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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 116, pp. 1--43.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/116\hfil Sufficient conditions of local solvability]
{Sufficient conditions of local solvability for partial
differential operators on spaces of Colombeau type}

\author[C. Garetto\hfil EJDE-2009/116\hfilneg]
{Claudia Garetto}

\address{Claudia Garetto \newline
Institut f\"ur Grundlagen der Bauingenieurwissenschaften\\
Leopold-Franzens Universit\"at Innsbruck\\
Technikerstrasse 13, 6020 Innsbruck, Austria}
\email{claudia@mat1.uibk.ac.at}

\thanks{Submitted June 22, 2009. Published September 21, 2009.}
\thanks{Supported by grant T305-N13 from FWF, Austria.}
\subjclass[2000]{46F30, 35D99}
\keywords{Algebras of generalized functions; \hfill\break\indent
 generalized solutions of partial differential equations}

\begin{abstract}
 We provide sufficient conditions of local solvability for partial
 differential operators with variable Colombeau coefficients. We
 mainly concentrate on operators which admit a right generalized
 pseudodifferential parame\-trix and on operators which are a bounded
 perturbation of a differential operator with constant Colombeau
 coefficients. The local solutions are intended in the Colombeau
 algebra $\mathcal{G}(\Omega)$ as well as in the dual
 $\mathcal{L}(\mathcal{G}_c(\Omega),\widetilde{\mathbb{C}})$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\newcommand {\lrangle}[1]{\langle#1\rangle}

\section{Introduction}

Colombeau algebras of generalized functions \cite{Colombeau:85,
GKOS:01} have proved to be a well-organized  and powerful
framework where to solve linear and nonlinear partial differential
equations involving non-smooth coefficients and strongly singular
data. So far, the purpose of many authors has been to find for a
specific problem of applicative relevance the most suitable
Colombeau framework where first to provide solvability and second
to give a qualitative description of the solutions. We recall that
several results of existence and uniqueness of the solution have
been obtained in this generalized context for hyperbolic Cauchy
problems with singular coefficients and initial data \cite{GH:03,
HdH:01, HdH:01c, LO:91, O:89}, for elliptic and hypoelliptic
equations \cite{GGO:03, HO:03, HOP:05} and for divergent type
quasilinear Dirichlet problems with singularities
\cite{PilSca:06}. The setting of generalized functions employed is
the Colombeau algebra $\mathcal{G}(\Omega)$ constructed on an open subset
$\Omega$ of $\mathbb{R}^n$, or more in general the Colombeau space $\mathcal{G}_E$ of
generalized functions based on a locally convex topological vector
space $E$ (see \cite{Garetto:05b, Garetto:05a} for definitions and
properties). For instance in \cite{BO:92, HO:03} solvability is
provided in the Colombeau space based on
$H^\infty(\mathbb{R}^n)=\cap_{s\in\mathbb{R}}H^s(\mathbb{R}^n)$. Recently, in order to
enlarge the family of generalized hyperbolic problems which can be
solved and in order to provide a more refined microlocal
investigation of the qualitative properties of the solution, the
dual $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ has replaced the classical Colombeau
setting $\mathcal{G}(\Omega)$ \cite{O:07}.

This paper is devoted to the longstanding general problem  of
solvability or more precisely local solvability in the Colombeau
context for partial differential operators with Colombeau
coefficients. Namely, it is the starting point of a challenging
project which aims to discuss and fully understand solvability and
local solvability of partial differential operators in the
Colombeau context.

Instead of dealing with a specific equation $P(x,D)u=v$ and
looking  for a new setting of generalized functions tailored to
this particular problem, we want to determine a class of locally
solvable partial differential operators. This will be done by
finding some sufficient conditions on $P$ of local solvability in
the Colombeau context $\mathcal{G}(\Omega)$ or $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$. As in
the classical theory of partial differential operators with smooth
coefficients, mainly developed by H\"ormander in
\cite{Hoermander:63, Hoermander:V1-4}, different mathematical
methods and level of technicalities concern the investigation of
solvability when the coefficients are constant or not. The
Malgrange-Ehrenpreis theorem essentially reduces the solvability
issue to the search for a fundamental solution in the constant
coefficients case but clearly this powerful tool loses efficiency
when the coefficients are variable. In this situation indeed, not
only the structural properties of the operator but also the
geometric features of the set $\Omega$ where we want to solve the
equation play a relevant role in stating existence theorems of
local or global solvability.

Differential operators with constant Colombeau coefficients; i.e.
coefficients in the ring $\widetilde{\mathbb{C}}$ of complex generalized numbers,
have been studied by various authors \cite{Garetto:08b,
Garetto:08c, HO:03}. In particular a notion of fundamental
solution has been introduced in \cite{Garetto:08b} as a functional
in the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ providing, by means of a
generalized version of the Malgrange-Ehrenpreis theorem, a
straightforward result of solvability in the Colombeau context. In
detail, a solution to the equation $P(D)u=v$,
$P(D)=\sum_{|\alpha|\le m}c_\alpha D^\alpha$ with
$c_\alpha\in\widetilde{\mathbb{C}}$ has been obtained via convolution of the
right hand side $v$ with a fundamental solution $E$ and certain
regularity qualities of the operator $P(D)$, the $\mathcal{G}$- and
$\mathcal{G}^\infty$-hypoellipticity for instance, have been proven to be
equivalent to some structural properties of its fundamental
solutions \cite[Theorems 3.6, 4.2]{Garetto:08c}.

In this paper we concentrate on differential operators with
variable Colombeau coefficients, i.e. $P(x,D)=\sum_{|\alpha|\le
m}c_\alpha(x)D^\alpha$. Being aware of the objective difficulty of
investigating solvability in wide generality, we fix our attention
on two classes of operators: the operators which are approximately
invertible, in the sense that they admit a right generalized
pseudodifferential parametrix, and the operators which are locally
a bounded perturbation of a differential operator with constant
Colombeau coefficients. In both these cases we will formulate
sufficient conditions of solvability which will require suitable
assumptions on the moderateness properties of the coefficients and
the right-hand side. Note that the symbolic calculus for
generalized pseudodifferential operators developed in
\cite{Garetto:04, Garetto:ISAAC07, GGO:03} is essential for
studying the first class of operators whereas the theory of
fundamental solutions in the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ is
heavily used in finding a local Colombeau solution for operators
of bounded perturbation type. This paper can therefore be
considered a natural follow-up of \cite{Garetto:08b}.

We now describe the contents of the paper in more detail.
Section \ref{sec_basic} collects the needed background of
Colombeau  theory and recalls, for the advantage of the reader,
the results of solvability obtained in the generalized constant
coefficients case. Definition and properties of a fundamental
solution in the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ are the topic of
Subsection \ref{subsec_fund}. Inspired by the work of H\"ormander
in \cite[Section 10.4]{Hoermander:V2}, in Subsection
\ref{subsec_comp} we introduce an order relation between operators
with constant Colombeau coefficients in terms of the corresponding
weight functions. In other words, we make use of the weight
function $\widetilde{P}(\xi)=\big(\sum_{|\alpha|\le m}|\partial^\alpha
P(\xi)|^2)^{1/2}$ (with values in $\widetilde{\mathbb{R}}$) in order to
determine the differential operators which are stronger (or
weaker, respectively) than $P(D)$. By stating this notion in few
equivalent ways (Proposition \ref{prop_basic_2}) we prove that an
$m$-oder differential operator $P(D)$ with coefficients in
$\widetilde{\mathbb{C}}$ is stronger than any differential operator with
coefficients in $\widetilde{\mathbb{C}}$ of order less than or equal to
$m$ if and only if it is
$\mathcal{G}$-elliptic. Analogously, we prove that if $P(D)$ is of
principal type then it is stronger or better it dominates
(Definition \ref{def_domin}) any differential operator with order
les than or equal to $m-1$. These results of comparison among differential
operators with constant Colombeau coefficients will be used in
Sections \ref{sec_bounded} and \ref{sec_local}.

In Section \ref{sec_parametrix} we begin our investigation of
local  solvability in the Colombeau context by considering
differential operators with coefficients in $\mathcal{G}$ that admit a
right generalized pseudodifferential operator parametrix. Given
$P(x,D)$ this means that there exists a pseudodifferential
operator $q(x,D)$ such that
$P_\varepsilon(x,D)q_\varepsilon(x,D)=I+r_\varepsilon(x,D)$ holds at the level of
representatives with $(r_\varepsilon)_\varepsilon$ a net of regularizing
operators. The moderateness properties of the reminder term
$r_\varepsilon$ are crucial in determining for each $x_0\in\mathbb{R}^n$ a
sufficiently small neighborhood $\Omega$ such that the equation
$P(x,D)T=F$ on $\Omega$ is solvable in $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ for any
$L^2_{\rm{loc}}$-moderate functional $F$. Different notions of a
generalized hypoelliptic symbol have been introduced in the recent
past in \cite{Garetto:ISAAC07, GGO:03, GH:05}. They all assure the
existence of a generalized parametrix $q(x,D)$ but in general do
not guarantee the moderateness properties on the regularizing
operator $r(x,D)$ which are essential for the previous result of
local solvability. For this reason in Propositions
\ref{prop_sol_hyp}, \ref{prop_sol_hyp_2} and \ref{prop_sol_hyp_3}
we make use of a definition of generalized hypoelliptic symbol,
first presented in \cite{Garetto:04}, which is less general than
the ones considered in  \cite{Garetto:ISAAC07, GGO:03, GH:05}, but
that combining the right moderateness and regularity properties,
provides local solvability in $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ as well as in
$\mathcal{G}(\Omega)$ and $\mathcal{G}^\infty (\Omega)$.

Section \ref{sec_elliptic} deals with a special class of
differential  operators: the operators which are $\mathcal{G}$-elliptic in
a neighborhood of a point $x_0$. Since they have a generalized
hypoelliptic symbol they admit a local generalized parametrix and
from the statements of Section \ref{sec_parametrix} we easily
obtain results of local solvability. The most interesting fact is
that this locally solvable operators are actually a perturbation
of a differential operator with constant Colombeau coefficients,
namely the same operator evaluated at $x=x_0$. Using the concepts
of Subsection \ref{subsec_comp} we prove that a differential
operator $P(x,D)$ which is $\mathcal{G}$-elliptic in $x_0$ can be written
in the form \begin{equation} \label{BP_intro} P_0(D)+\sum_{j=1}^r
c_j(x)P_j(D), \end{equation} where $P_0(D)=P(x_0,D)$, the operators $P_j(D)$
have coefficients in $\widetilde{\mathbb{C}}$ and are all weaker than $P_0(D)$
and the generalized functions $c_j$ belong to the Colombeau
algebra $\mathcal{G}(\mathbb{R}^n)$. This fact motivates our interest for the wider
class of generalized differential operators which are locally a
bounded perturbation of a differential operator with constant
Colombeau coefficients as in \eqref{BP_intro}. The precise
definition and some first examples are the topic of Section
\ref{sec_bounded}.

In Section \ref{sec_local} we provide some sufficient conditions
of  local solvability for operators of bounded perturbation type
as defined in Section \ref{sec_bounded}. The local solutions are
obtained by using a fundamental solution in
$\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ of $P_0(D)$, the comparison between the
operators $P_j(D)$ and $P_0(D)$ and, at the level of
representatives, suitable estimates of $B_{p,k}$-moderateness.
Theorem \ref{theo_locsolv_easy} and Theorem \ref{theo_locsolv}
have H\"ormander's theorem of local solvability for operators of
constant strength (\cite[Theorem 7.3.1]{Hoermander:63},
\cite[Theorem 13.3.3]{Hoermander:V2}) as a blueprint.

The paper ends with a sufficient condition of local solvability
for  operators which are not necessarily of bounded perturbation
type or do not have a generalized parametrix. In Section
\ref{sec_pseudo}, inspired by \cite[Chapter 4]{SaintRaymond:91},
we prove that a certain Sobolev estimate from below on the adjoint
of a generalized pseudodifferential operator is sufficient to
obtain local solvability in the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$. The
proof has the interesting feature of using the theory of
generalized Hilbert $\widetilde{\mathbb{C}}$-modules developed in
\cite{GarVer:08} and in particular the projection theorem on an
internal subset. We finally give some examples of generalized
differential and pseudodifferential operators fulfilling this
sufficient condition.

\section{Colombeau theory and partial differential operators with
constant Colombeau coefficients} \label{sec_basic}

In this section we recall some basic notions of Colombeau theory
and, for the advantage of the reader, what has been proved in
\cite{Garetto:08b, HO:03} about solvability in the Colombeau
context of partial differential operators with generalized
constant coefficients.

\subsection{Basic notions of Colombeau theory}
Main sources of this subsection are
\cite{Colombeau:85, Garetto:05b, Garetto:05a, GGO:03, GH:05, GKOS:01}.
\paragraph{Nets of numbers.}
Before dealing with the major points of the Colombeau construction
we begin by recalling some definitions concerning elements of
$\mathbb{C}^{(0,1]}$.

A net $(u_\varepsilon)_\varepsilon$ in $\mathbb{C}^{(0,1]}$ is said to be
\emph{strictly nonzero} if there exist $r>0$ and $\eta\in(0,1]$ such
that $|u_\varepsilon|\ge \varepsilon^r$ for all $\varepsilon\in(0,\eta]$. The regularity
issues discussed in this paper will make use of the following
concept of \emph{slow scale net (s.s.n)}. A slow scale net is a net
$(r_\varepsilon)_\varepsilon\in\mathbb{C}^{(0,1]}$ such that
\[
\forall q\ge 0\, \exists c_q>0\, \forall\varepsilon\in(0,1]
\quad|r_\varepsilon|^q\le c_q\varepsilon^{-1}.
\]
\paragraph{Colombeau spaces based on $E$.}
Let $E$ be a locally convex topological vector space topologized
through the family of seminorms $\{p_i\}_{i\in I}$. The elements of
\begin{gather*}
\mathcal{M}_E := \{(u_\varepsilon)_\varepsilon\in E^{(0,1]}:\, \forall i\in I\,
 \exists N\in\mathbb{N}\quad p_i(u_\varepsilon)=O(\varepsilon^{-N}) \text{ as } \varepsilon\to 0\},\\
\mathcal{M}^\mathrm{sc}_E :=\{(u_\varepsilon)_\varepsilon\in E^{(0,1]}:\, \forall i\in I\, \exists
(\omega_\varepsilon)_\varepsilon \text{ s.s.n. }\quad p_i(u_\varepsilon)=O(\omega_\varepsilon)
 \text{ as } \varepsilon\to 0\},\\
\mathcal{N}_E := \{(u_\varepsilon)_\varepsilon\in E^{(0,1]}:\, \forall i\in I\,
\forall q\in\mathbb{N}\quad p_i(u_\varepsilon)=O(\varepsilon^{q}) \text{ as } \varepsilon\to 0\},
\end{gather*}
are called $E$-moderate, $E$-moderate of slow scale type and
$E$-negligible, respectively. We define the space of
\emph{generalized functions based on $E$} as the factor space
$\mathcal{G}_E := \mathcal{M}_E / \mathcal{N}_E$. The expression \emph{``of slow scale
type''} is used for the generalized functions of the factor space
$\mathcal{G}^\mathrm{sc}_E:=\mathcal{M}^\mathrm{sc}_E/\mathcal{N}_E$.  The elements of $\mathcal{G}_E$ are
equivalence classes for which we use the notation
$u=[(u_\varepsilon)_\varepsilon]$.

Let $\Omega$ be an open subset of $\mathbb{R}^n$. For coherence with the
notations already in use, we set $\mathcal{E}_M(\Omega)=\mathcal{M}_{\mathcal{E}(\Omega)}$,
$\mathcal{N}(\Omega)=\mathcal{N}_{\mathcal{E}(\Omega)}$, $\mathcal{E}_M=\mathcal{M}_{\mathbb{C}}$ and $\mathcal{N}=\mathcal{N}_{\mathbb{C}}$.
The Colombeau algebra $\mathcal{G}(\Omega)$, as originally defined in its full
version by Colombeau in \cite{Colombeau:85}, is obtained as the
space $\mathcal{G}_E$ with $E=\mathcal{E}(\Omega)$. Analogously, the rings $\widetilde{\mathbb{C}}$
and $\widetilde{\mathbb{R}}$ of complex and real generalized numbers are the
Colombeau spaces $\mathcal{G}_\mathbb{C}$ and $\mathcal{G}_\mathbb{R}$ respectively. $\widetilde{\mathbb{C}}$ is
also the set of constants of $\mathcal{G}(\mathbb{R}^n)$. The space of
distributions $\mathcal{D}'(\Omega)$ is embedded into $\mathcal{G}(\Omega)$ via
convolution with a mollifier (see \cite{GKOS:01} for more
details). Since $\mathcal{G}(\Omega)$ is a sheaf with respect to $\Omega$ one has
a notion of support for $u\in\mathcal{G}(\Omega)$ and a subalgebra $\mathcal{G}_{\rm c}(\Omega)$
of compactly supported generalized functions.

\paragraph{Regularity theory.}
Regularity theory in the Colombeau context as initiated in \cite{O:92}
is based on the subalgebra $\mathcal{G}^\infty (\Omega)$ of all elements $u$
of $\mathcal{G}(\Omega)$ having a representative $(u_\varepsilon)_\varepsilon$ belonging
to the set
\begin{align*}
\mathcal{E}_M^\infty(\Omega):=\big\{&(u_\varepsilon)_\varepsilon\in\mathcal{E}[\Omega]:\forall K\Subset\Omega\,
\exists N\in\mathbb{N}\, \forall\alpha\in\mathbb{N}^n,\\
&\sup_{x\in K}|\partial^\alpha u_\varepsilon(x)|=O(\varepsilon^{-N}) \text{ as $\varepsilon\to
0$}\big\}.
\end{align*}
$\mathcal{G}^\infty (\Omega)$ coincides with the factor space $\mathcal{E}_M^\infty(\Omega)/\mathcal{N}(\Omega)$
and by construction has the intersection property
$\mathcal{G}^\infty (\Omega)\cap \mathcal{D}'(\Omega)=\mathcal{C}^\infty(\Omega)$.

\paragraph{Topological theory of Colombeau spaces.}
The family of seminorms $\{p_i\}_{i\in I}$ on $E$ determines
a \emph{locally convex $\widetilde{\mathbb{C}}$-linear} topology on $\mathcal{G}_E$
(see \cite[Definition 1.6]{Garetto:05b}) by means of the
\emph{valuations}
\[
\mathrm{v}_{p_i}([(u_\varepsilon)_\varepsilon]):=\mathrm{v}_{p_i}((u_\varepsilon)_\varepsilon):=\sup\{b\in\mathbb{R}:
p_i(u_\varepsilon)=O(\varepsilon^b)\, \text{as $\varepsilon\to 0$}\}
\]
and the corresponding \emph{ultra-pseudo-seminorms}
$\{\mathcal{P}_i\}_{i\in I}$, where $\mathcal{P}_i(u)=\mathrm{e}^{-\mathrm{v}_{p_i}(u)}$.
For the sake of brevity we omit to report definitions and properties
of valuations and ultra-pseudo-seminorms in the abstract context
of $\widetilde{\mathbb{C}}$-modules. Such a theoretical presentation can be
found in \cite[Subsections 1.1, 1.2]{Garetto:05b}.
More in general a theory of topological and locally convex topological
$\widetilde{\mathbb{C}}$-modules has been developed in \cite{Garetto:05a}.
The Colombeau algebra $\mathcal{G}(\Omega)$ has the structure of a
Fr\'echet $\widetilde{\mathbb{C}}$-modules and $\mathcal{G}_{\rm c}(\Omega)$ is the inductive
limit of a family of Fr\'echet $\widetilde{\mathbb{C}}$-modules.
We recall that on $\widetilde{\mathbb{C}}$ the valuation and the
ultra-pseudo-norm obtained through the absolute value in $\mathbb{C}$ are
denoted by $\mathrm{v}_{\widetilde{\mathbb{C}}}$ and $|\cdot|_{\mathrm{e}}$ respectively.
\paragraph{The dual $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ and its basic functionals.}
$\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ is the set of all continuous
$\widetilde{\mathbb{C}}$-linear functionals on $\mathcal{G}_{\rm c}(\Omega)$. As proven
in \cite{Garetto:05b} it contains (via continuous embedding)
 both the algebras $\mathcal{G}^\infty (\Omega)$ and $\mathcal{G}(\Omega)$;
 i.e., $\mathcal{G}^\infty (\Omega)\subseteq\mathcal{G}(\Omega)\subseteq\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$.
The inclusion $\mathcal{G}(\Omega)\subseteq\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ is given
via integration ($u\to\big( v\to\int_\Omega u(x)v(x)dx\big)$,
for definitions and properties of the integral of a Colombeau
generalized functions see \cite{GKOS:01}).
A special subset of $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ is obtained by
requiring the so-called ``basic'' structure. In detail,
we say that $T\in\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ is basic
(or equivalently $T\in\mathcal{L}_{\rm b}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$) if there
exists a net $(T_\varepsilon)_\varepsilon\in\mathcal{D}'(\Omega)^{(0,1]}$ fulfilling
the following condition: for all $K\Subset\Omega$ there exist
$j\in\mathbb{N}$, $c>0$, $N\in\mathbb{N}$ and $\eta\in(0,1]$ such that
\[
\forall f\in\mathcal{D}_K(\Omega)\, \forall\varepsilon\in(0,\eta]\quad |T_\varepsilon(f)|\le
c\varepsilon^{-N}\sup_{x\in K,|\alpha|\le j}|\partial^\alpha f(x)|
\]
and $Tu=[(T_\varepsilon u_\varepsilon)_\varepsilon]$ for all $u\in\mathcal{G}_{\rm c}(\Omega)$.

Analogously one can introduce the dual $\mathcal{L}(\mathcal{G}(\Omega),\widetilde{\mathbb{C}})$
and the corresponding set $\mathcal{L}_{\rm b}(\mathcal{G}(\Omega),\widetilde{\mathbb{C}})$ of basic
functionals. As in distribution theory,
Theorem 1.2 in \cite{Garetto:05b} proves that $\mathcal{L}(\mathcal{G}(\Omega),\widetilde{\mathbb{C}})$
can be identified with the set of functionals in $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$
having compact support.
\paragraph{Generalized differential operators.} $\mathcal{G}(\Omega)$
is a differential algebra, in the sense that derivatives of any
order can be defined extending the corresponding distributional
ones. We can therefore talk of differential operators in
the Colombeau context or, for simplicity, of
\emph{generalized differential operators}. Clearly, a
differential operator with singular distributional coefficients
generates a differential operator in the Colombeau context by
embedding its coefficients in the Colombeau algebra. Let
\[
P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha,
\]
with $c_\alpha\in\mathcal{G}(\Omega)$ for all $\alpha$. Its symbol
\[
P(x,\xi)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha
\]
is a polynomial of order $m$ with coefficients in $\mathcal{G}(\Omega)$
and representatives
\[
P_\varepsilon(x,\xi)=\sum_{|\alpha|\le m}c_{\alpha,\varepsilon}(x)D^\alpha.
\]
The operator $P(x,D)$ maps $\mathcal{G}_{\rm c}(\Omega)$, $\mathcal{G}(\Omega)$
and $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$ into themselves respectively
and $\mathcal{G}^\infty (\Omega)$ into $\mathcal{G}^\infty (\Omega)$ if the coefficients
are $\mathcal{G}^\infty $-regular. When the coefficients are constant
($c_\alpha\in\widetilde{\mathbb{C}}$ for all $\alpha$) we use the notation $P(D)$.

\subsection{Fundamental solutions in $\mathcal{L}_{\rm b}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$}
\label{subsec_fund}

Let $P(D)$ be a partial differential operator of order $m$ with
coefficients in $\widetilde{\mathbb{C}}$. Any net of polynomials $(P_\varepsilon)_\varepsilon$
determined by a choice of representatives of the coefficients of
$P(D)$ is called a representative of the polynomial $P$. Consider
the weight function $\widetilde{P}:\mathbb{R}^n\to\widetilde{\mathbb{R}}$ defined by
\[
\widetilde{P}^2(\xi)=\sum_{|\alpha|\le m}|\partial^\alpha P(\xi)|^2.
\]
The arguments in \cite[(2.1.10)]{Hoermander:63} yield the
following  assertion: there exists $C>0$ depending only on $m$ and
$n$ such that for all $(P_\varepsilon)_\varepsilon$ the inequality
\begin{equation}
\label{est_Hoer} \widetilde{P_\varepsilon}(\xi+\eta)\le
(1+C|\xi|)^m\widetilde{P_\varepsilon}(\eta)
\end{equation}
is valid for all
$\xi,\eta\in\mathbb{R}^n$ and all $\varepsilon\in(0,1]$. When the function
$\widetilde{P}:\mathbb{R}^n\to\widetilde{\mathbb{R}}$ is invertible in some point $\xi_0$ of
$\mathbb{R}^n$ Lemma 7.5 in \cite{HO:03} proves that for all
representative $(P_\varepsilon)_\varepsilon$ of $P$ there exist $N\in\mathbb{N}$ and
$\eta\in(0,1]$ such that \begin{equation} \label{est_inv} \widetilde{P_\varepsilon}(\xi)\ge
\varepsilon^N(1+C|\xi_0-\xi|)^{-m}, \end{equation} for all $\xi\in\mathbb{R}^n$ and
$\varepsilon\in(0,\eta]$. This means that $\widetilde{P}$ is invertible in any
$\xi$ once it is invertible in some $\xi_0$. Note that the
constant $C>0$ is the same appearing in \eqref{est_Hoer} and
$\varepsilon^N$ comes from the invertibility in $\widetilde{\mathbb{R}}$ of
$\widetilde{P}(\xi_0)$. It is not restrictive to assume for some strictly
non-zero net $(\lambda_\varepsilon)_\varepsilon$ that
\[
\widetilde{P_\varepsilon}(\xi)\ge \lambda_\varepsilon(1+C|\xi_0-\xi|)^{-m},
\]
for all $\varepsilon\in(0,1]$.

In the sequel $\mathcal{K}$ is the set of tempered weight functions
introduced by H\"ormander in
\cite[Definition 2.1.1]{Hoermander:63};
 i.e., the set of all positive functions $k$ on $\mathbb{R}^n$ such that for some
constants $C>0$ and $N\in\mathbb{N}$ the inequality
\[
\label{ineq_weight}
k(\xi+\eta)\le (1+C|\xi|)^N k(\eta)
\]
holds for all $\xi,\eta\in\mathbb{R}^n$. Concerning the H\"ormander spaces
$B_{p,k}$ which follow, main references are
\cite{Hoermander:63, Hoermander:V2}. Typical example of a weight
function is $k(\xi)=\lrangle{\xi}^s=(1+|\xi|^2)^{s/2}$, $s\in\mathbb{R}$.

\begin{definition} \label{def_cla_bpk} \rm
If $k\in\mathcal{K}$ and $p\in[1,+\infty]$ we denote by $B_{p,k}(\mathbb{R}^n)$ the set of all distributions $w\in\mathscr{S}'(\mathbb{R}^n)$ such that $\widehat{w}$ is a function and
\[
\Vert w\Vert_{p,k}=(2\pi)^{-n}\Vert k\widehat{w}\Vert_p <   \infty.
\]
\end{definition}

$B_{p,k}(\mathbb{R}^n)$ is a Banach space with the norm introduced in
Definition \ref{def_cla_bpk}. We have
$\mathscr{S}(\mathbb{R}^n)\subset B_{p,k}(\mathbb{R}^n)\subset\mathscr{S}'(\mathbb{R}^n)$
(in a topological sense) and that $\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$ is dense
in $B_{p,k}(\mathbb{R}^n)$ for $p<\infty$.

