# Normalization flow in the presence of a resonance

Dmitry Treschev
Steklov Mathematical Institute of Russian Academy of Sciences

###### Abstract

Following [18], we develop an approach to the Hamiltonian theory of normal
forms based on continuous averaging. We concentrate on the case of normal
forms near an elliptic singular point, but unlike [18] we do not assume that
frequences of the linearized system are nonresonant. We study analytic
properties of the normalization procedure. In particular we show that in the
case of a codimension one resonance an analytic Hamiltonian function may be
reduced to a normal form up to an exponentially small reminder with explicit
estimates of the reminder and the analyticity domain.

##  1 Introduction

Consider a Hamiltonian system with $n$ degrees of freedom in a neighborhood of
an elliptic singular point. In the linear approximation the dynamics is
determined by the Hamiltonian equations

$\displaystyle\dot{x}=\partial_{y}H_{2},\quad\dot{y}=-\partial_{x}H_{2},\qquad
x=(x_{1},\ldots,x_{n}),\quad y=(y_{1},\ldots,y_{n}),$ $\displaystyle
H_{2}=\sum_{j=1}^{n}\frac{\omega_{j}}{2}(y_{j}^{2}+x_{j}^{2}).$

The real numbers $\omega_{1},\ldots,\omega_{n}$ form the frequency vector
$\omega\in\mathbb{R}^{n}$.

According to Birkhoff [1] it is convenient to use complex coordinates

$(z,\overline{z})=(z_{1},\ldots,z_{n},\overline{z}_{1},\ldots,\overline{z}_{n})\in\mathbb{C}^{2n},\qquad
z_{j}=\frac{y_{j}+ix_{j}}{\sqrt{2}},\quad\overline{z}_{j}=\frac{y_{j}-ix_{j}}{\sqrt{2}}.$

Below all the phase variables including $x$ and $y$ may be complex. Hence the
overbar does not mean necessarily complex conjugation. In the other words, the
variables $z$ and $\overline{z}$ are assumed to be independent.

The variables $(z,\overline{z})$ are canonical i.e., for any two smooth
functions $F=F(z,\overline{z}),G=G(z,\overline{z})$ the Poisson bracket has
the form

$\\{F,G\\}=i\sum_{j=1}^{n}\big{(}\partial_{\overline{z}_{j}}F\partial_{z_{j}}G-\partial_{z_{j}}F\partial_{\overline{z}_{j}}G\big{)}.$

The same equation determines the bracket $\\{\,,\\}$ on the space of formal
power series in $z$ and $\overline{z}$. The function $H_{2}$ takes the form

$H_{2}=\sum_{j=1}^{n}\omega_{j}z_{j}\overline{z}_{j}.$

The Hamiltonian function is assumed to be $H_{2}+\widehat{H}$,

$\widehat{H}=\sum_{|k|+|\overline{k}|\geq
3}H_{k,\overline{k}}z^{k}\overline{z}^{\overline{k}}.$

Here $k,\overline{k}\in\mathbb{Z}_{+}^{n}$, $\mathbb{Z}_{+}:=\\{0,1,\ldots\\}$
are multiindices and $|k|=|k_{1}|+\ldots+|k_{n}|$ is the $l^{1}$-norm. Below
we use the shorter notation

${\bf z}=(z,\overline{z})\in\mathbb{C}^{2n},\quad{\bf
k}=(k,\overline{k})\in\mathbb{Z}_{+}^{2n},\quad{\bf z}^{\bf
k}=z^{k}\overline{z}^{\overline{k}},\qquad\widehat{H}=\sum_{|{\bf k}|\geq
3}\widehat{H}_{\bf k}{\bf z}^{\bf k}.$ (1.1)

The Hamiltonian equations are as follows:

$\dot{z}=i\partial_{\overline{z}}(H_{2}+\widehat{H}),\quad\dot{\overline{z}}=-i\partial_{z}(H_{2}+\widehat{H}).$
(1.2)

More generally, $\dot{F}=\\{\widehat{H},F\\}$ for any function (or formal
power series) $F=F({\bf z})$.

According to the theory of normal forms the function $\widehat{H}$ may be
simplified by passage to another coordinate system, [1].

The monomial $\widehat{H}_{\bf k}{\bf z}^{\bf k}$ in the expansion (1.1) is
said to be resonant if $\langle\omega,\overline{k}-k\rangle=0$, where
$\langle\,,\rangle$ is the standard inner product in $\mathbb{R}^{n}$. The
integer number $|\overline{k}-k|$ is said to be the order of the resonance.
Any integer vector ${\bf k}=(k,\overline{k})$ which determines a resonant
monomial ${\bf z}^{\bf k}$ satisfies $\overline{k}-k\in\mathbb{L}_{\omega}$,
where

$\mathbb{L}_{\omega}=\\{q\in\mathbb{Z}^{n}:\langle\omega,q\rangle=0\\}.$ (1.3)

If $\langle\omega,q\rangle=0$, $q\in\mathbb{Z}^{n}$ implies $q=0$, the
frequency vector $\omega$ is called nonresonant.

By using an appropriate coordinate change one gets rid of any finite set of
nonresonant monomials. To eliminate all nonresonant monomials, we have to use
coordinate changes in the form of, in general, divergent power series. The
Hamiltonian function $N$ obtained as a result of this (formal) normalization
is called the normal form of the original Hamiltonian $H_{2}+\widehat{H}$. The
normal form is unique as a formal series.

The problem of convergence/divergence of the normalizing transformation under
the assumption of analyticity of $\widehat{H}$ is central in the theory. H.
Eliasson attracted attention of specialists to another (harder) problem:
convergence/divergence of the normal form. If the normalization converges and
the lattice ${\cal L}_{\omega}$ is at most 1-dimensional then the system is
locally completely integrable. Various versions of the inverse statement are
proved in [19, 7, 4, 8].

Another corollary from convergence of the normalization is Lyapunov stability
of the equilibrium position in the case of a nonresonant frequency
vector111under some simple explicit additional conditions . Papers [5, 9]
contain examples of real-analytic Hamiltonians $H_{2}+\widehat{H}$ such that
the origin is Lyapunov unstable in the system (1.2) although in the linear
approximation the system is obviously stable.

Convergence of the normal form does not imply convergence of the
normalization. But it has interesting dynamical consequences: the measure of
the set covered by KAM-tori turns out to be noticeably bigger than in the case
when the normal form diverges [10].

Convergence of the normalizing transformation is an exceptional phenomenon in
any reasonable sense. At the moment this exceptionality is known in terms of
Baire category [14] and $\Gamma$-capacity [12, 10]. Explicit examples of real-
analytic systems with divergent normal form can be found in [6, 20, 5].

The case of the “trivial” normal form $N=H_{2}$ is special. According to the
Bruno-Russmann theorem [2, 13], see also [16], if $N=H_{2}$ and $\omega$
satisfies some (rather weak) Diophantine conditions,222 usually called the
Bruno conditions then the normalization converges.

The normalizing change of variables is traditionally constructed as a
composition of an infinite sequence of coordinate changes which normalize the
Hamiltonian function up to a remainder of a higher and higher degree [1]. In
further works (see for example [15]) this change of coordinates is represented
as a formal series in $z$ and $\overline{z}$. We propose another approach to
the normalization.

Let ${\cal F}$ be the space of all power series in the variables $z$ and
$\overline{z}$. Sometimes we refer to elements of ${\cal F}$ as functions
although they are only formal power series. Let ${\cal
F}_{\diamond}\subset{\cal F}$ be the subspace of series which start from terms
of degree at least 3. In this paper, assuming that $H_{0}$ is fixed, we study
various normalization flows $\phi_{\xi}^{\delta}$ on the space ${\cal
F}_{\diamond}$, $\delta\in[0,+\infty)$. Any shift

$H_{2}+\widehat{H}\mapsto
H_{2}+\phi_{\xi}^{\delta}(\widehat{H}),\qquad\widehat{H}\in{\cal
F}_{\diamond}$

is a transformation of the Hamiltonian function $H_{2}+\widehat{H}$ according
to a certain (depending on $\delta$ and on the initial Hamiltonian
$H_{2}+\widehat{H}$) canonical change of variables.

Any flow $\phi_{\xi}^{\delta}$ is determined by a certain ODE in ${\cal
F}_{\diamond}$:

$\partial_{\delta}H=-\\{\xi H,H_{2}+H\\},\qquad H|_{\delta=0}=\widehat{H}.$
(1.4)

Here $\xi$ is a linear operator on ${\cal F}_{\diamond}$. In fact, (1.4) is an
initial value problem (IVP) for a differential equation presented in the form
of Lax L-A pair. Usually such systems are considered in the theory of
integrable systems. However integrability of system (1.4) whatever it means
seems, is of no use to us.

Let ${\cal N}_{\diamond}\subset{\cal F}_{\diamond}$ be the subspace which
consists of series which contain only resonant monomials. We will choose
operators $\xi$ such that the space ${\cal N}_{\diamond}$ is an invariant
manifold with respect to $\phi^{\delta}_{\xi}$ and any point of ${\cal
N}_{\diamond}$ is fixed. We proposed such an approach in [18]. In this paper
we present further results. In particular here we do not assume that $\omega$
is a non-resonant vector.

In Section 3.1 we define the operator $\xi=\xi_{*}$. We represent the system
(1.4) in the form of an infinite ODE system (3.9) for the coefficients $H_{\bf
k}$. A more convenient equivalent form of it is the system (3.11). Because of
a special “nilpotent” structure of the system (3.11), the existence and
uniqueness of a solution for the corresponding IVP for any initial condition
turns out to be a simple fact (Section 4.1). In the product topology the
solution tends to the normal form as $\delta\to+\infty$.

We are particularly interested in the restriction of $\phi^{\delta}_{\xi_{*}}$
to the subspace ${\cal A}\subset{\cal F}$ of analytic Hamiltonian functions.
In Section 4.2 we prove that for any $\widehat{H}\in{\cal A}$ the solution
$H=H({\bf z},\delta)$ also lies in ${\cal A}$ for any $\delta\geq 0$. However
the polydisk of analyticity generically shrinks when $\delta$ grows. A rough
lower estimate for its (poly)radius gives a quantity of order $1/(1+\delta)$
(Theorem 2).

In Section 5 we assume that the normal form $H_{2}+N_{\diamond}$ of the
Hamiltonian $H_{2}+\widehat{H}$ satisfies the equation
$N_{\diamond}=O_{r}({\bf z})$, $r\geq 3$. For example, generically333 if the
frequency vector $\omega$ does not admit resonances of order 3 $r=4$. The case
$r>4$ is not generic but may happen for some values of parameters in
multiparametric families of Hamiltonians. Assuming that $\widehat{H}\in{\cal
A}$, $\omega$ satisfies the Bruno Condition, and $N_{\diamond}=O_{r}({\bf
z})$, Theorem 3 says that $H(\cdot,\delta)$ stays in ${\cal A}$ and the
polydisk of analyticity has radius of order (at least)
$(1+\delta)^{-1/(r-2)}$, $\delta>0$. In particular, generically this radius
$\sim(1+\delta)^{-1/2}$. This improves the corresponding estimate
$(1+\delta)^{-1}$ in [18].

If $\omega$ is collinear to a vector with rational components (codimention one
resonance) then Theorem 3 implies Corollary 5.1. This corollary says that
there exists a change of variables which reduces in a polydisk of radius
$\sim(1+\delta)^{-1/(r-2)}$ the Hamiltonian to the form $H_{2}+G^{0}+G^{*}$,
where $G^{0}\in{\cal N}_{\diamond}$ and $|G^{*}|\sim e^{-c\delta}$, $c>0$.

