\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 144, pp. 1--39.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2012/144\hfil Factorization of hyperbolic operators]
{Factorization of second-order strictly hyperbolic operators with
 non-smooth coefficients and microlocal diagonalization}

\author[M. Glogowatz \hfil EJDE-2012/144\hfilneg]
{Martina Glogowatz} 

\address{Martina Glogowatz \newline
Faculty of Mathematics, University of Vienna, Austria}
\email{martina.glogowatz@univie.ac.at}

\thanks{Submitted November 8, 2011. Published August 21, 2012.}
\thanks{Supported by project Y237 from the Austrian Science Fund (FWF)}
\subjclass[2000]{35S05, 46F30}
\keywords{Algebras of generalized functions; wave front sets;
\hfill\break\indent parameter dependent pseudodifferential operators}

\begin{abstract}
 We study strictly hyperbolic partial differential operators of second-order
 with non-smooth coefficients. After modeling them as semiclassical Colombeau
 equations of log-type we provide a factorization procedure on some
 time-space-frequency domain. As a result the operator is written as a
 product of two semiclassical first-order constituents of log-type which
 approximates the modelled operator microlocally at infinite points.
 We then present a diagonalization method so that microlocally at infinity
 the governing equation is equal to a coupled system of two semiclassical
 first-order strictly hyperbolic pseudodifferential equations.
 Furthermore we compute the coupling effect. We close with some remarks on
 the results and future directions.
\end{abstract}

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

\newcommand\MSym[2][]{\mathcal{M}^{(#2)}_{\mbox{$\mathcal{S}$}^{#1}}}
\newcommand\MhgSym[2][]{\mathcal{M}^{(#2)}_{\mbox{$\mathcal{S}$}^{#1}_{\rm hg}}}
\newcommand\MphgSym[2][]{\mathcal{M}^{(#2)}_{\mbox{$\mathcal{S}$}_{\rm phg}^{#1}}}
\newcommand\OPNerror[2]{{\mathcal{E}}^{(#1,#2)}_{\psi}}


\section{Introduction}\label{sec:Intro}

When studying strictly hyperbolic partial differential equations with
generalized coefficients the results commonly depend on the appropriate
choice of the asymptotic scale of the regularizing parameter.
In this paper we consider certain types of second order partial differential
equations with generalized coefficients of the form
\begin{equation}\label{eqn:Basic}
LU=F
\end{equation}
where $U, F$ are Colombeau generalized functions
(See section~\ref{sec:BasicNotions} for the precise
asymptotic behavior in this case.)
in $\mathcal{G}_{2,2}$ on $\mathbb{R}^{n+1}$ and on the level of representatives
the operator $L$ acts as
\begin{equation*}
(u_\varepsilon)_\varepsilon \mapsto (L_{\varepsilon}(x,z,D_t,D_x,D_z)
u_\varepsilon)_\varepsilon \quad \forall (u_\varepsilon)_\varepsilon
\in \mathcal{M}_{H^{\infty}}.
\end{equation*}
In detail the operator $L_\varepsilon$ is considered to be of the  form
\begin{equation}\label{eqn:Gov}
L_\varepsilon := \partial_{z}^2 + \sum_{j=1}^{n-1} b_{j,\varepsilon}(x,z)
\partial_{x_j}^2 -  c_{\varepsilon}(x,z) \partial_{t}^2.
\end{equation}
Here the coefficient $c_\varepsilon(x,z)$ is of log-type up to order
 $r \in \mathbb{N}$, i.e.\ for some constant $C > 0$ we have
 $\| \partial^{\alpha} c_\varepsilon \|_{L^{\infty}}
= \mathcal{O} (\log^{|\alpha|/r}(C/\varepsilon))$ as
$\varepsilon \to 0$, and strictly non-zero in the sense that there exists
$\varepsilon_1 \in (0,1]$ such that
 $\inf_{y \in \mathbb{R}^n} |c_\varepsilon(x,z)| \geq C$ for all
$\varepsilon \in (0,\varepsilon_1]$ and some constant $C>0$ independent of
 $\varepsilon$. Also $b_{j,\varepsilon}$ is of log-type up to order $r$ and
strictly non-zero. Moreover we assume that $c_\varepsilon \to c$ and
$b_{j,\varepsilon} \to b_{j}$ in the H\"{o}lder space
$\mathcal{C}^{0,\mu}(\mathbb{R}^n)$ with exponent $\mu \in (0,1)$ as
$\varepsilon \to 0$.

The aim of this paper is to provide a symbolic calculus  to explain
a diagonalization with respect to the parameter $z$ of the equation
in \eqref{eqn:Basic}. To do so we will reduce the diagonalization
problem to a factorization theorem which in the case of smooth coefficients
can be found in \cite[Appendix II]{Kumano-go:81},
\cite[Chapter 23]{Hoermander:3}. We note that in these references the results
are only valid for operators with simple characteristics on the phase space
with the zero section excluded. This is different to our approach as we are
 interested in factorizations with respect to the parameter $z$.

For recent contributions to a related topic we refer the reader to the work
 by Garetto and Oberguggenberger \cite{GaMoe:2011a}.
There the authors established existence and regularity results for solutions
 to strictly hyperbolic systems with Colombeau coefficients using symmetrisation
up to regularizing errors. In detail, they proved existence of generalized
 solutions in the case of slow scale coefficients and, in addition they showed
regularity in the case of logarithmic slow scale coefficients.
Note that a net $r_\varepsilon \in \mathbb{R}^{(0,1]}$ is a logarithmic slow
scale net if $|r_\varepsilon| = \mathcal{O}(\log^p(1/\varepsilon))$ for all
$p >0$ as $\varepsilon \to 0$ and a generalized coefficient
$(a_\varepsilon)_\varepsilon$ is said to be logarithmic slow scale regular if
for all $\alpha \in \mathbb{N}^n$ there exists  a logarithmic slow scale net
 $r_\varepsilon$ such that
$$
\| \partial^{\alpha} a_\varepsilon \|_{L^{\infty}(\mathbb{R}^n)}
= \mathcal{O} (r_\varepsilon) \quad \varepsilon \to 0.
$$
Again, the square roots of the principal symbol of the operator are assumed
to be simple on the phase space without the zero section in this case.

So when studying an equation of the form \eqref{eqn:Basic} with coefficients
that satisfy strongly positive logarithmic slow scale estimates one can adapt
the results in \cite{GaHo:05,Kumano-go:81} to obtain a diagonalization in
some microlocal subregion of the phase space and microlocal regularity has
to be understood in a $\mathcal{G}_{2,2}^\infty$ sense.
Recall that $\mathcal{G}_{2,2}^\infty$ is the space of
regular generalized functions in $\mathcal{G}_{2,2}$ and is characterized
by uniform $\varepsilon$-growth in all derivatives.
This is a generalization of the results in the smooth coefficient case
which will be briefly explained in the beginning of
Subsection~\ref{subsec:PreviousResults}.
Since in the Colombeau framework the evolution behavior of propagating
 singularities is not yet sufficiently understood it is not clear how one
can derive well-posed approximated Cauchy problems from the
resulting microlocal first-order equations due to the diagonalization.
For the theory for first-order hyperbolic pseudodifferential equations
with generalized symbols we refer to \cite{Hoermann:04, Moe:09, GaMoe:2011a}.

However, to analyze operators as in \eqref{eqn:Gov} with less regular
coefficients as in \cite{GaMoe:2011a} we try a different approach.
To overcome the necessity of the logarithmic slow scale assumption
(e.g. construction of an approximative inverse) we associate to the
 operator $L_\varepsilon$ in \eqref{eqn:Gov} a semiclassical modification
 $L_{\psi,\varepsilon}$ such that $L_{\psi,\varepsilon}=\varepsilon^2 L_\varepsilon$.
Here the $\psi=\psi(\varepsilon)$ refers to the phase function in which the
semiclassical scaling is kept to be retained and depend on the regularizing
parameter $\varepsilon \in (0,1]$. The detailed explanation of the correspondence
between the operators $L$ and $L_\psi$ is given in
 Subsection~\ref{subsec:ScColombeau}. Then instead of working with the
equation \eqref{eqn:Basic} we consider the corresponding semiclassical problem
\begin{equation}\label{eqn:ScBasic}
L_\psi U=F
\end{equation}
where $U, F$ are generalized functions in $\mathcal{G}_{2,2}$ and the
operator $L_\psi: \mathcal{G}_{2,2} \to \mathcal{G}_{2,2}$ acts on the
level of representatives as
\begin{equation*}
(u_\varepsilon)_\varepsilon \mapsto
(L_{\psi,\varepsilon}(x,z,D_t,D_x,D_z)u_\varepsilon)_\varepsilon
\quad \forall (u_\varepsilon)_\varepsilon \in \mathcal{M}_{H^{\infty}}.
\end{equation*}
Due to the additional $\varepsilon$-dependent semiclassical scaling parameter
we then apply the theory of generalized semiclassical pseudodifferential operators
which we call the $\psi$-pseudodifferential operators in the following for short.
This enables us to state a factorization theorem for $L_\psi$ in terms of two
first-order $\psi$-pseudodifferential operators with respect to the parameter
$z$ modulo two different types of error operators. As already mentioned above
the factorization is only valid when imposing certain restrictions on the
underlying time-space-frequency domain which we left $\varepsilon$-independent.
In detail the error operators are characterized by a semiclassical negligibility
condition outside a compact set. Since global semiclassical negligible operators
are easy to handle as they map moderate nets to negligible ones the interesting
 case is to understand the compactly supported error operators. But, due to the
semiclassical scaling, worse $\varepsilon$-asymptotics are introduced when
studying boundedness of such compactly supported operators. The basic notions
and a general calculus of $\psi$-pseudodifferential operators can be found in
Sections~\ref{sec:BasicNotions} and ~\ref{sec:PsiPseudoCalculus},
the factorization procedure is explained in Section~\ref{sec:Fact}.

To eliminate the remainder terms produced in the factorization procedure
we then proceed in Section~\ref{sec:WFInfty} by presenting an adapted notion
 of microlocal regularity which essentially corresponds to the semiclassical
 wave front set at infinite points within the Colombeau framework.
This allows us to overcome the difficulties that arise due to the compactly
supported error operators occurring in the factorization. Concepts of the wave
front set in Colombeau's theory have been explored in
\cite{NPS:98, GaHo:05, GaHo:06, Garetto:06}. For an introduction to semiclassical
analysis and microlocalization we refer to
\cite{Martinez:02, EvZw, GuSt:10, Alexandrova:08}. Again we remark that unlike
to the case of smooth coefficients we lose the property that singularities
are propagating on geometrically determined phase space trajectories.

In Section~\ref{sec:Diag} we then combine the factorization with our notion
of microlocal regularity in order to describe the desired microlocal diagonalization
 method. As a result we obtain a coupled system of two first-order
$\psi$-pseudodifferential equations that approximate equation
\eqref{eqn:ScBasic} microlocally at infinite points on an adequate subdomain
 of the phase space.

\subsection{The smooth background case}\label{subsec:PreviousResults}

A particular example of a strictly hyperbolic partial differential equation
of second order is the wave equation describing phenomena such as propagation
of elastic waves and vibrations. However, in this subsection of the introduction
we give a motivation for the present survey and is devoted to one-way wave
propagation in inhomogeneous acoustic media in the case of smooth background
coefficients. For more information on the relevance in seismic imaging and
migration models we refer the reader to \cite{Claerbout:85, BCS:01, SdH:05, SdH:06}.

To start with a real life problem we may ask whether and how one can establish a
tap-proof communication link between a source and a receiver in an underwater
environment. In the following we analyze how the correlation between the source
and the receiver location can help to detect a desired information in downward
continuation problems. The mathematical reformulation of such problems
corresponds to initial value problems of second-order partial differential
equations with a space-like direction as evolution parameter and is in general
not well-defined.

To overcome the ill-posedness the concept of wave-field decomposition enables
us to rewrite the full wave equation into a coupled system of approximative
one-way wave equations. Using microlocal techniques the system of one-way
equations can then be decoupled assuming that the wave field propagates in one
direction within certain propagation angles and prohibits propagation in the
opposite direction. Therefore the approximation allows to determine the
high-frequency components of the solution as they propagate along curved
trajectories in the phase space.

In the following we recall the crucial statements given by Stolk for the
directional wave field decomposition in the present of smooth background
data \cite{Stolk:04, Stolk:05}. For a discussion on the inverse scattering
problem with simultaneous consideration of possible reflection data see for
example \cite{Stolk:00, SdH:06, dHlRB:03} and the references therein.
Also we refer to \cite{LeRousseau:06, LeRH:06} for investigations of the
Cauchy problem of first-order pseudodifferential equations and to \cite{ZZB:03}
for numerical implementations.

As in \cite{Stolk:04} our basic type of model will be the wave equation for
inhomogeneous acoustic media in n-dimensions, $n \geq 2$. Since we are
interested in approximative one-way equations we allocate the vertical
direction $z$, which we call the depth, the lateral directions are denoted by $x$.
The medium itself is described by the wave speed $c = c(x,z)$ and the fluid
density $\rho = \rho(x,z)$. Also we let $U = U(t,x,z)$ denote the acoustic
 wave field and $F=F(t,x,z)$ a source which usually describes the initiation
of an acoustic wave. Then the acoustic wave equation is given by
\begin{equation} \label{eqn:Stolk}
PU:=\Big( -\frac{1}{\rho} \frac{1}{c^2} \partial_{t}^2
+ \sum_{j=1}^{n-1} \partial_{x_j} \frac{1}{\rho} \partial_{x_j}
+ \partial_{z} \frac{1}{\rho} \partial_{z} \Big) U = F
\end{equation}
where $U$ and $F$ are in the distribution space $\mathcal{D}'(\mathbb{R}^{n+1})$.
Further we assume smooth background data, i.e.\ the wave speed and the density
are functions in $\mathcal{C}^{\infty} (\mathbb{R}^n)$, which shall also satisfy
the boundedness conditions $0 < c(x,z), \rho(x,z) < \infty$ for all
$(x,z) \in \mathbb{R}^{n-1}\times \mathbb{R}$.

In the case of slow varying media and the presence of a source function
 we follow Stolk's approach and restrict the analysis to a microlocal
region $I_{\Theta_{2}}$ that is given by
\begin{align*}
I_{\Theta_{2}} &:= \{ (t,x,z,\tau,\xi,\zeta) \mid (x,z,\tau,\xi) \in I_{\Theta_{2}}',\; |\zeta| \leq C |\tau| \}\\
I_{\Theta_{2}}' &:= \{ (x,z,\tau,\xi) \mid \tau \neq 0,\ |\xi| \leq \sin\Theta_2 |c(x,z)^{-1}\tau| \}
\end{align*}
for some angle $\Theta_2  \in (0,\pi/2)$. An illustration
of the first of these domains is presented in Figure \ref{fig1};
see \cite{LeRousseau:06}.

\begin{figure}[h!]
\begin{center}
\includegraphics{fig1}
\end{center}
\caption{The shaded area corresponds to $I_{\Theta_2}$ at a given $(t,x,z)$ and
frequency $\tau$.
The bold dotted line designates the characteristic set $\text{Char}(P)$ and
$\theta$ is the propagation angle of the singularities}
\label{fig1}
\end{figure}

Then under the assumption that the wave front set of $U$ is contained
in $I_{\Theta_{2}}$, one can rewrite a microlocal equivalent model
to \eqref{eqn:Stolk} in terms of a coupled system of one-way wave
equations with the depth as evolution parameter.

The main result is then the following. The equation
\begin{equation*}
PU = F \quad \text{microlocally on $I_{\Theta_{2}}$}
\end{equation*}
is equivalent to the system of two first-order strictly hyperbolic
partial differential equations of the form
\begin{equation}\label{eqn:System}
P_{0,\pm} u_{\pm} := \big( \partial_z - i B_{\pm}(x,z,D_t,D_x) \big) u_{\pm}
 =  f_{\pm}  \quad \text{microlocally on $I_{\Theta_{2}}$},
\end{equation}
where the plus-minus sign refers to downward and upward migration.
Here $u_{\pm}, f_{\pm}$ are distributions and $B_{\pm}$ are pseudodifferential
operators of order 1 that can be chosen selfadjoint. Furthermore the coupling
effect of the counterpropagating constituents $u_{\pm}, f_{\pm}$ and the
original data $U,F$ can be computed by a Douglis Nirenberg elliptic
pseudodifferential operator transfer matrix.

To model wave propagation in the downward direction in the source-free
case one decouples \eqref{eqn:System} microlocally and we are interested
in the solutions to the problem
\begin{gather*}
PU = 0 \quad z > z_0\\
\text{WF}(U) \cap \{ z= z_0 , \; \zeta/\tau > 0 \} = \emptyset
\end{gather*}
for some initial depth $z_0 \in \mathbb{R}$ such that $U |_{z_0}$ is well-defined.

Using the geometry of the microlocal region we let the set
$J_{\Theta,+}(z_0) \subset T^*\mathbb{R}^{n+1} \setminus 0$ consisting of
points $(t,x,z,\tau,\xi,\zeta)$ so that the bicharacteristics
$(t(z),x(z),\tau(z),\xi(z))$ corresponding to $B_+$, parametrized by $z$,
and with propagation angle $\theta(z)$, pass through $(t_0,x_0,\tau_0,\xi_0)$
at initial depth $z=z_0$ and the points $(x(z),z, \tau(z),\xi(z))$ remain
in $I_{\Theta}'$ for all $z \in [0,Z]$. So $J_{\Theta,+}(z_0)$ is the set that
 can be reached from the initial depth $z=z_0$ while staying in $I_{\Theta,+}$
 and the propagation angle $\theta(z)$ along the bicharacteristic is always
smaller than $\Theta$ (cf. Figure 1).

Since the equation for downward migration in \eqref{eqn:System} holds
only microlocally, one obtains approximative solutions when studying a
perturbation $P_{+}$ of the operator $P_{0,+}$ including an additional damping
term $C = C(x,z,D_t,D_x)$ which vanishes in $I_{\Theta_{1}}$ for some fixed
positive angle $\Theta_1 < \Theta_2$ and suppresses singularities outside
$I_{\Theta_{2}}$. Also note that $u_{-}$ is vanishing on
$I_{\Theta_{2}} \cap \{ z = z_0 \}$ so that the perturbed initial value problem
for downward propagation now reads
\begin{align*}
P_{+}u_{+}  &= \left( \partial_z - i B_{+} + C \right) u_{+} = 0 \quad \mathbb{R}^{n} \times (z_0,Z)\\
u_{+} |_{z_0} &= Q^{-1}_{+} (z_0) U |_{z_0}
\end{align*}
where $Q^{-1}_+$ is the essential component of the transfer matrix mentioned
above. As a result the solution to the initial value problem for the perturbed
first-order pseudodifferential equation can be related to that of
$P_{0,+}u_+ = 0$ using the geometry in the region $J_{\Theta,+}$.
In detail one can recover the high frequency part of the original wave field
on some subset $J_{\Theta_1,+}$ of the phase space that can be reached from
an initial given depth $z_0$ while staying in $I_{\Theta_1}$ for propagation
angles $\theta(z) < \Theta_1 < \Theta_2$.

\section{Basic notions}\label{sec:BasicNotions}

In this section we specify the basic notions that are needed for our constructions.
As the problem is treated within the framework of Colombeau algebras we refer
to the literature \cite{Colombeau:85, Moe:92, NPS:98, GKOS:01, GGO:05, Garetto:08}
for a systematic treatment in this field.

One of the main objects in our setting are Colombeau generalized functions based
on $L^2$-norm estimates which were first introduced in \cite{BiMoe:92}.
The elements in this algebra are given by equivalence classes
$u := [(u_{\varepsilon})_{\varepsilon\in (0,1]}]$ of nets of regularizing
functions $u_\varepsilon$ in the Sobolev space
$H^{\infty}=\cap_{k \in \mathbb{Z}} H^k$ corresponding to certain asymptotic
 seminorm estimates. More precisely, we denote by $\mathcal{M}_{H^{\infty}}$
the nets of moderate growth whose elements are characterized by the property
\begin{equation*}
\forall \alpha \in \mathbb{N}^n \ \exists N \in \mathbb{N}: \;
 \| \partial^{\alpha} u_{\varepsilon} \|_{L^{2}(\mathbb{R}^n)}
= \mathcal{O}({\varepsilon}^{-N}) \quad  \text{as } {\varepsilon} \to 0.
\end{equation*}
Negligible nets are denoted by $\mathcal{N}_{H^{\infty}}$ and are nets in
$\mathcal{M}_{H^{\infty}}$ whose elements satisfy the  additional
 condition:
\begin{equation*}
\forall q \in \mathbb{N}: \; \| u_{\varepsilon} \|_{L^{2}(\mathbb{R}^n)}
= \mathcal{O}({\varepsilon}^{q}) \quad \text{ as } {\varepsilon} \to 0.
\end{equation*}
For properties of negligible nets see \cite[Proposition 3.4]{Garetto:05b}.
Then the algebra of generalized functions based on $L^{2}$-norm estimates
is defined as the factor space
$\mathcal{G}_{H^{\infty}} = \mathcal{M}_{H^{\infty}} / \mathcal{N}_{H^{\infty}}$
for which we continue to write $\mathcal{G}_{2,2} (\mathbb{R}^n)$ as in
\cite{BiMoe:92}. For simplicity, we shall also use the notation
$(u_\varepsilon)_{\varepsilon}$ instead of $(u_\varepsilon)_{\varepsilon \in (0,1]}$
throughout the paper.

Using \cite[Theorem 2.7]{BiMoe:92} we first note that the distributions
$H^{-\infty} = \cup_{k \in \mathbb{Z}} H^k$ are linearly embedded in
$\mathcal{G}_{2,2}(\mathbb{R}^n)$ by convolution with a mollifier
$\varphi_\varepsilon(x) = \varepsilon^{-n} \varphi(\varepsilon^{-1}x)$
where $\varphi \in \mathscr{S}(\mathbb{R}^n)$ is a Schwartz function such that
\begin{equation}\label{mollifier}
\int  \varphi(x) \, dx  =  1, \quad \int  x^{\alpha}  \varphi(x) \, dx  =  0
\quad \text{for all } |\alpha| \geq 1.
\end{equation}
Further, by the same result, $H^{\infty}(\mathbb{R}^n)$ is embedded as a
subalgebra of $\mathcal{G}_{2,2} (\mathbb{R}^n)$.

More generally, we introduce Colombeau algebras based on a locally convex
vector space $E$ topologized through a family of seminorms
$\{ p_i \}_{i \in I}$ as in \cite[Section 3]{Garetto:05a}, \cite[Section 1]{GHO:09}.
Again we call the elements of
\begin{align*}
\mathcal{M}_{E} = \{ (u_{\varepsilon})_{\varepsilon} \in E^{(0,1]}
\mid \forall i \in I  \ \exists N \in \mathbb{N}:  p_i( u_{\varepsilon})
 = \mathcal{O} ({\varepsilon}^{-N})  \mbox{ as }  {\varepsilon} \to 0\}
\end{align*}
$E$-moderate and the set
\begin{align*}
\mathcal{N}_E = \{ (u_{\varepsilon})_{\varepsilon} \in E^{(0,1]} \mid
 \forall i \in I  \ \forall q \in \mathbb{N}: p_i( u_{\varepsilon})
= \mathcal{O} ({\varepsilon}^{q}) \ \mbox{as} \ {\varepsilon} \to 0\}
\end{align*}
is said to be $E$-negligible. Then $\mathcal{N}_E$ is an ideal in
$\mathcal{M}_{E}$ and the space of Colombeau algebra based on $E$ is defined
 by the factor space $\mathcal{G}_E = \mathcal{M}_E / \mathcal{N}_E$ and
 possesses the structure of a $\widetilde{\mathbb{C}}$-module.
Here $\widetilde{\mathbb{C}}$ denotes the ring of complex generalized numbers
which is obtained by setting $E= \mathbb{C}$, this means
$\widetilde{\mathbb{C}} := \mathcal{M}_{\mathbb{C}} / \mathcal{N}_{\mathbb{C}}$.
For more information about the topological structure of $\widetilde{\mathbb{C}}$
we refer to \cite{Garetto:05a}.

