\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 244, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2013/244\hfil Involutive systems on the torus]
{Global solvability for involutive systems \\ on the torus}

\author[C.  Medeira \hfil EJDE-2013/244\hfilneg]
{Cleber de Medeira}  % in alphabetical order

\address{Cleber de Medeira \newline
Department of mathematics, Federal University of Paran\'a, 19081,
Curitiba, Brazil}
\email{clebermedeira@ufpr.br}

\thanks{Submitted April 19, 2013. Published November 8, 2013.}
\subjclass[2000]{35N10, 32M25}
\keywords{Global solvability; involutive systems;
complex vector fields; \hfill\break\indent Liouville number}

\begin{abstract}
 In this article, we consider a class of involutive systems of $n$
 smooth vector fields on the torus of dimension $n+1$. We prove that
 the global solvability of this class is related to an
 algebraic condition involving Liouville forms and the connectedness
 of all sublevel and superlevel sets of the primitive of a certain
 1-form associated with the system.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article we study  the global solvability of a  system of
vector fields on
$\mathbb{T}^{n+1}\simeq(\mathbb{R}/2\pi\mathbb{Z})^{n+1}$ given by
\begin{equation}\label{L}
  L_j=\frac{\partial}{\partial t_j}+(a_j+ib_j)(t)\frac{\partial}{\partial x},
\quad j=1,\ldots,n,
\end{equation}
where $(t_1,\ldots,t_n,x)=(t,x)$ denotes the coordinates on
$\mathbb{T}^{n+1}$,  $a_j,b_j\in {C}^{\infty}(\mathbb{T}^{n};\mathbb{R})$
and for each $j$ we consider $a_j$ or $b_j$ identically zero.

We assume that the system \eqref{L} is involutive
(see \cite{BCH,T2}) or equivalently that the 1-form
$c(t)=\sum_{j=1}^n (a_j+ib_j)(t)dt_j\in\wedge ^1{C}^{\infty}(\mathbb{T}_t^{n})$
is closed.

When the 1-form $c(t)$ is exact the problem was treated by Cardoso
and Hounie in \cite{CH}. Here, we will consider that only the
imaginary part of $c(t)$ is exact, that is, the real 1-form
 $b(t)=\sum_{j=1}^n b_j(t)dt_j$ is exact.

The system \eqref{L} gives rise to   a complex of differential
operators $\mathbb{L}$   which at the first level acts in the
following way
\begin{equation} \label{operator}
\mathbb{L}u=d_tu+c(t)\wedge\frac{\partial}{\partial x}u,\quad u\in
C^\infty(\mathbb T^{n+1}) \quad \text{or  }\mathcal D'(\mathbb{T}^{n+1})),
\end{equation}
where $d_t$ denotes the exterior
differential on the torus $\mathbb{T}^n_t$. Our aim is to carry out
a study of the global solvability at the first level of this
complex. In other words we study the global solvability of the
equation $\mathbb{L} u=f$ where $u\in\mathcal{D}'(\mathbb{T}^{n+1})$
and $f\in C^\infty(\mathbb T^n_t\times\mathbb T^1_x;\wedge^{1,0})$.

Note that if the equation $\mathbb{L}u=f$  has a solution $u$ then
$f$ must be  of the form
\[
f=\sum_{j=1}^n f_j(t,x)dt_j.
\]

The local solvability of this complex of operators was studied by
Treves in his seminal work \cite{T1}.

When each function $b_j\equiv0$, the global solvability was  treated
by Bergamasco and Petronilho in \cite{BP}. In this case the
system is globally solvable if and only if the real 1-form
$a(t)=\sum_{j=1}^n a_j(t)dt_j$ is either non-Liouville or rational
(see definition in \cite{BCM}).

When $c(t)$ is exact the problem was solved by Cardoso and Hounie in
\cite{CH}. In this case  the 1-form $c(t)$ has a global primitive
$C$ defined on $\mathbb{T}^n$ and  global solvability is equivalent
to the connectedness of all sublevels and superlevels of the real
function $Im(C)$.

