# Denjoy-Carleman solvability of Vekua-type periodic operators

Alexandre Kirilov<EMAIL_ADDRESS>Wagner Augusto Almeida de Moraes
<EMAIL_ADDRESS>Pedro Meyer Tokoro<EMAIL_ADDRESS>
###### Abstract

This paper explores the solvability and global hypoellipticity of Vekua-type
differential operators on the n-dimensional torus, within the framework of
Denjoy-Carleman ultradifferentiability. We provide the necessary and
sufficient conditions for achieving these global properties in the case of
constant-coefficient operators, along with applications to classical
operators. Additionally, we investigate a class of variable coefficients and
establish conditions for its solvability.

###### keywords:

Vekua-type operators , Solvability , Global hypoellipticity , Denjoy-Carleman
classes.

###### MSC:

[2020] Primary 35A01 , 35B10 , Secondary 30G20 , 46F05.

††journal: arXiv

[ufpr]organization= Departamento de Matemática, Universidade Federal do
Paraná, city=Curitiba, postcode=81531-990, state=Paraná, country=Brazil

[ppgm]organization=Programa de Pós-Graduação em Matemática, Universidade
Federal do Paraná, city=Curitiba, postcode=81531-990, state=Paraná,
country=Brazil

##  1 Introduction

The theory of generalized analytic functions, as introduced by Vekua in [18],
initially dealt with solutions to equations of the form:

$\partial_{\bar{z}}u+Au+B\bar{u}=F,$

where $\partial_{\bar{z}}$ stands for the Cauchy-Riemann operator, and the
coefficients $A$ and $B$ belong to a suitable function space in the complex
plane. This theory is closely linked to the theory of holomorphic functions
and relies on specific regularity conditions on the coefficients to apply the
similarity principle. Vekua used this framework to address problems in the
theory of shells and to explore problems of infinitesimal bendings of
surfaces.

In [16], V. Kravchenko extended this theory by replacing the Cauchy-Riemann
operator with more general complex-valued vector fields. This generalization
enabled the study of classical equations from mathematical physics, including
the Schrödinger, Dirac, and Maxwell equations, among others.

The work of Vekua, Kravchenko, Bers, and other researchers has found extensive
applications in various fields, such as boundary value problems in elasticity
theory, hydrodynamics, electric potential, mechanics, and more.

In the article [13], we investigated the solvability and global
hypoellipticity of the periodic Vekua-type operator $P$ defined by

$Pu=Lu-Au-B\bar{u},$ (1.1)

where $A$ and $B$ are complex constants, and
$L:\mathcal{C}^{\infty}(\mathbb{T}^{n})\to\mathcal{C}^{\infty}(\mathbb{T}^{n})$
is a general constant coefficient partial differential operator. In this case,
we obtained a complete characterization of these global properties for this
class of operators.

Now, in this article, we address the Denjoy-Carleman solvability problem for
the same class of constant coefficient operators. Additionally, we extend the
analysis to a class of operators with variable coefficients, building on the
results obtained in [3] and [7].

In this paper, we organize the content as follows. In Section 2, we revisit
fundamental notations and properties related to Denjoy-Carleman classes of
ultradifferentiable functions. We provide the necessary background and refer
the reader to [8] for more comprehensive details and proofs. Next, we delve
into the analysis of operators with constant coefficients, outlining the
conditions under which they are solvable in the ultradifferentiable sense.
This involves establishing a necessary and sufficient condition, linked to a
Diophantine condition, which relates the growth of a value associated with the
symbol of the operator to its solvability. We also demonstrate that global
hypoellipticity and solvability are equivalent properties.

In the latter part of the paper, we shift our focus to operators with variable
coefficients. Motivated by the work in [3] and [7], we consider a class of
complex vector fields with variable coefficients that satisfy the Nirenberg-
Treves condition $(P)$. This condition ensures the local solvability of this
class of vector fields. We derive conditions under which these operators are
solvable.

Further details on smooth and ultradifferentiable solvability and the
regularity of solutions on tori can be found in [1, 2, 4, 9]. In the wider
context of compact Lie groups, the reader can see some results in [10, 11, 12]
and the references cited therein.

##  2 Notations and Fourier analysis in Denjoy-Carleman classes

Let us begin by recalling the definition and some useful properties of the
spaces of ultradifferentiable functions of Roumieu type defined on the
$n$-dimensional torus $\mathbb{T}^{n}\simeq\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$.
For more details on the results and their proofs, we refer the reader to [8].

A sequence $\mathscr{M}=\\{m_{j}\\}_{j\in\mathbb{N}}$ of positive real numbers
is called a weight sequence if it satisfies the following conditions:

  1. $i.$

$m_{0}=m_{1}=1$;

  2. $ii.$

$m_{j}^{2}\leq m_{j-1}\,m_{j+1}$ for all $j\in\mathbb{N}$;

  3. $iii.$

$\displaystyle\sup_{j,k\in\mathbb{N}_{0}}\left(\dfrac{m_{j+k}}{m_{j}\,m_{k}}\right)^{\frac{1}{j+k}}\leq
H$, for some $H\geq 1$.

Given a weight sequence $\mathscr{M}=\\{m_{j}\\}_{j\in\mathbb{N}}$ and $h>0$,
consider the set $\mathcal{E}_{\mathscr{M},h}(\mathbb{T}^{n})$ of all smooth
functions $f:\mathbb{T}^{n}\to\mathbb{C}$ such that

$\|f\|_{\mathscr{M},h}\doteq\sup_{\alpha\in\mathbb{N}_{0}^{n}}\sup_{x\in\mathbb{T}^{n}}\,\dfrac{|\partial^{\alpha}f(x)|}{h^{|\alpha|}\,m_{|\alpha|}\,|\alpha|!}<\infty.$

We observe that $\mathcal{E}_{\mathscr{M},h}(\mathbb{T}^{n})$ is a Banach
space with respect to the norm $\|\cdot\|_{\mathscr{M},h}$, and we define the
space of periodic $\mathscr{M}$-ultradifferentiable functions of Roumieu type
by

$\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})=\underset{h\rightarrow+\infty}{\operatorname{ind}\lim}\,\mathcal{E}_{\mathscr{M},h}(\mathbb{T}^{n}).$

Moreover, its dual space, denoted by
$\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})$, is commonly referred to
as the space of periodic $\mathscr{M}$-ultradistributions.

Therefore, a smooth function $f:\mathbb{T}^{n}\to\mathbb{C}$ belongs to the
space $\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ if and only if there exist
constants $C,h>0$ such that, for all $\alpha\in\mathbb{N}_{0}^{n}$,

$\sup_{x\in\mathbb{T}^{n}}|\partial^{\alpha}f(x)|\leq
Ch^{|\alpha|}m_{|\alpha|}|\alpha|!.$

The best-known example of a space of ultradifferentiable functions is given by
the weight sequence $\mathscr{M}=\\{m_{j}\\}_{j\in\mathbb{N}_{0}}$, where
$m_{j}=(j!)^{s-1}$. For each fixed $s\geq 1$,
$\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})=\mathcal{G}^{s}(\mathbb{T}^{n})$ is
the space of periodic Gevrey functions of order $s$.

The Fourier coefficients of a function
$f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ are given by

$\widehat{f}(\xi)\doteq(2\pi)^{-n}\int_{\mathbb{T}^{n}}f(x)e^{-i\xi\cdot
x}\,dx,\quad\xi\in\mathbb{Z}^{n},$

and, as usual, the Fourier coefficients of an ultradistribution
$u\in\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})$ are defined by

$\widehat{u}(\xi)=(2\pi)^{-n}\langle u,e^{-i\xi\cdot
x}\rangle,\quad\xi\in\mathbb{Z}^{n}.$

The classes of ultradifferentiable functions and ultradistributions on tori
can be characterized by the rate of decay or growth of their Fourier
coefficients in the following way:

$\displaystyle
f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})\Longleftrightarrow$
$\displaystyle\ \text{there exist }C,\delta>0\text{ such that,}$ (2.1)
$\displaystyle\qquad\qquad|\widehat{f}(\xi)|\leq
C\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta^{j}\,(1+\|\xi\|)^{j}},\quad\xi\in\mathbb{Z}^{n};$
$\displaystyle
u\in\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})\Longleftrightarrow$
$\displaystyle\ \text{for all }\varepsilon>0,\text{ there is
}C_{\varepsilon}>0\text{ such that,}$ (2.2)
$\displaystyle\qquad\,\qquad|\widehat{u}(\xi)|\leq
C_{\varepsilon}\sup_{j\in\mathbb{N}_{0}}\dfrac{\varepsilon^{j}(1+\|\xi\|)^{j}}{m_{j}\,j!},\quad\xi\in\mathbb{Z}^{n}.$

Moreover, when $f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ and
$u\in\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})$, we have

$f(x)=\sum_{\xi\in\mathbb{Z}^{n}}\widehat{f}(\xi)e^{i\xi\cdot
x}\quad\text{and}\quad
u=\sum_{\xi\in\mathbb{Z}^{n}}\widehat{u}(\xi)e^{i\xi\cdot x},$

with convergences in $\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ and
$\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})$ respectively.

Another important characterization for the development of this work is given
by the partial Fourier series, which can be described as follows: let us write
$n=p+q$, with $p,q\in\mathbb{N}$, consider the direct sum
$\mathbb{R}^{n}=\mathbb{R}^{p}\oplus\mathbb{R}^{q}$ and write
$(t,x)\in\mathbb{T}^{n}$, meaning that $t\in\mathbb{T}^{p}$ and
$x\in\mathbb{T}^{q}$.

Given $f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$, for each
$t\in\mathbb{T}^{p}$, consider the function
$f_{t}:\mathbb{T}^{q}\to\mathbb{C}$ given by $f_{t}(x)=f(t,x)$. Clearly, for
each $t\in\mathbb{T}^{p}$, we have
$f_{t}\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{q})$ and by the
characterization (2.1), we can write

$f(t,x)=f_{t}(x)=\sum_{\xi\in\mathbb{Z}^{q}}\widehat{f}(t,\xi)e^{i\xi\cdot
x},$

with convergence in $\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{q})$, where

$\widehat{f}(t,\xi)\doteq(2\pi)^{-q}\int_{\mathbb{T}^{q}}f_{t}(x)e^{-i\xi\cdot
x}\,dx.$

Moreover, for each $\xi\in\mathbb{Z}^{q}$ fixed, we have that
$\widehat{f}(\cdot,\xi)\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{p})$.

Analogously, given an ultradistribution
$u\in\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})$, for each
$\xi\in\mathbb{Z}^{q}$, we define the partial Fourier coefficient of $u$ as
the functional
$\widehat{u}(t,\xi):\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{p})\to\mathbb{C}$
given by

$\langle\widehat{u}(t,\xi),f\rangle=(2\pi)^{-q}\langle u,f(t)\otimes
e^{-i\xi\cdot x}\rangle,\quad f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{p}).$

In this framework we have the following characterizations:

$\displaystyle
f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})\Longleftrightarrow\ $
$\displaystyle\exists C_{p},h_{p},h_{q}>0;\
\forall\alpha\in\mathbb{N}_{0}^{p},\ \forall\xi\in\mathbb{Z}^{q}$ (2.3)
$\displaystyle\
\sup_{t\in\mathbb{T}^{p}}|\partial^{\alpha}\widehat{f}(t,\xi)|\leq
C_{p}\,h_{p}^{|\alpha|}\,m_{|\alpha|}\,|\alpha|!\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{h_{q}^{j}\,(1+\|\xi\|)^{j}};$
$\displaystyle
u\in\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})\Longleftrightarrow\ $
$\displaystyle\forall\varepsilon>0,\ \exists C_{\varepsilon}>0;\ \forall
f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{p}),\ \forall\xi\in\mathbb{Z}^{q}$
(2.4) $\displaystyle\ \,|\langle\widehat{u}(t,\xi),f\rangle|\leq
C_{\varepsilon,h}\|f\|_{\mathscr{M},h}\,\sup_{j\in\mathbb{N}_{0}}\dfrac{\varepsilon^{j}\,(1+\|\xi\|)^{j}}{m_{j}\,j!}.$

##  3 Constant coefficient Operators

Consider the operator
$P:\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})\to\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$
defined by

$Pu=Lu-Au-B\bar{u},$ (3.1)

where $A,B\in\mathbb{C}$, and $L$ is an operator of the form

$L=\sum_{0<|\alpha|\leqslant m}c_{\alpha}\partial^{\alpha},$

with $c_{\alpha}\in\mathbb{C}$ for all $\alpha\in\mathbb{N}_{0}^{n}$
satisfying $0<|\alpha|\leqslant m$, and with symbol

$\sigma_{L}(\xi)=\sum_{0<|\alpha|\leqslant
m}i^{|\alpha|}c_{\alpha}\xi^{\alpha},\ \xi\in\mathbb{Z}^{n}.$

By the continuity of $P$, if
$u(x)=\sum_{\xi\in\mathbb{Z}^{n}}\widehat{u}(\xi)e^{i\xi\cdot
x}\in\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})$, we have

$\displaystyle Pu(x)=$
$\displaystyle\sum_{\xi\in\mathbb{Z}^{n}}\left(\sigma_{L}(\xi)-A\right)\widehat{u}(\xi)e^{i\xi\cdot
x}-B\sum_{\xi\in\mathbb{Z}^{n}}\overline{\widehat{u}(-\xi)}e^{-i\xi\cdot x}.$

Therefore, the Fourier coefficients of any solution of the equation $Pu=f$
must satisfy

$(\sigma_{L}(\xi)-A)\widehat{u}(\xi)-B\overline{\widehat{u}(-\xi)}=\widehat{f}(\xi).$

Taking the conjugate of the previous equation for $-\xi\in\mathbb{Z}^{n}$, we
obtain

$\overline{\widehat{f}(-\xi)}=(\overline{\sigma_{L}(-\xi)}-\bar{A})\overline{\widehat{u}(-\xi)}-\bar{B}\widehat{u}(\xi),$

which gives us, for each $\xi\in\mathbb{Z}^{n}$, the following linear system:

$\begin{cases}\left(\sigma_{L}(\xi)-A\right)\widehat{u}(\xi)-B\overline{\widehat{u}(-\xi)}=\widehat{f}(\xi),\\\
-\bar{B}\widehat{u}(\xi)+(\overline{\sigma_{L}(-\xi)}-\bar{A})\overline{\widehat{u}(-\xi)}=\overline{\widehat{f}(-\xi)}.\end{cases}$

Solving this system for $\widehat{u}(\xi)$, we obtain

$\Delta_{\xi}\widehat{u}(\xi)=(\overline{\sigma_{L}(-\xi)}-\bar{A})\widehat{f}(\xi)+B\overline{\widehat{f}(-\xi)},$
(3.2)

where $\Delta_{\xi}$ is the discriminant of the above system, that is,

$\Delta_{\xi}=\left(\sigma_{L}(\xi)-A\right)(\overline{\sigma_{L}(-\xi)}-\bar{A})-|B|^{2}.$
(3.3)

Observe that $\overline{\Delta_{\xi}}=\Delta_{-\xi}$ for all
$\xi\in\mathbb{Z}^{n}$. In particular, $\Delta_{\xi}=0$ if, and only if,
$\Delta_{-\xi}=0$.

