\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 $$\label{L} L_j=\frac{\partial}{\partial t_j}+(a_j+ib_j)(t)\frac{\partial}{\partial x}, \quad j=1,\ldots,n,$$ 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 $$\label{operator} \mathbb{L}u=d_tu+c(t)\wedge\frac{\partial}{\partial x}u,\quad u\in C^\infty(\mathbb T^{n+1}) \quad \text{or }\mathcal D'(\mathbb{T}^{n+1})),$$ 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 $$\label{condcompat} \hat{f}(t,\xi)e^{i\xi(a_0\cdot t+C(t))} \text{ is exact when \xi a_{0}\in\mathbb{Z}.}$$ 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 $\Omega_s=\{t\in\mathbb{T}^n,\;B(t)s\}$ 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 $$\label{La0} d_t+(a_0+ib(t))\wedge\frac{\partial}{\partial x}$$ 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, $$\label{23} \mathbb{L}=d_t+(a_0+ib(t))\wedge\frac{\partial}{\partial x}$$ and by $\mathbb{E}$ the corresponding space of compatibility conditions. The new operator $\mathbb{L}$ is associated with the vector fields $$\label{24} L_j=\frac{\partial}{\partial t_j}+(a_{j0}+ib_j(t))\frac{\partial}{\partial x},\quad j=1,\ldots,n.$$ \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 $$\label{naoliouville} \max_{j\in J}|\xi a_{j0}-\kappa_j|\geq \frac{C}{|\xi|^{N-1}},\quad \forall (\kappa,\xi)\in \mathbb{Z}^{m}\times\mathbb{N}.$$ 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 $$u(t,x)=\sum_{(\kappa,\xi)\in\mathbb{Z}^{m}\times\mathbb{Z}} \hat{u}(t_I,\kappa,\xi)e^{i(\kappa\cdot t_J+\xi x)}\label{coef uk}$$ and for each $j=1,\ldots,n$, $$f_j(t,x)=\sum_{(\kappa,\xi)\in\mathbb{Z}^{m}\times\mathbb{Z}}\hat{f}_j (t_I,\kappa,\xi)e^{i(\kappa\cdot t_J+\xi x)},\label{coef fk}$$ 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 $$\label{u sol} \hat{u}(t_I,\kappa,\xi)=\frac{1}{i(\kappa_M+\xi a_{M0})}\hat{f}_M(t_I,\kappa,\xi),\quad (\kappa,\xi)\neq(0,0),$$ 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 $$\label{comp1} i(\kappa_M+\xi a_{M0})\hat{f}_\ell(t_I,\kappa,\xi) =\frac{\partial}{\partial t_\ell}\hat{f}_M(t_I,\kappa,\xi)-\xi b_\ell(t)\hat{f}_M(t_I,\kappa,\xi).$$ 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: $$\label{relation2} T^{-1}\mathbb{L}_{\mathcal{A}} T=\mathbb{L}_{0,\mathcal{A}},$$ 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 $$L_j=\frac{\partial}{\partial t_j}+a_{j0}\frac{\partial}{\partial x}\label{LP}, \quad j\in J=\{j_1,\ldots,j_m\},$$ 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 $$\label{eq prov1}\frac{\partial}{\partial t_j}\hat{u}(t,\xi)+i\xi a_{j0}\hat{u}(t,\xi)=\hat{f}_j(t,\xi), \quad\text{j\in J}$$ and $$\label{eq prov2}\frac{\partial}{\partial t_\ell}\hat{u}(t,\xi)-\xi b_\ell(t)\hat{u}(t,\xi)=0, \quad\text{\ell\in I}$$ 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, $$\label{eq prov3} \begin{gathered} \hat{u}(t,\xi)e^{-\xi(B(t_I)-M)}:=\varphi_\xi(t_J),\quad \xi\geq0,\\ \hat{u}(t,\xi)e^{-\xi(B(t_I)-\mu)}:=\varphi_\xi(t_J),\quad \xi<0. \end{gathered}$$ 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 $$\label{eq solution} v(t_J,x):=\sum_{\xi\in \mathbb{Z}}\varphi_\xi(t_J)e^{i\xi x}\in\mathcal{D}'(\mathbb{T}^{m+1}).$$ 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} \begin{thebibliography}{00} \bibitem{BCH} S. Berhanu, P. D. Cordaro, J. 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