We are interested in global solvability when at least one of the
functions $b_j\not\equiv0$ and $c(t)$ is not exact. Moreover, we
suppose that $Im(c)$ is exact and for each $j$, $a_j\equiv 0$ or
$b_j\equiv 0$.

We prove that system \eqref{L} is globally solvable if and only
if the real 1-form $a(t)$ is either non-Liouville or rational and
any primitive of the 1-form $b(t)$ has only connected sublevels and
superlevels on $\mathbb{T}^n$ (see Theorem \ref{main theorem}).

The articles \cite{BCP,BdMZ,BK,BKNZ,BNZ,H} deal with similar questions.

\section{Preliminaries and statement of the main result}

There are natural compatibility conditions on the 1-form
$f$ for the existence of a solution $u$ to the equation
 $\mathbb{L} u=f$. We now move on to describing them.

If $f\in C^\infty(\mathbb T^n_x\times\mathbb T^1_t;\wedge^{
1,0})$  we consider the $x$-Fourier series
$$
f(t,x)=\sum_{\xi\in \mathbb{Z}} \hat{f}(t,\xi)e^{i\xi x},
$$
where $ \hat{f}(t,\xi)=\sum_{j=1}^{n}\hat{f}_j(t,\xi)dt_j $ and
$\hat{f}_j(t,\xi)$ denotes the Fourier transform with respect to
$x$.

Since $b$ is exact there exists a function
$B\in C^{\infty}(\mathbb{T}_t^n;\mathbb{R})$ such that $d_tB=b$. Moreover,
we may write $a=a_0+d_t A$ where
$A\in C^{\infty}(\mathbb{T}_t^n;\mathbb{R})$ and
$a_0\in\wedge ^1\mathbb{R}^n\simeq \mathbb{R}^n$. Thus, we may write
$c(t)=a_0+d_t C$ where $C(t)=A(t)+iB(t)$.


We will identify the 1-form $a_0\in\wedge^1\mathbb{R}^n$ with the
vector $a_0:=(a_{10},\ldots,a_{n0})$ in $\mathbb{R}^n$
consisting of the periods of the 1-form $a$ given by
$$
a_{j0}=\frac{1}{2\pi}\int_{0}^{2\pi}a_j(0,\ldots,\tau_j,\ldots,0)d\tau_j.
$$
Thus, if $f\in C^\infty(\mathbb T^n_t\times\mathbb T^1_x;\wedge^{
1,0})$ and if there exists $u\in\mathcal{D}'(\mathbb{T}^{n+1})$ such
that $\mathbb{L} u=f $ then, since $\mathbb{L}$ defines a
differential complex, $\mathbb{L} f=0$ or equivalently $L_jf_k=L_k
f_j$, $j,k=1,\ldots,n$; also
\begin{equation}\label{condcompat}
\hat{f}(t,\xi)e^{i\xi(a_0\cdot t+C(t))}
\text{ is exact when $\xi a_{0}\in\mathbb{Z}$.}
\end{equation}
We define now the  set
\[
\mathbb{E}=\big\{f\in C^\infty(\mathbb T^n_t\times
\mathbb{T}^1_x;\wedge^{ 1,0});\;\mathbb{L}  f=0\text{ and
\eqref{condcompat} holds} \big\}.
\]

\begin{definition} \rm
The operator $\mathbb{L}$   is said to be globally solvable on
$\mathbb{T}^{n+1}$ if for each $f\in\mathbb{E}$ there exists
$u\in\mathcal{D}'(\mathbb{T}^{n+1})$ satisfying $\mathbb{L} u=f$.
\end{definition}

Given $\alpha\notin\mathbb{Q}^n$ we say that $\alpha$ is
\emph{Liouville} when there exists a constant $C>0$ such that for each
$N\in\mathbb{N}$ the inequality
$$
\max_{j=1,\ldots,n}\Big|\alpha_j-\frac{p_j}{q}\Big|\leq\frac{C}{q^{N}},
$$
has infinitely many solutions
$(p_1,\ldots,p_n,q)\in\mathbb{Z}^n\times \mathbb{N}$.