Inspired by the references [3, 7, 13], we introduce the following notion of
solvability and the main result of this section.

######  Definition 3.1.

We say that the differential operator
$P:\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})\to\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$
defined in (3.1) is $\mathscr{M}$-solvable if there exists a subspace
$\mathcal{F}\subset\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ of finite
codimension such that, for all $f\in\mathcal{F}$, there exists
$u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ satisfying $Pu=f$.

######  Theorem 3.2.

The operator $P$ is $\mathscr{M}$-solvable if and only if for all
$\varepsilon>0$, there exist constants
$C_{\varepsilon},\gamma_{\varepsilon}>0$ such that

$\|\xi\|\geq\gamma_{\varepsilon}\ \Rightarrow\ |\Delta_{\xi}|\geq
C_{\varepsilon}\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|)^{j}}.$
(DC$\mathscr{M}$)

###### Proof.

Firstly, observe that

$\displaystyle\sup_{j\in\mathbb{N}_{0}}\dfrac{\varepsilon^{j}(1+\|\xi\|)^{j}}{m_{j}\,j!}\leq\sum_{j\in\mathbb{N}_{0}}\dfrac{\varepsilon^{j}(1+\|\xi\|)^{j}}{m_{j}\,j!}\leq\sum_{j\in\mathbb{N}_{0}}\dfrac{\varepsilon^{j}(1+\|\xi\|)^{j}}{j!}=e^{\varepsilon(1+\|\xi\|)}.$

Therefore, for any $\varepsilon>0$ we have

$\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|)^{j}}=\left(\sup_{j\in\mathbb{N}_{0}}\dfrac{\varepsilon^{j}(1+\|\xi\|)^{j}}{m_{j}\,j!}\right)^{-1}>0$
(3.4)

Let us assume that (DC$\mathscr{M}$) holds. Then, the set

$\Omega=\\{\xi\in\mathbb{Z}^{n}\,:\,\Delta_{\xi}=0\\}$

is finite. This implies that the subspace

$\mathcal{F}=\\{f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})\,:\,\widehat{f}(\xi)=0\text{
for all }\xi\in\Omega\\}$

has finite codimension in $\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$.

Given $f\in\mathcal{F}$, by (2.1), there exist $C,\delta>0$ such that, for all
$\xi\in\mathbb{Z}^{n}$,

$|\widehat{f}(\xi)|\leq
C\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta^{j}(1+\|\xi\|)^{j}}.$
(3.5)

Let $H>0$ be the value specified in condition iii. of the definition of a
weight sequence on page 2, and set $\varepsilon=\delta/H$ in condition
(DC$\mathscr{M}$). Then, there exist constants
$C_{\varepsilon},\gamma_{\varepsilon}>0$ such that

$\|\xi\|\geq\gamma_{\varepsilon}\Rightarrow|\Delta_{\xi}|\geq
C_{\varepsilon}\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|)^{j}}.$
(3.6)

Let $u\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})$ be a solution of
$Pu=f$. For $\|\xi\|\geq\gamma_{\varepsilon}$, we have
$\xi\in\mathbb{Z}^{n}\setminus\Omega$. Therefore, it follows from (3.2),
(3.5), and (3.6) that

$\displaystyle|\widehat{u}(\xi)|$
$\displaystyle=|\Delta_{\xi}|^{-1}\big{|}[\overline{\sigma_{L}(-\xi)}-\bar{A}]\widehat{f}(\xi)+B\overline{\widehat{f}(-\xi)}\big{|}$
$\displaystyle\leq\dfrac{C}{C_{\varepsilon}}\left(|\overline{\sigma_{L}(-\xi)}-\bar{A}|+|B|\right)\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta^{j}(1+\|\xi\|)^{j}}\sup_{j\in\mathbb{N}_{0}}\dfrac{(\delta/H)^{j}(1+\|\xi\|)^{j}}{m_{j}\,j!}.$

Notice that $\sigma_{L}$ is a polynomial of degree $N$, where $N$ is the order
of the operator $L$. Thus, there exists a constant $C>0$ such that

$|\overline{\sigma_{L}(-\xi)}-\bar{A}|+|B|\leq
C(1+\|\xi\|)^{N},\quad\xi\in\mathbb{Z}^{n}.$

Moreover, it follows from Lemma A.1 that

$\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta^{j}(1+\|\xi\|)^{j}}\
\sup_{j\in\mathbb{N}_{0}}\dfrac{(\delta/H)^{j}(1+\|\xi\|)^{j}}{m_{j}\,j!}$
$\displaystyle\leq\left(\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta/H)^{j}(1+\|\xi\|)^{j}}\right)^{2}\left(\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta/H)^{j}(1+\|\xi\|)^{j}}\right)^{-1}$
$\displaystyle=\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta/H)^{j}(1+\|\xi\|)^{j}}.$

Again, from Lemma A.1, we have

$\displaystyle|\widehat{u}(\xi)|\ \ \leq\ \ $ $\displaystyle
C(1+\|\xi\|)^{N}\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta/H)^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C(1+\|\xi\|)^{N}\left(\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta/H^{2})^{j}(1+\|\xi\|)^{j}}\right)^{2}$
$\displaystyle\leq\ \ $ $\displaystyle C(1+\|\xi\|)^{N}\
\dfrac{m_{N}\,N!}{(\delta/H^{2})^{N}(1+\|\xi\|)^{N}}\
\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta/H^{2})^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C\,h^{N}\,m_{N}\,N!\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{0}^{j}(1+\|\xi\|)^{j}},$

with $h=H/\delta$ and $\delta_{0}=\delta/H^{2}$.

Therefore, $u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ and $Pu=f$, that
is, $P$ is $\mathscr{M}$-solvable.

On the other hand, suppose that (DC$\mathscr{M}$) does not hold. Then, there
is $\varepsilon>0$ such that for each $\ell\in\mathbb{N}$, there is
$\xi_{\ell}\in\mathbb{Z}^{n}$ with $\|\xi_{\ell}\|\geq\ell$ and

$|\Delta_{\xi_{\ell}}|<\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi_{\ell}\|)^{j}}.$
(3.7)

Denoting $\xi_{m\ell}$ as the $m$-th coordinate of $\xi_{\ell}$, we can choose
$\\{\xi_{\ell}\\}_{\ell\in\mathbb{N}}$ such that all $\xi_{m\ell}$ are non-
zero and have the same sign for some $m\in\\{1,\dots,n\\}$.

Case 1: $\Delta_{\xi_{\ell}}=0$ for infinitely many $\ell\in\mathbb{N}$.

Passing to a subsequence, we may assume that $\Delta_{\xi_{\ell}}=0$ for all
$\ell\in\mathbb{N}$. Thus, from (3.2), we have
$(\overline{\sigma_{L}(-\xi)}-\bar{A})\widehat{f}(\xi)+B\overline{\widehat{f}(-\xi)}=0,$
for $\xi\in\Omega.$

If $B\neq 0$, from (3.3), we have
$(\sigma_{L}(\xi_{\ell})-A)\,(\overline{\sigma_{L}(-\xi_{\ell})}-\bar{A})=|B|^{2}$
for all $\ell\in\mathbb{N}$. Since $|B|^{2}\neq 0$, we conclude that
$\sigma_{L}(\xi_{\ell})-A\neq 0$ and
$\overline{\sigma_{L}(-\xi_{\ell})}-\bar{A}\neq 0$ for all
$\ell\in\mathbb{N}$. In this context, there are infinitely many compatibility
conditions for the Fourier coefficients of
$f\in\mathcal{C}^{\infty}(\mathbb{T}^{n})$ to satisfy $Pu=f$. Consequently,
$P$ is not $\mathscr{M}$-solvable.

If $B=0$, the Fourier coefficients of $u$ and $f$ must satisfy
$(\sigma_{L}(\xi)-A)\widehat{u}(\xi)=\widehat{f}(\xi)$ and
$(\overline{\sigma_{L}(-\xi)}-\bar{A})\overline{\widehat{u}(-\xi)}=\overline{\widehat{f}(-\xi)}$
for all $\xi\in\mathbb{Z}^{n}$. Since
$[\sigma_{L}(\xi_{\ell})-A]\,[\overline{\sigma_{L}(-\xi_{\ell})}-\bar{A}]=\Delta_{\xi_{\ell}}=0$
for all $\ell\in\mathbb{N}$, at least one of the following must hold:
$\sigma_{L}(\xi_{\ell})-A=0\quad\text{or}\quad\overline{\sigma_{L}(-\xi_{\ell})}-\bar{A}=0,$
for each $\ell\in\mathbb{N}$. This implies that for infinitely many indices
$\xi\in\mathbb{Z}^{n}$, either $\widehat{f}(\xi)=0$ or $\widehat{f}(-\xi)=0$.
Therefore, there are infinitely many compatibility conditions for the Fourier
coefficients of $f$ such that $Pu=f$ has a smooth solution. Consequently, $P$
is not solvable.

Case 2: $\Delta_{\xi_{\ell}}=0$ for a finite number of indices
$\ell\in\mathbb{N}$.

Passing to a subsequence, we may assume that $\Delta_{\xi_{\ell}}\neq 0$ for
all $\ell\in\mathbb{N}$, and consider the set
$\Omega=\\{\xi_{\ell}\in\mathbb{Z}^{n}:\ell\in\mathbb{N}\\}$. Observe that
$\xi\in\Omega\Leftrightarrow-\xi\notin\Omega$, and take an infinite subset
$\Omega_{0}\subset\Omega$.

Let us assume first that $B\neq 0$ and consider the function

$f(x)=\sum_{\xi\in\Omega_{0}}\Delta_{\xi}e^{i\xi\cdot x}.$

It follows from (3.7) and (2.1) that
$f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$. If
$u\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})$ is a solution of
$Pu=f$, proceeding as in [13], the projection of $u$ on the subspace
$\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})$ generated by the
frequencies $\pm\Omega_{0}$ is given by

$v(x)=\sum_{\xi\in\Omega_{0}}Be^{-i\xi\cdot
x}+\sum_{\xi\in\Omega_{0}}[\overline{\sigma_{L}(-\xi)}-\bar{A}]e^{i\xi\cdot
x}\in\mathscr{D}^{\prime}(\mathbb{T}^{n})\setminus
C^{\infty}(\mathbb{T}^{n}).$

Then,
$v\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})\setminus\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$,
which implies
$u\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})\setminus\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$.

Now consider the case where $B=0$. For all $\xi\in\Omega_{0}$, observe that

$\left(\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\varepsilon/H)^{j}(1+\|\xi\|)^{j}}\right)^{2}\geq\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|)^{j}}>|\Delta_{\xi}|=|\sigma_{L}(\xi)-A|\,|\sigma_{L}(-\xi)-A|.$

Therefore, either

$|\sigma_{L}(\xi)-A|<\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\varepsilon/H)^{j}(1+\|\xi\|)^{j}}\quad\text{or}\quad|\sigma_{L}(-\xi)-A|<\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\varepsilon/H)^{j}(1+\|\xi\|)^{j}}$

holds for infinitely many $\xi\in\Omega_{0}$.

If

$|\sigma_{L}(\xi)-A|<\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\varepsilon/H)^{j}(1+\|\xi\|)^{j}}$

holds for infinitely many $\xi\in\Omega_{0}$, we can pass to a subsequence and
assume that this inequality holds for all $\xi\in\Omega_{0}$. This implies
that the function

$f(x)=\sum_{\xi\in\Omega_{0}}[\sigma_{L}(\xi)-A]e^{i\xi\cdot x}$

belongs to $\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$. If
$u\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})$ is a solution of
$Pu=f$, proceeding as in [13], the projection of $u$ on the subspace
$\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})$ generated by the
frequencies in $\Omega_{0}$ is given by

$v(x)=\sum_{\xi\in\Omega_{0}}e^{i\xi\cdot
x}\in\mathscr{D}^{\prime}(\mathbb{T}^{n})\setminus
C^{\infty}(\mathbb{T}^{n}).$

Thus,
$v\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})\setminus\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$,
which implies $u\notin\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$.

The case where

$|\sigma_{L}(-\xi)-A|<\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\varepsilon/H)^{j}(1+\|\xi\|)^{j}}$

holds for infinitely many $\xi\in\Omega_{0}$ is analogous.