The inequality \eqref{est_Hoer} says that $\widetilde{P_\varepsilon}$ is a
tempered weight function for each $\varepsilon$ so it is meaningful to
consider the sets $B_{\infty,\widetilde{P_\varepsilon}}(\mathbb{R}^n)$ of distributions
as we will see in the next theorem, proven in \cite{Garetto:08b}.

\begin{theorem} \label{theo_fund_P}
To every differential operator $P(D)$ with coefficients in
$\widetilde{\mathbb{C}}$ such that $\widetilde{P}(\xi)$ is invertible in some
$\xi_0\in\mathbb{R}^n$ there exists a fundamental solution
$E\in\mathcal{L}_{\rm b}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$. More precisely, to every $c>0$
and $(P_\varepsilon)_\varepsilon$ representative of $P$ there exists a
fundamental solution $E$ given by a net of distributions
$(E_\varepsilon)_\varepsilon$ such that
$E_\varepsilon/\cosh(c|x|)\in B_{\infty,\widetilde{P_\varepsilon}}(\mathbb{R}^n)$
and for all $\varepsilon$
\[
\Bigl\Vert \frac{E_\varepsilon}{\cosh(c|x|)}\Bigr\Vert
_{\infty,\widetilde{P_\varepsilon}}\le C_0,
\]
where the constant $C_0$ depends only on $n,m$ and $c$.
\end{theorem}

One sees in the proof of Theorem \ref{theo_fund_P}
(Proposition 3.5 and Theorem 3.3 in \cite{Garetto:08c}) that for
each $\varepsilon$ the distribution $E_\varepsilon$ is a fundamental solution
of the operator $P_\varepsilon(D)$. Theorem \ref{theo_fund_P} entails
the following solvability result.

\begin{theorem} \label{theom_solv_dual_1}
Let $P(D)$ be a partial differential operator with coefficients
in $\widetilde{\mathbb{C}}$ such that $\widetilde{P}$ is invertible in some
$\xi_0\in\mathbb{R}^n$. Then the equation
\begin{equation} \label{eq_1_P}
P(D)u=v
\end{equation}
\begin{itemize}
\item[(i)] has a solution $u\in\mathcal{G}(\mathbb{R}^n)$ if $v\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$,
\item[(ii)] has a solution $u\in\mathcal{G}^\infty (\mathbb{R}^n)$ if $v\in\mathcal{G}^\infty_{\rm c}(\mathbb{R}^n)$,
\item[(iii)] has a solution $u\in\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$
 if $v\in\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$,
\item[(iv)] has a solution $u\in\mathcal{L}_{\rm b}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$
 if $v\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.
\end{itemize}
\end{theorem}

Theorem \ref{theom_solv_dual_1} extends to the dual the solvability
result obtained in $\mathcal{G}$ by H\"ormann and Oberguggenberger
in \cite{HO:03}. A more detailed investigation of the properties
of $u$, which heavily makes use of the theory of $B_{p,k}$-spaces,
can be found in \cite[Appendix]{Garetto:08b}.

\begin{remark} \label{rem_HO_inv} \rm
The condition of invertibility of $\widetilde{P}$ in a point
$\xi_0$ of $\mathbb{R}^n$ turns out to be equivalent to the solvability
statement (i) of Theorem \ref{theom_solv_dual_1}.
More precisely, Theorem 7.8 in \cite{HO:03} shows that if $v$
is invertible in some point of $\Omega$ and the equation $P(D)u=v$
is solvable in $\mathcal{G}(\Omega)$ then $\widetilde{P}$ is invertible in some point
of $\mathbb{R}^n$. In the same paper the authors prove that the
invertibility of the principal symbol $P_m$ in some $\xi_0$ implies
the invertibility of $\widetilde{P}(\xi_0)$. The converse does not
hold as one can see from $P_\varepsilon(\xi)=a_\varepsilon\xi+i$, with
$a=[(a_\varepsilon)_\varepsilon]\neq 0$ real valued and not invertible.
The principal symbol $P_1$ is not invertible (in any point
of $\mathbb{R}^n$) but $\widetilde{P}^2(0)=1+a^2$ is invertible in $\widetilde{\mathbb{R}}$.
In the same way we have that the existence of an invertible
coefficient in the principal part of $P(D)$ is a sufficient but
not necessary condition for the invertibility of the weight
function $\widetilde{P}$. Note that there exist differential operators
where all the coefficients are not invertible which still have an
invertible weight function. An example is given by
$P_\varepsilon(\xi_1,\xi_2)=a_\varepsilon\xi_1+ib_\varepsilon\xi_2$, where
$a_\varepsilon=1$ if $\varepsilon=n^{-1}$, $n\in\mathbb{N}$, and $0$ otherwise,
and $b_\varepsilon=0$ if $\varepsilon=n^{-1}$, $n\in\mathbb{N}$, and $1$ otherwise.
The coefficients generated by $(a_\varepsilon)_\varepsilon$ and $(b_\varepsilon)_\varepsilon$
are clearly not invertible but $\widetilde{P_\varepsilon}^2(1,1)=2(a_\varepsilon^2+b_\varepsilon^2)=2$.
\end{remark}

\subsection{Comparison of differential operators with constant Colombeau coefficients}
\label{subsec_comp}

Inspired by \cite[Section 10.4]{Hoermander:V2} we introduce an
 order relation between operators with constant Colombeau
coefficients by comparing the corresponding weight functions.

\begin{definition} \label{def_ord_rel} \rm
Let $P(D)$ and $Q(D)$ be partial differential operators with
coefficients in $\widetilde{\mathbb{C}}$. We say that $P(D)$ is stronger than
$Q(D)$ ($Q(D)\prec P(D)$) if there exist representatives
$(P_\varepsilon)_\varepsilon$ and $(Q_\varepsilon)_\varepsilon$ and a moderate
net $(\lambda_\varepsilon)_\varepsilon$ such that
\[
\widetilde{Q_\varepsilon}(\xi)\le \lambda_\varepsilon \widetilde{P_\varepsilon}(\xi)
\]
for all $\xi\in\mathbb{R}^n$ and $\varepsilon\in(0,1]$
\end{definition}

In the sequel we collect some estimates valid for polynomials
with coefficients in $\mathbb{C}$ proven in
\cite[Theorem 10.4.1, Lemma 10.4.2]{Hoermander:V2}.
We recall that $\widetilde{Q}(\xi,t)$ denotes the function
$(\sum_\alpha |Q^{(\alpha)}(\xi)|^2t^{2|\alpha|})^{1/2}$
for $Q$ polynomial of degree less than or equal to $m$ in 
$\mathbb{R}^n$ and $t$ positive
 real number. Clearly $\widetilde{Q}(\xi,1)=\widetilde{Q}(\xi)$.

\begin{proposition} \label{prop_dom}
\quad
\begin{itemize}
\item[(i)] There exists a constant $C>0$ such that for every
polynomial $Q$ of degree less than or equal to $m$ in $\mathbb{R}^n$,
\begin{equation}
\label{Hoer_1}
\frac{\widetilde{Q}(\xi,t)}{C}\le \sup_{|\eta|<t}|Q(\xi+\eta)|\le C\widetilde{Q}(\xi,t)
\end{equation}
for all $\xi\in\mathbb{R}^n$ and $t>0$.
\item[(ii)] There exist constants $C'$ and $C''$ such that for all polynomials $P$ and $Q$ 
of degree less than or equal to $m$,
\begin{equation}
\label{Hoer_2}
C'\widetilde{P}(\xi)\widetilde{Q}(\xi)\le \widetilde{PQ}(\xi)\le C''\widetilde{P}(\xi)\widetilde{Q}(\xi)
\end{equation}
for all $\xi\in\mathbb{R}^n$.
\end{itemize}
\end{proposition}

\begin{proof}
The first assertion is Lemma 10.4.2 in \cite{Hoermander:V2}.
Concerning (ii) the second inequality is clear from Leibniz'rule.
 From \eqref{Hoer_1} for any polynomial $Q$ we find $\eta$
with $|\eta|\le 1$ such that
\begin{equation}
\label{est_dom}
\widetilde{Q}(\xi)\le C|Q(\xi+\eta)|\le C\widetilde{Q}(\xi+\eta)
\end{equation}
for all $\xi$. Since from \cite[(2.1.10)]{Hoermander:63}
there exists a constant $C_0$ depending only on $m$ and $n$ such
that $\widetilde{Q}(\xi+\theta)\le (1+C_0|\theta|)^m\widetilde{Q}(\xi)$ for
all $\xi,\theta\in\mathbb{R}^n$, we get
\[
C\widetilde{Q}(\xi+\eta)\le C(1+C_0|\eta|)^m\widetilde{Q}(\xi)\le C_1\widetilde{Q}(\xi),
\]
where the constant $C_1$ does not depend on $Q$. Hence
\[
|Q(\xi+\eta)|\ge \frac{\widetilde{Q}(\xi)}{C}\ge \frac{\widetilde{Q}(\xi+\eta)}{C_1}.
\]
Taylor's formula gives $Q(\xi+\eta+\theta)=Q(\xi+\eta)
+\sum_{\alpha\neq 0}\frac{Q^{(\alpha)}(\xi+\eta)}{\alpha!}\theta^\alpha$
and then, for $\eta$ chosen as above,
\begin{align*}
|Q(\xi+\eta+\theta)|
&\ge |Q(\xi+\eta)|-\sum_{\alpha\neq 0}
 \frac{|Q^{(\alpha)}(\xi+\eta)|}{\alpha!}|\theta^\alpha|\\
&\ge \frac{\widetilde{Q}(\xi+\eta)}{C_1}-\widetilde{Q}(\xi+\eta)
 \sum_{\alpha\neq 0}\frac{1}{\alpha!}|\theta^\alpha|\\
&\ge \frac{\widetilde{Q}(\xi+\eta)}{C_1}-\widetilde{Q}(\xi+\eta)C_2|\theta|\\
&=\widetilde{Q}(\xi+\eta)(\frac{1}{C_1}-C_2|\theta|)
\end{align*}
for all $|\theta|\le 1$, with $C_2$ independent of $Q$. It follows that
\[
|Q(\xi+\eta+\theta)|\ge\frac{1}{2C_1}\widetilde{Q}(\xi+\eta)
\]
for all $\xi\in\mathbb{R}^n$, for $|\eta|\le 1$ depending on $Q$ and for
all $\theta$ with $|\theta|\le (2C_1C_2)^{-1}$.
Writing $P$ as $PQ/Q$ we obtain
$|P(\xi+\eta+\theta)|\le 2C_1|PQ(\xi+\eta+\theta)|/\widetilde{Q}(\xi+\eta)$.
Concluding, from the first assertion, the property \eqref{est_Hoer}
of the polynomial weight functions and the bound from
below \eqref{est_dom}, we have, for some $\theta$ suitably smaller
than $\min(1,(2C_1C_2)^{-1})$ and $\eta$ depending on $Q$, the inequality
\[
\widetilde{P}(\xi)\le (1+C_0|\eta|)^m\widetilde{P}(\xi+\eta)
\le C_3|P(\xi+\eta+\theta)|
\le C_4\frac{|PQ(\xi+\eta+\theta)|}{\widetilde{Q}(\xi+\eta)}
\le C_5\frac{\widetilde{PQ}(\xi)}{\widetilde{Q}(\xi)}
\]
where the constants involved depend only on the order of the
polynomials $P$ and $Q$ and the dimension $n$.
\end{proof}

Proposition \ref{prop_dom} clearly holds for representatives
$(P_\varepsilon)_\varepsilon$ and $(Q_\varepsilon)_\varepsilon$ of generalized polynomials
with the constants $C$, $C'$ and $C''$ independent of $\varepsilon$.

\begin{proposition} \label{prop_basic_1}
Let $P(D), P_1(D), P_2(D), Q(D), Q_1(D)$ and $Q_2(D)$ be differential
operators with constant Colombeau coefficients.
\begin{itemize}
\item[(i)] If $Q_1(D)\prec P(D)$ and $Q_2(D)\prec P(D)$ then $a_1 Q_1(D)+a_2 Q_2(D)\prec P(D)$ for all $a_1,a_2\in\widetilde{\mathbb{C}}$.
\item[(ii)] If $Q_1(D)\prec P_1(D)$ and $Q_2(D)\prec P_2(D)$ then $Q_1Q_2(D)\prec P_1P_2(D)$.
\item[(iii)] $P(D)+aQ(D)\prec P(D)$ for all $a\in\widetilde{\mathbb{C}}$ if and only if $Q(D)\prec P(D)$.
\end{itemize}
\end{proposition}

\begin{proof}
(i) The first assertion is trivial.
(ii) Working at the level of representatives from Proposition \ref{prop_dom}(ii) we can write $\widetilde{Q_{1,\varepsilon}Q_{2,\varepsilon}}\le C''\widetilde{Q_{1,\varepsilon}}\widetilde{Q_{2,\varepsilon}}$. It follows
\[
\widetilde{Q_{1,\varepsilon}Q_{2,\varepsilon}}\le C''\lambda_{1,\varepsilon}\lambda_{2,\varepsilon}\widetilde{P_{1,\varepsilon}}\widetilde{P_{2,\varepsilon}}\le \frac{C''}{C'}\lambda_{1,\varepsilon}\lambda_{2,\varepsilon}\widetilde{P_{1,\varepsilon}P_{2,\varepsilon}},
\]
with $(\frac{C''}{C'}\lambda_{1,\varepsilon}\lambda_{2,\varepsilon})_\varepsilon$ moderate net.

(iii) One direction is clear. Indeed, since $P\prec P$ from the first
assertion of this proposition we have that if
$Q\prec P$ then $P+aQ\prec P$ for all $a\in\widetilde{\mathbb{C}}$.
Conversely, let $P+aQ\prec P$. From $-P\prec P$ and (i) we have
that $aQ\prec P$. Finally, choosing $a=1$ we obtain $Q\prec P$.
\end{proof}

The following necessary and sufficient conditions for $Q\prec P$
are directly obtained from the first assertion of
Proposition \ref{prop_dom}.

\begin{proposition} \label{prop_basic_2}
Let $Q(D)$ and $P(D)$ be differential operators with
constant Co\-lombeau coefficients. The following statements are equivalent:
\begin{itemize}
\item[(i)] $Q(D)\prec P(D)$;
\item[(ii)] there exist representatives $(Q_\varepsilon)_\varepsilon$ and $(P_\varepsilon)_\varepsilon$ and a moderate net $(\lambda_\varepsilon)_\varepsilon$ such that
\[
|Q_\varepsilon(\xi)|\le \lambda_\varepsilon \widetilde{P_\varepsilon}(\xi)
\]
for all $\xi\in\mathbb{R}^n$ and $\varepsilon\in(0,1]$;
\item[(iii)] there exist representatives $(Q_\varepsilon)_\varepsilon$ and $(P_\varepsilon)_\varepsilon$ and a moderate net $(\lambda'_\varepsilon)_\varepsilon$ such that
\[
\widetilde{Q_\varepsilon}(\xi,t)\le \lambda'_\varepsilon \widetilde{P_\varepsilon}(\xi,t)
\]
for all $\xi\in\mathbb{R}^n$, for all $\varepsilon\in(0,1]$ and for all $t\ge 1$.
\end{itemize}
\end{proposition}

\begin{proof}
The implications $(i)\Rightarrow(ii)$ and $(iii)\Rightarrow(i)$
are trivial. We only have to prove that (ii) implies (iii).
Proposition \ref{prop_dom}(i) yields, for $t\ge 1$,
\begin{align*}
\widetilde{Q_\varepsilon}(\xi,t)&\le C\sup_{|\eta|<t}|Q_\varepsilon(\xi+\eta)|\\
&\le C\lambda_\varepsilon\sup_{|\eta|<t}\widetilde{P_\varepsilon}(\xi+\eta)\\
&=C\lambda_\varepsilon\sup_{|\eta|<t}\widetilde{P_\varepsilon}(\xi+\eta,1)\\
&\le C^2\lambda_\varepsilon\sup_{|\eta|<t+1}\widetilde{P_\varepsilon}(\xi+\eta)\\
&\le C^3\lambda_\varepsilon \widetilde{P_\varepsilon}(\xi,t+1)\\
&\le C^3\lambda_\varepsilon (1+t^{-1})^m \widetilde{P_\varepsilon}(\xi,t).
\end{align*}
Hence, $\widetilde{Q_\varepsilon}(\xi,t)\le \lambda'_\varepsilon\widetilde{P_\varepsilon}(\xi,t)$
for all $\xi\in\mathbb{R}^n$, $t\ge 1$ and $\varepsilon\in(0,1]$ with
$\lambda'_\varepsilon=C^3\lambda_\varepsilon 2^m$.
\end{proof}

The $\mathcal{G}$-elliptic polynomials (see \cite[Section 6]{Garetto:08c})
and their corresponding differential operators can be
 characterized by means of the order relation $\prec$.
We recall that a polynomial $P(\xi)$ with coefficients in
$\widetilde{\mathbb{C}}$ is $\mathcal{G}$-elliptic (or equivalently the operator
$P(D)$ is $\mathcal{G}$-elliptic) if there exists a representative
$(P_{m,\varepsilon})_\varepsilon$ of $P_m$, a constant $c>0$ and $a\in\mathbb{R}$ such that
\begin{equation} \label{est_ellip}
|P_{m,\varepsilon}(\xi)|\ge c\varepsilon^a|\xi|^m
\end{equation}
for all $\varepsilon\in(0,1]$ and for $\xi\in\mathbb{R}^n$.
 Estimate \eqref{est_ellip} is valid for any representative of $P$
with some other constant $c>0$ and on a smaller interval $(0,\varepsilon_0]$.
Due to the homogeneity of $P_{m,\varepsilon}$ it is not restrictive to
assume \eqref{est_ellip} valid only for all $\xi$ with $|\xi|=1$.

\begin{proposition} \label{prop_g_ellip}
Let $P(D)$ be a differential operator of order $m$ with coefficients
in $\widetilde{\mathbb{C}}$. $P(D)$ is stronger than any differential operator
with coefficients in $\widetilde{\mathbb{C}}$ of order less than or equal to
$m$ if and only if
it is $\mathcal{G}$-elliptic.
\end{proposition}

\begin{proof}
We assume that $P(D)$ is $\mathcal{G}$-elliptic and we prove that
the $\mathcal{G}$-ellipticity is a sufficient condition.
Let $(P_{m,\varepsilon})_\varepsilon$ be a representative of $P_m$ such that
$|P_{m,\varepsilon}(\xi)|\ge c\varepsilon^a|\xi|^m$ for some constants $c>0$
and $a\in\mathbb{R}$, for all $\xi\in\mathbb{R}^n$ and for all $\varepsilon\in(0,1]$.
It follows that $P_m(\xi)$ is invertible in any $\xi=\xi_0$ of $\mathbb{R}^n$
 and therefore from Remark \ref{rem_HO_inv} (and more precisely
from \cite[Proposition 7.6]{HO:03}) we have that $\widetilde{P}$ is
invertible in $\xi_0$.
The estimate $|P_{m,\varepsilon}(\xi)|\ge c\varepsilon^a|\xi|^m$ yields
\[
c\varepsilon^a|\xi|^m\le |P_{m,\varepsilon}(\xi)|\le |P_\varepsilon(\xi)|
+|P_\varepsilon(\xi)-P_{m,\varepsilon}(\xi)|\le |P_\varepsilon(\xi)|+C_\varepsilon(1+|\xi|^{m-1}),
\]
where $(c_\varepsilon)_\varepsilon$ is a strictly nonzero net. Assuming
$|\xi|\ge 2C_\varepsilon c^{-1}\varepsilon^{-a}$ we obtain the inequality
\[
c\varepsilon^a |\xi|^m\le 2|P_\varepsilon(\xi)|+2C_\varepsilon\le 2\widetilde{P_\varepsilon}(\xi)+2C_\varepsilon.
\]
For $|\xi|\ge R_\varepsilon$, where
 $R_\varepsilon=\max\{(4C_\varepsilon c^{-1}\varepsilon^{-a}+1)^{\frac{1}{m}},
 2C_\varepsilon c^{-1}\varepsilon^{-a}\}$, the following bound from below
\begin{equation} \label{est_Hoer_1}
\frac{c}{4}\varepsilon^{a}(1+|\xi|^m)\le \widetilde{P_\varepsilon}(\xi)
\end{equation}
holds. Since, from the invertibility of $\widetilde{P}$ in $\xi_0$
there exists a strictly nonzero net $(\lambda_\varepsilon)_\varepsilon$ such that
\[
\widetilde{P_\varepsilon}(\xi)\ge \lambda_\varepsilon(1+C|\xi_0-\xi|)^{-m},
\]
for all $\varepsilon\in(0,1]$, we can extend \eqref{est_Hoer_1}
to all $\xi\in\mathbb{R}^n$. More precisely,
\begin{align*}
&\lambda_\varepsilon(1+C|\xi_0|+CR_\varepsilon)^{-m}(1+R_\varepsilon)^{-m}(1+|\xi|)^m\\
&\le \lambda_\varepsilon (1+C|\xi_0-\xi|)^{-m}(1+|\xi|)^{-m}(1+|\xi|)^m
\le \widetilde{P_\varepsilon}(\xi)
\end{align*}
holds for $|\xi|\le R_\varepsilon$. The net $(R_\varepsilon)_\varepsilon$ is strictly nonzero.
Hence there exists a moderate net $\omega_\varepsilon$ such that
\[
(1+|\xi|^m)\le\omega_\varepsilon \widetilde{P_\varepsilon}(\xi)
\]
for all $\varepsilon\in(0,1]$ and $\xi\in\mathbb{R}^n$. Now if $Q(D)$
is a differential operator with coefficients in $\widetilde{\mathbb{C}}$ of
order $m'\le m$, \eqref{est_Hoer} yields
\[
\widetilde{Q_\varepsilon}(\xi)\le \widetilde{Q_\varepsilon}(0)(1+C|\xi|)^{m'}.
\]
Hence,
\[
\widetilde{Q_\varepsilon}(\xi)\le \widetilde{Q_\varepsilon}(0)c'(1+|\xi|^m)
\le c'\widetilde{Q_\varepsilon}(0)\omega_\varepsilon\,\widetilde{P_\varepsilon}(\xi),
\]
where $(c'\widetilde{Q_\varepsilon}(0)\omega_\varepsilon)_\varepsilon$ is moderate.
This means that $Q(D)\prec P(D)$.