Let us consider the following two sets
$$
J=\{j\in\{1,\ldots,n\}; \;b_j\equiv0\},\quad
K=\{k\in\{1,\ldots,n\}; \;a_k\equiv0\};
$$
and we will write $J=\{j_1,\ldots,j_m\}$ and $K=\{k_1,\ldots,k_p\}$.
Under the above notation, the main result of this work is the
following theorem.

\begin{theorem}\label{main theorem}
Let $B$ be a global primitive of the 1-form $b$.
If $J\cup K=\{1,\ldots,n\}$ then the operator $\mathbb{L}$ given in
\eqref{operator} is globally solvable if and only if one of the
following two conditions holds:
\begin{itemize}
\item[(I)] $J\neq\emptyset$ and $(a_{j_10},\ldots,a_{j_m0})\notin\mathbb{Q}^{m}$
is non-Liouville.
\item[(II)] The sublevels $\Omega_s=\{t\in\mathbb{T}^n,\;B(t)<s\}$ and
 superlevels $\Omega^s=\{t\in\mathbb{T}^n,\;B(t)>s\}$ are connected for every
$s\in\mathbb{R}$ and $(a_{j_10},\ldots,a_{j_m0})\in\mathbb{Q}^{m}$
if $J\neq\emptyset$.
\end{itemize}
\end{theorem}

Note that if $J=\emptyset$ then $K=\{1,\ldots,n\}$
(since $J\cup K=\{1,\ldots,n\}$ by hypothesis). In this case each
$a_k\equiv0$ and Theorem \ref{main theorem} says that $\mathbb{L}$ is globally
solvable if and only if all the sublevels and superlevels of $B$ are
connected in $\mathbb{T}^n$, which is according to \cite{CH}.

When $J=\{1,\ldots,n\}$ we have that $b=0$, hence any primitive of
$b$ has only connected subleves and superlevels on $\mathbb{T}^n$.
In this case Theorem \ref{main theorem} says that $\mathbb{L}$ is
globally solvable if and only if either $a_0\notin\mathbb{Q}^n$  is
non-Liouville or $a_0\in\mathbb{Q}^n$, which was proved in
\cite{BP}. Thus, in order to prove Theorem \ref{main theorem} it
suffices to consider the following situation
$\emptyset\neq J\neq\{1,\ldots,n\}$.

\begin{remark} \rm
 As in \cite{BP}, the differential operator $\mathbb{L}$  is
globally solvable if and only if the differential operator
\begin{equation}\label{La0}
d_t+(a_0+ib(t))\wedge\frac{\partial}{\partial x}
\end{equation}
is globally solvable.
\end{remark}

Indeed, consider the automorphism
\begin{gather*}
S:\mathcal{D}'(\mathbb{T}^{n+1})
\longrightarrow \mathcal{D}'(\mathbb{T}^{n+1})\\
\sum_{\xi\in \mathbb{Z}}\hat{u}(t,\xi)e^{i \xi x}
\longmapsto
\sum_{\xi\in \mathbb{Z}}\hat{u}(t,\xi)e^{i\xi A(t)}e^{i\xi x},
\end{gather*}
where $A$ is the previous smooth real valued
function satisfying $d_t A=a(t)-a_0$. Observe that following
relation holds:
\[ % \label{conjugation S}
S\mathbb{L}
S^{-1}=d_t+(a_0+ib(t))\wedge\frac{\partial}{\partial x},
\]
 which ensures the above statement.

Therefore, it is sufficient to prove Theorem \ref{main theorem} for the
operator \eqref{La0}. For the rest of this article, we will denote
by  $\mathbb{L}$ the operator \eqref{La0}; that is,
\begin{equation}\label{23}
\mathbb{L}=d_t+(a_0+ib(t))\wedge\frac{\partial}{\partial x}
\end{equation}
and by $\mathbb{E}$ the corresponding space of compatibility
conditions. The new operator $\mathbb{L}$ is associated with the
vector fields
\begin{equation}\label{24}
L_j=\frac{\partial}{\partial
t_j}+(a_{j0}+ib_j(t))\frac{\partial}{\partial x},\quad j=1,\ldots,n.
\end{equation}