Finally, note that this construction holds for any infinite subset
$\Omega_{0}\subset\Omega$, providing us with infinitely many linearly
independent functions $f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ such
that there are no $u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ satisfying
$Pu=f$. Therefore, $P$ is not $\mathscr{M}$-solvable. ∎

######  Definition 3.3.

We say that $P$ is $\mathscr{M}$-globally hypoelliptic when the conditions

$u\in\mathscr{D}^{\prime}_{\mathscr{M}}(\mathbb{T}^{n})\mbox{ and
}Pu\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})\mbox{ imply
}u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n}).$

######  Corollary 3.4.

$P$ is $\mathscr{M}$-solvable if, and only if, $P$ is $\mathscr{M}$-globally
hypoelliptic.

###### Proof.

If $P$ is $\mathscr{M}$-solvable, we have shown in Theorem 3.2 that for any
$f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$, if
$u\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})$ is a solution of
$Pu=f$, then $u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$. Therefore, $P$
is $\mathscr{M}$-globally hypoelliptic.

Conversely, if $P$ is not $\mathscr{M}$-solvable, according to Theorem 3.2, we
can find a function $f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ such that
if $u\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n})$ is a solution of
$Pu=f$, then $u\notin\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$. Therefore,
$P$ is not $\mathscr{M}$-globally hypoelliptic. ∎

In the papers [3] and [13], the notion of (smooth) solvability was introduced
as follows: An operator
$P:\mathcal{C}^{\infty}(\mathbb{T}^{n})\to\mathcal{C}^{\infty}(\mathbb{T}^{n})$
is said to be solvable if there exists a subspace
$\mathcal{F}\subset\mathcal{C}^{\infty}(\mathbb{T}^{n})$ of finite codimension
such that for all $f\in\mathcal{F}$, there exists
$u\in\mathcal{C}^{\infty}(\mathbb{T}^{n})$ such that $Pu=f$.

Moreover, it was shown that $P$ is solvable if and only if there exists
$\gamma>0$ such that

$\|\xi\|\geq\gamma\ \Rightarrow\ |\Delta_{\xi}|\geq(1+\|\xi\|)^{-\gamma}.$

Now, observe that for any $\varepsilon>0$ we have

$\dfrac{1}{(1+\|\xi\|)^{\gamma}}=\dfrac{\varepsilon^{\gamma}}{m_{\gamma}\,\gamma!}\,\dfrac{m_{\gamma}\,\gamma!}{\varepsilon^{\gamma}(1+\|\xi\|)^{\gamma}}\geq
C_{\varepsilon}\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|)^{j}},$

where $C_{\varepsilon}=\dfrac{\varepsilon^{\gamma}}{m_{\gamma}\,\gamma!}$.
Therefore, (DC$\mathscr{M}$) holds, and we have proved the following result:

######  Proposition 3.5.

If $P$ is solvable, then $P$ is $\mathscr{M}$-solvable.

However, the reciprocal of this statement is generally not true. In [7], the
authors provided a counterexample involving a constant-coefficient vector
field and continued fractions in the context of Gevrey spaces.

######  Corollary 3.6.

If $L$ is an elliptic differential operator, then $P$ is
$\mathscr{M}$-solvable and $\mathscr{M}$-globally hypoelliptic.

###### Proof.

It follows from [13, Theorem 4] that $P$ is solvable. Therefore, by Corollary
3.4 and Proposition 3.5, $P$ is $\mathscr{M}$-solvable and
$\mathscr{M}$-globally hypoelliptic. ∎

###  3.1 Examples

######  Example 3.7 (Laplace operator).

If $L=\sum_{j=1}^{n}{\partial^{2}}/{\partial x_{j}^{2}}$, then the operator
$P$ is $\mathscr{M}$-solvable and $\mathscr{M}$-globally hypoelliptic because
$L$ is elliptic.

######  Example 3.8 (Heat operator).

If $L={\partial}/{\partial t}-\eta^{2}\sum_{j=1}^{n}{\partial^{2}}/{\partial
x_{j}^{2}}$, where $\eta>0$, then $P$ is $\mathscr{M}$-solvable and
$\mathscr{M}$-globally hypoelliptic because $P$ is solvable (see [13, Theorem
5]).

######  Example 3.9 (Wave Operator).

Let $L={\partial^{2}}/{\partial
t^{2}}-\eta^{2}\sum_{j=1}^{n}{\partial^{2}}/{\partial x_{j}^{2}}$, with
$\eta>0$. The operator $P$ is $\mathscr{M}$-solvable and
$\mathscr{M}$-globally hypoelliptic if and only if one of the following
conditions holds:

  * (1)

$|B|<|\text{Im}(A)|$;

  * (2)

$|A|=|B|$, $\text{Re}(A)=0$, and $\eta$ is an irrational non-Liouville number;

  * (3)

(DC$\mathscr{M}$) holds.

This result is derived from Corollary 3.4, Proposition 3.5, and the smooth
solvability established in [13, Theorem 6].

######  Example 3.10 (Complex Vector Fields).

As a direct consequence of Theorem 3.2, we can extend the results involving
constant-coefficient complex vector fields obtained in [7, 3, 13] to the
framework of ultradifferentiable solvability and regularity. Consider

$L=\dfrac{\partial}{\partial t}+\sum_{j=1}^{n}C_{j}\dfrac{\partial}{\partial
x_{j}},$

where $C_{j}\in\mathbb{C}$ for $j=1,\ldots,n$. Then, $P$ is
$\mathscr{M}$-solvable if and only if, for all $\varepsilon>0$, there exist
$C_{\varepsilon},\gamma_{\varepsilon}>0$ such that

$\|\xi\|+|\tau|\geq\gamma_{\varepsilon}\ \Rightarrow\ |\Delta_{\xi,\tau}|\geq
C_{\varepsilon}\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|+|\tau|)^{j}},$

with

$\Delta_{\xi,\tau}=-|\tau+\xi\cdot
C|^{2}+|A|^{2}-|B|^{2}-2i\,\text{Re}(A(\tau+\xi\cdot\bar{C})).$

In particular, if $\text{Im}(C)=0$, then $P$ is $\mathscr{M}$-solvable if and
only if one of the following conditions holds:

  * (1)

$|B|>|A|$;

  * (2)

$|B|<|A|$ and $\text{Re}(A)\neq 0$;

  * (3)

(DCM) holds.

The proof of this particular case follows the same lines as [7, Theorem 4].

######  Remark 1.

The results for the Gevrey classes obtained by Almeida and Dattori da Silva in
[7] are particular cases of those developed in this paper for the Denjoy-
Carleman classes of Roumieu type. In fact, the Gevrey spaces are obtained when
we consider the weight sequence
$\mathscr{M}=\\{(j!)^{s-1}\\}_{j\in\mathbb{N}_{0}}$. It is possible to show
that

$t^{1/s}-s\,\log\left(\dfrac{1}{1-s^{-1}}\right)\leq\sup_{j\in\mathbb{N}_{0}}\log\left(\dfrac{t^{j}}{(j!)^{s}}\right)\leq
s\,t^{1/s},$

which allows us to demonstrate that the condition (DC$\mathscr{M}$) is
equivalent to the (DCs) condition obtained in [7].

##  4 A class of operators with variable coefficients

Motivated by references [3] and [7], we examine in this section the
$\mathscr{M}$-solvability of a class of Vekua-type operators with variable
coefficients.

First, we take the following complex vector field L into consideration:

$L=\frac{\partial}{\partial
t}-\sum_{j=1}^{n}\left(p_{j}(t)+iq_{j}(t)\right)\frac{\partial}{\partial
x_{j}},$ (4.1)

with $p_{j},q_{j}\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{1};\mathbb{R})$ for
$j=1,\dots,n$.

Assume that $L$ satisfies the Nirenberg-Treves condition $(P)$, implying that
$L$ is locally solvable (see [5, 17]). Furthermore, there exists a function
$q\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{1};\mathbb{R})$ that does not
change sign, and constants $\lambda_{1},\ldots,\lambda_{n}\in\mathbb{R}$ such
that $q_{j}(t)=\lambda_{j}q(t)$ for all $t\in\mathbb{T}^{1}$ (see [2]).
Consequently, we can rewrite $L$ as:

$\widetilde{L}=\frac{\partial}{\partial
t}-\sum_{j=1}^{n}\left(p_{j}(t)+i\lambda_{j}q(t)\right)\frac{\partial}{\partial
x_{j}}.$ (4.2)

and assume, without loss of generality, that $q\geq 0$.

Now, set $m(t)=(m_{1}(t),\dots,m_{n}(t))$ where

$\displaystyle m_{j}(t)=\int_{0}^{t}(p_{j}(\tau)-p_{0j})\,d\tau,\quad
p_{0j}=\frac{1}{2\pi}\int_{0}^{2\pi}p_{j}(\tau)\,d\tau,\text{ for }\
j=1,\dots,n,$

and define the operator
$T:\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n+1})\to\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n+1})$
by

$Tu(t,x)=\sum_{\xi\in\mathbb{Z}^{n}}\widehat{u}(t,\xi)e^{-im(t)\cdot\xi}e^{i\xi\cdot
x},$ (4.3)

where the ultradistribution
$u(t,x)=\sum_{\xi\in\mathbb{Z}^{n}}\widehat{u}(t,\xi)e^{i\xi\cdot
x}\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n+1})$ is expressed in its
partial Fourier series representation.

By Theorem 5.12 of [8], the operator $T$ and its restriction
$T|_{\mathcal{E}_{\mathscr{M}}}:\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})\to\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})$
are automorphisms that satisfy

$T\widetilde{L}T^{-1}=\dfrac{\partial}{\partial
t}-\sum_{j=1}^{n}(p_{0j}+i\lambda_{j}q(t))\dfrac{\partial}{\partial x_{j}}.$

###  4.1 The Vekua-type operator

Consider the operator
$P:\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})\to\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})$
given by

$Pu=Lu-(s(t)+i\delta q(t))u-\alpha q(t)\bar{u},$ (4.4)

with

$L=\dfrac{\partial}{\partial
t}-\sum_{j=1}^{n}(p_{0j}+i\lambda_{j}q(t))\dfrac{\partial}{\partial x_{j}},$
(4.5)

where the constants $\alpha\in\mathbb{C}\setminus\\{0\\}$ and
$\delta\in\mathbb{R}$, the vectors $\lambda=(\lambda_{1},\ldots,\lambda_{n})$
and $p_{0}=(p_{01},\ldots p_{0n})$ are in $\mathbb{R}^{n}$, and the functions
$q,s\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{1};\mathbb{R})$. Recall that $q$
does not change sign, and we can assume $q\geq 0$. Furthermore, we define

$\displaystyle A_{0}\doteq s_{0}+i\delta q_{0},\ B_{0}\doteq\alpha
q_{0},\text{ and }\ C_{0}\doteq 2\pi p_{0}+i\lambda q_{0}.$
$\displaystyle\text{where }\ q_{0}\doteq\int_{0}^{2\pi}q(\tau)\,d\tau>0\text{
and }s_{0}\doteq\int_{0}^{2\pi}s(\tau)\,d\tau,$

######  Theorem 4.1.

Assume that the following conditions hold:

  * (I)

$|\alpha|\neq|\delta|$;

  * (II)

There is no $(\xi,\tau)\in\mathbb{Z}^{n+1}$ solution to

$\begin{cases}\text{Re}(A_{0}(2\pi\tau+\xi\cdot\overline{C_{0}}))=0,\\\
|2\pi\tau+\xi\cdot C_{0}|^{2}=|A_{0}|^{2}-|B_{0}|^{2};\end{cases}$

  * (III)

For all $\varepsilon>0$, there exist $C_{\varepsilon},\gamma_{\varepsilon}>0$
such that if $\xi\in\mathbb{Z}^{n}$ and $\|\xi\|\geq\gamma_{\varepsilon}$,
then

$\min\left\\{|e^{-\rho_{\xi}q_{0}}-e^{s_{0}+i2\pi\xi\cdot
p_{0}}|,|1-e^{-\rho_{\xi}q_{0}+s_{0}+i2\pi\xi\cdot p_{0}}|\right\\}\geq
C_{\varepsilon}\,\inf_{j\in\mathbb{N}_{0}}\frac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|)^{j}},$

where
$\rho_{\xi}\in\\{\pm\sqrt{(\lambda\cdot\xi-i\delta)^{2}+|\alpha|^{2}}\\}$ we
choose to satisfy $\text{Re}(\rho_{\xi})\geq 0$.

Then, for all $f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})$, exists
$u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})$ such that $Pu=f$.

######  Remark 2.

The proof of this result is an adapted version of the proof of Theorem 7
presented in [3], tailored to our case. Due to the intricate nature of
ultradifferentiability, this proof is significantly more extensive and
challenging to follow. To assist the reader in understanding, we have chosen
to retain many intermediate steps rather than omitting them, as usual.

