j_{1},\ldots,j_{p+1},j_{p},k}^{\sigma_{1},\ldots,\sigma_{p+1},\sigma_{p},\sigma}\right)\,,$
(2.140)

namely they are obtained symmetrizing with respect to the last index the
coefficients $M_{\
\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$ of $M(U)$.

In particular they satisfy the symmetry condition: for any permutation $\pi$
of $\\{1,\ldots,p+1\\}$,

$X_{\
j_{\pi(1)},\ldots,j_{\pi(p+1)},k}^{\sigma_{\pi(1)},\ldots,\sigma_{\pi(p+1)},\sigma}=X_{\
j_{1},\ldots,j_{p+1},k}^{\sigma_{1},\ldots,\sigma_{p+1},\sigma}\,.$ (2.141)

In addition, if $X(U)$ is real-to-real, see (2.14), then one has

$\overline{X(U)^{+}_{k}}=X(U)^{-}_{k}\quad{\mathrm{i}.e.}\quad\overline{X_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},+}}=X_{\vec{\jmath}_{p+1},k}^{-\vec{\sigma}_{p+1},-}\,.$
(2.142)

By Lemma 2.9 we obtain the following characterization of vector fields.

######  Lemma 2.29.

(Characterization of vector fields in Fourier basis) Let $m\in\mathbb{R}$. A
real-to-real vector field $X(U)$ belongs to
$\widetilde{\mathfrak{X}}_{p+1}^{m}$ if and only if its coefficients
$X_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}$ (defined as in (2.139))
fulfill the symmetric and real-to-real conditions (2.141), (2.142) and: there
exist $\mu\geq 0$ and $C>0$ such that

$|X_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}|\leq C{\rm
max}_{2}\\{|j_{1}|,\dots,|j_{p+1}|\\}^{\mu}\
\max\\{|j_{1}|,\dots,|j_{p+1}|\\}^{m}$ (2.143)

for any
$(\vec{\jmath}_{p+1},k,\vec{\sigma}_{p+1},-\sigma)\in{\mathfrak{T}}_{p+2}$
(cfr. (2.10)).

The following lemma characterizes $0$, $1$ and $2$ forms.

######  Lemma 2.30.

(Operatorial characterization of Hamiltonians and $1$\- $2$ forms) Let
$p\in\mathbb{N}_{0}$. Then

  * $(i)$

A $0$-form $H(U)$ belongs to $\widetilde{\Uplambda}_{p+2}^{0}$ if and only if
there exists a matrix of $p$-homogeneous real-to-real operators $M(U)$ in
$\widetilde{\mathcal{M}}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$, such that,

$H(U)=\langle M(U)U,U\rangle_{r},\quad\forall
U\in\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2})\,.$ (2.144)

  * $(ii)$

A $1$-form $\theta(U)$ belongs to $\widetilde{\Uplambda}_{p+1}^{1}$ if and
only if there exists a matrix of pluri–homogeneous real-to-real operators
$M(U)$ in $\widetilde{\mathcal{M}}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$,
such that,

$\theta(U)[V]:=\langle M(U)U,V\rangle_{r}\,,\quad\forall
V\in\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2})\,.$ (2.145)

  * $(iii)$

A $2$-form $\Lambda(U)$ belongs to $\widetilde{\Uplambda}_{p}^{2}$ if and only
if there exists a matrix of pluri–homogeneous real-to-real operators $M(U)$ in
$\widetilde{\mathcal{M}}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$, satisfying
$M(U)^{\top}=-M(U)$, such that

$\Lambda(U)[V_{1},V_{2}]:=\langle
M(U)V_{1},V_{2}\rangle_{r}\,,\quad\forall(V_{1},V_{2})\in(\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2}))^{2}\,.$
(2.146)

###### Proof.

Proof of $(i)$: Identity (2.144) follows with an operator $M(U)$ which has
Fourier entries

$M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}=\frac{1}{p+2}H_{\vec{\jmath}_{p},j,-k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$
(2.147)

where the Fourier coefficients of $H$ are defined in (2.128). By (2.131) and
Lemma 2.9 the operator $M(U)$ defined by (2.147) is a matrix of $m$-operators
in $\widetilde{\mathcal{M}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$. Note
that, in view of (2.147), the entries of the operator $M(U)$ satisfy the
corresponding momentum condition
$\vec{\sigma}_{p}\cdot\vec{\jmath}_{p}+\sigma^{\prime}j=\sigma k$ thanks to
the restriction in (2.128). The reality condition (2.129) is equivalent to
(2.46).

Proof of $(ii)$: Identity (2.145) follows with an operator $M(U)$ which has
Fourier entries

$M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}=\Theta_{\vec{\jmath}_{p},j,-k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$
(2.148)

where the Fourier coefficients of $\Theta$ are defined in (2.132). By (2.134)
and Lemma 2.9 the operator $M(U)$ defined by (2.148) is a matrix of
$m$-operators in
$\widetilde{\mathcal{M}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$. Note
that, in view of (2.149), the entries of the operator $M(U)$ satisfy the
corresponding momentum condition
$\vec{\sigma}_{p}\cdot\vec{\jmath}_{p}+\sigma^{\prime}j=\sigma k$ thanks to
the restriction in (2.129). The reality condition (2.133) comes from (2.46).

Proof of $(iii)$: Identity (2.146) follows with an operator $M(U)$ which has
Fourier entries (cfr. (2.43))

$M_{\vec{\jmath}_{p},j,k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}=\Lambda_{\vec{\jmath}_{p},j,-k}^{\vec{\sigma}_{p},\sigma^{\prime},\sigma}$
(2.149)

where the Fourier coefficients of $\Lambda$ are defined in (2.135). By (2.138)
and Lemma 2.9 the operator $M(U)$ defined by (2.149) is a matrix of
$m$-operators in
$\widetilde{\mathcal{M}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$. Note
that, in view of (2.149), the entries of the operator $M(U)$ satisfy the
corresponding momentum condition
$\vec{\sigma}_{p}\cdot\vec{\jmath}_{p}+\sigma^{\prime}j=\sigma k$ thanks to
the restriction in (2.135). The antisymmetry of $\Lambda(U)$ amounts to
$M(U)^{\top}=-M(U)$ and the reality condition (2.137) comes from (2.46). ∎

We now extend to pluri-homogeneous $k$-forms the typical “operations” of
differential geometry.

######  Definition 2.31.

(Exterior derivative) We define the exterior derivative of a $r$-homogeneous
$k$-form $\Lambda(U)$ in $\widetilde{\Uplambda}_{r}^{k}$ as

${\rm d}\Lambda(U)[V_{1},\dots,V_{k+1}]:=\sum_{j=1}^{k+1}(-1)^{j-1}{\rm
d}_{U}\big{(}\Lambda(U)[V_{1},\dots,\widehat{V}_{j},\dots,V_{k+1}]\big{)}[V_{j}]$
(2.150)

where the notation $[V_{1},\dots,\widehat{V}_{j},\dots,V_{k+1}]$ denotes the
$k$-tuple obtained excluding the $j$-th component.

$\bullet$ If $H(U)$ is a $p+2$-homogeneous $0$-form in
$\widetilde{\Uplambda}^{0}_{p+2}$ then its exterior differential coincides
with the usual differential of functions, namely ${\rm d}H(U)[V]={\rm
d}_{U}H(U)[V]$. Moreover ${\rm d}H(U)$ is a $1$-form in
$\widetilde{\Uplambda}^{1}_{p+1}$ and we define the gradient $\nabla
H(U):=\nabla_{U}H(U)$ as the vector field in $\widetilde{\mathfrak{X}}_{p+1}$
such that, cfr. (2.145),

${\rm d}H(U)[V]:=\langle\nabla H(U),V\rangle_{r}\,,\quad\forall
V\in\dot{H}^{\infty}(\mathbb{T};\mathbb{C}^{2})\,.$ (2.151)

$\bullet$ If $\theta(U)$ is a $(p+1)$-homogeneous $1$-form in
$\widetilde{\Uplambda}^{1}_{p+1}$ written as in (2.145) then its exterior
differential is

${\rm d}\theta(U)[V_{1},V_{2}]=\Big{\langle}\big{(}{\rm d}_{U}X(U)-{\rm
d}_{U}X(U)^{\top}\big{)}V_{1},V_{2}\Big{\rangle}_{r},\quad X(U):=M(U)U$
(2.152)

where ${\rm d}_{U}X(U)$ and ${\rm d}_{U}X(U)^{\top}$ are, by the first remark
below Definition 2.5 and Lemma 2.10, matrices of operators in
${\widetilde{\mathcal{M}}}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$. Moreover
${\rm d}\theta(U)$ belongs to $\widetilde{\Uplambda}^{2}_{p}$.

######  Definition 2.32.

Let $p,p^{\prime}\in\mathbb{N}_{0}$ and set $r:=p+2-k$. Given a
$r$-homogeneous $k$-form $\Lambda(U)$ in $\widetilde{\Uplambda}_{r}^{k}$ and a
matrix of homogeneous operators $M(U)$ in
${\widetilde{\mathcal{M}}}_{p^{\prime}}\otimes\mathcal{M}_{2}(\mathbb{C})$ we
define the
$\bullet$ Pull back of $\Lambda(U)$ via the map $\varphi(U):=M(U)U$ as

$(\varphi^{*}\Lambda)(U)[V_{1},\dots,V_{k}]:=\Lambda(\varphi(U))[{\rm
d}_{U}\varphi(U)V_{1},\dots,{\rm d}_{U}\varphi(U)V_{k}]\,.$ (2.153)

$\bullet$ Lie derivative of $\Lambda(U)$ along the vector field $X(U):=M(U)U$
as

$(\mathcal{L}_{X}\Lambda)\,(U)[V_{1},\dots,V_{k}]:={\rm
d}_{U}\Lambda(U)[X(U)][V_{1},\dots,V_{k}]+\sum_{j=1}^{k}\Lambda(U)[V_{1},\dots,{\rm
d}_{U}X(U)[V_{j}],\dots,V_{k}].$ (2.154)

$\bullet$ Contraction of $\Lambda(U)$ with the vector field $X(U)=M(U)U$ as

$(i_{X}\Lambda)(U)[V_{1},\dots,V_{k-1}]:=\Lambda(U)[X(U),V_{1},\dots,V_{k-1}]\,.$
(2.155)

Let $\Lambda(U)$ is a $r$-homogeneous $k$-form in
$\widetilde{\Uplambda}_{r}^{k}$, $k=0,1,2$. Thanks to the first bullet below
Definition 2.5, Lemma 2.10 and $(i)$ and $(ii)$ of Proposition 2.15 (see also
(2.160)-(2.161)), one has the following:

$\bullet$ if $\varphi(U)=M(U)U$ is a map where $M(U)$ is a pluri-homogeneous
operator in
$\Sigma_{0}^{N}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
then $(\varphi^{*}\Lambda)(U)$ defined in (2.153) belongs to
$\Sigma_{r}\widetilde{\Uplambda}^{k}_{q}$;

$\bullet$ if $X(U)$ is a homogeneous vector field in
$\Sigma_{p^{\prime}+1}\mathfrak{\widetilde{X}}_{q}$ for some
$p^{\prime}\in\mathbb{N}_{0}$, then $(\mathcal{L}_{X}\Lambda)\,(U)$ defined in
(2.154) belongs to $\Sigma_{r+p^{\prime}}\widetilde{\Uplambda}^{k}_{q}$;

$\bullet$ if $k=1,2$ then $(i_{X}\Lambda)(U)$ defined in (2.155) belongs to
$\Sigma_{r+p^{\prime}+1}\widetilde{\Uplambda}^{k-1}_{q}$;

$\bullet$ the basic identities of differential geometry are directly verified
for pluri-homogeneous $k$-forms: Let $p\in\mathbb{N}_{0}$, $k=0,1,2$. Then for
any $\Lambda$ in $\widetilde{\Uplambda}_{p+2}^{k}$ it results

${\rm d}^{2}\Lambda=0\,.$ (2.156)

Given $\varphi(U):=M(U)U$ with $M(U)$ in
$\Sigma_{0}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
it results

${\rm d}(\varphi^{*}\Lambda)\,=(\varphi^{*}{\rm d}\Lambda)\,\,.$ (2.157)

Given also $\phi(U):=M^{\prime}(U)U$ with
$M^{\prime}(U)\in\Sigma_{0}\widetilde{\mathcal{M}}_{q}\otimes\mathcal{M}(\mathbb{C})$,
it results

$\phi^{*}\varphi^{*}\Lambda=(\varphi\circ\phi)^{*}\Lambda\ .$ (2.158)

Given $X(U)$ in $\Sigma_{1}\mathfrak{\widetilde{X}}_{q}$ then

$\mathcal{L}_{X}\Lambda={\rm d}\circ i_{X}\Lambda+i_{X}\circ{\rm d}\Lambda\,;$
(2.159)

$\bullet$ if $\varphi(U)=M^{\prime}(U)U$ is a map where $M^{\prime}(U)$ is a
pluri-homogeneous operator in
$\Sigma_{0}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
and $\theta(U)\in\widetilde{\Uplambda}^{1}_{p+1}$ and
$\Lambda(U)\in\widetilde{\Uplambda}^{2}_{p}$ are represented as in (2.145),
(2.146) then

$\displaystyle(\varphi^{*}\theta)(U)[V]=\langle{\rm
d}_{U}\varphi(U)^{\top}\,\,M(\varphi(U))\,\varphi(U),V\rangle\,;$ (2.160)
$\displaystyle(\varphi^{*}\Lambda)(U)[V_{1},V_{2}]=\langle{\rm
d}_{U}\varphi(U)^{\top}\,\,M(\varphi(U))\,\,{\rm
d}_{U}\varphi(U)V_{1},V_{2}\rangle\,.$ (2.161)

In Section 4 we shall use the following result about Lie derivatives and
approximate flows.

######  Lemma 2.33.

Let $p,N\in\mathbb{N}$ with $p\leq N$. Let $\theta^{\tau}$ be a
$\tau$-dependent family of $1$-forms in
$\Sigma_{1}\widetilde{\Uplambda}_{q}^{1}$ defined for $\tau\in[0,1]$. Let
$\mathcal{F}_{\leq N}^{\tau}$ be the approximate flow generated by a
pluri–homogeneous, $\tau$–dependent smoothing vector field $Y^{\tau}(U)$ in
$\Sigma_{p+1}\widetilde{\mathfrak{X}}_{q}^{-\varrho}$, defined for
$\tau\in[0,1]$ and some $\varrho\geq 0$ (cfr. Lemma 2.26). Then

$\frac{{\rm d}}{{\rm d}\tau}(\mathcal{F}_{\leq
N}^{\tau})^{*}\theta^{\tau}=(\mathcal{F}_{\leq
N}^{\tau})^{*}\big{[}\mathcal{L}_{Y^{\tau}}\theta^{\tau}+\partial_{\tau}\theta^{\tau}\big{]}+\theta_{>N+1}^{\tau}$
(2.162)

where $\theta_{>N+1}^{\tau}$ is a pluri–homogeneous $1$–form in
$\Sigma_{N+2}\widetilde{\Uplambda}_{q}^{1}$, with estimates uniform in
$\tau\in[0,1]$.