We now prove that the $\mathcal{G}$-ellipticity of $P(D)$ is necessary
in order to have $Q(D)\prec P(D)$ for all $Q$ of order less than or
equal to $m$. If $P(D)$ is not $\mathcal{G}$-elliptic then we can find
a representative $(P_\varepsilon)_\varepsilon$, a decreasing sequence
$\varepsilon_q\to 0$ and a sequence $\xi_{\varepsilon_q}$ with $|\xi_{\varepsilon_q}|=1$
such that
\[
|P_{m,\varepsilon_q}(\xi_{\varepsilon_q})|<\varepsilon_q^q
\]
for all $q\in\mathbb{N}$. We set $\xi_\varepsilon=\xi_{\varepsilon_q}$ for
$\varepsilon=\varepsilon_q$ and $0$ otherwise. By construction
$(|\xi_\varepsilon|)_\varepsilon\not\in\mathcal{N}$ and
\[
(P_{m,\varepsilon}(\xi_\varepsilon))_\varepsilon\in\mathcal{N}.
\]
Since there exists a moderate net $(c_\varepsilon)_\varepsilon$ such that
\[
\widetilde{P_\varepsilon}^2(t\xi)\le 2t^{2m}|P_{m,\varepsilon}(\xi)|^2
+c_\varepsilon t^{2m-2}\lrangle{\xi}^{2m-2}
\]
for all $\xi\in\mathbb{R}^n$ and $t\ge 1$, we obtain
\[
\widetilde{P_\varepsilon}(t\xi_\varepsilon)\le t^m n_\varepsilon+c'_\varepsilon t^{m-1},
\]
where $(n_\varepsilon)_\varepsilon\in\mathcal{N}$ and $(c'_\varepsilon)_\varepsilon\in\mathcal{E}_M$.
Since $(|\xi_\varepsilon|)_\varepsilon\in\mathcal{E}_M\setminus \mathcal{N}$ there exists
a component $\xi_{i,\varepsilon}$ such that
$(|\xi_{i,\varepsilon}|)_\varepsilon\in\mathcal{E}_M\setminus\mathcal{N}$. We take the homogeneous
polynomial of degree $m$ $Q(\xi)=\xi_i^m$ and we prove that
$Q(D)\not\prec P(D)$. By Proposition \ref{prop_basic_2} assume
that there exists representatives $(Q_\varepsilon)_\varepsilon$ and
$(P'_\varepsilon)_\varepsilon$ of $Q$ and $P$ respectively and a moderate
net $(\lambda_\varepsilon)_\varepsilon$ such that
\[
|Q_\varepsilon(\xi)|\le \lambda_\varepsilon\widetilde{P'_\varepsilon}(\xi)
\]
for all $\xi$ and $\varepsilon$. $Q_\varepsilon(\xi)$ is of the form
$(1+n_{1,\varepsilon})\xi_i^m$ where $(n_{1,\varepsilon})_\varepsilon\in\mathcal{N}$ and
concerning $\widetilde{P'_\varepsilon}(\xi)$ we have
$\widetilde{P'_\varepsilon}^2(\xi)\le 2\widetilde{P_\varepsilon}^2(\xi)+n_{2,\varepsilon}\lrangle{\xi}^{2m}$
with $(n_{2,\varepsilon})_\varepsilon\in\mathcal{N}$. This entails the estimate
\[
|(1+n_{1,\varepsilon})\xi_i^m|\le \lambda'_\varepsilon\widetilde{P_\varepsilon}(\xi)
+n_{3,\varepsilon}\lrangle{\xi}^m
\]
and for $\xi= t\xi_\varepsilon$, $t\ge 1$,
\[
|1+n_{1,\varepsilon}|t^m|\xi_{i,\varepsilon}^m|
\le \lambda'_\varepsilon\widetilde{P_\varepsilon}(t\xi_\varepsilon)+n_{3,\varepsilon}
\lrangle{t\xi_\varepsilon}^m\le n_{4,\varepsilon}t^m+c''_\varepsilon t^{m-1},
\]
where $(c''_\varepsilon)_\varepsilon$ is moderate and strictly nonzero. The inequality
\[
\frac{|1+n_{1,\varepsilon}||\xi_{i,\varepsilon}^m|-n_{4,\varepsilon}}{c''_\varepsilon}\le t^{-1}
\]
is valid for all $t\ge 1$. Hence
$|1+n_{1,\varepsilon}||\xi_{i,\varepsilon}^m|-n_{4,\varepsilon}\le 0$ and from
the invertibility of $(|1+n_{1,\varepsilon}|)_\varepsilon$ we get
\[
|\xi_{i,\varepsilon}|^m\le \frac{n_{4,\varepsilon}}{|1+n_{1,\varepsilon}|},
\]
with $(\frac{n_{4,\varepsilon}}{|1+n_{1,\varepsilon}|})_\varepsilon\in\mathcal{N}$.
Concluding the net $(|\xi_{i,\varepsilon}|)_\varepsilon$ is negligible in
contradiction with our assumptions.
\end{proof}

We introduce another order relation which is closely connected
with $Q(D)\prec P(D)$.

\begin{definition} \label{def_domin} \rm
Let $P(D)$ and $Q(D)$ be differential operators with coefficients
in $\widetilde{\mathbb{C}}$. We say that $P(D)$ dominates $Q(D)$
(and we write $P(D)\succ\succ Q(D)$ or $Q(D)\prec\prec P(D)$)
if there exist
\begin{itemize}
\item representatives $(P_\varepsilon)_\varepsilon$ and $(Q_\varepsilon)_\varepsilon$ of $P$
 and $Q$ respectively,
\item a moderate net $(\lambda_\varepsilon)_\varepsilon$,
\item a function $C(t)>0$ with $\lim_{t\to +\infty}C(t)=0$
 and the property
\[
\forall a\in\mathbb{R}\ \exists b\in\mathbb{R}\ \forall t\ge \varepsilon^b\quad C(t)\le
\varepsilon^a,
\]
\end{itemize}
such that
\[
\widetilde{Q_\varepsilon}(\xi,t)\le \lambda_\varepsilon C(t)\widetilde{P_\varepsilon}(\xi,t)
\]
for all $\xi\in\mathbb{R}^n$, $\varepsilon\in(0,1]$ and $t\ge 1$.
\end{definition}

Clearly $Q(D)\prec\prec P(D)$ implies $Q(D)\prec P(D)$ and $P(D)$
 dominates $P^{(\alpha)}(D)$ for all $\alpha\neq 0$.
Indeed, $\widetilde{P^{(\alpha)}_\varepsilon}(\xi,t)
\le t^{-|\alpha|}\widetilde{P_\varepsilon}(\xi,t)$.

\begin{proposition} \label{prop_basic_3}
Let $P(D), P_1(D), P_2(D), Q_1(D)$ and $Q_2(D)$ be differential
operators with constant Colombeau coefficients.
\begin{itemize}
\item[(i)] If $Q_1(D)\prec\prec P(D)$ and $Q_2(D)\prec\prec P(D)$
  then $a_1 Q_1(D)+a_2 Q_2(D)\prec\prec P(D)$
  for all $a_1,a_2\in\widetilde{\mathbb{C}}$.
\item[(ii)] If $Q_1(D)\prec\prec P_1(D)$ and $Q_2(D)\prec P_2(D)$
  then $Q_1Q_2(D)\prec\prec P_1P_2(D)$.
\end{itemize}
\end{proposition}

\begin{proof}
The first statement is trivial. By applying Proposition \ref{prop_dom}(ii)
to $P(t\xi)$ and $Q(t\xi)$ we obtain for any polynomials $P$ and $Q$
the estimate
\begin{equation} \label{est_dez}
C'\widetilde{P}(\xi,t)\widetilde{Q}(\xi,t)\le \widetilde{PQ}(\xi,t)
\le C''\widetilde{P}(\xi,t)\widetilde{Q}(\xi,t)
\end{equation}
where $C'$ and $C''$ depend only on the order of $P$ and $Q$.
Hence, for all $t\ge 1$ we have
\[
\widetilde{Q_{1,\varepsilon}Q_{2,\varepsilon}}(\xi,t)\le C''\widetilde{Q_{1,\varepsilon}}(\xi,t)\widetilde{Q_{2,\varepsilon}}(\xi,t)\le C''\lambda_{1,\varepsilon} C(t)\widetilde{P_{1,\varepsilon}}(\xi,t)\widetilde{Q_{2,\varepsilon}}(\xi,t).
\]
Proposition \ref{prop_basic_2}(iii) combined with the estimate
\eqref{est_dez} yields
\[
\widetilde{Q_{1,\varepsilon}Q_{2,\varepsilon}}(\xi,t)\le C''\lambda_{1,\varepsilon}
C(t)\widetilde{P_{1,\varepsilon}}(\xi,t)\lambda_{2,\varepsilon}\widetilde{P_{2,\varepsilon}}(\xi,t)
\le \lambda_\varepsilon C(t)\widetilde{P_{1,\varepsilon}P_{2,\varepsilon}}(\xi,t),
\]
valid for all $t\ge 1$, for all $\varepsilon\in(0,1]$ and for all $\xi\in\mathbb{R}^n$.
\end{proof}
The order relation $\prec\prec$ is used in comparing an operator
of principal type with a differential operator of order
strictly smaller.

\begin{definition} \label{def_princ_point} \rm
A partial differential operator $P(D)$ with constant Colombeau
coefficients is said to be of principal type if there exists
a representative $(P_{m,\varepsilon})_\varepsilon$ of the principal symbol
$P_{m}$, $a\in\mathbb{R}$ and $c>0$ such that
\[
|\nabla_\xi P_{m,\varepsilon}(\xi)|\ge c\varepsilon^a|\xi|^{m-1}
\]
for all $\varepsilon\in(0,1]$ and all $\xi\in\mathbb{R}^n$.
\end{definition}
As for $\mathcal{G}$-elliptic operators the previous estimate holds for
any representative $(P_\varepsilon)_\varepsilon$ of $P$, for some constant
$c$ and in an enough small interval $(0,\varepsilon_0]$.

\begin{proposition} \label{prop_princ_type}
Let $P(D)$ be a differential operator with coefficients
in $\widetilde{\mathbb{C}}$ of principal type and degree $m$ and let one
of the coefficients of $P_m(D)$ be invertible. Then,
\begin{itemize}
\item[(i)] $P(D)$ dominates any differential operator with
coefficients in $\widetilde{\mathbb{C}}$ of order less than or equal to 
$m-1$;
\item[(ii)] if $Q(D)$ has order $m$ and there exists a moderate
net $(\lambda_\varepsilon)_\varepsilon$ and representatives $(Q_{m,\varepsilon})_\varepsilon$
and $(P_{m,\varepsilon})_\varepsilon$ such that
\[
|Q_{m,\varepsilon}(\xi)|\le \lambda_\varepsilon |P_{m,\varepsilon}(\xi)|
\]
for all $\xi\in\mathbb{R}^n$ and $\varepsilon\in(0,1]$, then $Q(D)\prec P(D)$.
\end{itemize}
\end{proposition}

\begin{proof}
(i) Let $(P_{m,\varepsilon})_\varepsilon$ be a representative of $P_m$ such that
$|\nabla P_{m,\varepsilon}(\xi)|\ge c\varepsilon^a|\xi|^{m-1}$ for some constants
$c>0$ and $a\in\mathbb{R}$, for all $\xi\in\mathbb{R}^n$ and for all $\varepsilon\in(0,1]$.
We have, for some strictly nonzero net $(C_\varepsilon)_\varepsilon$ the inequality
\begin{align*}
|\nabla_\xi P_\varepsilon(\xi)|
&\ge |\nabla_\xi P_{m,\varepsilon}(\xi)| -|\nabla_\xi(P_\varepsilon-P_{m,\varepsilon})|\\
&\ge c'\varepsilon^a\lrangle{\xi}^{m-1}
-C_\varepsilon\lrangle{\xi}^{m-2}\ge \frac{1}{2}c'\varepsilon^a(1+|\xi|^{m-1}),
\end{align*}
valid for $|\xi|\ge R_\varepsilon$ with $(R_\varepsilon)_\varepsilon$ moderate and big enough.
Hence,
\[
\Bigl(\sum_{\alpha\neq 0}|P^{(\alpha)}_\varepsilon(\xi)|^2\Bigr)^{1/2}
\ge |\nabla_\xi P_\varepsilon(\xi)|\ge \frac{1}{2}c'\varepsilon^a(1+|\xi|^{m-1})
\]
for $|\xi|\ge R_\varepsilon$.
>From the invertibility of one of the coefficients of the principal
part we get the bound from below
\[
\omega_\varepsilon\le \sum_{\alpha\neq 0}|P^{(\alpha)}_\varepsilon(\xi)|^2,
\]
where $(\omega_\varepsilon)_\varepsilon$ is moderate. It follows, for $|\xi|\le R_\varepsilon$,
\[
\sum_{\alpha\neq 0}|P^{(\alpha)}_\varepsilon(\xi)|^2
\ge \omega_\varepsilon(1+|\xi|^{m-1})^{-2}(1+|\xi|^{m-1})^2
\ge \omega_\varepsilon ((1+(R_\varepsilon)^{m-1})^{-2}(1+|\xi|^{m-1})^2.
\]
Summarizing, we find a moderate and strictly nonzero net
$(\lambda_\varepsilon)_\varepsilon$ such that
\[
\lambda_\varepsilon^2(1+|\xi|^{2m-2})\le \sum_{\alpha\neq 0}
|P^{(\alpha)}_\varepsilon(\xi)|^2
\]
for all $\xi\in\mathbb{R}^n$ and $\varepsilon\in(0,1]$. This implies
\begin{equation} \label{est_oggi}
\lambda_\varepsilon^2 t^2(1+|\xi|^{2m-2})
\le \sum_{\alpha\neq 0}t^{2|\alpha|}|P^{(\alpha)}_\varepsilon(\xi)|^2
\le \widetilde{P_\varepsilon}(\xi,t),
\end{equation}
for all $t\ge 1$, $\xi\in\mathbb{R}^n$ and $\varepsilon\in(0,1]$.

Let $Q(D)$ be a differential operator with coefficients
in $\widetilde{\mathbb{C}}$ and order $m'\le m-1$. We have, for some moderate
net $(c_\varepsilon)_\varepsilon$ the inequality
\[
|Q_\varepsilon(\xi)|\le c_\varepsilon(1+|\xi|)^{m-1}
\]
and therefore from \eqref{est_oggi},
\[
|Q_\varepsilon(\xi)|\le (c_\varepsilon \lambda_\varepsilon^{-1}t^{-1})t^{1}
\lambda_\varepsilon(1+|\xi|^{m-1})\le (c'_\varepsilon
\lambda_\varepsilon^{-1}t^{-1})\widetilde{P_\varepsilon}(\xi,t).
\]
Arguing as in proof of Proposition \ref{prop_basic_2} and making
use of Proposition \ref{prop_dom} we obtain that there exists
a moderate net $(c''_\varepsilon)_\varepsilon$ such that
\[
\widetilde{Q_\varepsilon}(\xi,t)\le c''_\varepsilon t^{-1}\widetilde{P_\varepsilon}(\xi,t).
\]
Indeed
\begin{align*}
\widetilde{Q_\varepsilon}(\xi,t)
&\le C\sup_{|\eta|<t}|Q_\varepsilon(\xi+\eta)|\\
&\le Cc'_\varepsilon\lambda^{-1}_\varepsilon t^{-1}\sup_{|\eta|<t}
 \widetilde{P_\varepsilon}(\xi+\eta,t)\\
&\le C^2c'_\varepsilon\lambda^{-1}_\varepsilon t^{-1}
 \sup_{|\eta|<t}\sup_{|\theta|<t}|P_\varepsilon(\xi+\eta+\theta)|\\
&\le C^2c'_\varepsilon\lambda^{-1}_\varepsilon t^{-1}\sup_{|\eta|<2t}|
 P_\varepsilon(\xi+\eta)|\\
&\le C^3c'_\varepsilon\lambda^{-1}_\varepsilon t^{-1}\widetilde{P_\varepsilon}
 (\xi,2t)\le C^3 2^m c'_\varepsilon\lambda^{-1}_\varepsilon t^{-1}\widetilde{P_\varepsilon}(\xi,t).
\end{align*}
This means that $Q(D)\prec\prec P(D)$.

(ii) Let $Q(D)$ be a differential operator of order $m$ satisfying
the condition (ii). From the first statement we already know
that $Q(D)-Q_{m}(D)\prec\prec P(D)$ and thus
$Q(D)-Q_{m}(D)\prec P(D)$. It remains to prove that $P(D)$ is
stronger than $Q_m(D)$. Writing $\widetilde{Q_{m,\varepsilon}}^2$ as
$|Q_{m,\varepsilon}|^2+\sum_{\beta\neq 0}|Q^{(\beta)}_{m,\varepsilon}|^2$,
and since the second term has order less than or equal to $m-1$, we have
\begin{align*}
\widetilde{Q_{m,\varepsilon}}^2(\xi)
&\le \lambda_\varepsilon^2|P_{m,\varepsilon}(\xi)|^2+\lambda_{1,\varepsilon}
  \widetilde{P_{\varepsilon}}^2(\xi)\\
&\le 2\lambda_\varepsilon^2|P_{\varepsilon}(\xi)|^2+2\lambda_\varepsilon^2|
 (P_{\varepsilon}-P_{m,\varepsilon})(\xi)|^2+\lambda_{1,\varepsilon}\widetilde{P_{\varepsilon}}^2(\xi)\\
&\le(2\lambda_\varepsilon^2+\lambda_{2,\varepsilon}+\lambda_{1,\varepsilon})\widetilde{P_\varepsilon}^2(\xi).
\end{align*}
Hence $Q_m(D)\prec P(D)$.
\end{proof}

The following proposition determines a family of equally strong operators.

\begin{proposition} \label{prop_eq_strong}
Let $P(D)$ and $Q(D)$ be differential operators with coefficients
in $\widetilde{\mathbb{C}}$. If $Q(D)\prec\prec P(D)$ then
$P(D)\prec P(D)+aQ(D)\prec P(D)$ for all $a\in\widetilde{\mathbb{C}}$.
\end{proposition}

\begin{proof}
Since $Q(D)\prec\prec P(D)$ implies $Q(D)\prec P(D)$, from the
third assertion of Proposition \ref{prop_basic_1} we have that
$P(D)+aQ(D)\prec P(D)$ for all $a\in\widetilde{\mathbb{C}}$.
We now fix $a\in\widetilde{\mathbb{C}}$ and take $R(D)=P(D)+aQ(D)$. Arguing at
the level of representatives we obtain
\begin{align*}
\widetilde{P_\varepsilon}^2(\xi,t)
&=\sum_\alpha |R_\varepsilon^{(\alpha)} -a_\varepsilon
 Q_\varepsilon^{(\alpha)}|^2(\xi)t^{2|\alpha|}\\
&\le 2\widetilde{R_\varepsilon}^2(\xi,t)+2|a_\varepsilon|^2\widetilde{Q_\varepsilon}^2(\xi,t)\\
&\le 2\widetilde{R_\varepsilon}^2(\xi,t)+2|a_\varepsilon|^2\lambda_\varepsilon^2 C^2(t)
 \widetilde{P_\varepsilon}^2(\xi,t).
\end{align*}
By the moderateness assumption we have that
$|a_\varepsilon|^2\lambda_\varepsilon^2\le \varepsilon^{2a}$ for all $\varepsilon$ small enough.
Choosing $b\in\mathbb{R}$ such that $C(t)\le \varepsilon^{-a+1}$ for $t\ge\varepsilon^b$
we can write the inequality
\[
\widetilde{P_\varepsilon}^2(\xi,t)\le  2\widetilde{R_\varepsilon}^2(\xi,t)+2\varepsilon^{2a}
\varepsilon^{-2a+2}\widetilde{P_\varepsilon}^2(\xi,t)
\]
for $t\ge\varepsilon^b$, for $\varepsilon$ small enough and for all $\xi$.
It follows that
\[
\widetilde{P_\varepsilon}^2(\xi,t)\le  2\widetilde{R_\varepsilon}^2(\xi,t)
+\frac{1}{2}\widetilde{P_\varepsilon}^2(\xi,t)
\]
for $t\ge\max(1,\varepsilon^b)$ and $\varepsilon\in(0,\varepsilon_0]$ with
$\varepsilon_0\le 2^{-1}$. Hence,
\[
\widetilde{P_\varepsilon}(\xi,t)\le 2\widetilde{R_\varepsilon}(\xi,t),
\]
under the same conditions on $\varepsilon$ and $t$. Let
$(t_\varepsilon)_\varepsilon\in\mathcal{E}_M$ with $t_\varepsilon\ge \max(1,\varepsilon^b)$. We can write
\[
\widetilde{P_\varepsilon}(\xi)\le \widetilde{P_\varepsilon}(\xi,t_\varepsilon)
\le 2\widetilde{R_\varepsilon}(\xi,t_\varepsilon)\le 2t_\varepsilon^m\widetilde{R_\varepsilon}(\xi)
\le \lambda_\varepsilon\widetilde{R_\varepsilon}(\xi),
\]
valid for some moderate net $(\lambda_\varepsilon)_\varepsilon$ and for
$\varepsilon\in(0,\varepsilon_0]$. This means that $P(D)\prec R(D)=P(D)+aQ(D)$.
\end{proof}

The next corollary is straightforward from
$P^{(\alpha)}(D)\prec\prec P(D)$.

\begin{corollary} \label{corol_eq_strong}
For all $\alpha\in\mathbb{N}^n$,
\[
P(D)\prec P(D)+P^{(\alpha)}(D)\prec P(D).
\]
\end{corollary}

\section{Parametrices and local solvability}
\label{sec_parametrix}

We begin our investigation of locally solvable differential
operators  in the Colom\-beau framework, by showing that
differential operators which admits a generalized
pseudodifferential parametrix (at least a right generalized
parametrix) are locally solvable. Some needed notions of
generalized pseudodifferential operator theory are collected in
the following subsection.

\subsection{Preliminary notions of generalized pseudodifferential operator theory}
\paragraph{Symbols.}

Throughout this paper $S^m(\mathbb{R}^{2n})$ denotes the space of
H\"ormander symbols fulfilling global estimates on $\mathbb{R}^{2n}$. In
detail $|a|^{(m)}_{\alpha,\beta}$ is the seminorm
\[
\sup_{(x,\xi)\in\mathbb{R}^{2n}}\lrangle{\xi}^{-m+|\alpha|}
|\partial^\alpha_\xi\partial^\beta_x a(x,\xi)|.
\]
In a local context, that is on an open subset $\Omega$ of $\mathbb{R}^n$,
we work with symbols that satisfy uniform estimates on compact
subsets of $\Omega$. In this case we use the notation
$S^{m}(\Omega\times\mathbb{R}^n)$ and the seminorms
\[
|a|^{(m)}_{K,\alpha,\beta}:=\sup_{x\in K\Subset\Omega,
\xi\in\mathbb{R}^n}\lrangle{\xi}^{-m+|\alpha|}|
\partial^\alpha_\xi\partial^\beta_x a(x,\xi)|.
\]
The corresponding sets of generalized symbols are introduced by means
of the abstract models $\mathcal{G}_E$ and $\mathcal{G}^\mathrm{sc}_E$ introduced in
Section \ref{sec_basic} where $E=S^m(\mathbb{R}^{2n})$ or
$E=S^m(\Omega\times\mathbb{R}^n)$.

\paragraph{Mapping properties.}
A theory of generalized pseudodifferential operators has been
developed in \cite{Garetto:04, GGO:03} for symbols in
$\mathcal{G}_{{{S}}^m(\mathbb{R}^{2n})}$ and $\mathcal{G}^{\mathrm{sc}}_{{{S}}^m(\mathbb{R}^{2n})}$
and for more elaborated notions of generalized symbols and
amplitudes. We address the reader to the basic notions
section of \cite{Garetto:ISAAC07} for an elementary introduction
to the subject. In the sequel $\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n)$ is the Colombeau space
based on $E=\mathscr{S}(\mathbb{R}^n)$ and $\mathcal{G}^\infty_{\mathscr{S}}(\mathbb{R}^n)$ is the subspace of
$\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n)$ of those generalized functions $u$ having a
representative $(u_\varepsilon)_\varepsilon$ fulfilling the following condition:
\[
\exists N\in\mathbb{N}\ \forall\alpha,\beta\in\mathbb{N}^n\quad
\sup_{x\in\mathbb{R}^n}|x^\alpha\partial^\beta u_\varepsilon(x)|=O(\varepsilon^{-N}).
\]
Finally $\mathcal{L}(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ is the topological dual of $\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n)$.

Let now $p\in\mathcal{G}_{{{S}}^m(\mathbb{R}^{2n})}$. The pseudodifferential operator
\[
p(x,D)u=\int_{\mathbb{R}^n}\mathrm{e}^{ix\xi}p(x,\xi)\widehat{u}(\xi)\, d\xi
\]
\begin{itemize}
\item[(i)] maps $\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n)$ into $\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n)$,
\item[(ii)] can be continuously extended to a $\widetilde{\mathbb{C}}$-linear
  map on $\mathcal{L}(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})$,
\item[(iii)] maps basic functionals into basic functionals,
\item[(iv)] maps $\mathcal{G}^\infty_{\mathscr{S}}(\mathbb{R}^n)$ into itself if $p$ is of slow scale type.
\end{itemize}

\paragraph{Generalized symbols and asymptotic expansions}
The notion of asymptotic expansion for generalized symbols
in $\mathcal{G}_{S^m(\mathbb{R}^{2n})}$ is based on the following definition
at the level of representatives.

\begin{definition} \label{def_asymp} \rm
Let $\{m_j\}_{j\in\mathbb{N}}$ be sequences of real numbers with
$m_j\searrow -\infty$, $m_0=m$. Let
$\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$
be a sequence of elements
$(a_{j,\epsilon})_\epsilon\in\mathcal{M}_{S^{m_j}(\mathbb{R}^{2n})}$.
We say that the formal series
$\sum_{j=0}^\infty(a_{j,\epsilon})_\epsilon$ is the asymptotic
expansion of
$(a_\epsilon)_\epsilon\in\mathcal{E}[\mathbb{R}^{2n}]$,
$(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$
for short, if and only if for all $r\ge 1$
\[
\Bigl(a_\epsilon-\sum_{j=0}^{r-1}a_{j,\epsilon}\Bigr)_\epsilon
\in \mathcal{M}_{S^{m_r}(\mathbb{R}^{2n})}.
\]
\end{definition}

By arguing as in \cite[Theorem 2.2]{Garetto:ISAAC07}
one proves that there exists a net of symbols with a given
asymptotic expansion according to Definition \ref{def_asymp}.

\begin{theorem} \label{theo_asymp}
Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$
be a sequence of elements
$(a_{j,\epsilon})_\epsilon\in\mathcal{M}_{S^{m_j}(\mathbb{R}^{2n}}$
with $m_j\searrow -\infty$ and $m_0=m$. Then, there exists
$(a_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m}(\mathbb{R}^{2n})}$ such that
$(a_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$.
Moreover, if $(a'_\epsilon)_\epsilon\sim\sum_j(a_{j,\epsilon})_\epsilon$
then $(a_\varepsilon-a'_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n})}$.
\end{theorem}

We now take in consideration regular nets of symbols.
Inspired by the notations of \cite{Garetto:04} we say that
$(a_\varepsilon)_\varepsilon$ belongs to $\mathcal{M}_{S^m(\mathbb{R}^{2n}),b}$ if and only
if $|a_\varepsilon|^{(m)}_{\alpha,\beta}=O(\varepsilon^b)$ for all $\alpha$
and $\beta$. In other words we require the same kind of moderateness
for all orders of derivatives. A closer look to the proof of
Theorem 2.2 in \cite{Garetto:ISAAC07} yields the following corollary.

\begin{corollary} \label{corol_asymp}
Let $\{(a_{j,\epsilon})_\epsilon\}_{j\in\mathbb{N}}$ as in
Theorem \ref{theo_asymp}. If $(a_{j,\varepsilon})_\varepsilon
\in\mathcal{M}_{S^{m_j}(\mathbb{R}^{2n}),b}$ for each $j$ then there exists
$(a_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m}(\mathbb{R}^{2n}),b}$ such that
\[
\Bigl(a_\epsilon-\sum_{j=0}^{r-1}a_{j,\epsilon}\Bigr)_\epsilon\in \mathcal{M}_{S^{m_r}(\mathbb{R}^{2n}),b}.
\]
for every $r\ge 1$. This result is unique modulo
$\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),b}$.
\end{corollary}
Note that this statement recalls the first concept of asymptotic
expansion for nets of symbols studied in \cite{Garetto:04} but
avoids global estimates on the $\varepsilon$-interval $(0,1]$.

It is clear that when $p\in\mathcal{G}_{S^m(\mathbb{R}^{2n})}$ has a representative
 in $\cup_{b\in\mathbb{R}}\mathcal{M}_{S^m(\mathbb{R}^{2n}),b}$ then $p(x,D)$ maps
$\mathcal{G}^\infty_{\mathscr{S}}(\mathbb{R}^n)$ into $\mathcal{G}^\infty_{\mathscr{S}}(\mathbb{R}^n)$.

\paragraph{Kernels and regularizing operators.}
Any generalized pseudodifferential operator has a kernel
in $\mathcal{L}(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^{2n}),\widetilde{\mathbb{C}})$ but when $p\in\mathcal{G}_{{{S}}^{-\infty}(\mathbb{R}^{2n})}$
then $k_p\in\mathcal{G}_\tau(\mathbb{R}^{2n})$ with a representative
$(k_{p,\varepsilon})_\varepsilon$ fulfilling the following property:
\begin{equation} \label{prop_kern}
\begin{gathered}
\forall\alpha,\beta\in\mathbb{N}^n\, \forall d\in \mathbb{N}\,
\exists (\lambda_\varepsilon)_\varepsilon\in\mathcal{E}_M\, \forall\varepsilon\in(0,1]\\
 \sup_{(x,y)\in\mathbb{R}^{2n}}\lrangle{x}^{-d}\lrangle{y}^d
|\partial^{\alpha}_x\partial^{\beta}_y k_{p,\varepsilon}(x,y)|\le\lambda_\varepsilon,\\
\sup_{(x,y)\in\mathbb{R}^{2n}}\lrangle{x}^{d}\lrangle{y}^{-d}|
\partial^{\alpha}_x\partial^{\beta}_y k_{p,\varepsilon}(x,y)|\le\lambda_\varepsilon.
\end{gathered}
\end{equation}
If $p\in\mathcal{G}^\mathrm{sc}_{{{S}}^{-\infty}(\mathbb{R}^{2n})}$ then $(\lambda_\varepsilon)_\varepsilon$
in \eqref{prop_kern} is a slow scale net. With a symbol of order
$-\infty$ the pseudodifferential operator $p(x,D)$ can be written
in the form
\[
p(x,D)u=\int_{\mathbb{R}^n}k_p(x,y)u(y)\, dy.
\]
It maps $\mathcal{L}(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ into $\mathcal{G}_\tau(\mathbb{R}^n)$ and
$\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ into $\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n)$. If $p$ is of slow scale
 type then the previous mappings have image in $\mathcal{G}^{\infty}_\tau(\mathbb{R}^n)$ and
$\mathcal{G}^\infty_{\mathscr{S}}(\mathbb{R}^n)$ respectively.