###### Proof.

Given $u\in\mathscr{D}_{\mathscr{M}}^{\prime}(\mathbb{T}^{n+1})$ and
$f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1}),$ consider their partial
Fourier series representations:

$u(x,t)=\displaystyle\sum_{\xi\in\mathbb{Z}^{n}}\widehat{u}(t,\xi)e^{i\xi\cdot
x}\quad\text{ and }\quad
f(x,t)=\displaystyle\sum_{\xi\in\mathbb{Z}^{n}}\widehat{f}(t,\xi)e^{i\xi\cdot
x}.$

If $Pu=f$, then following the procedures in [3] and [7], conditions (I) and
(II) give us

$\widehat{u}(t,\xi)=\alpha e^{i(\xi\cdot
p_{0})t+S(t)}(z_{1\xi}(t)+z_{2\xi}(t)),\ \ \xi\in\mathbb{Z}^{n},$

with

$\displaystyle z_{1\xi}(t)=$
$\displaystyle-\int_{t}^{2\pi}\\!\\!e^{\rho_{\xi}(\widetilde{Q}(t)-\widetilde{Q}(\sigma))}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}G_{1\xi}(\sigma)\,d\sigma$ $\displaystyle\quad\
+\,e^{\rho_{\xi}\widetilde{Q}(t)}\\!\\!\displaystyle\int_{0}^{2\pi}\\!\dfrac{e^{-\rho_{\xi}(q_{0}+\widetilde{Q}(\sigma))}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}}{e^{-\rho_{\xi}q_{0}}-e^{i(\xi\cdot
p_{0})2\pi+s_{0}}}G_{1\xi}(\sigma)\,d\sigma,$ $\displaystyle z_{2\xi}(t)=$
$\displaystyle\displaystyle\int_{0}^{t}e^{\rho_{\xi}(Q(\sigma)-Q(t))}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}G_{2\xi}(\sigma)\,d\sigma$
$\displaystyle\qquad+\,e^{-\rho_{\xi}Q(t)}\\!\\!\displaystyle\int_{0}^{2\pi}\dfrac{e^{\rho_{\xi}(Q(\sigma)-q_{0})}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}}{1-e^{-\rho_{\xi}q_{0}+i(\xi\cdot
p_{0})2\pi+s_{0}}}G_{2\xi}(\sigma)\,d\sigma,$

where

$Q(t)=\int_{0}^{t}q(\tau)\,d\tau,\
\widetilde{Q}(t)=-\int_{t}^{2\pi}q(\tau)\,d\tau,\
S(t)=\int_{0}^{t}s(\tau)\,d\tau,$

and

$G_{\xi}(t)=\begin{bmatrix}G_{1\xi}(t)\\\
G_{2\xi}(t)\end{bmatrix}=\dfrac{-1}{2\alpha\rho_{\xi}}\begin{bmatrix}(\lambda\cdot\xi-i\delta)-\rho_{\xi}&-\alpha\\\
-(\lambda\cdot\xi-i\delta)+\rho_{\xi}&\alpha\end{bmatrix}\cdot\begin{bmatrix}\widehat{f}(t,\xi)\\\
\overline{\widehat{f}(t,-\xi)}\end{bmatrix}.$

Given that
$\rho_{\xi}\in\\{\pm\sqrt{(\lambda\cdot\xi-i\delta)^{2}+|\alpha|^{2}}\\}$ and
$\text{Re}(\rho_{\xi})>0$, we have

$|\rho_{\xi}|^{4}=(\lambda\cdot\xi)^{4}+2(|\alpha|^{2}+\delta^{2})(\lambda\cdot\xi)^{2}+(|\alpha|^{2}-\delta^{2})^{2}\leq
C^{\prime}(1+|\xi|)^{4},$

since
$(\lambda\cdot\xi)^{4}+2(|\alpha|^{2}+\delta^{2})(\lambda\cdot\xi)^{2}+(|\alpha|^{2}-\delta^{2})^{2}$
is a fourth-degree polynomial. Therefore, there exists a positive constant $C$
such that

$|\rho_{\xi}|\leq C(1+|\xi|),\quad\xi\in\mathbb{Z}^{n}.$ (4.6)

Moreover, we have that

$|\rho_{\xi}|^{4}=(\lambda\cdot\xi)^{4}+2(|\alpha|^{2}+\delta^{2})(\lambda\cdot\xi)^{2}+(|\alpha|^{2}+\delta^{2})^{2}\geq(|\alpha|^{2}+\delta^{2})^{2}>0,\quad\xi\in\mathbb{Z}^{n}.$

Consequently, there exists a positive constant $C_{\rho}$ such that

$|\rho_{\xi}|\geq C_{\rho},\quad\xi\in\mathbb{Z}^{n}.$

In particular,

$0<|\rho_{\xi}|^{-1}\leq C_{\rho}^{-1},\quad\xi\in\mathbb{Z}^{n}.$ (4.7)

Based on (4.7), it follows that the decay of $G_{1\xi}$ and $G_{2\xi}$ is the
same as that of $\widehat{f}(\cdot,\xi)$. Since
$f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})$, there exist positive
constants $C_{G}$, $h_{G_{1}}$, and $h_{G_{2}}$ such that for all
$N\in\mathbb{N}_{0}$, the following inequality holds:

$\left|\dfrac{d^{N}}{dt^{N}}G_{k\xi}(t)\right|\leq
C_{G}\,h_{G_{1}}^{N}\,m_{N}\,N!\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{h_{G_{2}}^{j}(1+\|\xi\|)^{j}},\quad\text{for
}k=1,2.$

Given $N\in\mathbb{N}_{0}$, we have

$\dfrac{d^{N}}{dt^{N}}e^{i(\xi\cdot p_{0})t}=i^{N}(\xi\cdot
p_{0})^{N}e^{i(\xi\cdot p_{0})t},$

which yields

$\left|\dfrac{d^{N}}{dt^{N}}e^{i(\xi\cdot p_{0})t}\right|\leq
h_{p}^{N}(1+|\xi|)^{N}|e^{i(\xi\cdot p_{0})t}|=h_{p}^{N}(1+|\xi|)^{N}$

for some constant $h_{p}>0$.

Similarly, we have

$\left|\dfrac{d^{N}}{dt^{N}}e^{-i(\xi\cdot p_{0})t}\right|\leq
h_{p}^{N}(1+|\xi|)^{N}.$

Now, employing de Faà di Bruno’s Formula, for $N\in\mathbb{N}_{0}$, we obtain

$\displaystyle\dfrac{d^{N}}{dt^{N}}e^{\rho_{\xi}\widetilde{Q}(t)}\ \ =\ \ $
$\displaystyle
e^{\rho_{\xi}\widetilde{Q}(t)}\\!\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(N)}\\!\\!\dfrac{N!}{\gamma!}\prod_{\ell=1}^{N}\left(\dfrac{1}{\ell!}\cdot\dfrac{d^{\ell}}{dt^{\ell}}\rho_{\xi}\widetilde{Q}(t)\right)^{\gamma_{\ell}}$
$\displaystyle=\ \ $ $\displaystyle
e^{\rho_{\xi}\widetilde{Q}(t)}\\!\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(N)}\\!\\!\dfrac{N!}{\gamma!}\prod_{\ell=1}^{N}\left(\dfrac{-\rho_{\xi}}{\ell!}\cdot\dfrac{d^{\ell-1}}{dt^{\ell-1}}q(t)\right)^{\gamma_{\ell}}.$

Since $q\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})$, exists
$C_{q0},h_{q0}>0$ such that

$\left|\dfrac{d^{k}}{dt^{k}}q(t)\right|\leq
C_{q0}\,h_{q0}^{k}\,m_{k}\,k!,\quad k\in\mathbb{N}_{0}.$

Then,

$\displaystyle\left|\dfrac{d^{N}}{dt^{N}}e^{\rho_{\xi}\widetilde{Q}(t)}\right|\
\ \leq\ \ $
$\displaystyle|e^{\rho_{\xi}\widetilde{Q}(t)}|\\!\sum_{\gamma\in\Delta(N)}\\!\\!\dfrac{N!}{\gamma!}|\rho_{\xi}|^{|\gamma|}\prod_{\ell=1}^{N}\left(\dfrac{1}{\ell!}\,C_{q0}\,h_{q0}^{\ell-1}\,m_{\ell-1}\,(\ell-1)!\right)^{\gamma_{\ell}}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{\rho_{\xi}\widetilde{Q}(t)}|\\!\sum_{\gamma\in\Delta(N)}\\!\dfrac{N!}{\gamma!}C^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\prod_{\ell=1}^{N}\left(\dfrac{1}{\ell}\,C_{q0}\,h_{q0}^{\ell}\,m_{\ell}\right)^{\gamma_{\ell}}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{\rho_{\xi}\widetilde{Q}(t)}|h_{q0}^{N}\\!\sum_{\gamma\in\Delta(N)}\\!\dfrac{1}{\gamma!}(CC_{q0})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\prod_{\ell=1}^{N}\left(\dfrac{1}{\ell}\,m_{\ell}\right)^{\gamma_{\ell}}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{\rho_{\xi}\widetilde{Q}(t)}|h_{q0}^{N}\\!\sum_{\gamma\in\Delta(N)}\\!\dfrac{1}{\gamma!}C_{q}^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{N}}{m_{|\gamma|}}.$

Notice that we have used the following facts: $\mathscr{M}$ is non-decreasing,
Lemma A.4 in A.4, the estimate (4.6), and that

$\prod_{\ell=1}^{N}\left(\dfrac{1}{\ell}\right)^{\gamma_{\ell}}\leq\prod_{\ell=1}^{N}\left(\dfrac{1}{\ell}\right)=\dfrac{1}{N!},$

Additionally, in the last inequality, we denoted $C_{q}=CC_{q0}$.

Since $s\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n+1})$, there exist
$C_{s0},h_{s0}>0$ such that,

$\left|\dfrac{d^{k}}{dt^{k}}s(t)\right|\leq
C_{s0}\,h_{s0}^{k}\,m_{k}\,k!,\quad k\in\mathbb{N}_{0}.$

Analogously,

$\displaystyle\left|\dfrac{d^{N}}{dt^{N}}e^{S(t)}\right|\leq|e^{S(t)}|h_{s0}^{N}\\!\\!\\!\sum_{\gamma\in\Delta(N)}\\!\\!\dfrac{1}{\gamma!}(C_{s0})^{|\gamma|}\dfrac{m_{N}}{m_{|\gamma|}},\quad
N\in\mathbb{N}_{0}.$

For $e^{-i(\xi\cdot p_{0})t}$, $e^{-\rho_{\xi}\widetilde{Q}(t)}$ and
$e^{-S(t)}$, the arguments are similar. Then, we have the following estimates,
for all $N\in\mathbb{N}_{0}$,

$\displaystyle\left|\dfrac{d^{N}}{dt^{N}}e^{\pm i\xi\cdot p_{0}t}\ \right|\leq
h_{p}^{N}(1+\|\xi\|)^{N};$ (4.8)
$\displaystyle\left|\dfrac{d^{N}}{dt^{N}}e^{\pm\rho_{\xi}\widetilde{Q}(t)}\right|\leq|e^{\pm\rho_{\xi}\widetilde{Q}(t)}|h_{q0}^{N}\\!\sum_{\gamma\in\Delta(N)}\\!\dfrac{1}{\gamma!}C_{q}^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{N}}{m_{|\gamma|}};$
(4.9) $\displaystyle\left|\dfrac{d^{N}}{dt^{N}}e^{\pm S(t)}\right|\leq|e^{\pm
S(t)}|h_{s0}^{N}\\!\\!\\!\sum_{\gamma\in\Delta(N)}\\!\\!\dfrac{1}{\gamma!}(C_{s0})^{|\gamma|}\dfrac{m_{N}}{m_{|\gamma|}}.$
(4.10)

After obtaining all the above estimates, we can now study the decay of the
derivatives of $e^{i\xi\cdot p_{0}t+S(t)}z_{1\xi}(t)$. For
$N\in\mathbb{N}_{0}$, we have

$\displaystyle\dfrac{d^{N}}{dt^{N}}\left(e^{i\xi\cdot
p_{0}t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)}\displaystyle\int_{t}^{2\pi}e^{-\rho_{\xi}\widetilde{Q}(\sigma)}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}G_{1\xi}(\sigma)\,d\sigma\right)$ $\displaystyle=\ \ $
$\displaystyle\displaystyle\sum_{k=0}^{N}\binom{N}{k}\dfrac{d^{N-k}}{dt^{N-k}}\left(e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)}\right)\dfrac{d^{k}}{dt^{k}}\left(\int_{t}^{2\pi}e^{-\rho_{\xi}\widetilde{Q}(\sigma)}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}G_{1\xi}(\sigma)\,d\sigma\right)$ $\displaystyle=\ \ $
$\displaystyle\dfrac{d^{N}}{dt^{N}}\left(e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)}\right)\left(\int_{t}^{2\pi}e^{-\rho_{\xi}\widetilde{Q}(\sigma)}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}G_{1\xi}(\sigma)\,d\sigma\right)$
$\displaystyle+\displaystyle\sum_{k=1}^{N}\binom{N}{k}\dfrac{d^{N-k}}{dt^{N-k}}\left(e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)}\right)\dfrac{d^{k-1}}{dt^{k-1}}\left(e^{-\rho_{\xi}\widetilde{Q}(t)}e^{-i(\xi\cdot
p_{0})t-S(t)}G_{1\xi}(t)\right).$

Given $k=1,\dots,N_{1}$ and $r=0,\dots,k-1$, it follows from (4.9) and (4.10)
that

$\displaystyle\left|\dfrac{d^{r}}{dt^{r}}\left(e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}\right)\right|\
\ \leq\ \ $
$\displaystyle\displaystyle\sum_{w=0}^{r}\binom{r}{w}\left|\dfrac{d^{w}}{dt^{w}}\left(e^{-\rho_{\xi}\widetilde{Q}(t)}\right)\right|\,\left|\dfrac{d^{r-w}}{dt^{r-w}}\left(e^{-S(t)}\right)\right|$
$\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\displaystyle\sum_{w=0}^{r}\binom{r}{w}\left[h_{q0}^{w}\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{1}{\gamma!}(C_{q})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{w}}{m_{|\gamma|}}\right]$
$\displaystyle\times\left[h_{s0}^{r-w}\\!\sum_{\eta\in\Delta(r-w)}\\!\dfrac{1}{\eta!}(C_{s0})^{|\eta|}\dfrac{m_{r-w}}{m_{|\eta|}}\right]$