###### Proof.

Recalling the definition of pullback (2.153) and using that $\mathcal{F}_{\leq
N}^{\tau}$ fulfills the approximate equation (2.123) (with $Y^{\tau}$
replacing $X^{\tau}$) we get

$\displaystyle\frac{{\rm d}}{{\rm d}\tau}(\mathcal{F}_{\leq
N}^{\tau})^{*}\theta^{\tau}\,(U)[\widehat{U}]=\frac{{\rm d}}{{\rm
d}\tau}\big{(}\theta^{\tau}(\mathcal{F}_{\leq N}^{\tau}(U))[{\rm
d}_{U}\mathcal{F}_{\leq N}^{\tau}(U)\widehat{U}]\big{)}$
$\displaystyle=\partial_{\tau}\theta^{\tau}(\mathcal{F}_{\leq
N}^{\tau}(U))[{\rm d}_{U}\mathcal{F}_{\leq N}^{\tau}(U)\widehat{U}]+{\rm
d}_{V}\theta^{\tau}(\mathcal{F}_{\leq N}^{\tau}(U))[Y^{\tau}(\mathcal{F}_{\leq
N}^{\tau}(U))+R_{>N}^{\tau}(U)U][{\rm d}_{U}\mathcal{F}_{\leq
N}^{\tau}(U)\widehat{U}]$ $\displaystyle\quad+\theta^{\tau}(\mathcal{F}_{\leq
N}^{\tau}(U))[{\rm d}_{V}Y^{\tau}(\mathcal{F}^{\tau}_{\leq N}(U)){\rm
d}_{U}\mathcal{F}^{\tau}_{\leq N}(U)\widehat{U}+{\rm
d}_{U}(R^{\tau}_{>N}(U)U)\widehat{U}]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\stackrel{{\scriptstyle\eqref{Lie},\eqref{pullback}}}{{=}}(\mathcal{F}_{\leq
N}^{\tau})^{*}\big{[}\mathcal{L}_{Y^{\tau}}\theta^{\tau}+\partial_{\tau}\theta^{\tau}\big{]}(U)[\widehat{U}]+\theta_{>N+1}^{\tau}(U)[\widehat{U}]$

where

$\theta_{>N+1}^{\tau}(U)[\widehat{U}]:={\rm
d}_{V}\theta^{\tau}(\mathcal{F}_{\leq N}^{\tau}(U))[R_{>N}^{\tau}(U)U][{\rm
d}_{U}\mathcal{F}_{\leq
N}^{\tau}(U)\widehat{U}]+\theta^{\tau}(\mathcal{F}_{\leq N}^{\tau}(U))[{\rm
d}_{U}(R^{\tau}_{>N}(U)U)\widehat{U}]\,.$ (2.163)

We now verify that $\theta_{>N+1}^{\tau}$ is a $1$-form in
$\Sigma_{N+2}\widetilde{\Uplambda}_{q}^{1}$. Representing, by Lemma 2.30,

$\theta^{\tau}(U)[\widehat{U}]=\langle
M^{\tau}(U)U,\widehat{U}\rangle_{r}\quad\text{with}\quad
M^{\tau}(U)\in\Sigma_{0}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,,$

and since $\mathcal{F}^{\tau}_{\leq N}(U)=({\rm Id}+R^{\tau}_{\leq N}(U))U$
with $R^{\tau}_{\leq N}(U)$ in
$\Sigma_{p}^{N}\widetilde{\mathcal{R}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
the $1$-form $\theta_{>N+1}^{\tau}$ in (2.163) reads

$\displaystyle\theta^{\tau}_{>N+1}(U)[\widehat{U}]=\langle
M^{\prime}(U)U,\widehat{U}\rangle_{r}\quad\text{where}\quad M^{\prime}(U):=$
$\displaystyle\left[{\rm d}_{U}\mathcal{F}_{\leq
N}^{\tau}(U)\right]^{\top}\,\left.{\rm
d}_{V}(M^{\tau}(V)V)\right|_{V=\mathcal{F}_{\leq
N}^{\tau}(U)}\,R_{>N}^{\tau}(U)$ $\displaystyle\\!\\!\\!+[{\rm
d}_{U}(R^{\tau}_{>N}(U)U)]^{\top}M^{\tau}(\mathcal{F}_{\leq N}^{\tau}(U))({\rm
Id}+R^{\tau}_{\leq N}(U))$

is a $\tau$-dependent matrix of operators in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
with estimates uniform in $\tau\in[0,1]$, because $R^{\tau}_{>N}(U)$ are
smoothing operators in
$\Sigma_{N+1}\widetilde{\mathcal{R}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
and using $(i)$, $(ii)$ of Proposition 2.15 and Lemma 2.10. Thus
$\theta_{>N+1}^{\tau}$ is a $1$-form in
$\Sigma_{N+2}\widetilde{\Uplambda}_{q}^{1}$, with estimates uniform in
$\tau\in[0,1]$. ∎

###  3 Hamiltonian formalism

Along the paper we consider real Hamiltonian systems and their symplectic
structures in real, complex and Fourier coordinates, that we describe in
Section 3.1. In Section 3.2 we introduce the notion of vector fields which are
Hamiltonian up to homogeneity $N$ and we prove that the classical Hamiltonian
theory is preserved “up to homogeneity $N$”. In Section 3.3 we present results
about linear symplectic flows. In Section 3.4 discuss Hamiltonian systems with
a paradifferential structure.

####  3.1 Hamiltonian and symplectic structures

##### Real Hamiltonian systems.

We equip the real phase space $\dot{L}^{2}_{r}\times\dot{L}^{2}_{r}$ with the
scalar product in (2.4) and the symplectic form

$\Omega_{0}\left(\begin{pmatrix}\eta_{1}\\\
\zeta_{1}\end{pmatrix},\begin{pmatrix}\eta_{2}\\\
\zeta_{2}\end{pmatrix}\right):=\Big{\langle}E_{0}\begin{pmatrix}\eta_{1}\\\
\zeta_{1}\end{pmatrix},\begin{pmatrix}\eta_{2}\\\
\zeta_{2}\end{pmatrix}\Big{\rangle}_{r}=-\langle\zeta_{1},\eta_{2}\rangle_{\dot{L}_{r}}+\langle\eta_{1},\zeta_{2}\rangle_{\dot{L}_{r}}$
(3.1)

where $E_{0}$ is the symplectic operator acting on
$\dot{L}^{2}_{r}\times\dot{L}^{2}_{r}$ defined by

$E_{0}:=\begin{pmatrix}0&-{\rm Id}\\\ {\rm Id}&0\end{pmatrix}\,.$ (3.2)

The Hamiltonian vector field $X_{H}$ associated to a (densely defined)
Hamiltonian function $H:\dot{L}^{2}_{r}\times\dot{L}^{2}_{r}\to\mathbb{R}$ is
characterized as the unique vector field satisfying

$\Omega_{0}\left(X_{H}(\eta,\zeta),\begin{pmatrix}\breve{\eta}\\\
\breve{\zeta}\end{pmatrix}\right)={\rm
d}H(\eta,\zeta)\left[\begin{pmatrix}\breve{\eta}\\\
\breve{\zeta}\end{pmatrix}\right],\quad\forall\begin{pmatrix}\breve{\eta}\\\
\breve{\zeta}\end{pmatrix}\in\dot{L}^{2}_{r}\times\dot{L}^{2}_{r}\,.$ (3.3)

As

$\displaystyle{\rm d}H(\eta,\zeta)\left[\begin{pmatrix}\breve{\eta}\\\
\breve{\zeta}\end{pmatrix}\right]$ $\displaystyle={\rm
d}_{\eta}H(\eta,\zeta)[\breve{\eta}]+{\rm
d}_{\zeta}H(\eta,\zeta)[\breve{\zeta}]=\Big{\langle}\begin{pmatrix}\nabla_{\eta}H\\\
\nabla_{\zeta}H\end{pmatrix},\begin{pmatrix}\breve{\eta}\\\
\breve{\zeta}\end{pmatrix}\Big{\rangle}_{r}$ (3.4)

where
$(\nabla_{\eta}H,\nabla_{\zeta}H)\in\dot{L}^{2}_{r}\times\dot{L}^{2}_{r}$
denote the $\dot{L}^{2}_{r}$\- gradients, the Hamiltonian vector field is
given by

$X_{H}=J\begin{pmatrix}\nabla_{\eta}H\\\
\nabla_{\zeta}H\end{pmatrix}=\begin{pmatrix}\nabla_{\zeta}H\\\
-\nabla_{\eta}H\end{pmatrix}\quad\text{where}\quad
J:=E_{0}^{-1}=\begin{pmatrix}0&{\rm Id}\\\ -{\rm Id}&0\end{pmatrix}.$ (3.5)

We also denote by

$\theta_{0}(\eta,\zeta)\Big{[}\begin{pmatrix}\breve{\eta}\\\
\breve{\zeta}\end{pmatrix}\Big{]}:=\frac{1}{2}\Big{\langle}E_{0}\begin{pmatrix}\eta\\\
\zeta\end{pmatrix},\begin{pmatrix}\breve{\eta}\\\
\breve{\zeta}\end{pmatrix}\Big{\rangle}_{r}$ (3.6)

the Liouville $1$–form. Note that ${\rm d}\theta_{0}=\Omega_{0}$, where the
exterior differential is recalled in Section 2.4.

##### Real linear Hamiltonian systems.

We now consider the most general quadratic real Hamiltonian

$H(\eta,\zeta)=\frac{1}{2}\Big{\langle}{\bf A}\begin{pmatrix}\eta\\\
\zeta\end{pmatrix},\begin{pmatrix}\eta\\\
\zeta\end{pmatrix}\Big{\rangle}_{r},\quad{\bf A}:=\begin{pmatrix}A&B\\\
B^{\top}&D\end{pmatrix}\,,\quad{\bf A}^{\top}={\bf A}\,,$ (3.7)

where $A,B,D$ are linear real operators acting on $\dot{L}^{2}_{r}$ and the
operators $A,D$ are symmetric, i.e. $A^{\top}=A$, $D^{\top}=D$, where
$A^{\top}$ denotes the transpose operator with respect to the real scalar
product $\langle\ ,\ \rangle_{\dot{L}^{2}_{r}}$. ${\bf A}^{\top}$ is the
transpose with respect to the scalar product $\langle\cdot,\cdot\rangle_{r}$
in (2.4).

######  Definition 3.1.

(Linear Hamiltonian operator) A linear operator $\mathcal{A}$ acting on (a
dense subspace) of $\dot{L}^{2}_{r}\times\dot{L}^{2}_{r}$ is Hamiltonian if it
has the form

$\mathcal{A}=J{\bf A}=J\begin{pmatrix}A&B\\\
B^{\top}&D\end{pmatrix},\qquad{\bf A}={\bf A}^{\top}\,,$ (3.8)

with $A,B,D$ real operators satisfying $A=A^{\top}$ and $D=D^{\top}$;
equivalently if $E_{0}\mathcal{A}={\bf A}$ is symmetric with respect to the
real scalar product $\langle\cdot,\cdot\rangle_{r}$ defined in (2.4).

We now provide the characterization of a real linear Hamiltonian
paradifferential operator. In view of (3.8) and (2.26) a matrix of
paradifferential operators is Hamiltonian if it has the form

$J{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\begin{bmatrix}a(x,\xi)&b(x,\xi)\\\
b(x,-\xi)&d(x,\xi)\end{bmatrix}\right)}={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\begin{bmatrix}b(x,-\xi)&d(x,\xi)\\\
-a(x,\xi)&-b(x,\xi)\end{bmatrix}\right)}$ (3.9)

with

$\displaystyle a(x,\xi)\in\mathbb{R},\ a(x,\xi)=a(x,-\xi)\,,\quad
d(x,\xi)\in\mathbb{R},\ d(x,\xi)=d(x,-\xi)\,,\quad
b(x,-\xi)=\overline{b(x,\xi)}\,.$ (3.10)

##### Real Hamiltonian systems in complex coordinates.

We now describe the above real Hamiltonian systems in the complex coordinates
defined by the change of variables

$\begin{pmatrix}\eta\\\ \zeta\end{pmatrix}=\mathcal{C}\begin{pmatrix}u\\\
\overline{u}\end{pmatrix}\,,\quad\mathcal{C}:=\frac{1}{\sqrt{2}}\left(\begin{matrix}{\rm
Id}&{\rm Id}\\\ -{\rm i}&{\rm i}\end{matrix}\right)\
,\quad\mathcal{C}^{-1}:=\frac{1}{\sqrt{2}}\left(\begin{matrix}{\rm Id}&{\rm
i}\\\ {\rm Id}&-{\rm
i}\end{matrix}\right)\,,\quad\mathcal{C}^{\top}:=\frac{1}{\sqrt{2}}\left(\begin{matrix}{\rm
Id}&-{\rm i}\\\ {\rm Id}&{\rm i}\end{matrix}\right)\,.$ (3.11)

Note that $\mathcal{C}$ is a map between the real subspace of vector functions
$\dot{L}^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})$ into
$\dot{L}^{2}_{r}\times\dot{L}^{2}_{r}$. In the sequel to save space we denote
$\begin{pmatrix}u\\\ \overline{u}\end{pmatrix}$ also as $(u,\overline{u})$.