\section{Sufficiency part of  Theorem \ref{main theorem}}

First assume  that $(a_{j_10},\ldots,a_{j_m0})\notin\mathbb{Q}^m$ is
non-Liouville where
\[
J=\{j_1,\ldots,j_m\}:=\{j\in\{1,\ldots,n\},\;b_j\equiv0\}.
\]
Then, there exist a constant $C>0$ and an integer $N>1$ such that
\begin{equation}\label{naoliouville}
\max_{j\in J}|\xi a_{j0}-\kappa_j|\geq
\frac{C}{|\xi|^{N-1}},\quad \forall (\kappa,\xi)\in
\mathbb{Z}^{m}\times\mathbb{N}.
\end{equation}

Consider the set $I$ where $I\cup J=\{1,\ldots,n\}$ and
$I\cap J=\emptyset$. Remember that $\emptyset \neq J\neq \{1,\ldots,n\}$
then $I\neq\emptyset$ and $b_\ell\not\equiv0$ if $\ell\in I$.

We denote by $t_J$ the variables $t_{j_1},\ldots,t_{j_m}$ and by
$t_I$ the other variables on $\mathbb{T}_{t}^n$.
Let $f(t,x)=\sum_{j=1}^n f_j(t,x)dt_j\in\mathbb{E}$. Consider the
$(t_J,x)$-Fourier series as follows
\begin{equation}
u(t,x)=\sum_{(\kappa,\xi)\in\mathbb{Z}^{m}\times\mathbb{Z}}
\hat{u}(t_I,\kappa,\xi)e^{i(\kappa\cdot t_J+\xi x)}\label{coef uk}
\end{equation}
and for each $j=1,\ldots,n$,
\begin{equation}
f_j(t,x)=\sum_{(\kappa,\xi)\in\mathbb{Z}^{m}\times\mathbb{Z}}\hat{f}_j
(t_I,\kappa,\xi)e^{i(\kappa\cdot t_J+\xi x)},\label{coef fk}
\end{equation}
where $\kappa=(\kappa_{j_1},\ldots,\kappa_{j_m})\in\mathbb{Z}^m$ and
$\hat{u}(t_I,\kappa,\xi)$ and $\hat{f}_j(t_I,\kappa,\xi)$ denote the
Fourier transform with res\-pect to variables
$(t_{j_1},\ldots,t_{j_m},x)$.

Substituting the formal series \eqref{coef uk} and
\eqref{coef fk} in the equations $L_ju=f_j$, $j\in J$, we have for each
$(\kappa,\xi)\neq (0,0)$
\[
i(\kappa_j+\xi
a_{j0})\hat{u}(t_I,\kappa,\xi)=\hat{f}_j(t_I,\kappa,\xi),\quad j\in J.
\]

Also, from the compatibility conditions $L_jf_{\ell}=L_{\ell}f_j$,
for all $j,\ell\in J$, we obtain the  equations
\[
 (\kappa_j+\xi
a_{j0})\hat{f}_{\ell}(t_I,\kappa,\xi)=(\kappa_\ell+\xi
a_{\ell0})\hat{f}_j(t_I,\kappa,\xi),\;\;j,\ell\in J.
\]
By the preceding equations we have
\begin{equation}\label{u sol}
\hat{u}(t_I,\kappa,\xi)=\frac{1}{i(\kappa_M+\xi
a_{M0})}\hat{f}_M(t_I,\kappa,\xi),\quad (\kappa,\xi)\neq(0,0),
\end{equation}
where  $M\in J$, $M=M(\xi)$ is such that
$$
 |\kappa_M+\xi a_{M0}|=\max_{j\in J}|\kappa_j
+\xi a_{j0}|\neq0.
$$
If $(\kappa,\xi)=(0,0)$, since $\hat{f}(t_I,0,0)$ is exact, there
exists $v\in{C}^{\infty}(\mathbb{T}^{n-m}_{t_I})$ such that
$dv=\hat{f}(\cdot,0,0)$. Thus, we choose $\hat{u}(t_I,0,0)=v(t_I)$.