Denoting $k_{r}\doteq k-1-r$, we have

$\displaystyle\left|\dfrac{d^{k_{r}}}{dt^{k_{r}}}\left(e^{-i(\xi\cdot
p_{0})t}G_{1\xi}(t)\right)\right|\ \leq\
\displaystyle\sum_{w=0}^{k_{r}}\binom{k_{r}}{w}\left|\dfrac{d^{k_{r}-w}}{dt^{k_{r}-w}}(G_{1\xi}(t))\right|\left|\dfrac{d^{w}}{dt^{w}}e^{-i(\xi\cdot
p_{0})t}\right|$ $\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{w=0}^{k_{r}}\binom{k_{r}}{w}\,h_{p}^{w}(1+\|\xi\|)^{w}C_{G}h_{G_{1}}^{k_{r}-w}m_{k_{r}-w}(k_{r}-w)!\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{h_{G_{2}}^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{w=0}^{k_{r}}\binom{k_{r}}{w}\,h_{p}^{w}(1+\|\xi\|)^{w}C_{G}h_{G_{1}}^{k_{r}-w}m_{k_{r}-w}(k_{r}-w)!\,\left(\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(h_{G_{2}}/H)^{j}(1+\|\xi\|)^{j}}\right)^{2}$
$\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{w=0}^{k_{r}}\binom{k_{r}}{w}\,h_{p}^{w}(1+\|\xi\|)^{w}C_{G}\,h_{G_{1}}^{k_{r}-w}m_{k_{r}-w}(k_{r}-w)!\,\dfrac{m_{w}\,w!}{(h_{G_{2}}/H)^{w}(1+\|\xi\|)^{w}}$
$\displaystyle\times\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(h_{G_{2}}/H)^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C_{G}\,h_{G0}^{k_{r}}\displaystyle\sum_{w=0}^{k_{r}}\binom{k_{r}}{w}\,m_{k_{r}-w}m_{w}(k_{r}-w)!w!\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{1}^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C_{G}\,h_{G0}^{k_{r}}\displaystyle\sum_{w=0}^{k_{r}}\binom{k_{r}}{w}\,m_{k_{r}}k_{r}!\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{1}^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C_{G}\,h_{G}^{k_{r}}\,m_{k_{r}}k_{r}!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{1}^{j}(1+\|\xi\|)^{j}},$

with $\delta_{1}=h_{G_{2}}/H$, $h_{G0}=\max\\{h_{p},h_{G_{1}}\\}$ and
$h_{G}=2h_{G0}$, noticing that

$\displaystyle\sum_{w=0}^{k_{r}}\binom{k_{r}}{w}=2^{k_{r}}.$

Then, given $k=1\dots,N-1$, using Lemmas A.1, A.3 and A.4 and the estimates
(4.8), (4.9) and (4.10), we have

$\displaystyle\left|\dfrac{d^{k-1}}{dt^{k-1}}\left(e^{-\rho_{\xi}\widetilde{Q}(t)}e^{-i(\xi\cdot
p_{0})t-S(t)}G_{1\xi}(t)\right)\right|$ $\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}\left|\dfrac{d^{r}}{dt^{r}}\left(e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}\right)\right|\left|\dfrac{d^{k-1-r}}{dt^{k-1-r}}\left(e^{-i(\xi\cdot
p_{0})t}G_{1\xi}(t)\right)\right|$ $\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}\Bigg{[}\displaystyle\sum_{w=0}^{r}\binom{r}{w}\left[h_{q0}^{w}\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{1}{\gamma!}(C_{q})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{w}}{m_{|\gamma|}}\right]$
$\displaystyle\times\left[h_{s0}^{r-w}\\!\\!\\!\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{1}{\eta!}(C_{s0})^{|\eta|}\dfrac{m_{r-w}}{m_{|\eta|}}\right]\,C_{G}\,h_{G}^{k_{r}}\,m_{k_{r}}k_{r}!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{1}^{j}(1+\|\xi\|)^{j}}\Bigg{]}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}\Bigg{[}\displaystyle\sum_{w=0}^{r}\binom{r}{w}\left[h_{q0}^{w}\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{1}{\gamma!}(C_{q})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{w}}{m_{|\gamma|}}\right]$
$\displaystyle\times\left[h_{s0}^{r-w}\\!\\!\\!\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{1}{\eta!}(C_{s0})^{|\eta|}\dfrac{m_{r-w}}{m_{|\eta|}}\right]\,C_{G}\,h_{G}^{k_{r}}\,m_{k_{r}}k_{r}!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta_{1}/H)^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\times\left(\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta_{1}/H^{2})^{j}(1+\|\xi\|)^{j}}\right)^{2}\Bigg{]}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}\Bigg{[}\displaystyle\sum_{w=0}^{r}\binom{r}{w}\left[h_{q0}^{w}\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{1}{\gamma!}(C_{q})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{w}}{m_{|\gamma|}}\right]$
$\displaystyle\times\dfrac{m_{|\gamma|}\,|\gamma|!}{\delta_{3}^{|\gamma|}(1+\|\xi\|)^{|\gamma|}}\,\left[h_{s0}^{r-w}\\!\\!\\!\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{1}{\eta!}(C_{s0})^{|\eta|}\dfrac{m_{r-w}}{m_{|\eta|}}\right]\,\dfrac{m_{|\eta|}\,|\eta|!}{\delta_{3}^{|\eta|}(1+\|\xi\|)^{|\eta|}}$
$\displaystyle\times
C_{G}\,h_{G}^{k_{r}}\,m_{k_{r}}\,k_{r}!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{2}^{j}(1+\|\xi\|)^{j}}\Bigg{]}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|C_{G}h_{0}^{k-1}\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}\Bigg{[}\displaystyle\sum_{w=0}^{r}\binom{r}{w}m_{w}m_{r-w}\left[\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{|\gamma|!}{\gamma!}\left(\dfrac{C_{q}}{\delta_{3}}\right)^{|\gamma|}\right]$
$\displaystyle\times\left[\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{|\eta|!}{\eta!}\left(\dfrac{C_{s0}}{\delta_{3}}\right)^{|\eta|}\right]\,m_{k_{r}}\,k_{r}!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{2}^{j}(1+\|\xi\|)^{j}}\Bigg{]}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|C_{G}h_{0}^{k-1}\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}m_{r}m_{k-1-r}\Bigg{[}\displaystyle\sum_{w=0}^{r}\binom{r}{w}C_{q1}(1+C_{q1})^{w-1}$
$\displaystyle\times
C_{s1}(1+C_{s1})^{r-w-1}\,(k-1-r)!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{2}^{j}(1+\|\xi\|)^{j}}\Bigg{]}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|C_{G}h_{0}^{k-1}m_{k-1}\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}\Bigg{[}\displaystyle\sum_{w=0}^{r}\binom{r}{w}C_{q2}^{w}C_{s2}^{r-w}(k-1-r)!$
$\displaystyle\times\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{2}^{j}(1+\|\xi\|)^{j}}\Bigg{]}$
$\displaystyle\leq\ \ $
$\displaystyle|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\,C_{G0}\,h_{00}^{k-1}\,m_{k-1}\,(k-1)!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{2}^{j}(1+\|\xi\|)^{j}},$

with $\delta_{2}=\delta_{1}/H$, $\delta_{3}=\delta_{1}/H^{2}$,
$h_{0}=\max\\{h_{q0},h_{s0}\\}$, $C_{q1}=C_{q}/\delta_{3}$,
$C_{s1}=C_{s0}/\delta_{3}$, $C_{q_{2}}=(1+C_{q1})$, $C_{s2}=1+C_{s1}$,
$h_{00}=2h_{0}$ and $C_{G0}=2C_{G}$, observing that

$\displaystyle\sum_{r=0}^{k-1}\binom{k-1}{r}\displaystyle\sum_{w=0}^{r}\binom{r}{w}\
=\ 2^{k}-\dfrac{1}{2}\ \leq\ 2^{k}\ =\ 2\cdot 2^{k-1}.$

Given $k=1,\dots,N$, it follows from the estimates (4.9), (4.9) and (4.10)
that

$\displaystyle\left|\dfrac{d^{N-k}}{dt^{N-k}}(e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)})\right|$ $\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{r=0}^{N-k}\binom{N-k}{r}\left|\dfrac{d^{r}}{dt^{r}}\left(e^{\rho_{\xi}\widetilde{Q}(t)+S(t)}\right)\right|\,\left|\dfrac{d^{N-k-r}}{dt^{N-k-r}}e^{i(\xi\cdot
p_{0})t}\right|$ $\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{r=0}^{N-k}\binom{N-k}{r}|e^{\rho_{\xi}\widetilde{Q}(t)+S(t)}|\displaystyle\sum_{w=0}^{r}\binom{r}{w}\left[h_{q0}^{w}\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{1}{\gamma!}(C_{q})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{w}}{m_{|\gamma|}}\right]$
$\displaystyle\times\left[h_{s0}^{r-w}\\!\\!\\!\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{1}{\eta!}(C_{s0})^{|\eta|}\dfrac{m_{r-w}}{m_{|\eta|}}\right]\,h_{p}^{N-k-r}\,(1+\|\xi\|)^{N-k-r}.$

Finally, we have

$\displaystyle\left|\displaystyle\sum_{k=1}^{N}\binom{N}{k}\dfrac{d^{N-k}}{dt^{N-k}}(e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)})\,\dfrac{d^{k-1}}{dt^{k-1}}\left(e^{-\rho_{\xi}\widetilde{Q}(t)}e^{-i(\xi\cdot
p_{0})t-S(t)}G_{1\xi}(t)\right)\right|$ $\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{k=1}^{N}\binom{N}{k}\left|\dfrac{d^{N-k}}{dt^{N-k}}(e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)})\right|\,\left|\dfrac{d^{k-1}}{dt^{k-1}}\left(e^{-\rho_{\xi}\widetilde{Q}(t)}e^{-i(\xi\cdot
p_{0})t-S(t)}G_{1\xi}(t)\right)\right|$ $\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{k=1}^{N}\binom{N}{k}\Bigg{[}\displaystyle\sum_{r=0}^{N-k}\binom{N-k}{r}|e^{\rho_{\xi}\widetilde{Q}(t)+S(t)}|\displaystyle\sum_{w=0}^{r}\binom{r}{w}\left[h_{q0}^{w}\\!\\!\\!\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{1}{\gamma!}(C_{q})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{w}}{m_{|\gamma|}}\right]$
$\displaystyle\times\left[h_{s0}^{r-w}\\!\\!\\!\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{1}{\eta!}(C_{s0})^{|\eta|}\dfrac{m_{r-w}}{m_{|\eta|}}\right]\,h_{p}^{N-k-r}\,(1+\|\xi\|)^{N-k-r}\Bigg{]}\,|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\,C_{G0}$
$\displaystyle\times
h_{00}^{k-1}\,m_{k-1}\,(k-1)!\,\displaystyle\left[\inf_{j\in\mathbb{N}_{0}}\left(\dfrac{m_{j}\,j!}{(\delta_{2}/H^{2})^{j}(1+\|\xi\|)^{j}}\right)\right]^{4}$
$\displaystyle\leq\ \ $
$\displaystyle\displaystyle\sum_{k=1}^{N}\binom{N}{k}\Bigg{[}\displaystyle\sum_{r=0}^{N-k}\binom{N-k}{r}h_{qs}^{r}h_{p}^{N-k-r}\displaystyle\sum_{w=0}^{r}\binom{r}{w}\Bigg{[}\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{1}{\gamma!}(C_{q})^{|\gamma|}(1+\|\xi\|)^{|\gamma|}\dfrac{m_{w}}{m_{|\gamma|}}$
$\displaystyle\times\dfrac{m_{|\gamma|}\,|\gamma|!}{\delta_{4}^{|\gamma|}(1+\|\xi\|)^{|\gamma|}}\Bigg{]}\,\left[\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{1}{\eta!}(C_{s0})^{|\eta|}\dfrac{m_{r-w}}{m_{|\eta|}}\,\dfrac{m_{|\eta|}\,|\eta|!}{\delta_{4}^{|\eta|}(1+\|\xi\|)^{|\eta|}}\right]\Bigg{]}$
$\displaystyle\times(1+\|\xi\|)^{N-k-r}\,\dfrac{m_{N-k-r}\,(N-k-r)!}{\delta_{4}^{N-k-r}(1+\|\xi\|)^{N-k-r}}\,C_{G0}\,h_{00}^{k-1}\,m_{k-1}\,(k-1)!$
$\displaystyle\times\displaystyle\inf_{j\in\mathbb{N}_{0}}\left(\dfrac{m_{j}\,j!}{\delta_{4}^{j}(1+\|\xi\|)^{j}}\right)$
$\displaystyle\leq\ \ $ $\displaystyle
C_{G0}\displaystyle\sum_{k=1}^{N}\binom{N}{k}h_{00}^{k-1}h_{pqs}^{N-k}\Bigg{[}\displaystyle\sum_{r=0}^{N-k}\binom{N-k}{r}\displaystyle\sum_{w=0}^{r}\binom{r}{w}m_{w}\Bigg{[}\displaystyle\sum_{\gamma\in\Delta(w)}\\!\\!\dfrac{|\gamma|!}{\gamma!}\left(\dfrac{C_{q}}{\delta_{4}}\right)^{|\gamma|}\Bigg{]}m_{r-w}$
$\displaystyle\times\Bigg{[}\displaystyle\sum_{\eta\in\Delta(r-w)}\\!\\!\dfrac{1}{\eta!}\left(\dfrac{C_{s0}}{\delta_{4}}\right)^{|\eta|}\Bigg{]}m_{N-k-r}\,\dfrac{1}{\delta_{4}^{N-k-r}}\,(N-k-r)!\Bigg{]}\,m_{k-1}\,(k-1)!$
$\displaystyle\times\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{4}^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C_{G0}\,h_{00}^{N}\displaystyle\sum_{k=1}^{N}\binom{N}{k}m_{k-1}\displaystyle\sum_{r=0}^{N-k}\binom{N-k}{r}m_{N-k-r}m_{r}\displaystyle\sum_{w=0}^{r}\binom{r}{w}[C_{q3}(1+C_{q3})^{w-1}]$
$\displaystyle\times[C_{s3}(1+C_{s3})^{r-w-1}](N-r-1)!\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{4}^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C_{G0}\,h_{00}^{N}\displaystyle\sum_{k=1}^{N}\binom{N}{k}m_{k-1}\displaystyle\sum_{r=0}^{N-k}\binom{N-k}{r}m_{N-k-r}m_{r}\displaystyle\sum_{w=0}^{r}\binom{r}{w}C_{q4}^{w}\,C_{s4}^{r-w}\,(N-r-1)!$
$\displaystyle\times\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{4}^{j}(1+\|\xi\|)^{j}}$
$\displaystyle\leq\ \ $ $\displaystyle
C\,h^{N}\,m_{N}\,N!\,\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta_{4}^{j}(1+\|\xi\|)^{j}},$

with $\delta_{4}=\delta_{2}/H^{2}$, $h_{qs}=\max\\{h_{q0},h_{s0}\\}$,
$h_{pqs}=\max\\{h_{p},h_{qs}\\}$, $C_{q3}=C_{q}/\delta_{4}$,
$C_{s3}=C_{s_{0}}/\delta_{4}$, $C_{q4}=1+C_{q3}$, $C_{s4}=1+C_{s3}$,
$h=2\max\\{h_{00},h_{pqs},C_{q4},C_{s4}\\}$ and $C=2C_{G0}$.