The pull-back $\Omega_{c}:=\mathcal{C}^{*}\Omega_{0}$ of the symplectic form
$\Omega_{0}$ in (3.1) is

$\Omega_{c}\left(\begin{pmatrix}u_{1}\\\
\overline{u}_{1}\end{pmatrix},\begin{pmatrix}u_{2}\\\
\overline{u}_{2}\end{pmatrix}\right)=\Big{\langle}\mathcal{C}^{\top}E_{0}\mathcal{C}\begin{pmatrix}u_{1}\\\
\overline{u}_{1}\end{pmatrix},\begin{pmatrix}u_{2}\\\
\overline{u}_{2}\end{pmatrix}\Big{\rangle}_{r}=\Big{\langle}E_{c}\begin{pmatrix}u_{1}\\\
\overline{u}_{1}\end{pmatrix},\begin{pmatrix}u_{2}\\\
\overline{u}_{2}\end{pmatrix}\Big{\rangle}_{r}\ $ (3.12)

where $E_{c}$ is the symplectic operator acting on
$\dot{L}^{2}_{\mathbb{R}}(\mathbb{T},\mathbb{C}^{2})$

$E_{c}:=\begin{pmatrix}0&-{\rm i}\\\ {\rm i}&0\end{pmatrix}={\rm i}E_{0}\,.$
(3.13)

Remark that $E_{c}^{\top}=-E_{c}$ and $E_{c}^{2}={\rm Id}$. Similarly the
Liouville $1$-form $\theta_{0}$ in (3.6) is transformed into the symplectic
form $\theta_{c}:=\mathcal{C}^{*}\theta_{0}$ given by

$\theta_{c}(U)\left[V\right]=\frac{1}{2}\langle
E_{c}U,V\rangle_{r}\,,\quad\forall\ U=\begin{pmatrix}u\\\
\overline{u}\end{pmatrix}\,,\ V=\begin{pmatrix}v\\\
\overline{v}\end{pmatrix}\,$ (3.14)

and it results

${\rm d}\theta_{c}=\Omega_{c}\,.$ (3.15)

Next we show how the differential and gradient of a Hamiltonian transform
under the complex change of coordinates. The pull-back under $\mathcal{C}$ of
the $1$-form (cfr. (3.4)) ${\rm
d}H(\eta,\zeta)[\cdot]=\Big{\langle}\begin{pmatrix}\nabla_{\eta}H\\\
\nabla_{\zeta}H\end{pmatrix},\cdot\Big{\rangle}_{r}$ is

$\displaystyle(\mathcal{C}^{*}{\rm
d}H)(u,\overline{u})[(v,\overline{v})]=\Big{\langle}\begin{pmatrix}\nabla_{u}H\\\
\nabla_{\overline{u}}H\end{pmatrix}\Big{|}_{\mathcal{C}(u,\overline{u})},\
\begin{pmatrix}v\\\ \overline{v}\end{pmatrix}\Big{\rangle}_{r}$ (3.16)

where

$\nabla_{u}H:=\frac{1}{\sqrt{2}}\left(\nabla_{\eta}H-{\rm
i}\nabla_{\zeta}H\right)|_{\mathcal{C}(u,\overline{u})}\,,\quad\nabla_{\overline{u}}H:=\frac{1}{\sqrt{2}}\left(\nabla_{\eta}H+{\rm
i}\nabla_{\zeta}H\right)|_{\mathcal{C}(u,\overline{u})}\,.$ (3.17)

Furthermore, by (3.11),

$\displaystyle(\mathcal{C}^{*}{\rm d}H)(u,\overline{u})[(v,\overline{v})]={\rm
d}_{u}H|_{\mathcal{C}(u,\overline{u})}[v]+{\rm
d}_{\overline{u}}H|_{\mathcal{C}(u,\overline{u})}[\overline{v}]$

having defined

${\rm d}_{u}H:=\frac{1}{\sqrt{2}}\left({\rm d}_{\eta}H-{\rm i}{\rm
d}_{\zeta}H\right)|_{\mathcal{C}(u,\overline{u})}\,,\quad{\rm
d}_{\overline{u}}H:=\frac{1}{\sqrt{2}}\left({\rm d}_{\eta}H+{\rm i}{\rm
d}_{\zeta}H\right)|_{\mathcal{C}(u,\overline{u})}\,.$

In the sequel we also use the compact notation, given $U=(u,\overline{u})$,

${\rm d}_{U}H(U)[\widehat{U}]:={\rm d}_{u}H(U)[\widehat{u}]+{\rm
d}_{\overline{u}}H(U)[\overline{\widehat{u}}]\,,\quad\forall\widehat{U}=\begin{pmatrix}\widehat{u}\\\
\overline{\widehat{u}}\end{pmatrix}\,.$

##### Real Hamiltonian vector fields in complex coordinates.

Given a real valued Hamiltonian $H(\eta,\zeta)$, consider the Hamiltonian in
complex coordinates $H_{c}:=H\circ{\mathcal{C}}$ which is a function of
$(u,\overline{u})$. Recalling the characterization (3.3) of the Hamiltonian
vector field and (3.16), the associated Hamiltonian vector field is

$X_{H_{c}}(U):=\mathcal{C}^{-1}(X_{H})_{|\mathcal{C}(U)}=J_{c}\begin{pmatrix}\nabla_{u}H_{c}\\\
\nabla_{\overline{u}}H_{c}\end{pmatrix}=J_{c}\nabla
H_{c}(U)=\begin{pmatrix}-{\rm i}\nabla_{\overline{u}}H_{c}\\\ {\rm
i}\nabla_{u}H_{c}\end{pmatrix}$ (3.18)

where $J_{c}:=E_{c}^{-1}=E_{c}$ is the Poisson tensor in (1.22). One has also
the characterization

$\Omega_{c}(X_{H_{c}},\cdot)={\rm d}_{U}H_{c}(U)[\cdot]\,.$ (3.19)

In case $H$ is the quadratic form (3.7), the transformed Hamiltonian
$H_{c}=H\circ\mathcal{C}$ is given by

$\displaystyle H_{c}(u,\overline{u})$
$\displaystyle=\frac{1}{2}\Big{\langle}{\bf R}\begin{pmatrix}u\\\
\overline{u}\end{pmatrix},\begin{pmatrix}u\\\
\overline{u}\end{pmatrix}\Big{\rangle}_{r},\quad{\bf
R}:=\mathcal{C}^{\top}{\bf A}\mathcal{C}:=\begin{pmatrix}R_{1}&R_{2}\\\
\overline{R}_{2}&\overline{R}_{1}\end{pmatrix}$ (3.20)

where $R_{1}:=(A-D)-{\rm i}(B+B^{\top})$, $R_{2}:=(A+D)+{\rm i}(B-B^{\top})$.
The operator ${\bf R}$ is real-to-real according to (2.12). In addition, since
${\bf A}$ is symmetric, cfr. (3.7), the operator ${\bf R}$ is symmetric with
respect to the real non-degenerate bilinear form
$\langle\cdot,\cdot\rangle_{r}$, namely

${\bf R}={\bf R}^{\top}\,,\quad\text{i.e.}\ \ R_{1}=R_{1}^{\top}\,,\ \
R_{2}^{*}=R_{2}\,.$ (3.21)

######  Definition 3.2.

(Linear Hamiltonian operator in complex coordinates) A real-to-real linear
operator $J_{c}{\bf M}$ is linearly Hamiltonian if ${\bf M}={\bf M}^{\top}$ is
symmetric with respect to the non-degenerate bilinear form
$\langle\cdot,\cdot\rangle_{r}$, cfr. (3.21).

In view of (2.26) and (3.21) a matrix of paradifferential real-to-real complex
operators is linearly Hamiltonian if

$J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\begin{matrix}b_{1}(U;t,x,\xi)&b_{2}(U;t,x,\xi)\\\
\overline{b_{2}(U;t,x,-\xi)}&\overline{b_{1}(U;t,x,-\xi)}\end{matrix}\right)}\,,\quad\begin{cases}b_{1}(U;t,x,-\xi)=b_{1}(U;t,x,\xi)\,,\\\
b_{2}(U;t,x,\xi)\in\mathbb{R}\,,\end{cases}$ (3.22)

namely $b_{1}$ is even in $\xi$ and $b_{2}$ is real valued.

######  Definition 3.3.

(Linearly symplectic map) A real-to-real linear transformation $\mathcal{A}$
is linearly symplectic if $\mathcal{A}^{*}\Omega_{c}=\Omega_{c}$ where
$\Omega_{c}$ is defined in (3.12), namely
$\mathcal{A}^{\top}\,E_{c}\,\mathcal{A}=E_{c}$, where $E_{c}$ is the
symplectic operator defined in (3.13).

Hamiltonian systems in Fourier basis. Given a Hamiltonian $H(U)$ expanded as
in (2.128) we characterize its Hamiltonian vector field. We decompose each
Fourier coefficients as $u_{j}=\frac{x_{j}+{\rm i}y_{j}}{\sqrt{2}}$,
$\overline{u_{j}}=\frac{x_{j}-{\rm i}y_{j}}{\sqrt{2}}$, where
$x_{j}:=\sqrt{2}\mathrm{Re}\,(u_{j})$ and
$y_{j}:=\sqrt{2}\mathrm{Im}\,(u_{j})$ and we define

$\partial_{u_{j}}:=\frac{1}{\sqrt{2}}\big{(}\partial_{x_{j}}-{\rm
i}\partial_{y_{j}}\big{)}\,,\quad\partial_{\overline{u_{j}}}:=\frac{1}{\sqrt{2}}\big{(}\partial_{x_{j}}+{\rm
i}\partial_{y_{j}}\big{)}\,,$ (3.23)

so that $\partial_{u_{j}^{\sigma}}u_{j}^{\sigma}=1$, for any $\sigma=\pm$, and
$\partial_{u_{j}^{\sigma}}u_{j}^{\sigma^{\prime}}=0$, for any
$\sigma\sigma^{\prime}=-1$. For a real valued Hamiltonian $H$ it results

$\overline{\partial_{u_{j}}H}=\partial_{\overline{u_{j}}}H\,.$ (3.24)

We now write a Hamiltonian vector field (3.18) in the coordinates
$(u_{j})_{j\in\mathbb{Z}\setminus\\{0\\}}$. For notational simplicity we also
denote $u\equiv(u_{j})_{j\in\mathbb{Z}\setminus\\{0\\}}$. We first note that,
by (3.12) and (2.5), the symplectic form (3.12) reads, for any
$U=(u,\overline{u})$, $V=(v,\overline{v})$,

$\Omega_{c}\big{(}U,V\big{)}=\sum_{j\in\mathbb{Z}\setminus\\{0\\}}-{\rm
i}\overline{u_{j}}v_{j}+{\rm i}u_{j}\overline{v_{j}}=-{\rm
i}\sum_{j\in\mathbb{Z}\setminus\\{0\\}\atop\sigma\in\\{\pm\\}}\sigma
u_{j}^{-\sigma}v_{j}^{\sigma}\,.$ (3.25)

######  Lemma 3.4.

(Fourier expansion of a Hamiltonian vector field) The Fourier components of
the Hamiltonian vector field associated to a real Hamiltonian $H(U)$ are, for
any $\sigma=\pm$, $k\in\mathbb{Z}\setminus\\{0\\}$,

$\big{(}J_{c}\nabla H(U)\big{)}^{\sigma}_{k}=-{\rm
i}\sigma\partial_{u_{k}^{-\sigma}}H(U)\,.$ (3.26)

In particular, if the Hamiltonian $H$ is expanded as in (2.128), then

$\big{(}J_{c}\nabla H(U)\big{)}^{\sigma}_{k}=-{\rm
i}\sigma\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(\vec{\jmath}_{p+1},k,\vec{\sigma}_{p+1},-\sigma)\in{\mathfrak{T}}_{p+2}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},-\sigma}u_{\vec{\jmath}_{p+1}}^{\vec{\sigma}_{p+1}}\,.$
(3.27)

###### Proof.

The expression (3.26) is a consequence of (3.25) and the definition (3.19) of
a Hamiltonian vector field, using (3.24). Finally (3.27) is a consequence of
(3.26), (2.129), (2.130). ∎

By (3.27) and (2.139) we deduce the following characterization of Hamiltonian
vector fields:

######  Lemma 3.5.

(Characterization of $(p+1)$–homogeneous Hamiltonian vector fields) A
$(p+1)$–homogeneous real-to-real vector field
$X(U)\in\widetilde{\mathfrak{X}}_{p+1}$ of the form (2.139) is Hamiltonian if
and only if the coefficients

$H_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},\sigma}:=-{\rm i}\sigma
X_{\vec{\jmath}_{p+1},k}^{\vec{\sigma}_{p+1},-\sigma},\qquad\forall(\vec{\jmath}_{p+1},k,\vec{\sigma}_{p+1},\sigma)\in{\mathfrak{T}}_{p+2}\,,$
(3.28)

satisfy (2.130), (2.129) and (2.131). In such a case $X(U)$ is the Hamiltonian
vector field generated by

$H(U)=\frac{1}{p+2}\sum_{(\vec{\jmath}_{p+2},\vec{\sigma}_{p+2})\in{\mathfrak{T}}_{p+2}}H_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}u_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}\,,\quad
H_{\vec{\jmath}_{p+2}}^{\vec{\sigma}_{p+2}}:=-{\rm
i}\sigma_{p+2}X_{\vec{\jmath}_{p+1},j_{p+2}}^{\vec{\sigma}_{p+1},-\sigma_{p+2}}\,.$

The Poisson bracket between two real functions $F,G$ is, by (3.19), (3.25),
(3.26), (3.24),

$\\{F,G\\}:={\rm
d}F(X_{G})=\Omega_{c}(X_{F},X_{G})=\sum_{j\in\mathbb{Z}\setminus\\{0\\}}{\rm
i}\Big{(}\partial_{\overline{u_{j}}}F\partial_{u_{j}}G-\partial_{u_{j}}F\partial_{\overline{u_{j}}}G\Big{)}\,.$
(3.29)

Note that the right hand side of (3.29) is well–defined also for complex
valued functions $F$ and $G$ and, with a small abuse of notation, we shall
still refer to it as the Poisson bracket between $F$ and $G$.

####  3.2 Hamiltonian systems up to homogeneity $N$

Along the paper we encounter vector fields which are Hamiltonian up to
homogeneity $N$. We distinguish between linear and nonlinear ones.

##### Linear Hamiltonian operators.

In the sequel let $p,N,K,K^{\prime}\in\mathbb{N}_{0}$ and $K^{\prime}\leq K$,
$r>0$.

######  Definition 3.6.

(Linearly Hamiltonian operator up to homogeneity $N$) A real-to-real matrix of
spectrally localized maps $J_{c}{\bf B}(U;t)$ in
$\Sigma{\mathcal{S}}_{K,K^{\prime},p}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$
is linearly Hamiltonian up to homogeneity $N$ if the pluri-homogeneous
component $\mathcal{P}_{\leq N}({\bf B}(U;t))$ (defined in (2.36)) is
symmetric, namely

$\mathcal{P}_{\leq N}({\bf B}(U;t))=\mathcal{P}_{\leq N}({\bf
B}(U;t)^{\top})\,.$ (3.30)

In particular, a matrix of paradifferential real-to-real complex operators is
linearly Hamiltonian up to homogeneity $N$ if it has the form (cfr. (3.22))

$\\!\\!\\!J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(\begin{matrix}b_{1}(U;t,x,\xi)&b_{2}(U;t,x,\xi)\\\
\overline{b_{2}(U;t,x,-\xi)}&\overline{b_{1}(U;t,x,-\xi)}\end{matrix}\right)}\,,\
\begin{cases}b_{1}(U;t,x,-\xi)-b_{1}(U;t,x,\xi)\in\Gamma^{m}_{K,K^{\prime},N+1}[r]\\\
\mathrm{Im}\,b_{2}(U;t,x,\xi)\in\Gamma^{m^{\prime}}_{K,K^{\prime},N+1}[r]\end{cases}$
(3.31)

for some $m,m^{\prime}$ in $\mathbb{R}$.