Given $\alpha\in\mathbb{Z}_+^{n-m}$  we obtain from
\eqref{naoliouville} and \eqref{u sol} the
inequality
\[
|\partial^{\alpha}\hat{u}(t_I,\kappa,\xi)|
\leq \frac{1}{C}|\xi|^{N-1}|
\partial^{\alpha}\hat{f}_M(t_I,\kappa,\xi)|.
\]
Since each $f_j$  is a smooth function we conclude that
$$
u(t,x)=\sum_{(\kappa,\xi)\in\mathbb{Z}^{m}\times
\mathbb{Z}}\hat{u}(t_I,\kappa,\xi)e^{i(\kappa\cdot
t_J+\xi x)}\in{C}^{\infty}(\mathbb{T}^{n+1}).
$$
By construction  $u$ is a  solution of
$$
L_{j}u=f_{j},\quad j\in J.
$$

Now, we will prove that $u$ is also a solution to the equations
$$
L_{\ell} u=f_\ell, \quad \ell\in I.
$$
Let $\ell\in I$. Given $(\kappa,\xi)\neq(0,0)$ by the compatibility
condition $L_M f_\ell=L_\ell f_M$ we have
\begin{equation}\label{comp1}
 i(\kappa_M+\xi a_{M0})\hat{f}_\ell(t_I,\kappa,\xi)
=\frac{\partial}{\partial t_\ell}\hat{f}_M(t_I,\kappa,\xi)-\xi
b_\ell(t)\hat{f}_M(t_I,\kappa,\xi).
\end{equation}
Therefore, \eqref{u sol} and \eqref{comp1} imply
\begin{align*}
&\frac{\partial}{\partial t_\ell}\hat{u}(t_I,\kappa,\xi)-\xi
 b_{\ell}(t)\hat{u}(t_I,\kappa,\xi)\\
&= \frac{1}{i(\kappa_M+\xi
 a_{M0})}\frac{\partial} {\partial
 t_\ell}\hat{f}_M(t_I,\kappa,\xi)-\xi
 b_\ell(t)\frac{1}{i(\kappa_M+\xi
 a_{M0})}\hat{f}_M(t_I,\kappa,\xi)\\
&= \frac{1}{i(\kappa_M+\xi
 a_{M0})}\Big(\frac{\partial}{\partial
 t_\ell}\hat{f}_M(t_I,\kappa,\xi)-\xi
 b_\ell(t)\hat{f}_M(t_I,\kappa,\xi)\Big)\\
&=  \hat{f}_\ell(t_I,\kappa,\xi).
\end{align*}

If $(\kappa,\xi)=(0,0)$ then $\frac{\partial}{\partial
t_\ell}\hat{u}(t_I,0,0)=\hat{f}_\ell(t_I,0,0)$.

We have thus proved that condition (I) implies global solvability.

Suppose now that the condition (II) holds. Let $q_J$ be the
smallest positive integer such that
$q_J(a_{j_10},\ldots,a_{j_m0})\in\mathbb{Z}^m$.

We denote by  $\mathcal{A}:= {{q}_{J}}\mathbb{Z}$ and
$\mathcal{B}:= \mathbb{Z}\backslash \mathcal{A}$  and define
\[
\mathcal{D}_\mathcal{A}'(\mathbb{T}^{n+1}):= \big\{u\in
\mathcal{D}'(\mathbb{T}^{n+1});\quad u(t,x)=\sum_{\xi\in
\mathcal{A}}\hat{u}(t,\xi)e^{i\xi x}\big\}.
\]
Let $\mathbb{L}_{\mathcal{A}}$ be the operator $\mathbb{L}$ acting
on $\mathcal{D}_\mathcal{A}'(\mathbb{T}^{n+1})  $. Similarly, we
define $\mathcal{D}_\mathcal{B}'(\mathbb{T}^{n+1})$ and
$\mathbb{L}_{\mathcal{B}}$.