To realize the log-type up to order $r \in \mathbb{N}$ conditions on the coefficients
of the model operator \eqref{eqn:Gov} a rescaling in the mollification is required,
 see \cite{HdH:06, Moe:92}. In detail, we are going to consider regularized functions
where the asymptotic growth is estimated in powers of
$\omega_\varepsilon := (\log(C/\varepsilon))^{1/r}$ for some $r \in \mathbb{N}$ and
a constant $C >0$ such that $(\omega_\varepsilon)_\varepsilon$ is strongly positive.
Here the exponent $1/r$ is not essential in the further considerations but we give
some remarks in this regard in Section~\ref{sec:Diag}.
In this case the regularization is obtained by convolution with the logarithmically
scaled mollifier
$\varphi_{\omega_\varepsilon^{-1}}(.) := \omega_\varepsilon^n
\varphi(\omega_\varepsilon .)$ with $\varphi \in \mathscr{S}(\mathbb{R}^n)$
as in \eqref{mollifier}.

\subsection*{Further terminology}

To establish a factorization theorem for generalized semiclassical partial
differential operator as in \eqref{eqn:ScBasic} we will have to give a meaning
to the square root of such operators. For this reason we introduce generalized
semiclassical pseudodifferential operators which are characterized by symbols
with respect to a certain phase function. Because the phase function is of simple
type we shall initially discuss the main notions of generalized symbols which
satisfy asymptotic growth conditions with respect to
$\omega_\varepsilon = (\log(C/\varepsilon))^{1/r}$ as already mentioned above.
 As usual, we use the notation $\langle \xi \rangle := (1 + |\xi|^2)^{1/2}$.

First, we let $m \in \mathbb{R}$ and denote by
$S^{m} = S^{m}(\mathbb{R}^n \times \mathbb{R}^n)$ the set of symbols of order $m$
as introduced by H\"{o}rmander in \cite[Definition 18.1.1.]{Hoermander:3}.
Furthermore the space $S^{ m}_{ \rm hg} = S^{ m}_{ \rm hg}(\mathbb{R}^n
 \times (\mathbb{R}^n \setminus 0))$ consists of homogeneous functions
$a \in \mathcal{C}^{\infty} (\mathbb{R}^n \times (\mathbb{R}^n \setminus 0))$
of order $m$; i.e., $a(x,\lambda \xi) = \lambda^{m}a(x,\xi)$ for all $\lambda > 0$
and $\xi \neq 0$, such that
\begin{equation*}
\forall \alpha,\beta \in \mathbb{N}^n \ \exists C >0: \; 
 |\partial^{\alpha}_\xi \partial^{\beta}_x a(x,\xi)| \leq C|\xi|^{m-|\alpha|}
\quad \text{for }(x,\xi) \in \mathbb{R}^n \times (\mathbb{R}^n \setminus 0).
\end{equation*}
Since the symbol class $S^{ m}$ satisfies global estimates we remark that
$\mathcal{M}_{S^m}$ is different to the symbol
classes in \cite[Section 1.4]{GHO:09}. We note that
$\mathcal{M}_{S^{m}}$ is the space $\mathcal{M}_E$ by setting $E = S^{ m}$.
In the following we will typically encounter subspaces of
$\mathcal{M}_{S^m}$ subjected to two different asymptotic scales.
This extends the definition of $\mathcal{M}_{S^m}$ and
$\mathcal{N}_{S^m}$ in the following way:
%
\begin{definition}\label{defn:MSymb} \rm
Let $\nu$ be a non-negative real number and $l,k \in \mathbb{R}$.
For $m \in \mathbb{R}$ we let $(a_\varepsilon)_\varepsilon$ be a family
of symbols $a_\varepsilon \in S^{ m}$. We then say that
$(a_\varepsilon)_\varepsilon$ is in the generalized symbol class
$\MSym[m,k]{\nu,l}$ if and only if
\begin{equation}\label{TwoScaleSymb}
\begin{gathered}
\exists \eta \in (0,1] \ \forall \alpha, \beta \in \mathbb{N}^n \
\exists C > 0 \ \forall \varepsilon \in (0,\eta]:\\
 q^{(m)}_{\alpha, \beta} (a_{\varepsilon})  :=
\sup_{(x,\xi) \in  \mathbb{R}^{2n}} | \partial_{\xi}^{\alpha} \partial_x^{\beta}
a_\varepsilon (x,\xi) | \langle \xi \rangle^{-m + |\alpha|}
\leq C \varepsilon^{k} \omega_{\varepsilon}^{\nu |\beta| +l}.
\end{gathered}
\end{equation}
We call $m, k$ and $(\nu,l)$ the order, the growth type and the log-type
respectively of $\MSym[m,k]{\nu,l}$.
\end{definition}

This definition is similar to that for generalized symbols
$\mathcal{S}^{m,\mu}_{\rho,\delta,\omega}$ in \cite[Definition 4.1]{GGO:05}.
Also, since $(\omega_\varepsilon)_\varepsilon$ is a strongly positive
slow scale net we note that a generalized symbol $(a_\varepsilon)_\varepsilon$
in $\MSym[m,k]{\nu,l}$ can always be estimated in the following way:
\begin{equation}\label{RegSymb}
\begin{gathered}
\exists \eta \in (0,1] \ \forall
\alpha, \beta \in \mathbb{N}^n \ \exists C > 0 \ \text{such that}\\
q^{(m)}_{\alpha, \beta} (a_{\varepsilon}) \leq C \varepsilon^{k-1}
\qquad \forall \varepsilon \in (0,\eta].
\end{gathered}
\end{equation}
Later in Lemma~\ref{lem:psiparametrix} we give the construction scheme
for approximative inverse operators and the advantage of using
\eqref{TwoScaleSymb} rather than \eqref{RegSymb} will be clear then.
Hereafter we typically work in spaces $\MSym[m,k]{\nu,l}$ with
$\nu \in \{ 0,1 \}$. Analogously to Definition~\ref{defn:MSymb} one
introduces the space
$\MhgSym[m,k]{\nu,l}(\mathbb{R}^n \times (\mathbb{R}^n \setminus 0))$.

Furthermore the notion of negligibility in $\MSym[m,k]{\nu,l}$ is defined
as follows:

\begin{definition} \rm
An element of $\MSym[m,k]{\nu,l}$ is said to be negligible, denoted
by $\mathcal{N}_{\mbox{$\mathcal{S}$}^m}$, if the following condition is fulfilled:
\begin{equation*}
\begin{gathered}
\exists \eta \in (0,1] \ \forall q \in \mathbb{N} \
\forall \alpha, \beta \in \mathbb{N}^n \ \exists C > 0 \ \text{such that}\\
q^{(m)}_{\alpha, \beta} (a_{\varepsilon}) \leq C \varepsilon^{q}
\quad \forall \varepsilon \in (0,\eta].
\end{gathered}
\end{equation*}
\end{definition}

To give an example, let $P(x,D_x) = \sum_{|\alpha| \leq m} a_{\alpha}(x) D^{\alpha}_x$
be a partial differential operator with bounded and measurable coefficients.
Then the logarithmically scaled regularized coefficients
$a_{\alpha,\varepsilon} := a_\alpha \mathbin{*} \varphi_{\omega_\varepsilon^{-1}}$
satisfy the  estimate
\begin{equation*}
\| \partial^{\alpha} a_{\alpha,\varepsilon} \|_{L^{\infty}(\mathbb{R}^n)}
= \mathcal{O}(\omega_\varepsilon^{|\alpha|})
= \mathcal{O}(\log^{|\alpha|/r}(C/\varepsilon)).
\end{equation*}
Therefore $(a_{\alpha,\varepsilon})_{\varepsilon}$ is of log-type up to order $r$,
that is $\| \partial^{\alpha} a_{\alpha,\varepsilon} \|_{L^{\infty}(\mathbb{R}^n)}
= \mathcal{O}(\log(C/\varepsilon))$ for all $|\alpha| \leq r$, but it is not
logarithmic slow scale regular.
Recall that the asymptotic norm estimate (of any order) of a logarithmic slow
scale coefficient has a bound $\mathcal{O}(r_\varepsilon)$ where
$|r_\varepsilon| = \mathcal{O}(\log^p(1/\varepsilon))$ for all $p > 0$
as $\varepsilon$ tends to 0.
In the same manner we obtain for the logarithmically scaled regularization
 of the operator a symbol $(p_\varepsilon)_\varepsilon$ where
$p_\varepsilon (x,\xi) := (p(.,\xi) \mathbin{*} \varphi_{\omega_\varepsilon^{-1}})(x)$
which is of class $\MSym[m,0]{1,0}$ and of log-type up to order $(\infty,r)$,
 cf. \cite[Remark 3.2]{Hoermann:04}.

Also we will  use  the following microlocal symbol classes:

\begin{definition}\label{defn:SymbOpenSet} \rm
Let $U \subset \mathbb{R}^n \times \mathbb{R}^n$ be open and conic with
 respect to the second variable. We say that a generalized symbol
$(a_\varepsilon)_\varepsilon$ is in $\MSym[m,k]{\nu,l}(U)$
if $a_\varepsilon \in \mathcal C^{\infty} (U)$ for fixed $\varepsilon \in (0,1]$
and there is a constant $K>0$ independent of $\varepsilon$ such that:
\begin{equation*}
\begin{gathered}
\exists \eta \in (0,1] \ \forall \alpha,\beta \in \mathbb{N}^n \
\exists C>0 \ \text{for which}\\
|\partial_{\xi}^{\alpha} \partial_x^{\beta} a_\varepsilon(x,\xi)|
\leq C \varepsilon^{k} \omega_\varepsilon^{\nu |\beta|+l}
\langle \xi\rangle^{m-|\alpha|} \quad \text{for } (x,\xi) \in V_{K},
 \ \varepsilon \in (0,\eta],
\end{gathered}
\end{equation*}
where $V_{K}:= \{ (x,\xi) \in U \ | \; d(\xi,\zeta) \geq K,\,
 \zeta \in \partial \text{pr}_2(U) \neq \emptyset \}$.
 We observe that, if $(x,\xi) \in V_{K}$ then
$(x,\lambda \xi)\in V_{K}$ for all $\lambda \geq 1$.
\end{definition}
Note that Definition \ref{defn:MSymb} is equivalent to Definition
\ref{defn:SymbOpenSet} in the case that $U=\mathbb{R}^n \times \mathbb{R}^n$.
Also, Definition~\ref{defn:SymbOpenSet} is different to a straightforward
Colombeau generalization of global symbols $S^m(U)$. Also note the difference
to the definition of the classical symbols which in the case of local symbols
 is given in \cite[Definition 2.3, page 141]{ChPi:82}.

In the following subsection we give some motivating remarks for the microlocal
constructions used in Section~\ref{sec:WFInfty}.

\subsection{Remarks on the governing equation in a semiclassical Colombeau setting}
\label{subsec:ScColombeau}

As this is a first investigation we will concentrate on operators as given
in \eqref{eqn:Gov} which are characterized by homogeneity. Also note that our
 model operators have less regular log-type coefficients than logarithmic slow scale.

As pointed out in the motivation the main aim is to provide a diagonalization
 procedure using factorization theorems and microlocal analysis. Because it turns
out that the log-type condition is not strong enough for our constructions we
will introduce certain pseudodifferential operators defined on the Heisenberg group,
so-called semiclassical generalized pseudodifferential operators.

In the usual theory of semiclassical pseudodifferential operators in the
 Kohn-Nirenberg or standard quantization one associates to a symbol $a \in S^m$
an operator $A: \ensuremath{{\mathcal S}} \to \ensuremath{{\mathcal S}}$ in the
following way
\begin{align*}
A_\psi(x,D_x) u(x)
&:= (2\pi)^{-n} \int (\mathcal{F}a)(q,p) e^{-ipq \hbar /2}
 \pi_\hbar(q,p) u (x) \,d q \,d p \\
&= (2\pi \hbar)^{-n} \int e^{i (x-y) \xi/ \hbar} a(x,\xi) u (y) \,dy \,d \xi
\end{align*}
where in the first line $\mathcal{F}(a)$ denotes the Fourier transform
of $a$ on the projected Heisenberg group and $\pi_\hbar(q,p)$ is the
projected Schr\"{o}dinger representation of the Heisenberg group on
$\mathbb{R}^{2n}$ with parameter $\hbar$ and the second line corresponds
to the more familiar presentation of the first one. In detail we have
\begin{equation*}
\bigl(\pi_\hbar (q,p) u \bigr)(x) := e^{i(qx + \hbar pq/2)} u(x+ \hbar p)
\end{equation*}
and $\hbar$ denotes the normalized Planck constant. Note that for
 $\hbar = 1$ this representation leads to the usual Kohn-Nirenberg calculus.
 Also, using representation theory we observe for $\hbar \neq \hbar'$ that
$\pi_\hbar$ and $\pi_{\hbar'}$ are inequivalent representations on
$L^2(\mathbb{R}^n)$ and moreover for $\hbar \neq 0$ the representation
 $\pi_{\hbar}$ is irreducible and unitary on $L^2(\mathbb{R}^n)$.

Once introduced such an additional artificial parameter $\hbar$ gives us now
the opportunity to apply the theory of $\hbar$-pseudodifferential operators
used in semiclassical analysis which is a formalism in an asymptotic regime
$\hbar \ll 1$. As we want to combine this with the theory of generalized
pseudodifferential operators we start writing $\hbar := \hbar(\varepsilon)$
with the restriction $\hbar(\varepsilon) \to 0$ in the case that $\varepsilon \to 0$.

For technical reasons (e.g. construction of an approximative inverse) we also
have to prevent other possible values of the function $\hbar(\varepsilon)$.
 As this is a first approach in this field we have chosen
$\hbar(\varepsilon):= \varepsilon$ at this point.

So when given a family $(a_\varepsilon)_\varepsilon \in \mathcal{M}_{S^m}$
 we define the corresponding semiclassical pseudodifferential operator of
Colombeau type to be the linear operator
$A_\psi: \mathcal{G}_{2,2} \to \mathcal{G}_{2,2}$ which acts on the level of
representatives as
\begin{equation}\label{modifiedOP}
(u_\varepsilon)_\varepsilon \mapsto (A_{\psi,\varepsilon}(x,D_x)
u_\varepsilon)_\varepsilon \quad \forall (u_\varepsilon)_\varepsilon
\in \mathcal{M}_{H^{\infty}}.
\end{equation}
Here for fixed $\varepsilon \in (0,1]$ the right hand side corresponds to
the standard quantization of the symbol $(a_\varepsilon)_\varepsilon$
with parameter $\hbar(\varepsilon) = \varepsilon$; that is,
%
\begin{equation*}
\begin{split}
A_{\psi,\varepsilon}(x,D_x) u_\varepsilon(x)
&:= (2\pi)^{-n} \int_{\mathbb{R}^n} (\mathcal{F}
 a_\varepsilon)(q,p) e^{-ipq/2} \pi_\hbar(q,p)u_\varepsilon (x) \,dq \,dp \\
&= (2\pi \varepsilon)^{-n} \int e^{i (x-y) \xi/ \varepsilon}
 a_\varepsilon(x,\xi) u_\varepsilon (y) \,dy \,d \xi.
\end{split}
\end{equation*}
Then with $L_\varepsilon$ as in \eqref{eqn:Gov} the corresponding
$\psi$-pseudodifferential operator $L_\psi$ is essentially given by the
 oscillatory integral
\begin{equation*}
L_{\psi,\varepsilon} u_\varepsilon(t,y)
=  (2\pi \varepsilon)^{-n-1} \int e^{i t \tau/\varepsilon
+ iy \eta/\varepsilon} l_\varepsilon(y, \tau,\eta) \hat{u}_\varepsilon
(\tau/\varepsilon,\eta/\varepsilon) \, d\tau \, d\eta
\end{equation*}
for fixed $\varepsilon \in (0,1]$ and $(l_\varepsilon)_\varepsilon$
is called the generalized symbol of $L_\psi$ and is given by
\begin{equation*}
l_{\varepsilon}(y,\tau,\xi,\zeta) :=  - \zeta^2
- \langle b_{\varepsilon}(y) \xi,\xi \rangle + c_\varepsilon(y)  \tau^2
\end{equation*}
where we have set
$b_\varepsilon(y):= \text{diag} \left( b_{1,\varepsilon}(y),\ldots,b_{n-1,
\varepsilon}(y) \right)$, $y:=(x,z)$ and $\eta:=(\xi,\zeta)$.
Hence $(l_\varepsilon)_\varepsilon \in \MSym[2,0]{1,0}$ and the operator
$L_\varepsilon$ can easily be reconstructed from $L_{\psi,\varepsilon}$; i.e.,
$L_\varepsilon =  \varepsilon^{-2} L_{\psi,\varepsilon}$, because of the
homogeneity of the operators.

Given such a scaled generalized operator as above we carry out all transformations
within algebras of generalized functions from now on. More explicitly we will
study the action of the linear operator
 $L_\psi = \operatorname{OP}_{\psi} (l_\varepsilon)$ from $\mathcal{G}_{2,2}$
into itself in the following sense: on the level of representatives $L_\psi$
acts as in \eqref{modifiedOP}. This explains our governing equation
\begin{equation*}
L_\psi U = F
\end{equation*}
for which we will now present a microlocal diagonalization method.

\section{\texorpdfstring{$\psi$}{psi}-Pseudodifferential calculus}
\label{sec:PsiPseudoCalculus}

In this section we introduce a general calculus for $\psi$-pseudodifferential
operators which are certain semiclassical standard quantizations of generalized
symbols as demonstrated in the previous section. For an introduction in semiclassical
analysis we refer the reader to \cite{Martinez:02,EvZw,GuSt:10}.
Since most of the techniques are similar to the classical theory of
 pseudodifferential operators we also want to give \cite{Hoermander:3,Taylor:81}
as references. Moreover a detailed discussion on pseudodifferential operators
with Colombeau generalized symbols can be found in \cite{GGO:05,Garetto:08}.

To continue, given a generalized symbol $(a_\varepsilon(x,\xi))_\varepsilon$
one can assign the semiclassical standard quantization which we denote by
$\operatorname{OP}_\psi(a_\varepsilon):=a_\varepsilon(x,\varepsilon D_x)$.
In this sense $\psi= \psi(\varepsilon)$ can be thought of as a scaled phase
function on phase space. Therefore, we let
\begin{equation*}
\psi_\varepsilon(x,\xi) := \langle x, \xi \rangle/ \varepsilon
:= x \xi /\varepsilon  \quad \varepsilon \in (0,1].
\end{equation*}
throughout this article.

Moreover we choose the following convention for defining the Fourier transform
$\mathcal{F}$ of a function $u \in L^2(\mathbb{R}^n)$:
\[
\mathcal{F} u(\xi) := \hat{u}(\xi) :=  \int e^{-i x \xi} u(x) \,dx
:= \lim_{\sigma \to 0_+} \int e^{-i x \xi - \sigma \langle x \rangle} u(x) \,dx.
\]
Then the Fourier transform is an isomorphism on $L^2$ and the inverse Fourier
transform of $u \in L^2(\mathbb{R}^n)$ is given by the following formula
\[
\mathcal{F}^{-1} u(x) :=  (2\pi)^{-n} \int e^{i x \xi} u(\xi) \,d\xi
 := \lim_{\sigma \to 0_+} (2\pi)^{-n} \int e^{i x \xi
- \sigma \langle \xi \rangle} u(\xi) \,d\xi.
\]
More detailed pieces of information of the Fourier transform on
$\mathcal{G}_{2,2}$ can be found in \cite{Anton:99}.
As already mentioned above we will focus on generalized pseudodifferential
operators having the following phase-amplitude representation.

\begin{definition}\label{defn:psi-PsiDO} \rm
Let $(a_\varepsilon)_\varepsilon \in \MSym[m,k]{\nu,l}(\mathbb{R}^n
\times \mathbb{R}^{n})$. We define the corresponding linear operator
 $A_\psi : \mathcal{G}_{2,2} \to \mathcal{G}_{2,2}$ such that on the
level of representatives we have
\begin{equation*}
(u_\varepsilon)_\varepsilon \mapsto (A_{\psi,\varepsilon}(x,D_x)
u_\varepsilon)_\varepsilon \quad \forall (u_\varepsilon)_\varepsilon
\in \mathcal{M}_{H^{\infty}}
\end{equation*}
and
\begin{equation}\label{psiPsiDO}
\begin{split}
A_{\psi,\varepsilon}(x,D_x) u_\varepsilon(x)
& := (2\pi \varepsilon)^{-n} \int e^{i (x-y) \xi/
\varepsilon} a_\varepsilon(x,\xi) u_\varepsilon (y) \,dy \,d \xi \\
& \phantom{:}= \varepsilon^{-n} \mathcal{F}^{-1}_{\xi \to x/\varepsilon}
\Bigl( a_\varepsilon(x,\xi) \hat{u}_\varepsilon (\xi/\varepsilon) \Bigr)
\end{split}
\end{equation}
where the above integral is interpreted as an oscillatory integral.
We call $A_\psi$ the $\psi$-pseudodifferential operator with generalized
symbol $(a_\varepsilon)_\varepsilon = (a_\varepsilon(x,\xi))_\varepsilon$.
 Also we will often write $A_\psi \in \operatorname{OP}_\psi \MSym[m,k]{\nu,l}$
when the generalized symbol of $A_\psi$ belongs to the class $\MSym[m,k]{\nu,l}$.
\end{definition}

\begin{remark} \rm
 (a) Note that the map in the above definition preserves moderateness and
negligibility, respectively, so that  $A_\psi$ is well-defined on equivalence
classes and continuous
(cf. \cite[Section 1.2]{Hoermann:04} and \cite[Theorem 2.7]{Kumano-go:81}).
Further, we remark that $(A_{\psi,\varepsilon}(x,D_x)u_\varepsilon)_\varepsilon
= (A_{\varepsilon}(x, \varepsilon D_x)u_\varepsilon)_\varepsilon$ since formally
we can always perform the rescaling. The problem of rescaling will be discussed
in the next subsection.

(b) Note that Definition~\ref{defn:psi-PsiDO} agrees with the notion of the
semiclassical standard quantization of a generalized symbol
$(a_\varepsilon)_\varepsilon \in \MSym[m,k]{\nu,l} (\mathbb{R}^n \times \mathbb{R}^n)$
 and can simply written as in \eqref{psiPsiDO} using scaled Fourier transforms.
 Another commonly used quantization is the Weyl quantization $a^W(x,\varepsilon D)$
 which has the nice property that real symbols correspond to self-adjoint
 Weyl quantizations. Since we will exclusively deal with the standard quantization
of generalized symbols we decided to call them $\psi$-pseudodifferential operators.
\end{remark}

To prepare the factorization theorem of Section~\ref{sec:Fact} we will have
to consider products of $\psi$-pseudodifferential operators. We therefore start
with some general observations concerning the notion of asymptotic expansion
of a generalized symbol in $\MSym[m,k]{\nu,l}$.