######  Definition 3.7.

(Linearly symplectic map up to homogeneity $N$) A real-to-real matrix of
spectrally localized maps ${\bf S}(U;t)$ in
$\Sigma\mathcal{S}_{K,K^{\prime},0}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$ is
linearly symplectic up to homogeneity $N$ if

${\bf S}(U;t)^{\top}\,E_{c}\,{\bf S}(U;t)=E_{c}+S_{>N}(U;t)$ (3.32)

where $E_{c}$ is the symplectic operator defined in (3.13) and $S_{>N}(U;t)$
is a matrix of spectrally localized maps in
$\mathcal{S}_{K,K^{\prime},N+1}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$.

The approximate inverse up to homogeneity $N$ of a linearly symplectic map up
to homogeneity $N$ is still linearly symplectic up to homogeneity $N$.

######  Lemma 3.8.

Let $p,N\in\mathbb{N}$ with $p\leq N$. Let $\Phi_{\leq N}(U):={\bf B}_{\leq
N}(U)U$ be such that ${\bf B}_{\leq N}(U)-{\rm
Id}\in\Sigma_{p}^{N}\widetilde{\mathcal{S}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
and ${\bf B}_{\leq N}(U)$ is linearly symplectic up to homogeneity $N$
(Definition 3.7). Then its approximate inverse $\Psi_{\leq N}(V)$, constructed
in Lemma 2.24, has the form $\Psi_{\leq N}(V)={\bf A}_{\leq N}(V)V$ where
${\bf A}_{{\leq N}}(V)-{\rm Id}$ is in
$\Sigma_{p}^{N}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
and ${\bf A}_{{\leq N}}(V)$ is linearly symplectic up to homogeneity $N$, more
precisely

${\bf A}_{{\leq N}}(V)^{\top}\,E_{c}\,{\bf A}_{{\leq N}}(V)=E_{c}+S_{>N}(V)$
(3.33)

where $S_{>N}(V)$ is a matrix of pluri–homogeneous spectrally localized maps
in
$\Sigma_{N+1}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.

###### Proof.

As ${\bf B}_{\leq N}(U)$ is symplectic up to homogeneity $N$, one has

${\bf B}_{\leq N}(U)^{\top}\,E_{c}\,{\bf B}_{\leq N}(U)=E_{c}+S_{>N}(U)$
(3.34)

where $S_{>N}(U)$ is a pluri-homogeneous operator in
$\Sigma_{N+1}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
being the left hand side above a pluri-homogeneous operator). Then we evaluate
(3.34) at $U=\Psi_{\leq N}(V)$, apply ${\bf A}_{\leq N}(V)$ to the right and
${\bf A}_{\leq N}(V)^{\top}$ to the left and use (2.110) and the composition
properties in Proposition 2.19. The operator $S_{>N}(V)$ is pluri-homogeneous
as the left-hand side of (3.33). ∎

The class of linearly Hamiltonian operators up to homogeneity $N$ is closed
under conjugation under a linearly symplectic up to homogeneity $N$ map.

######  Lemma 3.9.

Let $J_{c}{\bf B}(U;t)$ be a linearly Hamiltonian operator up to homogeneity
$N$ (Definition 3.6) and $\mathcal{G}(U;t)$ be an invertible map, linearly
symplectic to homogeneity $N$ (Definition 3.7). Then the operators
$\mathcal{G}(U;t)J_{c}{\bf B}(U;t)\mathcal{G}(U;t)^{-1}$ and
$(\partial_{t}\mathcal{G}(U;t))\mathcal{G}^{-1}(U;t)$ are linearly Hamiltonian
up to homogeneity $N$.

###### Proof.

Set ${\bf B}:={\bf B}(U;t)$ and $\mathcal{G}:=\mathcal{G}(U;t)$ for brevity.
As $\mathcal{G}$ is invertible, we deduce from (3.32) that $\mathcal{P}_{\leq
N}\left(\mathcal{G}J_{c}\right)=\mathcal{P}_{\leq
N}\left(J_{c}[\mathcal{G}^{-1}]^{\top}\right)$. Then

$\displaystyle\mathcal{P}_{\leq N}\left(\mathcal{G}J_{c}{\bf
B}\mathcal{G}^{-1}\right)$ $\displaystyle=\mathcal{P}_{\leq
N}\left(\mathcal{G}J_{c}\mathcal{P}_{\leq N}[{\bf
B}]\mathcal{G}^{-1}\right)\stackrel{{\scriptstyle\eqref{A:sym23}}}{{=}}\mathcal{P}_{\leq
N}\left(J_{c}[\mathcal{G}^{-1}]^{\top}\mathcal{P}_{\leq N}[{\bf
B}]\mathcal{G}^{-1}\right)=J_{c}{\bf M}$

where ${\bf M}:=\mathcal{P}_{\leq
N}\left([\mathcal{G}^{-1}]^{\top}\mathcal{P}_{\leq N}[{\bf
B}]\mathcal{G}^{-1}\right)$ is symmetric since $\mathcal{P}_{\leq N}[{\bf
B}^{\top}]=\mathcal{P}_{\leq N}[{\bf B}]$. This proves that
$\mathcal{G}J_{c}{\bf B}\mathcal{G}^{-1}$ is linearly Hamiltonian up to
homogeneity $N$.

Next, differentiating (3.32) (with $\mathcal{G}(U;t)$ replacing ${\bf
S}(U;t)$), we get

$\mathcal{P}_{\leq
N}\left[E_{c}(\partial_{t}\mathcal{G})\mathcal{G}^{-1}\right]=-\mathcal{P}_{\leq
N}\left[(\mathcal{G}^{-1})^{\top}(\partial_{t}\mathcal{G})^{\top}E_{c}\right]=\mathcal{P}_{\leq
N}\left[\big{(}E_{c}(\partial_{t}\mathcal{G})\mathcal{G}^{-1}\big{)}^{\top}\right]$

showing that $(\partial_{t}\mathcal{G})\mathcal{G}^{-1}$ is linearly
Hamiltonian up to homogeneity $N$. ∎

##### Nonlinear Hamiltonian systems up to homogeneity $N$.

Let $K,K^{\prime}\in\mathbb{N}_{0}$ with $K^{\prime}\leq K$, $r>0$ and $U\in
B_{s_{0}}^{K}(I;r)$. Let

$Z:={\bf M}_{0}(U;t)U\quad\text{ with }\quad{\bf
M}_{0}(U;t)\in\mathcal{M}^{0}_{K,K^{\prime},0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})\,.$
(3.35)

######  Definition 3.10.

(Hamiltonian system up to homogeneity $N$) Let
$N,K,K^{\prime}\in\mathbb{N}_{0}$ with $K\geq K^{\prime}+1$ and assume (3.35).
A $U$–dependent system

$\partial_{t}Z=J_{c}\nabla H(Z)+M_{>N}(U;t)[U]$ (3.36)

is Hamiltonian up to homogeneity $N$ if

$\bullet$ $H(Z)$ is a pluri-homogeneous Hamiltonian in
$\Sigma_{2}^{N+2}\widetilde{\Uplambda}_{q}^{0}$;

$\bullet$ $M_{>N}(U;t)$ is a matrix of non-homogeneous operators in
$\mathcal{M}_{K,K^{\prime}+1,N+1}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$.

In view of the first bullet after Definition 2.31 the Hamiltonian vector field
$J_{c}\nabla H(Z)$ is in $\Sigma_{1}^{N+1}\widetilde{\mathfrak{X}}_{q}$.

We shall perform nonlinear changes of variables which are symplectic up to
homogeneity $N$ according to the following definition.

######  Definition 3.11.

(Symplectic map up to homogeneity $N$) Let $p,N\in\mathbb{N}$ with $p\leq N$.
We say that

$\mathcal{D}(Z;t)=M(Z;t)Z\qquad\text{with}\qquad M(Z;t)-{\rm
Id}\in\Sigma\mathcal{M}_{K,K^{\prime},p}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})\,,$
(3.37)

is symplectic up to homogeneity $N$, if its pluri-homogeneous component
$\mathcal{D}_{\leq N}(Z):=\big{(}\mathcal{P}_{\leq N}M(Z;t)\big{)}Z$ satisfies

$(\mathcal{D}_{\leq N})^{*}\Omega_{c}\,=\Omega_{c}+\Omega_{>N}$ (3.38)

where $\Omega_{>N}$ is a pluri-homogeneous 2-form in
$\Sigma_{N+1}\widetilde{\Uplambda}^{2}_{q}$.

Equivalently, by (3.12) and the operatorial representation (2.146) of
$2$-forms, the nonlinear map $\mathcal{D}(Z;t)$ is symplectic up to
homogeneity $N$, if

$\left[{\rm d}_{Z}\mathcal{D}_{\leq N}(Z)\right]^{\top}\,E_{c}\,{\rm
d}_{Z}\mathcal{D}_{\leq N}(Z)=E_{c}+E_{>N}(Z)\quad\text{with}\quad
E_{>N}(Z)\in\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,.$
(3.39)

######  Remark 3.12.

In the real setting we say that a map $\mathcal{D}(\eta,\zeta)$ is symplectic
up to homogeneity if its pluri-homogeneous component $\mathcal{D}_{\leq
N}(\eta,\zeta)$ satisfies

$\big{[}{\rm d}_{(\eta,\zeta)}\mathcal{D}_{\leq
N}(\eta,\zeta)\big{]}^{\top}\,E_{0}\,\big{[}{\rm
d}_{(\eta,\zeta)}\mathcal{D}_{\leq
N}(\eta,\zeta)\big{]}=E_{0}+E_{>N}(\eta,\zeta)$ (3.40)

where $E_{0}$ is the real symplectic tensor defined in (3.2) and $E_{>N}$ is
matrix of real operators in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.

We now show that the usual properties of symplectic maps still hold, up to
homogeneity $N$. For example the approximate inverse of a symplectic up to
homogeneity $N$ map is symplectic up to homogeneity $N$ as well.

######  Lemma 3.13.

Let $p,N\in\mathbb{N}$ with $p\leq N$. Let $\mathcal{D}_{\leq N}(Z)=Z+M_{\leq
N}(Z)Z$ as in (2.108) be symplectic up to homogeneity $N$. Then its
approximate inverse $\mathcal{E}_{\leq N}(V)=V+\breve{M}_{\leq N}(V)V$ up to
homogeneity $N$ as in (2.109) (provided by Lemma 2.24) is symplectic up to
homogeneity $N$ as well. Moreover

$\big{[}{\rm d}_{Z}\mathcal{D}_{\leq N}(Z)\big{]}J_{c}\big{[}{\rm
d}_{Z}\mathcal{D}_{\leq N}(Z)\big{]}^{\top}=J_{c}+J_{>N}(Z),\quad
J_{>N}(Z)\in\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\
.$ (3.41)

###### Proof.

As $\mathcal{D}_{\leq N}(Z)$ is symplectic up to homogeneity $N$, we get that,
using also the first bullet after Definition 2.32,

$(\mathcal{E}_{\leq N})^{*}(\mathcal{D}_{\leq
N})^{*}\Omega_{c}=(\mathcal{E}_{\leq
N})^{*}\left[\Omega_{c}+\Omega_{>N}\right]=(\mathcal{E}_{\leq
N})^{*}\Omega_{c}+\widetilde{\Omega}_{>N}$ (3.42)

for some pluri-homogeneous 2-forms $\Omega_{>N},\widetilde{\Omega}_{>N}$ in
$\Sigma_{N+1}\widetilde{\Uplambda}^{2}_{q}$. Now recall that, being
$\mathcal{E}_{\leq N}(V)$ the approximate inverse of $\mathcal{D}_{\leq N}(Z)$
up to homogeneity $N$, by (2.111) one has $\mathcal{D}_{\leq
N}\circ\mathcal{E}_{\leq N}={\rm Id}+F_{>N}$ for some $F_{>N}(V)=M_{>N}(V)V$
with $M_{>N}(V)$ in
$\Sigma_{N+1}\widetilde{\mathcal{M}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.
Thus we can also write

$(\mathcal{E}_{\leq N})^{*}(\mathcal{D}_{\leq
N})^{*}\Omega_{c}\stackrel{{\scriptstyle\eqref{pullback2}}}{{=}}\left(\mathcal{D}_{\leq
N}\circ\mathcal{E}_{\leq N}\right)^{*}\Omega_{c}=\left({\rm
Id}+F_{>N}\right)^{*}\Omega_{c}\stackrel{{\scriptstyle\eqref{pullback.form2}}}{{=}}\Omega_{c}+\Omega_{>N}^{\prime}(V)$
(3.43)

for some pluri-homogeneous 2-form $\Omega_{>N}^{\prime}$ in
$\Sigma_{N+1}\widetilde{\Uplambda}^{2}_{q}$ (by the first bullet below
Definition 2.5, Lemma 2.10 and Proposition 2.15). Then (3.42)-(3.43) prove
that $\mathcal{E}_{\leq N}$ is symplectic up to homogeneity $N$.

Next we prove (3.41). We start from (3.39) for $\mathcal{E}_{\leq N}$
evaluated at $\mathcal{D}_{\leq N}(Z)$, i.e.

$\big{[}{\rm d}_{V}\mathcal{E}_{\leq N}(\mathcal{D}_{\leq
N}(Z))\big{]}^{\top}\,E_{c}\,{\rm d}_{V}\mathcal{E}_{\leq N}(\mathcal{D}_{\leq
N}(Z))=E_{c}+E_{>N}(Z)$ (3.44)

with $E_{>N}(Z)$ in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$,
by Proposition 2.15 and Lemma 2.10. Then apply $J_{c}$ to the left of (3.44)
and $\big{[}{\rm d}_{Z}\mathcal{D}_{\leq N}(Z)\big{]}J_{c}\big{[}{\rm
d}_{Z}\mathcal{D}_{\leq N}(Z)\big{]}^{\top}$ to the right of it and use the
first and last of (2.112), and Proposition 2.15 to deduce (3.41). ∎

The approximate flow of a Hamiltonian smoothing vector field is symplectic up
to homogeneity $N$.

######  Lemma 3.14.

Let $p,N\in\mathbb{N}$ with $p\leq N$. Let $Y(U)$ be a homogeneous Hamiltonian
smoothing vector field in $\widetilde{\mathfrak{X}}_{p+1}^{-\varrho}$ for some
$\varrho\geq 0$. Then its approximate flow $\mathcal{F}_{\leq N}^{\tau}$
(provided by Lemma 2.26) is symplectic up to homogeneity $N$ (Definition
3.11).