Theorem 4 from Section 6 is an auxiliary statement used in the proof of
Theorem 3. However, it probably has an independent value. It has the same
assumptions: $\widehat{H}\in{\cal A}$, $\omega$ satisfies the Bruno Condition,
$N_{\diamond}=O_{r}({\bf z})$. Then there exists a change of the variables
which reduces the Hamiltonian function to the form $H_{2}+G$, $G=O_{r}({\bf
z})$. Theorem 4 presents upper estimates for Taylor coefficients of the
function $G$ in terms of analytic properties of $\widehat{H}$. As a corollary
we obtain the Bruno-Sigel theorem about convergence of the normalization in
the case of the trivial normal form (Corollary 6.1).

Section 7 contains technical statements concerning majorants, Bruno sequences
etc.

##  2 Basic construction

For any $q=(q_{1},\ldots,q_{n})\in\mathbb{Z}^{n}$ let
$|q|=|q_{1}|+\cdots+|q_{n}|$ be its $l^{1}$-norm. For any
$k,\overline{k}\in\mathbb{Z}_{+}^{n}$ we put

${\bf k}=(k,\overline{k})\in\mathbb{Z}_{+}^{2n},\quad{\bf
k}^{\prime}=\overline{k}-k\in\mathbb{Z}^{n},\quad{\bf
k}^{*}=(\overline{k},k),\quad|{\bf k}|=|k|+|\overline{k}|.$

Then ${\bf k}^{*}-{\bf k}=({\bf k}^{\prime},-{\bf k}^{\prime})$. In particular
${\bf k}^{\prime}=0$ if and only if ${\bf k}={\bf k}^{*}$.

We will use the notation

$\mathbb{Z}_{\diamond}^{2n}=\\{{\bf k}\in\mathbb{Z}_{+}^{2n}:|{\bf k}|\geq
3\\}.$

###  2.1 Spaces

Let ${\cal F}$ be the vector space of all series

$H=\sum_{{\bf k}\in\mathbb{Z}_{+}^{2n}}H_{{\bf k}}{\bf z}^{\bf k},\qquad
H_{{\bf k}}\in\mathbb{C}.$ (2.1)

The series (2.1) are assumed to be formal i.e., there is no restriction on the
values of the coefficients $H_{{\bf k}}$. So, ${\cal F}$ coincides with the
ring
$\mathbb{C}[[z_{1},\ldots,z_{n},\overline{z}_{1},\ldots,\overline{z}_{n}]]$.
The Poisson bracket determines on ${\cal F}$ the structure of a Lie algebra.
Below we use on ${\cal F}$ the product topology i.e., a sequence
$H^{(1)},H^{(2)},\ldots\in{\cal F}$ is said to be convergent if for any ${\bf
k}\in\mathbb{Z}_{+}^{2n}$ the sequence of coefficients $H^{(1)}_{{\bf
k}},H^{(2)}_{{\bf k}},\ldots$ converges.

For any $H$ satisfying (2.1) we define $p_{{\bf k}}(H)=H_{{\bf k}}$. So
$p_{{\bf k}}:{\cal F}\to\mathbb{C}$ is a canonical projection, corresponding
to ${\bf k}\in\mathbb{Z}_{+}^{2n}$. Suppose $F\in{\cal F}$ depends on a
parameter $\delta\in I$, where $I\subset\mathbb{R}$ is an interval. In other
words, we consider a map

$f:I\to{\cal F},\quad I\ni\delta\mapsto F(\cdot,\delta).$

We say that $F$ is smooth in $\delta$ if all the maps $p_{\bf k}\circ f$ are
smooth.

Let ${\cal F}_{r}\subset{\cal F}$ be the space of “real” series:

${\cal F}_{r}=\\{H\in{\cal F}:\overline{H}_{{\bf k}}=H_{{\bf k}^{*}}\;\mbox{
for any ${\bf k}\in\mathbb{Z}_{+}^{n}$}\\}.$

We define ${\cal A}\subset{\cal F}$ as the space of analytic series:

${\cal A}=\\{H\in{\cal F}:\mbox{ there exist $c,a$ such that $|H_{{\bf
k}}|\leq ce^{a|{\bf k}|}$ for any ${\bf k}\in\mathbb{Z}_{+}^{2n}$}\\}.$

The product topology may be restricted from ${\cal F}$ to ${\cal A}$, but,
being a scale of Banach spaces, ${\cal A}$ may be endowed with a more natural
topology. We have ${\cal A}=\cup_{\rho>0}{\cal A}^{\rho}$, where ${\cal
A}^{\rho}$ is a Banach space with the norm $\|\cdot\|_{\rho}$,

$\|H\|_{\rho}=\sup_{{\bf z}\in D_{\rho}}|H({\bf z})|,\quad D_{\rho}=\\{{\bf
z}\in\mathbb{C}^{2n}:|z_{j}|<\rho,\;|\overline{z}_{j}|<\rho,\;j=1,\ldots,n\\}.$
(2.2)

Then we have: ${\cal A}^{\rho^{\prime}}\subset{\cal A}^{\rho}$,
$\|\cdot\|_{\rho}\leq\|\cdot\|_{\rho^{\prime}}$ for any
$0<\rho<\rho^{\prime}$.

###### Lemma 2.1

If $\|H\|_{\rho}\leq c$ then

$|H_{{\bf k}}|\leq c\rho^{-|{\bf k}|}.$ (2.3)

The proof is standard. It follows from the Cauchy formula: for any positive
$\rho_{0}<\rho$ and any $H\in{\cal A}^{\rho}$

$H_{{\bf k}}=\frac{1}{(2\pi i)^{2n}}\oint dz_{1}\ldots\oint dz_{n}\oint
d\overline{z}_{1}\ldots\oint\frac{H({\bf z})\,d\overline{z}_{n}}{{\bf z}^{{\bf
k}+{\bf{1}}}},$

where ${\bf{1}}=(1,\ldots,1)\in\mathbb{Z}_{+}^{2n}$ and the integration is
performed along the circles

$|z_{1}|=\rho_{0},\ldots,\quad|z_{n}|=\rho_{0},\quad|\overline{z}_{1}|=\rho_{0},\ldots,\quad|\overline{z}_{n}|=\rho_{0}.$

This implies $|H_{\bf k}|\leq c\rho_{0}^{-|{\bf k}|}$. Since $\rho_{0}<\rho$
is arbitrary, we obtain (2.3).

###### Corollary 2.1

Topology determined on ${\cal A}^{\rho}$ by the norm $\|\cdot\|_{\rho}$ is
stronger than the product topology induced from ${\cal F}$ i.e., if a sequence
$\\{H^{(j)}\\}$ converges in $({\cal A}^{\rho},\|\cdot\|_{\rho})$ then it
converges in ${\cal F}$.

We define the resonant sublattice (1.3). The number of generators in the group
$(\mathbb{L}_{\omega},+)$ is said to be the rank of $\mathbb{L}_{\omega}$. In
particular if $\operatorname{rank}\mathbb{L}_{\omega}=0$ (equivalently,
$\mathbb{L}_{\omega}=\\{0\\}$) then the frequency vector is called
nonresonant. If $\operatorname{rank}\mathbb{L}_{\omega}=1$, we have a simple
resonance if $\operatorname{rank}\mathbb{L}_{\omega}>1$, the resonance is
multiple, if $\operatorname{rank}\mathbb{L}_{\omega}=n-1$ then we have a
codimension one resonance. In the latter case all trajectories of the
linearized system

$\dot{z}=i\partial_{\overline{z}}H_{0},\quad\dot{\overline{z}}=i\partial_{z}H_{0}$

are periodic.

Let ${\cal N}\subset{\cal F}$ be the space of “normal forms”:

${\cal N}=\\{H\in{\cal F}:H_{{\bf k}}\neq 0\;\mbox{ implies }{\bf
k}^{\prime}\in\mathbb{L}_{\omega}\\}.$

It is easy to check that ${\cal N}$ is a subalgebra in ${\cal F}$ with respect
to the operations of multiplication and the Poisson bracket.

We define the subspace (the subring) ${\cal F}_{\diamond}\subset{\cal F}$ of
the series

$H=\sum_{{\bf k}\in\mathbb{Z}_{\diamond}^{2n}}H_{{\bf k}}{\bf z}^{\bf
k},\qquad H_{{\bf k}}\in\mathbb{C}$ (2.4)

i.e., for any $H\in{\cal F}_{\diamond}$ we have $H=O_{3}({\bf z})$. By
definition we also put ${\cal N}_{\diamond}={\cal N}\cap{\cal F}_{\diamond}$.

##  3 Normalization flow

Consider the change of variables in the form of a shift

${\bf z}=(z,\overline{z})\mapsto{\bf
z}_{\delta}=(z_{\delta},\overline{z}_{\delta})$ (3.1)

along solutions of the Hamiltonian system444 The so-called, Lie method. with
Hamiltonian $F=F({\bf z},\delta)=O_{3}({\bf z})$ and independent variable
$\delta$:

$z^{\prime}=i\partial_{\overline{z}}F,\quad\overline{z}^{\prime}=-i\partial_{z}F,\qquad(\cdot)^{\prime}=d/d\delta.$
(3.2)

Suppose the function $H_{2}+\widehat{H}$ expressed in the variables ${\bf
z}_{\delta}$ takes the form $H_{2}+H$:

$H_{2}({\bf z})+\widehat{H}({\bf z})=H_{2}({\bf z}_{\delta})+H({\bf
z}_{\delta},\delta).$ (3.3)

Differentiating equation (3.3) in $\delta$, we obtain:

$\partial_{\delta}H=-\\{F,H_{2}+H\\},\qquad H|_{\delta=0}=\widehat{H}.$

The main idea of the continuous averaging is to take $F$ in the form $\xi H$,
where $\xi$ is a certain operator555This idea is similar to the idea of
Moser’s homotopy method [11]. , depending on the problem we deal with. Then we
obtain an initial value problem in ${\cal F}_{\diamond}$:

$\partial_{\delta}H=-\\{\xi H,H_{2}+H\\},\qquad H|_{\delta=0}=\widehat{H}.$
(3.4)

###  3.1 Operator $\xi_{*}$

To explain the idea, we start with the “simplest” operator $\xi=\xi_{*}$. We
put

$\sigma_{q}=\operatorname{sign}(\langle\omega,q\rangle),\quad\omega_{q}=|\langle\omega,q\rangle|,\qquad
q\in\mathbb{Z}^{n}.$ (3.5)

For any $H$ satisfying (2.4) we put

$\xi_{*}H=-i\sum_{{\bf k}\in\mathbb{Z}^{2n}_{\diamond}}\sigma_{{\bf
k}^{\prime}}H_{{\bf k}}{\bf z}^{\bf k}=i(H^{-}-H^{+}),\qquad
H^{\pm}=\sum_{\pm\sigma_{{\bf k}^{\prime}}>0}H_{{\bf k}}{\bf z}^{\bf k}.$
(3.6)

We also put

$H^{0}=\sum_{\langle\omega,{\bf k}^{\prime}\rangle=0}H_{{\bf k}}{\bf z}^{\bf
k}.$

So that $H=H^{-}+H^{0}+H^{+}$.

We obtain a more detailed form of the IVP (3.4):

$\displaystyle\partial_{\delta}H=v_{0}(H)+v_{1}(H)+v_{2}(H),\qquad
H|_{\delta=0}=\widehat{H},$ (3.7) $\displaystyle
v_{0}=-\\{\xi_{*}H,H_{2}\\},\quad v_{1}=-\\{\xi_{*}H,H^{0}\\},\quad
v_{2}=-\\{\xi_{*}H,H^{-}+H^{+}\\}.$

Informal explanation for the choice (3.6) of the operator $\xi=\xi_{*}$ is as
follows. Removing nonlinear terms $v_{1}+v_{2}$ in (3.7), we obtain the
equation $\partial_{\delta}H=v_{0}(H)$ or, in more detail

$\partial_{\delta}H_{{\bf k}}=-\omega_{{\bf k}^{\prime}}H_{{\bf k}},\qquad
H_{{\bf k}}|_{\delta=0}=\widehat{H}_{{\bf k}}$ (3.8)

which can be easily solved:

$H_{{\bf k}}=e^{-\omega_{{\bf k}^{\prime}}\delta}\widehat{H}_{{\bf k}}.$

Thus when $\delta\to+\infty$, solution $H$ of the truncated problem (3.8)
tends to $H^{0}\in{\cal N}_{\diamond}$.