\subsection{Asymptotic expansion of the first kind}

First we present the asymptotic expansion of the first kind which is inspired
by the results given in \cite[Section 2.5]{Garetto:08} but also uses a relevant
technical aspect from the semiclassical approach. Since this first notion
of an asymptotic expansion turns out to have no suitable invariant character
 under rescaling we will further introduce the asymptotic expansion of second
kind which behaves slightly different when performing the rescaling.
We should emphasize here that this invariance property of the asymptotic
expansion of the second kind will be essential from Section~\ref{sec:WFInfty}
onwards, where we discuss microlocal regularity at infinite points.

The definition of the asymptotic expansion of the first kind is now the following:

\begin{definition}\label{defn:AE1} \rm
For $j \in \mathbb{N}$ let $\{m_j\}_j$ be a strictly decreasing sequence of real
numbers with $m_j \searrow -\infty$ as $j \to \infty$, $m_0 = m$ and
$\{l_j\}_j$ be a sequence of the form $l_j = \sigma j + l$ for some fixed
 $\sigma,l \in \mathbb{R}$ and $\sigma \geq 0$. Further let
$\{ (a_{j,{\varepsilon}})_{\varepsilon} \}_{j}$ be a sequence with
 $( a_{j,{\varepsilon}} )_{\varepsilon} \in \MSym[m_j,k]{\nu,l_j}$ so that
the following uniform growth type condition is satisfied:
\begin{equation}\label{UniformCond}
\begin{gathered}
\exists \eta \in (0,1] \ \forall j \in \mathbb{N} \ \forall
\alpha, \beta \in  \mathbb{N}^n \ \exists C >0 \ \text{for which}\\
q^{(m_j)}_{\alpha,\beta} (a_{j,\varepsilon})
\leq C \varepsilon^{k} \omega_\varepsilon^{\nu |\beta|+l_j} \quad
 \forall \varepsilon \in (0,\eta].
 \end{gathered}
\end{equation}
We say that $\sum_{j=0}^{\infty} (\varepsilon^{j} a_{j,\varepsilon})_\varepsilon$
is the asymptotic expansion of the first kind for
$(a_\varepsilon)_\varepsilon \in \MSym[m,k]{\nu,l}$, denoted by
$(a_\varepsilon)_\varepsilon \sim \sum_{j} (\varepsilon^j
a_{j,\varepsilon})_\varepsilon$, if and only if
\begin{equation*}
(a_\varepsilon - \sum_{j=0}^{N-1} \varepsilon^j a_{j,\varepsilon})_\varepsilon
\in \MSym[m_N, N+k-1]{\nu,l} \quad \forall N \geq 1.
\end{equation*}
Moreover $(a_{0,\varepsilon})_\varepsilon$ is said to be the principal
symbol of $(a_\varepsilon)_\varepsilon$ if $a_{0,\varepsilon}$ is not
 identically vanishing for any fixed $\varepsilon \in (0,\eta]$.
\end{definition}

We are now in a position to state our first result.

\begin{lemma}\label{lem:AE1}
Let $\{ l_j\}_j$, $\{ m_j \}_j$ and $\{(a_{j,\varepsilon})_\varepsilon\}_j$
be as in Definition~\ref{defn:AE1}. Then there exists a generalized symbol
 $(a_\varepsilon)_\varepsilon \in \MSym[m,k]{\nu,l}$ such that
$(a_\varepsilon)_\varepsilon \sim \sum_{j} (\varepsilon^j
a_{j,\varepsilon})_\varepsilon$ in $\MSym[m,k]{\nu,l}$.
Moreover the asymptotic expansion of the first kind determines
$(a_\varepsilon)_\varepsilon$ uniquely modulo $\mathcal{N}_{\mbox{$\mathcal{S}$}^{-\infty}}$.
\end{lemma}

Before presenting the proof of Lemma~\ref{lem:AE1} we need a technical auxiliary
 result.

\begin{lemma}\label{lem:ChooseMu}
Let $\{ (a_{j,\varepsilon})_\varepsilon \}_j$ be as in Definition~\ref{defn:AE1}
 and $\chi$ a smooth function such that $\chi \equiv 0$ on $[0,1]$,
$\chi \equiv 1$ on $[2,\infty)$ and $0 \leq \chi \leq 1$. Then there exists a
zero sequence $\{\mu_j\}_j$ such that for some fixed $\eta \in (0,1]$ we
have for every $j \in \mathbb{N}$ and $\alpha, \beta \in \mathbb{N}^n$ with
 $|\alpha + \beta|\leq j$:
\begin{equation*}
|\chi(\mu_j (\varepsilon \omega_\varepsilon^{\sigma})^{-1})
q^{(m_j)}_{\alpha,\beta}(a_{j,\varepsilon})| \leq 2^{-j-1} \varepsilon^{k}
\omega_\varepsilon^{\nu |\beta|+l_j} (\varepsilon \omega_\varepsilon^{\sigma})^{-1}
\quad \forall \varepsilon \in(0,\eta],
\end{equation*}
with the same $\sigma$ as in Definition~\ref{defn:AE1}.
\end{lemma}

\begin{proof} Using the uniformity condition \eqref{UniformCond} of the sequence
$\{ (a_{j,{\varepsilon}})_{\varepsilon} \}_{j}$ we obtain that for some
$\eta \in (0,1]$ and $\forall j \in \mathbb{N}$, $\forall
\alpha,\beta \in \mathbb{N}^n$, $\exists C^{(1)}_{j,\alpha,\beta} >0$ and
 for all $\varepsilon \in (0,\eta]$ we have
\begin{align*}
\varepsilon^{-k} \omega_\varepsilon^{- \nu |\beta|- l_j}|\chi
(\mu_j (\varepsilon \omega_\varepsilon^{\sigma})^{-1}) q^{(m_j)}_{\alpha,\beta}
(a_{j,\varepsilon}) |
& \leq C^{(1)}_{j,\alpha,\beta}  \chi(\mu_j
  (\varepsilon \omega_\varepsilon^{\sigma})^{-1}) \\
& = C^{(1)}_{j,\alpha,\beta}  \frac{\mu_j}{\varepsilon \omega_\varepsilon^{\sigma}}
 \chi(\mu_j (\varepsilon \omega_\varepsilon^{\sigma})^{-1})
 \frac{\varepsilon \omega_\varepsilon^{\sigma}}{\mu_j} \\
&\leq C^{(1)}_{j,\alpha,\beta}  \frac{\mu_j}{\varepsilon \omega_\varepsilon^{\sigma}}
\end{align*}
since $\chi(\mu_j (\varepsilon \omega_\varepsilon^{\sigma})^{-1}) = 0$ for
$\mu_j \leq \varepsilon \omega_\varepsilon^{\sigma}$ and $0 \leq \chi \leq 1$.
 We now choose the sequence $\{\mu_j\}_j$ in the following way:
for all $j \in \mathbb{N}$ and all $\alpha, \beta \in \mathbb{N}^n$  with
$|\alpha + \beta| \leq j$  we have
\begin{equation}\label{ChooseMu}
 C^{(1)}_{j,\alpha,\beta} \mu_j \leq 2^{-j-1}
\end{equation}

and the proof is complete.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:AE1}]
In the following proof we basically combine techniques from the theory
of non-linear generalized functions, \cite[Theorem 2.2]{Garetto:08}, and Borel's
Theorem from the semiclassical approach as in \cite[Theorem 4.11]{EvZw} or
alternatively \cite[Proposition 2.3.2]{Martinez:02}. To avoid overload calculations,
we may assume without loss of generality that $k = 0$ in Lemma~\ref{lem:AE1}
and therefore also in the uniformity condition \eqref{UniformCond}.

We let $\phi \in \mathcal C^{\infty}(\mathbb{R}^n)$, $0 \leq \phi \leq 1$
such that $\phi(\xi) = 0$ for $|\xi| \leq 1$ and $\phi(\xi) = 1$ for
$|\xi| \geq 2$. Further let $\chi$ and $\{\mu_j \}_j$ be as in
Lemma~\ref{lem:ChooseMu}. We introduce functions
\begin{align*}
b^{(1)}_{j,\varepsilon}(x,\xi)
 &:= \chi(\mu_j(\varepsilon \omega_\varepsilon^{\sigma})^{-1})a_{j,\varepsilon}(x,\xi)\\
b^{(2)}_{j,\varepsilon}(x,\xi)
 &:= \left(1-\chi(\mu_j(\varepsilon \omega_\varepsilon^{\sigma})^{-1})\right)\phi(\lambda_j \xi) a_{j,\varepsilon}(x,\xi)
\end{align*}
where $\{ \lambda_j \}_j$ is a strictly decreasing zero sequence such that
 for all $j: \lambda_j \leq 1$ and will be specified later. For fixed
$\varepsilon \in (0,1]$ we also define
\begin{equation*}
a_{\varepsilon}(x,\xi):= \sum_{j \geq 0} \varepsilon^j b_{j,\varepsilon}^{(1)}(x,\xi)
 +\sum_{j \geq 0} \varepsilon^j b_{j,\varepsilon}^{(2)}(x,\xi).
\end{equation*}
By construction the first sum consists of at most finitely many nonzero
terms (depending on $\varepsilon \in (0,1]$ fixed), since $\mu_j \to 0$ as
$j \to \infty$. Moreover the second sum is locally finite and we conclude
that $(a_\varepsilon)_\varepsilon \in \mathcal{E} (\mathbb{R}^n
 \times \mathbb{R}^n)^{(0,1]}$.

\textbf{Step 1:} In the first part of the proof we show that
$(a_\varepsilon)_\varepsilon \in \MSym[m,0]{\nu,l}$.
Since $(a_{j,\varepsilon})_\varepsilon \in \MSym[m_j,0]{\nu,l_j}$
satisfy the uniform growth type condition with $k=0$ we obtain that  there
exists $\eta \in (0,1]$ such that for all $j \in \mathbb{N}$, and all
$\alpha, \beta \in \mathbb{N}^n$ we have
\begin{equation}\label{Est1}
|\partial^{\alpha}_\xi \partial^{\beta}_x b^{(1)}_{j,\varepsilon}(x,\xi)|
\leq C^{(1)}_{j,\alpha,\beta}  \omega_\varepsilon^{\nu |\beta| + l_j}
\langle \xi \rangle^{m_j-|\alpha|} \quad \forall \varepsilon \in(0,\eta]
\end{equation}
and the constant $C^{(1)}_{j,\alpha,\beta} > 0$ is the same as in the proof
of Lemma~\ref{lem:ChooseMu}.

Concerning $b^{(2)}_{j,\varepsilon}$ we observe that
$\operatorname{supp} (\partial^{\alpha} \phi)(\lambda_j\xi)
\subset \{ \xi: \lambda_j^{-1} \leq |\xi| \leq 2 \lambda_j^{-1} \}$
 for every $|\alpha| \geq 1$. Accordingly, if $|\alpha| \geq 1$,
 we can assume $\lambda_j \leq 2/ |\xi| \leq 4 \langle \xi \rangle^{-1}$
 on $\operatorname{supp} (\partial^{\alpha} \phi)(\lambda_j\xi)$ and
 $b^{(2)}_{j,\varepsilon}$ can be estimated as follows: $\exists \eta \in (0,1]$
independent of $\alpha, \beta$ and $j$ such that
\begin{equation}\label{Est2}
\begin{split}
|\partial^{\alpha}_\xi \partial^{\beta}_x b^{(2)}_{j,\varepsilon}(x,\xi)|
 & \leq \langle \xi \rangle^{m_j - |\alpha|} \sum_{\gamma \leq \alpha}
 c(\chi,\phi,\gamma) 4^{|\alpha - \gamma|} q^{(m_j)}_{\gamma,\beta}(a_{j,\varepsilon})
\\
& \leq C^{(2)}_{j,\alpha,\beta} \omega_\varepsilon^{\nu |\beta|+l_j}
\langle \xi \rangle^{m_j - |\alpha|}
\end{split}
\end{equation}
for all $\varepsilon \in (0,\eta]$. At this point we choose the sequence
$\{ \lambda_j \}_j$ strictly decreasing and so that
\begin{equation}\label{ChooseLambda}
\forall j \in \mathbb{N} \ \forall \alpha, \beta \in \mathbb{N}^n \text{ with } |\alpha+ \beta|\leq j \text{ we have:} \quad C^{(2)}_{j,\alpha,\beta} \lambda_j \leq 2^{-j-1}.
\end{equation}
To show that $(a_\varepsilon)_\varepsilon \in \MSym[m,0]{\nu,l}$ we note that
\begin{equation}\label{Decompose1}
\forall \alpha, \beta \in \mathbb{N}^n \ \exists j_0 \in \mathbb{N}: \; |\alpha + \beta| \leq j_0, \ m_{j_0}+ 1\leq m 
\end{equation}
and decompose $(a_\varepsilon)_\varepsilon$ in an appropriate way, that now is
\begin{equation}\label{Decompose2}
a_\varepsilon(x,\xi)= \sum_{j\leq j_0 -1} \varepsilon^j b_{j,\varepsilon}(x,\xi)
 + \sum_{j \geq j_0} \varepsilon^j b_{j,\varepsilon}(x,\xi) =: f_\varepsilon(x,\xi)
+ s_\varepsilon(x,\xi)
\end{equation}
where we have set $b_{j,\varepsilon} (x,\xi):=b^{(1)}_{j,\varepsilon}(x,\xi)
+ b^{(2)}_{j,\varepsilon}(x,\xi)$. Then $\exists \eta \in (0,1]$ such that
for all $\alpha, \beta \in \mathbb{N}^n$
\begin{align*}
|\partial^{\alpha}_\xi \partial^{\beta}_x f_{\varepsilon}(x,\xi)|
 & =  |\sum_{j \leq j_0 - 1} \varepsilon^j \partial^{\alpha}_\xi \partial^{\beta}_x
 \bigl\{ b^{(1)}_{j,\varepsilon}(x,\xi) + b^{(2)}_{j,\varepsilon}(x,\xi) \bigr\} |
 \\
&\leq \omega_\varepsilon^{\nu |\beta| +l} \langle \xi \rangle^{m-|\alpha|}
\sum_{j \leq j_0 - 1} (\varepsilon \omega_\varepsilon^{\sigma})^j
\bigl(C^{(1)}_{j,\alpha,\beta} + C^{(2)}_{j,\alpha,\beta}\bigr) \\
&\leq  C_{j_0} \omega_\varepsilon^{\nu |\beta|+l} \langle \xi \rangle^{m-|\alpha|}
\end{align*}
for all $\varepsilon \in (0,\eta]$ by \eqref{Est1} and \eqref{Est2}.
 We next estimate the remainder $s_\varepsilon$. For this purpose we
 apply \eqref{ChooseMu} from Lemma~\ref{lem:ChooseMu} and use
 \eqref{ChooseLambda} to obtain
\begin{align*}
\omega_\varepsilon^{- \nu |\beta|-l} |\partial^{\alpha}_\xi
 \partial^{\beta}_x s_{\varepsilon}(x,\xi)|
& \leq \sum_{j \geq j_0} (\varepsilon \omega_\varepsilon^{\sigma})^{j}
 \langle \xi \rangle^{m_j - |\alpha|} 2^{-j-1}
 \left\{ \mu_j^{-1} + \lambda_j^{-1} \right\}  \\
& \leq (\varepsilon \omega_\varepsilon^{\sigma})^{j_0 -1}
 \langle \xi \rangle^{m_{j_0}+1-|\alpha|} \sum_{j \geq j_0}  2^{-j}
 (\varepsilon \omega_\varepsilon^{\sigma})^{j-j_0} \\
& \leq (\varepsilon \omega_\varepsilon^{\sigma})^{j_0 -1}
 \langle \xi \rangle^{m_{j_0}+1 - |\alpha|} \leq \langle  \xi
 \rangle^{m - |\alpha|} \ \quad  \forall \varepsilon \in (0,\eta]
\end{align*}
for some $\eta \in (0,1]$ independent of the order of differentiation.
Note that in the second inequality we used that
$\mu_j^{-1} \leq (\varepsilon \omega_\varepsilon^{\sigma})^{-1}$ on
$\operatorname{supp} (b_{j,\varepsilon}^{(1)})$ and
$\lambda_j^{-1} \leq \langle \xi \rangle$ on
$\operatorname{supp} (b_{j,\varepsilon}^{(2)})$. And so,
$(a_\varepsilon)_\varepsilon \in \MSym[m,0]{\nu,l}$ as required.

\textbf{Step 2:} We complete the proof by showing that for every $N \geq 1$ we have
\begin{equation}\label{Step2}
\quad (a_\varepsilon -  \sum_{j \leq N-1} \varepsilon^j
a_{j,\varepsilon} )_\varepsilon \in \MSym[m_N,N]{\nu,l_N}(\mathbb{R}^n
\times \mathbb{R}^n).
\end{equation}
Therefore, we let $N \geq 1$ be a fixed natural number and write
\begin{align*}
a_\varepsilon(x,\xi)
-  \sum_{j \leq N-1} \varepsilon^j a_{j,\varepsilon}(x,\xi)
&= \sum_{j \leq N-1} \varepsilon^j \bigl( 1-\chi(\mu_j
 (\varepsilon \omega_\varepsilon^{\sigma})^{-1}) \bigr)
  \bigl( \phi (\lambda_j \xi) -1 \bigr) a_{j,\varepsilon}(x,\xi)  \\
&\quad + \sum_{j \geq N} \varepsilon^j b_{j,\varepsilon}(x,\xi) =:
 g_\varepsilon(x,\xi)+ t_\varepsilon(x,\xi).
\end{align*}
We first calculate the contribution of $g_\varepsilon$. For this note
that $|\xi| \leq 2 \lambda_j^{-1}$ on the support of $(\phi (\lambda_j \xi) -1)$
and we obtain that $\exists \eta \in (0,1]$ such that $\forall j \leq N-1$ we have
\begin{align*}
|\partial^{\alpha}_\xi \partial^{\beta}_x
\bigl( \phi (\lambda_j \xi) -1 \bigr) & a_{j,\varepsilon} (x,\xi) | \\
& \leq \sum_{\gamma \leq \alpha}  c(\phi,\gamma) \lambda_j^{|\gamma|}
q^{(m_j)}_{\alpha-\gamma,\beta}(a_{j,\varepsilon}) \langle \xi \rangle^{m_j
- |\alpha-\gamma|} \\
& \leq \langle \xi \rangle^{m_N - |\alpha|} \sum_{\gamma \leq \alpha}
c(\phi,\gamma) \lambda_j^{|\gamma|} \langle 2 \lambda_{N-1}^{-1}
 \rangle^{m_j-m_N+|\gamma|} q^{(m_j)}_{\alpha-\gamma,\beta}(a_{j,\varepsilon}) \\
& \leq C_{N,\alpha,\beta} \omega_\varepsilon^{\nu |\beta| +l_j} \langle \xi
\rangle^{m_N - |\alpha|} \ \quad  \forall \varepsilon \in (0,\eta].
\end{align*}
Now taking into account that
$1-\chi(\mu_j (\varepsilon \omega_\varepsilon^{\sigma})^{-1}) \neq 0$ only
if $\mu_j \leq 2 \varepsilon \omega_\varepsilon^{\sigma}$ we compute
\begin{align*}
\omega_\varepsilon^{- \nu |\beta|-l} & |\partial^{\alpha}_\xi \partial^{\beta}_x
 g_\varepsilon(x,\xi) | \\
& \leq C_{N,\alpha,\beta} \langle \xi \rangle^{m_N - |\alpha|}
 \sum_{j \leq N-1} \hspace{1pt} (\varepsilon \omega_\varepsilon^{\sigma})^{j}
 \left( 1-\chi(\mu_j (\varepsilon \omega_\varepsilon^{\sigma})^{-1}) \right) \\
& = C_{N,\alpha,\beta} (\varepsilon \omega_\varepsilon^{\sigma})^{N}
 \langle \xi \rangle^{m_N - |\alpha|}   \sum_{j \leq N-1} \mu_j^{j-N}
\Bigl( \frac{\mu_j}{\varepsilon \omega_\varepsilon^{\sigma}} \Bigr)^{ N - j}
 \left( 1-\chi(\mu_j (\varepsilon \omega_\varepsilon^{\sigma})^{-1}) \right) \\
&\leq  \widetilde{C}_{N,\alpha,\beta} (\varepsilon \omega_\varepsilon^{\sigma})^{N}
\langle \xi \rangle^{m_N - |\alpha|} \ \quad  \forall \varepsilon \in (0,\eta].
\end{align*}
We proceed with the estimation of $t_\varepsilon$ in which we again use a
decomposition as in \eqref{Decompose1} and \eqref{Decompose2}.
Therefore we let $N \in \mathbb{N}$, $N \geq 1$ be fixed. Then
%
\[
\forall \alpha, \beta \in \mathbb{N}^n \ \exists j_0 \in \mathbb{N}: \ |\alpha + \beta| \leq j_0, \ m_{j_0}+ 1\leq m_N \text{ and } j_0-1 \geq N;
\]
we write
\begin{equation}\label{Decompose:t}
t_{\varepsilon}(x,\xi) = \sum_{j = N}^{j_0-1} \varepsilon^j b_{j,\varepsilon}(x,\xi)
 + s_\varepsilon(x,\xi)
\end{equation}
with $s_\varepsilon$ as in \eqref{Decompose2}. As already shown in the estimate
for $s_\varepsilon$ we also have
\begin{equation*}
|\partial^{\alpha}_\xi \partial^{\beta}_x s_{\varepsilon}(x,\xi)|
\leq \omega_\varepsilon^{\nu |\beta|+l} (\varepsilon \omega_\varepsilon^{\sigma})^{j_0
-1} \langle \xi \rangle^{m_{j_0} +1- |\alpha|} .
\end{equation*}
Furthermore we obtain for the first term on the right hand side of
\eqref{Decompose:t}
\begin{align*}
|\partial^{\alpha}_\xi \partial^{\beta}_x \sum_{j = N}^{j_0-1}
\varepsilon^j b_{j,\varepsilon}(x,\xi)|
& \leq \omega_\varepsilon^{\nu |\beta|+l} \langle \xi \rangle^{m_N-|\alpha|}
 \sum_{j=N}^{j_0 - 1} (\varepsilon \omega_\varepsilon^{\sigma})^{j}
  \bigl(C^{(1)}_{j,\alpha,\beta} + C^{(2)}_{j,\alpha,\beta}\bigr) \\
&\leq  C_{j_0} (\varepsilon \omega_\varepsilon^{\sigma})^{N}
 \omega_\varepsilon^{\nu |\beta|+l} \langle \xi \rangle^{m_N-|\alpha|}.
\end{align*}
Putting this together we get \eqref{Step2} which completes the proof.
\end{proof}

\begin{remark}\label{rem:AE1} \rm
Before we proceed we make some observations concerning the construction
of the generalized symbol $(a_\varepsilon)_\varepsilon$ in Lemma~\ref{lem:AE1}
which was given by the expression
\begin{equation*}
a_{\varepsilon}(x,\xi) = \sum_{j \geq 0} \varepsilon^j \chi
\Bigl(\frac{\mu_j}{\varepsilon \omega_\varepsilon^{\sigma}}\Bigr)
a_{j,\varepsilon}(x,\xi) + \sum_{j \geq 0} \varepsilon^j
\Bigl\{1-\chi\Bigl(\frac{\mu_j}{\varepsilon \omega_\varepsilon^{\sigma}}
 \Bigr)\Bigr\}\phi(\lambda_j \xi) a_{j,\varepsilon}(x,\xi).
\end{equation*}
Here the first sum is inspired from the semiclassical approach whereas the
second sum is also used in the usual Colombeau framework. Then it is may
worth to mention the following facts:

(a) To construct the square root of the homogeneous operator $L_\psi$ we also
 need to discuss polyhomogeneous symbols. Then the second sum in the definition
of $a_\varepsilon$ also makes sense for homogeneous symbols $a_{j,\varepsilon}$,
but the first sum would induce a finite number of singularities (at $\xi = 0$)
for fixed $\varepsilon \in (0,1]$. To overcome this problem we will introduce
generalized symbols that are smoothed off at the zero section so that we can
require a symbol to have an asymptotic expansion in homogeneous terms if we
allow some additional error terms in the asymptotic expansion. For the precise
description of these so-called polyhomogeneous generalized symbols we refer
to Definition~\ref{defn:PhgSymb}.