###### Proof.

Recalling that $\Omega_{c}={\rm d}\theta_{c}$ we have

$\displaystyle\frac{{\rm d}}{{\rm d}\tau}(\mathcal{F}_{\leq
N}^{\tau})^{*}\Omega_{c}\,$
$\displaystyle\stackrel{{\scriptstyle\eqref{commu}}}{{=}}{\rm d}\frac{{\rm
d}}{{\rm d}\tau}(\mathcal{F}_{\leq N}^{\tau})^{*}\theta_{c}\,$
$\displaystyle\stackrel{{\scriptstyle\eqref{deriv:pullback}}}{{=}}{\rm
d}(\mathcal{F}_{\leq N}^{\tau})^{*}\mathcal{L}_{Y}\theta_{c}\,+{\rm
d}\theta_{>N+1}^{\tau}\,$
$\displaystyle\stackrel{{\scriptstyle\eqref{cartan},\eqref{commu},\eqref{d2=0}}}{{=}}(\mathcal{F}_{\leq
N}^{\tau})^{*}{\rm d}(i_{Y}\Omega_{c})\,+{\rm d}\theta_{>N+1}^{\tau}\,$
$\displaystyle\stackrel{{\scriptstyle\eqref{complexvecham}}}{{=}}(\mathcal{F}_{\leq
N}^{\tau})^{*}{\rm d}^{2}H_{p+2}\,+{\rm
d}\theta_{>N+1}^{\tau}\,\stackrel{{\scriptstyle\eqref{d2=0}}}{{=}}{\rm
d}\theta_{>N+1}^{\tau}\,$ (3.45)

where $H_{p+2}$ is the Hamiltonian of $Y(U)$ and $\theta_{>N+1}^{\tau}$ is a
pluri-homogeneous $1$-form in $\Sigma_{N+2}\widetilde{\Uplambda}^{1}_{q}$.
Integrating (3.45) from $0$ to $\tau$, and using that $\mathcal{F}_{\leq
N}^{0}={\rm Id}$, we get

$(\mathcal{F}_{\leq
N}^{\tau})^{*}\Omega_{c}\,=\Omega_{c}+\Omega_{>N}^{\tau}\,,\quad\Omega_{>N}^{\tau}:=\int_{0}^{\tau}{\rm
d}\theta_{>N+1}^{t}\,{\rm d}t$

where $\Omega_{>N}$ is in $\Sigma_{N+1}\widetilde{\Uplambda}^{2}_{q}$. This
proves that $\mathcal{F}_{\leq N}^{\tau}$ is symplectic up to homogeneity $N$.
∎

A symplectic map up to homogeneity $N$ transforms a Hamiltonian system up to
homogeneity $N$ into another Hamiltonian system up to homogeneity $N$.

######  Lemma 3.15.

Let $p,N\in\mathbb{N}$ with $p\leq N$, $K,K^{\prime}\in\mathbb{N}_{0}$ with
$K\geq K^{\prime}+1$. Let $Z:={\bf M}_{0}(U;t)U$ as in (3.35). Assume
$\mathcal{D}(Z;t)=M(Z;t)Z$ is a symplectic map up to homogeneity $N$
(Definition 3.11) such that

$M(Z;t)-{\rm
Id}\in\begin{cases}\Sigma\mathcal{M}_{K,K^{\prime},p}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})\qquad\qquad\text{if}\quad{\bf
M}_{0}(U;t)={\rm Id}\,,\\\
\Sigma\mathcal{M}_{K,0,p}[\breve{r},N]\otimes\mathcal{M}_{2}(\mathbb{C})\,,\,\forall\breve{r}>0\quad\text{otherwise}\,.\end{cases}$
(3.46)

If $Z(t)$ solves a $U$-dependent Hamiltonian system up to homogeneity $N$
(Definition 3.10), then the variable $W:=\mathcal{D}(Z;t)$ solves another
$U$-dependent Hamiltonian system up to homogeneity $N$ (generated by the
transformed Hamiltonian).

###### Proof.

Decompose $\mathcal{D}(Z;t)=\mathcal{D}_{\leq
N}(Z)+M_{>N}^{\mathcal{D}}(Z;t)Z$ where $\mathcal{D}_{\leq
N}(Z):=\mathcal{P}_{\leq N}[M(Z;t)]Z$ is its pluri-homogeneous component and

$M_{>N}^{\mathcal{D}}(Z;t)\in\begin{cases}\mathcal{M}_{K,{K^{\prime}},N+1}[r]\otimes\mathcal{M}_{2}(\mathbb{C})\qquad\qquad\text{if}\quad{\bf
M}_{0}(U;t)={\rm Id}\,,\\\
\mathcal{M}_{K,0,N+1}[\breve{r}]\otimes\mathcal{M}_{2}(\mathbb{C})\,,\,\forall\breve{r}>0\quad\text{otherwise}\,.\end{cases}$

By Definition 3.11 the map $\mathcal{D}_{\leq N}(Z)$ satisfies (3.38). If
$Z(t)$ solves (3.36) then $W=\mathcal{D}(Z;t)$ solves

$\displaystyle\partial_{t}W$ $\displaystyle=\big{(}{\rm
d}_{Z}\mathcal{D}_{\leq
N}(Z)+M_{>N}^{\mathcal{D}}(Z;t)\big{)}\left[J_{c}\nabla_{Z}H(Z)+M_{>N}(U;t)U\right]+(\partial_{t}M_{>N}^{\mathcal{D}}(Z;t))Z$
$\displaystyle={\rm d}_{Z}\mathcal{D}_{\leq
N}(Z)J_{c}\nabla_{Z}H(Z)+M^{\prime}_{>N}(U;t)U$ (3.47)

where, by the first bullet below Definition 2.5 and Proposition 2.15,

$M^{\prime}_{>N}(U;t)\in\mathcal{M}_{K,{K^{\prime}+1},N+1}[r]\otimes\mathcal{M}_{2}(\mathbb{C})\
.$ (3.48)

Denote by $\breve{\mathcal{D}}_{\leq N}(W)$ the approximate inverse up to
homogeneity $N$ of $\mathcal{D}_{\leq N}(Z)$ (see Lemma 2.24). Then

$\breve{\mathcal{D}}_{\leq N}(W)=\breve{\mathcal{D}}_{\leq
N}\big{(}\mathcal{D}_{\leq
N}(Z)+M_{>N}^{\mathcal{D}}(Z;t)Z\big{)}=Z+\breve{M}^{\prime}_{>N}(Z;t)Z$

where, by (2.111) and Proposition 2.15,

$\breve{M}^{\prime}_{>N}(Z;t)\in\begin{cases}\mathcal{M}_{K,{K^{\prime}},N+1}[r]\otimes\mathcal{M}_{2}(\mathbb{C})\qquad\qquad\text{if}\quad{\bf
M}_{0}(U;t)={\rm Id}\,,\\\
\mathcal{M}_{K,0,N+1}[\breve{r}]\otimes\mathcal{M}_{2}(\mathbb{C})\,,\,\forall\breve{r}>0\quad\text{otherwise}\,.\end{cases}$

Finally we substitute $Z={\bf M}_{0}(U;t)U$, cfr. (3.35), in the
non–homogeneous term $\breve{M}^{\prime}_{>N}(Z;t)Z$ and using $(iii)$ and
$(i)$ Proposition 2.15 we get

$Z=\breve{\mathcal{D}}_{\leq N}(W)+\breve{M}_{>N}(U;t)U$ (3.49)

with
$\breve{M}_{>N}(U;t)\in\mathcal{M}_{K,K^{\prime},N+1}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$.
We substitute (3.49) in the term $\nabla_{Z}H(Z)$ in (3.47) to obtain

$\displaystyle\partial_{t}W$ $\displaystyle={\rm d}_{Z}\mathcal{D}_{\leq
N}(Z)J_{c}\nabla_{Z}H(\breve{\mathcal{D}}_{\leq
N}(W))+M^{\prime\prime}_{>N}(U;t)U$
$\displaystyle\stackrel{{\scriptstyle\eqref{diinv2},\eqref{Z.M0}}}{{=}}{\rm
d}_{Z}\mathcal{D}_{\leq N}(Z)J_{c}[{\rm d}_{Z}\mathcal{D}_{\leq
N}(Z)]^{\top}[{\rm d}_{W}\breve{\mathcal{D}}_{\leq N}(\mathcal{D}_{\leq
N}(Z))]^{\top}\nabla_{Z}H(\breve{\mathcal{D}}_{\leq
N}(W))+M^{\prime\prime\prime}_{>N}(U;t)U$
$\displaystyle\stackrel{{\scriptstyle\eqref{lasolita},\eqref{Z.M0}}}{{=}}J_{c}[{\rm
d}_{W}\breve{\mathcal{D}}_{\leq
N}(W)]^{\top}\nabla_{Z}H(\breve{\mathcal{D}}_{\leq
N}(W))+M^{\prime\prime\prime\prime}_{>N}(U;t)U$
$\displaystyle=J_{c}\nabla_{W}(H\circ\breve{\mathcal{D}}_{\leq
N})(W)+M^{\prime\prime\prime\prime}_{>N}(U;t)U$ (3.50)

where $M^{\prime\prime}_{>N}(U;t)$, $M^{\prime\prime\prime}_{>N}(U;t),$
$M^{\prime\prime\prime\prime}_{>N}(U;t)$ are matrices of operators as in
(3.48). Note that in the very last passage we also substituted
$\mathcal{D}_{\leq N}(Z)=W+M_{>N}(U;t)U$ where $M_{>N}(U;t)$ is a matrix of
operators as in (3.48). This proves that system (3.50) is Hamiltonian up to
homogeneity $N$. ∎

####  3.3 Linear symplectic flows

We consider the flow of a linearly Hamiltonian up to homogeneity $N$
paradifferential operator.

######  Lemma 3.16.

(Linear symplectic flow) Let $p\in\mathbb{N}$,
$N,K,K^{\prime}\in\mathbb{N}_{0}$ with $K^{\prime}\leq K$, $m\leq 1$, $r>0$.
Let $J_{c}{{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(B\right)}$ be a
linearly Hamiltonian operator up to homogeneity $N$ (Definition 3.6) where
$B(\tau,U;t,x,\xi)$ is a matrix of symbols

$B(\tau,U;t,x,\xi):=\begin{pmatrix}b_{1}(\tau,U;t,x,\xi)&b_{2}(\tau,U;t,x,\xi)\\\
\overline{b_{2}(\tau,U;t,x,-\xi)}&\overline{b_{1}(\tau,U;t,x,-\xi)}\end{pmatrix},\quad\begin{cases}b_{1}\in\Sigma\Gamma^{0}_{K,K^{\prime},p}[r,N]\\\
b_{2}\in\Sigma\Gamma^{m}_{K,K^{\prime},p}[r,N]\,,\end{cases}$ (3.51)

with $b_{1}^{\vee}-b_{1}$ in $\Gamma^{0}_{K,K^{\prime},N+1}[r]$ and the
imaginary part $\textup{ Im }b_{2}$ in $\Gamma^{0}_{K,K^{\prime},N+1}[r]$
(cfr. (3.31)) uniformly in $|\tau|\leq 1$. Then there exists $s_{0}>0$ such
that, for any $U\in B_{s_{0},\mathbb{R}}^{K}(I;r)$, the system

$\begin{cases}\partial_{\tau}\mathcal{G}^{\tau}_{B}(U;t)=J_{c}\,{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\big{(}B(\tau,U;t,x,\xi)\big{)}\mathcal{G}^{\tau}_{B}(U;t)\\\
\mathcal{G}_{B}^{0}(U;t)={\rm Id}\,,\end{cases}$ (3.52)

has a unique solution $\mathcal{G}^{\tau}_{B}(U)$ defined for all $|\tau|\leq
1$, satisfying the following properties:

  * (i)

Boundedness: For any $s\in\mathbb{R}$ the linear map
$\mathcal{G}^{\tau}_{B}(U;t)$ is invertible and there is $r(s)\in]0,r[$ such
that for any $U\in B_{s_{0},\mathbb{R}}^{K}(I;r(s))$ for any $0\leq k\leq
K-K^{\prime}$, $V\in
C_{*}^{K-K^{\prime}}(I;\dot{H}^{s}(\mathbb{T},\mathbb{C}^{2}))$,

$\|\partial_{t}^{k}(\mathcal{G}^{\tau}_{B}(U;t)V)\|_{\dot{H}^{s-\frac{3}{2}k}}+\|\partial_{t}^{k}(\mathcal{G}^{\tau}_{B}(U;t)^{-1}V)\|_{\dot{H}^{s-\frac{3}{2}k}}\leq\big{(}1+C_{s,k}\|U\|_{k+K^{\prime},s_{0}}\big{)}\|V\|_{k,s}$
(3.53)

uniformly in $|\tau|\leq 1$.

In particular $\mathcal{G}_{B}^{\tau}(U;t)$ and
$\mathcal{G}_{B}^{\tau}(U;t)^{-1}$ are non–homogeneous spectrally localized
maps in
$\mathcal{S}_{K,K^{\prime},0}^{0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$
according to Definition 2.16.

  * (ii)

Linear symplecticity: The map $\mathcal{G}^{\tau}_{B}(U;t)$ is linearly
symplectic up to homogeneity $N$ (Definition 3.7). If $J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(B\right)}$ is linearly
Hamiltonian (Definition 3.2), then $\mathcal{G}^{\tau}_{B}(U;t)$ is linearly
symplectic (Definition 3.3).

  * (iii)

Homogeneous expansion: $\mathcal{G}_{B}^{\tau}(U;t)$ and its inverse are
spectrally localized maps and $\mathcal{G}_{B}^{\tau}(U;t)^{\pm}-{\rm Id}$
belong to
$\Sigma\mathcal{S}_{K,K^{\prime},p}^{(N+1)m_{0}}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$
with $m_{0}:=\max(m,0)$, uniformly in $|\tau|\leq 1$.

###### Proof.

Since the symbols $b_{1}$ and $\textup{ Im }b_{2}$ have order $0$ and
$\textup{ Re }b_{2}$ has order $m\leq 1$, the existence of the flow
$\mathcal{G}^{\tau}_{B}(U;t)$ and the estimates (3.53) (actually with loss of
$k$ derivatives instead of $\tfrac{3}{2}k$) are classical and follow as in
Lemma 3.22 of [16]. In view of (2.72), the bounds (3.53) imply that
$\mathcal{G}_{B}^{\tau}(U;t)$ is in
$\mathcal{S}_{K,K^{\prime},0}^{0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$. The
inverse $\mathcal{G}_{B}^{\tau}(U;t)^{-1}$ satisfies the same estimates
regarding it as the time $\tau$-flow
$\mathcal{G}^{\tau^{\prime}}_{B^{-}}(U;t)|_{\tau^{\prime}=\tau}$ of the system

$\partial_{\tau^{\prime}}\mathcal{G}_{B^{-}}^{\tau^{\prime}}(U;t)=J_{c}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B^{-}(\tau,\tau^{\prime},U;t,x,\xi))\mathcal{G}_{B^{-}}^{\tau^{\prime}}(U;t)\,,\quad\mathcal{G}_{B^{-}}^{0}(U;t)={\rm
Id}\,,$ (3.54)

where $B^{-}(\tau,U;t,x,\xi):=-B(\tau-\tau^{\prime},U;t,x,\xi)$.