Let $e_{j}=(0,\ldots,1,\ldots,0)$ be the $j$-th unit vector in
$\mathbb{Z}_{+}^{n}$ and let ${\bf
e}_{j}=(e_{j},e_{j})\in\mathbb{Z}_{+}^{2n}$. Equation (3.7) written for each
Taylor coefficient $H_{{\bf k}}$ has the form

$\displaystyle\partial_{\delta}H_{{\bf k}}$ $\displaystyle=$
$\displaystyle-\omega_{{\bf k}^{\prime}}H_{{\bf k}}+v_{1,{\bf k}}(H)+v_{2,{\bf
k}}(H),\qquad H_{\bf k}|_{\delta=0}=\widehat{H}_{\bf k},$ (3.9) $\displaystyle
v_{1,{\bf k}}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{n}\sum_{\sigma_{{\bf m}^{\prime}}=0,\,{\bf l}+{\bf
m}-{\bf k}={\bf e}_{j}}\sigma_{{\bf
k}^{\prime}}(\overline{l}_{j}m_{j}-l_{j}\overline{m}_{j})H_{{\bf l}}H_{{\bf
m}},$ (3.10) $\displaystyle v_{2,{\bf k}}$ $\displaystyle=$ $\displaystyle
2\sum_{j=1}^{n}\sum_{\sigma_{{\bf l}^{\prime}}<0<\sigma_{{\bf
m}^{\prime}},\,{\bf l}+{\bf m}-{\bf k}={\bf
e}_{j}}(\overline{l}_{j}m_{j}-l_{j}\overline{m}_{j})H_{{\bf l}}H_{{\bf m}}.$

By using the change of variables

$H_{{\bf k}}={\cal H}_{{\bf k}}e^{-\omega_{{\bf k}^{\prime}}\delta},$

we reduce the equations (3.9) to the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\partial_{\delta}{\cal H}_{{\bf k}}$
$\displaystyle=$ $\displaystyle v_{1,{\bf k}}({\cal H})+{\bf v}_{2,{\bf
k}}({\cal H}),\qquad{\cal H}_{{\bf k}}|_{\delta=0}=\widehat{H}_{{\bf k}},$
(3.11) $\displaystyle\\!\\!\\!\\!\\!\\!\\!{\bf v}_{2,{\bf k}}$
$\displaystyle=$ $\displaystyle 2\sum_{j=1}^{n}\sum_{\sigma_{{\bf
l}^{\prime}}<0<\sigma_{{\bf m}^{\prime}},\,{\bf l}+{\bf m}-{\bf k}={\bf
e}_{j}}\\!(\overline{l}_{j}m_{j}-l_{j}\overline{m}_{j}){\cal H}_{{\bf l}}{\cal
H}_{{\bf m}}e^{-\omega_{{\bf l}^{\prime},{\bf m}^{\prime}}\delta},$ (3.12)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\omega_{{\bf l}^{\prime},{\bf m}^{\prime}}$
$\displaystyle=$ $\displaystyle\omega_{{\bf l}^{\prime}}+\omega_{{\bf
m}^{\prime}}-\omega_{{\bf l}^{\prime}+{\bf m}^{\prime}}$

###### Remark 3.1

1\. The functions $v_{1,{\bf k}}$ and $v_{2,{\bf k}}$ are quadratic
polynomials in the variables $H_{{\bf k}}$. The functions ${\bf v}_{2,{\bf
k}}$ are quadratic polynomials in ${\cal H}_{{\bf k}}$ with coefficients,
depending on $\delta$.

2\. The polynomial $v_{1,{\bf k}}$ vanishes if ${\bf
k}^{\prime}\in\mathbb{L}_{\omega}$.

3\. The polynomials $v_{1,{\bf k}},v_{2,{\bf k}}$ and ${\bf v}_{2,{\bf k}}$ do
not depend on the variables $H_{\bf s},{\cal H}_{{\bf s}}$, for any ${\bf
s}\in\mathbb{Z}_{\diamond}^{2n}$ such that $|{\bf s}|>|{\bf k}|-2$.

4\. The polynomials $v_{2,{\bf k}}$ and ${\bf v}_{2,{\bf k}}$ do not depend on
the variables $H_{\bf s},{\cal H}_{{\bf s}}$ for any ${\bf
s}\in\mathbb{Z}_{\diamond}^{2n}$ such that $0\leq\sigma_{{\bf
k}^{\prime}}\langle\omega,{\bf s}^{\prime}\rangle\leq\omega_{{\bf
k}^{\prime}}$.

5\. For any $\sigma_{{\bf l}^{\prime}}<0<\sigma_{{\bf m}^{\prime}}$ the
quantity $\omega_{{\bf l}^{\prime},{\bf m}^{\prime}}$ is positive:

$\omega_{{\bf l}^{\prime},{\bf
m}^{\prime}}=\left\\{\begin{array}[]{ccc}2\omega_{{\bf l}^{\prime}}&\mbox{ if
}&\sigma_{{\bf k}^{\prime}}>0,\\\ 2\omega_{{\bf m}^{\prime}}&\mbox{ if
}&\sigma_{{\bf k}^{\prime}}<0.\end{array}\right.$

##  4 Resonant normalization exists

###  4.1 Formal aspect

Let $I\subset\mathbb{R}$ be an interval containing the point $0$. We say that
the path $\gamma:I\to{\cal F}_{\diamond}$ is a solution of the system (3.4) on
the interval $I$ if $\gamma(0)=\widehat{H}$ and the coefficients $H_{{\bf
k}}(\delta)=p_{{\bf k}}\gamma(\delta)$ satisfy (3.9).

If the solution $\gamma:I\to{\cal F}_{\diamond}$ exists and is unique then for
any $\delta\in I$ we put
$\gamma(\delta)=\phi^{\delta}_{\xi_{*}}(\widehat{H})$.

###### Definition 4.1

The ordinary differential equation

$F^{\prime}=\Phi(F),\qquad F=\sum_{|{\bf k}|\geq 3}F_{\bf k}{\bf z}^{\bf k}$

on ${\cal F}$ has a nilpotent form if for any ${\bf
k}\in\mathbb{Z}_{\diamond}^{2n}$ we have: $F^{\prime}_{\bf k}=\Phi_{\bf k}$,
where $\Phi_{\bf k}=p_{\bf k}\circ\Phi$ is a function of the variables $F_{\bf
m}$, $|{\bf m}|<|{\bf k}|$.

According to item 3 of Remark 3.1 the system (3.11) has a nilpotent form. In
particular, $\partial_{\delta}{\cal H}_{{\bf k}}=0$ for $|{\bf k}|=3$.

###### Theorem 1

For any $\widehat{H}\in{\cal F}_{\diamond}$ and any $\delta\in\mathbb{R}$ the
element $H(\cdot,\delta)=\phi^{\delta}_{\xi_{*}}(\widehat{H})\in{\cal F}$ is
well-defined. For any ${\bf k}\in\mathbb{Z}_{\diamond}^{2n}$ the function
${\cal H}_{\bf k}(\delta)=H_{\bf k}(\delta)e^{\omega_{{\bf
k}^{\prime}}\delta}$ equals

${\cal H}_{\bf k}(\delta)=\widehat{H}_{\bf k}+P_{\bf k}(\widehat{H},\delta),$
(4.1)

where $P_{\bf k}$ is a polynomial in $\widehat{H}_{\bf m}$, $|{\bf m}|<|{\bf
k}|$ with coefficients in the form of (finite) linear combinations of terms
$\delta^{s}e^{-\nu\delta}$, $s\in\mathbb{Z}_{+}$. Here $\nu\geq 0$ and
moreover, if the term $\delta^{s}e^{-\nu\delta}$ with $\nu=0$ appears in a
coefficient of $P_{\bf k}$, ${\bf k}^{\prime}\in\mathbb{L}_{\omega}$ then in
this term $s=0$.

Proof of Theorem 1. For $|{\bf k}|=3$ we have: ${\cal H}_{\bf
k}(\delta)=\widehat{H}_{\bf k}$. Suppose equations (4.1) hold for all vectors
${\bf k}\in\mathbb{Z}_{\diamond}^{2n}$ with $|{\bf k}|<K$. Take any ${\bf
k}\in\mathbb{Z}^{2n}_{\diamond}$ with $|{\bf k}|=K$. By (3.11)

${\cal H}_{\bf k}=\widehat{H}_{\bf k}+I_{1}+I_{2},\qquad
I_{1}=\int_{0}^{\delta}v_{1,{\bf k}}({\cal H}(\lambda))\,d\lambda,\quad
I_{2}=\int_{0}^{\delta}{\bf v}_{2,{\bf k}}({\cal
H}(\lambda),\lambda)\,d\lambda.$

By using (3.10), (3.12), and the induction assumption, we obtain (4.1).

If ${\bf k}^{\prime}\in\mathbb{L}_{\omega}$ then $v_{1,{\bf k}}=0$ while each
term in ${\bf v}_{2,{\bf k}}$ has the form $\lambda^{s}e^{-\lambda\delta}$
with $\lambda>0$. This implies the last statement of Theorem 1.

###### Corollary 4.1

The limit $\lim_{\delta\to+\infty}\phi^{\delta}(\widehat{H})$ exists in the
product topology on ${\cal F}_{\diamond}$ and lies in ${\cal N}_{\diamond}$.

Indeed, by Theorem 1 for any ${\bf k}\in\mathbb{Z}_{\diamond}^{2n}$ there
exists the limit

$\lim_{\delta\to+\infty}H_{{\bf k}}(\delta)=H_{{\bf k}}(+\infty).$

The convergence is exponential and $H_{{\bf k}}(+\infty)$ vanishes if ${\bf
k}^{\prime}\not\in\mathbb{L}_{\omega}$. This is equivalent to the existence of
$\lim_{\delta\to+\infty}\phi^{\delta}_{\xi_{*}}(\widehat{H})=\phi^{+\infty}_{\xi_{*}}(\widehat{H})$
together with the statement $\phi^{+\infty}_{\xi_{*}}(\widehat{H})\in{\cal
N}_{\diamond}$.

###### Corollary 4.2

Suppose $\widehat{H}\in{\cal F}_{r}\cap{\cal F}_{\diamond}$. Then
$\phi^{\delta}_{\xi_{*}}(\widehat{H})\in{\cal F}_{r}\cap{\cal F}_{\diamond}$
for any $\delta\geq 0$.

Indeed, the required identities $\overline{H}_{{\bf k}}(\delta)=H_{{\bf
k}^{*}}(\delta)$, $\delta\geq 0$ can be proved by induction in $|{\bf k}|$ by
using the equation (3.11).