(b) Another important property concerns the effect of rescaling. This will be
used in Section~\ref{sec:WFInfty} where we test microlocal regularity of
generalized functions using $L^2$-boundedness results of zeroth-order
$\psi$-pseudodifferential operators. To exemplify this let
$\{(a_{j,\varepsilon})_\varepsilon\}_j$ be a sequence with
$(a_{j,\varepsilon})_\varepsilon \in \MSym[-j,k]{\nu,l_j}$ and $l_j$
as in Definition~\ref{defn:AE1}, $j \geq 0$. By Lemma~\ref{lem:AE1}
there exists $(a_\varepsilon)_\varepsilon \in \MSym[0,k]{\nu,l}$ such that
for all $N \geq 1$ we have
\begin{equation}\label{AE:Order0}
(r_\varepsilon)_\varepsilon := (a_\varepsilon - \sum_{j \leq N-1}
 \varepsilon^j a_{j,\varepsilon} )_\varepsilon \in \MSym[-N,N+k-1]{\nu,l}.
\end{equation}
Now using the rescaling
$(x,\xi) \mapsto (r_\varepsilon(x,\varepsilon \xi))_\varepsilon$ we obtain
for every $N \geq 1$
\begin{equation*}
(x,\xi) \mapsto (r_\varepsilon(x,\varepsilon \xi) )_\varepsilon
\in \MSym[-N,k-1]{\nu,l}
\end{equation*}
concluding that the asymptotic expansion of the first kind keeps the order
of the error operator invariant under rescaling but not its growth type.
According to the $L^2$-boundedness it seems suitable to have an asymptotic
expansion which preserves the growth type of the error operator in
\eqref{AE:Order0} under rescaling rather than the order of the same.

Therefore, note that $(r_\varepsilon)_\varepsilon$ in \eqref{AE:Order0}
can also be considered as a symbol of class $\mathcal{N}_{\mbox{$\mathcal{S}$}^0}$.
Then the rescaling does not influence the negligibility of the symbol
 $(r_\varepsilon)_\varepsilon$ since
\begin{equation*}
(x,\xi) \mapsto (r_\varepsilon(x,\varepsilon \xi) )_\varepsilon
\in \mathcal{N}_{\mbox{$\mathcal{S}$}^0}.
\end{equation*}
Using this we now introduce the asymptotic expansion of the second kind
to achieve an invariant growth type under rescaling at least when studying
 zeroth-order operators. In detail we will study asymptotic expansions of
the form $(\sum_{j\geq0} \varepsilon^j a_{j,\varepsilon})_\varepsilon$ which
are similar those in Definition~\ref{defn:AE1} but where the sequence
 $\{ (a_{j,\varepsilon})_\varepsilon \}_j$ is uniformly bounded in the
generalized symbol class of order $m=m_0$.
\end{remark}

\subsection{Asymptotic expansion of the second kind}\label{subsec:AE2}

To eliminate the rescaling problem arising in Remark~\ref{rem:AE1}(b) we now present the
asymptotic expansion of the second kind.

\begin{definition}\label{defn:AE2} \rm
Let $m, k \in \mathbb{R}$ and $\{l_j\}_j$ be a sequence of real numbers of
the form $l_j = \sigma j + l$ for some fixed $\sigma,l \in \mathbb{R}$ and
$\sigma \geq 0$, $j \in \mathbb{N}$. Further we let
$\{ (a_{j,{\varepsilon}})_{\varepsilon} \}_{j}$ be a sequence with
$( a_{j,{\varepsilon}} )_{\varepsilon} \in \MSym[m,k]{\nu,l_j}$
satisfying the uniformity condition:
\[
\begin{gathered}
\quad \exists \eta \in (0,1] \ \forall j \in \mathbb{N} \ \forall \alpha, \beta \in \mathbb{N}^n \ \exists C >0 \ \text{such that} \\
\quad  q^{(m)}_{\alpha,\beta} (a_{j,\varepsilon}) \leq C \varepsilon^k \omega_\varepsilon^{\nu |\beta| +l_j} \qquad \forall \varepsilon \in (0,\eta].
\end{gathered}
\]
We say that the $\sum_{j=0}^{\infty} (\varepsilon^j a_{j,\varepsilon})_\varepsilon$
is the asymptotic expansion of the second kind for the symbol
$(a_\varepsilon)_\varepsilon \in \MSym[m,k]{\nu,l}$, denoted by
 $(a_\varepsilon)_\varepsilon \approx \sum_{j}
(\varepsilon^j a_{j,\varepsilon})_\varepsilon$, if
\begin{equation*}
(a_\varepsilon - \sum_{j=0}^{N-1} \varepsilon^j a_{j,\varepsilon})_\varepsilon
\in \MSym[m,N]{\nu,l_N} \quad \forall N \geq 1.
\end{equation*}
Again we call $(a_{0,\varepsilon})_\varepsilon$ the principal symbol of
$(a_\varepsilon)_\varepsilon$ if $a_{0,\varepsilon}$ is not identically
zero for every fixed $\varepsilon >0$ sufficiently small.
\end{definition}

Comparing  Definition~\ref{defn:AE1} and \ref{defn:AE2} we observe that
any generalized symbol $(a_\varepsilon)_\varepsilon$ from Lemma~\ref{lem:AE1}
also admits an asymptotic expansion of the second kind.

Furthermore we note that our notions of an asymptotic expansion are different
from the one used in the semiclassical approach. A crucial difference is
that in the semiclassical setting one uses the H\"{o}rmander symbol class
 $S^m_{0,0}$ as the underlying classical space and not $S^m := S^m_{1,0}$
as in our case, cf. \cite{Hoermander:66}. As above, the semiclassical
 asymptotic expansion provides an invariant character of zeroth-order operators
under rescaling. For more details on the semiclassical asymptotic expansion we
refer to \cite{DiSj:99,Martinez:02,GuSt:10}.

\subsection{Composition of \texorpdfstring{$\psi$}{psi}-pseudodifferential 
operators}

In this subsection we establish the composition law of two
 $\psi$-pseudodifferential operators utilizing the method of stationary phase
 to determine the asymptotic behavior in the occurring oscillatory integrals.
 Because of the specific construction of the $\psi$-pseudodifferential operators
it suffices to study a corollary of the stationary phase formula which can be
found in \cite[Corollary 2.6.3]{Martinez:02} and \cite[Example 2.2]{GiSj:94}.

In detail, this simplified formula says that given a function
$a \in \mathcal{C}^{\infty}_{\rm c} (\mathbb{R}^{2n})$ then for every
$N \geq 1$ one has
\begin{equation*}
\left( \frac{\lambda}{2 \pi} \right)^n \int e^{-i \lambda xy } a(x,y) \, d x \, d y
= \sum_{|\alpha| \leq N-1} \frac{1}{ \alpha! \lambda^{|\alpha|}}
\left(  D_x^{\alpha} \partial_y^{\alpha} a \right) (0,0) + S_N (a;\lambda)
\end{equation*}
where the parameter $\lambda$ is considered in the limit $\lambda \to \infty$.
 Moreover the remainder term $S_N (a;\lambda)$ can be estimated as follows
\begin{equation*}
|S_{N}(a; \lambda)| \leq \frac{C}{N! \lambda^N} \sum_{|\alpha+\beta|
 \leq 2n+1} \| \partial_x^{\alpha} \partial_y^{\beta}
( \partial_x \cdot \partial_y )^N a \|_{L^1(\mathbb{R}^{2n})} \quad \lambda \geq 1
\end{equation*}
where $\partial_x \cdot \partial_y = \sum_{j=1}^{n} \partial_{x_j} \partial_{y_j}$
and the constant $C > 0$ is independent of $\lambda$. In the application we will
 work with smooth functions $a$ which are not compactly supported but are well-behaved
at infinity and also depend on the parameter $\lambda$. For a more general
discussion on the method of stationary phase we refer the reader
to \cite[Section 2.6]{Martinez:02} or alternatively to \cite[Section 2]{GiSj:94}.

Using the above formula we can show the following result.

\begin{proposition}\label{prop:Product}
Let $A_\psi(x,D_x)$ and $B_\psi(x,D_x)$ be two $\psi$-pseudodifferential operators
with generalized symbols $(a_\varepsilon)_\varepsilon \in \MSym[m_1,k_1]{\nu,l_1}$
and $(b_\varepsilon)_\varepsilon \in \MSym[m_2,k_2]{\nu,l_2}$ respectively.
Then the product $A_\psi B_\psi$ is well-defined and maps $\mathcal{G}_{2,2}$
into itself. Moreover $A_\psi B_\psi$ is a $\psi$-pseudodifferential operator
with generalized symbol $(a_\varepsilon \# b_\varepsilon)_\varepsilon$ in
$\MSym[m_1 + m_2,k_1 + k_2]{\nu,l_1+l_2}$ and we have the following representation
\begin{equation}\label{AE:Product}
(a_{\varepsilon} \# b_{\varepsilon})_\varepsilon \sim \sum_{|\alpha| \ge 0}
\left( \frac{{\varepsilon}^{|\alpha|}}{\alpha!} D^{\alpha}_{\xi} a_{\varepsilon}(x,\xi) \partial^{\alpha}_{x} b_{\varepsilon}(x,\xi) \right)_{\varepsilon}.
\end{equation}
\end{proposition}

Note that in Proposition~\ref{prop:Product} the generalized symbol
$(a_\varepsilon \# b_\varepsilon)_\varepsilon$ is given by its asymptotic
expansion of the first kind. Hence it can also be reinterpreted as a symbol
having an asymptotic expansion of the second kind.

The following proof is an adaption of \cite[Theorem 2.6.5]{Martinez:02} and
\cite[Theorem 9.13]{EvZw}.

\begin{proof}[Proof of Proposition \ref{prop:Product}]
Without loss of generality we assume that $(a_\varepsilon)_\varepsilon$
and $(b_\varepsilon)_\varepsilon$ are of growth type $0$; i.e.,
 $k_1 = k_2 = 0$, as the proof for more general growth type assumptions requires
only slight changes in the argumentation.
Let $u \in \mathcal{G}_{2,2}(\mathbb{R}^{n})$ having $(u_\varepsilon)_\varepsilon$
as representative and let $\varepsilon \in (0,1]$ be fixed and arbitrary.
Then $A_{\psi} B_{\psi} u$ makes sense as an oscillatory integral and we write
\begin{align} \label{ABproduct}
A_{\psi,\varepsilon} B_{\psi,\varepsilon} u_\varepsilon(x)
& = \frac{1}{(2 \pi \varepsilon)^{n}} \lim_{\sigma\to 0_{+}} \int e^{i(x-y)
 \eta/\varepsilon - \sigma \langle y \rangle - \sigma \langle \eta \rangle}
 a_\varepsilon(x,\eta) (B_{\psi,\varepsilon} u_\varepsilon)(y) \, dy \, d \eta \nonumber  \\
& = \frac{1}{(2 \pi \varepsilon)^{n}}
\lim_{\sigma\to 0_{+},\,\tau \to 0_{+}}
\int e^{ix \xi/\varepsilon - \tau \langle \xi \rangle} c_{\sigma,\varepsilon}(x,\xi)
\hat{u}_\varepsilon(\xi/\varepsilon) \, d \xi
\end{align}
where we have set
\begin{equation*}
c_{\sigma,\varepsilon} (x,\xi) = (2 \pi \varepsilon)^{-n}
\int e^{i(x-y)(\eta - \xi) /\varepsilon - \sigma \langle y \rangle
- \sigma \langle \eta \rangle} a_\varepsilon(x,\eta) b_{\varepsilon}(y,\xi)
\, dy \, d \eta.
\end{equation*}
In what follows we show that $c_{\sigma,\varepsilon}(x,\xi)$ corresponds to a
generalized symbol in $\MSym[m_1+m_2,0]{\nu,l_1+l_2}$ uniformly for all
 $\sigma >0$. Here the relevant asymptotic behavior of $c_{\sigma,\varepsilon}$
as $\varepsilon \to 0$ is described by using the method of stationary phase.
Finally, by passing to the limit $\sigma \to 0_{+}$, we obtain by Lebesgue's
dominated convergence theorem that the limit of $(c_{\sigma,\varepsilon})_\varepsilon$
exists in $\MSym[m_1+m_2,0]{\nu,l_1+l_2}$ and admits a representation as stated
in \eqref{AE:Product}.

For that purpose we first split the integral representation of
$c_{\sigma,\varepsilon}(x,\xi)$ into three parts.
Therefore, let $\chi \in \mathcal{C}^{\infty}_{\rm c} (\mathbb{R})$ such that
 $\chi(s) = 1$ for $|s| \leq 1/4$ and $\chi(s) = 0$ for $|s| \geq 1/2$ and set
$\chi_1 (x,y) = \chi(|x-y|)$ for $x,y \in \mathbb{R}^n$. For $|\xi| \geq 1$ we write
\begin{equation} \label{Splitting}
c_{\sigma,\varepsilon} (x,\xi) = d_{\sigma,\varepsilon} (x,\xi) + e_{\sigma,\varepsilon} (x,\xi) + f_{\sigma,\varepsilon} (x,\xi),
\end{equation}
where $d_{\sigma,\varepsilon}, e_{\sigma,\varepsilon}$ and
$f_{\sigma,\varepsilon}$ are defined by the additional factors
$1-\chi(|\xi-\eta| / |\xi|)$, $\chi(|\xi-\eta| / |\xi|)(1-\chi_1(x,y))$ and
$\chi(|\xi-\eta| / |\xi|)\chi_1(x,y)$ in the integrand of $c_{\sigma,\varepsilon}$.
Also we note that $f_{\sigma,\varepsilon}$ represents the part of
 $c_{\sigma,\varepsilon}$ that will determine its asymptotic behavior as we meet
the critical points of the phase.

In the following we give the proof under the assumption that
$|\xi| \geq 1$. In the case that $|\xi| \leq 1$ a likewise decomposition as
in \eqref{Splitting} will lead to the desired result. Again, this is similar as
 in the proof of \cite[Theorem 2.6.5]{Martinez:02}.

As a first step we check that the integrand of $d_{\sigma,\varepsilon}$ is in
$L^1(\mathbb{R}^n_y \times \mathbb{R}^n_\eta)$ uniformly in $\sigma > 0$.
Therefore, we introduce the operator
\begin{equation*}
L_\varepsilon := \Big( 1+ \frac{|\xi - \eta|^2}{\varepsilon^2}
+ \frac{|x-y|^2}{\varepsilon^2} \Big)^{-1}
\Big( 1 + \frac{\xi - \eta}{\varepsilon}D_y + \frac{x-y}{\varepsilon} D_\eta \Big)
\end{equation*}
satisfying the  equality
\begin{equation*}
L_\varepsilon e^{i(x-y)(\eta - \xi) /\varepsilon} = e^{i(x-y)(\eta - \xi) /\varepsilon}.
\end{equation*}
Using integration by parts we obtain
\begin{align*}
&(2 \pi \varepsilon)^{n} d_{\sigma,\varepsilon} (x,\xi) \\
&= \int e^{i(x-y)(\eta - \xi) /\varepsilon} ( {}^t \hspace{-1pt} L_\varepsilon )^k
\Bigl\{ (1-\chi(|\xi-\eta|/|\xi|)) e^{- \sigma \langle y \rangle
- \sigma \langle \eta \rangle} a_\varepsilon(x,\eta) b_{\varepsilon}(y,\xi) \Bigr\}
\, dy \, d \eta.
\end{align*}
Further, for every $k \in \mathbb{N}$ with $k > n+1/2 $ we obtain
\begin{align*}
(2 \pi \varepsilon)^{n} & d_{\sigma,\varepsilon}(x,\xi) \\
&= \int_{|\xi - \eta| \geq |\xi|/4} \mathcal{O}
\Big( \frac{\omega_\varepsilon^{k \nu+l_1+l_2} \langle
 \eta \rangle^{m_1} \langle \xi \rangle^{m_2}}{(1 + |\xi - \eta|/\varepsilon
 + |x-y|/\varepsilon)^k}\Big) \, dy \, d \eta \\
& = \int_{|\xi - \eta| \geq |\xi|/4} \mathcal{O}
\Big( \varepsilon^n \omega_\varepsilon^{k \nu+l_1+l_2} \langle \eta \rangle^{m_1}
 \langle \xi \rangle^{m_2}
 \Bigl\{1 + \frac{|\xi - \eta|}{\varepsilon} \Bigr\}^{n+1/2-k} \Big) \, d \eta \\
& = \int_{|\xi - \eta| \geq |\xi|/4} \mathcal{O} \Big( \varepsilon^{k-1/2}
 \omega_\varepsilon^{k \nu +l_1+l_2} \frac{ \langle \eta \rangle^{m_1}
  \langle \xi \rangle^{m_2}}{ ( 1+|\xi| + |\eta| )^{k-n-1/2}}\Big) \, d \eta
\quad \text{as } \varepsilon \to 0.
\end{align*}
Then, by a straightforward calculation one shows that for every $k \geq |m_1| + 2n+1$
we have
\begin{equation*}%\label{dsigma}
(2 \pi \varepsilon)^{n} d_{\sigma,\varepsilon}(x,\xi)
= \mathcal{O} \left( \varepsilon^{k-1/2} \omega_\varepsilon^{k \nu+l_1+l_2} \langle
\xi \rangle^{m_1+m_2-k+2n+1+|m_1|} \right)
\end{equation*}
for $\varepsilon$ sufficiently small. Since $(\omega_\varepsilon)_\varepsilon$
is a slow scale net we conclude that
\begin{equation*}
(2 \pi \varepsilon)^{n} d_{\sigma,\varepsilon}(x,\xi)
= \mathcal{O} \left( \varepsilon^{N} \omega_\varepsilon^{l_1+l_2}
\langle \xi \rangle^{m_1+m_2-N} \right) \quad \forall \ N \geq 0
\end{equation*}
uniformly for every $(x,\xi) \in \mathbb{R}^{2n}$ with $|\xi| \geq 1$ and
$\sigma > 0$ as $\varepsilon \to 0$.

We next estimate $e_{\sigma,\varepsilon}(x,\xi)$. Using the coordinate change
 $z=y-x$ and integration by parts we can write
\begin{equation*}
(2 \pi \varepsilon)^{n} e_{\sigma,\varepsilon} (x,\xi)
=\int_{\substack{|z|\geq 1/4 \\ |\xi-\eta| \leq |\xi|/2}}
 e^{-iz(\eta - \xi) /\varepsilon} ( {}^t \hspace{-1pt}
L_{1,\varepsilon} )^k r_{\sigma,\varepsilon}(x,z,\xi,\eta) \, dz \, d \eta
\end{equation*}
where we have set $L_{1,\varepsilon} := -\varepsilon z/|z|^2 D_{\eta}$ and
\begin{equation*}
r_{\sigma,\varepsilon}(x,z,\xi,\eta)
:= \chi(|\xi-\eta|/|\xi|)(1-\chi(|z|)) e^{- \sigma \langle x+z \rangle
- \sigma \langle \eta \rangle} a_\varepsilon(x,\eta) b_{\varepsilon}(x+z,\xi).
\end{equation*}
Note that the operator $L_{1,\varepsilon}$ satisfies
\begin{equation*}
{}^t L_{1,\varepsilon} = -L_{1,\varepsilon} \quad \text{and} \quad
 L_{1,\varepsilon} e^{-iz(\eta - \xi) /\varepsilon} = e^{-iz(\eta - \xi) /\varepsilon}.
\end{equation*}
Also, the integrand of $e_{\sigma,\varepsilon} (x,\xi)$ is integrable with respect
to $z$ if we take $k > n$. To check the integrability with respect
to the $\eta$-variable we partition the domain of integration into the two regions
\[
\Omega_1 := \{ \eta \in \mathbb{R}^{n} : |\xi-\eta| \le 1/4 \}, \quad
\Omega_2 := \{ \eta \in \mathbb{R}^{n} : |\xi-\eta| \ge 1/4\}
\]
and write
\begin{equation*}
(2 \pi \varepsilon)^{n} e_{\sigma,\varepsilon} (x,\xi)
 = \sum_{j=1}^{2} \int_{\Omega_j} e^{-iz(\eta - \xi) /\varepsilon}
( {}^t \hspace{-1pt} L_{1,\varepsilon} )^k r_{\sigma,\varepsilon}(x,z,\xi,\eta)
\, dz \, d \eta =: \sum_{j=1}^{2} I_j .
\end{equation*}
Concerning $I_1$ we use the coordinate transformation
$\zeta = \eta- \xi$ and by Peetre's inequality
 $\langle \xi +\zeta \rangle^s \leq 2^{|s|/2} \langle \xi \rangle^s
\langle \zeta \rangle^{|s|}$ for every $s \in \mathbb{R}$ we obtain
\begin{equation*}
I_1 = \int_{\substack{|z|\geq 1/4 \\ |\zeta| \leq 1/4}} \mathcal{O}
(\varepsilon^k \omega_\varepsilon^{l_1+l_2} |z|^{-k} \langle \xi \rangle^{m_2}
\langle \xi + \zeta \rangle^{m_1-k}) \, dz \, d \zeta
 = \mathcal{O} (\varepsilon^k \omega_\varepsilon^{l_1+l_2}
\langle \xi \rangle^{m_1+m_2-k} )
\end{equation*}
for every $k > n$ as $\varepsilon \to 0$. For the estimation of the second part
 $I_2$ we introduce
\begin{equation*}
L_{2,\varepsilon} := \frac{\varepsilon^2}{|\xi-\eta|^2}
\frac{\xi-\eta}{\varepsilon} D_{z}
\end{equation*}
and let $j \in \mathbb{N}$. Again using integration by parts we obtain for
$|\xi| \geq 1$
\begin{align*}
I_2 &= \int e^{-iz(\eta - \xi) /\varepsilon} ( {}^t \hspace{-1pt} L_{2,\varepsilon} )^j
 ( {}^t \hspace{-1pt} L_{1,\varepsilon} )^k r_{\sigma,\varepsilon}(x,z,\xi,\eta) \, dz
  \, d \eta \\
&= \int_{\substack{ |z| \geq 1/4 \\ 1/4 \leq |\xi-\eta| \leq | \xi |/2}} \mathcal{O}
(\varepsilon^{j + k} \omega_\varepsilon^{j \nu +l_1+l_2} |z|^{-k}
\langle \xi \rangle^{m_2} \langle \eta \rangle^{m_1 - k} |\xi-\eta|^{-j})
\, dz \, d \eta
\end{align*}
and is integrable with respect to $z$ if we take $k > n$ for sufficiently small
$\varepsilon$. Then using the coordinate change $\eta=\xi+\zeta$ and applying
Peetre's inequality gives
\begin{align*}
I_2
& = \int_{1/4 \leq |\zeta| \leq |\xi|/2} \mathcal{O} (\varepsilon^{j + k}
 \omega_\varepsilon^{j \nu +l_1+l_2}  \langle \xi \rangle^{m_1+m_2-k}
 \langle \zeta \rangle^{|m_1|+k -j} ) \, d \zeta \\
& = \mathcal{O} (\varepsilon^{j + k} \omega_\varepsilon^{j \nu +l_1+l_2}
 \langle \xi \rangle^{m_1+m_2-k} )
\end{align*}
for every $j,k \in \mathbb{N}$ with $k> n$ and $j > |m_1| + k +n$ as
$\varepsilon \to 0$. Combining the above yields to
\[
(2 \pi \varepsilon)^{n} e_{\sigma,\varepsilon}(x,\xi)
= \mathcal{O} \left( \varepsilon^{N} \omega_\varepsilon^{l_1+l_2}\langle
\xi \rangle^{m_1+m_2-N} \right) \quad \forall N \geq 0
\]
uniformly for all $\sigma > 0$ and $(x,\xi) \in \mathbb{R}^{2n}$,
$|\xi| \geq 1$ as $\varepsilon \to 0$. By the same arguments as above one can
also show that for every $\alpha,\beta \in \mathbb{N}^n$, $|\xi| \geq 1$
 and $N \in \mathbb{N}$ we have
\begin{equation*}
|\partial_{\xi}^{\alpha} \partial_x^{\beta} d_{\sigma,\varepsilon}(x,\xi)|
 + |\partial_{\xi}^{\alpha} \partial_x^{\beta} e_{\sigma,\varepsilon}(x,\xi)|
= \mathcal{O} \left( \varepsilon^N \omega_\varepsilon^{\nu |\beta|
+l_1+l_2}\langle \xi \rangle^{m_1+m_2-|\alpha| - N} \right)
\end{equation*}
uniformly for $\sigma > 0$ as $\varepsilon \to 0$. So we deduce that
$(d_{\sigma,\varepsilon})_\varepsilon$ and $(e_{\sigma,\varepsilon})_\varepsilon$
are contained in $\MSym[m_1+m_2-N,N]{\nu,l_1+l_2}$ for every $N \in \mathbb{N}$
uniformly with respect to $\sigma>0$ and $(x,\xi) \in \mathbb{R}^{2n}$,
$|\xi| \geq 1$.