Let us prove item $(ii)$. Set ${\bf B}:={{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}{(B(\tau,U;t,x,\xi))}$ and
$\mathcal{G}^{\tau}:=\mathcal{G}^{\tau}_{B}(U;t)$ for brevity. By (3.52) we
get, for any $\tau$,

$\displaystyle\partial_{\tau}(\mathcal{G}^{\tau})^{\top}\,E_{c}\,\mathcal{G}^{\tau}$
$\displaystyle=-(\mathcal{G}^{\tau})^{\top}{\bf
B}^{\top}J_{c}E_{c}\mathcal{G}^{\tau}+(\mathcal{G}^{\tau})^{\top}E_{c}J_{c}{\bf
B}\mathcal{G}^{\tau}$ $\displaystyle\stackrel{{\scriptstyle J_{c}E_{c}={\rm
Id},\eqref{Ecci}}}{{=}}(\mathcal{G}^{\tau})^{\top}({\bf B}-{\bf
B}^{\top})\mathcal{G}^{\tau}\stackrel{{\scriptstyle\eqref{ipoBnBtN}}}{{=}}(\mathcal{G}^{\tau})^{\top}({\bf
B}_{>N}-{\bf B}_{>N}^{\top})\mathcal{G}^{\tau}\,.$

Therefore

$(\mathcal{G}^{\tau})^{\top}\,E_{c}\,\mathcal{G}^{\tau}=E_{c}+S_{>N}\quad\text{where}\quad
S_{>N}:=\int_{0}^{\tau}(\mathcal{G}^{\tau^{\prime}})^{\top}({\bf B}_{>N}-{\bf
B}_{>N}^{\top})\mathcal{G}^{\tau^{\prime}}\,{\rm d}\tau^{\prime}$

is a matrix of spectrally localized maps in
$\mathcal{S}_{K,K^{\prime},N+1}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$ because
$\mathcal{G}_{B}^{\tau}(U;t)$ and $\mathcal{G}_{B}^{\tau}(U;t)^{-1}$ are in
$\mathcal{S}_{K,K^{\prime},0}^{0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$, the
paradifferential operator ${\bf B}$ belongs to
$\Sigma\mathcal{S}^{m_{0}}_{K,K^{\prime},p}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$
(see the fourth bullet after Definition 2.16), and $(ii)$ of Proposition 2.19.
This proves that the $\mathcal{G}^{\tau}_{B}(U;t)$ is linearly symplectic up
to homogeneity $N$ according to Definition 3.7. The same proof shows that, if
$J_{c}{{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(B\right)}$ is
linearly Hamiltonian, then $\mathcal{G}^{\tau}_{B}(U;t)$ is linearly
symplectic.

Let us prove item $(iii)$. By (3.52), iterating $N$–times the fundamental
theorem of calculus we get the expansion

$\mathcal{G}_{B}^{\tau}(U;t)={\rm
Id}+\sum_{j=1}^{N}S_{j}^{\tau}(U)+S_{>(pN)}^{\tau}(U;t)$ (3.55)

where

$S_{j}^{\tau}(U):=\int_{0}^{\tau}\\!\int_{0}^{\tau_{1}}\\!\\!\\!\cdots\\!\\!\int_{0}^{\tau_{j-1}}J_{c}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B(\tau_{1},U;t,x,\xi))\cdots
J_{c}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B(\tau_{j},U;t,x,\xi))\,{\rm
d}\tau_{1}\cdots{\rm d}\tau_{j}$

and, writing for brevity ${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B(\tau_{j},U)):={{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B(\tau_{j},U;t,x,\xi))$,

$S_{>(pN)}^{\tau}(U;t)=\int_{0}^{\tau}\\!\int_{0}^{\tau_{1}}\\!\\!\\!\cdots\\!\\!\int_{0}^{\tau_{N}}J_{c}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B(\tau_{1},U))\cdots J_{c}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B(\tau_{N+1},U))\mathcal{G}_{B}^{\tau_{N+1}}(U;t)\,{\rm
d}\tau_{1}\cdots{\rm d}\tau_{N+1}.$

Since each ${{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(B(\tau_{j},U))$
belongs to
$\Sigma\mathcal{S}^{m_{0}}_{K,K^{\prime},p}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$
and $\mathcal{G}_{B}^{\tau_{N+1}}(U)$ is in
$\mathcal{S}_{K,K^{\prime},0}^{0}\otimes\mathcal{M}_{2}(\mathbb{C})$ we
deduce, by $(ii)$ of Proposition 2.19, that $\mathcal{G}_{B}^{\tau}(U;t)-{\rm
Id}$ in (3.55) is a matrix of spectrally localized maps in
$\Sigma\mathcal{S}_{K,K^{\prime},p}^{(N+1)m_{0}}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$,
uniformly in $|\tau|\leq 1$. The analogous statement for
$\mathcal{G}_{B}^{\tau}(U;t)^{-1}-{\rm Id}$ follows by (3.54). ∎

The flow generated by a Fourier multiplier satisfies similar properties.

######  Lemma 3.17.

(Flow of a Fourier multiplier) Let $p\in\mathbb{N}$ and $g_{p}(Z;\xi)$ be a
$p$–homogeneous, $x$-independent, real symbol in
$\widetilde{\Gamma}^{\frac{3}{2}}_{p}$. Then the flow
$\mathcal{G}_{g_{p}}^{\tau}(Z)$ defined by

$\partial_{\tau}\mathcal{G}_{g_{p}}^{\tau}(Z)={{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left({\rm
i}g_{p}(Z;\xi)\right)}\mathcal{G}_{g_{p}}^{\tau}(Z)\,,\quad\mathcal{G}_{g_{p}}^{0}(Z)={\rm
Id}\,,$ (3.56)

is well defined for any $|\tau|\leq 1$ and satisfies the following properties:

  * $(i)$

Boundedness: For any $K\in\mathbb{N}$ and $r>0$ the flow
$\mathcal{G}^{\tau}_{g_{p}}(Z)$ is a real-to-real diagonal matrix of
spectrally localized maps in
$\mathcal{S}_{K,0,0}^{0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$. Moreover there
is $s_{0}>0$ such that for any $s\in\mathbb{R}$, there is $r(s)\in(0,r)$ such
that for any functions $Z\in B^{K}_{s_{0},\mathbb{R}}(I;r(s))$ and $W\in
C^{K}_{*}(I;\dot{H}^{s}(\mathbb{T},\mathbb{C}^{2}))$, it results, for any
$0\leq k\leq K$,

$\|\partial_{t}^{k}(\mathcal{G}_{g_{p}}^{\tau}(Z)W)\|_{\dot{H}^{s-\frac{3}{2}k}}+\|\partial_{t}^{k}(\mathcal{G}_{g_{p}}^{\tau}(Z)^{-1}W)\|_{\dot{H}^{s-\frac{3}{2}k}}\leq\big{(}1+C_{s,k}\|Z\|_{k,s_{0}}\big{)}\|W\|_{k,s}$
(3.57)

uniformly in $|\tau|\leq 1$.

  * $(ii)$

Linear symplecticity: The flow map $\mathcal{G}^{\tau}_{g_{p}}(Z)$ is linearly
symplectic (Definition 3.3).

  * $(iii)$

Homogeneous expansion: The flow map $\mathcal{G}^{\tau}_{g_{p}}(Z)$ and its
inverse $\mathcal{G}^{-\tau}_{g_{p}}(Z)$ are matrices of spectrally localized
maps such that $\mathcal{G}^{\pm\tau}_{g_{p}}(Z)-{\rm Id}$ belong to
$\Sigma\mathcal{S}^{\frac{3}{2}(N+1)}_{K,0,p}[r,N]\otimes\mathcal{M}_{2}(\mathbb{C})$,
uniformly in $|\tau|\leq 1$.

###### Proof.

Since $g_{p}(Z;\xi)$ is real and independent of $x$, then the flow
$\mathcal{G}^{\tau}_{g_{p}}(Z)$ is well defined in $\dot{H}^{s}$ and it is
unitary, namely
$\|\mathcal{G}^{\tau}_{g_{p}}(Z)W\|_{\dot{H}^{s}}=\|W\|_{\dot{H}^{s}}$.
Moreover, since $g_{p}$ is a Fourier multiplier of order $\frac{3}{2}$, we
have

$\displaystyle\|\partial_{t}(\mathcal{G}^{\tau}_{g_{p}}(Z)W)\|_{\dot{H}^{s-\frac{3}{2}}}$
$\displaystyle=\|\mathcal{G}^{\tau}_{g_{p}}(Z)\partial_{t}W\|_{\dot{H}^{s-\frac{3}{2}}}+\|{{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left({\rm
i}\partial_{t}g_{p}(Z;\xi)\right)}\mathcal{G}^{\tau}_{g_{p}}(Z)W\|_{\dot{H}^{s-\frac{3}{2}}}$
$\displaystyle\leq\|W\|_{1,s}+C\|Z\|_{1,s_{0}}^{p}\|W\|_{0,s}\,.$

The estimates for the $k$–th derivative follow similarly using also that
$\mathcal{G}^{\tau}_{g_{p}}(Z)^{-1}=\mathcal{G}^{-\tau}_{g_{p}}(Z)$.

To prove $(ii)$ we use that, in view of (2.24), (3.22) and since
$g_{p}(Z;\xi)$ is real valued, the operator ${{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left({\rm
i}g_{p}(Z;\xi)\right)}$ is linearly Hamiltonian, according to Definition 3.2.
Then, as for item ($ii$) of Lemma 3.16, the flow
$\mathcal{G}^{\tau}_{g_{p}}(Z)$ is linearly symplectic. Finally also item
$(iii)$ follows as for item ($iii$) of Lemma 3.16, since ${{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left({\rm
i}g_{p}(Z;\xi)\right)}^{k}$ is in
$\widetilde{\mathcal{S}}_{kp}^{\frac{3}{2}k}\otimes\mathcal{M}_{2}(\mathbb{C})$
and $\mathcal{G}^{\tau}_{g_{p}}(Z)$ is in
$\mathcal{S}_{K,0,0}^{0}[r]\otimes\mathcal{M}_{2}(\mathbb{C})$, uniformly in
$|\tau|\leq 1$. ∎

####  3.4 Paradifferential Hamiltonian structure

In order to compute the Hamiltonian vector field associated to a
paradifferential Hamiltonian we provide the following result.

######  Lemma 3.18.

Let $p\in\mathbb{N}$, $m\in\mathbb{R}$. Let $S(U)$ be a real-to-real symmetric
matrix of $p$-homogeneous spectrally localized maps in
$\widetilde{\mathcal{S}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$ and define
the Hamiltonian function

$H(U):=\frac{1}{2}\big{\langle}S(U)U,U\big{\rangle}_{r}\ .$ (3.58)

Then its gradient

$\nabla H(U)=S(U)U+R(U)U$ (3.59)

where $R(U)$ is a real-to-real matrix of homogeneous smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ for
any $\varrho\geq 0$.

###### Proof.

By the definition (2.151), the gradient $\nabla H(U)$ is the vector field

$\nabla H(U)=S(U)U+L(U)^{\top}U\quad\text{where}\quad L(U)W:=\frac{1}{2}{\rm
d}_{U}S(U)[W]U\,.$ (3.60)

As $S(U)$ is a spectrally localized map in
$\widetilde{\mathcal{S}}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$, by Lemma
2.21 the transposed of its internal differential, namely $L(U)^{\top}$, is a
smoothing operator in
$\widetilde{\mathcal{R}}_{p}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$ for
any $\varrho\geq 0$. Then (3.59) follows from (3.60). ∎

As a corollary we obtain the Hamiltonian vector field associated to a
paradifferential Hamiltonian.

######  Lemma 3.19.

Let $p\in\mathbb{N}$, $m\in\mathbb{R}$ and $a(U;x,\xi)$ a real valued
homogeneous symbol in $\widetilde{\Gamma}^{m}_{p}$. Then the Hamiltonian
vector field generated by the Hamiltonian

$H(U):=\mathrm{Re}\,\langle
A(U)u,\overline{u}\rangle_{\dot{L}^{2}_{r}}=\frac{1}{2}\Big{\langle}\begin{pmatrix}0&\overline{A(U)}\\\
A(U)&0\end{pmatrix}U,U\Big{\rangle}_{r},\quad A(U):={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(a(U;x,\xi)\right)}\,,$

is

$J_{c}\nabla H(U)={{{\rm
Op}_{\mathtt{vec}}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(-{\rm
i}a(U;x,\xi)\right)}U+R(U)U$

where $R(U)$ is a real-to-real matrix of homogeneous smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ for
any $\varrho\geq 0$.

We now prove that if a homogeneous Hamiltonian vector field $X(U)=J_{c}\nabla
H(U)$ can be written in paradifferential form

$X(U)=J_{c}{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))U+R(U)U$

where $A(U)$ is a matrix of symbols and $R(U)$ is a smoothing operator, then
${{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}={{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}^{\top}$ up to a
smoothing operator. As a consequence we may always assume, up to modifying the
smoothing operator, that the paradifferential operator ${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))$ is symmetric, namely that
$J_{c}{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))$ is linearly
Hamiltonian.

######  Lemma 3.20.

Let $p\in\mathbb{N}$, $m\in\mathbb{R}$ and $\varrho\geq 0$. Let

$X(U)=J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U;x,\xi)\right)}U+R(U)U=J_{c}\nabla
H(U)$ (3.61)

be a $(p+1)$-homogeneous Hamiltonian vector field, where (cfr. (2.28))

$A(U;x,\xi)=\begin{pmatrix}a(U;x,\xi)&b(U;x,\xi)\\\
\overline{b(U;x,-\xi)}&\overline{a(U;x,-\xi)}\end{pmatrix}$ (3.62)

is matrix of symbols in
$\widetilde{\Gamma}_{p}^{m}\otimes\mathcal{M}_{2}(\mathbb{C})$ and $R(U)$ is a
real-to-real matrix of smoothing operators in
$\widetilde{\mathcal{R}}_{p}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$.
Then we may write

$X(U)=J_{c}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A_{1}(U;x,\xi))U+R_{1}(U)U$ (3.63)

where the matrix of paradifferential operators ${{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A_{1}(U;x,\xi)\right)}$ is
symmetric, with matrix of symbols

$A_{1}(U;x,\xi)=\frac{1}{2}\begin{pmatrix}a+a^{\vee}&b+\overline{b}\\\
\overline{b}^{\vee}+b^{\vee}&\overline{a+a^{\vee}}\end{pmatrix}$ (3.64)

and $R_{1}(U)$ is another real-to-real matrix of smoothing operators in
$\widetilde{\mathcal{R}}_{p}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$.