\paragraph{$L^2$-continuity.}
We finally discuss some $L^2$-continuity. From the well-known
estimate (see \cite[Chapter2, Theorem 4.1]{Kumano-go:81})
\[
\Vert a(x,D)u\Vert_2\le C_0\max_{|\alpha+\beta|\le
l_0}|a|^{(0)}_{\alpha,\beta}\Vert u\Vert_2,\quad\quad \text{for}\
u\in\mathscr{S}(\mathbb{R}^n)
\]
valid for $a\in S^0(\mathbb{R}^n)$, for some $l_0>0$ and for a constant
$C_0$ depending on the space dimension $n$, one easily has that
a generalized pseudodifferential operator $p(x,D)$ with
symbol $p\in\mathcal{G}_{{{S}}^0(\mathbb{R}^{2n})}$ maps $\mathcal{G}_{L^2(\mathbb{R}^n)}$
continuously into itself. If we now consider a basic
functional $T$ of $\mathcal{L}(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ given by a net
$(T_\varepsilon)_\varepsilon\in\mathcal{M}_{L^2(\mathbb{R}^n)}$, we have that $p(x,D)T$
is a basic functional in $\mathcal{L}(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ with the
same ${L^2}$-structure. We introduce the notation
$\mathcal{L}_2(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ for the set of basic functionals in
$\mathcal{L}(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ with a representative in $\mathcal{M}_{L^2(\mathbb{R}^n)}$.
Hence, a pseudodifferential operator with symbol in
$p\in\mathcal{G}_{{{S}}^0(\mathbb{R}^{2n})}$ has the mapping property
\begin{equation} \label{L_2_cont}
p(x,D):\mathcal{L}_2(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}})\to \mathcal{L}_2(\mathcal{G}_{\mathscr{S}}(\mathbb{R}^n),\widetilde{\mathbb{C}}).
\end{equation}
Analogously, in the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ one can define
the subset $\mathcal{L}_{2,{\rm{loc}}}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ of those
basic functionals $T$ defined by a net
$(T_\varepsilon)_\varepsilon\in\mathcal{M}_{L^2_{\rm{loc}}(\mathbb{R}^n)}$, i.e.
$(\phi T_\varepsilon)_\varepsilon\in\mathcal{M}_{L^2(\mathbb{R}^n)}$ for all $\phi\in\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$.

If $P(x,D)$ is a differential operator with Colombeau coefficients
and $P(x,\xi)\in\mathcal{G}_{S^{m}(\mathbb{R}^{2n})}$ then in addition to the
mapping properties as a pseudodifferential operator we have that
the restriction to any open subset $\Omega$ maps $\mathcal{G}_{\rm c}(\Omega)$,
$\mathcal{G}(\Omega)$, $\mathcal{L}(\mathcal{G}(\Omega),\widetilde{\mathbb{C}})$ and $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$
into themselves respectively. Typical example is obtained by taking
the coefficients $c_\alpha$ of
$P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ in the algebra
$\mathcal{G}_E$ with $E={\cap_s W^{s,\infty}(\mathbb{R}^n)}$. In this case one
can use the notation $\mathcal{G}_\infty(\mathbb{R}^n)$ for simplicity.

\subsection{A first sufficient condition of local solvability}

\begin{theorem} \label{theo_sol_par}
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ be a differential
operator with coefficients $c_\alpha\in\mathcal{G}_\infty(\mathbb{R}^n)$.
Let $(P_\varepsilon)_\varepsilon$ a representative of $P$. If
\begin{itemize}
\item[(i)] there exist $(q_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m'}(\mathbb{R}^{2n})}$
with $m'\le 0$ and $(r)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n})}$ such that
\[
P_\varepsilon(x,D)q_\varepsilon(x,D)=I+r_\varepsilon(x,D)
\]
on $\mathscr{S}(\mathbb{R}^n)$ for all $\varepsilon\in(0,1]$,
\item[(ii)] there exists $l<-n$ such that
$|r_\varepsilon|^{(l)}_{0,0}=O(1)$,
\end{itemize}
then for all $x_0\in\mathbb{R}^n$ there exists a neighborhood $\Omega$ of
$x_0$ and a cut-off function $\phi$, identically $1$ near $x_0$,
such that the following solvability result holds: \begin{equation}
\label{first_solv} \forall
F\in\mathcal{L}_{2,{\rm{loc}}}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})\ \exists
T\in\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})\quad P(x,D)T=\phi F\quad \text{on $\Omega$}.
\end{equation}
\end{theorem}

\begin{proof}
We begin by dealing with the regularizing operator $r_\varepsilon$.
 From (ii) it follows that
\[
r_\varepsilon(x,D)u=\int_{\mathbb{R}^n}k_{r_\varepsilon}(x,y)u(y)\,
dy=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathrm{e}^{i(x-y)\xi}r_\varepsilon(x,\xi)\,
d\xi\, u(y)\, dy,
\]
with $u\in\mathscr{S}(\mathbb{R}^n)$,
\[
k_{r_\varepsilon}(x,y)=\int_{\mathbb{R}^n}\mathrm{e}^{i(x-y)\xi}r_\varepsilon(x,\xi)\, d\xi
\]
and
\begin{equation} \label{est_k_eps}
\sup_{x\in \mathbb{R}^n, y\in\mathbb{R}^n} |k_{r_\varepsilon}(x,y)|=O(1).
\end{equation}
We now take a neighborhood $\Omega$ of $x_0$ and a cut-off
$\phi\in\mathcal{C}^\infty_{\rm c}(\Omega)$ and investigate the properties of the net
of operators $r_\varepsilon(x,D)\phi$ on $\Omega$.
For all $g\in L^2(\Omega)$ we have that $\phi g\in L^2(\mathbb{R}^n)$ and
therefore
\[
r_\varepsilon(x,D)(\phi g)|_\Omega
=\Bigl(\int_\Omega k_{r_\varepsilon}(x,y)\phi(y)g(y)\, dy\Bigr) \Big|_\Omega.
\]
This net of distributions actually belongs to $L^2(\Omega)$. Indeed,
\begin{align*}
\Vert r_\varepsilon(x,D)(\phi g)|_\Omega \Vert_{2}
&\le \int_\Omega \Bigl(\int_\Omega |k_{r_\varepsilon}(x,y)|^2\, dx\Bigr)
^{1/2}|\phi(y) g(y)|\, dy\\
&\le |\Omega|\sup_{\Omega\times\Omega}|k_{r_\varepsilon}(x,y)|
\Vert \phi\Vert_2 \Vert g\Vert_2.
\end{align*}
 From \eqref{est_k_eps} by choosing $\Omega$ small enough and a
suitable $\phi\in\mathcal{C}^\infty_{\rm c}(\Omega)$, we obtain that
\[
\Vert r_\varepsilon(x,D)(\phi g)|_\Omega \Vert_{2}\le \frac{1}{2}\Vert g\Vert_2
\]
for all $g\in L^2(\Omega)$ uniformly on an interval $(0,\varepsilon_0]$.
In other words the net of operators
\[
\widetilde{r}_{\varepsilon}:L^2(\Omega)\to L^2(\Omega),\quad
\widetilde{r}_\varepsilon(g)=r_\varepsilon(x,D)(\phi g)|_\Omega
\]
has operator norm less than or equal to $\frac{1}{2}$ for all
$\varepsilon\in(0,\varepsilon_0]$.
In the same way we define
\begin{gather*}
\widetilde{I}:L^2(\Omega)\to L^2(\Omega),\quad \widetilde{I}(g)=\phi g,\\
\widetilde{q}_\varepsilon:L^2(\Omega)\to L^2(\Omega),\quad
\widetilde{q}_\varepsilon(g)=q_\varepsilon(x,D)(\phi g)|_\Omega.
\end{gather*}
This last mapping property follows from
$q_\varepsilon(x,D):L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ valid because $m'\le 0$.
One can choose $\phi$ such that $\Vert \widetilde{I}-I\Vert$ is
very small and in particular $\Vert \widetilde{I}-I+\widetilde{r}_\varepsilon\Vert <1$
uniformly on $(0,\varepsilon_0]$.
The series $\sum_{n=0}^\infty\Vert \widetilde{I}-I+\widetilde{r}_\varepsilon\Vert^n$ is
convergent. Hence, from Theorem 2 in \cite[Chapter 2]{Yosida:80}
we have that
$\widetilde{I}+\widetilde{r}_\varepsilon$ has a continuous linear inverse on $L^2(\Omega)$
for all $\varepsilon\in(0,\varepsilon_0]$ with operator norm uniformly bounded
in $\varepsilon$.

Let now $(F_\varepsilon)_\varepsilon$ be a net in
$\mathcal{M}_{L^2_{\rm{loc}}(\mathbb{R}^n)}$ representing
$F\in\mathcal{L}_{2,{\rm{loc}}}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$. We have that
$(\phi F_\varepsilon)_\varepsilon\in\mathcal{M}_{L^2(\Omega)}$ and we can define for
$\varepsilon\in(0,\varepsilon_0]$ the net
\begin{equation}
\label{def_sol}
T_\varepsilon:=\widetilde{q}_\varepsilon(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}(\phi F_\varepsilon).
\end{equation}
$T_\varepsilon$ belongs to $L^2(\Omega)$ for all $\varepsilon\in(0,\varepsilon_0]$ and
the properties of the operators involved in \eqref{def_sol} yield
\begin{align*}
\Vert T_\varepsilon\Vert_2
&=\Vert\widetilde{q}_\varepsilon(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}(\phi F_\varepsilon)\Vert_2\\
&=\Vert q_\varepsilon(x,D)(\phi(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}
 (\phi F_\varepsilon))|_\Omega\Vert_2\\
&\le c|q_\varepsilon|^{(0)}_{l_0}\Vert\phi(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}(\phi F_\varepsilon)
\Vert_2\le c'|q_\varepsilon|^{(0)}_{l_0}\Vert(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}\Vert\,
 \Vert\phi F_\varepsilon\Vert_2.
\end{align*}
This means that, restricting $\varepsilon$ on the interval $(0,\varepsilon_0]$,
the net $(T_\varepsilon)_\varepsilon$ is $L^2(\Omega)$-moderate and therefore generates
a basic functional $T$ in $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$.
$T$ solves the equation $P(x,D)T=\phi F$ on $\Omega$. Indeed, working
at the level of the representatives we have
\begin{align*}
P_\varepsilon(x,D)|_\Omega(\widetilde{q}_\varepsilon(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}(\phi F_\varepsilon))
&=P_\varepsilon(x,D)|_\Omega(q_\varepsilon(x,D)\phi(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}(\phi F_\varepsilon))\\
&=P_\varepsilon(x,D)q_\varepsilon(x,D)|_\Omega(\phi(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}(\phi F_\varepsilon))\\
&=(\widetilde{I}+\widetilde{r}_\varepsilon)(\widetilde{I}+\widetilde{r}_\varepsilon)^{-1}(\phi F_\varepsilon)=\phi F_\varepsilon.
\end{align*}
\end{proof}

\begin{remark} \label{rem_set_up} \rm
It is not restrictive to consider differential operators with
coefficients in $\mathcal{G}_\infty(\mathbb{R}^n)$ when one wants to investigate
local solvability in the Colombeau context. Indeed, if we assume
to work on an open subset $\Omega'$ and we take
$P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ with
$c_\alpha\in\mathcal{G}(\Omega')$, by choosing the neighborhood
$\Omega$ of $x_0$ small enough the equation $P(x,D)T=\phi F$ on $\Omega$
is equivalent to $P_1(x,D)T=\phi F$ with
\[
P_1(x,D)=\sum_{|\alpha|\le m}\varphi(x)c_\alpha(x)D^\alpha
\]
and $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega')$ identically $1$ on $\Omega$.
It follows that $\varphi c_\alpha\in\mathcal{G}_{\rm c}(\Omega')\subseteq\mathcal{G}_\infty(\mathbb{R}^n)$
and therefore we are in the mathematical set-up of
Theorem \ref{theo_sol_par}.
\end{remark}

In the next proposition we find a family of differential operators
which satisfy the hypotheses of Theorem \ref{theo_sol_par},
in other words a condition on the symbol which assures the
existence of a parametrix $q$ with regularizing term $r$ as above.
We go back to some definition of generalized hypoelliptic symbol
introduced for pseudodifferential operators in \cite{Garetto:04}.
Here the attention is focused not so much on the parametrix $q$ but
on the required boundedness in $\varepsilon$ of the regularizing operator $r$.
This makes us to avoid some more general definitions of hypoelliptic
symbol already employed in Colombeau theory,
see \cite{Garetto:ISAAC07, GGO:03, GH:05}, which have less
restrictive assumptions on the scales in $\varepsilon$, guarantee the
existence of a parametrix but not the desired behaviour of $r$.

\begin{proposition} \label{prop_sol_hyp}
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ be a
differential operator with coefficients $c_\alpha\in\mathcal{G}_\infty(\mathbb{R}^n)$.
We assume that there exists $a,a'\in\mathbb{R}$, $a\le a'$, $0\le m'\le m$,
$R>0$ and a representative $(P_\varepsilon)_\varepsilon$ of $P$ fulfilling
the following conditions:
\begin{itemize}
\item[(i)] $|P_\varepsilon|^{(m)}_{\alpha,\beta}=O(\varepsilon^{a})$ for all
 $\alpha,\beta\in\mathbb{N}^n$;
\item[(ii)] there exists $c>0$ such that
$|P_\varepsilon(x,\xi)|\ge c\,\varepsilon^{a'}\lrangle{\xi}^{m'}$
for all $x\in\mathbb{R}^n$, for $|\xi|\ge R$ and for all $\varepsilon\in(0,1]$;
\item[(iii)] for all $\alpha,\beta\in\mathbb{N}^n$ there exists
$(c_{\alpha,\beta,\varepsilon})_\varepsilon$ with $c_{\alpha,\beta,\varepsilon}=O(1)$
such that
\[
|\partial^\alpha_\xi\partial^\beta_x P_\varepsilon(x,\xi)|\le c_{\alpha,\beta,\varepsilon}|P_\varepsilon(x,\xi)|\lrangle{\xi}^{-|\alpha|}
\]
for all $x\in\mathbb{R}^n$, for $|\xi|\ge R$ and for all $\varepsilon\in(0,1]$.
\end{itemize}
Then there exists $(q_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-{m'}}(\mathbb{R}^{2n}),-a'}$
and $(r_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),a-a'}$ such that
\[
P_\varepsilon(x,D)q_\varepsilon(x,D)=I+r_\varepsilon(x,D)
\]
for all $\varepsilon\in(0,1]$. Moreover there exists $s_\varepsilon(x,D)$ with
$(s_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),2a-2a'}$ such that
\[
q_\varepsilon(x,D)P_\varepsilon(x,D)=I+s_\varepsilon(x,D)
\]
for all $\varepsilon\in(0,1]$.
\end{proposition}

\begin{proof}
Let $\psi$ be a smooth function in the variable $\xi$ such that
$\psi(\xi)=0$ for $|\xi|\le R$ and $\psi(\xi)=1$ for $|\xi|\ge 2$.
By adapting the proof of
\cite[Proposition 8.1 and Theorem 8.1]{Garetto:04} to our situation
one easily obtains from the hypotheses (i) and (ii) that
\begin{itemize}
\item $q_{0,\varepsilon}:=\psi(\xi)P_\varepsilon^{-1}(x,\xi)$ defines a net
in $\mathcal{M}_{S^{-m'}(\mathbb{R}^{2n}),-a'}$,
\item $(q_{0,\varepsilon}\partial^\alpha_\xi\partial^\beta_x P_\varepsilon)_\varepsilon\in \mathcal{M}_{S^{-|\alpha|}(\mathbb{R}^{2n}),0}$ for all $\alpha,\beta\in\mathbb{N}^n$,
\item for each $j\ge 1$, the net
\[
q_{j,\varepsilon}:=-\Bigl\{\sum_{|\gamma|+l=j,\, l<j}\frac{(-i)^{|\gamma|}}{\gamma !}\partial^\gamma_\xi P_\varepsilon\partial^\gamma_x q_{l,\varepsilon}\Bigr\}q_{0,\varepsilon}
\]
belongs to $\mathcal{M}_{S^{-m'-j}(\mathbb{R}^{2n}),-a'}$.
\end{itemize}

Corollary \ref{corol_asymp} implies that there exists
$(q_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-m'}(\mathbb{R}^{2n}),-a'}$ having
$\{(q_{j,\varepsilon})_\varepsilon\}_j$ as asymptotic expansion with fixed
moderateness $\varepsilon^{-a'}$. Let us now consider the composition
$P_\varepsilon(x,D)q_\varepsilon(x,D)=\lambda_\varepsilon(x,D)$. Basic properties
of symbolic calculus show that
\[
\Bigl(\lambda_\varepsilon-\sum_{|\gamma|<r}\frac{(-i)^{|\gamma|}}{\gamma !}
\partial^\gamma_\xi P_\varepsilon\partial^\gamma_x q_\varepsilon\Bigr)_\varepsilon
\in\mathcal{M}_{S^{m-m'-r}(\mathbb{R}^{2n}),a-a'}
\]
for all $r\ge 1$. Making use of
$(q_\varepsilon-\sum_{l=0}^{r-1}q_{l,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{-m'-r}(\mathbb{R}^{2n}),-a'}$
we can write
\begin{align*}
\sum_{|\gamma|<r}\frac{(-i)^{|\gamma|}}{\gamma !}
 \partial^\gamma_\xi P_\varepsilon\partial^\gamma_x q_\varepsilon
&= \sum_{|\gamma|<r}\sum_{l=0}^{r-1}\frac{(-i)^{|\gamma|}}{\gamma !}
 \partial^\gamma_\xi P_\varepsilon\partial^\gamma_x q_{l,\varepsilon} +s_\varepsilon\\
&= P_\varepsilon q_{0,\varepsilon}+\sum_{j=1}^{r-1}P_\varepsilon q_{j,\varepsilon}
+\sum_{j=1}^{r-1}\sum_{|\gamma|+l=j,\, l<j}
 \frac{(-i)^{|\gamma|}}{\gamma !}\partial^\gamma_\xi
P_\varepsilon\partial^\gamma_x q_{l,\varepsilon} \\
&\quad +\sum_{|\gamma|+l\ge r,\, |\gamma|<r,
\, l<r}\frac{(-i)^{|\gamma|}}{\gamma !}\partial^\gamma_\xi P_\varepsilon
 \partial^\gamma_x q_{l,\varepsilon}+s_\varepsilon,
\end{align*}
where $(s_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{m-m'-r}(\mathbb{R}^{2n}),a-a'}$.
By definition of $q_{0,\varepsilon}$ and $q_{j,\varepsilon}$ we have that the
right-hand side of the previous formula equals
\[
1+\sum_{|\gamma|+l\ge r,\, |\gamma|<r,\, l<r}\frac{(-i)^{|\gamma|}}{\gamma !}\partial^\gamma_\xi P_\varepsilon\partial^\gamma_x q_{l,\varepsilon}+s_\varepsilon
\]
when $|\xi|\ge 2R$. Hence the net $(\lambda_\varepsilon-1)_\varepsilon$ belongs
to $\mathcal{M}_{S^{m-m'-r}(\mathbb{R}^{2n}),a-a'}$ for $|\xi|\ge 2R$.
Since $q_{0,\varepsilon}P_\varepsilon(x,\xi)-1=\psi(\xi)-1\in\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$ the
domain restriction can be dropped. Concluding,
$(\lambda_\varepsilon-1)_\varepsilon:=(r_\varepsilon)_\varepsilon$ is an element of
$\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),a-a'}$. Analogously one can construct
a net of symbols $(q'_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-m'}(\mathbb{R}^{2n}),-a'}$
such that $q'_\varepsilon(x,D)P_\varepsilon(x,D)=I+r'_\varepsilon(x,D)$
 with $(r'_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),a-a'}$.
By applying $q'_\varepsilon(x,D)P_\varepsilon(x,D)$ to $q_\varepsilon(x,D)$ we get
\[
(I+r'_\varepsilon(x,D))q_\varepsilon(x,D)=q'_\varepsilon(x,D)P_\varepsilon(x,D)q_\varepsilon(x,D)
=q'_\varepsilon(x,D)(I+r_\varepsilon(x,D))
\]
which at the level of symbols means
\[
q_\varepsilon+r'_\varepsilon\sharp q_\varepsilon=q'_\varepsilon+q'_\varepsilon\sharp r_\varepsilon.
\]
Thus, $(q_\varepsilon-q'_\varepsilon)\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),a-2a'}$ and
since $a-a'\ge 2a-2a'$ the equality
$q_\varepsilon(x,D)P_\varepsilon(x,D)=I+s_\varepsilon(x,D)$ holds with
$(s_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),2a-2a'}$.
\end{proof}
A straightforward combination of Proposition \ref{prop_sol_hyp}
with Theorem \ref{theo_sol_par} entails the following result of
local solvability.

\begin{proposition} \label{prop_sol_hyp_2}
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ be a differential
operator with coefficients $c_\alpha\in\mathcal{G}_\infty(\mathbb{R}^n)$.
Let $(P_\varepsilon)_\varepsilon$ be a representative of $P$ fulfilling the
hypotheses of Proposition \ref{prop_sol_hyp} with $a=a'$.
Then for all $x_0\in\mathbb{R}^n$ there exists a neighborhood $\Omega$ of $x_0$
and a cut-off function $\phi$, identically $1$ near $x_0$, such that
\[
\forall F\in\mathcal{L}_{2,{\rm{loc}}}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})\ \exists
T\in\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})\quad P(x,D)T=\phi F \text{on $\Omega$}.
\]
\end{proposition}

\begin{proof}
If $a=a'$ from Proposition \ref{prop_sol_hyp} we have that there
exists a parametrix $q_\varepsilon(x,D)$ with
$(q_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-m'}(\mathbb{R}^{2n}),-a}$, $-m'\le 0$,
and a regularizing operator $r_\varepsilon(x,D)$ with
$(r_\varepsilon)_\varepsilon\in\mathcal{M}_{S^{-\infty}(\mathbb{R}^{2n}),0}$. This means that
$|r_\varepsilon|^{(l)}_{0,0}=O(1)$ for all $l\in\mathbb{R}$. The conditions under
which Theorem \ref{theo_sol_par} holds are therefore fulfilled.
\end{proof}

\begin{example} \rm
As an explanatory example we consider the operator generated by
\[
P_\varepsilon(x,D)=-\varepsilon^a\Delta+\sum_{|\alpha|\le 1}c_{\alpha,\varepsilon}(x)D^\alpha,
\]
where
\[
\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial_{x_i^2}},\quad
c_{\alpha,\varepsilon}=c_\alpha\ast\varphi_{\omega(\varepsilon)},\quad
c_\alpha\in L^\infty(\mathbb{R}^n),
\]
$\varphi$ is a mollifier in $\mathscr{S}(\mathbb{R}^n)$ and
$(\omega^{-1}(\varepsilon))_\varepsilon$ a slow scale net. It follows that
$[(c_{\alpha,\varepsilon})_\varepsilon]$ belongs to $\mathcal{G}_\infty(\mathbb{R}^n)$ with
\[
\Vert\partial^\beta c_{\alpha,\varepsilon}\Vert_\infty
\le \Vert c_\alpha\Vert_\infty\, \omega(\varepsilon)^{-|\beta|}
\Vert\partial^\beta\varphi\Vert_1\le c\varepsilon^{-b}
\]
for all $b>0$. For any $a<0$ this operator is locally solvable
in the sense of Theorem \ref{theo_sol_par} because it fulfills
the conditions (i), (ii), (iii) of Proposition \ref{prop_sol_hyp}
with $a=a'$.
\end{example}

Due to the existence of a generalized parametrix for the operator
$P(x,D)$ of Proposition \ref{prop_sol_hyp_2}, the local solution
inherits the regularity properties of the right hand-side.

\begin{proposition} \label{prop_sol_hyp_3}
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ be a
differential operator with coefficients $c_\alpha\in\mathcal{G}_\infty(\mathbb{R}^n)$.
Let $(P_\varepsilon)_\varepsilon$ be a representative of $P$ fulfilling the
hypotheses of Proposition \ref{prop_sol_hyp} with $a=a'$.
Then for all $x_0\in\mathbb{R}^n$ there exists a neighborhood $\Omega$ of $x_0$
and a cut-off function $\phi$, identically $1$ near $x_0$, such that
\begin{gather*}
\forall f\in\mathcal{G}(\mathbb{R}^n)\ \exists u\in\mathcal{G}(\Omega)\quad P(x,D)u=\phi f\quad
\text{on $\Omega$}, \\
\forall f\in\mathcal{G}^\infty (\mathbb{R}^n)\ \exists u\in\mathcal{G}^\infty (\Omega)\quad P(x,D)u=\phi
f\quad \text{on $\Omega$}.
\end{gather*}
\end{proposition}

\begin{proof}
First $f\in\mathcal{G}(\mathbb{R}^n)$ can be regarded as an element of
$\mathcal{L}_{2,{\rm{loc}}}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$. Let $(f_\varepsilon)_\varepsilon$ be
a representative of $f$. From Theorem \ref{theo_sol_par} we
can construct a local solution $u\in\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$
having a representative
\[
u_\varepsilon=\widetilde{q_\varepsilon}(\widetilde{I}+\widetilde{r_\varepsilon})^{-1}(\phi f_\varepsilon)
\]
in $\mathcal{M}_{L^2(\Omega)}$. The equality
\[
q_\varepsilon(x,D)P_\varepsilon(x,D)=I+s_\varepsilon(x,D)
\]
holds on $\mathscr{S}'(\mathbb{R}^n)$. Taking the restrictions of the previous
operators to the open set $\Omega$, since they all map $L^2(\Omega)$
into $L^2(\Omega)$ we have that
\begin{equation} \label{formula}
q_\varepsilon(x,D)P_\varepsilon(x,D)v = v+s_\varepsilon(x,D)v
\end{equation}
holds on $\Omega$ for all $v\in L^2(\Omega)$. Here and in the sequel
we omit the restriction notation $|_\Omega$ for the sake of simplicity.
 From \eqref{formula} follows
\[
u_\varepsilon+s_\varepsilon(x,D)u_\varepsilon=q_\varepsilon(x,D)(\phi f_\varepsilon).
\]
Since $s_\varepsilon(x,D)$ is a regularizing operator and $(f_\varepsilon)_\varepsilon$
is a net of smooth functions we already see that $(u_\varepsilon)_\varepsilon$
is a net of smooth functions as well. In particular,
from the mapping properties of generalized pseudodifferential
operators we know that
$(q_\varepsilon(x,D)(\phi f_\varepsilon))_\varepsilon\in\mathcal{M}_{\mathcal{C}^\infty(\Omega)}=\mathcal{E}_M(\Omega)$.
Finally we write $s_\varepsilon(x,D)u_\varepsilon$ as
\[
\int_\Omega k_{s_\varepsilon}(x,y)u_\varepsilon(y)\, dy.
\]
Combining the boundedness of the open set $\Omega$ with the following
kernel property
\[
\forall\alpha\in\mathbb{N}^n\, \forall d\in \mathbb{N}\quad
\sup_{(x,y)\in\mathbb{R}^{2n}}\lrangle{x}^{-d}\lrangle{y}^d
|\partial^{\alpha}_x\partial^{\beta}_y
k_{s_\varepsilon}(x,y)|=O(1)
\]
we obtain
\begin{equation}
\label{regularizing}
\sup_{x\in\Omega}|\partial^\alpha s_\varepsilon(x,D)u_\varepsilon|
\le \sup_{x\in\Omega}\Vert \partial^\alpha_x k_{s_\varepsilon}(x,\cdot)
\Vert_{L^2(\Omega)}\, \Vert u_\varepsilon\Vert_{L^2(\Omega)}
\le c\Vert u_\varepsilon\Vert_{L^2(\Omega)}
\end{equation}
for $\varepsilon$ small enough. Hence $(s_\varepsilon(x,D)u_\varepsilon)_\varepsilon\in\mathcal{E}_M(\Omega)$.
 Concluding the net $(u_\varepsilon)_\varepsilon$ belongs to $\mathcal{E}_M(\Omega)$ and generates
a solution $u$ in $\mathcal{G}(\Omega)$ to $P(x,D)u=\phi f$.