Using the same arguments, we can obtain similar estimates for the derivatives
of

$\dfrac{d^{N}}{dt^{N}}\left(e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)}\right)\left(\displaystyle\int_{t}^{2\pi}e^{-\rho_{\xi}\widetilde{Q}(\sigma)}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}G_{1\xi}(\sigma)\,d\sigma\right)$

and of

$e^{i(\xi\cdot
p_{0})t+S(t)}e^{\rho_{\xi}\widetilde{Q}(t)}\\!\\!\displaystyle\int_{0}^{2\pi}\\!\dfrac{e^{-\rho_{\xi}(q_{0}+\widetilde{Q}(\sigma))}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}}{e^{-\rho_{\xi}q_{0}}-e^{i(\xi\cdot
p_{0})2\pi+s_{0}}}G_{1\xi}(\sigma)\,d\sigma,$

observing that

$\displaystyle\left|\displaystyle\int_{t}^{2\pi}\\!\\!\\!e^{-\rho_{\xi}\widetilde{Q}(\sigma)}e^{-i(\xi\cdot
p_{0})\sigma-S(\sigma)}G_{1\xi}(\sigma)\,d\sigma\right|\ \ \leq\ \ $
$\displaystyle\displaystyle\int_{t}^{2\pi}|e^{-\rho_{\xi}\widetilde{Q}(\sigma)-S(\sigma)}G_{1\xi}(\sigma)|\,d\sigma$
$\displaystyle\leq\ \ $ $\displaystyle
2\pi\,\displaystyle\sup|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\sup|G_{1\xi}(t)|$
$\displaystyle\leq\ \ $ $\displaystyle
Ch_{G_{1}}\\!\,\displaystyle\sup|e^{-\rho_{\xi}\widetilde{Q}(t)-S(t)}|\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{h_{G_{2}}^{j}(1+\|\xi\|)^{j}}$

and that the condition (III) holds.

The study of the decay of the derivatives of $e^{i(\xi\cdot
p_{0})t+S(t)}z_{2\xi}(t)$ is analogous. Therefore,
$u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ and
$Pu=f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ by construction. ∎

######  Remark 3.

For the bidimensional torus $\mathbb{T}^{2}$, conditions (I) and (II) combined
with $\lambda\neq 0$ imply condition (III) (see [3]). However, in general,
this implication does not hold (see [7], Example 2).

######  Theorem 4.2.

Let $P$ the Vekua-type operator defined in (4.4) with $\lambda=0$. Suppose
that one of the following conditions holds:

  * (1)

$|B_{0}|>|A_{0}|$;

  * (2)

$|B_{0}|\leq|A_{0}|$, $|\alpha|>|\delta|$ and there is no
$(\xi,\tau)\in\mathbb{Z}^{n+1}$ solution of

$\begin{cases}\text{Re}(A_{0}(\tau+(\xi\cdot p_{0})))=0\\\
4\pi^{2}|\tau+(\xi\cdot p_{0})|^{2}=|A_{0}|^{2}-|B_{0}|^{2}\end{cases};$
(4.11)

  * (3)

$|\alpha|<|\delta|$ and $s_{0}\neq 0$;

  * (4)

$|\alpha|<|\delta|$, $s_{0}=0$, there is no $(\xi,\tau)\in\mathbb{Z}^{n+1}$
satisfying (4.11) and the following diophantine condition holds:
For all $\varepsilon>0$, exist $\gamma_{\varepsilon},C_{\varepsilon}>0$ such
that $(\xi,\tau)\in\mathbb{Z}^{n+1}$, $\ \|\xi\|\geq\gamma_{\varepsilon}$
implies

$|2\tau\pi+(\xi\cdot p_{0})2\pi-q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}|\geq
C_{\varepsilon}\,\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi\|)^{j}}.$
($DC^{\prime}_{\mathscr{M}}$)

Then, for all $f\in C^{\infty}(\mathbb{T}^{n+1})$, exists $u\in
C^{\infty}(\mathbb{T}^{n+1})$ such that $Pu=f$.

###### Proof.

If (1), (2), (3), or (4) holds, it is not difficult to see that conditions (I)
and (II) in Theorem 4.1 hold. Moreover, if (1), (2), or (3) holds, then there
exists $C>0$ such that

$|e^{-\rho_{\xi}q_{0}}-e^{s_{0}+i2\pi(\xi\cdot p_{0})}|\geq
C\quad\text{and}\quad|1-e^{-\rho_{\xi}q_{0}+s_{0}+i2\pi(\xi\cdot p_{0})}|\geq
C,\quad\xi\in\mathbb{Z}^{n}.$

In fact, first notice that $\lambda=0$ gives us that, for all
$\xi\in\mathbb{Z}$, we have
$\rho_{\xi}=\sqrt{|\alpha|^{2}-\delta^{2}}\doteq\rho$.

Suppose that (1) holds. Notice that
$|A_{0}|^{2}=s_{0}^{2}+|\delta|^{2}q_{0}^{2}$ and
$|B_{0}|^{2}=|\alpha|^{2}q_{0}^{2}$. Since $|A_{0}|<|B_{0}|$, we have

$|\alpha|^{2}q_{0}>s_{0}^{2}+|\delta|^{2}q_{0}^{2}\geq|\delta|^{2}q_{0}^{2}\Rightarrow|\alpha|>|\delta|,$

which implies $\rho\in\mathbb{R}_{>0}$. Suppose there exists
$\xi\in\mathbb{Z}^{n}$ such that $e^{-\rho q_{0}}-e^{i(\xi\cdot
p_{0})2\pi+s_{0}}=0$. Then, there exists $\tau\in\mathbb{Z}$ such that

$\rho q_{0}+i(\xi\cdot p_{0})2\pi+s_{0}=-i2\pi\tau.$

Comparing the real parts, we obtain $\rho q_{0}+s_{0}=0$. Since
$\rho=\sqrt{|\alpha|^{2}-\delta^{2}}$, we have

$\displaystyle q_{0}\sqrt{|\alpha|^{2}-\delta^{2}}+s_{0}=0$
$\displaystyle\Rightarrow(|\alpha|^{2}-\delta^{2})q_{0}^{2}=s_{0}^{2}$
$\displaystyle\Rightarrow|\alpha|^{2}q_{0}^{2}=s_{0}^{2}+\delta^{2}q_{0}^{2}$
$\displaystyle\Rightarrow|A_{0}|^{2}=|B_{0}|^{2},$

which contradicts (1). Therefore, $e^{-\rho q_{0}}-e^{i(\xi\cdot
p_{0})2\pi+s_{0}}\neq 0$ for all $\xi\in\mathbb{Z}^{n}$. Analogously, we show
that $1-e^{-\rho q_{0}+i(\xi\cdot p_{0})2\pi+s_{0}}\neq 0$ for all
$\xi\in\mathbb{Z}^{n}$. This implies

$-\rho q_{0}\neq i(\xi\cdot p_{0})2\pi+s_{0}+i2\pi k,\quad\text{for
}k\in\mathbb{N}.$

Notice that, since $|\alpha|>|\delta|$, we have $\rho\in\mathbb{R}$. Then,
given $\xi\in\mathbb{Z}^{n}$, we have

$|e^{-\rho q_{0}}-e^{i(\xi\cdot p_{0})2\pi}|\geq|e^{-\rho
q_{0}}-e^{s_{0}}|\quad\text{and}\quad|1-e^{-\rho q_{0}+i(\xi\cdot
p_{0})2\pi+s_{0}}|\geq|1-e^{-\rho q_{0}+s_{0}}|.$

Choosing $C=\min\\{|e^{-\rho q_{0}}-e^{s_{0}}|,|1-e^{-\rho
q_{0}+s_{0}}|\\}>0$, we have

$|e^{-\rho q_{0}}-e^{i(\xi\cdot p_{0})2\pi+s_{0}}|\geq
C\quad\text{and}\quad|1-e^{-\rho q_{0}+i(\xi\cdot p_{0})2\pi+s_{0}}|\geq
C,\quad\xi\in\mathbb{Z}^{n}.$

If (2) or (3) holds, the arguments are similar. Notice that

$C=C\frac{m_{0}\cdot 0!}{\varepsilon^{0}(1+\|\xi\|)^{0}}\geq
C\inf_{j\in\mathbb{N}_{0}}\frac{m_{j}\cdot
j!}{\varepsilon^{j}(1+\|\xi\|)^{j}}$

for all $\varepsilon>0$. Proceeding as in Theorem 4.1, we show that for all
$f\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$, there exists
$u\in\mathcal{E}_{\mathscr{M}}(\mathbb{T}^{n})$ such that $Pu=f$.

If (4) holds, we can not prove that there exists $C>0$ like in the other
cases. In this case, we use the arguments of Theorem 4.1 and the Diophantine
condition ($DC^{\prime\prime}_{\mathscr{M}}$) which is equivalent to
($DC^{\prime}_{\mathscr{M}}$), as proved in the following lemma. ∎

######  Lemma 4.3.

The condition ($DC^{\prime}_{\mathscr{M}}$) is equivalent to the following
condition:
For all $\varepsilon>0$, there exist $\gamma_{\varepsilon},C_{\varepsilon}>0$
such that for $\xi\in\mathbb{Z}^{n}$, $\|\xi\|\geq\gamma_{\varepsilon}$, we
have

$\left|e^{i((\xi\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1\right|\geq
C_{\varepsilon}\inf_{j\in\mathbb{N}_{0}}\frac{m_{j}\cdot
j!}{\varepsilon^{j}(1+\|\xi\|)^{j}}.$ ($DC^{\prime\prime}_{\mathscr{M}}$)

###### Proof.

This proof is an adaptation of the proof of Lemma 13 in [3]. Suppose that
($DC^{\prime}_{\mathscr{M}}$) does not hold. Then, there exists
$\varepsilon>0$ such that for each $\ell\in\mathbb{N}$, there exists
$(\xi_{\ell},\tau_{\ell})\in\mathbb{Z}^{n+1}$ with $\|\xi_{\ell}\|\geq\ell$
and

$\left|2\pi\tau_{\ell}+(\xi_{\ell}\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}\right|<\frac{1}{2}\inf_{j\in\mathbb{N}_{0}}\frac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi_{\ell}\|)^{j}}.$

Notice that

$\displaystyle|e^{i((\xi_{\ell}\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1|^{2}\ \ =\ \ $
$\displaystyle\left|e^{i(2\pi\tau_{\ell}+(\xi_{\ell}\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1\right|^{2}$ $\displaystyle=\ \ $
$\displaystyle 2(1-\cos(2\pi\tau_{\ell}+(\xi_{\ell}\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}))$ $\displaystyle=\ \ $ $\displaystyle
2(2\pi\tau_{\ell}+(\xi_{\ell}\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})\sin(\xi_{\ell})$ $\displaystyle\leq\ \ $
$\displaystyle\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi_{\ell}\|)^{j}}$
$\displaystyle\leq\ \ $
$\displaystyle\left[\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\varepsilon/H)^{j}(1+\|\xi_{\ell}\|)^{j}}\right]^{2},$

for some $\xi_{\ell}\in\mathbb{R}$. Then,

$|e^{i((\xi_{\ell}\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1|<\inf_{j\in\mathbb{N}_{0}}\left(\dfrac{m_{j}\,j!}{(\varepsilon/H)^{j}(1+\|\xi_{\ell}\|)^{j}}\right).$

Therefore, (DC${}_{\mathscr{M}}^{\prime\prime}$) does not hold.

On the other hand, suppose that ($DC^{\prime\prime}_{\mathscr{M}}$) does not
hold. Then, there exists $\varepsilon>0$ such that for each
$\ell\in\mathbb{N}$, there exists $\xi_{\ell}\in\mathbb{Z}^{n}$ with
$\|\xi_{\ell}\|\geq\ell$ and

$|e^{i((\xi_{\ell}\cdot p_{0})2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1|<\inf_{j\in\mathbb{N}_{0}}\frac{m_{j}\cdot
j!}{\varepsilon^{j}(1+\|\xi_{\ell}\|)^{j}}.$

For each $\ell\in\mathbb{Z}$,

$-\tau_{\ell}=\left\lfloor\xi_{\ell}\,p_{0}-\frac{q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}}{2\pi}\right\rfloor\in\mathbb{Z},$

that is, $-\tau_{\ell}$ is the integer part of
$\xi_{\ell}\,p_{0}-\frac{1}{2\pi}q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}$.