###### Proof.

The linear vector field ${\rm d}_{U}X(U)$, obtained linearizing a Hamiltonian
vector field $X(U)=J_{c}\nabla H(U)$, is Hamiltonian, namely

${\rm d}_{U}X(U)=J_{c}S(U)\quad\text{where}\quad S(U)=S(U)^{\top}\quad\text{is
symmetric}\,.$

On the other hand, by linearizing (3.61),

$S(U)={{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))+{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}({\rm
d}_{U}A(U)[\cdot])U+J_{c}\breve{R}(U)$

where $\breve{R}(U):=R(U)+{\rm d}_{U}R(U)[\cdot]U$ is a matrix of smoothing
operators in
$\widetilde{\mathcal{R}}_{p}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})$
(see the remark after Definition 2.5). Then, since $S(U)$ is symmetric, it
results, writing for brevity $A:=A(U)$,

${{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A)-{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A)^{\top}=\big{(}(J_{c}\breve{R}(U))^{\top}-{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}({\rm
d}_{U}A(U)[\cdot])U\big{)}+\big{(}\big{[}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}({\rm
d}_{U}A(U)[\cdot])U\big{]}^{\top}-J_{c}\breve{R}(U)\big{)}.$

We now apply Lemma 2.22 to the spectrally localized map ${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))-{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))^{\top}$ which has the form
(2.106) with

$L:=\big{(}(J_{c}\breve{R}(U))^{\top}-{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}({\rm
d}_{U}A(U)[\cdot])U\big{)}\,,\quad R:=\big{(}\big{[}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}({\rm
d}_{U}A(U)[\cdot])U\big{]}^{\top}-J_{c}\breve{R}(U)\big{)}\,.$ (3.65)

By Lemma 2.21, the operator $\big{[}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}({\rm
d}_{U}A(U)[\cdot])U\big{]}^{\top}$ is in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$,
and therefore both $L^{\top}$ and $R$ in (3.65) are $p$-homogeneous smoothing
operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$.
The assumptions of Lemma 2.22 are satisfied, implying that

${{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))-{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}(A(U))^{\top}=:R^{\prime}(U)\in\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})\,.$

In conclusion we deduce (3.63) with

${{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}{(A_{1}(U))}:=\frac{1}{2}\Big{(}{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}{(A(U))}+{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}{(A(U))}^{\top}\Big{)}\,,\quad
R_{1}(U):=\frac{1}{2}J_{c}R^{\prime}(U)+R(U)\,,$

and (3.64) follows recalling (2.26). ∎

Another consequence of Lemma 2.22 is the following.

######  Lemma 3.21.

Let $p\in\mathbb{N}$, $m\in\mathbb{R}$ and $\varrho\geq 0$. Let $S(U)$ be a
matrix of spectrally localized homogeneous maps in
$\widetilde{\mathcal{S}}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ which is
linearly Hamiltonian (Definition 3.2) of the form

$S(U)=J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U;x,\xi)\right)}+R(U)\,,$
(3.66)

where $A(U;x,\xi)$ is a real-to-real matrix of symbols in
$\widetilde{\Gamma}^{m}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ as in (3.62),
and $R(U)$ is a real-to-real matrix of smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$.
Then we may write

$S(U)=J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A_{1}(U;x,\xi)\right)}+R_{1}(U)\,$

where the matrix of symbols $A_{1}(U;x,\xi)$ in
$\widetilde{\Gamma}^{m}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$ has the form
(3.64) and $R_{1}(U)$ is another matrix of real-to-real smoothing operators in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$. In
particular the homogeneous operator $J_{c}{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A_{1}(U)\right)}$ is linearly
Hamiltonian.

###### Proof.

It is enough to prove that the operator ${{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}$ is equal to
${{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}^{\top}$ up
to a matrix of smoothing operators. To prove this claim, recall that $S(U)$
linearly Hamiltonian means that $E_{c}S(U)$ is symmetric, so that by (3.66)
one gets

${{{\rm Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}-{{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}^{\top}=-R(U)^{\top}E_{c}-E_{c}R(U)\,.$

Now, since $S(U)$ and ${{{\rm
Op}^{{\scriptscriptstyle{\mathrm{B}W}}}}\left(A(U)\right)}$ are spectrally
localized maps, so is $R(U)$ in (3.66). By Remark 2.11 the transpose
$R(U)^{\top}$ is also a smoothing operator in
$\widetilde{\mathcal{R}}^{-\varrho}_{p}\otimes\mathcal{M}_{2}(\mathbb{C})$,
proving the claim. ∎

###  4 Construction of a Darboux symplectic corrector

If ${\bf B}(U;t)$ is a spectrally localized map which is linearly symplectic
up to homogeneity $N$, then the associated nonlinear map $\Phi_{\leq
N}(U):={\bf B}_{\leq N}(U)U$, where ${\bf B}_{\leq N}(U)=\mathcal{P}_{\leq
N}({\bf B}(U;t))$, is not symplectic up to homogeneity $N$. In this section we
provide a systematic procedure to construct a nearby nonlinear map which is
symplectic up to homogeneity $N$ according to Definition 3.11.

######  Theorem 4.1.

(Symplectic correction up to homogeneity $N$) Let $p,N\in\mathbb{N}$ with
$p\leq N$. Consider a nonlinear map

$\Phi_{\leq N}(U):={\bf B}_{\leq N}(U)U\,,$ (4.1)

where

  1. (i)

${\bf B}_{\leq N}(U)-{\rm Id}$ is a matrix of pluri–homogeneous spectrally
localized maps in
$\Sigma_{p}^{N}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$;

  2. (ii)

${\bf B}_{\leq N}(U)$ is linearly symplectic up to homogeneity $N$ (Definition
3.7).

Then there exists a real-to-real map

$\mathcal{C}_{\leq N}(W)=W+R_{\leq N}(W)W\quad\text{with}\quad R_{\leq
N}(W)\in\Sigma_{p}^{N}{\widetilde{\mathcal{R}}}_{q}^{-\varrho}\otimes\mathcal{M}_{2}(\mathbb{C})\,,\
\mbox{ for any }\varrho\geq 0\,,$ (4.2)

such that the Darboux correction

$\mathcal{D}_{N}(U):=\big{(}\mathcal{C}_{\leq N}\circ\Phi_{\leq
N}\big{)}(U)=\big{(}{\rm Id}+R_{\leq N}(\Phi_{\leq N}(U))\big{)}\Phi_{\leq
N}(U)$ (4.3)

is symplectic up to homogeneity $N$, according to Definition 3.11.

######  Remark 4.2.

The first assumption implies that the operator in (4.7) is smoothing for any
$\varrho>0$. This fact and the second assumption allow to deduce that the
vector field representing the perturbed symplectic $1$-form $\theta_{\leq N}$
in (4.10) is a smoothing perturbation of $E_{c}V$, see (4.11). These
properties are crucial to guarantee that the vector field $Y^{\tau}(V)$
solving the Darboux equation (4.26) is smoothing (see Lemma 4.9), which in
turn implies that the Darboux corrector $\mathcal{C}_{\leq N}(W)$ in (4.2) is
a smoothing perturbation of the identity.

The rest of this section is devoted to the proof of Theorem 4.1.

In order to correct the nonlinear map $\Phi_{\leq N}$ defined in (4.1) we
develop a perturbative Darboux procedure to construct a nearby symplectic map
up to homogeneity $N$. The map $\Phi_{{\leq N}}$ induces the nonstandard
symplectic $2$-form

$\Omega_{\leq N}:=\Psi_{\leq N}^{*}\ \Omega_{c}$ (4.4)

where $\Psi_{{\leq N}}$ is the approximate inverse of $\Phi_{{\leq N}}$
defined by Lemma 2.24 and $\Omega_{c}$ is the standard symplectic form in
(3.12). The next lemma describes properties of the approximate inverse
$\Psi_{\leq N}$.

######  Lemma 4.3.

(Approximate inverse) The approximate inverse up to homogeneity $N$ of the map
$\Phi_{\leq N}(U)={\bf B}_{\leq N}(U)U$ defined in (4.1) has the form

$\Psi_{\leq N}(V)={\bf A}_{\leq N}(V)V$ (4.5)

where

  * (i)

${\bf A}_{{\leq N}}(V)-{\rm Id}$ is a matrix of pluri-homogeneous spectrally
localized maps in
$\Sigma_{p}^{N}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$;

  * (ii)

${\bf A}_{{\leq N}}(V)$ is linearly symplectic up to homogeneity $N$
(Definition 3.7), more precisely (3.33) holds.

In addition

${\rm d}_{V}\Psi_{{\leq N}}(V)={\bf A}_{{\leq N}}(V)+{\bf G}_{{\leq
N}}(V)\,,\qquad{\bf G}_{{\leq N}}(V)\widehat{V}:={\rm d}_{V}{\bf A}_{{\leq
N}}(V)[\widehat{V}]V\,,$ (4.6)

and

  * (iii)

${\bf G}_{{\leq N}}(V)$ is a matrix of pluri–homogeneous operators in
$\Sigma_{p}^{N}\widetilde{\mathcal{M}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$;

  * (iv)

the transposed operator

${\bf G}_{{\leq N}}^{\top}(V):=[{\bf G}_{{\leq
N}}(V)]^{\top}\in\Sigma_{p}^{N}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
(4.7)

is a matrix of $\varrho$-smoothing operators for arbitrary $\varrho\geq 0$ .

###### Proof.

Items $(i)$ and $(ii)$ are proved in Lemma 3.8, and $(iii)$ follows by the
fifth bullet after Definition 2.16. Finally $(iv)$ follows applying Lemma 2.21
to each spectrally localized map $\mathcal{P}_{q}({\bf A}_{\leq N}(V))$ for
$q=p,\ldots,N$ (with $U\leadsto V$ and $V\leadsto\widehat{V}$). ∎

We now compute $\Omega_{{\leq N}}$.

######  Lemma 4.4.

(Non-standard symplectic form $\Omega_{\leq N}$) The symplectic $2$-form
$\Omega_{{\leq N}}=\Psi_{\leq N}^{*}\ \Omega_{c}$ in (4.4) is represented as
$\Omega_{{\leq N}}(V)=\langle E_{{\leq N}}(V)\cdot,\cdot\rangle_{r}$ with
symplectic tensor

$E_{{\leq N}}(V)=E_{c}+{\bf A}_{\leq N}^{\top}(V)E_{c}{\bf G}_{\leq N}(V)+{\bf
G}_{\leq N}^{\top}(V)E_{c}{\bf A}_{\leq N}(V)+{\bf G}_{\leq
N}^{\top}(V)E_{c}{\bf G}_{\leq N}(V)+{{\bf S}}_{>N}(V)$ (4.8)

where

  * (i)

${\bf G}_{{\leq N}}^{\top}(V)E_{c}{\bf A}_{{\leq N}}(V)$ and ${\bf G}_{{\leq
N}}^{\top}(V)E_{c}{\bf G}_{{\leq N}}(V)$ are matrices of pluri–homogeneous
smoothing operators in
$\Sigma_{p}^{2N}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$;

  * (ii)

${{\bf S}}_{>N}(V):={\bf A}_{\leq N}^{\top}(V)E_{c}{\bf A}_{\leq N}(V)-E_{c}$
is a matrix of pluri–homogeneous spectrally localized maps in
$\Sigma_{N+1}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.

Moreover

$\Omega_{{\leq N}}={\rm d}\theta_{\leq N}\,,$ (4.9)

where the $1$-form

$\theta_{\leq N}:=\Psi_{\leq N}^{*}\ \theta_{c}\
,\qquad\theta_{c}(V):=\frac{1}{2}\big{\langle}E_{c}V,\cdot\big{\rangle}_{r}\,,$
(4.10)

has the form

$\theta_{{\leq N}}(V)=\frac{1}{2}\Big{\langle}[Z_{{\leq N}}(V)+{{\bf
S}}_{>N}(V)]V,\cdot\Big{\rangle}_{r}\quad\text{with}\quad Z_{{\leq
N}}(V):=E_{c}+{\bf G}_{{\leq N}}^{\top}(V)\ E_{c}\ {\bf A}_{{\leq N}}(V)\,.$
(4.11)

###### Proof.

By (2.161) we have that

$\Omega_{{\leq N}}(V)[X,Y]=\big{\langle}{\rm d}\Psi_{{\leq
N}}(V)^{\top}\,E_{c}\,{\rm d}\Psi_{{\leq N}}(V)X,Y\big{\rangle}_{r}$

which, using (4.6) and the fact that ${\bf A}_{\leq N}(V)$ is linearly
symplectic up to homogeneity $N$ (cfr. (3.33)), provides formula (4.8). Then
items ($i$)-($ii$) follow by (4.6), (4.7) and Proposition 2.15. The identity
(4.9) follows by (3.15) and (2.157). Finally (4.11) follows similarly
computing $\Psi_{\leq N}^{*}\theta_{c}$ by (2.160),

$\theta_{\leq N}(V)[X]=\frac{1}{2}\big{\langle}{\rm d}_{V}\Psi_{\leq
N}(V)^{\top}\,\,E_{c}\,\Psi_{\leq N}(V),X\big{\rangle}\,,$

and using (4.5), (4.6) and ${\bf A}_{\leq N}^{\top}(V)E_{c}{\bf A}_{\leq
N}(V)=E_{c}+{{\bf S}}_{>N}(V)$. ∎

The key step is to implement a Darboux–type procedure to transform the
symplectic form $\Omega_{{\leq N}}$ back to the standard symplectic form
$\Omega_{c}$ up to arbitrary high degree of homogeneity. It turns out that the
required transformation is a smoothing perturbation of the identity as claimed
in Theorem 4.1, see Proposition 4.7. This is not at all obvious, since in the
expression (4.8) of $E_{\leq N}(V)$ the second operator ${\bf A}_{\leq
N}^{\top}(V)E_{c}{\bf G}_{\leq N}(V)$ is not smoothing. However it has a nice
structure that we now describe.

######  Lemma 4.5.

Let $X(V)$ be a pluri-homogeneous vector field in
$\Sigma_{p^{\prime}+1}\widetilde{\mathfrak{X}}_{q}$ for some
$p^{\prime}\in\mathbb{N}_{0}$. Then

${\bf A}^{\top}_{\leq N}(V)E_{c}{\bf G}_{\leq
N}(V)[X(V)]=\nabla\mathcal{W}(V)+R(V)V+M_{>N}(V)V$ (4.12)

where

  * •

$\mathcal{W}(V)$ is $0$-form in
$\Sigma_{p+p^{\prime}+2}\widetilde{\Uplambda}^{0}_{q}$;

  * •

$R(V)$ is matrix of pluri–homogeneous smoothing operators in
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$;

  * •

$M_{>N}(V)$ is a matrix of pluri–homogeneous operators in
$\Sigma_{N+1}{\widetilde{\mathcal{M}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.