###  4.2 Analytic aspect

###### Theorem 2

Suppose $\widehat{H}\in{\cal A}^{\rho}\cap{\cal F}_{\diamond}$,
$\|\widehat{H}\|_{\rho}=h\rho^{3}$. Then for any $\delta\geq 0$

${\cal H}=\sum{\cal H}_{\bf k}(\delta){\bf z}^{\bf k}\in{\cal
A}^{g(\delta)}\cap{\cal F}_{\diamond},$

where for some constants $A,B>0$, depending on $\rho,h$, and $n$

$g(\delta)\geq\frac{A}{1+B\delta}.$ (4.2)

More precisely,

$|{\cal H}|\leq\frac{h\rho^{3}}{4(1+32nh\rho\delta)^{3}}\quad\mbox{in the
domain}\quad\Big{\\{}{\bf z}\in\mathbb{C}^{2n}:|{\bf
z}|\leq\frac{\rho}{4(1+32nh\rho\delta)}\Big{\\}}.$

Proof. We use the majorant method. We remind definitions and basic facts
concerning majorants in Section 7.1. We have:

$\widehat{H}\ll f(\zeta)=\sum_{{\bf
k}\in\mathbb{Z}_{\diamond}^{2n}}\widehat{\bf H}_{{\bf k}}{\bf z}^{\bf
k},\qquad\zeta=\sum_{j=1}^{n}(z_{j}+\overline{z}_{j}).$

The function $f(\zeta)=O_{3}(\zeta)$, a majorant for the initial condition,
will be chosen later.

Consider together with (3.11) the following majorant system

$\displaystyle\\!\\!\\!\\!\\!\\!\partial_{\delta}{\bf H}_{{\bf k}}$
$\displaystyle=$ $\displaystyle{\bf V}_{{\bf k}},\qquad{\bf H}_{{\bf
k}}|_{\delta=0}=\widehat{\bf H}_{{\bf k}},$ (4.3)
$\displaystyle\\!\\!\\!\\!\\!\\!{\bf V}_{{\bf k}}$ $\displaystyle=$
$\displaystyle 2\sum_{j=1}^{n}\sum_{{\bf l}+{\bf m}-{\bf k}={\bf
e}_{j}}(\overline{l}_{j}m_{j}+l_{j}\overline{m}_{j}){\bf H}_{{\bf l}}{\bf
H}_{{\bf m}}.$ (4.4)

To obtain equation (4.4), we have replaced in $v_{1,{\bf k}}$ and ${\bf
v}_{2,{\bf k}}$ minuses by pluses, dropped the exponential multipliers, and
added some new positive (for positive ${\bf H}_{{\bf l}}$ and ${\bf H}_{{\bf
m}}$) terms. The main property of the system (4.3)–(4.4) is such that if for
any ${\bf k}\in\mathbb{Z}_{\diamond}^{2n}$ we have $|\widehat{H}_{{\bf
k}}|\leq\widehat{\bf H}_{{\bf k}}$ then for any $\delta\geq 0$ the inequality
$|v_{1,{\bf k}}+{\bf v}_{2,{\bf k}}|\leq{\bf V}_{\bf k}$ holds i.e.,
conditions (a) and (b) from Definition 7.2 hold.

The system (4.3)–(4.4) has a nilpotent form. Therefore by Theorem 5 we may use
Majorant principle: if ${\bf H}$ is a solution of (4.3)–(4.4) then (3.11) has
a solution ${\cal H}$ and ${\cal H}\ll{\bf H}$ for any $\delta\geq 0$.

The system (4.3)–(4.4) may be written in a shorter form:

$\partial_{\delta}{\bf H}=4\sum_{j=1}^{n}\partial_{z_{j}}{\bf
H}\,\partial_{\overline{z}_{j}}{\bf H},\qquad{\bf H}|_{\delta=0}=f(\zeta).$
(4.5)

Since the initial condition ${\bf H}|_{\delta=0}$ depends on $z$ and
$\overline{z}$ only through the variable $\zeta$, we may look for a solution
of (4.5) in the form ${\bf H}(z,\overline{z},\delta)=F(\zeta,\delta)$. The
function $F$ satisfies the equation

$\partial_{\delta}F=4n(\partial_{\zeta}F)^{2},\qquad F|_{\delta=0}=f(\zeta).$

The function $G=\partial_{\zeta}F$ satisfies the inviscid Burgers’ equation

$\partial_{\delta}G=8nG\partial_{\zeta}G,\qquad
G|_{\delta=0}=\partial_{\zeta}f(\zeta)=O_{2}(\zeta).$ (4.6)

By using the method of characteristics we obtain that the function $G=G(x,t)$
which solves (4.6), satisfies the equation

$G=f^{\prime}(\zeta+8n\delta G).$ (4.7)

By Lemma 7.3 we can take $f(\zeta)=h\rho\zeta^{3}/(\rho-\zeta)$. Then by Lemma
7.4

$f^{\prime}(\zeta)\ll a\zeta^{2}/(b-\zeta),\qquad a=2h\rho,\quad b=\rho/2.$

Putting $\tau=8n\delta$, we obtain from (4.7)

$G=\frac{a(\zeta+\tau G)^{2}}{b-\zeta-\tau G}.$

This is a quadratic equation w.r.t. $G$. The solution is

$G=\frac{2a\zeta^{2}}{b-\zeta-2a\tau\zeta+\sqrt{(b-\zeta-2a\tau\zeta)^{2}-4a\tau\zeta^{2}(1+a\tau)}}.$

It is analytic for

$|\zeta|<\frac{b}{1+2a\tau+2\sqrt{a\tau(1+a\tau)}}.$

If $\displaystyle|\zeta|\leq\frac{b}{2(1+2a\tau)}$ then
$\displaystyle|G|\leq\frac{ab}{(1+2a\tau)^{2}}$. This implies that

$|{\cal H}|\leq|F|\leq\frac{ab^{2}}{2(1+2a\tau)^{3}}\quad\mbox{in the
domain}\quad\Big{\\{}|{\bf z}|\leq\frac{b}{2(1+2a\tau)}\Big{\\}}.$

###### Corollary 4.3

Suppose $\widehat{H}\in{\cal A}\cap{\cal F}_{r}$. Then
$\phi^{\delta}_{\xi_{*}}(\widehat{H})\in{\cal A}\cap{\cal F}_{r}$ for any
$\delta\geq 0$.

Note that for a typical $\widehat{H}\in{\cal A}\cap{\cal F}_{\diamond}$ one
should expect that $\phi^{+\infty}_{\xi_{*}}(\widehat{H})\in{\cal
N}_{\diamond}$ does not belong to ${\cal A}$, [10].

##  5 Degenerate normal form

###  5.1 Normalization theorem

Suppose the normal form $H_{2}+N_{\diamond}$ is such that
$N_{\diamond}:=\lim_{\delta\to+\infty}\phi_{\xi_{*}}^{\delta}(\widehat{H})\in{\cal
N}$ vanishes in several first orders in ${\bf z}$. In this section we obtain
more precise estimates (in comparison with those from Theorem 2) for the
analyticity domain of the Hamiltonian function ${\cal H}(\cdot,\delta)$ when
$\delta$ increases.

For any $j\in\mathbb{N}$ we put

$\Omega_{s}=\max\Big{\\{}\frac{1}{|\langle\omega,q\rangle|}:q\in\mathbb{Z}^{n}\setminus\mathbb{L}_{\omega},\,|q|\leq
s\Big{\\}}.$ (5.1)

###### Definition 5.1

We say that $\\{a_{j}\\}_{j\in\mathbb{Z}_{+}}$, $a_{j}\geq 1$ is a Bruno
sequence if $\\{a_{j}\\}$ is monotone non-decreasing and

$\sum_{j=0}^{\infty}2^{-j}\ln a_{j}<\infty.$

Obviously if $\\{a_{j}\\}$ and $\\{b_{j}\\}$ are two Bruno sequences then the
product $\\{a_{j}b_{j}\\}$ is also a Bruno sequence.

###### Definition 5.2

We say that the vector $\omega\in\mathbb{R}^{n}$ satisfies the Bruno condition
if the sequence $\\{a_{j}\\}$, $a_{j}=\max\\{1,\Omega_{2^{j}+2}\\}$ is Bruno.

###### Theorem 3

Suppose $\widehat{H}\in{\cal A}^{\rho}\cap{\cal F}_{\diamond}$, $\omega$
satisfies the Bruno condition, and $N_{\diamond}=O_{r}({\bf z})$. Then for any
$\delta\geq 0$ the solution ${\cal H}({\bf z},\delta)=\sum{\cal H}_{\bf
k}(\delta){\bf z}^{\bf k}$ of (3.11) belongs to the space ${\cal
A}^{g(\delta)}$, where for some constant $C$, depending on
$\rho,\|\widehat{H}\|_{\rho},\\{\Omega_{j}\\}$, and $n$

$g(\delta)\geq\frac{C}{1+\delta^{1/(r-2)}}.$ (5.2)

More precisely, for some
$C_{H}=C_{H}(\rho,\|\widehat{H}\|_{\rho},\\{\Omega_{j}\\},n)$ and
$C_{z}=C_{z}(\rho,\|\widehat{H}\|_{\rho},\\{\Omega_{j}\\},n)$

$|{\cal H}|\leq\frac{C_{H}}{1+\delta^{r/(r-2)}}\quad\mbox{in the
domain}\quad\Big{\\{}{\bf z}\in\mathbb{C}^{2n}:|{\bf
z}|\leq\frac{C_{z}}{1+\delta^{1/(r-2)}}\Big{\\}}.$ (5.3)

In particular, if $r=3$, Theorem 3 essentially coincides with Theorem 2. If
$r\geq 4$, (5.2) improves the estimate (4.2).

Proof of Theorem 3 is presented in Section 6.

###  5.2 Resonance of codimension one

Suppose $\operatorname{rank}\mathbb{L}_{\omega}=n-1$. Then there exists a
unique collection of integers $q_{1},\ldots,q_{n},p$, and $\lambda>0$ such
that

$p>0,\quad GCD(|q_{1}|,\ldots,|q_{n}|,p)=1,\quad\mbox{and}\quad\omega=\lambda
q/p.$

Hence in (3.12) we have $\omega_{{\bf l}^{\prime},{\bf
m}^{\prime}}\geq\lambda/p$.

If the normal form of $H_{2}+\widehat{H}$ is of order $r$ in ${\bf z}$, we may
use Theorem 3 to obtain the following corollary.

###### Corollary 5.1

Suppose $\operatorname{rank}\mathbb{L}_{\omega}=n-1$, $\widehat{H}\in{\cal
A}^{\rho}\cap{\cal F}_{\diamond}$, and the normal form of $H_{2}+\widehat{H}$
is of order $O_{r}({\bf z})$.

Then there exists a canonical change of variables ${\bf w}\mapsto{\bf z}$
which transforms the Hamiltonian to $H_{2}({\bf w})+G({\bf w})$, $G=O_{r}({\bf
w})$, where

$G=G^{0}+G^{*},\qquad G^{0}\in{\cal N}_{\diamond}$

and in the domain

$\displaystyle\Big{\\{}{\bf w}\in\mathbb{C}^{2n}:|{\bf
w}|\leq\frac{C_{z}}{1+\delta^{1/(r-2)}}\Big{\\}}$ (5.4)

the following estimate holds:

$|G^{0}|\leq\frac{C_{H}}{1+\delta^{r/(r-2)}}\quad\mbox{and}\quad|G^{*}|\leq
e^{-\lambda\delta/p}\frac{C_{H}}{1+\delta^{r/(r-2)}}.$ (5.5)

Estimates (5.5) mean that the nonresonant part $G^{*}$ may be reduced to an
exponentially small quantity of order $e^{-\lambda\delta/p}$. The price is a
little size ($\sim\delta^{-1/(r-2)}$) of the domain in which the Hamiltonian
function $G$ is defined.