So it remains to study the term
$(2 \pi \varepsilon)^{n} f_{\sigma,\varepsilon}(x,\xi)$
which was given by
\begin{equation*}
\int e^{i(x-y)(\eta - \xi) /\varepsilon} \chi(|\xi-\eta|/|\xi|)\chi_1(x,y)
 e^{- \sigma (\langle y \rangle + \langle \eta \rangle)} a_\varepsilon(x,\eta)
 b_{\varepsilon}(y,\xi) \, dy \, d \eta.
\end{equation*}
Now writing $\xi = \lambda \nu$ where $\lambda = |\xi|$ and using the coordinate
transformations
\begin{equation*}
  \zeta = (\eta - \xi) / \lambda, \quad   z=y-x     									 
\end{equation*}
we obtain
\begin{equation*}
f_{\sigma,\varepsilon} (x,\xi) =\frac{\lambda^n}{(2 \pi \varepsilon)^n}
\int_{|z| \leq 1/2,\,  |\zeta| \leq 1/2} e^{-i \lambda z \zeta/\varepsilon}
t_{\sigma,\varepsilon}(z,\zeta; x,\xi) \, dz \, d\zeta
\end{equation*}
where $t_{\sigma,\varepsilon}$ is a function in
$\mathcal{C}_{\rm c}^{\infty} (\mathbb{R}^n_z \times \mathbb{R}^n_\zeta)$
containing $(x,\xi)$ as parameters and is given by
\begin{equation*}
t_{\sigma,\varepsilon}(z,\zeta; x,\xi) = \chi(|\zeta|) \chi(|z|)
e^{- \sigma \langle x+z \rangle - \sigma \langle \lambda(\nu + \zeta) \rangle}
a_\varepsilon(x,\lambda(\nu+ \zeta)) b_{\varepsilon}(x+z,\lambda \nu).
\end{equation*}
Then the method of stationary phase gives for every $N \in \mathbb{N}$, $N \geq 1$
\begin{equation*}
f_{\sigma,\varepsilon}(x,\xi)
= \sum_{|\alpha| \leq N-1} \frac{\varepsilon^{|\alpha|}}{\alpha !
 \lambda^{|\alpha|}} D^{\alpha}_\zeta \partial_z^{\alpha}
t_{\sigma,\varepsilon}(z,\zeta;x,\xi)  \Big\vert _{\substack{z=0
\\ \zeta = 0}} + S_N(t_{\sigma,\varepsilon};\lambda/\varepsilon)
\end{equation*}
and the remainder can be estimated as follows
\begin{align*}
|S_N(t_{\sigma,\varepsilon};\lambda / \varepsilon)|
&\leq \frac{C\varepsilon^N}{N! \lambda^N} \sum_{|\alpha + \beta| \leq 2n+1}
 \| \partial_\zeta^{\alpha} \partial_z^{\beta}
 (\partial_\zeta \cdot \partial_z)^N t_{\sigma,\varepsilon} \|_{L^1(\mathbb{R}_z^n
 \times \mathbb{R}_\zeta^n)} \\
&= \mathcal{O} \left( \varepsilon^{N} \omega_\varepsilon^{(2n+1+N)\nu+l_1+l_2}
 \lambda^{m_1 +m_2 -N} \right)\\
& = \mathcal{O} \left( \varepsilon^{N-1} \omega_\varepsilon^{l_1+l_2}
 \langle \xi \rangle^{m_1 +m_2 -N} \right)
\end{align*}
for $|\xi| \geq 1$ and $\varepsilon$ sufficiently small. Finally, since similar
estimates hold true for the derivatives and we conclude that
$(S_N(t_{\sigma,\varepsilon}))_\varepsilon$ is contained in
$\MSym[m_1+m_2-N,N-1]{\nu,l_1+l_2}$ for every $N \in \mathbb{N}$ uniformly
with respect to $\sigma>0$ for every $(x,\xi) \in \mathbb{R}^{2n}$ with
$|\xi| \geq 1$.

As already mentioned in the beginning of the proof we now pass to the limit
$\sigma \to 0_{+}$.
Then by Lebesgue's dominated convergence theorem we have
\begin{equation*}
c_{\sigma,\varepsilon}(x,\xi) \to c_\varepsilon(x,\xi) \quad \text{as } \sigma \to 0_+
\end{equation*}
so that equation \eqref{ABproduct} reads
\begin{align*}
A_{\psi,\varepsilon} B_{\psi,\varepsilon} u_\varepsilon(x)
&= (2 \pi \varepsilon)^{-n} \lim_{\tau \to 0_{+}} \int e^{ix \xi/\varepsilon
 - \tau \langle \xi \rangle} c_{\varepsilon}(x,\xi) \hat{u}_\varepsilon
 (\xi/\varepsilon) \, d \xi \\
& = (2 \pi \varepsilon)^{-n} \int e^{ix \xi/\varepsilon} c_{\varepsilon}(x,\xi)
\hat{u}_\varepsilon(\xi/\varepsilon) \, d \xi
\end{align*}
where the last equation holds in the sense of oscillatory integrals.
Moreover the generalized symbol $(c_\varepsilon)_\varepsilon$ is equal
to $(a_\varepsilon \# b_{\varepsilon})_\varepsilon$ as described in
\eqref{AE:Product} and in particular
$(c_\varepsilon)_\varepsilon \in \MSym[m_1+m_2,0]{\nu,l_1+l_2}$.
This completes the proof.
\end{proof}

\section{A factorization procedure for \texorpdfstring{$L_\psi$}{L-psi}} \label{sec:Fact}

Concerning products of $\psi$-pseudodifferential operators that approximate
$L_\psi$ we make similar considerations as in the smooth setting which can
be found in \cite[Appendix II]{Kumano-go:81} and \cite[Chapter 23]{Hoermander:3}.
First we write for $L_\psi = \varepsilon^2 L_\varepsilon$ with $L_\varepsilon$
from the beginning
\begin{equation}\label{GovOp}
L_{\psi}(y,D_t,D_x,D_z) =: (\partial_z^2 + A(y,D_t,D_x))_{\psi}
\end{equation}
where $A_\psi = A_\psi(y,D_t,D_x)$ is the $\psi$-pseudodifferential operator
with generalized symbol $(a_\varepsilon)_\varepsilon$ and
\begin{equation}\label{SymbolA}
a_\varepsilon(y,\tau,\xi) = c_\varepsilon(y)\tau^2 - \langle b_\varepsilon(y)
\xi, \xi \rangle \quad \varepsilon \in (0,1].
\end{equation}
To recall the requirements made on the coefficients $c_\varepsilon(y)$
and $b_{j,\varepsilon}(y), 1 \leq j \leq n-1$ we also refer to
 Section~\ref{sec:Intro}.

Before stating the main theorem of this section we give a few more details
 about notation. In the following we will study operators on
$\mathbb{R}^{n+1}$ of the form
\begin{equation*}
S_\psi = \sum_{j=0}^2 S_{j,\psi} (y,D_t,D_x) (\partial_z^{2-j})_\psi
\end{equation*}
with coefficients $S_{j,\psi} \in \operatorname{OP}_\psi \MSym[k_j,0]{1,l_j}$
for some $k_j,l_j\in \mathbb{R}$, $j=0,1,2$. Further we write
\begin{equation*}
S_\psi = \sum_{j=0}^2 S_{j,\psi} (y,D_t,D_x) (\partial_z^{2-j})_\psi
\quad \text{ on } I'
\end{equation*}
when the symbols of the coefficients $S_{j,\psi}$ are restricted to a set $I'$.

Concerning the factorization domain we make similar considerations as
 in \cite{Stolk:04} and introduce the set
\begin{equation}\label{BasicSet}
I'_{\theta_1} := \bigl\{ (y,\tau,\xi) \mid \tau \neq 0, \,
 \langle b(y)\xi,\xi \rangle < \sin^2 (\theta_1) c(y) \tau^2 \bigr\}
\end{equation}
for some $\theta_1 \in (0, \pi/2)$. Then $I'_{\theta_1}$ is an open subset
 of $\mathbb{R}^n \times (\mathbb{R}^n \setminus 0)$ and conic with respect to
the second variable. Moreover we have chosen $I'_{\theta_1}$ independent of
the parameter $\varepsilon \in (0,1]$ for simplicity. Note that the main
reason for restricting the analysis to the domain $I'_{\theta_1}$ is that
$(a_\varepsilon)_\varepsilon$ is non-negative there.

Further, as already mentioned in Section~\ref{sec:PsiPseudoCalculus}
it is necessary to introduce the notion of polyhomogeneous generalized symbols
that are smoothed off at the origin.

\begin{definition}\label{defn:PhgSymb}
We say that a generalized symbol $(p_\varepsilon)_\varepsilon$ is polyhomogeneous
in $\MSym[m,k]{\nu,l}$, denoted by
$(p_\varepsilon)_\varepsilon \in \MphgSym[m,k]{\nu,l}$, if there exist a sequence
of symbols $(\overline{p}_{m-j,\varepsilon})_{\varepsilon}$ in
$\MhgSym[m-j,k]{\nu,\nu j+l}$, $j \geq 0$ and a cut-off
$\varphi \in \mathcal{C}_{\rm c}^{\infty}(\mathbb{R}^n)$ equal to $1$ near the
 origin so that $\forall N \geq 1$ we have
\begin{equation}\label{AE:PhgSymb}
\bigl(p_\varepsilon - \sum_{j = 0}^{N-1} \varepsilon^j (1-\varphi)
\overline{p}_{m-j,\varepsilon}\bigr)_\varepsilon \in \MSym[m-N,N+k]{\nu,\nu N+l}
\quad \text{ for } \xi \text{ sufficiently large.}
\end{equation}
For this we will use the following notation
\begin{equation*}
(p_\varepsilon)_\varepsilon \dot{\sim} \ \sum_{j \geq 0}
\bigl(\varepsilon^j \overline{p}_{m-j,\varepsilon}\bigr)_\varepsilon
\quad \text{in } \MSym[m,k]{\nu,l}.
\end{equation*}
Here the homogeneous part $(\overline{p}_{m,\varepsilon})_\varepsilon$
is called the principal symbol of $(p_\varepsilon)_\varepsilon$ if there
exist an $\eta \in (0,1]$ and a constant $ K > 0$ such that
 $\overline{p}_{m,\varepsilon} \not\equiv 0$ for all $|\xi| \geq K$ and any
fixed $\varepsilon \in (0,\eta]$. In particular a $\psi$-pseudodifferential
operator is polyhomogeneous if its generalized symbol is polyhomogeneous.
\end{definition}

\begin{remark} \rm
First note that the principal symbol is uniquely determined if the dual variable
is sufficiently large. Also $\eqref{AE:PhgSymb}$ can be interpreted in the
following sense: $\forall N \geq 1$ we have
\begin{equation}\label{AE:PhgSymbError}
\bigl(p_\varepsilon - \sum_{j = 0}^{N-1} \varepsilon^j (1-\varphi)
\overline{p}_{m-j,\varepsilon}\bigr)_\varepsilon = (r_\varepsilon)_\varepsilon
+ (s_\varepsilon)_\varepsilon
\end{equation}
for some  $(s_\varepsilon)_\varepsilon \in \MSym[m-N,N+k]{\nu,\nu N+l}$ and
 $(r_\varepsilon)_\varepsilon$ is some compactly supported regular generalized
symbol of order $-\infty$ in the following sense:
\[
\begin{gathered}
\exists \eta \in (0,1] \ \exists k \in \mathbb{R} \ \forall m \in \mathbb{R} \ \forall \alpha, \beta \in \mathbb{N}^n \ \exists C > 0 \text{ such that}\\
| \partial_{\xi}^{\alpha} \partial_x^{\beta} r_\varepsilon (x,\xi) |  \leq C \varepsilon^{k} \langle \xi \rangle^{m - |\alpha|} \qquad \forall \varepsilon \in (0,\eta].
\end{gathered}
\]
We refer to Section~\ref{sec:WFInfty} where it is essential to distinguish
between the two different types of errors on the right hand side of
\eqref{AE:PhgSymbError}.
\end{remark}

The aim of this section is now to factorize $L_\psi$ on $I'_{\theta_1}$ in
terms of two first-order operators of the form
 $L_{j,\psi} = (\partial_z + A_{j})_\psi$ on $I'_{\theta_1}$ with $A_{j,\psi}$
a polyhomogeneous $\psi$-pseudodifferential operator with generalized symbol in
 $\MphgSym[1,0]{1,0}$ on $I'_{\theta_1}$, $j=1,2$. Here a first rough approximation
 would suggest $A_{j,\psi}$ to be the square root of the operator
$A_\psi = \operatorname{OP}_\psi ((a_\varepsilon)_\varepsilon)$ given in
 \eqref{GovOp} on the set $I'_{\theta_1}$ where
 $\pm (i \sqrt{a_\varepsilon})_\varepsilon$ is described explicitly, $j=1,2$.

We are now going to show the following result.

\begin{theorem}\label{thm:Fact}
Let $L_\psi = (\partial^2_z +A)_\psi$ and $I'_{\theta_1}$ be as in \eqref{GovOp}
and \eqref{BasicSet} respectively.
Then the operator $L_\psi$ can be factorized into
\begin{equation}\label{eqn:Fact}
L_\psi = L_{1,\psi} L_{2,\psi} + R_\psi \quad \text{ on } I'_{\theta_1}
\end{equation}
where $L_{j,\psi} = (\partial_z + A_{j})_\psi$ and
$A_{j,\psi} \in \operatorname{OP}_\psi \MphgSym[1,0]{1,0}$ on
$I'_{\theta_1}$ whose principal symbol is equal to
$\pm (i \sqrt{a_\varepsilon})_\varepsilon$, $j=1,2$.
Moreover the remainder is given by
$R_\psi = \Gamma_{0,\psi} + \Gamma_{1,\psi}(\partial_z)_\psi$ for some
$\psi$-pseudodifferential operators $\Gamma_{j,\psi}$ with generalized symbol
 $(\gamma_{j,\varepsilon})_\varepsilon$ in $\mathcal{N}_{\mbox{$\mathcal{S}$}^{2-j}}$
on $I'_{\theta_1}$, $j=0,1$.
\end{theorem}

We remark that the product $L_{1,\psi} L_{2,\psi}$ in \eqref{eqn:Fact} is not a
$\psi$-pseudodifferential operator on $\mathbb{R}^{n+1}$. But one can overcome
this by introducing a generalized cut-off $g_\psi (D_t,D_y)$ such that the
difference $L_{1,\psi} L_{2,\psi} g_\psi - L_{1,\psi} L_{2,\psi}$ is insignificant
on some adequate subdomain of the phase space $T^* \mathbb{R}^{n+1}\setminus 0$
under a microlocal point of view.
For the specification for microlocal analysis we refer to the next section.

\subsection*{Technical preliminaries}

Note that $a_\varepsilon$ as in \eqref{SymbolA} is a homogeneous function
with respect to the dual variables $(\tau,\xi)$ and so that there exists
 $\eta \in (0,1]$ such that
\begin{equation*}
|a_\varepsilon(y,\tau,\xi)| \geq C |(\tau,\xi)|^2 \quad
\text{ on } I'_{\theta_2} \text{ for every } \varepsilon \in (0,\eta]
\end{equation*}
for some fixed $\theta_2 \in (0,\pi/2)$ with $\theta_1 < \theta_2$ and the
constant $C>0$ independent of $\varepsilon$. Moreover we set for some
$\varepsilon_0 \in (0,1]$ which will be specified later
\begin{equation}\label{RootA}
\overline{b}_{j,\varepsilon}^{(0)}(y,\tau,\xi) :=
\begin{cases}
\pm i \sqrt{a_\varepsilon(y,\tau,\xi)}, & \varepsilon \in (0,\varepsilon_0] \\
0, & \varepsilon \in (\varepsilon_0,1]
\end{cases} \quad \text{on } I'_{\theta_2}
\end{equation}
which in turn gives
$(\overline{b}^{(0)}_{j,\varepsilon})_\varepsilon \in \MhgSym[1,0]{1,0}$ on
$I'_{\theta_2}$. We now associate $(\overline{b}^{(0)}_{j,\varepsilon})_\varepsilon$
to a generalized symbol
$(\overline{c}^{(0)}_{j,\varepsilon})_\varepsilon \in
\MSym[1,0]{1,0}(\mathbb{R}^n \times (\mathbb{R}\setminus 0) \times \mathbb{R}^{n-1})$
so that the difference $(\overline{c}^{(0)}_{j,\varepsilon} -
\overline{b}^{(0)}_{j,\varepsilon})_\varepsilon$ is in $\mathcal{N}_{\mbox{$\mathcal{S}$}^1}$
on $I_{\theta_1}'$ in the sense of Definition~\ref{defn:SymbOpenSet}.

Therefore, since $(c_\varepsilon(y))_\varepsilon$ is strictly non-zero we can
define for fixed $\varepsilon \in(0,\varepsilon_1]$
\begin{equation*}
f_\varepsilon(y,\tau,\xi) := \frac{\langle b_\varepsilon(y) \xi,
\xi\rangle}{c_\varepsilon(y) \tau^2} \quad \text{on }
\mathbb{R}^n \times (\mathbb{R} \setminus 0) \times \mathbb{R}^{n-1}.
\end{equation*}
Recall that $\varepsilon_1 \in (0,1]$ is so that
$\inf_{y \in \mathbb{R}^n} |c_\varepsilon(y)| \geq C$ for some $C>0$ and every
 $\varepsilon \in (0,\varepsilon_1]$. Furthermore for every fixed
$\varepsilon \in (0,\varepsilon_1]$ we let $\widetilde{h}_\varepsilon$ be
the smooth function defined on
$\mathbb{R}^n \times (\mathbb{R} \setminus 0) \times \mathbb{R}^{n-1}$,
$0 \leq \widetilde{h}_\varepsilon \leq 1$, which is given by
\begin{equation}\label{CutOff}
\widetilde{h}_\varepsilon(y,\tau,\xi) := \begin{cases}
0, & |f_\varepsilon| \geq \sin^2 \gamma_2 \\
1, & |f_\varepsilon| \leq \sin^2 \gamma_1\\
\left({1 + e^{\frac{1}{|f_\varepsilon| - \sin^2 \gamma_1} - \frac{1}{\sin^2 \gamma_2 - |f_\varepsilon|}}}\right)^{-1}, & \sin^2 \gamma_1 < |f_\varepsilon| < \sin^2 \gamma_2
\end{cases}
\end{equation}
for some fixed $\gamma_1$ and $\gamma_2$ with $0 < \gamma_1 < \gamma_2 < \pi/2$.
For all other $\varepsilon \in (0,1]$ we may set $\widetilde{h}_\varepsilon \equiv 0$
and we obtain that
$(\widetilde{h}_\varepsilon)_\varepsilon \in \MSym[0,0]{1,0}(\mathbb{R}^n
\times (\mathbb{R}\setminus 0) \times \mathbb{R}^{n-1})$ and also
$(\widetilde{h}_\varepsilon)_\varepsilon \in \MhgSym[0,0]{1,0}$ on $I'_{\theta_2}$.

Here and hereafter we let $\theta_1, \theta_2$ be fixed so that
$0 < \theta_1 < \gamma_1 < \gamma_2 < \theta_2 < \pi/2$. Then there is an
$\varepsilon_2 \in(0,1]$ such that
\begin{equation*}
I'_{\theta_1} \subset \{ (y,\tau,\xi) \mid \tau \neq 0
\text{ and } |f_\varepsilon| \leq \sin^2 \gamma_1 \} \quad \varepsilon
\in (0,\varepsilon_2]
\end{equation*}
because of the assumptions made on the coefficients
$c_\varepsilon(y), b_{j,\varepsilon}(y), y \in \mathbb{R}^n, 1 \leq j \leq n-1$.

Similar observations yield to the following:
there exists $\varepsilon_3 \in (0,1]$ such that
\begin{equation*}
\{ (y,\tau,\xi) \mid \tau \neq 0, |f| \geq \sin^2 \theta_2 \}
 \subset \{(y,\tau,\xi) \mid \tau \neq 0, |f_\varepsilon| \geq \sin^2 \gamma_2 \}
\end{equation*}
for all $\varepsilon \in (0,\varepsilon_3]$ where we have set
 $f := \lim_{\varepsilon \to 0} f_\varepsilon$.