When $f\in\mathcal{G}^\infty (\mathbb{R}^n)$ since the net of symbols $(q_\varepsilon)_\varepsilon$ is
regular we have that $(q_\varepsilon(x,D)(\phi f_\varepsilon))_\varepsilon$ generates
an element of $\mathcal{G}^\infty (\Omega)$. Clearly as one sees in \eqref{regularizing}
also $s(x,D)u$ belongs to $\mathcal{G}^\infty (\Omega)$. Hence $u\in\mathcal{G}^\infty (\Omega)$.
\end{proof}

\section{Local solvability of partial differential operators
$\mathcal{G}$-elliptic in a neighborhood of a point}
\label{sec_elliptic}

In this section we concentrate on a special type of partial
differential operators with coefficients in $\mathcal{G}(\mathbb{R}^n)$.
Their properties will inspire the more general model introduced
in Section 4. In the sequel we often refer to the work on
 generalized hypoelliptic and elliptic symbols
in \cite{Garetto:ISAAC07, Garetto:08c, GGO:03}.

\begin{definition} \label{def_ellip_point} \rm
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ be a
partial differential operator with coefficients in $\mathcal{G}(\mathbb{R}^n)$.
We say that $P(x,D)$ is $\mathcal{G}$-elliptic in a neighborhood of $x_0$
if there exists a representative $(P_{m,\varepsilon})_\varepsilon$ of the
principal symbol $P_{m}$, a neighborhood $\Omega$ of $x_0$, $a\in\mathbb{R}$
 and $c>0$ such that
\begin{equation} \label{def_ellip}
|P_{m,\varepsilon}(x,\xi)|\ge c\varepsilon^a
\end{equation}
for all $x\in\Omega$, for $|\xi|=1$ and for all $\varepsilon\in(0,1]$.
\end{definition}

\begin{remark} \label{rem_ellip} \rm
It is clear that \eqref{def_ellip} holds for an arbitrary
representative $(P_\varepsilon)_\varepsilon$ of $P$ on a smaller interval $(0,\eta]$
and with some smaller constant $c>0$. In addition if $P(x,D)$
is $\mathcal{G}$-elliptic in a neighborhood of $x_0$ then $P(x_0,D)$ is
$\mathcal{G}$-elliptic. The converse does not hold.
Indeed, let $\varphi\in\mathcal{C}^\infty_{\rm c}(\mathbb{R})$ with $\varphi(0)=1$
and $\varphi(x)=0$ for $|x|\ge 2$. The differential operator
$P(x,D)$ with representative $P_\varepsilon(x,D)=\varphi(x/\varepsilon)D^2$
is $\mathcal{G}$-elliptic in $0$ but not in a neighborhood $\{|x|<r\}$ of $0$.
This is due to the fact that $P_\varepsilon(x,\xi)=0$ for $x\neq 0$ and
$\varepsilon<2^{-1}|x|$.
\end{remark}

As for $\mathcal{G}$-elliptic operators with constant Colombeau coefficients
(see \cite[Section6]{Garetto:08c}) the following estimates hold
in a neighborhood of $x_0$.

\begin{proposition} \label{prop_ellip_point}
Let $P(x,D)$ be $\mathcal{G}$-elliptic in a neighborhood of $x_0$. Then there
exists a representative $(P_\varepsilon)_\varepsilon$ of $P$, a neighborhood
$\Omega$ of $x_0$, moderate strictly nonzero nets $(R_\varepsilon)_\varepsilon$
and $(c_{\alpha,\beta,\varepsilon})_\varepsilon$ and a constant $c_0>0$ such that
\begin{gather*}
|P_{\varepsilon}(x,\xi)|\ge c_0\varepsilon^a\lrangle{\xi}^m,\\
|\partial^\alpha_\xi\partial^\beta_x P_{\varepsilon}(x,\xi)|
\le c_{\alpha,\beta,\varepsilon}|P_\varepsilon(x,\xi)|\lrangle{\xi}^{-|\alpha|}
\end{gather*}
for $x\in\Omega$, $|\xi|\ge R_\varepsilon$ and for all $\varepsilon\in(0,1]$.
\end{proposition}

\begin{proof}
>From Definition \ref{def_ellip_point} we have
\[
|P_\varepsilon(x,\xi)|\ge |P_{m,\varepsilon}(x,\xi)|
-|P_{\varepsilon}(x,\xi)-P_{m,\varepsilon}(x,\xi)|\ge c\varepsilon^a|\xi|^m
 -c_{m-1,\varepsilon}\lrangle{\xi}^{m-1}
\]
for all $\xi\in\mathbb{R}^n$, $x\in\Omega$, $\varepsilon\in(0,1]$ and with
$(c_{m-1,\varepsilon})_\varepsilon$ a moderate and strictly nonzero net.
Defining the radius $R_\varepsilon=\max\{1,2^m c_{m-1,\varepsilon}c^{-1}\varepsilon^{-a}\}$
we get for $x\in\Omega$, $|\xi|\ge R_\varepsilon$ and for all $\varepsilon$, the inequality
\[
|P_\varepsilon(x,\xi)|\ge |\xi|^m(c\varepsilon^a-c_{m-1,\varepsilon}2^{m-1}|\xi|^{-1})
\ge \frac{c}{2}\varepsilon^a|\xi|^m\ge c_0\varepsilon^a\lrangle{\xi}^m.
\]
Concerning the derivatives we have, always for $|\xi|\ge R_\varepsilon$,
\begin{align*}
|\partial^\alpha_\xi\partial^\beta_x P_{\varepsilon}(x,\xi)|
&\le \lambda_{\alpha,\beta,\varepsilon}\lrangle{\xi}^{m-|\alpha|}\\
&\le \lambda_{\alpha,\beta,\varepsilon}c_0^{-1}\varepsilon^{-a}|P_\varepsilon(x,\xi)
|\lrangle{\xi}^{-|\alpha|}\\
&=c_{\alpha,\beta,\varepsilon}|P_\varepsilon(x,\xi)|
\lrangle{\xi}^{-|\alpha|}.
\end{align*}
\end{proof}

By adapting the arguments of Proposition \ref{prop_sol_hyp} to this
kind of nets of symbols, and in analogy with \cite[Theorem 6.8]{GGO:03},
\cite[Propositions 2.7, 2.8]{Garetto:ISAAC07}, we obtain that a
differential operator $\mathcal{G}$-elliptic in a neighborhood of $x_0$
admits a local parametrix.

\begin{proposition} \label{prop_par_point}
Let $P(x,D)$ be $\mathcal{G}$-elliptic in a neighborhood of $x_0$.
Then there exists a neighborhood $\Omega$ of $x_0$ and generalized
symbols $q\in\mathcal{G}_{S^{-m}(\Omega\times\mathbb{R}^n)}$ and
$r,s\in\mathcal{G}_{S^{-\infty}(\Omega\times\mathbb{R}^n)}$ such that
\begin{equation} \label{formula_1}
\begin{gathered}
P(x,D)q(x,D)=I+r(x,D),\\
q(x,D)P(x,D)=I+s(x,D)
\end{gathered}
\end{equation}
as operators acting on $\mathcal{G}_{\rm c}(\Omega)$ with values in $\mathcal{G}(\Omega)$.
\end{proposition}

Note that if we take $\Omega$ bounded we can assume that the estimates involving the symbols in \eqref{formula_1} are global in $\xi$ and $x$ as well. The equalities between generalized operators followed from the corresponding equalities at the level of representatives. More precisely,
\[
\begin{split}
P_\varepsilon(x,D)q_\varepsilon(x,D)&=I+r_\varepsilon(x,D),\\
q_\varepsilon(x,D)P_\varepsilon(x,D)&=I+s_\varepsilon(x,D)
\end{split}
\]
for all $\varepsilon\in(0,1]$. Since $P(x,D)$ is properly supported
$q(x,D)P(x,D)=I+s(x,D)$ holds on $\mathcal{G}(\Omega)$ and $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$
as well. It follows that if $P(x,D)$ is $\mathcal{G}$-elliptic in a
neighborhood of $x_0$ and locally solvable then it inherits
the regularity of the right-hand side, in the sense that if
$P(x,D)T=v$ on $\Omega$ with $T\in\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}}$ and
$v\in\mathcal{G}(\Omega)$ then $T\in\mathcal{G}(\Omega)$. The problem is that
Definition \ref{def_ellip_point} in general does not guarantee
the assumption on the parametrix $(q_\varepsilon)_\varepsilon$ and the
regularizing term $(r_\varepsilon)_\varepsilon$ which allow to apply
Theorem \ref{theo_sol_par} and obtain local solvability.
In the following particular case an operator which is $\mathcal{G}$-elliptic
in a neighborhood of $x_0$ is also locally solvable at $x_0$.

\begin{proposition}\label{prop_loc_sol_ellip_point}
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$
be $\mathcal{G}$-elliptic in a neighborhood of $x_0$ with
\[
|P_{m,\varepsilon}(x,\xi)|\ge c\varepsilon^a
\]
in a neighborhood $\Omega_1$ of $x_0$, for all $\xi\in\mathbb{R}^n$
with $|\xi|=1$ and for all $\varepsilon\in(0,1]$. If the coefficients
$c_\alpha$ are $\mathcal{G}^\infty $-regular in $x_0$ of order $a$;
 i.e. on a neighborhood $\Omega_2$ of $x_0$ the following
\[
\forall\beta\in\mathbb{N}^n,\quad \sup_{x\in\Omega_2}|\partial^\beta
c_{\alpha,\varepsilon}(x)|=O(\varepsilon^a)
\]
holds, then there exist a neighborhood $\Omega$ of $x_0$ and
a cut-off function $\phi\in\mathcal{C}^\infty_{\rm c}(\Omega)$ identically $1$
near  $x_0$ such that:
\begin{itemize}
\item[(i)] for all $F\in\mathcal{L}_{2,{\rm{loc}}}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$
  there exist $T\in\mathcal{L}(\mathcal{G}_{\rm c}(\Omega),\widetilde{\mathbb{C}})$, which is a solution of
$P(x,D)T=\phi F$   on $\Omega$;
\item[(ii)] for all $f\in\mathcal{G}(\mathbb{R}^n)$ there exists $u\in\mathcal{G}(\Omega)$
  solving $P(x,D)u=\phi f$ on $\Omega$;
\item[(iii)] for all $f\in\mathcal{G}^\infty (\mathbb{R}^n)$ there exists $u\in\mathcal{G}^\infty (\Omega)$
  solving $P(x,D)u=\phi f$ on $\Omega$;
\end{itemize}
\end{proposition}

\begin{proof}
We can choose a representative $(P_\varepsilon)_\varepsilon$ such that the
inequalities $|P_{m,\varepsilon}(x,\xi)|\le c\varepsilon^a$ and
$|P_{\varepsilon}(x,\xi)-P_{m,\varepsilon}(x,\xi)|\le c_{m-1}\varepsilon^a\lrangle{\xi}^{m-1}$
hold on the interval $(0,1]$, for all $x$ in a  neighborhood of $x_0$
and all $\xi\in\mathbb{R}^n$. By following the proof of Proposition
\ref{prop_ellip_point} we see that the radius does not depend
on $\varepsilon$ and that the nets $(c_{\alpha,\beta,\varepsilon})$ are $O(1)$
as $\varepsilon$ tends to $0$. We are under the hypotheses of
Proposition \ref{prop_sol_hyp_2}. This yields the first assertion.
Proposition \ref{prop_par_point} and the considerations above on
the regularity of $P(x,D)$ prove assertion (ii).
Finally, assertion (iii) is clear for Proposition \ref{prop_sol_hyp_3}.
\end{proof}

We have found a class of partial differential operators with
coefficients in $\mathcal{G}(\mathbb{R}^n)$, that under the hypothesis of
$\mathcal{G}$-ellipticity in a neighborhood $x_0$ and under suitable
assumptions on the moderateness of the coefficients, are
locally solvable at $x_0$. These locally solvable operators
belong to the wider family of operators which can be written
in the form
\begin{equation}
\label{const_str_form}
P_0(D)+\sum_{j=1}^r c_j(x)P_j(D),
\end{equation}
in a neighborhood of $x_0$. Here the operators
$P_0(D)$, $P_j(D)$, $j=1,\dots ,r$ have constant Colombeau coefficients
and each $c_j$ is a Colombeau generalized function.

We conclude this section by proving that a differential operator
$P(x,D)$ with coefficients in $\mathcal{G}$ which is $\mathcal{G}$-elliptic in
$x_0$, i.e. $P(x_0,D)$ is $\mathcal{G}$-elliptic, can be written in the
form \eqref{const_str_form}.

\begin{proposition} \label{prop_ellip_cs}
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ be a differential
operator with coefficients in $\mathcal{G}(\mathbb{R}^n)$ which is $\mathcal{G}$-elliptic
in $x_0$. Then $P(x,D)$ can be written in the
 form \eqref{const_str_form} with
\begin{itemize}
\item $c_j\in\mathcal{G}(\mathbb{R}^n)$, $c_j(x_0)=0$, $P_0(D)$ and $P_j(D)$
operators with constant Colom\-beau coefficients,
\item $\widetilde{P_0}$ invertible in some point of $\mathbb{R}^n$,
\item $P_0(D)$ stronger than any $P_j(D)$.
\end{itemize}
\end{proposition}

\begin{proof}
We set $P_0(D)=P(x_0,D)$ and we have
\[
P(x,D)=P_0(D)+\sum_{|\alpha|\le m}(c_\alpha(x)-c_\alpha(x_0))D^\alpha.
\]
Clearly the coefficients $c_\alpha(x)-c_\alpha(x_0)$ belong
to $\mathcal{G}(\mathbb{R}^n)$ and vanish for $x=x_0$. For each $\alpha\in\mathbb{N}^n$
we find an operator $P_{j(\alpha)}(D)=D^\alpha$. By hypothesis
$P_0(D)$ is $\mathcal{G}$-elliptic in $x_0$. Hence from
Proposition \ref{prop_g_ellip} we have that $P_0(D)$ is stronger
than any $P_{j(\alpha)}(D)$ with $|\alpha|\le m$ and the weight
function $\widetilde{P_{0}}$ is invertible in any point of $\mathbb{R}^n$.
\end{proof}

A differential operator which is $\mathcal{G}$-elliptic in a neighborhood
$\Omega$ of $x_0$ is in particular $\mathcal{G}$-elliptic in $x_0$ and therefore
it can be written in the form \eqref{const_str_form} on the
whole of $\mathbb{R}^n$. The special structure \eqref{const_str_form}
of the $\mathcal{G}$-elliptic operators motivates the investigations
of Section \ref{sec_bounded}.

\section{Bounded perturbations of differential operators with
constant Colombeau coefficients: definition and examples}
\label{sec_bounded}

In this section we concentrate on operators with coefficients
in $\mathcal{G}(\mathbb{R}^n)$ which are locally a bounded perturbation of a
differential operator with constant Colombeau coefficients
as in \eqref{const_str_form}. More precisely, we say that
$P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ is of
\emph{bounded perturbation type}, or of \emph{BP-type},
in a neighborhood $\Omega$ of $x_0$ if it has the form
\[
P_0(D)+\sum_{j=1}^r c_j(x)P_j(D),
\]
when restricted to $\Omega$, with
\begin{itemize}
\item[(H1)] $c_j\in\mathcal{G}(\Omega)$, $c_j(x_0)=0$, $P_0(D)=P(x_0,D)$ and
  $P_j(D)$, $j=1,\dots ,r$, operators with constant Colombeau coefficients
\item[(H2)] $\widetilde{P_0}$ invertible in some point of $\mathbb{R}^n$,
\item[(H3)] $P_0(D)$ stronger than any $P_j(D)$.
\end{itemize}

\begin{remark} \rm
Our definition of BP-type is clearly inspired by the classical
theory of operators of constant strength
(see \cite{Hoermander:63, Hoermander:V2}). The direct generalization
of this concept to the Colombeau setting would mean to require
$P(x_0,D)\prec P(x,D)\prec P(x_0,D)$ for all $x$ in neighborhood
$\Omega$ of $x_0$ with $\prec$ the order relation introduced in Section 1.
 However, due to some some structural and technical constraints of our
framework, it is not clear at the moment if one can obtain from this
general definition a local bounded perturbation property as above.
This is related to the fact that one can not use the properties of
a linear space on the set of differential operators with coefficients
in $\widetilde{\mathbb{C}}$ weaker than $P_0(D)$. Indeed this set has the algebraic
structure of a module over $\widetilde{\mathbb{C}}$ and $\widetilde{\mathbb{C}}$ is only a ring
and not a field.
\end{remark}

As for the operators with constant Colombeau coefficients in
\cite[Theorem 7.8]{HO:03}, the local solvability of $P(x,D)$
in the Colombeau algebra $\mathcal{G}(\Omega)$, where $\Omega$ is an open
neighborhood of $x_0$, implies the invertibility of the weight
function $\widetilde{P_0}$ in some point of $\mathbb{R}^n$.

\begin{proposition} \label{prop_nec_cond}
Let $P(x,D)$ be a differential operator with coefficients
in $\mathcal{G}(\mathbb{R}^n)$ such that has the form
\[
P_0(D)+\sum_{j=1}^r c_j(x)P_j(D),
\]
in a neighborhood $\Omega$ of $x_0$ and fulfills the hypothesis {\rm (H1)}.
Let $v\in\mathcal{G}(\Omega)$ with $v(x_0)$ invertible in $\widetilde{\mathbb{C}}$.
If the equation $P(x,D)u=v$ is solvable in $\mathcal{G}(\Omega)$ then $\widetilde{P_0}$
is invertible in some point of $\mathbb{R}^n$.
\end{proposition}

\begin{proof}
We begin by observing that
\[
v(x_0)=P(x_0,D)u(x_0)=P_0(D)u(x_0).
\]
 From \eqref{est_inv} we see that $\widetilde{P_0}$ is invertible
in some point of $\mathbb{R}^n$ if and only if it is invertible in
any point of $\mathbb{R}^n$ and then in particular in $\xi=0$.
We assume that $\widetilde{P_0}(0)$ is not invertible. It follows
that for all $q$ there exists $\varepsilon_q\in(0,q^{-1}]$ such that
\[
\widetilde{P_{\varepsilon_q}}^2(0)=\sum_{|\alpha|\le m}|
c_{\alpha,\varepsilon_q}(x_0)|^2(\alpha !)^2<\varepsilon_q^q.
\]
Choosing $\varepsilon_q\searrow 0$ we have
\[
|c_{\alpha,\varepsilon_\nu}(x_0)|^2\le (\alpha!)^{-2}\varepsilon_\nu^\nu\le
(\alpha!)^{-2}\varepsilon_\nu^q
\]
for all $\nu\ge q$. Hence, for $c_\varepsilon=1$ for $\varepsilon=\varepsilon_q$,
 $q\in\mathbb{N}$, and $c_\varepsilon=0$ otherwise, all the nets
$(c_\varepsilon\cdot c_{\alpha,\varepsilon}(x_0))_\varepsilon$ are negligible.
 Concluding, for $c=[(c_\varepsilon)_\varepsilon]\in\widetilde{\mathbb{R}}$ the equality
$v(x_0)=P_0(D)u(x_0)$ implies
\[
c\cdot v(x_0)=c\cdot P_0(D)u(x_0)=\sum_{|\alpha|\le m}
(c\cdot c_\alpha(x_0))D^\alpha u(x_0)=0,
\]
in contradiction with $c\cdot v(x_0)\neq 0$.
\end{proof}

We now collect some examples of operators of BP-type.
It is clear by Proposition \ref{prop_ellip_cs} that the differential
operators which are $\mathcal{G}$-elliptic in $x_0$ are of BP-type
in any neighborhood of $x_0$. For the advantage of the reader
we write two explicit examples.

\begin{example} \rm

(i) Let $c_i\in\mathcal{G}(\mathbb{R})$, $i=0,\dots ,3$, with $c_2(0)$
invertible in $\widetilde{\mathbb{C}}$, $\mathop{\rm supp}c_3\subseteq(-3/2,-1/2)$
and $\mathop{\rm supp} c_2\subseteq(-1,1)$. The operator
\[
P(x,D)=c_3(x)D^3+c_2(x)D^2+c_1(x)D+c_0(x)
\]
is a bounded perturbation of $P(0,D)$ in the neighborhood
$\Omega:=(-1/4,1/4)$. Indeed, $P(x,D)|_\Omega = c_2|_\Omega D^2
+c_1|_\Omega D+c_0|_\Omega$ and $P(x,D)|_\Omega$ is
$\mathcal{G}$-elliptic in $0$.

(ii) For $i=1,2$ let $\varphi_{i}\in\mathcal{C}^\infty_{\rm c}(\mathbb{R})$,
$\varphi_i(0)=1$ and $\varphi_{i,\varepsilon}(x)=\varphi_i(x/\varepsilon)$.
Let $c_\alpha\in\mathcal{G}(\mathbb{R})$ for $|\alpha|\le 1$.
The operator $P(x,D)$ with representative
\[
P_\varepsilon(x,D)=\varphi_{1,\varepsilon}(x)D^2_{x_1}+\varphi_{2,\varepsilon}(x)D^2_{x_2}
+\sum_{|\alpha|\le 1}c_{\alpha,\varepsilon}(x)D^\alpha
\]
is a bounded perturbation of $P(0,D)$. More precisely we can write
\begin{align*}
P_\varepsilon(x,D)&=D^2_{x_1}+D^2_{x_2}+(\varphi_{1,\varepsilon}(x)-1)D_{x_1}^2
+(\varphi_{2,\varepsilon}(x)-1)D_{x_2}^2\\
&\quad +\sum_{|\alpha|\le 1}(c_{\alpha,\varepsilon}(x)-c_{\alpha,\varepsilon}(0))D^\alpha.
\end{align*}
\end{example}

A statement analogous to Proposition \ref{prop_ellip_cs} holds
for operators of principal type.

\begin{proposition} \label{prop_princ_cs}
Let $P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ be a
differential operator with coefficients in $\mathcal{G}(\mathbb{R}^n)$.
Let the coefficients of the principal part be constant with
at least one of them invertible in $\widetilde{\mathbb{C}}$. If $P(x,D)$ is of
 principal type then it is of BP-type in any neighborhood $\Omega$
of any point $x_0\in\mathbb{R}^n$.
\end{proposition}

\begin{proof}
We take $P_0(D)=P(x_0,D)=P_m(D)+\sum_{|\alpha|
\le m-1}c_\alpha(x_0)D^\alpha$. By hypothesis we have that
$P_0(D)$ is of principal type. Moreover, the invertibility of
one of the coefficients of $P_m(D)$ entails the invertibility
of $\widetilde{P_0}(\xi)$. Now we write
\[
P(x,D)=P_0(D)+\sum_{|\alpha|\le m-1}(c_\alpha(x)-c_\alpha(x_0))D^\alpha.
\]
The operators $P_{j(\alpha)}(D)=D^\alpha$ are all of order less than or
equal to $m-1$ and therefore by Proposition \ref{prop_princ_type}(i)
 $P_0(D)$ dominates any $P_{j(\alpha)}$.
\end{proof}

In the next proposition we see an example of operator of BP-type
where the decomposition is obtained by deriving with respect to $\xi$.

\begin{proposition} \label{prop_deriv}
Let $P(x,\xi)=c_{2,0}\xi_1^2+c_{1,1}\xi_1\xi_2+c_{0,2}\xi_2^2
+c_{1,0}(x)\xi_1+c_{0,1}(x)\xi_2+c_{0,0}(x)$
be a polynomial in the $\mathbb{R}^2$-variable $(\xi_1,\xi_2)$ with
coefficients in $\mathcal{G}(\mathbb{R}^2)$. Let the coefficients of the principal
part be constant with at least one invertible. Let $x_0\in\mathbb{R}^2$
and $P_0(D)=P(x_0,D)$. If $4c_{2,0}c_{0,2}-c_{1,1}^2$ and
$2c_{2,0}+2c_{0,2}+c_{1,1}$ are invertible in $\widetilde{\mathbb{C}}$ then
for $j=1,2,3$ there exists $c_j\in\mathcal{G}(\mathbb{R}^2)$ with $c_j(x_0)=0$ such that
\[
P(x,D)=P_0(D)+c_1(x)P_1(D)+c_2(x)P_2(D)+c_3(x)P_3(D),
\]
where
\begin{gather*}
P_1(D)=P_0^{(1,0)}(D),\quad
P_2(D)= P_0^{(0,1)}(D),\\
P_3(D)= P_0^{(2,0)}(D)+P^{(1,1)}_0(D)+P^{(0,2)}_0(D).
\end{gather*}
Hence, $P(x,D)$ is of BP-type in any neighborhood of $x_0$.
\end{proposition}

\begin{proof}
We argue at the level of symbols. By fixing $x=x_0$ we have
\begin{gather*}
P_{0}(\xi)=P(x_0,\xi)=c_{2,0}\xi_1^2+c_{1,1}\xi_1\xi_2+c_{0,2}\xi_2^2+c_{1,0}(x_0)\xi_1+c_{0,1}(x_0)\xi_2+c_{0,0}(x_0)\\
P_{0}^{(1,0)}(\xi)= 2c_{2,0}\xi_1+c_{1,1}\xi_2+c_{1,0}(x_0),\\
P_{0}^{(0,1)}(\xi)= c_{1,1}\xi_1+2c_{0,2}\xi_2+c_{0,1}(x_0),\\
P_{0}^{(1,1)}(\xi)=c_{1,1},\\
P_{0}^{(2,0)}(\xi)= 2c_{2,0},\\
P_{0}^{(0,2)}(\xi)=2c_{0,2}.
\end{gather*}
 From the invertibility of one of the principal part's coefficients
we have that $\widetilde{P_0}$ is invertible in any point $\xi_0$ of $\mathbb{R}^n$.
Moreover, $P_1:=P_{0}^{(0,1)}\prec\prec P_0$,
$P_2:=P_{0}^{(0,1)}\prec\prec P_0$ and
$P_3:=P_0^{(2,0)}+P^{(1,1)}_0+P^{(0,2)}_0\prec\prec P_0$.
The operator
\begin{align*}
P(x,D)&=P_0(D)+(c_{1,0}(x)-c_{1,0}(x_0))D_{x_1}+(c_{0,1}(x)
-c_{0,1}(x_0))D_{x_2}\\
&\quad +(c_{0,0}(x)-c_{0,0}(x_0))
\end{align*}
can be written as
\[
P_0(D)+c_1(x)P_1(D)+c_2(x)P_2(D)+c_3(x)P_3(D),
\]
where the generalized functions $c_1(x)$, $c_2(x)$ and $c_3(x)$
are solutions of the  system:
\begin{gather*}
2c_{2,0}\,c_1(x) + c_{1,1}\,c_2(x) = c_{1,0}(x)-c_{1,0}(x_0),\\
c_{1,1}\,c_1(x)  + 2c_{0,2}\,c_2(x)= c_{0,1}(x)-c_{0,1}(x_0),\\
c_{1,0}(x_0)\,c_1(x) + c_{0,1}(x_0)\,c_2(x) +(2c_{2,0}+2c_{0,2}
+c_{1,1})c_3(x)= c_{0,0}(x)-c_{0,0}(x_0).
\end{gather*}
In detail,
\begin{gather*}
c_1(x)=\frac{2c_{0,2}(c_{1,0}(x)-c_{1,0}(x_0))
 -c_{1,1}(c_{0,1}(x)-c_{0,1}(x_0))}{4c_{2,0}c_{0,2}-c_{1,1}^2},\\
c_2(x)=\frac{2c_{2,0}(c_{0,1}(x)-c_{0,1}(x_0))-c_{1,1}(c_{1,0}(x)
 -c_{1,0}(x_0))}{4c_{2,0}c_{0,2}-c_{1,1}^2},\\
c_3(x)=\frac{(c_{0,0}(x)-c_{0,0}(x_0))-c_{1,0}(x_0)c_1(x)
-c_{0,1}(x_0)c_2(x)}{2c_{2,0}+2c_{0,2}+c_{1,1}}.
\end{gather*}
 It is clear that $c_1(x)$, $c_2(x)$ and $c_3(x)$ vanish at $x=x_0$.
\end{proof}

\section{Conditions of local solvability for operators of bounded
perturbation type in the Colombeau context}
\label{sec_local}

Purpose of this section is to provide sufficient conditions of local
solvability for an operator
$P(x,D)=\sum_{|\alpha|\le m}c_\alpha(x)D^\alpha$ with coefficients
in $\mathcal{G}(\mathbb{R}^n)$ which is of bounded perturbation type in a
neighborhood $\Omega$ of $x_0$, that is
\[
P(x,D)=P_0(D)+\sum_{j=1}^r c_j(x)P_j(D)
\]
on $\Omega$ with,
\begin{itemize}
\item[(H1)] for all $j=1,\dots ,n$, $c_j\in\mathcal{G}(\Omega)$, $c_j(x_0)=0$,
 $P_0(D)$ and $P_j(D)$ operators with constant Colombeau coefficients
\item[(H2)] $\widetilde{P_0}$ invertible in some point of $\mathbb{R}^n$,
\item[(H3)] $P_0(D)$ stronger than any $P_j(D)$.
\end{itemize}
This requires some specific properties of the spaces $B_{p,k}$
which are proven in \cite[Chapters X, XIII]{Hoermander:V2} and
collected in the sequel.