To obtain estimates (5.5) in the domain (5.4), we split the function ${\cal
H}({\bf z},\delta)$, obtained in Theorem 3 into the resonant, ${\cal H}^{0}$,
and non-resonant, ${\cal H}^{*}$, parts

${\cal H}^{0}=\sum_{|{\bf k}|\geq 3,\,{\bf
k}^{\prime}\in\mathbb{L}_{\omega}}{\cal H}_{\bf k}{\bf z}^{\bf k},\quad{\cal
H}^{*}=\sum_{|{\bf k}|\geq 3,\,{\bf
k}^{\prime}\in\mathbb{Z}^{n}\setminus\mathbb{L}_{\omega}}{\cal H}_{\bf k}{\bf
z}^{\bf k}.$

The corresponding function $H({\bf z},\delta)$ equals $G^{0}+G^{*}$,

$G^{0}={\cal H}^{0},\quad G^{*}=\sum_{|{\bf k}|\geq 3,\,{\bf
k}^{\prime}\in\mathbb{Z}^{n}\setminus\mathbb{L}_{\omega}}{\cal H}_{\bf
k}e^{-\omega_{{\bf k}^{\prime}}\delta}{\bf z}^{\bf k}.$

Estimates (5.5) follow from (5.3) and the inequality $\omega_{{\bf
k}^{\prime}}\geq\lambda/p$ for any ${\bf
k}^{\prime}\in\mathbb{Z}^{n}\setminus\mathbb{L}_{\omega}$.

##  6 Proof of Theorem 3

###  6.1 Inductive step

###### Definition 6.1

The sequence $\\{b_{j}\\}_{j\in\mathbb{N}}$ is said to be sublinear if
$\lim_{j\to+\infty}b_{j}/j=0$. The sequence $\\{b_{j}\\}_{j\in\mathbb{N}}$ is
said to be convex if $b_{j-1}-2b_{j}+b_{j+1}\geq 0$ for all $j\geq 2$.

###### Lemma 6.1

Suppose $\widehat{H}\in{\cal A}\cap{\cal F}_{\diamond}$ is such that

$(1)$ $\displaystyle\widehat{H}=\sum_{|{\bf k}|\geq s}\widehat{H}_{\bf k}{\bf
z}^{\bf k}=O_{s}({\bf z})$, $|\widehat{H}_{\bf k}|\leq ce^{b_{|{\bf
k}|}+\alpha|{\bf k}|}$, $\alpha\geq 0$, $s\geq 3$,

$(2)$ the sequence $\\{b_{j}\\}$ is nonincreasing, sublinear and convex666 We
will use sequences $\\{b_{j}\\}$ with $b_{j}<0$.

Then an analytic canonical transformation ${\bf w}\mapsto{\bf
z}=\vartheta({\bf w})$ of coordinates reduces the Hamiltonian
$H_{2}+\widehat{H}$ to the form
$H_{2}\circ\vartheta+\widehat{H}\circ\vartheta=H_{2}+G^{0}+G$, where

$\displaystyle\displaystyle G^{0}=\sum_{s\leq|{\bf k}|\leq
2s-3,\,\langle\omega,{\bf k}^{\prime}\rangle=0}\widehat{H}_{\bf k}{\bf w}^{\bf
k}\in{\cal N}_{\diamond},$ (6.1) $\displaystyle\displaystyle G=\sum_{|{\bf
k}|\geq 2s-2}G_{\bf k}{\bf w}^{\bf k},\qquad|G_{\bf k}|\leq ce^{b_{|{\bf
k}|}+(\alpha+\varepsilon)|{\bf k}|},$ (6.2)
$\displaystyle\displaystyle\varepsilon=c\Lambda\Omega_{2s-2},\qquad\Lambda=\frac{1}{2}ne^{2\alpha}(2s)^{2n+1}e^{2b_{s}-b_{2s-2}},$
(6.3)

Moreover, for any $0<\rho\leq e^{-\alpha}$ and $\rho^{\prime}$ satisfying

$0<\rho^{\prime}\leq\rho-\frac{\varepsilon}{nse^{\alpha}}(\rho
e^{\alpha})^{s-1}$ (6.4)

we have: $\vartheta:D_{\rho^{\prime}}\to D_{\rho}$.

Proof. We construct the transformation $\vartheta$ by using continuous
averaging with the following operator $\xi_{s}$. Suppose $H_{2}({\bf
z}_{\delta})+H({\bf z}_{\delta},\delta)$,

$H({\bf z},\delta)=\sum_{|{\bf k}|\geq s}H_{\bf k}(\delta){\bf z}^{\bf k}$

is the Hamiltonian function obtained as a result of the coordinate
transformation ${\bf z}_{\delta}\mapsto{\bf z}$. We define

$\xi_{s}H({\bf z},\delta)=-i\sum_{{\bf k}\in{\cal Z}_{s}}\sigma_{{\bf
k}^{\prime}}H_{\bf k}(\delta){\bf z}^{\bf k},\qquad{\cal Z}_{s}=\\{{\bf
k}\in\mathbb{Z}_{\diamond}^{2n}:s\leq|{\bf k}|\leq 2s-3\\},$

where $\sigma_{{\bf k}^{\prime}}$ is determined by (3.5). We have:
$H=H^{-}+H^{+}+H^{0}+H^{*}$:

$\displaystyle H^{\pm}=\sum_{{\bf k}\in{\cal Z}_{s}^{\pm}}H_{\bf
k}(\delta){\bf z}^{\bf k},\quad H^{0}=\sum_{{\bf k}\in{\cal Z}_{s}^{0}}H_{\bf
k}(\delta){\bf z}^{\bf k},\quad H^{*}=\sum_{|{\bf k}|\geq 2s-2}H_{\bf
k}(\delta){\bf z}^{\bf k},$ $\displaystyle{\cal Z}_{s}^{\pm}=\\{{\bf
k}\in{\cal Z}_{s}:\pm\sigma_{{\bf k}^{\prime}}>0\\},\quad{\cal
Z}_{s}^{0}=\\{{\bf k}\in{\cal Z}_{s}:\sigma_{{\bf k}^{\prime}}=0\\}.$

Then by definition $\xi_{s}H=i(H^{-}-H^{+})$. Equation (3.4) takes the form

$\displaystyle\partial_{\delta}H^{\pm}=\pm
i\\{H^{\pm},H_{2}\\},\quad\partial_{\delta}H^{0}=0,$ (6.5)
$\displaystyle\partial_{\delta}H^{*}=-i\\{H^{-}-H^{+},H^{-}+H^{+}+H^{0}+H^{*}\\}.$
(6.6)

Here we used the fact that for any two monomials of degree at least $s$ their
Poisson bracket has degree at least $2s-2$.

We present system (6.5)–(6.6) in a more detailed form:

$\displaystyle\partial_{\delta}H_{\bf k}$ $\displaystyle=$
$\displaystyle-\omega_{{\bf k}^{\prime}}H_{\bf
k},\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\mbox{for}\quad{\bf k}\in{\cal
Z}_{s},$ (6.7) $\displaystyle\partial_{\delta}H_{\bf k}$ $\displaystyle=$
$\displaystyle-i\\{H^{-}-H^{+},H^{-}+H^{+}+H^{0}+H^{*}\\}_{\bf
k}\quad\mbox{for}\quad|{\bf k}|\geq 2s-2,$ (6.8) $\displaystyle H_{\bf
k}|_{\delta=0}$ $\displaystyle=$ $\displaystyle\widehat{H}_{\bf k}.$ (6.9)

Here $\omega_{{\bf k}^{\prime}}$ is determined by (3.5), and $\\{\,,\\}_{\bf
k}$ denotes the Taylor coefficient with the number ${\bf k}$ in the expansion
of the bracket. We solve the IVP (6.7)–(6.9) on the interval
$\delta\in[0,+\infty)$. The required change of variables is obtained when
$\delta=+\infty$.

Equations (6.7) can be easily solved: $H_{\bf k}=e^{-\omega_{{\bf
k}^{\prime}}\delta}\widehat{H}_{\bf k}$ for ${\bf k}\in{\cal Z}_{s}$. This
implies (6.1).

Now we associate with system (6.7)–(6.8) a majorant system for the functions

${\bf H}^{\pm}=\sum_{{\bf k}\in{\cal Z}_{s}^{\pm}}{\bf H}_{\bf k}{\bf z}^{\bf
k}\gg H^{\pm},\quad{\bf H}^{0}=\sum_{{\bf k}\in{\cal Z}_{s}^{0}}{\bf H}_{\bf
k}{\bf z}^{\bf k}\gg H^{0}\equiv\widehat{H}^{0},\quad{\bf H}^{*}=\sum_{|{\bf
k}|\geq 2s-2}{\bf H}_{\bf k}{\bf z}^{\bf k}\gg H^{*}.$

By condition (1) of Lemma 6.1 we may choose

$\displaystyle\widehat{\bf H}_{\bf k}$ $\displaystyle=$ $\displaystyle{\bf
H}_{\bf k}|_{\delta=0}=ce^{b_{|{\bf k}|}+\alpha|{\bf k}|},\qquad|{\bf k}|\geq
s,$ $\displaystyle{\bf H}_{\bf k}(\delta)$ $\displaystyle=$ $\displaystyle
e^{-\delta/\Omega_{2s-2}}ce^{b_{|{\bf k}|}+\alpha|{\bf k}|},\qquad\;{\bf
k}\in{\cal Z}_{s}^{-}\cup{\cal Z}_{s}^{+},\quad\delta\geq 0.$

We take as a majorant system for (6.8) the following one:

$\displaystyle\partial_{\delta}{\bf H}_{\bf k}\geq
e^{-\delta/\Omega_{2s-2}}\Sigma_{1}({\bf k}),\qquad|{\bf k}|\geq 2s-2,$ (6.10)
$\displaystyle\Sigma_{1}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\sum_{{\bf l}\in{\cal Z}_{s}^{-}\cup{\cal
Z}_{s}^{+},\,|{\bf m}|\geq s,\,{\bf l}+{\bf m}={\bf k}+{\bf
e}_{j}}(\overline{l}_{j}m_{j}+l_{j}\overline{m}_{j})\widehat{\bf H}_{\bf
l}{\bf H}_{\bf m}.$

For any $|{\bf k}|\geq 2s-2$ we put $\;{\bf H}_{\bf k}=c\,e^{b_{|{\bf
k}|}+\alpha|{\bf k}|}{\bf h}_{\bf k},\quad{\bf h}_{\bf k}(\delta)\geq{\bf
h}_{\bf k}(0)=1$.

We use the notation

$\chi_{K}(\delta)=\max_{|{\bf k}|\leq K}{\bf h}_{\bf
k}(\delta),\qquad\delta\geq 0.$

###### Lemma 6.2

Suppose the sequence $\\{b_{j}\\}_{j\in\mathbb{N}}$ is nonincreasing and
convex. Then for any ${\bf k}\in\mathbb{Z}_{+}^{2n}$, $|{\bf k}|\geq 2s-2$ the
estimate

$\Sigma_{1}({\bf k})\leq c^{2}|{\bf k}|e^{\alpha|{\bf k}|}\Lambda\chi_{|{\bf
k}|}$ (6.11)

holds, where $\Lambda$ is determined by (6.3).

We prove Lemma 6.2 in the end of this section.

By Lemma 6.2 the differential inequalities (6.10) follow from (and may be
replaced by) the system

$\partial_{\delta}{\bf h}_{\bf k}=c|{\bf
k}|e^{-\delta/\Omega_{2s-2}}\Lambda\chi_{|{\bf k}|},\qquad{\bf h}_{\bf
k}(0)=1.$

Since the multiplier $ce^{-\delta/\Omega_{2s-2}}\Lambda$ does not depend on
${\bf k}$, we conclude777for example, by induction in $|{\bf k}|$ that
$\chi_{|{\bf k}|}={\bf h}_{\bf k}$ and we may take

${\bf h}_{\bf k}(\delta)=\exp\bigg{(}c|{\bf
k}|\Lambda\int_{0}^{\delta}e^{-\widetilde{\delta}/\Omega_{2s-2}}\,d\widetilde{\delta}\bigg{)}.$

In particular, ${\bf h}_{\bf k}={\bf h}_{\bf k}(\delta)$ is an increasing
function of $\delta\in[0,+\infty)$. Taking $\delta=+\infty$, we obtain:

${\bf h}_{\bf k}(+\infty)=\exp\big{(}c\Lambda\Omega_{2s-2}|{\bf k}|\big{)}.$

This implies that we may use in (6.2) $\varepsilon$, determined by (6.3).