Then with $(\overline{b}^{(0)}_{j,\varepsilon})_\varepsilon$ as in \eqref{RootA}
 we set $\varepsilon_0 := \min_{1 \leq k \leq 3} \varepsilon_k$ and define
its extension $(\overline{c}^{(0)}_{j,\varepsilon})_\varepsilon$ by
\begin{equation*}
\bigl(\overline{c}^{(0)}_{j,\varepsilon}\bigr)_\varepsilon
:= \bigl(\widetilde{h}_\varepsilon \overline{b}^{(0)}_{j,\varepsilon}\bigr)_\varepsilon
 \in \MSym[1,0]{1,0} (\mathbb{R}^n \times (\mathbb{R}\setminus 0)
 \times \mathbb{R}^{n-1}).
\end{equation*}
Indeed we have that
$(\overline{c}^{(0)}_{j,\varepsilon} - \overline{b}^{(0)}_{j,\varepsilon})_\varepsilon
 \in \mathcal{N}_{\mbox{$\mathcal{S}$}^1}$ on $I_{\theta_1}'$ since
 $\widetilde{h}_\varepsilon \equiv 1$ on $I_{\theta_1}'$.

\subsection*{Factorization procedure}

Our aim here is to decompose the operator $L_\psi$ as announced in \eqref{eqn:Fact}.
Therefore we give a construction scheme for the generalized symbols
$(a_{j,\varepsilon})_\varepsilon$ of the operators $A_{j,\psi}$, $j=1,2$
 by means of their polyhomogeneous asymptotic expansions; i.e., on the
set $I'_{\theta_1}$ we have
\begin{equation}\label{AE:aj}
(a_{j,\varepsilon}(y,\tau,\xi))_\varepsilon  \dot{\sim}
 \sum_{\mu \geq 0} \bigl(\varepsilon^{\mu} \overline{c}_{j,\varepsilon}^{(\mu)}
(y,\tau,\xi) \bigr)_\varepsilon \quad \text{ in } \MSym[1,0]{1,0}
\end{equation}
and the sequence
$\{ (\overline{c}_{j,\varepsilon}^{(\mu)})_\varepsilon\}_{\mu \in \mathbb{N}}$
consists of elements $(\overline{c}_{j,\varepsilon}^{(\mu)})_\varepsilon
\in \MSym[1-\mu,0]{1,\mu}(\mathbb{R}^n \times (\mathbb{R}\setminus 0)
\times \mathbb{R}^{n-1})$ and satisfies the uniformity condition:
\[
\begin{gathered}
\exists \eta \in (0,1] \ \exists K > 0 \ \forall \mu \in \mathbb{N} \ \forall \alpha,\beta \in \mathbb{N}^n \ \exists C>0 \ \text{for which} \\
|\partial_{(\tau,\xi)}^{\alpha} \partial_y^{\beta} \overline{c}_{j,\varepsilon}^{(\mu)}(y,\tau,\xi)| \leq C \omega_\varepsilon^{|\beta|} \langle \xi\rangle^{1-\mu-|\alpha|} \qquad \text{for } |\tau| \geq K, \ \varepsilon \in (0,\eta]
\end{gathered}
\]
for $j=1,2$. More precisely we will recursively construct a sequence
$\{(\overline{b}_{j,\varepsilon}^{(\mu)})_\varepsilon\}_{\mu}$ of symbols
$(\overline{b}_{j,\varepsilon}^{(\mu)})_\varepsilon$ in $\MhgSym[1-\mu,\mu]{1,\mu}$
on $I'_{\theta_2}$, $j=1,2$ such that
\begin{equation*}
\bigl( \overline{c}_{j,\varepsilon}^{(\mu)}(y,\tau,\xi) \bigr)_\varepsilon
= \bigl( \varepsilon^{-\mu} \overline{b}_{j,\varepsilon}^{(\mu)}(y,\tau,\xi)
\bigr)_\varepsilon \quad \text{on } I'_{\theta_2}.
\end{equation*}
Recall that on $I'_{\theta_1}$ \eqref{AE:aj} is equivalent to the following:
there exists a smooth cut-off equal to $1$ near the origin such that for every
$N \geq 1$ we have on $I'_{\theta_1}$
\begin{equation*}
\bigl(a_{j,\varepsilon} - \sum_{\mu = 0}^{N-1} \varepsilon^\mu (1-\varphi)
\overline{c}_{j,\varepsilon}^{(\mu)} \bigr)_\varepsilon \in \MSym[1-N,N]{1,N}
\quad \text{for } |(\tau,\xi)| \text{ sufficiently large.}
\end{equation*}

\begin{proof}[Proof of Theorem~\ref{thm:Fact}]
To begin with, we set for $j=1,2$ and $\varepsilon \in (0,\varepsilon_0]$
\[
\overline{a}_{j,\varepsilon}^{(1)}:= \overline{b}_{j,\varepsilon}^{(0)}, \quad
l_{j,\varepsilon}^{(1)} := i\zeta + a_{j,\varepsilon}^{(1)}
\]
where
\[
\overline{b}_{j,\varepsilon}^{(0)}(y,\tau,\xi)
= \pm i\sqrt{ a_\varepsilon(y,\tau,\xi) }\quad \text{on }I'_{\theta_2},\;
j=1,2.
\]
 Moreover $(a^{(1)}_{j,\varepsilon})_\varepsilon \in \MphgSym[1,0]{1,0}$
with polyhomogeneous asymptotic expansion
$(\widetilde{h}_\varepsilon \overline{b}_{j,\varepsilon}^{(0)})_\varepsilon$
with $\widetilde{h}_\varepsilon$ as in \eqref{CutOff} and
$(\overline{b}_{j,\varepsilon}^{(0)})_\varepsilon$ only prescribed on
$I'_{\theta_2}$, $j=1,2$. As in \eqref{RootA} we set
$\overline{b}_{j,\varepsilon}^{(0)}$ and $l_{j,\varepsilon}^{(1)}$ equal to zero
for all other $\varepsilon \in (0,1]$. Furthermore we define the first-order
operator $L_{j,\psi}^{(1)}$ through $(\partial_z + A_{j}^{(1)})_\psi$ where
$A^{(1)}_{j,\psi} = \operatorname{OP}_\psi ((a_{j,\varepsilon}^{(1)})_\varepsilon)$
for $j=1,2$.

Now taking $L_{1,\psi}^{(1)} L_{2,\psi}^{(1)}$ as a first approximation for
$L_\psi$ we can compute the error as follows:
\begin{align*}
L_{1,\psi}^{(1)} L_{2,\psi}^{(1)} - L_{\psi}
&=\phantom{:} (\partial_z + A_1^{(1)})_{\psi} (\partial_z + A_2^{(1)})_{\psi}
 - (\partial^2_z + A)_{\psi} \\
& =\phantom{:}  (A^{(1)}_{1} + A^{(1)}_{2})_{\psi} (\partial_z)_{\psi}
 + \operatorname{OP}_{\psi}\bigl( \bigl(\varepsilon \hspace{1pt} \partial_z a_{2,\varepsilon}^{(1)}\bigr)_\varepsilon \bigr) + A_{1,\psi}^{(1)} A_{2,\psi}^{(1)} - A_{\psi} \\
& =: \Gamma_{0,\psi}^{(1)} + \Gamma_{1,\psi}^{(1)}(\partial_z)_{\psi}
\end{align*}
where we have used the Leibniz composition rule for
$\psi$-pseudodifferential operators; that is,
$(\partial_z)_{\psi} A_{2,\psi}^{(1)} = A_{2,\psi}^{(1)} (\partial_z)_{\psi}
+ \operatorname{OP}_{\psi} \bigl(
(\varepsilon \partial_z a_{2,\varepsilon}^{(1)})_\varepsilon \bigr)$.
Then $\Gamma_{1,\psi}^{(1)} = (A^{(1)}_{1} + A^{(1)}_{2})_{\psi}
\in \operatorname{OP}_\psi \mathcal{N}_{\mbox{$\mathcal{S}$}^1}$ on $I'_{\theta_1}$.
 Moreover $\Gamma_{0,\psi}^{(1)}$ is equal to
$\operatorname{OP}_{\psi}\bigl( (\varepsilon \hspace{1pt}
\partial_z a_{2,\varepsilon}^{(1)})_\varepsilon \bigr)
 + A_{1,\psi}^{(1)} A_{2,\psi}^{(1)} - A_{\psi}$ and is a polyhomogeneous
$\psi$-pseudodifferential operator with generalized symbol
 $(\gamma_{0,\varepsilon}^{(1)})_\varepsilon$ in $\MphgSym[1,1]{1,1}(I'_{\theta_1})$
by Proposition~\ref{prop:Product}.
 In detail $(\gamma_{0,\varepsilon}^{(1)})_\varepsilon$ admits the asymptotic
expansion
\begin{equation*}
\bigl(\gamma_{0,\varepsilon}^{(1)}\bigr)_\varepsilon  \dot{\sim}
 \Bigl(\varepsilon \partial_z \overline{a}_{2,\varepsilon}^{(1)}
+ \sum_{|\alpha| \ge 1} \frac{{\varepsilon}^{|\alpha|}}{\alpha!}
\bigl( D_{\xi}^{\alpha} \overline{a}_{1,\varepsilon}^{(1)} \bigr)
\bigl( \partial_{x}^{\alpha} \overline{a}_{2,\varepsilon}^{(1)} \bigr)
\Bigr)_\varepsilon  \quad \text{on } I'_{\theta_1}.
\end{equation*}
To improve this argument we will construct recursively the $N$-th approximation:
 $L^{(N)}_{1,\psi}L^{(N)}_{2,\psi}-L_\psi$. Before, for convenience of the reader,
we compute the second order approximation for $L_\psi$.

Therefore we let $\varepsilon \in (0,\varepsilon_0]$ be fixed and define
\begin{equation*}
\overline{a}_{j,\varepsilon}^{(2)}:= \overline{a}_{j,\varepsilon}^{(1)}
+ \overline{b}_{j,\varepsilon}^{(1)},\quad
l_{j,\varepsilon}^{(2)} := i\zeta + a_{j,\varepsilon}^{(2)}
\end{equation*}
where $(a_{j,\varepsilon}^{(2)})_\varepsilon$ denotes the symbol
with polyhomogeneous asymptotic expansion
$(\widetilde{h}_\varepsilon \overline{a}_{j,\varepsilon}^{(2)})_\varepsilon$
 and the existence of the generalized symbol
$(\overline{b}_{j,\varepsilon}^{(1)})_\varepsilon$ will be clarified immediately.
 Again we set $\overline{b}_{j,\varepsilon}^{(1)}$ and $l_{j,\varepsilon}^{(2)}$
equal zero for all other $\varepsilon \in (0,1]$. From the preceding observations
we have the approximation
\begin{equation*}
L_{1,\psi}^{(1)} L_{2,\psi}^{(1)}  - L_{\psi} = \Gamma_{0,\psi}^{(1)}
+ \Gamma_{1,\psi}^{(1)} (\partial_z)_\psi \quad \text{ on } I_{\theta_1}'.
 \end{equation*}
To obtain a suitable second order approximation we would have to keep the
following expression small
\begin{align}\label{SecApprox}
%\begin{split}
L_{1,\psi}^{(2)} L_{2,\psi}^{(2)} - L_{\psi}
&= ( L_{1}^{(1)} + B_{1}^{(1)} )_\psi ( L_{2}^{(1)} +  B_{2}^{(1)} )_\psi - L_{\psi} \nonumber \\
& =\Gamma_{0,\psi}^{(1)} + \Gamma_{1,\psi}^{(1)}(\partial_z)_{\psi}
 + ( B_{1}^{(1)} + B_{2}^{(1)} )_\psi ( \partial_z )_{\psi} \\
&\quad  + \operatorname{OP}_{\psi}\bigl( \bigl(\varepsilon
 \partial_z b_{2,\varepsilon}^{(1)}\bigr)_\varepsilon \bigr)
 + A_{1,\psi}^{(1)} B_{2,\psi}^{(1)} + B_{1,\psi}^{(1)} A_{2,\psi}^{(1)}
 + B_{1,\psi}^{(1)} B_{2,\psi}^{(1)} \nonumber
%\end{split}
\end{align}
where in the above step we have used the Leibniz rule.

We now specify
$(\overline{b}_{1,\varepsilon}^{(1)})_\varepsilon,
(\overline{b}_{2,\varepsilon}^{(1)})_\varepsilon$ on $I'_{\theta_2}$
as follows: because the the generalized symbol
$(\overline{a}_{j,\varepsilon}^{(1)})_\varepsilon$ satisfies:
\begin{equation}\label{Ell:ps}
\exists \eta \in (0,1]: \; |\overline{a}_{j,\varepsilon}^{(1)}(y,\tau,\xi)|
\geq C |(\tau,\xi)| \quad \text{ on } I'_{\theta_2}, \; \forall
\varepsilon \in (0,\eta]
\end{equation}
for some constant $C>0$ that is independent of $\varepsilon \in (0,\eta]$, $j=1,2$,
the matrix
\begin{equation}\label{CoeffMatrix}
\begin{pmatrix} 1 & 1 \\ \overline{a}_{2,\varepsilon}^{(1)}
& \overline{a}_{1,\varepsilon}^{(1)} \end{pmatrix}
\end{equation}
is invertible on $I'_{\theta_2}$. We denote by
 $\bigl(\widetilde{\gamma}_{0,\varepsilon}^{(1)}\bigr)_\varepsilon$ and
$\bigl(\widetilde{\gamma}_{1,\varepsilon}^{(1)}\bigr)_\varepsilon$ the
naturally extended top order symbols on $I'_{\theta_2}$ of
$\bigl(\gamma_{0,\varepsilon}^{(1)}\bigr)_\varepsilon$ and
$\bigl(\gamma_{1,\varepsilon}^{(1)}\bigr)_\varepsilon$ on $I'_{\theta_1}$
respectively. Thus the system
\begin{align*}
-\widetilde{\gamma}_{1,\varepsilon}^{(1)}
 &=  \overline{b}_{1,\varepsilon}^{(1)} + \overline{b}_{2,\varepsilon}^{(1)} \\
-\widetilde{\gamma}_{0,\varepsilon}^{(1)}
  &=  \overline{b}_{1,\varepsilon}^{(1)} \overline{a}_{2,\varepsilon}^{(1)}
  + \overline{a}_{1,\varepsilon}^{(1)} \overline{b}_{2,\varepsilon}^{(1)}
\end{align*}
is uniquely solvable for $(\overline{b}_{j,\varepsilon}^{(1)})_\varepsilon $
in $\MhgSym[0,1]{1,1}$ on $I'_{\theta_2}$, $j=1,2$.

With this choice of $(\overline{b}_{j,\varepsilon}^{(1)})_\varepsilon$
the second-order approximation for the operator $L_\psi$ in \eqref{SecApprox}
now reads
\begin{equation*}
L_{1,\psi}^{(2)} L_{2,\psi}^{(2)}  - L_{\psi}
= \Gamma_{0,\psi}^{(2)} + \Gamma_{1,\psi}^{(2)}(\partial_z)_{\psi}
\quad \text{on } I'_{\theta_1}
\end{equation*}
and $\Gamma_{0,\psi}^{(2)} = \Gamma_{0,\psi}^{(2)}(y,D_t,D_x)$ and
$\Gamma_{1,\psi}^{(2)} = \Gamma_{1,\psi}^{(2)}(y,D_t,D_x)$ are the
$\psi$-pseudodifferential operators with generalized symbols
$(\gamma_{0,\varepsilon}^{(2)})_{\varepsilon}$ in $\MphgSym[0,2]{1,2}$ on
$I'_{\theta_1}$ and $(\gamma_{1,\varepsilon}^{(2)})_{\varepsilon}$ is in
 $\mathcal{N}_{\mbox{$\mathcal{S}$}^1}$ on $I'_{\theta_1}$, respectively.

To continue the proof, we assume for $N \ge 1$ that
$(\overline{b}_{j,\varepsilon}^{(\nu)})_\varepsilon \in \MhgSym[1-\nu,\nu]{1,\nu}$
 on $I'_{\theta_2}$ is determined for all $\nu \le N-1$ and $j=1,2$. For fixed
$\varepsilon \in (0,\varepsilon_0]$ and $j=1,2$ we set
\begin{equation*}
\overline{a}_{j,\varepsilon}^{(N)} := {\sum_{\nu = 0}^{N-1}}
\overline{b}_{j,\varepsilon}^{(\nu)},\quad
 L_{j,\psi}^{(N)} = (\partial_z + A_j^{(N)})_{\psi}
\end{equation*}
with $A_{j,\psi}^{(N)} (y,D_t,D_x)
= \operatorname{OP}_{\psi}\bigl( (a_{j,\varepsilon}^{(N)})_\varepsilon \bigr)$
 with $(a_{j,\varepsilon}^{(N)})_\varepsilon$ the symbol with polyhomogeneous
 expansion $(\widetilde{h}_\varepsilon \overline{a}_{j,\varepsilon}^{(N)})_\varepsilon$.
Again for all other $\varepsilon \in (0,1]$ we set $a_{j,\varepsilon}^{(N)}$
and $L_{j,\varepsilon}^{(N)}$ equal to zero, $j=1,2$. Furthermore, we suppose
an $N$-th order approximation for $L_\psi$ of the form
\begin{equation*}
L_{1,\psi}^{(N)} L_{2,\psi}^{(N)} - L_{\psi}
= \Gamma_{0,\psi}^{(N)} + \Gamma_{1,\psi}^{(N)}(\partial_z)_\psi \quad \text{on }
 I'_{\theta_1}
\end{equation*}
where $\Gamma_{0,\psi}^{(N)}$ and $\Gamma_{1,\psi}^{(N)}$ are the
$\psi$-pseudodifferential operators with generalized symbols
$(\gamma_{0,\varepsilon}^{(N)})_\varepsilon$ in $\MphgSym[2-N,N]{1,N}$ on
$I'_{\theta_1}$ and $(\gamma_{1,\varepsilon}^{(N)})_\varepsilon$ in
$\mathcal{N}_{\mbox{$\mathcal{S}$}^1}$ on $I'_{\theta_1}$, respectively.

Again we denote by $(\widetilde{\gamma}_{0,\varepsilon}^{(N)})_\varepsilon$
and $(\widetilde{\gamma}_{1,\varepsilon}^{(N)})_\varepsilon$ the naturally
extended top order terms on $I'_{\theta_2}$ of
 $(\gamma_{0,\varepsilon}^{(N)})_\varepsilon$ and
 $(\gamma_{1,\varepsilon}^{(N)})_\varepsilon$ on $I'_{\theta_1}$.
For the induction step we specify $(b_{1,\varepsilon}^{(N)})_\varepsilon,
(b_{2,\varepsilon}^{(N)})_\varepsilon$ on $I'_{\theta_2}$ as follows: since
$(\overline{a}_{j,\varepsilon}^{(1)})_\varepsilon$ satisfies \eqref{Ell:ps}
the matrix in \eqref{CoeffMatrix} is invertible on $I'_{\theta_2}$
and thus the system
\begin{align*}
-\widetilde{\gamma}_{1,\varepsilon}^{(N)}
 &=  \overline{b}_{1,\varepsilon}^{(N)} + \overline{b}_{2,\varepsilon}^{(N)} \\
-\widetilde{\gamma}_{0,\varepsilon}^{(N)}
 &=  \overline{b}_{1,\varepsilon}^{(N)} \overline{a}_{2,\varepsilon}^{(1)}
  + \overline{a}_{1,\varepsilon}^{(1)}  \overline{b}_{2,\varepsilon}^{(N)}
\end{align*}
is uniquely solvable for $(\overline{b}_{j,\varepsilon}^{(N)})_\varepsilon$ 
in $\MhgSym[1-N,N]{1,N}$ on $I'_{\theta_2}$, $j=1,2$.

We write $B_{j,\psi}^{(N)}$ for the $\psi$-pseudodifferential operator
with polyhomogeneous generalized symbol
$(b_{1,\varepsilon}^{(N)})_\varepsilon:=(\widetilde{h}_\varepsilon
\overline{b}_{j,\varepsilon}^{(N)})_\varepsilon$ and set
$L_{j,\psi}^{(N+1)} := (L_{j}^{(N)} + B_j^{(N)})_\psi$. Then the following is valid
\begin{align*}
L_{1,\psi}^{(N+1)} L_{2,\psi}^{(N+1)} - L_{\psi}
& = \bigl( L_{1}^{(N)} + B_{1}^{(N)} \bigr)_\psi \bigl( L_{2}^{(N)}
+ B_{2}^{(N)} \bigr)_\psi - L_{\psi}  \\
&=  \Gamma_{0,\psi}^{(N)} + \Gamma_{1,\psi}^{(N)}(\partial_z)_\psi
 + \bigl( B_{1}^{(N)} + B_{2}^{(N)} \bigr)_\psi \bigl( \partial_z \bigr)_{\psi}  \\
&\quad + \operatorname{OP}_{\psi} \bigl( \bigl(\varepsilon \partial_z
 b_{2,\varepsilon}^{(N)}\bigr)_\varepsilon \bigr)
+ A_{1,\psi}^{(N)} B_{2,\psi}^{(N)} + B_{1,\psi}^{(N)} A_{2,\psi}^{(N)}
 + B_{1,\psi}^{(N)} B_{2,\psi}^{(N)}.
\end{align*}
Indeed we have an ($N$+1)-th order approximation of the  form
\begin{equation*}
L_{1,\psi}^{(N+1)} L_{2,\psi}^{(N+1)} - L_{\psi}
= \Gamma_{0,\psi}^{(N+1)} + \Gamma_{1,\psi}^{(N+1)}(\partial_z)_\psi
\end{equation*}
and $\Gamma_{0,\psi}^{(N+1)}$ and $\Gamma_{1,\psi}^{(N+1)}$ are the
$\psi$-pseudodifferential operators with generalized symbols
$(\gamma_{0,\varepsilon}^{(N+1)})_\varepsilon \in \MphgSym[1-N,N+1]{1,N+1}$
and $(\gamma_{1,\varepsilon}^{(N+1)})_\varepsilon \in \mathcal{N}_{\mbox{$\mathcal{S}$}^1}$
on $I'_{\theta_1}$, respectively. This completes the induction step.
\end{proof}

\section{The generalized infinite wave front set}\label{sec:WFInfty}

In this section we will discuss an alternative description of microlocality
and regularity theory in Colombeau algebras of generalized functions.
 We refer to \cite{NPS:98, GaHo:05, GaHo:06} for more details on the commonly
used notion of a generalized wave front set. Therein the generalized wave front
set is explained by replacing the standard $\mathcal{C}^{\infty}$-regularity
by $\mathcal{G}^{\infty}$-regularity. In the same manner we can reformulate
local $H^{\infty}$-regularity used in the Sobolev based wave front set in terms
of $\mathcal{G}^{\infty}$-regularity.

In the case of generalized pseudodifferential operators that satisfies logarithmic
slow scale estimates this is a straightforward modification of the regularity
results in \cite{GaHo:05} within the Colombeau theory in a Sobolev based context
as in \cite{Wunsch:08}. But the situation changes dramatically when working with
$\psi$-pseudodifferential operators with symbols of log-type.

For this reason we will introduce the notion of generalized infinite wave front
set describing negligibility of a generalized function at infinite points.
 For further studies about the infinite wave front set and the more refined
semiclassical wave front set we refer to \cite{CoMa:03, Alexandrova:08} and
 \cite{Martinez:02, EvZw, GuSt:10}.