Then  $\mathbb{L}$ is globally solvable if and only if
$\mathbb{L}_{\mathcal{A}} $ and $\mathbb{L}_{\mathcal{B}} $ are
globally solvable (see \cite{BCP}).

\begin{lemma}\label{lemma q=qJ}
The operator $\mathbb{L}_{\mathcal{A}}$ is globally solvable.
\end{lemma}

\begin{proof}
 Since ${{q}_{J}}a_0\in\mathbb{Z}^n$, we define
\begin{gather*}
T:\mathcal{D}_\mathcal{A}'(\mathbb{T}^{n+1})
\longrightarrow \mathcal{D}_\mathcal{A}'(\mathbb{T}^{n+1})\\
\sum_{\xi \in \mathcal{A}}\hat{u}(t,\xi )e^{i \xi x}
 \longmapsto \sum_{\xi \in \mathcal{A}}\hat{u}(t,\xi )
 e^{-i\xi a_0\cdot t}e^{i \xi x}.
\end{gather*}
Note that $T$ is an  automorphism of
$\mathcal{D}_\mathcal{A}'(\mathbb{T}^{n+1})$ (and of
${C}^{\infty}_{\mathcal{A}}(\mathbb{T}^{n+1})$). Furthermore the
following relation holds:
\begin{equation} \label{relation2}
T^{-1}\mathbb{L}_{\mathcal{A}} T=\mathbb{L}_{0,\mathcal{A}},
\end{equation}
where $\mathbb{L}_0:= d_t+ib(t)\wedge{\frac{\partial}{\partial x}}$.

Let $B$ be a global primitive of $b$ on $\mathbb T^n$. Since all the
sublevels and superlevels of $B$ are connected in $\mathbb{T}^n$, by
work \cite{BP} we have $\mathbb{L}_0$ globally solvable, hence
$\mathbb{L}_{0,\mathcal{A}}$ is globally solvable. Since $T$ is an
automorphism, from equality \eqref{relation2} we obtain that
$\mathbb{L}_\mathcal{A}$ is globally solvable.
\end{proof}

If ${q}_J=1$ then $\mathcal{A}=\mathbb Z$ and the proof is complete.
Otherwise we have:
\begin{lemma}
The operator $\mathbb{L}_{\mathcal{B}}$ is globally solvable.
\end{lemma}

\begin{proof}
Let $(\kappa,\xi)\in \mathbb{Z}^m\times \mathcal{B}$. Since
${{q}_{J}}$ is defined as the smallest natural such that
${{q}_{J}}(a_{j_10},\ldots,a_{j_m0})\in\mathbb{Z}^m$, there exists
$\ell\in J$ such that
\[
\big|a_{\ell0}-\frac{\kappa_\ell}{\xi}\big|\geq \frac{C}{{|\xi|}},
\]
 where $C=1/{q}_{J}$. Therefore
$$
\max_{j\in
J}\big|a_{j0}-\frac{\kappa_j}{\xi}\big|
\geq\big|a_{\ell0}-\frac{\kappa_\ell}{\xi}\big|
\geq \frac{C}{|\xi|},\quad  (\kappa,\xi)\in \mathbb{Z}^m\times
\mathcal{B}.
$$
Note that if the denominators $\xi\in \mathcal{B}$ then
$(a_{j_10},\ldots,a_{j_m0})$ behaves as  non-Liouville. Thus, the
rest of the proof is analogous to the case where
$(a_{j_10},\ldots,a_{j_m0})$  is non-Liouville.
\end{proof}

\section{Necessity part of Theorem \ref{main theorem}}

Assume first that $(a_{j_10},\ldots,a_{j_m0})\in\mathbb{Q}^m$ and
the global primitive $B:\mathbb{T}^n\rightarrow \mathbb{R}$ of $b$
has a disconnected sublevel or superlevel on $\mathbb{T}^n$.

By Lemma \ref{lemma q=qJ} we have that $\mathbb{L}_{\mathcal{A}}$
is globally solvable if and only if $\mathbb{L}_{0,\mathcal{A}}$ is
globally solvable, where $\mathcal{A}={{q}_{J}}\mathbb{Z}$ and
$\mathbb{L}_{0}=d_t+ib(t)\wedge\frac{\partial}{\partial x}$. Since
$B$ has a disconnected  sublevel or superlevel, we have
$\mathbb{L}_{0,\mathcal{A}}$ not globally solvable by \cite{CH}.
Therefore $\mathbb{L}$ is not globally solvable.