Let $\vartheta^{-1}_{\delta}$ be the $\delta$-shift along solutions of the
Hamiltonian system

$\dot{z}_{j}=i\partial_{\overline{z}}\xi_{s}H,\quad\dot{\overline{z}_{j}}=-i\partial_{z}\xi_{s}H,\qquad\xi_{s}H=-i\sum_{{\bf
k}\in{\cal Z}_{s}}\sigma_{{\bf k}^{\prime}}\widehat{H}_{\bf k}e^{-\omega_{{\bf
k}^{\prime}}\delta}{\bf z}^{\bf k}.$

We estimate $\rho^{\prime}$ such that for any ${\bf z}\in D_{\rho^{\prime}}$
and $\delta\in[0,+\infty)$ the point ${\bf
z}(\delta)=\vartheta^{-1}_{\delta}({\bf z})$ does not leave $D_{\rho}$. While
${\bf z}(\delta)=(z_{1}(\delta),\ldots,\overline{z}_{n})\in D_{\rho}$, $\rho
e^{\alpha}\leq 1$, we have the estimate

$|\dot{z}_{j}(\delta)|\leq\sum_{{\bf k}\in{\cal Z}_{s}}|{\bf k}|ce^{b_{|{\bf
k}|}+\alpha|{\bf k}|-\delta/\Omega_{2s-2}}\rho^{|{\bf
k}|-1}\leq(2s)^{2n}ce^{b_{s}-\delta/\Omega_{2s-2}}(\rho
e^{\alpha})^{s-1}e^{\alpha},$ (6.12)

where $j$ is any number from the set $\\{1,\ldots,n\\}$. Here we have used
that the sequence $b_{s}$ does not increase, $s\leq|{\bf k}|\leq 2s$, and the
sum contains less than $(2s)^{2n-1}$ terms. Estimate (6.12) implies

$|z_{j}(\delta)-z_{j}(0)|\leq(2s)^{2n}ce^{b_{s}}\Omega_{2s-2}(\rho
e^{\alpha})^{s-1}e^{\alpha},\qquad j=1,\ldots,n.$

Since the sequence $\\{b_{j}\\}$ does not increase, we have:
$b_{s}-b_{2s-2}\geq 0$. Hence

$|z_{j}(\delta)-z_{j}(0)|\leq(2s)^{2n}ce^{2b_{s}-b_{2s-2}}\Omega_{2s-2}(\rho
e^{\alpha})^{s-1}e^{\alpha}=\frac{1}{nse^{\alpha}}(\rho
e^{\alpha})^{s-1}\varepsilon.$

The differences $\overline{z}_{j}(\delta)-\overline{z}_{j}(0)$ can be
estimated in the same way. This implies (6.4).

Proof of Lemma 6.2. For any ${\bf l},{\bf m},{\bf k}\in\mathbb{Z}_{+}^{2n}$
the equation ${\bf l}+{\bf m}={\bf k}+{\bf e}_{j}$ implies $|{\bf l}|+|{\bf
m}|=|{\bf k}|+2$. This implies the estimate

$\displaystyle\Sigma_{1}({\bf k})\leq e^{\alpha|{\bf k}|+2\alpha}\chi_{|{\bf
k}|}\Sigma_{2}({\bf k}),$ (6.13) $\displaystyle\Sigma_{2}({\bf k})$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{n}\sum_{{\bf l}\in{\cal
Z}_{s}^{-}\cup{\cal Z}_{s}^{+},\,|{\bf m}|\geq s,\,{\bf l}+{\bf m}={\bf
k}+{\bf e}_{j}}c^{2}\,|{\bf l}|\cdot|{\bf m}|e^{b_{|{\bf l}|}+b_{|{\bf
m}|}-b_{|{\bf k}|}}.$

Since in $\Sigma_{2}$ $|{\bf m}|\leq|{\bf k}|$, we have: $\Sigma_{2}({\bf
k})\leq c^{2}|{\bf k}|\Sigma_{3}({\bf k})$, where

$\Sigma_{3}({\bf k})=\sum_{j=1}^{n}S_{j}({\bf k}),\qquad S_{j}({\bf
k})=\sum_{{\bf l}\in{\cal Z}_{s}^{-}\cup{\cal Z}_{s}^{+},\,|{\bf m}|\geq
s,\,{\bf l}+{\bf m}={\bf k}+{\bf e}_{j}}|{\bf l}|\cdot e^{b_{|{\bf
l}|}+b_{|{\bf m}|}-b_{|{\bf k}|}}.$

The number of terms in the sum $S_{j}({\bf k})$ with fixed $|{\bf l}|\leq
2s-3$ does not exceed

$\\#\\{{\bf l}^{\prime}\in\mathbb{Z}_{+}^{2n}:|{\bf l}^{\prime}|=|{\bf
l}|\\}<(2s)^{2n-1}.$

Therefore

$S_{j}({\bf k})\leq\sum_{p+q=|{\bf k}|+2,\,s\leq p\leq 2s-3,\,q\geq
s}(2s)^{2n}e^{b_{p}+b_{q}-b_{|{\bf k}|}}.$

The sequence $\\{b_{j}\\}$ is convex. Therefore by Lemma 7.5, if $|{\bf
k}|\geq 2s-2$, $p+q=|{\bf k}|+2$, and $p,q\geq s$, we have:

$b_{p}+b_{q}\leq b_{|{\bf k}|+2-s}+b_{s}\quad\mbox{and}\quad b_{|{\bf
k}|+2-s}-b_{|{\bf k}|}\leq b_{s}-b_{2s-2}.$

This implies $b_{p}+b_{q}-b_{|{\bf k}|}\leq 2b_{s}-b_{2s-2}$. Hence

$S_{j}({\bf k})\leq\frac{1}{2}(2s)^{2n+1}e^{2b_{s}-b_{2s}-2}=\frac{\Lambda
e^{-2\alpha}}{n},\quad\Sigma_{3}({\bf k})\leq e^{-2\alpha}\Lambda$

and we obtain (6.11).

###  6.2 Low order normalization

In this section, assuming that the normal form of $H_{2}+\widehat{H}$ is
$H_{2}+N_{\diamond}$, $N_{\diamond}=O_{r}({\bf z})$, we apply Lemma 6.1
inductively to reduce the Hamiltonian to the form $H_{2}+G$, $G=O_{r}$.
Assuming $\widehat{H}\in{\cal A}$, we estimate $G$ in terms of $\widehat{H}$.
As a by-product we obtain another proof of the Bruno-Russmann theorem on
convergence of the normalization in the case of zero normal form.

Motivated by conditions of Theorem 3 we will assume that

(1) $\omega\in\mathbb{R}^{n}$ satisfies the Bruno condition,

(2) $\widehat{H}\in{\cal F}_{\diamond}\cap{\cal A}$,

(3) the normal form of the Hamiltonian $H_{2}+\widehat{H}$ is
$H_{2}+N_{\diamond}$ $N_{\diamond}=O_{r}({\bf z})$, $r\geq 3$.

Then by Lemma 7.6 there exists a sublinear convex negative sequence
$\\{b_{s}\\}$ such that

$a_{j}=\exp\big{(}b_{2^{j+1}+2}-2b_{2^{j}+2}\big{)}$ (6.14)

where $\\{a_{j}\\}$ is the following Bruno sequence

$a_{j}=n2^{2n+j}(2^{j}+2)^{2n+1}\Omega_{2^{j}+2},\qquad j\in\mathbb{N}.$
(6.15)

By Lemma 7.1 there exist $\widehat{c}_{0}$ and $\widehat{\alpha}_{0}$ such
that

$\widehat{H}=\sum_{|{\bf k}|\geq 3}\widehat{H}_{\bf k}{\bf z}^{\bf
k},\qquad|\widehat{H}_{\bf k}|\leq\widehat{c}_{0}e^{b_{|{\bf
k}|}+\widehat{\alpha}_{0}|{\bf k}|}.$

Then, choosing $\alpha_{0}\geq\widehat{\alpha}_{0}$, we have:

$|\widehat{H}_{\bf k}|\leq c_{0}e^{b_{|{\bf k}|}+\alpha_{0}|{\bf
k}|},\qquad|{\bf k}|\geq 3,$ (6.16)

where $c_{0}=\widehat{c}_{0}e^{3(\widehat{\alpha}_{0}-\alpha_{0})}$. This
means that choosing $\alpha_{0}$ sufficiently big, we may assume that

$c_{0}e^{2\alpha_{0}}\leq 1/16,\quad ne^{2\alpha_{0}}\geq 2.$ (6.17)

###### Theorem 4

Suppose that $\omega$ and $\widehat{H}$ satisfy the above conditions (1)–(3)
and (6.14)–(6.17).

Then an analytic canonical transformation of coordinates

$\vartheta:D_{\rho_{*}}\to D_{\rho_{0}},\quad{\bf w}\mapsto{\bf
z}=\vartheta({\bf
w}),\qquad\rho_{0}=e^{-\alpha_{0}},\quad\rho_{*}=\rho_{0}e^{-1/2}.$ (6.18)

reduces the Hamiltonian $H_{2}+\widehat{H}$ to the form
$H_{2}\circ\vartheta+\widehat{H}\circ\vartheta=H_{2}+G$, where

$G({\bf w})=\sum_{|{\bf k}|\geq r}G_{\bf k}{\bf w}^{\bf k}=O_{r}({\bf
w}),\quad|G_{\bf k}|\leq c_{0}e^{b_{|{\bf k}|}+\beta|{\bf
k}|},\qquad\beta\leq\alpha_{0}+1/2.$ (6.19)

Remark. Normalization we perform in the proof of Theorem 4 deals with a finite
number of “small divisors” $\langle\omega,{\bf k}^{\prime}\rangle$. Hence the
assumption that $\omega$ satisfies the Bruno condition looks too strong.888For
$|{\bf k}|>r$ we do not need any conditions on small divisors. We use this
assumption to obtain uniform in $r$ estimates (6.18)–(6.19) for $\rho_{*}$ and
$G_{\bf k}$.

###### Corollary 6.1

(The Bruno-Russmann theorem). Suppose that

(a) $\omega\in\mathbb{R}^{n}$ satisfies the Bruno condition,

(b) $\widehat{H}\in{\cal A}\cap{\cal F}_{\diamond}$,

(c) the normal form of the Hamiltonian $H_{2}+\widehat{H}$ equals $H_{2}$.

Then an analytic canonical transformation of coordinates ${\bf w}\mapsto{\bf
z}=\vartheta({\bf w})$ reduces the Hamiltonian $H_{2}+\widehat{H}$ to the
normal form $H_{2}\circ\vartheta+\widehat{H}\circ\vartheta=H_{2}$.

To prove Corollary 6.1 it is sufficient to pass in Theorem 4 to the limit
$r\to+\infty$.