\subsection{The spaces $B_{p,k}$: properties and calculus}

Given $k\in\mathcal{K}$ we define for any $\nu>0$ the functions
\begin{equation}
\label{def_k_delta}
k_\nu(\xi)=\sup_\eta \mathrm{e}^{-\nu|\eta|}k(\xi-\eta),\quad
M_k(\xi)=\sup_\eta k(\xi+\eta)/k(\eta).
\end{equation}
One easily proves that $k_\nu$ and $M_k$ are both tempered
weight functions. More precisely there exists a constant
$C_\nu>0$ such that for all $\xi\in\mathbb{R}^n$,
\[
1\le k_\nu(\xi)/k(\xi)\le C_\nu.
\]
If $k(\xi+\eta)\le(1+C|\xi|)^Nk(\eta)$ then
\[
M_{k_\nu}(\xi)\le (1+C|\xi|)^N,
\]
for all $\nu>0$. In particular $M_{k_\nu}\to 1$ uniformly on
compact subsets of $\mathbb{R}^n$ when $\nu\to 0$. The following
theorem collects some important properties of the spaces
$B_{p,k}$ which are proven in \cite[Chapters X, XIII]{Hoermander:V2}.

\begin{theorem} \label{theo_Bpk}
\begin{itemize}
\item[(i)] If $u_1\in B_{p,k_1}\cap\mathcal{E}'$ and $u_2\in B_{\infty,k_2}$
then $u_1\ast u_2\in B_{p,k_1k_2}$ and
\[
\Vert u_1\ast u_2\Vert_{p,k_1k_2} \le \Vert u_1\Vert_{p,k_1}\,
 \Vert u_2\Vert_{\infty,k_2}.
\]
\item[(ii)] If $u\in B_{p,k}$ and $\phi\in\mathscr{S}(\mathbb{R}^n)$ then
$\phi u\in B_{p,k}$ and
\[
\Vert \phi u\Vert_{p,k}\le \Vert\phi\Vert_{1,M_k}\, \Vert u\Vert_{p,k}.
\]
\item[(iii)] For every $\phi\in\mathscr{S}(\mathbb{R}^n)$ there exists $\nu_0>0$ such that
\[
\Vert \phi u\Vert_{p,k_{\nu}}\le 2\Vert \phi\Vert_{1,1}\Vert u\Vert_{p,k_{\nu}}
\]
for all $\nu\in(0,\nu_0)$.
\item[(iv)] If $\psi\in\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$, $x_0\in\mathbb{R}^n$ and $h$ is a $\mathcal{C}^\infty$-function with $h(x_0)=0$ then for $\psi_{\delta,x_0}(x)=\psi((x-x_0)/\delta)$ one has
\[
\Vert \psi_{\delta,x_0}h\Vert_{1,1}=O(\delta)
\]
as $\delta\to 0$.
\end{itemize}
\end{theorem}

Note that the number $\nu_0$ in (iii) depends on $\phi$ and the
weight function $k$ while the norm of the operator $u\to\phi u$ does
not depend on $k$.

The next proposition concerns nets of distributions and nets
of $B_{p,k}$-elements.

\begin{proposition} \label{prop_basic}
\quad
\begin{itemize}
\item[(i)] If $(g_\varepsilon)_\varepsilon\in\mathcal{E}'(\mathbb{R}^n)^{(0,1]}$ generates a basic
functional $\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ then for all $p\in[1,+\infty]$
there exists $k\in\mathcal{K}$ such that
$(g_\varepsilon)_\varepsilon\in\mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$; in particular if $(g_\varepsilon)_\varepsilon$
is the representative of a generalized function in $\mathcal{G}_{\rm c}(\mathbb{R}^n)$ then
$(g_\varepsilon)_\varepsilon\in\mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$ for all $k$.
\item[(ii)] If $(g_\varepsilon)_\varepsilon\in\mathcal{E}'(\mathbb{R}^n)^{(0,1]}$ with
$\mathop{\rm supp} g_\varepsilon\subseteq K\Subset\mathbb{R}^n$ for all $\varepsilon$ and
$(g_\varepsilon)_\varepsilon\in\mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$ then $(g_\varepsilon)_\varepsilon$
generates a basic functional in $\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.
\item[(iii)] If $(g_\varepsilon)_\varepsilon\in\mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$ and
$\lrangle{\xi}^j k^{-1}(\xi)\in L^{q}(\mathbb{R}^n)$ with $1/p+1/q=1$
then $(g_\varepsilon)_\varepsilon\in\mathcal{M}_{\mathcal{C}^j(\mathbb{R}^n)}$.
\item[(iv)] If $(S_\varepsilon)_\varepsilon$ and $(T_\varepsilon)_\varepsilon$ generate
basic functionals in $\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ then
$(S_\varepsilon\ast T_\varepsilon)_\varepsilon$ defines a basic functional too.
\end{itemize}
\end{proposition}

\begin{proof}
(i) The definition of a basic functional implies that there exists
a moderate net $(c_\varepsilon)_\varepsilon$, a compact set $K\Subset\mathbb{R}^n$ and
 a number $m\in\mathbb{N}$ such that the estimate
\[
|\widehat{g_\varepsilon}(\xi)|=|g_\varepsilon(\mathrm{e}^{-i\cdot\xi})|
\le c_\varepsilon\sup_{x\in K, |\alpha|\le m}|\partial^\alpha_x\mathrm{e}^{-ix\xi}|
\le c_\varepsilon\lrangle{\xi}^m
\]
holds for all $\varepsilon\in(0,1]$ and $\xi\in\mathbb{R}^n$.
For $k(\xi):=\lrangle{\xi}^{\frac{-mp-n-1}{p}}$ we obtain
\[
\Vert g_\varepsilon\Vert_{p,k}^p=\int_{\mathbb{R}^n}\lrangle{\xi}^{-mp-n-1}|
\widehat{g_\varepsilon}(\xi)|^p\, d\xi \le {c'_\varepsilon}
\int_{\mathbb{R}^n}\lrangle{\xi}^{-mp-n-1}\lrangle{\xi}^{mp}\, d\xi.
\]
Thus $(g_\varepsilon)_\varepsilon\in\mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$. The second assertion
of (i) is clear since if $(g_\varepsilon)_\varepsilon$ is the representative
of a generalized function in $\mathcal{G}_{\rm c}(\mathbb{R}^n)$ then
$(\widehat{g_\varepsilon})_\varepsilon$ is a moderate net of functions in
$\mathscr{S}(\mathbb{R}^n)$ and clearly a moderate net of $B_{p,k}$-functions
for all $k$.

(ii) Let $f\in\mathcal{C}^\infty(\mathbb{R}^n)$ and $\psi$ be a cut-off function
identically $1$ in a neighborhood of $K$. We can write $g_\varepsilon(f)$
as $\widehat{g_\varepsilon}((\psi f)\check\,)=k\widehat{g_\varepsilon}(k^{-1}
(\psi f)\check\,)$. Hence,
\[
|g_\varepsilon(f)|\le \Vert k\widehat{g_\varepsilon}\Vert_p\Vert k^{-1}(\psi f)\check\,
\Vert_q,
\]
with $1/p+1/q=1$. By choosing $h$ large enough, using the bound from
below $k(\xi)\ge k(0)(1+C|\xi|)^{-N}$ and the continuity of the
inverse Fourier transform on $\mathscr{S}(\mathbb{R}^n)$ we are led to
\[
|g_\varepsilon(f)|\le c\Vert k\widehat{g_\varepsilon}\Vert_p\,
\sup_{\xi\in\mathbb{R}^n, |\alpha|\le h}\lrangle{\xi}^h|
\partial^\alpha((\psi f)\check\,)|
\]
and, for some $h'\in\mathbb{N}$ and $c'>0$, to
\[
|g_\varepsilon(f)|\le c'\Vert k\widehat{g_\varepsilon}\Vert_p\sup_{x\in
\mathop{\rm supp}\psi, |\beta|\le h'}|\partial^\beta f(x)|.
\]
Since $(g_\varepsilon)_\varepsilon$ is $B_{p,k}$-moderate it follows that
$(g_\varepsilon)_\varepsilon$ defines a basic functional in $\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.

(iii) From the hypothesis $\lrangle{\xi}^j k^{-1}(\xi)\in L^{q}(\mathbb{R}^n)$
and $(g_\varepsilon)_\varepsilon\in\mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$ it follows that
$(\xi^\alpha \widehat{g_\varepsilon})_\varepsilon\in\mathcal{M}_{L^1(\mathbb{R}^n)}$ for all
$\alpha$ with $|\alpha|\le j$. Therefore,
$g_\varepsilon(x)=\int_{\mathbb{R}^n}\mathrm{e}^{ix\xi}\widehat{g_\varepsilon}(\xi)\, d\xi$
is a moderate net of $\mathcal{C}^j$-functions.

$(iv)$ Combining the property of basic functionals with the definition
of convolution we get
\begin{align*}
|S_\varepsilon\ast T_\varepsilon(f)|
&=|S_{\varepsilon,x}T_{\varepsilon,y}(f(x+y))|\\
&\le c_\varepsilon\sup_{x\in K, |\alpha|\le m}|\partial^\alpha_x(T_{\varepsilon,y}
 (f(x+y)))|\\
&=c_\varepsilon\sup_{x\in K, |\alpha|\le m}|T_{\varepsilon,y}(\partial^\alpha f(x+y))|\\
&\le c_\varepsilon\sup_{x\in K, |\alpha|\le m}c'_\varepsilon\sup_{y\in K',
|\beta|\le m'}|\partial^{\alpha+\beta}f(x+y)|\\
&\le c_\varepsilon c'_\varepsilon\sup_{z\in K+K', |\gamma|\le m+m'}|\partial^\gamma f(z)|,
\end{align*}
valid for all $f\in\mathcal{C}^\infty(\mathbb{R}^n)$ and for all $\varepsilon\in(0,1]$ with
$(c_\varepsilon)_\varepsilon$ and $(c'_\varepsilon)_\varepsilon$ moderate nets.
\end{proof}

In the course of the paper we will use the expression basic functional
$T\in\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ of order $N$ for a functional $T$ defined
by a net of distributions $(T_\varepsilon)_\varepsilon\in\mathcal{E}'(\mathbb{R}^n)^{(0,1]}$ such that
\[
|T_\varepsilon(f)|\le \lambda_\varepsilon\sup_{x\in K\Subset\mathbb{R}^n,
|\alpha|\le N}|\partial^\alpha f(x)|
\]
holds for all $f\in\mathcal{C}^\infty(\mathbb{R}^n)$, for all $\varepsilon\in(0,1]$ and for some
moderate net $(\lambda_\varepsilon)_\varepsilon$. It follows from
Proposition \ref{prop_basic}(i) that $(T_\varepsilon)_\varepsilon$ is a moderate
net in $B_{p,k}(\mathbb{R}^n)^{(0,1]}$ with
$k(\xi)=\lrangle{\xi}^{\frac{-Np-n-1}{p}}$.

In our investigation of the local solvability of $P(x,D)$
we distinguish between
\begin{enumerate}
\item[(1)] the coefficients $c_j$ are standard smooth functions,
\item[(2)] the coefficients $c_j$ are Colombeau generalized functions.
\end{enumerate}
In both these cases we will adapt the proof of
\cite[Theorem 13.3.3]{Hoermander:V2}
(or \cite[Theorem 7.3.1]{Hoermander:63}) to our generalized context.
>From the assumption of BP-type in $\Omega$ it is not restrictive
in the following statements to take
$P(x,D)=P_0(D)+\sum_{j=1}^r c_j(x)P_j(D)$ with coefficients
$c_j\in\mathcal{G}(\Omega)$.

\subsection{Local solvability: case $c_j\in\mathcal{C}^\infty$}

\begin{theorem} \label{theo_locsolv_easy}
Let $\Omega$ be a neighborhood of $x_0$ and let
\[
P(x,D)=P_0(D)+\sum_{j=1}^r c_j(x)P_j(D)
\]
 with $c_j\in\mathcal{C}^\infty(\Omega)$
for all $j$. If the hypotheses $(H1)$, $(H2)$, $(H3)$ are fulfilled
and in addition
\begin{itemize}
\item[(H4)] $\widetilde{P_{j,\varepsilon}}(\xi)\le\lambda_{j,\varepsilon}\widetilde{P_{0,\varepsilon}}(\xi)$,
where $\lambda_{j,\varepsilon}=O(1)$
\end{itemize}
holds for all $j$ and for a certain choice of representatives, then there exists a sufficiently small neighborhood $\Omega_{\delta}:=\{x:\, |x-x_0|<\delta\}$ of $x_0$ such that
\begin{itemize}
\item[(i)] for all $F\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ there exists $T\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ solving $P(x,D)T=F$ on $\Omega_\delta$,
\item[(ii)] for all $v\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$ there exists $u\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$ solving $P(x,D)u=v$ on $\Omega_{\delta}$.
\end{itemize}
\end{theorem}

\begin{proof}
We organize the proof in a few steps.

\textbf{Step 1: the operator $P_0(D)$.}
Since $\widetilde{P_0}$ is invertible in some point of $\mathbb{R}^n$ from
Theorem \ref{theo_fund_P} we know that there exists a fundamental
solution $E_0\in\mathcal{L}_{\rm b}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ having a representative
$(E_{0,\varepsilon})_\varepsilon$ such that
\[
\Bigl\Vert \frac{E_{0,\varepsilon}}{\cosh(c|x|)}\Bigr\Vert_{\infty,
\widetilde{P_{0,\varepsilon}}}\le C_0,
\]
for all $\varepsilon\in(0,1]$. Note that the constant $C_0$ does not
depend on $\varepsilon$. It follows from Theorem \ref{theo_Bpk}(ii),
the inequality \eqref{est_Hoer} and the definition of
$M_{\widetilde{P_{0,\varepsilon}}}$ the estimate
\begin{align*}
\Vert\varphi E_{0,\varepsilon}\Vert_{\infty,\widetilde{P_{0,\varepsilon}}}
&\le\Vert\varphi\cosh(c|x|)\Vert_{1,M_{\widetilde{P_{0,\varepsilon}}}}
\Bigl\Vert \frac{E_{0,\varepsilon}}{\cosh(c|x|)}\Bigr\Vert_{\infty,
 \widetilde{P_{0,\varepsilon}}}\\
&\le \Vert (1+C|\xi|)^m\mathcal{F}(\varphi\cosh(c|\cdot|))
 (\xi)\Vert_1\, \cdot C_0\le C_1
\end{align*}
valid for all $\varphi\in\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$ and $\varepsilon\in(0,1]$ with $m$
order of the polynomial $P_0$.

\textbf{Step 2: the equation $P_{0,\varepsilon}(D)u=f$ when $f\in\mathcal{E}'(\mathbb{R}^n)$
with $\mathop{\rm supp} f\subseteq \Omega_{\delta_0}$.}
Let $\delta_0>0$ and let $\chi$ be a function in $\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$
identically $1$ in a neighborhood of $\{x:\, |x|\le 2\delta_0\}$.
>From the previous considerations we have that
$F_{0,\varepsilon}:=\chi E_{0,\varepsilon}\in B_{\infty,\widetilde{P_{0,\varepsilon}}}$
with $\Vert F_{0,\varepsilon}\Vert_{\infty,\widetilde{P_{0,\varepsilon}}}\le C_1$
for all $\varepsilon$. Moreover, for all $f\in\mathcal{E}'(\mathbb{R}^n)$ with
$\mathop{\rm supp} f\subseteq \Omega_{\delta_0}$ we have that
\[
E_{0,\varepsilon}\ast f = F_{0,\varepsilon}\ast f
\]
on $\Omega_{\delta_0}$. Hence, by definition of a fundamental solution of $P_0(D)$, we can write on $\Omega_{\delta_0}$,
\begin{equation} \label{first_part}
P_{0,\varepsilon}(D)(F_{0,\varepsilon}\ast f)=f.
\end{equation}

\textbf{Step 3: the operator $\sum_{j=1}^r c_j(x)P_j(D)$ on
$\Omega_{\delta}\subseteq\Omega_{\delta_0}$.}
We now study the operator
\[
\sum_{j=1}^r c_j(x)P_j(D).
\]
Let $\psi\in\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$ such that $\psi(x)=1$ when $|x|\le 1$
and $\psi(x)=0$ when $|x|\ge 2$. We set
$\psi_{\delta,x_0}(x)=\psi((x-x_0)/\delta)$. We fixed the
representatives $(\widetilde{P_{0,\varepsilon}})_\varepsilon$, $(\widetilde{P_{j,\varepsilon}})_\varepsilon$
and $(\lambda_{j,\varepsilon})_\varepsilon$ fulfilling $(H4)$ and we study the
net of operators
\[
A_{\delta,\varepsilon}(g)=\sum_{j=1}^r \psi_{\delta,x_0}c_j
P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)
\]
defined for $g\in\mathcal{D}'(\mathbb{R}^n)$. More precisely for $k\in\mathcal{K}$,
$1\le p\le\infty$ and $k_\nu$ as in \eqref{def_k_delta} we want to
estimate $A_{\delta,\varepsilon}$ on $B_{p,k_\nu}$. Since $\psi_{\delta,x_0}c_j$
belongs to $\mathscr{S}(\mathbb{R}^n)$ by Theorem \ref{theo_Bpk}(iii) we find a
sufficiently small $\nu_\delta$, depending on the coefficients $c_j$
and on $\psi_{\delta,x_0}$, such that
\[
\Vert \psi_{\delta,x_0}c_j P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)
\Vert_{p,k_\nu}\le 2\Vert\psi_{\delta,x_0}c_j\Vert_{1,1}\,
\Vert P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)\Vert_{p,k_\nu}
\]
holds for all $\nu<\nu_\delta$. Hence, from Theorem \ref{theo_Bpk}(i),
the properties of the net $(F_{0,\varepsilon})_\varepsilon$ and (H4) we have
\begin{align*}
&\Vert A_{\delta,\varepsilon}(g)\Vert_{p,k_\nu}\\
&\le \sum_{j=1}^r 2\Vert\psi_{\delta,x_0}c_j\Vert_{1,1}\,
\Vert P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)\Vert_{p,k_\nu}\\
&\le \sum_{j=1}^r 2\Vert\psi_{\delta,x_0}c_j\Vert_{1,1}\Vert P_{j,\varepsilon}
 (D)F_{0,\varepsilon}\Vert_{\infty,1}\Vert g\Vert_{p,k_\nu}\\
&=\sum_{j=1}^r 2\Vert\psi_{\delta,x_0}c_j\Vert_{1,1}
 \Vert P_{j,\varepsilon}\widehat{F_{0,\varepsilon}}\Vert_{\infty}
 \Vert g\Vert_{p,k_\nu}\le \sum_{j=1}^r 2\Vert\psi_{\delta,x_0}c_j
 \Vert_{1,1}\Vert \widetilde{P_{j,\varepsilon}}\widehat{F_{0,\varepsilon}}\Vert_{\infty}\Vert
  g\Vert_{p,k_\nu}\\
&\le 2\sum_{j=1}^r\Vert\psi_{\delta,x_0}c_j\Vert_{1,1}\,\lambda_{j,\varepsilon}\,
\Vert F_{0,\varepsilon}\Vert_{\infty,\widetilde{P_{0,\varepsilon}}}
 \Vert g\Vert_{p,k_\nu}\le 2C_1\sum_{j=1}^r\Vert\psi_{\delta,x_0}c_j
 \Vert_{1,1}\,\lambda_{j,\varepsilon}\, \Vert g\Vert_{p,k_\nu}
\end{align*}
Since $c_j(x_0)=0$ the assumptions of Theorem \ref{theo_Bpk}(iv)
are satisfied. Hence, we have $\Vert\psi_{\delta,x_0}c_j\Vert_{1,1}=O(\delta)$.
Combining this fact with $|\lambda_{j,\varepsilon}|=O(1)$ we conclude that
there exist $\delta_1$ and $\epsilon_1$ small enough such that
\[
\Vert A_{\delta,\varepsilon}(g)\Vert_{p,k_\nu}\le 2^{-1}\Vert g\Vert_{p,k_\nu}
\]
is valid for all $\delta<\delta_1$, for all $\nu<\nu_\delta$, for all
$g\in B_{p,k_\nu}(\mathbb{R}^n)$ and for all $\varepsilon\in(0,\epsilon_1)$.
Since $B_{p,k}=B_{p,k_\nu}$ it follows that for all
$f\in B_{p,k}(\mathbb{R}^n)$ there exists a unique solution
$(g_\varepsilon)_{\varepsilon\in(0,\varepsilon_1)}$ with $g_\varepsilon\in B_{p,k}(\mathbb{R}^n)$
of the equation
\[
g+A_{\delta,\varepsilon}(g)=\psi_{\delta,x_0}f
\]
for $\varepsilon\in(0,\varepsilon_1)$. Note that both $\delta_1$ and $\varepsilon_1$ do not
depend on the weight function $k$ and that this is possible thanks
to the equivalent norm $\Vert\cdot\Vert_{p,k_\nu}$.

\textbf{Step 4: the equation $P(x,D)T=F$ on $\Omega_\delta$ with
$F\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.}
Let $(F_\varepsilon)_\varepsilon$ be a net in $\mathcal{E}'(\mathbb{R}^n)^{(0,1]}$ which defines $F$.
By Proposition \ref{prop_basic}(i) we know that
$(F_\varepsilon)_\varepsilon\in \mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$ for some $k$. Let us take
$\Omega_\delta$ with $\delta<\delta_1<\delta_0$ such that the previous
arguments are valid for $B_{p,k_\nu}=B_{p,k}$ with $\nu<\nu_\delta$.
We study the equation at the level of representatives on $\Omega_\delta$
which can be written as
\[
P_{0,\varepsilon}(D)(T_\varepsilon)+\sum_{j=1}^r \psi_{\delta,x_0}c_j P_{j,\varepsilon}
(D)T_\varepsilon = \psi_{\delta,x_0}F_\varepsilon.
\]
Let $(g_\varepsilon)_\varepsilon$ be the unique solution of the equation
\[
g_\varepsilon+A_{\delta,\varepsilon}(g_\varepsilon)=\psi_{\delta,x_0}F_\varepsilon
\]
on the interval $(0,\varepsilon_1)$. We have that $(g_\varepsilon)_\varepsilon$ is
$B_{p,k}$-moderate (for simplicity we can set $g_\varepsilon=0$ for
$\varepsilon\in[\varepsilon_1,1]$). Indeed,
\[
\Vert g_\varepsilon\Vert_{p,k_\nu}\le \Vert A_{\delta,\varepsilon}(g_\varepsilon)
\Vert_{p,k_\nu}+\Vert\psi_{\delta,x_0}F_\varepsilon\Vert_{p,k_\nu}\le 2^{-1}
\Vert g_\varepsilon\Vert_{p,k_\nu}+\Vert\psi_{\delta,x_0}F_\varepsilon\Vert_{p,k_\nu}
\]
and
\begin{equation} \label{est_g}
\Vert g_\varepsilon\Vert_{p,k}\le c\Vert\psi_{\delta,x_0}F_\varepsilon\Vert_{p,k}
\le c\Vert\psi_{\delta,x_0}\Vert_{1,M_{k}}\Vert F_\varepsilon\Vert_{p,k}
\le \lambda_\varepsilon,
\end{equation}
with $(\lambda_\varepsilon)_\varepsilon\in\mathcal{E}_M$. Since  
$\mathop{\rm supp}{A_{\delta,\varepsilon}(g)}
\subseteq\mathop{\rm supp}\psi_{\delta,x_0}\subseteq \Omega_{2\delta}$ 
for all $g\in\mathcal{D}'(\mathbb{R}^n)$ we conclude that 
$\mathop{\rm supp} g_\varepsilon$ is contained in a compact 
set uniformly with respect to $\varepsilon$ 
(and therefore $\mathop{\rm supp} g_\varepsilon\subseteq\Omega_{\delta_0}$ 
for some $\delta_0$). From Proposition \ref{prop_basic}(ii) it follows 
that $(g_\varepsilon)_\varepsilon$ generates a basic functional 
in $\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$. Let now
\[
T_\varepsilon= F_{0,\varepsilon}\ast g_\varepsilon.
\]
The fourth assertion of Proposition \ref{prop_basic} yields that
the net $(T_\varepsilon)_\varepsilon$ defines $T\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.
By construction (steps 2 and 3) and for $\varepsilon$ small enough, we have
\begin{align*}
P_{0,\varepsilon}(D)(T_\varepsilon)|_{\Omega_\delta}+\sum_{j=1}^r
(\psi_{\delta,x_0}c_j P_{j,\varepsilon}(D)T_\varepsilon)|_{\Omega_\delta}
& = P_{0,\varepsilon}(D)(F_{0,\varepsilon}\ast g_\varepsilon)|_{\Omega_\delta}
 +A_{\delta,\varepsilon}(g_\varepsilon)|_{\Omega_\delta}\\
&=g_\varepsilon|_{\Omega_\delta}+A_{\delta,\varepsilon}(g_\varepsilon)|_{\Omega_\delta}\\
&=\psi_{\delta,x_0}F_\varepsilon|_{\Omega_\delta}=F_\varepsilon|_{\Omega_\delta}.
\end{align*}
Hence $P(x,D)T|_{\Omega_\delta}=F|_{\Omega_\delta}$.