\subsection{Microlocal behavior at infinity}

Here we introduce a suitable notion of asymptotic negligibility of a generalized
function $u \in \mathcal{G}_{2,2} (\mathbb{R}^{n})$ with respect to certain
regularities on the phase space.

To do so we use the following notation which is similar to \cite{CoMa:03}.
 For a non-zero vector $\xi_0 \in \mathbb{R}^n$ we write for the projection
onto the unit sphere $\xi_0/|\xi_0|$. Moreover, given such $\xi_0$ we say
that $\Gamma_{\infty \xi_0} \subset \mathbb{R}^n$ is a conic neighborhood of
the direction $\xi_0/|\xi_0|$ if $\Gamma_{\infty \xi_0}$ is the intersection
of the complement of some open ball centered at the origin with an open cone
containing the direction.

Furthermore, for $(x_0,\xi_0)$ in $T^*\mathbb{R}^{n} \setminus 0$ we say that
a generalized symbol $p:=(p_\varepsilon)_\varepsilon \in \MphgSym[m,0]{\nu,0}$
is elliptic at $(x_0,\infty \xi_0)$ or also elliptic at infinity at $(x_0,\xi_0)$,
if there is an open neighborhood $U$ of $x_0$ and a constant $C > 0$ such that
\begin{equation*}
|p_\varepsilon(x,\xi)| \geq C \langle \xi \rangle^m \quad
\forall (x,\xi) \in U \times \Gamma_{\infty \xi_0} \text{ as } \varepsilon \to 0.
\end{equation*}
We denote by ${\mathop{\rm Ell}}^{\rm i}(p)$ the set of all points
 $(x_0,\xi_0) \in T^* \mathbb{R}^n \setminus 0$ where $p$ is elliptic at infinity.

Then the generalized infinite wave front set is defined as follows.

\begin{definition} \rm
For $u \in \mathcal{G}_{2,2}(\mathbb{R}^n)$ we denote by
${\mathop{\rm WF}}^{\rm i}(u) \subset T^* \mathbb{R}^n \setminus 0$
the generalized infinite wave front set of $u$ which is characterized as follows.
 We say that $(x_0,\xi_0) \notin {\mathop{\rm WF}}^{\rm i}(u)$ at infinity,
denoted by $(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i}(u)$, if
there exists $\chi := (\chi_\varepsilon)_\varepsilon \in \MphgSym[\infty,0]{0,0}$
elliptic at $(x_0,\infty \xi_0)$ so that
\begin{equation*}
\operatorname{OP}_{\psi}(\chi) u = 0 \quad \text{in }
\mathcal{G}_{2,2}(\mathbb{R}^n).
\end{equation*}
\end{definition}

\begin{remark} \rm
Let $(u_\varepsilon)_\varepsilon \in \mathcal{M}_{H^{\infty}}$ and suppose that
there exists $\chi_{m} := (\chi_{m,\varepsilon})_\varepsilon \in \MphgSym[m,0]{0,0}$
for some $m \in \mathbb{R}$ elliptic at $(x_0,\infty \xi_0)$ with
\begin{equation*}
\forall q \in \mathbb{N}: \; \| \operatorname{OP}_{\psi,\varepsilon}
(\chi_{m,\varepsilon}) u_\varepsilon \|_{L^2(\mathbb{R}^{n})}
= \mathcal O (\varepsilon^q) \quad \text{as } \varepsilon \to 0.
\end{equation*}
Using \cite[Proposition 3.4]{Garetto:05b} then the above line is equivalent
to $\operatorname{OP}_{\psi}(\chi_{m}) u  = 0$ in $\mathcal{G}_{2,2}$.
\end{remark}

Furthermore we introduce the notion of an infinite conic support to describe
singularities of $\psi$-pseudodifferential operators. Therefore let
$p:=(p_\varepsilon)_\varepsilon \in \MphgSym[m,k]{\nu,l}$ be the generalized
symbol of $P_\psi$. Then the infinite conic support of $P_\psi$ is the
set $\operatorname{cone\,supp}^{\rm i}(p) \subset T^* \mathbb{R}^n \setminus 0$
 and is defined as the complement (in $T^* \mathbb{R}^n \setminus 0$) of
the set of points $(x_0,\xi_0) \in T^* \mathbb{R}^n \setminus 0$ such that
there exist an open neighborhood $U$ of $x_0$, a conic open neighborhood
 $\Gamma_{\xi_0}$ of the direction $\xi_0$ and a constant $K > 0$ such that
the following is satisfied:
\begin{equation}\label{Iconesupp}
\forall \alpha, \beta \in \mathbb{N}^{n} \ \forall N \in \mathbb{N}: \;
|\partial^{\alpha}_{\xi} \partial^{\beta}_{x} p_\varepsilon(x,\xi)|
\langle \xi \rangle^{-m+N+|\alpha|} = \mathcal{O}(\varepsilon^{N})
\quad \text{as } \varepsilon \to 0,
\end{equation}
uniformly in $(x,\xi) \in U \times \bigl(\Gamma_{\xi_0}
\cap \{ \xi \in \mathbb{R}^n : |\xi| \geq K \} \bigr)$.
This is again a condition on the behavior for a generalized symbol at infinity
and we therefore say that $(x_0, \xi_0) \notin \operatorname{cone\,supp}^{\rm i} (p)$
at infinity and write $(x_0,\infty \xi_0) \notin \operatorname{cone\,supp}^{\rm i} (p)$
whenever \eqref{Iconesupp} is fulfilled.

The idea here is that $\operatorname{cone\,supp}^{\rm i}(p)^c$ are the directions
on the phase space in which $P_\psi$ annihilates singularities as they are
contained in $\mathcal{N}_{\mbox{$\mathcal{S}$}^m}(U \times \Gamma)$.
To give a connection to these two notions of wave front sets we state the
following theorem.

\begin{theorem}\label{thm:InftyWF}
Given $u \in \mathcal{G}_{2,2}$ and $P_\psi$ a $\psi$-pseudodifferential
 operator with generalized symbol
$p=(p_\varepsilon)_\varepsilon \in \MphgSym[m,k]{\nu,l}$. Then the following
statement is valid
\begin{equation*}
{\mathop{\rm WF}}^{\rm i}(P_\psi u) \subset {\mathop{\rm WF}}^{\rm i}(u)
\cap \operatorname{cone\,supp}^{\rm i}(p)
\end{equation*}
and we say that the $\psi$-pseudodifferential operator $P_\psi$ is microlocal at infinity.
\end{theorem}

We remark that most of the properties of the infinite wave front set of a
generalized function in $\mathcal{G}_{2,2}$ can be derived from the theorem
of Calder\'{o}n-Vaillancourt for the class pseudodifferential operators with
 symbols in $S^0_{0,0}$.

\begin{proof}
We first show the inclusion relation ${\mathop{\rm WF}}^{\rm i}(P_\psi u)
\subset \operatorname{cone\,supp}^{\rm i}(p)$.
Therefore we let $(x_0,\xi_0) \in T^* \mathbb{R}^n \setminus 0$ such that
$(x_0,\infty \xi_0) \notin \operatorname{cone\,supp}^{\rm i}(p)$ which in turn
implies that $(p_\varepsilon)_\varepsilon$ is in
 $\mathcal{N}_{\mbox{$\mathcal{S}$}^m} (U \times \Gamma_{\xi_0})$ for some open
 neighborhood $U$ of $x_0$ and some conic open neighborhood $\Gamma_{\xi_0}$
of $\xi_0$. More precisely, we have that $\exists K_1 > 0$ such that
for all $\alpha, \beta \in \mathbb{N}^{n}$ and all $N \in \mathbb{N}$
\begin{equation*}
 |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} p_\varepsilon(x,\xi)|
\langle \xi \rangle^{-m+N+|\alpha|} = \mathcal{O}(\varepsilon^{N})
 \quad \text{as } \varepsilon \to 0,
\end{equation*}
uniformly in $(x,\xi) \in U \times \Gamma_{\infty \xi_0}$ with
$\Gamma_{\infty \xi_0} = \Gamma_{\xi_0}
\cap \{ \xi \in \mathbb{R}^n : |\xi| \geq K_1 \}$.
We will now construct a symbol
$\bigl(\chi^{(-m)}_{J,K} \bigr)_\varepsilon \in \MphgSym[-m,0]{0,0}$
elliptic at $(x_0,\infty \xi_0)$ such that
\begin{equation*}
\forall q \in \mathbb{N}: \ \| \operatorname{OP}_{\psi,\varepsilon}(\chi^{(-m)}_{J,K}) P_{\psi,\varepsilon}
u_\varepsilon \|_{L^2(\mathbb{R}^n)}
= \mathcal O (\varepsilon^q) \quad \text{as } \varepsilon \to 0.
\end{equation*}
For that reason let $\phi \in \mathcal{C}^{\infty}_{\rm c}(\mathbb{R}^{n})$
such that $\phi(z) = 1$ for $|z| \leq 1/2$ and $\phi(z) = 0$ for $|z| \geq 1$.
 Now, given $(x_0,\xi_0) \in T^* \mathbb{R}^n \setminus 0$ we define for fixed
$J >0$ the function
\begin{equation*}
\lambda_{J}^{(-m)}(x,\xi) := \phi \Bigl(\frac{x-x_0}{J} \Bigr)
\phi\Big( \Bigl\{ \frac{\xi}{|\xi|} - \frac{\xi_0}{|\xi_0|} \Bigr\} \frac{1}{J} \Big)
|\xi|^{-m}   \quad \quad \text{for } \xi \neq 0.
\end{equation*}
Further for some fixed $K >0$ let
\begin{equation}\label{ConstructSymb}
\chi^{(-m)}_{J,K}(x,\xi) := (1-\phi)\Bigl( \frac{\xi}{2K} \Bigr)
\lambda_{J}^{(-m)}(x,\xi).
\end{equation}
Then for $J,K > 0$ fixed $(\chi^{(-m)}_{J,K}(x,\xi) )_\varepsilon
 \in \MphgSym[-m,0]{0,0}$ is elliptic at $(x_0,\infty \xi_0)$ and supported in
\[
|x - x_0| \leq J,  \quad \Bigl| \frac{\xi}{|\xi|} - \frac{\xi_0}{|\xi_0|} \Bigr|
\leq J,  \quad |\xi| \geq K.
\]
So $\chi^{(-m)}_{J,K}$ is a cut-off in a conic neighborhood of $\xi_0$ and
is supported in a cone of directions near $\xi_0$. In particular we have
$\chi^{(-m)}_{J,K} \# p_\varepsilon = 0$ for $|\xi| \leq K$. We now choose
$J, K>0$ such that
\begin{equation*}
\operatorname{supp} (\chi^{(-m)}_{J,K}) \subset U \times \Gamma_{\infty \xi_0}.
\end{equation*}
Then by the $L^2$-boundedness theorem of Calder\'{o}n-Vaillancourt for symbols
of class $S^0_{0,0}$, see \cite[Chapter 13, Theorem 1.3]{Taylor:81},
there are constants $j_0, j_1 \in \mathbb{N}$ and $C > 0$ each of which
depend on $n$ but independent of $\varepsilon > 0$ such that
\begin{align*}
\| \operatorname{OP}_{\psi,\varepsilon} (\chi^{(-m)}_{J,K})
P_{\psi,\varepsilon} u_{\varepsilon} \|_{L^2}
&= \| \operatorname{OP}_{\psi,\varepsilon}(\chi^{(-m)}_{J,K} \#_2 p_\varepsilon)
 u_\varepsilon \|_{L^2} \\
&\leq C \sup_{|\alpha| \leq j_0, |\beta| \leq j_1} \|
\partial^{\alpha}_{\xi} \partial_{x}^{\beta} (\chi^{(-m)}_{J,K} \#_2 p_\varepsilon)
\|_{L^\infty(\mathbb{R}^{2n})} \| u_\varepsilon \|_{L^2}
\end{align*}
where in the last step we performed the rescaling by means of the asymptotic
expansion of the second kind for the composition formula of the pseudodifferential
operators expressed by using the notation $\#_2$,
see Subsection~\ref{subsec:AE2}. Moreover we used the following estimation
\begin{align*}
|\partial^{\alpha}_\xi \partial_x^{\beta} (\chi^{(-m)}_{J,K}
 \#_2 p_\varepsilon  (x,\varepsilon \xi))|
& = |\varepsilon^{|\alpha|} (\partial^{\alpha}_\xi \partial_x^{\beta}
 \chi^{(-m)}_{J,K} \#_2 p_\varepsilon)(x,\varepsilon \xi)| \\
&\leq \varepsilon^{|\alpha|} \sup_{(x,\xi) \in \mathbb{R}^{2n}}
 |(\partial^{\alpha}_\xi \partial_x^{\beta}  \chi^{(-m)}_{J,K}
\#_2 p_\varepsilon)(x, \varepsilon \xi)| \\
&= \varepsilon^{|\alpha|} \| \partial^{\alpha}_\xi \partial_x^{\beta}
\chi^{(-m)}_{J,K} \#_2 p_\varepsilon \|_{L^\infty (\mathbb{R}^{2n})}.
\end{align*}
Now using the fact that $(p_\varepsilon)_\varepsilon
\in \mathcal{N}_{\mbox{$\mathcal{S}$}^m} (U \times \Gamma_{\xi_0})$
one has for every $q \in \mathbb{N}$ that
\begin{equation*}
\sup_{|\alpha| \leq j_0, |\beta| \leq j_1}
\| \partial^{\alpha}_{\xi} \partial_{x}^{\beta} (\chi^{(-m)}_{J,K} \#_2
 p_\varepsilon) \|_{L^\infty(\mathbb{R}^{2n})}
= \mathcal O (\varepsilon^q) \quad \text{ as } \varepsilon \to 0
\end{equation*}
by Proposition~\ref{prop:Product} and we deduce that
$(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i}(P_\psi u)$
which completes the first part of the proof.

We continue the proof by showing the following inclusion:


\noindent\textbf{Claim.}
${\mathop{\rm WF}}^{\rm i} (P_\psi u ) \subset {\mathop{\rm WF}}^{\rm i}(u)$.
\\
\emph{Proof of the Claim.}
We take $(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i}(u)$ and let $U$
be some open neighborhood of $x_0$ and $\Gamma_{\xi_0}$ a conic neighborhood
of $\xi_0$. Then, for some $j \in \mathbb{N}$ there exists
$\chi_{j} := (\chi_{j,\varepsilon})_\varepsilon \in \MphgSym[j,0]{0,0}$
elliptic at $(x_0,\infty \xi_0)$ such that for some constant $K_1 > 0$ we have 
\begin{equation}\label{InftyEll}
\exists C>0 \ \exists \eta \in (0,1]: \  |\chi_{j,\varepsilon}(x,\xi)| \geq C \langle \xi \rangle^j \quad
\forall \varepsilon \in (0,\eta]
\end{equation}
uniformly on $U \times \Gamma_{\infty {\xi_0}}$ with
$\Gamma_{\infty {\xi_0}}=\Gamma_{\xi_0} \cap \{(x,\xi)  :  |\xi| \geq K_1 \}$ and
for all $q \in \mathbb{N}$,
\[ %\label{NotInFSu}
 \| \operatorname{OP}_{\psi,\varepsilon}(\chi_{j,\varepsilon})
 u_\varepsilon \|_{L^2}  =  \mathcal O (\varepsilon^q) \quad \text{as }
\varepsilon \to 0 .
\]
Then the assumptions in Lemma~\ref{lem:psiparametrix} from the next section
are satisfied on $U \times \Gamma_{\xi_0}$ and there is a
$\widetilde{\chi}_{-j}:= (\widetilde{\chi}_{-j,\varepsilon})_\varepsilon
\in \MphgSym[-j,0]{0,0}$ such that
\begin{equation*}
\operatorname{OP}_{\psi} (\widetilde{\chi}_{-j}) \operatorname{OP}_\psi (\chi_{j})
 = I_{\psi} + R_{\psi} \quad \text{ on } U \times \Gamma_{\xi_0}
\end{equation*}
where the generalized symbol $(r_\varepsilon)_\varepsilon$ of
$R_\psi$ is in $\MSym[-N,N]{0,0}$ on $U \times \Gamma_{\xi_0}$ for every
$N \in \mathbb{N}$.

As in \eqref{ConstructSymb} one constructs a generalized symbol
 $(\kappa_{j-m})_\varepsilon \in \MphgSym[j-m,0]{0,0}$ elliptic at
$(x_0, \infty \xi_0)$ such that the $\operatorname{supp}(\kappa_{j-m})$
is contained in the set where \eqref{InftyEll} is valid. We then write
\begin{align*}
&\operatorname{OP}_{\psi,\varepsilon} (\kappa_{j-m} \#_2 p_\varepsilon)\\
&= \operatorname{OP}_{\psi,\varepsilon} (\kappa_{j-m} \#_2
 p_\varepsilon \#_2 \widetilde{\chi}_{-j,\varepsilon})
\operatorname{OP}_{\psi,\varepsilon} (\chi_{j,\varepsilon})
- \operatorname{OP}_{\psi,\varepsilon} (\kappa_{j-m} \#_2
p_\varepsilon \#_2 r_\varepsilon).
\end{align*}
Concerning the first term on the right hand side we rescale and obtain
by the Calder\'{o}n-Vaillancourt theorem that there are constants
$j_0,j_1 \in \mathbb{N} $ and $C >0$ such that
\begin{align*}
&\| \operatorname{OP}_{\psi,\varepsilon} (\kappa_{j-m}
\#_2 p_\varepsilon \#_2 \widetilde{\chi}_{-j,\varepsilon})
\operatorname{OP}_{\psi,\varepsilon} (\chi_{j,\varepsilon})
 u_\varepsilon \|_{L^2}  \\
& \leq C \sup_{|\alpha| \leq j_0, |\beta| \leq j_1}
\| \partial_{\xi}^{\alpha} \partial_x^{\beta} \kappa_{j-m}
\#_2 p_\varepsilon \#_2 \widetilde{\chi}_{-j,\varepsilon})
 \|_{L^{\infty}(\mathbb{R}^{2n})} \| \operatorname{OP}_{\psi,\varepsilon}
 (\chi_{j,\varepsilon}) u_\varepsilon \|_{L^2}
\end{align*}
and the latter expression is $\mathcal{O}(\varepsilon^q)$ as
$\varepsilon \to 0$ for every $q \in \mathbb{N}$ by assumption.
Similarly we obtain for every $q \in \mathbb{N}$:
\begin{eqnarray*}
\| \operatorname{OP}_{\psi,\varepsilon} (\kappa_{j-m}
 \#_2 p_\varepsilon \#_2 r_\varepsilon) u_\varepsilon \|_{L^2}
= \mathcal{O} (\varepsilon^q) \quad \text{as } \varepsilon \to 0
\end{eqnarray*}
since $(r_\varepsilon)_\varepsilon \in \mathcal{N}_{\mbox{$\mathcal{S}$}^0}$
on the support of $\kappa_{j-m}$. Therefore
 $(x_0, \infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i} (P_\psi u)$.
\end{proof}

As a next result we reformulate the infinite wave front set in terms
of the Fourier transform of a localized function.

For this purpose we introduce the space $\mathcal{G}_{\rm c}(\mathbb{R}^n)$
of compactly supported generalized functions consisting of those
$u \in \mathcal{G}(\mathbb{R}^n)$ such that for some $K \Subset \mathbb{R}^n$
 the restriction $u$ to $\mathbb{R}^n \setminus K$ is equal to $0$ as an element
of $\mathcal{G}(\mathbb{R}^n \setminus K)$. Note that for an open subset $\Omega$
of $\mathbb{R}^n$ the space $\mathcal{G}(\Omega)$ is defined by the space
$\mathcal{G}_{E}$ by setting $E= \mathcal{E}(\Omega)$ the space of smooth
functions on $\Omega$ topologized through the family of seminorms
$p_{K_n,j}(f):= \sup_{x \in K_n,|\alpha| \leq j} |\partial^{\alpha}f(x)|$
where $(K_n)_n$ is an exhausting sequence of compact subsets of $\Omega$.

\begin{proposition} \label{prop5.3}
Let $u \in \mathcal{G}_{\rm c}(\mathbb{R}^n)$. If
$(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i} (u)$ then there exists
 $\gamma \in \mathcal{C}^{\infty}_{\rm c}(\mathbb{R}^n)$ with $\gamma(x_0) \neq 0$
and a conic neighborhood $\Gamma$ of $\xi_0$ such that for all
$N,q \in \mathbb{N}$ there exists $C >0$ satisfying
\begin{equation}\label{EquivIWF}
| \mathcal{F}(\gamma u_\varepsilon) (\xi) |
\leq C \varepsilon^q \langle \xi \rangle^{-N} \quad \text{as } \varepsilon \to 0
\end{equation}
for all $\xi \in \Gamma$ with $|\xi| \geq K/\varepsilon$ for some
$K > 0$ independent of $\varepsilon$.
\end{proposition}

Note that our notion of regularity is derived from the infinite wave
front set and therefore differs from \cite[Section 6]{HoMoe:04}
and \cite[Sections 2,3]{GaHo:05}. It is also different to
\cite[Definitions 3.13 and 3.14, Section 3.2.2]{NPS:98}
if one replaces there the condition of $\mathcal{G}^{\infty}$ rapid decrease
by rapid decrease. We remark that a generalized function
 $u \in \mathcal{G}(\mathbb{R}^n)$ is of rapid decrease if it has a
representative $(u_\varepsilon)_\varepsilon$ with the property:
\begin{equation*}
\begin{gathered}
\exists N \in \mathbb{N} \ \forall p \in \mathbb{N} \ \exists C>0 \ \exists \eta \in (0,1]:\\
  | u_{\varepsilon}(x) | \leq C \varepsilon^{-N}  \langle x \rangle^{-p}
\quad \forall x \in \mathbb{R}^n, \varepsilon \in (0,\eta].
\end{gathered}
\end{equation*}

\begin{proof}
The following proof is similar to \cite[Proposition 7.4]{GiSj:94}.
First note that $\mathcal{G}_{\rm c} \subset \mathcal{G}_{2,2}$.
Since $(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i}(u)$ we can
find a $\chi:=(\chi_{\varepsilon})_\varepsilon \in \MphgSym[-m,0]{0,0}$
elliptic at $(x_0,\infty \xi_0)$ such that $\operatorname{OP}_\psi (\chi) u = 0$
in $\mathcal{G}_{2,2}$ for some $m \in \mathbb{R}$. Furthermore we let
$\gamma \in \mathcal{C}^{\infty}_{\rm c} (\mathbb{R}^n)$ with $\gamma (x_0) \neq 0$.
Then there exists a symbol
 $\phi \in S^0 (\mathbb{R}^n), \operatorname{supp}(\phi) \subset
\Gamma_{\infty \xi_0}= \Gamma_{\xi_0} \cap \{ \xi: |\xi| \geq K \}$
for some $K>0$ and $\phi(t \xi) = 1$ for $t \geq 1, \xi \in \Gamma_{\infty \xi_0}$
so that
\begin{equation*}
\phi_\psi (D) \gamma (x) = A_\psi \operatorname{OP}_\psi(\chi) + R_\psi
\end{equation*}
where $A_\psi$ and $R_\psi$ is a $\psi$-pseudodifferential operator with
generalized symbol in $\MSym[m,0]{0,0}$ and $\mathcal{N}_{\mbox{$\mathcal{S}$}^0}$,
respectively. Then by assumption we deduce that $\phi_\psi (D) \gamma u = 0$
in $\mathcal{G}_{2,2}$. In particular we deduce that $\phi_\psi (D) \gamma u = 0$
in $\mathcal{G}_{\mathscr{S}}$ since
$\mathcal{G}_{\mathscr{S}} \hookrightarrow \mathcal{G}_{2,2}$,
see \cite[Proposition 3.5]{Garetto:05b}.
Using the fact that the Fourier transform is an isomorphism on
$\mathcal{G}_{\mathscr{S}}$ we obtain
$\xi \mapsto (\phi(\varepsilon \xi) \widehat{\gamma u}_\varepsilon (\xi))_\varepsilon$
is in $\mathcal{N}_{\mathscr{S}}$ under consideration of the scaling.
 Hence we also have that
\[
\forall N \in \mathbb{N} \ \forall q \in \mathbb{N}: \; \| \langle \xi / \varepsilon \rangle^N \phi (\xi) \widehat{\gamma u}_\varepsilon
(\xi / \varepsilon) \|_{L^{\infty}} = \mathcal{O} (\varepsilon^q)
\]
showing \eqref{EquivIWF}.
\end{proof}

We also study the behavior of the infinite wave front set of a function under
the action of $\psi$-pseudodifferential operators that are elliptic at infinity.
%
\begin{theorem} \label{thm5.4}
Let $P_\psi$ be a $\psi$-pseudodifferential operator with symbol
$p:=(p_\varepsilon)_\varepsilon$ in $\MphgSym[m,0]{\nu,0}$ that is elliptic
at $(x_0, \infty \xi_0) \in T^* \mathbb{R}^n \setminus 0$.
Then the following holds:
\begin{equation*}
{\mathop{\rm WF}}^{\rm i} (u) \subset {\mathop{\rm WF}}^{\rm i}(P_\psi u)
\cup {\mathop{\rm Ell}}^{\rm i}(p)^{\rm c}.
\end{equation*}
\end{theorem}

\begin{proof}
Suppose $(x_0, \infty \xi_0)$ is not contained in the right hand side of the
claimed inclusion relation. Since
$(x_0, \infty \xi_0) \in {\mathop{\rm Ell}}^{\rm i}(p)$ the symbol
 $(p_\varepsilon)_\varepsilon$ is elliptic at $(x_0,\infty \xi_0)$
 and therefore there is an open neighborhood $U$ of $x_0$ and conic
neighborhood $\Gamma_{\xi_0}$ containing $\xi_0$ such that
\begin{equation*}
|p_\varepsilon(x,\xi)| \geq C \langle \xi \rangle^m \quad
\forall (x,\xi) \in U \times \Gamma_{\infty \xi_0} \text{ as } \varepsilon \to 0.
\end{equation*}
By Lemma~\ref{lem:psiparametrix} from the next section there exists
an approximative inverse $Q_\psi$ so that
\begin{equation}\label{IMLInverse}
I_\psi = Q_\psi P_\psi  + R_\psi  \quad \text{on } U \times \Gamma_{\xi_0}
\end{equation}
and $(x_0,\infty \xi_0) \notin \operatorname{cone\,supp}^{\rm i} (r)$
and $r:= (r_\varepsilon)_\varepsilon$ is the generalized symbol of
$R_\psi$. Hence $(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i}(R_\psi u)$
by Theorem~\ref{thm:InftyWF}.