###  6.3 Proof of Theorem 4

The initial Hamiltonian satisfies estimate (6.16). We apply Lemma 6.1
inductively. Let

$N=\max\\{m\in\mathbb{Z}:2^{m-1}+2<r\\}.$

Then $s$ in Lemma 6.1 takes the values

$s_{1},s_{2},\ldots,s_{N}\qquad s_{m}=2^{m-1}+2.$

We obtain the sequences $\alpha_{0},\alpha_{1},\ldots,\alpha_{N}$,
$\varepsilon_{1},\ldots,\varepsilon_{N}$, $\Lambda_{1},\ldots,\Lambda_{N}$,
which estimate the Hamiltonian on $m$-th step:

$\alpha_{m+1}=\alpha_{m}+\varepsilon_{m+1},\quad\varepsilon_{m}=c_{0}\Lambda_{m}\Omega_{s_{m}},\quad\Lambda_{m}=\frac{ne^{2\alpha_{m-1}}}{2}(2s_{m})^{2n+1}e^{2b_{s_{m}}-b_{s_{m+1}}}.$
(6.20)

By (6.14)–(6.15) with $j=m-1$

$e^{2b_{s_{m}}-b_{s_{m+1}}}n2^{m-2}(2s_{m})^{2n+1}\Omega_{s_{m}}=1.$ (6.21)

Equation (6.21) combined with (6.20) implies

$\\!\\!\\!\varepsilon_{m}=\frac{c_{0}ne^{2\alpha_{m-1}}}{2}(2s_{m})^{2n+1}e^{2b_{s_{m}}-b_{s_{m+1}}}\Omega_{s_{m}}=\frac{c_{0}e^{2\alpha_{m-1}}}{2^{m-1}}=\frac{2c_{0}e^{2\alpha_{0}+2(\varepsilon_{0}+\ldots+\varepsilon_{m-1})}}{2^{m}}.$
(6.22)

Now we prove by induction that for $c_{0}e^{2\alpha_{0}}\leq 1/16$ (see
(6.17)) we have $\varepsilon_{m}\leq 2^{-m-2}$. Indeed, for $m=0$ the estimate
holds: by (6.22) $\varepsilon_{0}=2c_{0}^{2}e^{2\alpha_{0}}\leq 1/8$. Suppose
it is valid for $m\leq m_{0}$. Then

$\varepsilon_{m_{0}+1}\leq\frac{1}{8}\frac{e^{2^{-2}+\ldots+2^{m-3}}}{2^{m_{0}+1}}<\frac{e^{1/2}}{2^{m_{0}+2}}<2^{-m_{0}-3}.$

Putting $\beta=\alpha_{N}$, we obtain:
$\beta-\alpha_{0}\leq\sum_{m=1}^{\infty}\varepsilon_{m}\leq 1/2$. This implies
(6.19).

To estimate $\rho_{*}$, we define the sequence
$\vartheta_{1},\ldots,\vartheta_{N}$, where $\vartheta_{m}:D_{\rho_{m-1}}\to
D_{\rho_{m}}$ is the map from Lemma 6.1 on the $m$-th step. We also define

$\rho_{0},\ldots,\rho_{N},\qquad\rho_{0}=e^{-\alpha_{0}},\quad\rho_{N}=\rho_{*}.$
(6.23)

This implies that the map
$\vartheta=\vartheta_{1}\circ\ldots\circ\vartheta_{N}:D_{\rho_{N}}\to
D_{\rho_{0}}$ is well-defined.

Inequality (6.4) on $m$-th step reads

$\rho_{m+1}\leq\rho_{m}-\frac{\varepsilon_{m+1}}{nse^{\alpha}_{m}}(\rho_{m}e^{\alpha_{m}})^{s_{m+1}-1}.$
(6.24)

Now we show that $\rho_{m}$ $(m\geq 1)$, determined by the equation
$\rho_{m}e^{\alpha_{m}}=1$, satisfies (6.24). Indeed,
$\rho_{0}e^{\alpha_{0}}=1$ by (6.23). Assuming $\rho_{m}e^{\alpha_{m}}=1$ for
some $m\geq 0$, we have:

$\bigg{(}\rho_{m}-\frac{\varepsilon_{m+1}}{nse^{\alpha_{m}}}(\rho_{m}e^{\alpha_{m}})^{s_{m+1}}\bigg{)}=e^{\varepsilon_{m+1}}-\frac{\varepsilon_{m+1}}{nse^{\alpha_{m}}}e^{\alpha_{m+1}}=e^{\varepsilon_{m+1}}\Big{(}1-\frac{\varepsilon_{m+1}}{ns}\Big{)}.$

The last quantity exceeds 1 because $\varepsilon_{m}\leq\varepsilon_{0}\leq
1/8$. Hence $\rho_{m+1}=e^{-\alpha_{m+1}}$ satisfies (6.24).

Therefore
$\rho_{*}=\rho_{N}=\rho_{0}e^{-\sum_{1}^{N}\varepsilon_{m}}\geq\rho_{0}e^{-1/2}$.

###  6.4 Proof of Theorem 3

Let $\\{b_{s}\\}$ be a sublinear convex negative sequence, satisfying (6.14)
with the Bruno sequence (6.15). Then by Theorem 4 there exists an analytic
canonical transformation of coordinates which reduces the Hamiltonian
$H_{2}+\widehat{H}$ to the form $H_{2}+G$, where the function $G$ satisfies
(6.19). Since $b_{|{\bf k}|}$ are negative, we have:

$|G_{\bf k}|\leq c_{0}e^{\beta|{\bf k}|},\qquad|{\bf k}|\geq r.$

Then we start another continuous averaging procedure with the operator
$\xi=\xi_{*}$. We repeat the proof of Theorem 2 until equation (4.7). In our
case $f^{\prime}=O_{r-1}({\bf z})$. By Lemma 7.3

$f^{\prime}(\zeta)\ll\frac{a\zeta^{r-1}}{b-\zeta},\qquad
a=a(c_{0},\beta),\quad b=b(c_{0},\beta).$

Putting $\tau=8n\delta$, we obtain

$G=\frac{a(\zeta+\tau G)^{r-1}}{b-(\zeta+\tau G)}.$ (6.25)

To estimate the function $G=G(\zeta)$, we have to use a quantitative version
of the implicit function theorem (Lemma 6.3 below). First, we introduce the
function $\widetilde{G}(\zeta)=\zeta+\tau G(\zeta)$. It satisfies the
equation, obtained from (6.25):

$\zeta=\widetilde{G}+\varphi(\widetilde{G}),\qquad\varphi(\widetilde{G})=-\frac{a\tau\widetilde{G}^{r-1}}{b-\widetilde{G}}.$
(6.26)

###### Lemma 6.3

(see [3]). Let $x=y+\varphi(y)$ in the complex ball $\\{y\in\mathbb{C}:|y|\leq
6\rho\\}$, where the function $\varphi$ is analytic and $|\varphi|\leq\rho/2$.
Then there is a unique analytic function $\psi$ defined for $|x|\leq\rho$ such
that

$y=x+\psi(x),\quad|\psi|\leq|\varphi|\quad\mbox{for }|x|\leq\rho.$

Applying Lemma 6.3 to equation (6.26) with $x=\zeta$ and $y=\widetilde{G}$, we
take

$\rho=\min\Big{\\{}\frac{b}{12},\frac{1}{6}\Big{(}\frac{b}{24a\tau}\Big{)}^{1/(r-2)}\Big{\\}}.$

Then in the ball $|\widetilde{G}|\leq 6\rho$ we have

$|\varphi|\leq\frac{a\tau(6\rho)^{r-1}}{b-6\rho}\leq\frac{2a\tau(6\rho)^{r-1}}{b}\leq\frac{\rho}{2}.$

This implies that $\widetilde{G}=\zeta+\tau G=\zeta+\psi(\zeta)$, where
$|\psi|\leq\rho/2$ for $|\zeta|\leq\rho$.

If $\tau=8n\delta$ is sufficiently large then

$\rho=\frac{1}{6}\Big{(}\frac{b}{24a\tau}\Big{)}^{1/(r-2)}\geq\frac{c_{z}}{1+\delta^{1/(r-2)}}$

for some constant $c_{z}$. Hence in the domain $\\{|\zeta|\leq\rho\\}$

$|G|=\Big{|}\frac{\widetilde{G}-\zeta}{\tau}\Big{|}\leq\frac{\rho}{2\tau}\leq\frac{C}{1+\delta^{1+1/(r-2)}}$

for some constant $C$. The function
$\int_{0}^{\zeta}G(\widehat{\zeta})\,d\widehat{\zeta}$ is a majorant for
$|{\cal H}|$. This implies $|{\cal H}|\leq|\rho G|\leq
C_{H}/(1+\delta^{1+2/(r-2)})$.

Generically $\omega$ does not admit resonances of order less than 4 (this
includes the case of nonresonant $\omega$). Then $r=4$ and diameter of the
analyticity domain for ${\cal H}$ is of order $\delta^{-1/2}$. This estimate
is better than $\delta^{-1}$ declared in [18].

##  7 Technical part

###  7.1 Majorants

###### Lemma 7.1

Suppose $F=\sum_{{\bf k}\in\mathbb{Z}_{+}^{2n}}F_{\bf k}{\bf z}^{\bf
k}\in{\cal A}^{\rho}$, and the sequence $\\{b_{j}\\}_{j\in\mathbb{Z}_{+}}$ is
sublinear. Then for any $\alpha>-\ln\rho$ there exists $c>0$ such that

$|F_{\bf k}|\leq ce^{b_{|{\bf k}|}+\alpha|{\bf k}|}.$ (7.1)

Proof. By Lemma 2.1 we have $|F_{\bf k}|\leq c_{F}\rho^{-|{\bf k}|}$,
$c_{F}=\|F\|_{\rho}$. Hence in (7.1) we have to choose $c$ such that

$c_{F}e^{-b_{q}-(\alpha+\ln\rho)q}<c\quad\mbox{for any }\;q\in\mathbb{Z}_{+}.$
(7.2)

For any $\alpha>-\ln\rho$ the function $q\mapsto
e^{-b_{q}-(\alpha+\ln\rho)q}$, $q\in\mathbb{Z}_{+}$ is bounded because $b_{q}$
is sublinear.

For any $F,{\bf F}\in{\cal F}$ we say that $F\ll{\bf F}$ iff for their Taylor
coefficients we have the inequalities $|F_{\bf k}|\leq{\bf F}_{\bf k}$, ${\bf
k}\in\mathbb{Z}_{+}^{2n}$. In this case we say that ${\bf F}$ is a majorant
for $F$.

###### Lemma 7.2

Suppose $F\ll{\bf F}$ and $\hat{F}\ll\hat{\bf F}$. Then

1. $F+\hat{F}\ll{\bf F}+\hat{\bf F}$, $F\hat{F}\ll{\bf F}\hat{\bf F}$, $\partial_{z_{s}}F\ll\partial_{z_{s}}{\bf F}$, and $\partial_{\overline{z}_{s}}F\ll\partial_{\overline{z}_{s}}{\bf F}$, $s=1,\ldots,n$.

2. If $F$ and ${\bf F}$ depend on the parameter $\delta\in[\delta_{1},\delta_{2}]$ then

$\int_{\delta_{1}}^{\delta_{2}}F\,d\delta\ll\int_{\delta_{1}}^{\delta_{2}}{\bf
F}\,d\delta.$

We skip an obvious proof.