\textbf{Step 5: the equation $P(x,D)u=v$ on $\Omega_\delta$ with
$v\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$.}
Let $(v_\varepsilon)_\varepsilon$ be a representative of $v$. By Proposition
\ref{prop_basic}(i) we know that we can work in the space
$B_{p,k}(\mathbb{R}^n)$ with $k$ arbitrary. Moreover, the interval
$(0,\varepsilon_1)$ and the neighborhood $\Omega_\delta$ do not depend on $k$.
This means that we can write
\[
g_\varepsilon+A_{\delta,\varepsilon}(g_\varepsilon)=\psi_{\delta,x_0}v_\varepsilon,
\]
where, combining the moderateness of $\Vert g_\varepsilon\Vert_{p,k}$
in \eqref{est_g} for any $k$ with Proposition \ref{prop_basic}(iii),
we have that $(g_\varepsilon)_\varepsilon\in \mathcal{M}_{\mathcal{C}^\infty(\mathbb{R}^n)}=\mathcal{E}_M(\mathbb{R}^n)$.
The convolution between a basic functional in $\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$
and a Colombeau generalized function in $\mathcal{G}_{\rm c}(\mathbb{R}^n)$ gives a
generalized function in $\mathcal{G}_{\rm c}(\mathbb{R}^n)$
(see \cite[Propositions 1.12, 1.14 and Remark 1.16]{Garetto:06a}).
Hence, $u$ with representative
\[
u_\varepsilon:=F_{0,\varepsilon}\ast g_\varepsilon
\]
belongs to $\mathcal{G}_{\rm c}(\mathbb{R}^n)$ and
$P_\varepsilon(x,D)u_\varepsilon=\psi_{\delta,x_0}v_\varepsilon =v_\varepsilon$ on $\Omega_\delta$
by construction.
\end{proof}

\begin{example} \rm
On $\mathbb{R}^2$ we define the operator
\[
\varepsilon^a D^2_x-\varepsilon^bD^2_y+c_1(x,y)\varepsilon^a D_x+c_2(x,y)
\varepsilon^b D_y+c_3(x,y)\varepsilon^c,
\]
where $c_1$, $c_2$ and $c_3$ are smooth functions and $a,b,c\in\mathbb{R}$
with $c\ge\min\{a,b\}$. We set $P_{0,\varepsilon}(D)=\varepsilon^a D^2_x-\varepsilon^bD^2_y$,
$P_{1,\varepsilon}(D)=\varepsilon^a D_x$, $P_{2,\varepsilon}(D)=\varepsilon^b D_y$ and
$P_{3,\varepsilon}(D)=\varepsilon^c$. Assume that all the coefficients $c_j$
vanish in a point $(x_0,y_0)$. The hypotheses of
Theorem \ref{theo_locsolv_easy} are fulfilled. Indeed,
\[
\widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2)=(\varepsilon^a\xi_1^2
-\varepsilon^b\xi_2^2)^2+(2\varepsilon^a\xi_1)^2+(2\varepsilon^b\xi_2)^2
+4\varepsilon^{2a}+4\varepsilon^{2b}
\]
is invertible in $(1,0)$ and concerning the functions
\[
\widetilde{P_{1,\varepsilon}}^2(\xi_1)=\varepsilon^{2a}\xi_1^2+\varepsilon^{2a},\quad
\widetilde{P_{2,\varepsilon}}^2(\xi_2)=\varepsilon^{2b}\xi_2^2+\varepsilon^{2b},\quad
\widetilde{P_{3,\varepsilon}}^2=\varepsilon^{2c}
\]
the following inequalities hold:
\begin{gather*}
\widetilde{P_{1,\varepsilon}}^2(\xi_1)=\varepsilon^{2a}\xi_1^2+\varepsilon^{2a}
 \le 4\varepsilon^{2a}\xi_1^2+4\varepsilon^{2a}\le \widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2),\\
\widetilde{P_{2,\varepsilon}}^2(\xi_2)=\varepsilon^{2b}\xi_2^2+\varepsilon^{2b}
 \le 4\varepsilon^{2b}\xi_2^2+4\varepsilon^{2b}\le \widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2),\\
\widetilde{P_{3,\varepsilon}}^2=\varepsilon^{2c}\le 4\varepsilon^{2a}+4\varepsilon^{2b}
 \le\widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2).
\end{gather*}
\end{example}


\subsection{Local solvability: case $c_j\in\mathcal{G}$.}


\begin{theorem} \label{theo_locsolv}
Let $\Omega$ be a neighborhood of $x_0$ and let
\[
P(x,D)=P_0(D)+\sum_{j=1}^r c_j(x)P_j(D)
\]
 with $c_j\in\mathcal{G}(\Omega)$ for all $j$.

If, for a certain choice of representatives, the hypotheses
{\rm (H1)--(H3)} are fulfilled and in addition,
\begin{itemize}
\item[(H5)] $\widetilde{P_{j,\varepsilon}}(\xi)\le\lambda_{j,\varepsilon}\widetilde{P_{0,\varepsilon}}(\xi)$
with
\[
\sup_{|\alpha|\le n+1+\frac{n+1}{p}+N}\sup_{x\in\Omega}|\partial^\alpha c_{j,\varepsilon}(x)|\, \lambda_{j,\varepsilon}=O(1)
\]
for some $p\in[1,\infty)$, for some $N\in\mathbb{N}$ and for all $j=1,\dots ,r$,
\end{itemize}
then there exists a sufficiently small neighborhood
$\Omega_{\delta}:=\{x:\, |x-x_0|<\delta\}$ of $x_0$ such that
\begin{itemize}
\item[(i)] for all $F\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ 
of order $N$ there exists $T\in\mathcal{L}_{\rm b}
(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ solving 
$P(x,D)T=F$ on $\Omega_\delta$.
\end{itemize}

If, for a certain choice of representatives, the hypotheses
{\rm (H1)--(H3)} are fulfilled and in addition,
\begin{itemize}
\item[(H6)] $\widetilde{P_{j,\varepsilon}}(\xi)\le\lambda_{j,\varepsilon}\widetilde{P_{0,\varepsilon}}(\xi)$
with
\[
\forall N\in\mathbb{N}\ \exists a>0\quad\sup_{|\alpha|\le
n+1+N}\sup_{x\in\Omega}|\partial^\alpha c_{j,\varepsilon}(x)|\,
\lambda_{j,\varepsilon}=O(\varepsilon^a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\]
for all $j=1,\dots ,r$,
\end{itemize}
then there exists a sufficiently small neighborhood
$\Omega_{\delta}:=\{x:\, |x-x_0|<\delta\}$ of $x_0$ such that
\begin{itemize}
\item[(ii)] for all $F\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ there exists
$T\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ solving $P(x,D)T=F$ on $\Omega_\delta$.
\item[(iii)] for all $v\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$ there exists $u\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$
solving $P(x,D)u=v$ on $\Omega_{\delta}$.
\end{itemize}
\end{theorem}

\begin{proof}
As for Theorem \ref{theo_locsolv_easy} we organize the proof in
few steps. Clearly the steps 1 and 2 are as in the proof of
Theorem \ref{theo_locsolv_easy} since $P_0(D)$ is an operator
with constant Colombeau coefficients. New methods have to be
applied to $\sum_{j=1}^r c_j(x)P_j(D)$ since the coefficients
$c_j$ are Colombeau generalized functions. We work on a
neighborhood $\Omega=\Omega_{2\delta_0}$ of $x_0$ and we use
the notation introduced in proving Theorem \ref{theo_locsolv_easy}.

\textbf{First set of hypotheses.}
We begin with the assumptions (H1), (H2), (H3) and (H5).

\textbf{Step 3: the operator $\sum_{j=1}^r c_j(x)P_j(D)$ on
$\Omega_{\delta}\subseteq\Omega_{\delta_0}$.}
We fix a choice of representatives $(c_{j,\varepsilon})$ and the
representatives $P_{j,\varepsilon}$ and $P_{0,\varepsilon}$ of the hypothesis.
Let $k\in\mathcal{K}$ and $p\in[1,\infty)$ as in $(H5)$.
We define the operator
\[
A_{\delta,\varepsilon}(g)=\sum_{j=1}^r \psi_{\delta,x_0}
\psi_{\delta_0,x_0}^2c_{j,\varepsilon}P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g).
\]
It maps $\mathcal{D}'(\mathbb{R}^n)$ into $\mathcal{E}'(\mathbb{R}^n)$. We want to estimate the
$B_{p,k_\nu}$-norm of $A_{\delta,\varepsilon}(g)$. We begin by observing
that by Theorem \ref{theo_Bpk}$(iv)$ there exists $\delta_1<\delta_0$
such that
\[
\Vert \psi_{\delta,x_0}\psi_{\delta_0,x_0}\Vert_{1,1}\le c\delta
\]
for all $\delta<\delta_1$. Choosing $\delta<\delta_1$ and
$\phi=\psi_{\delta,x_0}\psi_{\delta_0,x_0}\in\mathscr{S}(\mathbb{R}^n)$ from
Theorem \ref{theo_Bpk}(iii) we obtain a certain $\nu_\delta$
such that for all $\nu\in(0,\nu_\delta)$ and for all $\varepsilon\in(0,1]$
the following estimate holds:
\begin{align*}
\Vert A_{\delta,\varepsilon}(g)\Vert_{p,k_\nu}
&\le 2\sum_{j=1}^r\Vert \psi_{\delta,x_0}\psi_{\delta_0,x_0}\Vert_{1,1}\,
\Vert\psi_{\delta_0,x_0}c_{j,\varepsilon}P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)
\Vert_{p,k_\nu}\\
&\le 2c\delta\sum_{j=1}^r\Vert\psi_{\delta_0,x_0}c_{j,\varepsilon}P_{j,\varepsilon}
(D)(F_{0,\varepsilon}\ast g)\Vert_{p,k_\nu}.
\end{align*}
We assume $k(\xi+\eta)\le (1+|\xi|)^{N+\frac{n+1}{p}}k(\eta)$
for all $\xi$. It follows that
$M_{k_\nu}(\xi)\le (1+|\xi|)^{N+\frac{n+1}{p}}$ for all $\nu$.
An application of the first two assertions of Theorem \ref{theo_Bpk}
combined with $(H5)$ and the properties of $F_{0,\varepsilon}$ entails,
on a certain interval $(0,\varepsilon_k)$ depending on $k$, the inequality
\begin{align*}
&\Vert A_{\delta,\varepsilon}(g)\Vert_{p,k_\nu}\\
&\le 2c\delta\sum_{j=1}^r\Vert\psi_{\delta_0,x_0}c_{j,\varepsilon}
 \Vert_{1,M_{k_\nu}}\Vert P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)\Vert_{p,k_\nu}\\
&\le 2c\delta\sum_{j=1}^r\Vert(1+C|\xi|)^{N+\frac{n+1}{p}}
 \mathcal{F}(\psi_{\delta_0,x_0}c_{j,\varepsilon})(\xi)
 \Vert_1\Vert P_{j,\varepsilon}(D)(F_{0,\varepsilon})\Vert_{\infty,1}\Vert g
 \Vert_{p,k_\nu}\\
&\le 2c\delta\sum_{j=1}^r c(\psi_{\delta_0,x_0})
 \sup_{|\alpha|\le n+1+\frac{n+1}{p}+N}
 \sup_{x\in\Omega}|\partial^\alpha c_{j,\varepsilon}(x)|\lambda_{j,\varepsilon}
 \Vert F_{0,\varepsilon}\Vert_{\infty,\widetilde{P_{0,\varepsilon}}}\Vert g\Vert_{p,k_\nu}\\
&\le 2 c_1(k,\delta_0,\psi)\delta \Vert g\Vert_{p,k_\nu}.
\end{align*}
This means that for any $k$ there exist $\delta=\delta_k$ small enough
and $\varepsilon_k$ small enough such that
\[
\Vert A_{\delta,\varepsilon}(g)\Vert_{p,k_\delta}\le 2^{-1}
\Vert g\Vert_{p,k_\delta}
\]
holds for all $\varepsilon\in(0,\varepsilon_k)$ and for all
$g\in B_{p,k_\delta}(\mathbb{R}^n)=B_{p,k}(\mathbb{R}^n)$. It follows that
for all $f\in B_{p,k}(\mathbb{R}^n)$ there exists a unique solution
$(g_\varepsilon)_{\varepsilon\in(0,\varepsilon_k)}$ with $g_\varepsilon\in B_{p,k}(\mathbb{R}^n)$
 of the equation
\[
g+A_{\delta,\varepsilon}(g)=\psi_{\delta,x_0} f
\]
for $\varepsilon\in(0,\varepsilon_k)$.

\textbf{Step 4: the equation $P(x,D)T=F$ on $\Omega_\delta$ with
$F\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.}
Let $(F_\varepsilon)_\varepsilon$ be a net in $\mathcal{E}'(\mathbb{R}^n)^{(0,1]}$ which defines
$F$ of order $N$. By Proposition \ref{prop_basic}(i) we know
that $(F_\varepsilon)_\varepsilon\in \mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$ for
$k(\xi)=\lrangle{\xi}^{-N-\frac{n+1}{p}}$. Hence, for all $\nu>0$
we have $M_{k_\nu}(\xi)\le (1+|\xi|)^{N+\frac{n+1}{p}}$ and
we are under the hypothesis of the previous step. Let us take
$\Omega_\delta$ with $\delta=\delta_k$ such that the previous arguments
are valid for $B_{p,k_\delta}=B_{p,k}$. At the level of
representatives we can write
\[
P_{0,\varepsilon}(D)(T_\varepsilon)+\sum_{j=1}^r \psi_{\delta,x_0}
\psi_{\delta_0,x_0}^2c_{j,\varepsilon} P_{j,\varepsilon}(D)T_\varepsilon
= \psi_{\delta,x_0}F_\varepsilon
\]
on $\Omega_\delta$. Let $(g_\varepsilon)_\varepsilon$ be the unique solution of
the equation
\[
g_\varepsilon+A_{\delta,\varepsilon}(g_\varepsilon)=\psi_{\delta,x_0}F_\varepsilon
\]
on the interval $(0,\varepsilon_k)$. As in Theorem \ref{theo_locsolv_easy}
one easily sees that $(g_\varepsilon)_\varepsilon$ is $B_{p,k}$-moderate.
Since  $\mathop{\rm supp}{A_{\delta,\varepsilon}(g)}\subseteq\mathop{\rm supp}
\psi_{\delta,x_0}\subseteq \Omega_{2\delta}$ for all
$g\in\mathcal{D}'(\mathbb{R}^n)$ we conclude that $\mathop{\rm supp} g_\varepsilon$ is
contained in a compact set uniformly with respect to $\varepsilon$
(and therefore $\mathop{\rm supp} g_\varepsilon\subseteq\Omega_{\delta_0}$
for some $\delta_0$). From Proposition \ref{prop_basic}(ii)
it follows that $(g_\varepsilon)_\varepsilon$ generates a basic functional
in $\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$. Let now
\[
T_\varepsilon= F_{0,\varepsilon}\ast g_\varepsilon.
\]
The fourth assertion of Proposition \ref{prop_basic} yields that
the net $(T_\varepsilon)_\varepsilon$ defines $T\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.
By construction, for all $\varepsilon\in(0,\varepsilon_k)$ we have
\begin{align*}
&P_{0,\varepsilon}(D)(T_\varepsilon)|_{\Omega_\delta}+\sum_{j=1}^r
(\psi_{\delta,x_0}\psi_{\delta_0,x_0}^2c_{j,\varepsilon} P_{j,\varepsilon}
(D)T_\varepsilon)|_{\Omega_\delta} \\
&= P_{0,\varepsilon}(D)(F_{0,\varepsilon}\ast g_\varepsilon)|_{\Omega_\delta}+A_{\delta,\varepsilon}
 (g_\varepsilon)|_{\Omega_\delta}\\
&=g_\varepsilon|_{\Omega_\delta}+A_{\delta,\varepsilon}(g_\varepsilon)|_{\Omega_\delta}\\
&=\psi_{\delta,x_0}F_\varepsilon|_{\Omega_\delta}=F_\varepsilon|_{\Omega_\delta}.
\end{align*}
We have solved the equation $P(x,D)T=F$ on a neighborhood of $x_0$
which depends on the weight function $k$ or in other words on the
order of the functional $F$.

\textbf{Second set of hypotheses.}
We now assume that $(H1)$, $(H2)$, $(H3)$ and $(H6)$ hold and we
prove that in this case one can find a neighborhood $\Omega_\delta$
which does not depend on the weight function $k$.

\textbf{Step 3: the operator $\sum_{j=1}^r c_j(x)P_j(D)$ on
$\Omega_{\delta}\subseteq\Omega_{\delta_0}$.}
Let $k$ be an arbitrary weight function. Choosing $\delta<\delta_1$
and any $\nu\in(0,\nu_\delta)$ our set of hypotheses combined with
the properties of $F_{0,\varepsilon}$ yields on an interval $(0,\varepsilon_k)$
depending on $k$ the estimates
\begin{align*}
\Vert A_{\delta,\varepsilon}(g)\Vert_{p,k_\nu}
&\le 2c\delta\sum_{j=1}^r\Vert\psi_{\delta_0,x_0}c_{j,\varepsilon}
  \Vert_{1,M_{k_\nu}}\Vert P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)\Vert_{p,k_\nu}\\
&\le 2c\delta\sum_{j=1}^r\Vert(1+C|\xi|)^{N_k}\mathcal{F}(\psi_{\delta_0,x_0}c_{j,\varepsilon})(\xi)\Vert_1\Vert P_{j,\varepsilon}(D)(F_{0,\varepsilon})\Vert_{\infty,1}\Vert g\Vert_{p,k_\nu}\\
&\le 2c\delta\sum_{j=1}^r c(\psi_{\delta_0},x_0)\sup_{|\alpha|\le n+1+N_k}\sup_{x\in\Omega}|\partial^\alpha c_{j,\varepsilon}(x)|\lambda_{j,\varepsilon}\Vert F_{0,\varepsilon}\Vert_{\infty,\widetilde{P_{0,\varepsilon}}}\Vert g\Vert_{p,k_\nu}\\
&\le 2\delta c_1(k,\delta_0,\psi)\varepsilon^{a_k} \Vert g\Vert_{p,k_\nu}.
\end{align*}
At this point taking $\varepsilon_k$ so small that
$c_1(k,\delta_0,\psi)\varepsilon^{a_k}_k<1$ and by requiring $\delta<1/4$
we have that for all $\delta<\min({\delta_1,1/4})$ there exist
$\nu_\delta$ such that for all $\nu<\nu_\delta$ the inequality
\[
\Vert A_{\varepsilon,\delta}(g)\Vert_{p,k_\nu}\le 2^{-1}\Vert g\Vert_{p,k_\nu}
\]
holds in a sufficiently small interval $\varepsilon\in(0,\varepsilon_k)$.
Note that $\delta$ does not depend on $k$ while $\varepsilon_k$ does.
Clearly for all for all $f\in B_{p,k}(\mathbb{R}^n)$ there exists a unique
solution $(g_\varepsilon)_{\varepsilon\in(0,\varepsilon_k)}$ with
$g_\varepsilon\in B_{p,k}(\mathbb{R}^n)$ of the equation $g+A_{\delta,\varepsilon}(g)
=\psi_{\delta,x_0} f$ for $\varepsilon\in(0,\varepsilon_k)$.

\textbf{Step 4: the equation $P(x,D)T=F$ on $\Omega_\delta$ with
$F\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.}
Let $(F_\varepsilon)_\varepsilon$ be a net in $\mathcal{E}'(\mathbb{R}^n)^{(0,1]}$ which defines
$F$. By Proposition \ref{prop_basic}(i) we know that
$(F_\varepsilon)_\varepsilon\in \mathcal{M}_{B_{p,k}(\mathbb{R}^n)}$ for some weight function
$k$. The previous arguments are valid in a neighborhood
$\Omega_\delta$ of $x_0$, with $\delta$ independent of $k$
and $(g_\varepsilon)_\varepsilon$ is the unique solution of the equation
\[
g_\varepsilon+A_{\delta,\varepsilon}(g_\varepsilon)=\psi_{\delta,x_0}F_\varepsilon
\]
on the interval $(0,\varepsilon_k)$. The functional generated by
\[
T_\varepsilon= F_{0,\varepsilon}\ast g_\varepsilon
\]
is the solution $T$ of the equation $P(x,D)T=F$ on $\Omega_\delta$.

\textbf{Step 5: the equation $\sum_{j=1}^r c_j(x)P_j(D)=v\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$
on $\Omega_{\delta}$.}
Let $(v_\varepsilon)_\varepsilon\in\mathcal{E}_M(\mathbb{R}^n)$ be a representative of $v\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$.
For $\varepsilon\in(0,\varepsilon_k)$ the equation
\begin{equation} \label{eq_import}
A_{\varepsilon,\delta}g+g=\psi_{\delta,x_0}v_\varepsilon
\end{equation}
has a unique solution $g_\varepsilon$ in $B_{p,k}$. Moreover,
\begin{equation}
\label{est_import}
\Vert g_\varepsilon\Vert_{p,k}\le C\Vert\psi_{\delta_0,x_0}v_\varepsilon\Vert_{p,k}
\end{equation}
for all $\varepsilon\in(0,\varepsilon_k)$. Since $\psi_{\delta,x_0}v_\varepsilon$ belongs to any $B_{p,k}$ space we can conclude that there exists a net of distributions $(g_\varepsilon)_\varepsilon$ which solve the equation \eqref{eq_import} in $\mathcal{D}'$ and such that for all $k\in\mathcal{K}$ the estimate \eqref{est_import} holds on a sufficiently small interval $(0,\varepsilon_k)$. From the embedding $B_{p,k}\subset \mathcal{C}^j$ when $(1+|\xi|)^j/k(\xi)\in L^{q}$, $1/p+1/q=1$ the inequality \eqref{est_import} and the fact that $(\Vert\psi_{\delta,x_0}v_\varepsilon\Vert_{p,k})_\varepsilon$ is moderate we have the following:
\[
\forall j\in\mathbb{N}\, \exists \varepsilon_j\in(0,1], \varepsilon_j\searrow 0,\quad
(g_\varepsilon)_{\varepsilon\in(0,\varepsilon_j)}\in\mathcal{M}_{\mathcal{C}^j(\mathbb{R}^n)}.
\]
In other words the net of distributions $(g_\varepsilon)_\varepsilon$ solves
the equation \eqref{eq_import} in $\mathcal{D}'$ and has more and more
moderate derivatives as $\varepsilon$ goes to $0$. $(g_\varepsilon)_\varepsilon$ is
the representative of a basic functional $g$ in $\mathcal{L}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$.
We already know that $u=F_0\ast g\in\mathcal{L}_{\rm b}(\mathcal{G}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ solves
the equation $P(x,D)u=v$ in $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega_\delta),\widetilde{\mathbb{C}})$.