###### Proof.

For simplicity of notation we set ${\bf A}(V):={\bf A}_{\leq N}(V)$ and ${\bf
G}(V):={\bf G}_{\leq N}(V)$.

Step 1: For any vector $W$, the linear operator

${\bf K}_{W}(V):={\bf A}(V)^{\top}\,E_{c}\,\big{[}{\rm d}_{V}{\bf
A}(V)[W]\big{]}$ (4.13)

is symmetric up to homogeneity $N$, precisely

${\bf K}_{W}(V)-{\bf K}_{W}(V)^{\top}={\rm d}_{V}{{\bf S}}_{>N}(V)[W]$ (4.14)

where ${{\bf S}}_{>N}(V)$ is the spectrally localized map in
$\Sigma_{N+1}\widetilde{\mathcal{S}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$ of
Lemma 4.4.
Indeed, differentiating the relation ${\bf A}(V)^{\top}E_{c}{\bf
A}(V)=E_{c}+{{\bf S}}_{>N}(V)$ (see Lemma 4.4 $(ii)$), in direction $W$, we
get

$\displaystyle{\rm d}_{V}{{\bf S}}_{>N}(V)[W]$ $\displaystyle={\bf
A}(V)^{\top}E_{c}\,\big{[}{\rm d}_{V}{\bf A}(V)[W]\big{]}+\big{[}{\rm
d}_{V}{\bf A}(V)[W]\big{]}^{\top}E_{c}{\bf A}(V)$ $\displaystyle={\bf
A}(V)^{\top}E_{c}\,\big{[}{\rm d}_{V}{\bf A}(V)[W]\big{]}-\Big{(}{\bf
A}(V)^{\top}E_{c}\,\big{[}{\rm d}_{V}{\bf A}(V)[W]\big{]}\Big{)}^{\top}$

proving, in view of (4.13), (4.14).

Step 2: The linear operator

${\bf K}(V):={\bf K}_{X(V)}(V)={\bf A}(V)^{\top}E_{c}\big{[}{\rm d}_{V}{\bf
A}(V)[X(V)]\big{]}$ (4.15)

can be decomposed as

${\bf K}(V)=S(V)+\breve{R}(V)+M_{>N}(V)$ (4.16)

where
$\bullet$ $S(V)$ is a symmetric matrix of spectrally localized
pluri–homogeneous maps in
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$;
$\bullet$ $\breve{R}(V)$ is a symmetric matrix of pluri–homogeneous smoothing
operators in
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$ as well as its transpose;
$\bullet$ $M_{>N}(V):=\frac{1}{2}{\rm d}_{V}{{\bf S}}_{>N}(V)[X(V)]$ is a
matrix of pluri–homogeneous operators in
$\Sigma_{N+1}\widetilde{\mathcal{M}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$.
We apply Lemma 2.20 to each component ${\bf A}_{q}(V):=\mathcal{P}_{q}[{\bf
A}(V)]$, $q=p,\ldots,N$, each of which is a map in
$\widetilde{\mathcal{S}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$, see Lemma 4.3
$(i)$. Lemma 2.20 (with $M(U)U\leadsto X(V)$) gives the decomposition

${\rm d}_{V}{\bf A}(V)[X(V)]=\sum_{q=p}^{N}q{\bf
A}_{q}\big{(}X(V),V,\dots,V\big{)}=S^{\prime}(V)+R^{\prime}(V)\,,$ (4.17)

where $S^{\prime}(V)$ is a matrix of spectrally localized pluri–homogeneous
maps in
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
and $R^{\prime}(V)$ is a matrix of pluri–homogeneous smoothing operators in
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$, as well as its transpose. Then we obtain by (4.14)
(with $W=X(V)$),

$\displaystyle{\bf K}(V)=$ $\displaystyle\,\frac{1}{2}({\bf K}(V)+{\bf
K}(V)^{\top})+\frac{1}{2}{\rm d}_{V}{{\bf S}}_{>N}(V)[X(V)]$
$\displaystyle\stackrel{{\scriptstyle\eqref{KV},\eqref{decompl0}}}{{=}}$
$\displaystyle\underbrace{\frac{1}{2}\big{[}{\bf
A}^{\top}E_{c}S^{\prime}-S^{\prime\top}E_{c}{\bf A}\big{]}(V)}_{:=S(V),\
S(V)=S(V)^{\top}}+\underbrace{\frac{1}{2}\big{[}{\bf
A}^{\top}E_{c}R^{\prime}-R^{\prime\top}E_{c}{\bf
A}\big{]}(V)}_{:=\breve{R}(V),\
\breve{R}(V)=\breve{R}(V)^{\top}}+M_{>N}(V)\,,$

where $M_{>N}(V)=\frac{1}{2}{\rm d}_{V}{{\bf S}}_{>N}(V)[X(V)]$ is in
$\Sigma_{N+1}\widetilde{\mathcal{M}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$ by
Proposition 2.15. Since the maps $S^{\prime}(V)$ and ${\bf A}(V)$ are
spectrally localized then $S(V)$ belongs to
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
by Proposition 2.19 $(ii)$ and Lemma 2.18. The operator $\breve{R}(V)$ is in
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$ by Proposition 2.19 $(i)$, Lemma 2.18 and the fact
that $R^{\prime}(V)$ belongs
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$ as well as its transpose.

Conclusion: The identity (4.12) follows by Step 2 defining

$\mathcal{W}(V):=\frac{1}{2}\big{\langle}S(V)V,V\big{\rangle}_{r}\,,\quad
S(V)\mbox{ in }\eqref{decompl}\ .$

Indeed by Lemma 3.18, $\nabla\mathcal{W}(V)=S(V)V+R^{\prime\prime}(V)V$ for
some smoothing operator $R^{\prime\prime}(V)$ belonging to
$\widetilde{\mathcal{R}}^{-\varrho}_{p+p^{\prime}}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho>0$. Then we get

$\displaystyle{\bf A}^{\top}_{\leq N}(V)E_{c}{\bf G}_{\leq N}(V)[X(V)]$
$\displaystyle\stackrel{{\scriptstyle\eqref{selfad},\eqref{sommabella2}}}{{=}}{\bf
K}(V)V\stackrel{{\scriptstyle\eqref{decompl}}}{{=}}S(V)V+\breve{R}(V)V+M_{>N}(V)V$
$\displaystyle=\nabla\mathcal{W}(V)-R^{\prime\prime}(V)V+\breve{R}(V)V+M_{>N}(V)V\
,$

proving (4.12) with $R(V):=\breve{R}(V)-R^{\prime\prime}(V)$ which belongs to
$\Sigma_{p+p^{\prime}}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$. ∎

######  Remark 4.6.

The vector field $R(V)V$ in (4.12) depends on $X(V)$ and its differential
${\rm d}_{V}X(V)$. This is because $S(V)$ depends linearly on $X(V)$ (actually
it is the term $S^{\prime}(V)$ that depends linearly on $X(V)$, see (4.17)).
Hence the smoothing vector field $R^{\prime\prime}(V)V$ coming from the
gradient $\nabla\mathcal{W}(V)$, given explicitly by
$R^{\prime\prime}(V)V:=L(V)^{\top}V$ where $L(V)W=\frac{1}{2}{\rm
d}_{V}S(V)[W]V$ (see (3.60)), depends on the differential of $X(V)$.

Now we present the main Darboux procedure.

######  Proposition 4.7.

(Darboux procedure) There exists a $\tau$–dependent pluri-homogeneous
smoothing vector field $Y^{\tau}(V)$ in
$\Sigma_{p+1}\widetilde{\mathfrak{X}}^{-\varrho}_{p+1}$, for any $\varrho\geq
0$, defined for $\tau\in[0,1]$, such that its approximate time $1$–flow

$\mathcal{F}_{\leq N}(V)=V+R_{\leq N}(V)V\quad\text{with}\quad R_{\leq
N}(V)\in\Sigma_{p}^{N}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})\,,\
\ \forall\varrho\geq 0\,,$ (4.18)

(given by Lemma 2.26) satisfies

$\mathcal{F}_{\leq N}^{*}\Omega_{{\leq N}}=\Omega_{c}+\Omega_{>N}$ (4.19)

where $\Omega_{>N}$ is a pluri-homogeneous $2$-form in
$\Sigma_{N+1}\widetilde{\Uplambda}_{q}^{2}$.

###### Proof.

We follow the famous deformation argument by Moser. We define the homothety
between the symplectic $2$-forms $\Omega_{c}$ and $\Omega_{\leq N}$ defined in
(4.4) by setting

$\Omega^{\tau}:=\Omega_{c}+\tau(\Omega_{{\leq
N}}-\Omega_{c})\,,\quad\forall\tau\in[0,1]\,.$ (4.20)

Equivalently $\Omega^{\tau}=\langle E^{\tau}(V)\cdot,\cdot\rangle_{r}$ with
associated symplectic tensor

$\displaystyle E^{\tau}(V)$ $\displaystyle=E_{c}+\tau(E_{{\leq N}}(V)-E_{c})$
(4.21) $\displaystyle\stackrel{{\scriptstyle\eqref{EB}}}{{=}}E_{c}+\tau{\bf
R}_{{\leq N}}(V)+\tau{\bf A}_{{\leq N}}^{\top}(V)\ E_{c}\ {\bf G}_{{\leq
N}}(V)+\tau{{\bf S}}_{>N}(V)$ (4.22)

where

$\bullet$ ${\bf R}_{{\leq N}}(V)$ is the matrix of pluri–homogeneous smoothing
operators

${\bf R}_{{\leq N}}(V):={\bf G}_{{\leq N}}^{\top}(V)\ E_{c}{\bf A}_{{\leq
N}}(V)+{\bf G}_{{\leq N}}^{\top}(V)\ E_{c}\ {\bf G}_{{\leq N}}(V)$

belonging to
$\Sigma_{p}{\widetilde{\mathcal{R}}}^{-\varrho}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
for any $\varrho\geq 0$;

$\bullet$ ${{\bf S}}_{>N}(V)$ is the map in
$\Sigma_{N+1}{\widetilde{\mathcal{S}}}_{q}\otimes\mathcal{M}_{2}(\mathbb{C})$
of Lemma 4.4.

In addition, by (4.9) and (3.15), we have

$\Omega^{\tau}={\rm
d}\theta^{\tau}\,,\quad\theta^{\tau}:=\theta_{c}+\tau(\theta_{\leq
N}-\theta_{c})$ (4.23)

where $\theta_{\leq N}$ is given in (4.11).

We look for a $\tau$–dependent pluri-homogeneous smoothing vector field
$Y^{\tau}(V)$ in $\Sigma_{p+1}\widetilde{\mathfrak{X}}^{-\varrho}_{q}$, for
any $\varrho\geq 0$, such that its approximate flow $\mathcal{F}^{\tau}_{\leq
N}$ up to homogeneity $N$ (defined by Lemma 2.26), satisfies

$\frac{{\rm d}}{{\rm d}\tau}(\mathcal{F}^{\tau}_{\leq
N})^{*}\Omega^{\tau}={\rm
d}\theta^{\tau}_{>N+1}\,,\quad\forall\tau\in[0,1]\,,$ (4.24)

for a certain $1$-form $\theta^{\tau}_{>N+1}$ in
$\Sigma_{N+2}\widetilde{\Uplambda}^{1}_{q}$. Then, integrating (4.24) and
recalling (4.20), we deduce

$(\mathcal{F}^{1}_{\leq N})^{*}\Omega_{{\leq N}}=\Omega_{c}+\int_{0}^{1}{\rm
d}\theta^{\tau}_{>N+1}\,{\rm d}\tau\,,$

which proves (4.19) with $\mathcal{F}_{\leq N}:=\mathcal{F}^{1}_{\leq N}$ and
$\Omega_{>N}:=\int_{0}^{1}{\rm d}\theta^{\tau}_{>N+1}\,{\rm d}\tau$.

We now construct the vector field $Y^{\tau}(V)$. Using the definition of Lie
derivative and the Cartan magic formula, we derive the chain of identities

$\displaystyle\frac{{\rm d}}{{\rm d}\tau}(\mathcal{F}_{\leq
N}^{\tau})^{*}\Omega^{\tau}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Omegadtheta},\eqref{commu}}}{{=}}{\rm
d}\Big{(}\frac{{\rm d}}{{\rm d}\tau}(\mathcal{F}_{\leq
N}^{\tau})^{*}\theta^{\tau}\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{deriv:pullback}}}{{=}}{\rm
d}\Big{(}(\mathcal{F}_{\leq
N}^{\tau})^{*}(\mathcal{L}_{Y^{\tau}}\theta^{\tau}+\frac{{\rm d}}{{\rm
d}\tau}\theta^{\tau})\Big{)}+{\rm d}\tilde{\theta}^{\tau}_{>N+1}$
$\displaystyle\stackrel{{\scriptstyle\eqref{cartan},\eqref{commu}}}{{=}}(\mathcal{F}^{\tau}_{\leq
N})^{*}{\rm d}\big{(}(i_{Y^{\tau}}\circ{\rm d}+{\rm d}\circ
i_{Y^{\tau}})\theta^{\tau}+\theta_{{\leq N}}-\theta_{c}\big{)}+{\rm
d}\tilde{\theta}^{\tau}_{>N+1}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Omegadtheta},\eqref{d2=0}}}{{=}}(\mathcal{F}^{\tau}_{\leq
N})^{*}{\rm d}\big{(}i_{Y^{\tau}}\Omega^{\tau}+\theta_{{\leq
N}}-\theta_{c}\big{)}+{\rm d}\tilde{\theta}^{\tau}_{>N+1}$ (4.25)

where $\tilde{\theta}^{\tau}_{>N+1}$ is a $1$-form in
$\Sigma_{N+2}\widetilde{\Uplambda}_{q}^{1}$ by Lemma 2.33. We look for a
vector field $Y^{\tau}(V)$ and a $0$-form $\mathcal{W}^{\tau}(V)$ such that

$i_{Y^{\tau}}\Omega^{\tau}+\theta_{{\leq
N}}-\theta_{c}=\breve{\theta}^{\tau}_{>N+1}+{\rm d}\mathcal{W}^{\tau},$ (4.26)

for some pluri–homogeneous $1$–form $\breve{\theta}^{\tau}_{>N+1}$ in
$\Sigma_{N+2}\widetilde{\Uplambda}^{1}_{q}$. If (4.26) holds, then, in view of
(4.25), equation (4.24) is satisfied with

$\theta_{>N+1}^{\tau}:=(\mathcal{F}^{\tau}_{\leq
N})^{*}\breve{\theta}^{\tau}_{>N+1}+\tilde{\theta}^{\tau}_{>N+1}\in\Sigma_{N+2}\widetilde{\Uplambda}^{1}_{q}\,.$

We turn to solve equation (4.26). Using (4.21), (4.11), recalling that