Notice that

$0\leq\xi_{\ell}\,p_{0}-\frac{q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}}{2\pi}+\tau_{\ell}<1.$

We have

$|e^{i(\xi_{\ell}\,p_{0}2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1|<\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi_{\ell}\|)^{j}}\leq\dfrac{1}{\varepsilon(1+\|\xi_{\ell}\|)}\to
0,\quad\ell\to\infty,$

since $\|\xi_{\ell}\|\geq\ell\to\infty$ with $\ell\to\infty$. Then,

$\xi_{\ell}\,p_{0}-\dfrac{q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}}{2\pi}+\tau_{\ell}\to
0\quad\text{or}\quad\xi_{\ell}\,p_{0}-\dfrac{q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}}{2\pi}+\tau_{\ell}\to
1,\quad\ell\to\infty.$

Then, taking $\ell\in\mathbb{N}$ sufficiently large, we have

$\displaystyle\left|e^{i(2\pi\tau_{\ell}+\xi_{\ell}\,p_{0}2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1\right|^{2}$
$\displaystyle=2\left(1-\cos(2\pi\tau_{\ell}+\xi_{\ell}\,p_{0}2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})\right)$
$\displaystyle\geq\frac{1}{2}\left|2\pi\tau_{\ell}+\xi_{\ell}\,p_{0}2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}\right|^{2},$

using the fact that $1-\cos(\theta)\geq\theta^{2}/4$ for all $\theta$ in a
neighborhood of the origin. This follows from the Taylor expansion of
$1-\cos(\theta)$. Since

$\left|e^{i(2\pi\tau_{\ell}+\xi_{\ell}\,p_{0}2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}})}-1\right|<\inf_{j\in\mathbb{N}_{0}}\frac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi_{\ell}\|)^{j}},\quad\ell\in\mathbb{N},$

we have

$\left|2\pi\tau_{\ell}+\xi_{\ell}\,p_{0}2\pi-
q_{0}\sqrt{\delta^{2}-|\alpha|^{2}}\right|<2\inf_{j\in\mathbb{N}_{0}}\frac{m_{j}\,j!}{\varepsilon^{j}(1+\|\xi_{\ell}\|)^{j}},\quad\ell\in\mathbb{N}.$

Therefore, ($DC^{\prime}_{\mathscr{M}}$) does not hold. We conclude that
($DC^{\prime}_{\mathscr{M}}$) and ($DC^{\prime\prime}_{\mathscr{M}}$) are
equivalent. ∎

##  Appendix A Technical results

######  Lemma A.1.

Let $\mathscr{M}=\\{m_{j}\\}_{j\in\mathbb{N}_{0}}$ a weight sequence and $H>0$
from property (W3). Then, for all $\rho>0$, we have

$\left(\sup_{j\in\mathbb{N}_{0}}\dfrac{\rho^{j}}{m_{j}\,j!}\right)^{2}\leq\sup_{j\in\mathbb{N}_{0}}\dfrac{\rho^{j}H^{j}}{m_{j}\,j!}.$

###### Proof.

[14, Proposition 3.6]. ∎

######  Remark 4.

Equivalently, we have

$\left(\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\rho^{j}}\right)^{2}\geq\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\rho^{j}H^{j}}.$

In particular, given $\delta>0$, taking $\rho=\delta/H$, we obtain

$\left(\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{(\delta/H)^{j}}\right)^{2}\geq\inf_{j\in\mathbb{N}_{0}}\dfrac{m_{j}\,j!}{\delta^{j}}.$

######  Lemma A.2 (Faà di Bruno Formula).

Given $f\in C^{\infty}(\mathbb{R})$ and $k\in\mathbb{N}$, we have that

$\dfrac{d^{k}}{dt^{k}}e^{f(t)}=e^{f(t)}\displaystyle\sum_{\gamma\in\Delta(k)}\dfrac{k!}{\gamma!}\prod_{\ell=1}^{k}\left(\dfrac{1}{\ell!}\dfrac{d^{\ell}}{dt^{\ell}}f(t)\right)^{\gamma_{\ell}},$

with
$\Delta(k)=\left\\{\gamma=(\gamma_{1},\dots,\gamma_{k})\in\mathbb{N}_{0}^{k}\,:\,\sum_{\ell=1}^{k}\ell\gamma_{\ell}=k\right\\}$.

######  Lemma A.3.

Given $k\in\mathbb{N}_{0}$ and $R>0$, we have that

$\sum_{\gamma\in\Delta(k)}\dfrac{|\gamma|!}{\gamma!}R^{|\gamma|}=R(1+R)^{k-1},$

with
$\Delta(k)=\left\\{\gamma=(\gamma_{1},\dots,\gamma_{k})\in\mathbb{N}_{0}^{k}\,:\,\sum_{\ell=1}^{k}\ell\gamma_{\ell}=k\right\\}$.

###### Proof.

[15, Lemma 1.4.1]. ∎

######  Lemma A.4.

Let $\mathscr{M}$ a weight sequence and $\gamma\in\Delta(k)$. Denote
$|\gamma|=\gamma_{1}+\cdots+\gamma_{k}$. Then,

$m_{|\gamma|}m_{1}^{\gamma_{1}}\cdots m_{k}^{\gamma_{k}}\leq m_{k}.$

###### Proof.

[6, Proposition 4.4]. ∎

Acknowledgments. The first and second authors were supported in part by CNPq -
Brasil (grants 316850/2021-7 and 423458/2021-3). This study was financed in
part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brasil (CAPES) - Finance Code 001.

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# Phase-stable limited relativistic acceleration or unlimited relativistic
acceleration in the laser-thin-foil interactions

Yongsheng Huang http://www.anianet.com/adward<EMAIL_ADDRESS>China
Institute of Atomic Energy, Beijing 102413, China.  Naiyan Wang  Xiuzhang Tang
China Institute of Atomic Energy, Beijing 102413, China.  Yan Xueqing  State
Key Laboratory of Nuclear Physics and Technology, Institute of Heavy Ion
Physics, Peking University, Beijing 100871, China

###### Abstract

To clarify the relationship between phase-stable acceleration (PSA) and
unlimited relativistic acceleration (URA) (Phys. Rev. Lett. 104, 135003
(2010)), an analytical relativistic model is proposed in the interactions of
the ultra-intense laser and nanometer foils, based on hydrodynamic equations.
The dependence of the ion momentum on time is consistent with the previous
results and checked by PIC simulations. Depending on the initial ion momentum,
relativistic RPA contains two acceleration processes: phase-stable limited
relativistic acceleration (PS-LRA) and URA. In PS-LRA, the potential is a deep
well trapping the ions. The ion front, i.e., the bottom, separates it into two
parts: the left half region is PSA region; the right half region is PSD
region, where the ions climb up and are decelerated to return back. In PS-LRA,
the maximum ion energy is limited. If the initial ion momentum large enough,
ions will experience a potential downhill and drop into a bottomless abyss,
which is called phase-lock-like position. URA is not phase-stable any more. At
the phase-lock-like position the ions can obtain unlimited energy gain and the
ion density is non-zero. You cannot have both PSA and URA.

###### pacs:

52.38.Kd,41.75.Jv,52.40.Kh,52.65.-y

Laser-ion acceleration has been an international research focusMakoTajima ;
Machnisms ; Esirkepov ; Yin , however it is still a challenge to obtain mono-
energetic proton beams larger than $100\mathrm{MeV}$. As a promising method to
generate relativistic mono-energetic protons, radiation pressure acceleration
(RPA) has attracted more attentionEsirkepov ; Yin ; EsirkepovPRL96 ; Henig
and becomes dominant in the interaction of the ultra-intense laser pulse with
nanometer foils. The phase-stable accelerationYanxqPRL  predicted that the
energy spread can be improved in the interactions of nanometer-foils with
circular-polarized laser pulses. With thin-shell modelunlimitedRPA , Bulanov
and coworkersunlimitedRPA  pointed out that the ions can obtain unlimited
energy gain by RPA in the relativistic limit. Yan and coworkers tried to
predict the ion energy distribution with a self-similar hydrodynamic
theoryYanxq . However, it is nonrelativistic and under the plasma
approximation which allows $\nabla\bullet E\neq 0$ and $n_{i}-n_{e}=0$ satisfy
together, where $E$ is the acceleration field and $n_{i}$ (or $n_{e}$) is the
ion (or electron) density. However, can the ions obtain unlimited energy gain
in the phase-stable region or is the phase-stable acceleration still possible
in the relativistic limit? How about the relationship between them or what are
the critical conditions for them? To give clear answers of the questions, an
analytic self-consistent relativistic fluid model is proposed to describe the
relativistic radiation pressure acceleration and to recheck the unlimited ion-
acceleration region.

The ion acceleration in the interaction of ultra-intense laser and nanometer
foils contains two stages: the hole-boring process and the radiation pressure
acceleration. Here the transition of them is assumed steady and the
instability of the acceleration sheath is suppressed well, which can be
realized for specially designed targets stableRPA1 ; stableRPA . In the hole-
boring process, the ion velocity can reach $u_{hb}$Qiao2009 ; breaktime ,
which is the hole-boring velocity and also the initial velocity of the ions in
the radiation pressure acceleration. The initial time, $t_{0}$ is when the
compression layer is detached from the foil. It is decided by the target
thickness and $u_{hb}$. With the initial conditions: $u_{hb}$ and $t_{0}$, the
dependence of the ion energy on time can be obtained and consists with the
results of thin-shell modelEsirkepov  and has been checked by PIC
simulationsEsirkepov . Depending on the laser intensity, the initial ion
momentum will determine two different acceleration processes: the phase-stable
limited relativistic acceleration (PS-LRA) and the unlimited relativistic
acceleration (URA). When the initial ion momentum is smaller than the critical
one, the potential is a deep well trapping the ions. The acceleration mode is
PS-LRA and contains the phase-stable acceleration region (PSA) and the phase-
stable deceleration region (PSD). The well is separated into two half-regions:
the left half-region and the right half-region by the bottom, which is the ion
front. The left half-region is PSA region, where the electric field is
positive and ions coast down to the bottom. While the right half-region is PSD
region, where the electric field is negative and ions climb up the potential
uphill and are decelerated to return to the bottom. The deceleration is also
phase-stable. No matter in PSA or PSD region, the maximum ion energy is
limited and ascertained by an analytical formulation. Since PSA and PSD are
separated by the ion front where the ion density is zero, the ions in two
regions can not exchange from each other. If the initial ion momentum is large
enough, the ions can get across the potential uphill and experience a
potential downhill and then drop into the bottomless abyss which is the phase-
lock-like position. If the ions can reach the phase-lock-like position, they
can obtain unlimited energy gain as Bulanov and coworkers pointed
outunlimitedRPA . The acceleration mode is URA and not phase-stable any more.
The electron density is smaller than the ion density and the electron front
increases with time. The phase-lock-like position of URA is the limiting ion
front and the ion density is non-zero as time tends to zero. You cannot have
both PSA and URA in the same acceleration process. Therefore, the maximum ion
energy is finite in the phase-stable region and URA is not phase-stable any
more. However, no matter in PS-LRA or URA, the plasma tends to neutral as time
tends to infinite.

For convenience, the physical parameters: the time, $t$, the ion position,
$x$, the ion velocity, $v$, the electron field, $E$, the electric potential,
$\varphi$, the plasma density, $n$, and the light speed, $c$, are normalized
as follows: ${\tau}={\omega
t},\hat{x}=xk,u=v/c,\hat{E}={E}/{E_{0}},{\phi}=\varphi/\varphi_{0},\hat{n}={n}/{n_{0}},$
where $n$ represents $n_{i}$ (or $n_{e}$), $n_{0}$ is the reference density,
$\omega$ is the light frequency, $k=\omega/c$ is the wave number, $c$ is the
light speed, $E_{0}={k\varphi_{0}}$, $e\varphi_{0}=\gamma_{em}m_{e}c^{2}$ and
$\gamma_{em}$ is the maximum electron energy. Here $e$ is the elemental
charge.

With reference to the results given by Mako and Tajima in Ref. MakoTajima , in
the self-similar state, the density distribution of ions is assumed as:

$\hat{n}_{k}=\frac{1}{\Sigma Q_{k}}(1+\phi)^{\alpha},k=1,...,N,$ (1)

where the subscribe k stands for the ion species, $Q_{k}$ is the charge number
of the $k$th species ion, the index $\alpha$ depends on the laser intensity
and the target thickness and discussed in the Ref. Yanxq .

With the transformation: $\xi=\hat{x}/\tau$, the normalized continuity and
motion equation of ions and Poisson’s equation are given as:

$\displaystyle\left(u_{k}-\xi\right)\frac{\partial\ln\hat{n}_{k}}{\partial\xi}=-\frac{\partial
u_{k}}{\partial\xi},$ (2)
$\displaystyle\left(u_{k}-\xi\right)\frac{\partial\gamma_{k}u_{k}}{\partial\xi}=-\beta_{k}\frac{\partial\phi}{\partial\xi},$
$\frac{1}{\tau^{2}}\frac{\partial^{2}\phi}{\partial\xi^{2}}=-\rho\left(\Sigma
Q_{k}\hat{n}_{k}-\hat{n}_{e}\right)$ (3)

where $\beta_{k}=\frac{Q_{k}\gamma_{em}m_{e}}{M_{k}}$, $M_{k}$ is the mass of
certain ions, $\gamma_{k}=(1-u_{k}^{2})^{-1/2}$,
$\rho=\frac{\omega_{pe}^{2}}{\gamma_{em}\omega^{2}}$, and
$\omega_{pe}^{2}=\frac{n_{0}e^{2}}{\epsilon_{0}m_{e}}$.