\textbf{Step 6: $g$ belongs to $\mathcal{G}_{\rm c}(\mathbb{R}^n)$ then $u\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$.}
We finally prove that $g$ is an element of $\mathcal{G}_{\rm c}(\mathbb{R}^n)$.
Since we already know that $g$ has compact support we just have
to prove that $g$ belongs to $\mathcal{G}(\mathbb{R}^n)$. This will imply that
$u\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$. We generate a representative of $g$ which
belongs to $\mathcal{E}_M(\mathbb{R}^n)$. Let $(n_\varepsilon)_\varepsilon\in\mathcal{N}$ with
$n_\varepsilon\neq 0$ for all $\varepsilon$ and $\rho\in\mathcal{C}^\infty_{\rm c}(\mathbb{R}^n)$ with $\int\rho=1$.
For $\rho_{n_\varepsilon}(x)=\rho(x/n_\varepsilon)n_\varepsilon^{-n}$,
the net $g_\varepsilon\ast \rho_{n_\varepsilon}$ belongs to $\mathcal{E}_M(\mathbb{R}^n)$.
Indeed, $g_\varepsilon\ast\rho_{n_\varepsilon}\in\mathcal{C}^\infty$ for each $\varepsilon$ and
taking $\varepsilon$ small enough
\[
\sup_{x\in K}|\partial^\alpha (g_\varepsilon\ast\rho_{n_\varepsilon})(x)|
= \sup_{x\in K}|\partial^\alpha (g_\varepsilon)\ast\rho_{n_\varepsilon}(x)|
=\sup_{x\in K}\Bigl|\int\partial^\alpha g_\varepsilon(x-n_\varepsilon z)\rho(z)\,
dz\Bigr|\le \varepsilon^{-N}.
\]
Now, since
\[
g_\varepsilon-g_\varepsilon\ast\rho_{n_\varepsilon}(x)
=\int (g_\varepsilon(x)-g_\varepsilon(x-n_\varepsilon z))\rho(z)\, dz
\]
and for all $q\in\mathbb{N}$,
\[
\sup_{x\in K}|g_\varepsilon-g_\varepsilon\ast\rho_{n_\varepsilon}(x)|=O(\varepsilon^q),
\]
we conclude that for all $u\in\mathcal{G}_{\rm c}(\mathbb{R}^n)$,
$g(u)=[(g_\varepsilon\ast\rho_{n_\varepsilon})_\varepsilon](u)$ where
$[(g_\varepsilon\ast\rho_{n_\varepsilon})_\varepsilon]\in\mathcal{G}(\mathbb{R}^n)$. This means that $g$
is an element of $\mathcal{G}(\mathbb{R}^n)$.
\end{proof}

\begin{remark} \label{rem_tech} \rm
Note that in the proof we do not make use of the assertion (iii)
of Theorem \ref{theo_Bpk} with $\phi=\psi_{\delta,x_0}c_{j,\varepsilon}$
and $u=P_{j,\varepsilon}(D)(F_{0,\varepsilon}\ast g)$ because this would generate
some $\nu_0$ depending on $\phi$ and therefore on $\varepsilon$.
This would also lead to a neighborhood $\Omega_\delta$ of $x_0$
whose radius depends on the parameter $\varepsilon$. Finally, note that
the condition $(H6)$ is an assumption of $\mathcal{G}^\infty $-regularity for
 the coefficients $c_j$.
\end{remark}

\begin{example} \label{example_2} \rm
On $\mathbb{R}^2$ we define the operator
\[
D^2_x-\varepsilon^{-1}D^2_y+c_{1,\varepsilon}(x,y)D_x+c_{2,\varepsilon}
(x,y)D_y+c_{3,\varepsilon}(x,y),
\]
where $(c_{1,\varepsilon})_\varepsilon$, $(c_{2,\varepsilon})_\varepsilon$ and
$(c_{3,\varepsilon})_\varepsilon$ are moderate nets of smooth functions.
We set $P_{0,\varepsilon}(D)=D^2_x-\varepsilon^{-1}D^2_y$, $P_{1}(D)=D_x$,
$P_{2}(D)=D_y$ and $P_{3}(D)=I$. Assume that the coefficients
$c_{j,\varepsilon}$ vanish in a point $(x_0,y_0)$ for all $\varepsilon$.
The weight function
\[
\widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2)=(\xi_1^2-\varepsilon^{-1}\xi_2^2)^2+(2\xi_1)^2
+(2\varepsilon^{-1}\xi_2)^2+4+4\varepsilon^{-2}
\]
is invertible in $(1,0)$ and concerning the functions
$\widetilde{P_{1}}^2(\xi_1)=\xi_1^2+1$, $\widetilde{P_{2}}^2(\xi_2)=\xi_2^2+1$
and $\widetilde{P_{3}}^2=1$ the following inequalities hold:
\begin{gather*}
\widetilde{P_{1}}^2(\xi_1)=\xi_1^2+1\le 4\xi_1^2+4
 \le \widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2),\\
\widetilde{P_{2}}^2(\xi_2)=\xi_2^2+1=\frac{1}{4}\varepsilon^2(4\varepsilon^{-2}(\xi_2^2+1))
 \le \frac{1}{4}\varepsilon^2\widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2),\\
\widetilde{P_{3}}^2=1\le 4\le \widetilde{P_{0,\varepsilon}}^2(\xi_1,\xi_2).
\end{gather*}
One easily sees that for $\lambda_{1}=1$,
$\lambda_{2,\varepsilon}=\frac{1}{2}\varepsilon$ and $\lambda_3=1$ the assumptions
(H1)--(H3) and (H6) of Theorem \ref{theo_locsolv} are fulfilled if,
 on a certain neighborhood $\Omega$ of $(x_0,y_0)$ and for all
$N\in\mathbb{N}$, the following holds:
\begin{gather*}
\sup_{|\alpha|\le N}\sup_{x\in\Omega}|\partial^\alpha c_{1,\varepsilon}(x)|=O(\varepsilon),
\quad
\sup_{|\alpha|\le N}\sup_{x\in\Omega}|\partial^\alpha c_{2,\varepsilon}(x)|=O(1),
\\
\sup_{|\alpha|\le N}\sup_{x\in\Omega}|\partial^\alpha c_{3,\varepsilon}(x)|=O(\varepsilon).
\end{gather*}
Hence the operator
\[
D^2_x-[(\varepsilon^{-1})_\varepsilon]D^2_y+c_1(x,y)D_x+c_2(x,y)D_y+c_3(x,y),
\]
with $\mathcal{G}^\infty $-coefficients $c_1=[(c_{1,\varepsilon})_\varepsilon]$,
$c_2=[(c_{2,\varepsilon})_\varepsilon]$ and $c_3=[(c_{3,\varepsilon})_\varepsilon]$
is locally solvable at $(x_0,y_0)$ in both the Colombeau algebra
$\mathcal{G}(\mathbb{R}^2)$ and the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^2),\widetilde{\mathbb{C}})$.
\end{example}

\section{A sufficient condition of local solvability for generalized
pseudodifferential operators}
\label{sec_pseudo}

We conclude this article by providing a sufficient condition of
local solvability for generalized pseudodifferential operators.
The generalization from differential to pseudodifferential
operators obliges us to find suitable functional analytic methods
able to generate a local solution which are not a simple
convolution between the right-hand side term and a fundamental
solution. In particular, we will make use of the Sobolev mapping
properties of a generalized pseudodifferential operator $a(x,D)$.
In the sequel we use the notation $H^0(\mathbb{R}^n)=L^2(\mathbb{R}^n)$ and
$\Vert\cdot\Vert_s$ for the Sobolev $H^s$-norm.

\begin{proposition} \label{prop_Sob}
Let $a\in\mathcal{G}_{S^m(\mathbb{R}^{2n})}$. There exist $l_0\in\mathbb{N}$ and a constant
$C_0>0$ such that the inequality
\begin{equation} \label{Sob_ineq}
\Vert a_\varepsilon(x,D)u\Vert_{s-m}\le C_0\max_{|\alpha+\beta|
\le l_0}|\lrangle{\xi}^{s-m}\sharp a_\varepsilon\sharp\lrangle{\xi}^{-s}|^{(0)}
_{\alpha,\beta}\,\Vert u\Vert_s
\end{equation}
holds for all $s\in\mathbb{R}$, for all $u\in H^s(\mathbb{R}^n)$, for all
representatives $(a_\varepsilon)_\varepsilon$ of $a$ and for all $\varepsilon\in(0,1]$.
\end{proposition}

It is clear from \eqref{Sob_ineq} that the moderateness properties
of $(a_\varepsilon)_\varepsilon$ are the same of $(a_\varepsilon(x,D)u)_\varepsilon$ as a
net in $H^{s-m}(\mathbb{R}^n)^{(0,1]}$.

\begin{remark} \label{rem_Sob} \rm
The Colombeau space $\mathcal{G}_{L^2(\mathbb{R}^n)}$ based on $L^2(\mathbb{R}^n)$ is not
contained in the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$. Indeed, for
$f\in L^2(\mathbb{R})\cap L^1(\mathbb{R})$, $f\neq 0$, and $(n_\varepsilon)_\varepsilon\in\mathcal{N}$
we have that $f=[(n_\varepsilon^{-1/2}f(\cdot/n_\varepsilon))_\varepsilon]$ is not
$0$ in $\mathcal{G}_{L^2(\mathbb{R})}$ but
\[
\int_\mathbb{R} u(x)f(x)\, dx= 0
\]
for all $u\in\mathcal{G}_{\rm c}(\mathbb{R})$. Analogously, the embedding
$H^{s_1}(\mathbb{R}^n)\subseteq H^{s_2}(\mathbb{R}^n)$ cannot be reproduced at
the level of the corresponding Colombeau spaces for $s_1\ge s_2$.
This is due to the fact that there exist nets in
$\mathcal{N}_{H^{s_2}(\mathbb{R}^n)}\cap\mathcal{M}_{H^{s_1}(\mathbb{R}^n)}$ which do not belong
to $\mathcal{N}_{H^{s_1}(\mathbb{R}^n)}$. For example, for $f\in H^1(\mathbb{R}^n)$ with
$\Vert f'\Vert_0\neq 0$ and $(n_\varepsilon)_\varepsilon\in\mathcal{N}$ we have that
$(n_\varepsilon^{1/2}f(\cdot/n_\varepsilon))_\varepsilon\in\mathcal{M}_{H^1(\mathbb{R})}\cap\mathcal{N}_{H^0(\mathbb{R})}$
but $(n_\varepsilon^{-1/2}f'(\cdot/n_\varepsilon))_\varepsilon\not\in\mathcal{N}_{H^0(\mathbb{R})}$ and
therefore $(n_\varepsilon^{1/2}f(\cdot/n_\varepsilon))_\varepsilon\not\in\mathcal{N}_{H^1(\mathbb{R})}$.
\end{remark}

The embedding issues of Remark \ref{rem_Sob} lead us to study the
equation $a(x,D)T=F$ in the dual $\mathcal{L}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}}$ even
when $F$ belongs to a Colombeau space based on a Sobolev space.

\begin{theorem} \label{theo_loc_Sob}
Let $a\in\mathcal{G}_{S^m(\mathbb{R}^{2n})}$. Assume that there exist a representative
$(a_\varepsilon^\ast)_\varepsilon$ of $a^\ast$, a strictly non-zero net
$(\lambda_\varepsilon)_\varepsilon$, a positive number $\delta>0$ and
$0\le s\le m$ such that
\begin{equation}
\label{inv_Sob}
\Vert\varphi\Vert_s\le \lambda_\varepsilon\Vert
a^\ast_\varepsilon(x,D)|_{\Omega_\delta}\varphi\Vert_0
\end{equation}
for all $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$, $\Omega_\delta:=\{|x|<\delta\}$,
and for all $\varepsilon\in(0,1]$.\\
Then, for all $F\in\mathcal{L}_{\rm b}(\mathcal{G}_{\rm c}(\mathbb{R}^n),\widetilde{\mathbb{C}})$ generated by a net
in $\mathcal{M}_{H^{-s}(\mathbb{R}^n)}$ there exists $T\in\mathcal{L}_{\rm b}(\mathcal{G}_{\rm c}(\Omega_\delta),\widetilde{\mathbb{C}})$
generated by a net in $\mathcal{M}_{L^2(\Omega_\delta)}$ such that
\[
a(x,D)|_{\Omega_\delta}T=F|_{\Omega_\delta}
\]
in $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega_\delta),\widetilde{\mathbb{C}})$.
\end{theorem}

The proof of Theorem \ref{theo_loc_Sob}  uses  the theory of
Hilbert $\widetilde{\mathbb{C}}$-modules (see \cite{GarVer:08}) and in particular
of the projection operators defined on the Hilbert
 $\widetilde{\mathbb{C}}$-module $\mathcal{G}_{L^2(\Omega_\delta)}$.
>From \cite[Proposition 2.21]{GarVer:08} we have that if $E$ is
 a nonempty subset in $\mathcal{G}_{L^2(\Omega_\delta)}$ generated by a
net $(E_\varepsilon)_\varepsilon$ of nonempty convex subsets of $L^2(\Omega_\delta)$
($E=[(E_\varepsilon)_\varepsilon]:=\{u\in\mathcal{G}_{L^2(\Omega_\delta)}: \exists\, {\rm{repr.}}
(u_\varepsilon)_\varepsilon\, \forall\varepsilon\in(0,1]\  u_\varepsilon\in E_\varepsilon\}$) then
there exists a map $P_E:\mathcal{G}_{L^2(\Omega_\delta)}\to E$ called
projection on $E$ such that
$\Vert u-P_E(u)\Vert=\inf_{w\in E}\Vert u-w\Vert$ for all
$u\in\mathcal{G}_{L^2(\Omega_\delta)}$. In this particular case we have
in addition that the net $(P_{\overline{E}_\varepsilon}\,(u_\varepsilon))_\varepsilon$
is moderate in $L^2(\Omega_\delta)$ and the property
$P_E(u)=[(P_{\overline{E}_\varepsilon}\,(u_\varepsilon))_\varepsilon]$.
If $E_\varepsilon$ is a subspace of $L^2(\Omega_\delta)$ we easily have
for all $v\in L^2(\Omega_\delta)$ the following inequality in the
norm of $L^2(\Omega_\delta)$:
$\Vert P_{\overline{E}_\varepsilon}v\Vert
\le \Vert v- P_{\overline{E}_\varepsilon}v\Vert+\Vert v\Vert
\le \inf_{w\in\overline{E}_\varepsilon}\Vert v-w\Vert+\Vert v\Vert\le 2\Vert
v\Vert$.

\begin{proof}[Proof of Theorem \ref{theo_loc_Sob}]
Let $a_\varepsilon^\ast$ be the representative of $a^\ast$ fulfilling
the hypotheses of Theorem \ref{theo_loc_Sob}.
Let $E_\varepsilon:=\{\psi\in L^2(\Omega_\delta):
 \exists\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)\quad \psi=a_\varepsilon^\ast(x,D)
|_{\Omega_\delta}(\varphi)\}$. $E_\varepsilon$ is a nonempty subspace
 of $L^2(\Omega_\delta)$ and therefore we can define the projection
operator $P_E:\mathcal{G}_{L^2(\Omega_\delta)}\to E$ on
$E:=[(E_\varepsilon)_\varepsilon]$ $\widetilde{\mathbb{C}}$-submodule of $\mathcal{G}_{L^2(\Omega_\delta)}$.
We use for the operator $\mathcal{C}^\infty_{\rm c}(\Omega_\delta)\to L^2(\Omega_\delta):
\varphi\to a_\varepsilon^\ast(x,D)|_{\Omega_\delta}(\varphi)$ the notation
$A^\ast_\varepsilon$. The condition \eqref{inv_Sob} means that
$A^\ast_\varepsilon:\mathcal{C}^\infty_{\rm c}(\Omega_\delta)\to E_\varepsilon$ is invertible.
Combining \eqref{inv_Sob} with the Sobolev embedding properties
we have that
\begin{equation} \label{ext_A_eps}
\Vert (A^\ast_\varepsilon)^{-1}v\Vert_{L^2(\Omega_\delta)}\le
\Vert (A^\ast_\varepsilon)^{-1}v\Vert_s\le \lambda_\varepsilon\Vert v
\Vert_{L^2(\Omega_\delta)}
\end{equation}
holds for all $v\in E_\varepsilon$. Taking the closure $\overline{E}_\varepsilon$
 of $E_\varepsilon$ in $L^2(\Omega_\delta)$ the inequality \eqref{ext_A_eps}
allows us to extend $(A^\ast_\varepsilon)^{-1}$ to a continuous operator
from $\overline{E}_\varepsilon$ to $H^s(\mathbb{R}^n)\subseteq L^2(\Omega_\delta)$,
with \eqref{ext_A_eps} valid for all $v\in\overline{E}_\varepsilon$.

Let $u$ be an element of $E$ defined by the net
$(u_\varepsilon)_\varepsilon$, $u_\varepsilon\in E_\varepsilon$. Clearly
$(A^\ast_\varepsilon)^{-1}u_\varepsilon\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$  with
$\Vert(A^\ast_\varepsilon)^{-1}u_\varepsilon\Vert_s
\le \lambda_\varepsilon\Vert u_\varepsilon\Vert_{L^2(\Omega_\delta)}$ and
if $u'_\varepsilon\in E_\varepsilon$ is another net generating $u$ we obtain
\[
\Vert(A^\ast_\varepsilon)^{-1}(u_\varepsilon-u'_\varepsilon)\Vert_s
\le \lambda_\varepsilon\Vert u_\varepsilon-u'_\varepsilon\Vert_{L^2(\Omega_\delta)}.
\]
This means that we can define the $\widetilde{\mathbb{C}}$-linear functional
\[
S:E\to\widetilde{\mathbb{C}}:u=[(u_\varepsilon)_\varepsilon]\to[(((A^\ast_\varepsilon)^{-1}u_\varepsilon
|F_\varepsilon)_{L^2(\mathbb{R}^n)})_\varepsilon]
\]
where
\begin{align*}
|((A^\ast_\varepsilon)^{-1}u_\varepsilon|F_\varepsilon)_{L^2(\mathbb{R}^n)}|
&=(2\pi)^{-n}|(\lrangle{\xi}^{s}
\widehat{(A^\ast_\varepsilon)^{-1}u_\varepsilon}|\lrangle{\xi}^{-s}\widehat{F_\varepsilon}
)_{L^2(\mathbb{R}^n)}|\\
&\le \Vert (A^\ast_\varepsilon)^{-1}u_\varepsilon\Vert_{s} \Vert F_\varepsilon\Vert_{-s}\\
&\le \lambda_\varepsilon\Vert u_\varepsilon\Vert_{L^2(\Omega_\delta)}\Vert F_\varepsilon\Vert_{-s}.
\end{align*}
 From the previous inequality we also have that the functional $S$
is continuous. Since $E$ is a $\widetilde{\mathbb{C}}$-submodule the projection $P_E$
is continuous and $\widetilde{\mathbb{C}}$-linear. It follows that
\[
S\circ P_E:\mathcal{G}_{L^2(\Omega_\delta)}\to \widetilde{\mathbb{C}}:u\to S(P_E(u))
\]
is a continuous $\widetilde{\mathbb{C}}$-linear functional on $\mathcal{G}_{L^2(\Omega_\delta)}$
with basic structure. Indeed, it is defined by the net
$L^2(\Omega_\delta)\to \mathbb{C}: v\to ((A^\ast_\varepsilon)^{-1}P_{\overline{E}_\varepsilon}\
 v|F_\varepsilon)_{L^2(\mathbb{R}^n)}$ such that
\[
|((A^\ast_\varepsilon)^{-1}P_{\overline{E}_\varepsilon}\ v|F_\varepsilon)_{L^2(\mathbb{R}^n)}|
\le \lambda_\varepsilon\Vert P_{\overline{E}_\varepsilon}\ v\Vert_{L^2(\Omega_\delta)}
\Vert F_\varepsilon\Vert_{-s}\le 2\lambda_\varepsilon\Vert F_\varepsilon\Vert_{-s}\Vert v
\Vert_{L^2(\Omega_\delta)},
\]
where the nets $(\lambda_\varepsilon)_\varepsilon$ and $(\Vert F_\varepsilon\Vert_{-s})_\varepsilon$
are moderate. By the Riesz representation theorem for Hilbert
$\widetilde{\mathbb{C}}$-modules and $\widetilde{\mathbb{C}}$-linear functionals
\cite[Theorem 4.1 and Proposition 4.4]{GarVer:08}) we have that
there exists a unique $t\in \mathcal{G}_{L^2(\Omega_\delta)}$ such that
\[
(S\circ P_E)(u)=(u|t)_{L^2(\Omega_\delta)}
\]
for all $u\in\mathcal{G}_{L^2(\Omega_\delta)}$. More precisely there exists
a representative $(t_\varepsilon)_\varepsilon$ of $t$ such that the equality
\[
((A^\ast_\varepsilon)^{-1}P_{\overline{E}_\varepsilon}\ v|F_\varepsilon)_{L^2(\mathbb{R}^n)}
=(v|t_\varepsilon)_{L^2(\Omega_\delta)}
\]
holds for all $v\in L^2(\Omega_\delta)$. Let $T$ be the basic
functional in $\mathcal{L}(\mathcal{G}_{\rm c}(\Omega_\delta),\widetilde{\mathbb{C}})$ generated by the
net $(t_\varepsilon)_\varepsilon$. $T$ solves the equation
$a(x,D)|_{\Omega_\delta}T=F|_{\Omega_\delta}$.
Indeed, since $A^\ast_\varepsilon(x,D)\varphi\in E_\varepsilon\subseteq L^2(\Omega_\delta)$
for all $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$ we can write
\begin{align*}
((A^\ast_\varepsilon)^{-1}P_{\overline{E}_\varepsilon}A^\ast_\varepsilon(x,D)
\varphi|F_\varepsilon)_{L^2(\mathbb{R}^n)}
&=((A^\ast_\varepsilon)^{-1}A^\ast_\varepsilon(x,D)
\varphi|F_\varepsilon)_{L^2(\mathbb{R}^n)}=(\varphi|F_\varepsilon)_{L^2(\mathbb{R}^n)}\\
&=(A^\ast_\varepsilon(x,D)\varphi|t_\varepsilon)_{L^2(\Omega_\delta)}.
\end{align*}
Thus,
\[
(\varphi|F_\varepsilon)_{L^2(\mathbb{R}^n)}=(\varphi|a_\varepsilon(x,D)|_{\Omega_\delta}t_\varepsilon)
_{L^2(\Omega_\delta)}
\]
for all $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$ or in other words,
$a_\varepsilon(x,D)|_{\Omega_\delta}t_\varepsilon = F_\varepsilon|_{\Omega_\delta}$
in $\mathcal{D}'(\Omega_\delta)$.
\end{proof}

Theorem \ref{theo_loc_Sob} tells us that classical operators
which satisfy the condition \eqref{inv_Sob} are locally solvable
in the Colombeau context, in the sense that under suitable moderateness
conditions on the right hand side we will find a local generalized
solution. An example is given by differential operators which
are at the same time principally normal and of principal type at $0$.
More precisely, Proposition 4.3 in \cite{SaintRaymond:91} proves
that a differential operator $a(x,D)$ which is both principally normal
and of principal type at $0$ fulfills the inequality
$\Vert \varphi\Vert_{m-1}\le \Vert a^\ast(x,D)\varphi\Vert_0$ for
all $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$ with $\delta$ small enough.

We now go back to the case of differential operators with
generalized Colombeau coefficients. In other words we assume that
the symbol $a$ is of the type
$a=[(a_\varepsilon)_\varepsilon]=\sum_{|\alpha|\le m}c_\alpha(x)\xi^\alpha$ with
$c_\alpha\in\mathcal{G}_\infty(\mathbb{R}^n)$. The next proposition shows that the
local solvability condition \eqref{inv_Sob} holds under an ellipticity
assumption on the real part of the symbol $a_\varepsilon$. We recall that
for all $m\in\mathbb{N}$ and for all $\delta>0$ the inequality
\begin{equation} \label{ineq_SR}
\Vert\varphi\Vert_m\le 2\delta\Vert\varphi\Vert_{m+1}
\end{equation}
is true for all $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$
(see \cite[Lemma 4.2]{SaintRaymond:91}).

\begin{proposition} \label{prop_SR}
Let $a(x,D)$ be a generalized differential operator with symbol
$a\in\mathcal{G}_{S^{m}(\mathbb{R}^{2n})}$. If there exists $b\in\mathbb{R}$,
a representative $(a_\varepsilon)_\varepsilon\in \mathcal{M}_{S^{m}(\mathbb{R}^{2n}),b}$,
 a constant $c_0>0$ and a net $(c_\varepsilon)_\varepsilon\in \mathcal{M}_{S^{m-1}(\mathbb{R}^{2n}),b}$
such that
\[
\Re\, a_\varepsilon(x,\xi)= c_0\varepsilon^b\lrangle{\xi}^{m}+c_\varepsilon(x,\xi)
\]
for all $(x,\xi)$ and all $\varepsilon\in(0,1]$, then there exist a
sufficiently small $\delta>0$ and a constant $C>0$ such that
\[
\Vert \varphi\Vert_{\frac{m}{2}}\le C\varepsilon^{-b}\Vert a_\varepsilon^\ast(x,D)\varphi\Vert_0
\]
for all $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$ and for all $\varepsilon\in(0,1]$.
\end{proposition}

\begin{proof}
We begin by writing $2\Re\, (\varphi|a_\varepsilon^\ast(x,D)\varphi)$ as
$(\varphi|(a_\varepsilon+a_\varepsilon^\ast)(x,D)\varphi)$. Recalling that
$a_\varepsilon^\ast=\overline{a_\varepsilon}$ modulo $\mathcal{M}_{S^{m-1}(\mathbb{R}^n),b}$ we have
\[
2\Re\, (\varphi|a_\varepsilon^\ast(x,D)\varphi)=(\varphi|2c_0\varepsilon^b
\lambda^m(D)\varphi)+(\varphi|c_{1,\varepsilon}(x,D)\varphi),
\]
where $\lambda^m(D)$ has symbol $\lrangle{\xi}^m$ and
$(c_{1,\varepsilon})_\varepsilon\in\mathcal{M}_{S^{m-1}(\mathbb{R}^{2n}),b}$. Hence,
\begin{align*}
2\Re\, (\varphi|a_\varepsilon^\ast(x,D)\varphi)
&\ge c_1\varepsilon^b \Vert\varphi\Vert^2_{\frac{m}{2}}
 -(\varphi|c_{1,\varepsilon}(x,D)\varphi)\\
&\ge c_1\varepsilon^b\Vert\varphi\Vert^2_{\frac{m}{2}}-\Vert\varphi
\Vert_{\frac{m}{2}-1}\Vert c_{1,\varepsilon}(x,D)\varphi\Vert_{-\frac{m}{2}}
\end{align*}
Combining Proposition \ref{prop_Sob} with \eqref{ineq_SR} we obtain
\[
2\Re\, (\varphi|a_\varepsilon^\ast(x,D)\varphi)\ge c_1\varepsilon^b\Vert
\varphi\Vert^2_{\frac{m}{2}}-\Vert\varphi\Vert_{\frac{m}{2}-1}c_2
\varepsilon^b\Vert\varphi\Vert^2_{-\frac{m}{2}}\ge c_1\varepsilon^b\Vert
\varphi\Vert^2_{\frac{m}{2}}-2\delta c_2\varepsilon^b\Vert\varphi
\Vert^2_{\frac{m}{2}}.
\]
Concluding for $\delta$ small enough there exists a constant
$C>0$ such that
\[
\Vert\varphi\Vert^2_{\frac{m}{2}}\le C\varepsilon^{-b}\Vert\varphi
\Vert_{\frac{m}{2}}\Vert a_\varepsilon^\ast(x,D)\varphi\Vert_{-\frac{m}{2}}
\le C\varepsilon^{-b}\Vert\varphi\Vert_{\frac{m}{2}}\Vert a_\varepsilon^\ast(x,D)
\varphi\Vert_{0}
\]
holds for all $\varphi\in\mathcal{C}^\infty_{\rm c}(\Omega_\delta)$.
\end{proof}

\begin{example} \rm
Note that the condition \eqref{inv_Sob} can be fulfilled by
differential operators which are not a bounded perturbation of
a differential operator with constant Colombeau coefficients.
This means that the results of this section enlarge the family
of generalized differential operators whose local solvability
we are able to investigate in the Colombeau context.
As an explanatory example in $\mathbb{R}^2$ consider
\[
a_\varepsilon(x,D)=D_1+b_\varepsilon(x)D_2,
\]
where $(b_\varepsilon)_\varepsilon$ is the representative of a generalized function.
The generalized differential operator $a(x,D)$ generated by
$(a_\varepsilon)_\varepsilon$ is not a bounded perturbation of the operator
at $0$ if we take $b_\varepsilon(0)=0$. However if $(b_\varepsilon)_\varepsilon$
is real valued and suitable moderateness conditions are satisfied
(for instance $(b_\varepsilon)_\varepsilon$ bounded in $\varepsilon$ together with
all its derivatives), the arguments of
\cite[Proposition 4.3]{SaintRaymond:91} lead us to an estimate
from below of the type considered by Theorem \ref{theo_loc_Sob}.
\end{example}

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\end{document}