Suppose now that $(a_{j_10},\ldots,a_{j_m0})\notin\mathbb{Q}^m$ is
Liouville. Therefore, by  work \cite{BP} the involutive system
$\mathbb{L}_{J}$  generated by the vector fields
\begin{equation}
 L_j=\frac{\partial}{\partial t_j}+a_{j0}\frac{\partial}{\partial
 x}\label{LP}, \quad j\in J=\{j_1,\ldots,j_m\},
\end{equation}
is not globally solvable on $\mathbb{T}^{m+1}$.

As in the sufficiency part, we will consider the set $I$ such that
 $J\cup I=\{1,\ldots,n\}$ and $J\cap I=\emptyset$.

Consider the space of compatibility conditions $\mathbb{E}_{J}$
associated to $\mathbb{L}_J$. Since \eqref{LP} is not globally
solvable on $\mathbb{T}^{m+1}$ there exists $g(t_J,x)=\sum_{j\in J}
g_j(t_J,x)dt_j\in\mathbb{E}_{J}$ such that
$$
\mathbb{L}_{J} v=g
$$
 has no solution  $v\in \mathcal{D}'(\mathbb{T}^{m+1})$.

Now, we  define smooth functions $f_1,\ldots,f_n$ on
$\mathbb{T}^{n+1}$ such that $f=\sum_{j=1}^nf_jdt_j\in\mathbb{E}$
and $\mathbb{L} u= f$ has no solution
$u\in\mathcal{D}'(\mathbb{T}^{n+1})$.

Let $B$ be a primitive of the 1-form $b$. Thus, we have
$\frac{\partial}{\partial t_j}B=b_j$. Since for each $j\in J$ the
function $b_j\equiv0$ then $B$ depends only on  the variables $t_I$;
that is, $B=B(t_I)$.

For $\ell\in I$ we choose $f_\ell\equiv 0$  and for $j\in J$ we
define
$$
f_j(t,x):=\sum_{\xi\in\mathbb{Z}}\hat{f}_j(t,\xi)e^{i\xi x},\;
$$
where
\[
\hat{f}_j(t,\xi):=
\begin{cases}
\hat{g}_j(t_J,\xi)e^{\xi(B(t_I)-M)} & \text{if $\xi\geq0$}\\
\hat{g}_j(t_J,\xi)e^{\xi(B(t_I)-\mu)} & \text{if $\xi<0$},
\end{cases}
\]
where $M$ and $\mu$ are, respectively, the maximum and minimum of $B$ over
$\mathbb{T}^n$.


Given $\alpha\in\mathbb{Z}_+^n$, for each $j\in J$ we obtain
$$
\partial^{\alpha}\hat{f}_j(t,\xi)=[\partial^{\alpha_J}
g_j(t_J,\xi) ]\xi^{|\alpha_I|}[\partial^{\alpha_I}B(t_I)]e^{\xi(B(t_I)-M)},
\quad \xi\geq0,
$$
and
$$
\partial^{\alpha}\hat{f}_j(t,\xi)=[\partial^{\alpha_J}
g_j(t_J,\xi) ]\xi^{|\alpha_I|}[\partial^{\alpha_I}B(t_I)]e^{\xi(B(t_I)-\mu)},
\quad\xi<0,
$$
where $|\alpha_I|:=\sum_{i\in I}\alpha_i$. Since the derivatives
of $B$ are bounded on $\mathbb{T}^{n}$ then, there exists a constant
$C_{\alpha}>0$ such that $|\partial^{\alpha_I}B(t_I)|\leq
C_{\alpha}$ for all $t_I\in\mathbb{T}_{t_I}^{n-m}$. Therefore,
$$
|\partial^{\alpha}\hat{f}_j(t,\xi)|\leq
C_{\alpha}|\xi|^{|\alpha_I|}|\partial^{\alpha_J} g_j(t_J,\xi)
|,\quad \xi\in\mathbb{Z}.
$$

Since $g_j$ are smooth functions it is possible to conclude by the
above inequality  that  $f_j$, $j\in J$, are smooth functions.
Moreover, it is easy to check that
$f=\sum_{j=1}^{n}f_jdt_j\in\mathbb{E}$.