Solving Eq. (2), the ion velocity satisfies:

$\xi+\left(\frac{\gamma_{k}}{\gamma_{k,0}}\right)^{-3/2}u_{k,0}=u_{k}+\frac{\gamma_{k}^{-3/2}}{2\alpha}\left(\chi-\chi_{0}\right),$
(4)

and the potential in the ion region is given by:

$\phi_{1}=\frac{\left(\chi-\chi_{0}-2\alpha
u_{k,0}\gamma_{k,0}^{3/2}\right)^{2}}{4\alpha\beta_{k}}-1,$ (5)

where $\gamma_{k,0}=(1-u_{k,0}^{2})^{-1/2}$, $\chi_{0}=\chi(u_{k,0})$,

$\chi=\int^{u_{k}}_{0}\gamma_{k}^{3/2}du_{k},$ (6)

and $u_{k,0}$ is the hole-boring velocity given byQiao2009 ; breaktime :

$u_{k,0}=\frac{u_{hb}}{c}=\sqrt{\frac{Z}{A}\frac{m_{e}}{\gamma_{hb}M_{k}}\frac{n_{c}}{2n_{0}}}a$
(7)

where $a^{2}=0.732I_{10^{18}\mathrm{W/cm^{2}}}\lambda^{2}_{\mathrm{\mu m}}$,
$I_{10^{18}\mathrm{W/cm^{2}}}$ is the laser intensity in unit of
$10^{18}\mathrm{W/cm^{2}}$, and $\gamma_{hb}={(1-(u_{hb}/c)^{2})^{-1/2}}$. The
beginning time $\tau_{0}$ is when the compressed ion and electron layer is
detached from the foil and given bybreaktime :

$\tau_{0}=\frac{d}{\lambda}\frac{2\pi}{u_{k,0}},$ (8)

at $\xi=0$.

In order to obtain the dependence of the ion velocity on time, it is need to
solve Eq. (4). From Eq. (4), the following differential equations of two
variables are given:

$\left\\{\begin{aligned} \frac{dp_{k}}{dt}=\frac{2\alpha
V_{k}(1+p_{k}^{2})^{3/2}}{t\left[3\alpha
p_{k}V_{k}\sqrt{1+p_{k}^{2}}-(2\alpha+1)t\right]},\\\
\frac{dV_{k}}{dt}=\frac{-2\alpha V_{k}}{3\alpha
p_{k}V_{k}\sqrt{1+p_{k}^{2}}-(2\alpha+1)t}.\end{aligned}\right.$ (9)

where $p_{k}=u_{k}\gamma_{k}$ is the normalized ion momentum and $V_{k}$
satisfies:

$V_{k}=\int^{t}_{0}\frac{p_{k}}{\sqrt{1+p_{k}^{2}}}dt-t\frac{p_{k}}{\sqrt{1+p_{k}^{2}}}.$
(10)

Using matlab function $ode113$ to solve Eq. (9) with the initial time
$\tau_{0}$ and initial velocity, $u_{k,0}$, the dependence of the ion energy
on time has been calculated matlabfile . As an example, Figure 1 shows the
comparison of our analytical fluid model with the thin-shell model and PIC
simulationsEsirkepov . The results of our model consist well with that of the
PIC simulations. When $\tau$ is larger than $40$ times of the laser cycle, our
results are a little larger than that of the PIC simulations. One of the
reason is the loss of laser energy and the decreasing of the electron
temperature for large $\tau$ in the simulations, while the electron
temperature is assumed to be a constant in our model.

Figure 1:  (Color online) Comparison of our analytical fluid model with thin-
shell model and Esirkepov et al.’s simulations for $\sigma/a\approx 0.1$,
$a=316$, $d=\lambda=1\mu m$, $n_{0}=49n_{c}=5.5\times 10^{22}/cm^{3}$, and
$\alpha=1.8$Yanxq .

Combing Eqs. (1) and (5), it is obtained:

$\hat{n}_{k}=\frac{1}{\Sigma Q_{k}}\frac{\left(\chi-\chi_{0}-2\alpha
u_{k,0}\gamma_{k,0}^{3/2}\right)^{2\alpha}}{\left(4\alpha\beta_{k}\right)^{\alpha}},$
(11)

With Eqs. (3), (4) and (5), the electron density in the ion region is written
as:

$\hat{n}_{e}=\left(1+\phi_{1}\right)^{\alpha}+\frac{1}{\rho\tau^{2}}\frac{\partial^{2}\phi_{1}}{\partial\xi^{2}},$
(12)

Equation (12) shows the plasma can not be quasi-neutral at a finite time.
However, the plasma tends to neutral as the time tends to infinite. It is
obtained:

$\lim_{\tau\rightarrow+\infty}n_{e}=\Sigma Q_{k}n_{k},$ (13)

With Eq. (11) and $\hat{n}_{k}(\xi=\xi_{i,f})=0$, the possible maximum ion
energy at the ion front satisfies:

$\int_{0}^{p_{k,m}}\gamma_{k}^{-\frac{3}{2}}dp_{k}=\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}},$ (14)

where $\gamma_{k}^{2}=1+p_{k}^{2}$, $p_{k,m}=u_{k,m}/\sqrt{1-u_{k,m}^{2}}$ is
the normalized limiting momentum of ions at the ion front: $\xi=\xi_{i,f}$ and
$p_{k,0}=u_{k,0}/\sqrt{1-u_{k,0}^{2}}$.

Different from the nonrelativistic case, it contains two acceleration modes
and depends on the initial conditions: $u_{k,0}$ and $\alpha$ in the
relativistic case. The critical condition is decided by:

$\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}}=\int_{0}^{+\infty}(1+x^{2})^{-3/4}dx\approx
2.622.$ (15)

First, $\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}}\lneq 2.622$ for $p_{k,0}\lneq 0.5064$ and
$\alpha=2$, which requires $a\lneq 203$ for $n_{0}=49n_{c}$, $d=\lambda=1\mu
m$. Figure 2 shows the dependence of the plasma density, electric field and
potential on $\xi$ for $n_{0}=10^{22}\mathrm{/cm^{3}}$, $\alpha=2$ and $a=70$.
In this case, it is divided into two regions: the phase-stable acceleration
region for $0\leq\xi\leq\xi_{i,f}$ and the phase-stable deceleration region
$\xi_{i,f}\leq\xi\leq 1$.

(I) In the phase-stable acceleration region, $0\leq\xi\leq\xi_{i,f}$, the
electric field $E\geq 0$ and the electron density is larger than the ion
density. The maximum ion momentum $p_{k,m1}$ is limited and given Eq. (14) at
the ion front $\xi_{i,f}$ ascertained by Eq. (4) with $u_{k}=u_{k,m1}$, where
$u_{k,m1}=p_{k,m1}/\sqrt{1+p_{k,m1}^{2}}$. At the ion front, the ion density
is zero. The difference of the electron density and ion density decreases with
time and tends to zero.

The potential shown by Figure 2 (b) in $0\leq\xi\leq\xi_{i,f}$ gives an
intuitionistic explanation of PSA. The ions coast down the slope of the
potential, and the gradient, i.e., the electric field, becomes gently as the
ions come to the bottom of the potential although few can reach there.
Therefore the ions at higher potential will obtain more acceleration and the
energy spread is improved. That is PSA. Different from the real gliding
process, the ions can not pass through the bottom and climb up since the ion
front is the limiting point and the ion density tends to zero at the bottom.

(II) In the phase-stable deceleration region, $\xi_{i,f}\leq\xi\leq 1$, the
eletric field $E\leq 0$ and the electron density is larger than the ion
density too as shown in Figure 2. In this region, the ion momentum $p_{k}$
satisfies: $p_{k,m1}\leq p_{k}\leq p_{k,m2}$, where $p_{k,m2}=p_{k}(\xi=1)$
and given by Eq. (4) with $\xi=1$. All the ions in this region are decelerated
to $\xi=\xi_{i,f}$. The absolute value of the electric field decreases with
the decreasing $\xi$ and is zero at the ion front. Therefore the deceleration
is also phase-stable.

With Figure 2 (b), for ions, the potential in $\xi_{i,f}\leq\xi\leq 1$ is a
mountain with height of about $35\mathrm{GeV}$ which is far larger than the
maximum ion energy of about $\sqrt{p_{m,2}^{2}+1}-1\approx 7\mathrm{GeV}$ at
$\xi=1$. Therefore, the ions will be decelerated at the potential uphill.
Since the gradient, i.e., the value of electric field increases with the
potential height, the deceleration is also phase-stable. As point out above,
the limiting point is still the ion front $\xi=\xi_{i,f}$, and the ions in PSD
region can also not pass through the bottom into PSA region. The ions in PSA
and PSD region can not exchange from each other because of the zero-density
dividing point $\xi=\xi_{i,f}$.

In this case, the maximum ion momentum is $p_{k,m1}$ at the ion front
$\xi_{i,f}\lneq 1$. Therefore, it is called phase-stable limited relativistic
acceleration (PS-LRA). The ions in the two phase-stable regions can not
exchange from each other and the ion front $\xi_{i,f}$ is the dividing line.
Combing the condition of PS-LRA and the above discussion about PSA and PSD,
the ions with an initial momentum of $p_{k,0}$ not large enough to get across
the potential at $\xi=1$ will drop in PSA region or PSD region and obtain a
finite maximum energy ascertained by Eq. (14).

Figure 2:  (Color online) Phase-stable limited relativistic acceleration (PS-
LRA) contains two regions: phase-stable acceleration region for
$0\leq\xi\leq\xi_{i,f}\approx 0.87$ and phase-stable deceleration region for
$0.87\leq\xi\leq 1$. (a)The density of ions and electrons for different time
and the ion momentum VS the self-similar variable $\xi$. At the ion front
$\xi_{i,f}=0.87$, the ion density is zero. (b)The potential VS $\xi$. (c) and
(d) The electric field $\hat{E}$ VS $\xi$. $\hat{E}\geq 0$ for
$0\leq\xi\leq\xi_{i,f}$ and $\hat{E}\leq 0$ for $\xi_{i,f}\leq\xi\leq 1$.
Here, $n_{0}=10^{22}\mathrm{/cm^{3}}$, $\alpha=2$ and $a=70$.

If the initial ion momentum satisfies:

$\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}}\geq 2.622.$ (16)

the ions will not experience a potential well in PS-LRA and will coast down
from the potential slope and drop into the bottomless abyss at $\xi=1$ as
shown by Figure 3 (d). $\xi=1$ is called phase-lock-like position and the ions
can obtain unlimited energy gain as shown in Figure 3 (b). It is called
unlimited relativistic acceleration (URA), which is not phase-stable any more.
Eq. (16) satisfies for $p_{k,0}\geq 0.5064$, i.e., $a\geq 203$ from Eq. (7)
for $\alpha=2$, $n_{0}=49n_{c}=5.5\times 10^{22}\mathrm{/cm^{3}}$ and
$d=\lambda=1\mathrm{\mu m}$. Figure 3 shows the dependence of the plasma
density, electric field and potential on $\xi$ for $a=316$, $d=\lambda=1\mu
m$, $n_{0}=49n_{c}=5.5\times 10^{22}/cm^{3}$, and $\alpha=2$. In this case,
the electron density is smaller than the ion density and the acceleration is
not phase-stable any more. In all the region, the electric field increases
with $\xi$ and is larger than zero. At a finite time, the electron front,
where $n_{e}=0$, $\xi_{e,f}\lneq 1$, which is 0.971, 0.994, 0.99997 at
$\tau=5,20,2000$ separately. Therefore the possible maximum the ion momentum
is shown by Figure 3 (a) and (b).

As discussed above, due to the initial ion momentum $p_{k,0}$ large enough,
the ions can reach URA region. Therefore the ion momentum and the field have a
sharp increase and tend to infinite, the ion density becomes a non-zero
constant in $\xi\in(1-\delta,1]$ as $\tau\rightarrow\infty$, where $\delta$ is
an infinitesimal. As shown in Figure 3 (b), at the phase-lock-like position
and the limiting ion front $\xi=1$, the ion can obtain unlimited energy gain
and the ion density is non-zero. It is similar with the unlimited phase-lock
ion acceleration as pointed out by Bulanov and coworkersunlimitedRPA  in the
relativistic limit. In URA region, it is found that (I) the unlimited ion
acceleration requires the initial ion momentum is large enough and should meet
Equation (16); (II) it is not phase-stable any more; (III) the phase-lock-like
position is $\xi=1$, the limiting ion front; (IV) the ion density at the
limiting ion front is non-zero.

Figure 3:  (Color online) Unlimited relativistic acceleration (URA) with the
phase-lock-like position $\xi=1$.(a)The density of ions and electrons for
different time and the ion momentum VS $\xi$. The ion density is no-zero at
the limiting ion front: $\xi=1$. The electron density is smaller than that of
ions at any finite time. (b)The enlargement of (a) near $\xi=1$. $\xi=1$ is
the phase-lock-like position where the ion momentum tends to infinity. (c) and
(d) The electric field $\hat{E}$ and the potential $\phi$ VS $\xi$.
$\hat{E}\geq 0$ for all the region and the acceleration is not phase-stable
any more. Here $a=316$, $d=\lambda=1\mathrm{\mu m}$, $n_{0}=49n_{c}=5.5\times
10^{22}\mathrm{/cm^{3}}$, and $\alpha=2$.

In the conclusion, it has been given an analytical relativistic fluid model to
describe the relativistic radiation pressure acceleration with the initial
parameters from the hole-boring stage. The dependence of the ion velocity on
the acceleration time can be obtained and is consistent with that of thin-
shell model and PIC simulations. There are two acceleration modes: PS-LRA and
URA with a critical initial ion momentum ascertained by an explicit
formulation. In PS-LRA, the ions are trapped in a deep potential well and the
maximum ion energy is limited and the ion front is the well bottom and
$\xi_{i,f}\lneq 1$. URA is not phase-stable any more and there is a phase-
lock-like position in it. At the phase-lock-like position, corresponding to
the relativistic limit, the ions can obtain unlimited energy gain and the ion
density is non-zero as time tends infinite. Although the unlimited ion
acceleration can not be reached at any finite time, the ions can be
accelerated to any large energy if the laser pulse is long enough. As an
important result, you cannot obtain both PSA and URA. Therefore, if the laser
parameters are large enough to obtain URA, the energy spread must be lost. If
one wants to improve the energy spread with PSA, the maximum ion energy is
limited.

###### Acknowledgements.

This work was supported by the Key Project of Chinese National Programs for
Fundamental Research (973 Program) under contract No. $2011CB808104$ and the
Chinese National Natural Science Foundation under contract No. $10834008$.

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