###### Lemma 7.3

Suppose $F\in{\cal A}^{\rho}$, $F=\sum_{|{\bf k}|\geq s}F_{\bf k}{\bf z}^{\bf
k}=O_{s}({\bf z})$, and $|F_{\bf k}|\leq a\rho^{s-|{\bf k}|}$. Then

$F\ll\frac{a\rho\zeta^{s}}{\rho-\zeta},\qquad\zeta=z_{1}+\ldots+z_{n}+\overline{z}_{1}+\ldots+\overline{z}_{n}.$
(7.3)

Proof. Note that $\sum_{|{\bf k}|=j}{\bf z}^{\bf k}\ll\zeta^{j}$ for any
$j\in\mathbb{Z}_{+}$. Then

$F\ll\sum_{|{\bf k}|\geq s}a\rho^{s-|{\bf k}|}{\bf z}^{\bf k}\ll
a\rho^{s}\sum_{j=s}^{\infty}\frac{\zeta^{j}}{\rho^{j}}=\frac{a\rho\zeta^{s}}{\rho-\zeta}.$

###### Lemma 7.4

For any $\rho>0$

$\partial_{\zeta}\frac{\rho\zeta^{3}}{\rho-\zeta}\ll\frac{2\rho\zeta^{2}}{\rho/2-\zeta}.$

Proof. Since

$\partial_{\zeta}\frac{\rho\zeta^{3}}{\rho-\zeta}=\frac{3\rho\zeta^{2}}{\rho-\zeta}+\frac{\rho\zeta^{3}}{(\rho-\zeta)^{2}},$

the lemma follows from two estimates:

$\displaystyle\frac{3\rho\zeta^{2}}{\rho-\zeta}$ $\displaystyle\ll$
$\displaystyle\frac{3\rho\zeta^{2}}{2(\rho/2-\zeta)},$
$\displaystyle\frac{\rho\zeta}{(\rho-\zeta)^{2}}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}k\Big{(}\frac{\zeta}{\rho}\Big{)}^{k}=\sum_{k=0}^{\infty}\frac{k}{2^{k}}\Big{(}\frac{\zeta}{\rho/2}\Big{)}^{k}\ll\sum_{k=0}^{\infty}\frac{1}{2}\Big{(}\frac{\zeta}{\rho/2}\Big{)}^{k}=\frac{\rho}{2(\rho/2-\zeta)^{2}}.$

###  7.2 Majorant principle

We use majorant method to obtain estimates for solutions of initial value
problems (IVP) in ${\cal F}$.

As an example consider the IVP

$\partial_{\delta}F=\Phi(F,\delta),\qquad F|_{\delta=0}=\widehat{F}.$ (7.4)

Here $F\in{\cal F}$ depends on the parameter $\delta$ and $\Phi$ is a map from
${\cal F}\times\mathbb{R}_{+}$ to ${\cal F}$.

###### Definition 7.1

IVP (7.4) is said to be power regular if for any $\widehat{F}\in{\cal F}$
equation (7.4) has a unique solution $F=F({\bf z},\delta)\in{\cal F}$ for all
$\delta>0$

We associate with (7.4) the so-called majorant system

$\partial_{\delta}{\bf F}=\Psi({\bf F},\delta),\qquad{\bf
F}|_{\delta=0}=\widehat{\bf F}.$ (7.5)

We put $\Phi_{\bf k}=p_{\bf k}\circ\Phi$ and $\Psi_{\bf k}=p_{\bf
k}\circ\Psi$.

###### Definition 7.2

IVP (7.5) is said to be a majorant IVP for (7.4) if the following two
properties hold:

(a) $\widehat{F}\ll\widehat{\bf F}$.

(b) For any $F\ll{\bf F}$, ${\bf k}\in\mathbb{Z}_{+}^{2n}$, and $\delta\geq 0$
we have: $\Phi_{\bf k}(F,\delta)\ll\Psi_{\bf k}({\bf F},\delta)$.

Majorant principle. Suppose the IVP (7.4) is power regular. Suppose also that
there exists a solution ${\bf F}={\bf F}(\cdot,\delta)\in{\cal A}$ of (7.5) on
the interval $\delta\in[0,\delta_{0}]$. Then (7.4) has a unique analytic
solution $F$ on $[0,\delta_{0}]$ and $F(\cdot,\delta)\ll{\bf
F}(\cdot,\delta)$.

Remarks. 1. Definitions 7.1 and 7.2 as well as the Majorant principle
obviously extend to systems of equations, where $F,\widehat{F}\in{\cal F}^{m}$
and $\Phi:{\cal F}^{m}\times\mathbb{R}_{+}\to{\cal F}$.

2\. One may replace the first equation (7.5) by the inequality
$\partial_{\delta}{\bf F}\gg\Psi({\bf F},\delta)$.

The majorant principle presented here differs from the majorant argument used
since Cauchy times. Traditionally the evolution variable (in our case
$\delta$) is regarded complex as well and Taylor expansions in it are used. In
our approach this variable is a real parameter in both exact solution and a
majorant. Due to this we are able to obtain majorant estimates for solutions
of (7.4) on large (even infinite) intervals of $\delta$.

###### Theorem 5

Suppose both systems (7.4) and (7.5) have nilpotent structure (see Definition
4.1). Then Majorant principle holds true.

We expect that Majorant principle is valid in a much wider generality. But in
this paper we are only interested in the case of systems having nilpotent
form.

Proof of Theorem 5. Let ${\bf k}^{0}$ be an index with minimal possible degree
$|{\bf k}^{0}|$. For example, in the system (3.11) $|{\bf k}^{0}|=3$.
Nilpotent form of (7.4) implies that

$0=\partial_{\delta}F_{{\bf k}^{0}}\ll\partial_{\delta}{\bf F}_{{\bf k}^{0}}.$

Hence $F_{{\bf k}^{0}}(\delta)\ll{\bf F}_{{\bf k}^{0}}(\delta)$ for
$\delta\geq 0$.

We proceed by induction in $|{\bf k}|$. Suppose $F_{{\bf k}}(\delta)\ll{\bf
F}_{{\bf k}}(\delta)$, $\delta\geq 0$ provided $|{\bf k}|<K$. For any ${\bf
k}$ such that $|{\bf k}|=K$ we have by induction assumption and item (b) of
Definition 7.2:

$\partial_{\delta}({\bf F}_{\bf k}-F_{\bf k})=\Psi_{\bf k}({\bf
F}(\cdot,\delta),\delta)-\Phi_{\bf k}({\bf F}(\cdot,\delta),\delta)\gg 0.$

Therefore

${\bf F}_{\bf k}(\delta)=\widehat{\bf F}_{\bf
k}(\delta)+\int_{0}^{\delta}\Psi_{\bf k}({\bf
F}(\cdot,\lambda),\lambda)\,d\lambda\gg F_{\bf k}(\delta)=\widehat{F}_{\bf
k}(\delta)+\int_{0}^{\delta}\Phi_{\bf k}(F(\cdot,\lambda),\lambda)\,d\lambda.$

Here we used that arguments of $\Psi_{\bf k}$ and $\Phi_{\bf k}$ are known by
the induction assumption.

This majorant inequality makes sense if the left-hand side is defined i.e.,
for any $\delta\in[0,\delta_{0}]$.

###  7.3 On convex sequences

###### Lemma 7.5

Suppose the sequence $\\{b_{j}\\}_{j\in\mathbb{N}}$ is convex. Then

(1) for any $1\leq m<k<l$

$(l-k)b_{m}+(m-l)b_{k}+(k-m)b_{l}\geq 0.$ (7.6)

(2) for any $1\leq m<k\leq l$

$b_{k}+b_{l}\leq b_{k-m}+b_{l+m},$ (7.7)

Proof. (1) First, consider the case $l-k=1$. If $k-m=1$ then (7.6) coincides
with definition of convexity. We use induction in $k-m$. Suppose (7.6) holds
for $k-m=q$. Then inequality (7.6) is obtained if we add the inequalities

$\displaystyle(l-k)b_{m}+(m-l+1)b_{k-1}+(k-1-m)b_{l-1}\geq 0,$
$\displaystyle\qquad\qquad\;-(m-l+1)b_{k-1}+2(m-l+1)b_{l-1}-(m-l+1)b_{l}\geq
0$

which hold by the induction assumption.

The case $l-k>1$ follows from the case $l-k=1$ also by induction.

(2) Equation (7.6) implies the inequalities

$\displaystyle(l-k)b_{k-m}+(k-m-l)b_{k}+mb_{l}$ $\displaystyle\geq$
$\displaystyle 0,$ $\displaystyle mb_{k}+(k-l-m)b_{l}+(l-k)b_{l+m}$
$\displaystyle\geq$ $\displaystyle 0$

from which (7.7) follows.

###### Lemma 7.6

1. For any convex sublinear sequence $\\{b_{s}\\}$ the sequence $\\{a_{j}\\}$, satisfying (6.14) is Bruno.

2. For any Bruno sequence $\\{a_{j}\\}$ there exists a sublinear convex negative non-increasing sequence $\\{b_{s}\\}$, satisfying (6.14).

Proof. 1. Suppose $\\{b_{s}\\}$ is convex and sublinear, and $\\{a_{j}\\}$
satisfies (6.14). First, note that

$\ln a_{j}-\ln a_{j-1}=-b_{2^{j+1}+2}+3b_{2^{j}+2}-2b_{2^{j-1}+2}.$

The latter expression is nonnegative by convexity of $\\{b_{s}\\}$ (see Lemma
7.5). Hence, $a_{j}$ is non-decreasing.

Equation (6.14) implies

$\frac{1}{2}\sum_{j=0}^{J-1}2^{-j}\ln
a_{j}=\sum_{j=0}^{J-1}\Big{(}2^{-j-1}b_{2^{j+1}+2}-2^{-j}b_{2^{j}+2}\Big{)}=2^{-J}b_{2^{J}+2}-b_{3}.$

Since $\\{b_{j}\\}$ is sublinear, $2^{-J}b_{2^{J}+2}\to 0$ as $J\to\infty$.
Therefore $\\{a_{j}\\}$ is a Bruno sequence.

2. Suppose $\\{a_{j}\\}$ is a Bruno sequence. We compute $\\{b_{2^{j}+2}\\}$ from the equation

$\frac{b_{2^{J}+2}}{2^{J}}=b_{3}+\frac{1}{2}\sum_{j=1}^{J-1}\frac{\ln
a_{j}}{2^{j}}$

which implies (6.14). To have sublinearity, we need
$\lim_{J\to+\infty}2^{-J}b_{2^{J}+2}=0$. Hence, we take
$b_{3}=-\frac{1}{2}\sum_{j=1}^{\infty}2^{-j}\ln a_{j}$. Such a choice of
$b_{3}$ implies that $b_{2^{J}+2}<0$, $J=0,1,\ldots$

The equation

$b_{2^{J}+2}=-2^{J-1}\sum_{j=J}^{\infty}\frac{\ln
a_{j}}{2^{j}}=-\frac{1}{2}\sum_{j=0}^{\infty}\frac{\ln a_{J+j}}{2^{j}}$

implies

$b_{2^{J+1}+2}-b_{2^{J}+2}=\frac{1}{2}\ln
a_{J}-\frac{1}{2}\sum_{j=1}^{\infty}\frac{\ln
a_{J+j}}{2^{j}}\leq\frac{1}{2}\ln
a_{J}\Big{(}1-\sum_{j=1}^{\infty}2^{-j}\Big{)}=0.$

Hence the sequence $\\{b^{2^{j}+2}\\}$ does not increase.

For any integer $s\in(2^{j}+2,2^{j+1}+2)$ we define $b_{s}$ by linear
interpolation:

$b_{s}=\frac{s-2^{j}-2}{2^{j}}b_{2^{j}+2}+\frac{2^{j+1}+2-s}{2^{j}}b_{2^{j+1}+2}.$

Such $\\{b_{s}\\}$ is obviously sublinear.

We check convexity of $\\{b_{s}\\}$ first, on the subsequence
$\\{b_{2^{j}+2}\\}_{j\in\mathbb{Z}_{+}}$. We have to prove the inequality

$2^{J}b_{2^{J-1}+2}-(2^{J}+2^{J-1})b_{2^{J}+2}+2^{J-1}b_{2^{J+1}+2}\geq 0.$
(7.8)

By (6.14) the left-hand side of (7.8) equals $2^{J-1}(\ln a_{J}-\ln a_{J-1})$.
For any $J\geq 1$ this expression is nonnegative because $\\{a_{j}\\}$ is non-
decreasing.

If $b_{s}$, $s\neq 2^{j}+2$ are determined by linear interpolation, the
convexity, negativity, and monotonicity remain true for all the sequence
$\\{b_{s}\\}_{s\in\mathbb{N}}$.

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