Suppose that there exists $u\in\mathcal{D}'(\mathbb{T}^{n+1})$ such
that $\mathbb{L} u=f$. Then, if
$u(t,x)=\sum_{\xi\in\mathbb{Z}}\hat{u}(t,\xi)e^{i\xi x}$, for
each $\xi\in\mathbb{Z}$  we have
\begin{equation}\label{eq
prov1}\frac{\partial}{\partial t_j}\hat{u}(t,\xi)+i\xi
a_{j0}\hat{u}(t,\xi)=\hat{f}_j(t,\xi), \quad\text{$j\in J$}
\end{equation}
and
\begin{equation} \label{eq
prov2}\frac{\partial}{\partial t_\ell}\hat{u}(t,\xi)-\xi
b_\ell(t)\hat{u}(t,\xi)=0, \quad\text{$\ell\in I$}
\end{equation}
Thus, for each $\ell\in I$  we may write \eqref{eq prov2} as follows
\begin{gather*}
\frac{\partial}{\partial t_\ell}\big(\hat{u}(t,\xi)e^{-\xi(B(t_I)-M)}\big)
= 0,\quad \text{if }\xi\geq0,\\
\frac{\partial}{\partial
t_\ell}\big(\hat{u}(t,\xi)e^{-\xi(B(t_I)-\mu)}\big)
= 0,\quad \text{if }\xi<0.
\end{gather*}
Therefore,
\begin{equation}\label{eq prov3}
\begin{gathered}
\hat{u}(t,\xi)e^{-\xi(B(t_I)-M)}:=\varphi_\xi(t_J),\quad
\xi\geq0,\\
\hat{u}(t,\xi)e^{-\xi(B(t_I)-\mu)}:=\varphi_\xi(t_J),\quad
\xi<0.
\end{gathered}
\end{equation}

Let ${t_I}^{*}$ and ${t_I}_*$ such that $B({t_I}^{*})=M$ and
$B({t_I}_*)=\mu$. Thus, $\varphi_\xi(t_J)=\hat{u}(t_J,{t_I}^*,\xi)$
if $\xi\geq0$ and $\varphi_\xi(t_J)=\hat{u}(t_J,{t_I}_*,\xi)$ if
$\xi<0$ for all $t_J$. Since $u\in\mathcal{D}'(\mathbb{T}^{n+1})$
we have
\begin{equation}\label{eq solution}
v(t_J,x):=\sum_{\xi\in \mathbb{Z}}\varphi_\xi(t_J)e^{i\xi
x}\in\mathcal{D}'(\mathbb{T}^{m+1}).
\end{equation}
On the other hand, by \eqref{eq prov1} and \eqref{eq prov3} we have
for each $j\in J$
\begin{gather*}
\frac{\partial}{\partial t_j}(\varphi_\xi(t_J)e^{\xi(B(t_I)-M)})+i\xi
a_{j0}(\varphi_\xi(t_J)e^{\xi(B(t_I)-M)})
=\hat{f}_j(t,\xi),\quad\xi\geq0,
\\
\frac{\partial}{\partial
t_j}(\varphi_\xi(t_J)e^{\xi(B(t_I)-\mu)})+i\xi
a_{j0}(\varphi_\xi(t_J)e^{\xi(B(t_I)-\mu)})
=\hat{f}_j(t,\xi),\quad\xi<0,
\end{gather*}
thus
\begin{gather*}
\frac{\partial}{\partial t_j}\varphi_\xi(t_J)+i\xi
a_{j0}\varphi_\xi(t_J) =\hat{g}_j(t_J,\xi),\quad\xi\in\mathbb{Z},
\;j\in J.
\end{gather*}
We conclude that the $v$ given by \eqref{eq solution} is a solution of
$\mathbb{L}_{J}v=g$, which is a contradiction.
%\end{proof}

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\end{document}