Furthermore, since
$(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i} (P_\psi u)$
we obtain that
\[
(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i}(Q_\psi P_\psi u)
\subset {\mathop{\rm WF}}^{\rm i}(P_\psi u)
\]
and by \eqref{IMLInverse}
we deduce that $(x_0,\infty \xi_0) \notin {\mathop{\rm WF}}^{\rm i}(u)$.
\end{proof}

\subsection{Microlocal factorization}

In this section we use the notion of microlocal behavior at infinite points
of a given generalized function in $\mathcal{G}_{2,2}(\mathbb{R}^n)$ to give
a microlocal interpretation of Theorem~\ref{thm:Fact}. To do so let $I$ be a
conic subset of $\mathbb{R}^n \times (\mathbb{R}^n \setminus 0)$.
We then say that two generalized functions $u,v \in \mathcal{G}_{2,2}$
are microlocally equivalent at infinity on $I$ if and only if there exists $
(\chi_\varepsilon)_\varepsilon \in \MphgSym[\infty,0]{0,0}$ elliptic at
$(x_0,\infty \xi_0)$ for every $(x_0,\xi_0) \in I$ such that $\chi_\psi (u-v) = 0$
in $\mathcal{G}_{2,2}$.

Similar to \cite{Stolk:04} we introduce a subset of the phase space associated
to $I'_{\theta_1}$ which is given by
\begin{equation}\label{IMLRegion}
I_{\theta_1}:= \{ (t,y,\tau,\eta) \mid (y,\tau,\xi)
\in I'_{\theta_1}, \ |\zeta| \leq \sqrt{c_1} |\tau| \}
\end{equation}
and $c_1>0$ is the upper bound of the H\"{o}lder continuous coefficient $c(y)$.
 Note that in the following $I_{\theta_1}$ will serve as the adequate
space-frequency domain on which we establish the microlocal factorization
theorem at infinite points.

As already mentioned in the preceding section
$L_{j,\psi}= (\partial_z + A_j(y,D_t,D_x))_\psi$, $j=1,2$ is not a
 $\psi$-pseudodifferential operator on $\mathbb{R}^{n+1}$. To overcome this we
are going to introduce a microlocal cut-off $g_\psi(D_t,D_y)$ for $I_{\theta_1}$
such that
\begin{equation}\label{EquivIML}
g_\psi L_{j,\psi} u = L_{j,\psi} u \quad \text{microlocally at infinity on }
I_{\theta_1}
\end{equation}
where $u \in \mathcal{G}_{2,2}$ and $g_\psi L_{j,\psi}$ is a
$\psi$-pseudodifferential operator with generalized symbol in $\MSym[1,0]{1,0}$.
 Here the microlocal cut-off $g_\psi$ is constructed in the following way.

First let $K_0, K_1$ be some fixed constants so that $0<K_0<K_1<\infty$.
Then $g \in \mathcal{C}^{\infty} (\mathbb{R}^{n+1})$, $0 \leq g \leq 1$ is
defined by
\begin{equation*}
g(\tau,\eta) := \begin{cases}
0, & |\zeta| \geq 3 \sqrt{c_1} |(\tau,\xi)| \text{ or } |(\tau,\xi)| \leq K_0\\[3pt]
\tilde{\sigma} (\tau,\xi), & |\zeta| \leq 2 \sqrt{c_1} |(\tau,\xi)|\\[3pt]
\frac{\tilde{\sigma} (\tau,\xi)}{1 + e^{\frac{1}{|\zeta|/|(\tau,\xi)|
- 2\sqrt{c_1}} - \frac{1}{3\sqrt{c_1} - |\zeta|/|(\tau,\xi)|}}},
 & 2\sqrt{c_1}|(\tau,\xi)| < |\zeta| < 3 \sqrt{c_1}|(\tau,\xi)|
\end{cases}
\end{equation*}
and the function $\tilde{\sigma}$ is a cut-off near the origin given by
\begin{equation*}
\tilde{\sigma}(\tau,\xi) := \begin{cases}
0, & |(\tau,\xi)| \leq K_0\\
1, & |(\tau,\xi)| \geq K_1\\
\frac{1}{{1 + e^{\frac{K_1-K_0}{|(\tau,\xi)|
- K_0} - \frac{K_1-K_0}{K_1 - |(\tau,\xi)|}}}}, & K_0 < |(\tau,\xi)| < K_1.
\end{cases}
\end{equation*}
Then $g \in S^0(\mathbb{R}^{n+1})$ has the form shown in Figure \ref{fig2}:

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.9]{fig2}
\end{center}
\caption{}
\label{fig2}
\end{figure}


 Moreover, $g_\psi L_{j,\psi}$ acts as a $\psi$-pseudodifferential operator
in $(t,x,z)$ with generalized symbol in $\MSym[1,0]{1,0}$ which can be shown
by an adaption of \cite[Theorem 18.1.35]{Hoermander:3}.

Furthermore \eqref{EquivIML} is satisfied, since
$\operatorname{cone\,supp}^{\rm i}(g - 1) \cap I_{\theta_1} = \emptyset$.
In the same manner one shows that
$g_\psi L_{1,\psi} L_{2,\psi} = L_{1,\psi} L_{2,\psi}$ microlocally at infinity
on $I_{\theta_1}$.

Summarizing the observations from above we obtain the following main theorem:

\begin{theorem}\label{thm:IMLFact}
Let $L_\psi$ and $I_{\theta_1}$ be as in \eqref{GovOp} and \eqref{IMLRegion}.
Then the operator $L_\psi$ can be factorized into a product of two first-order
 $\psi$-pseudodifferential operators as follows
\begin{equation*}
L_\psi = L_{1,\psi} L_{2,\psi} \quad \text{microlocally at infinity on} \ I_{\theta_1}
\end{equation*}
where $L_{j,\psi} = \bigl(\partial_z + A_j\bigr)_\psi$ and
 $A_{j,\psi}$ is the $\psi$-pseudodifferential operator as in
Theorem~\ref{thm:Fact}.
\end{theorem}

\section{Microlocal diagonalization for \texorpdfstring{$L_\psi$}{L-psi}}\label{sec:Diag}

The main issue in this section is to diagonalize the microlocal equation
 $L_\psi U = F$ using the refined factorization theorem~\ref{thm:IMLFact}.
In detail, we will rewrite the equation
\begin{equation*}
L_{\psi} U = F \quad \text{microlocally at infinity on } I_{\theta_1}
\end{equation*}
into an equivalent system of the form
\begin{equation*}
\big( \partial_z - i B_{\pm}(x,z,D_t,D_x) \big)_\psi u_{\pm}
 =   f_{\pm}  \quad \text{microlocally at infinity on } I_{\theta_1}.
\end{equation*}
To show this we will discuss a different approach to the one given by Stolk
in \cite{Stolk:04} as it turns out that the factorization theorem
already contains all the ingredients for the diagonalization.
Let us begin with the following lemma which was already used in the previous section.

\begin{lemma}\label{lem:psiparametrix}
Let $m\in \mathbb{R}$ and $P_\psi$ be a polyhomogeneous $\psi$-pseudodifferential
operator whose generalized symbol is given by
$(p_\varepsilon)_\varepsilon \dot{\sim}
\sum_{j=0}^{\infty}( \varepsilon^j \overline{p}_{m-j,\varepsilon})_\varepsilon$
in $\MSym[m,0]{\nu,0}$ on $I'_{\theta_1}$ for some
$(\overline{p}_{m-j,\varepsilon})_\varepsilon \in \MhgSym[m-j,0]{\nu,\nu j}$ on
$I'_{\theta_1}$, $j \geq 0$. Furthermore suppose that the principal
symbol $(\overline{p}_{m,\varepsilon})_\varepsilon$ satisfies an estimate of
the form
\begin{equation}\label{StrongEll}
\exists C >0 \ \exists \eta \in (0,1] : \; |\overline{p}_{m,\varepsilon}(y,\tau,\xi)|
\geq C |(\tau,\xi)|^{m} \quad \text{on } I'_{\theta_1}, \; \varepsilon \in (0,\eta].
\end{equation}
Then there exists a $\psi$-pseudodifferential operator $Q_\psi$ with
generalized symbol in $\MphgSym[-m,0]{\nu,0}$ such that the following is valid
\begin{equation}\label{psiparametrix}
Q_\psi P_\psi = I_\psi + R_\psi \quad \text{ on } I'_{\theta_1},
\end{equation}
where $I$ is the identity and the generalized symbol of
$R_\psi$ is in $\mathcal{N}_{\mbox{$\mathcal{S}$}^0}$ on $I'_{\theta_1}$.
More precisely, the polyhomogeneous generalized symbol
$(q_\varepsilon)_\varepsilon$ of $Q_\psi$ is written in terms of its
asymptotic expansion as
\begin{equation}\label{AE:parametrix}
(q_\varepsilon)_\varepsilon \dot{\sim} \sum_{k \geq 0}
\bigl(\varepsilon^k \overline{q}_{-m-k,\varepsilon}\bigr)_\varepsilon \quad \text{in }
\MSym[-m,0]{\nu,0} \text{ on } I'_{\theta_1}
\end{equation}
for some $(\overline{q}_{-m-k,\varepsilon})_\varepsilon
\in \MhgSym[-m-k,0]{\nu,\nu k} \text{ on } I'_{\theta_1},\ k \geq 0$.
\end{lemma}

\begin{proof}
The proof follows the classical arguments given in
 \cite[Chapter 2]{Kumano-go:81} for $\varepsilon$-dependent symbols.
We construct the generalized symbol of $Q_\psi$ by means of its asymptotic expansion.
Therefore we will recursively define symbols
$(\overline{q}_{-m-k,\varepsilon})_\varepsilon \in \MhgSym[-m-k,0]{\nu,\nu k}$ on
$I'_{\theta_1}$, $k \geq 0$, so that the symbol of $Q_\psi$ is given by
\eqref{AE:parametrix} and satisfies \eqref{psiparametrix}.

Because of \eqref{StrongEll} we may define for $(y,\tau,\xi) \in I'_{\theta_1}$
and some $\eta \in (0,1]$,
\begin{equation*}
\overline{q}_{-m,\varepsilon}(y,\tau,\xi) :=
\begin{cases}
\overline{p}_{m,\varepsilon}(y,\tau,\xi)^{-1}, & \varepsilon \in (0,\eta]\\
0, & \text{otherwise.}
\end{cases}
\end{equation*}
Concerning the asymptotic behavior of the derivatives of
$\overline{q}_{-m,\varepsilon}$ we use the Leibniz rule and obtain that
there exists $C > 0$ such that for every $\varepsilon > 0$ sufficiently small,
\begin{align*}
|\partial^{\alpha}_{(\tau,\xi)} \partial^{\beta}_{y} \overline{q}_{-m,\varepsilon}|
&= \bigl| \sum_{\substack{\alpha_1 + \ldots + \alpha_\mu = \alpha \\ \beta_1
 + \ldots + \beta_\mu = \beta} } \left( \partial^{\alpha_1}_{(\tau,\xi)}
  \partial^{\beta_1}_{y} \overline{p}_{m,\varepsilon} \right) \ldots
 \left( \partial^{\alpha_\mu}_{(\tau,\xi)} \partial^{\beta_\mu}_{y}
 \overline{p}_{m,\varepsilon} \right) \frac{1}{\overline{p}_{m,\varepsilon}^{1+\mu}}
 \bigr| \\
& \leq C \hspace{1pt} \omega_\varepsilon^{\nu |\beta|} |( \tau, \xi)|^{-m-|\alpha|}
\quad \text{on } I'_{\theta_1}
\end{align*}
from which we deduce that
$(\overline{q}_{-m,\varepsilon})_\varepsilon \in \MhgSym[-m,0]{\nu,0}$ on
$I'_{\theta_1}$. We now proceed by induction. Therefore we define
$(\overline{q}_{-m-k,\varepsilon})_\varepsilon$ on $I'_{\theta_1}$ by
\begin{equation}\label{ConstParametrix}
\overline{q}_{-m-k,\varepsilon}
:= - \Bigl\{ \sum_{|\gamma| + j + l =k,\, l <k}
\frac{1}{\gamma!} D^{\gamma}_\xi \overline{q}_{-m-l,\varepsilon}
\partial^{\gamma}_x \overline{p}_{m-j,\varepsilon} \Bigr\}
\frac{1}{\overline{p}_{m,\varepsilon}} \quad \quad  k \geq 1
\text{ on } I'_{\theta_1}.
\end{equation}
Assuming that $(\overline{q}_{-m-N,\varepsilon})_\varepsilon
\in \MhgSym[-m-N,0]{\nu,\nu N}(I'_{\theta_1})$ for $N < k$ then from
the construction given in \eqref{ConstParametrix} one easily verifies that
$(\overline{q}_{-m-k,\varepsilon})_\varepsilon \in
\MhgSym[-m-k,0]{\nu,\nu k}(I'_{\theta_1})$.

Moreover by Lemma~\ref{lem:AE1} we get the existence of a polyhomogeneous
generalized symbol $(q_\varepsilon)_\varepsilon \in \MphgSym[-m,0]{\nu,0}$
having the following asymptotic expansion
\begin{equation*}
(q_\varepsilon)_\varepsilon \dot{\sim} \sum_{k \ge 0}
\bigl(\varepsilon^k \overline{q}_{-m-k,\varepsilon}\bigr)_\varepsilon \quad \text{in }
\MSym[-m,0]{\nu,0} \text{ on } I_{\theta_1}'.
\end{equation*}
Let $Q_\psi$ be the $\psi$-pseudodifferential operator with generalized
symbol $(q_\varepsilon)_\varepsilon$. Then \eqref{AE:parametrix} is satisfied
and it remains to show equation \eqref{psiparametrix}. Concerning the asymptotic
expansion of $Q_\psi P_\psi$ one has for every $N \geq 1$
\begin{equation}\label{Check}
\begin{split}
\sum_{|\gamma| < N} \frac{\varepsilon^{|\gamma|}}{\gamma!} D^{\gamma}_\xi
q_\varepsilon \partial^{\gamma}_x p_\varepsilon
&=  \overline{q}_{-m,\varepsilon}
\overline{p}_{m,\varepsilon} + \sum_{k=1}^{N-1} \sum_{\substack{|\gamma|
+ l +j = k}} \frac{\varepsilon^{k}}{\gamma!} D^{\gamma}_\xi
\overline{q}_{-m-l,\varepsilon} \partial^{\gamma}_x  \overline{p}_{m-j,\varepsilon}\\
&\quad + \ \sum_{k \geq N}\sum_{\substack{|\gamma| +l+ j =k \\ |\gamma| < N}} \frac{\varepsilon^{k}}{\gamma!} D^{\gamma}_\xi \overline{q}_{-m-l,\varepsilon} \partial^{\gamma}_x \overline{p}_{m-j,\varepsilon} % \quad \text{on } J'_{\theta_1,\varepsilon},
\end{split}
\end{equation}
on $V_K :=I'_{\theta_1} \cap (\mathbb{R}^n \times \{ (\tau,\xi) :
|(\tau,\xi)| \geq K \})$ for some $K > 0$ independent of $\varepsilon$.
Here the second term on the right hand side vanishes by \eqref{ConstParametrix}
on $V_K$. Furthermore the last expression of \eqref{Check} is in
 $\mathcal{N}_{\mbox{$\mathcal{S}$}^0} (I'_{\theta_1})$ and
$\overline{q}_{-m,\varepsilon} \overline{p}_{m,\varepsilon} = 1$ on
$V_K$ which establishes the statement made in \eqref{psiparametrix}.
\end{proof}

Similarly one can construct a $\psi$-pseudodifferential operator
$\widetilde{Q}_\psi$ with generalized symbol in $\MphgSym[-m,0]{\nu,0}$ having
a representation of the form \eqref{AE:parametrix} and satisfies
\begin{equation*}
P_\psi \widetilde{Q}_\psi = I_\psi + \widetilde{R}_\psi \quad
\text{ on } I'_{\theta_1}
\end{equation*}
where the generalized symbol of $\widetilde{R}_\psi$ is in
$\mathcal{N}_{\mbox{$\mathcal{S}$}^0}$ on $I'_{\theta_1}$. Furthermore $Q_\psi$ is
related to $\widetilde{Q}_\psi$ in the following way
\begin{equation*}
Q_\psi = Q_\psi (P_\psi \widetilde{Q}_\psi)
= (Q_\psi P_\psi) \widetilde{Q}_\psi = \widetilde{Q}_\psi \quad \mod{} \operatorname{OP}_\psi \mathcal{N}_{\mbox{$\mathcal{S}$}^m} \text{ on } I'_{\theta_1}.
\end{equation*}

\subsection*{Diagonalization}

To get an idea we start rewriting the inhomogeneous equation $L_{\psi}U = F$
 into a first-order system with respect to the parameter $z$:
\begin{equation} \label{EquivSystem}
\Big[ \left(\text{Id} \, \partial_z\right)_{\psi} -
\begin{pmatrix} 0 & 1 \\ -A & 0  \end{pmatrix}_{\psi} \Big]
\begin{pmatrix} U \\ \left(\partial_z\right)_{\psi} U \end{pmatrix}
= \begin{pmatrix} 0 \\ F  \end{pmatrix}
\end{equation}
with $U,F \in \mathcal{G}_{2,2}(\mathbb{R}^{n+1})$, $\text{Id}$ the 2$\times$2
 identity matrix and $A_\psi$ the $\psi$-pseudodifferential operator with
generalized symbol $(a_\varepsilon)_\varepsilon$ as in \eqref{SymbolA}.
For brevity we will drop the identity matrix $\text{Id}$ in the equations
from now on. Hereafter we are going to reduce the operator in \eqref{EquivSystem}
to diagonal form. To make this notion rigorous we make the following arrangements.

First let
\begin{align*}
Q_\psi = \begin{pmatrix} Q^{11} & Q^{12} \\ Q^{21} & Q^{22}
\end{pmatrix}_{\psi }, & \quad P_\psi
= \begin{pmatrix} P^{11} & P^{12} \\ P^{21} & P^{22}
\end{pmatrix}_{\psi }
\end{align*}
be 2$\times$2 matrices of $\psi$-pseudodifferential operators whose entries
 satisfy the requirements of Lemma~\ref{lem:psiparametrix}.
Also we choose the top order of the symbols of $Q^{11}_\psi$, $Q^{21}_\psi$
equal to $1-m$ and those of $Q^{12}_\psi$, $Q^{22}_\psi$ equal to $-m$ for
some fixed $m \in \mathbb{R}$.

Using Lemma~\ref{lem:psiparametrix} we choose $P_\psi$ to be the approximative
inverse of $Q_\psi$ on $I'_{\theta_1}$ in the following sense:
with $P^{11}_\psi$, $P^{12}_\psi$ being operators of order $m-1$
and $P^{21}_\psi$, $P^{22}_\psi$ of order $m$ and one has
$P_\psi Q_\psi = \text{Id}_\psi + E_\psi$ on $I'_{\theta_1}$ for some
$ 2\times2$  $\psi$-pseudodifferential operator matrix $E_\psi$ of the form
\begin{equation*}
E_\psi = \begin{pmatrix} E^{11} & E^{12} \\ E^{21} & E^{22}
\end{pmatrix}_{\psi}
\end{equation*}
%
and $E^{11}_\psi$, $E^{22}_\psi \in \operatorname{OP}_\psi \mathcal{N}_{\mbox{$\mathcal{S}$}^0}$, $E^{12}_\psi \in \operatorname{OP}_\psi \mathcal{N}_{\mbox{$\mathcal{S}$}^{-1}}$ and $E^{21}_\psi \in \operatorname{OP}_\psi \mathcal{N}_{\mbox{$\mathcal{S}$}^1}$ on $I'_{\theta_1}$.

Then by \eqref{EquivIML} and \eqref{EquivSystem} the equation
\begin{equation*}
L_\psi U = F \quad \text{ microlocally at infinity on } I_{\theta_1}
\end{equation*}
holds if and only if
%
\begin{equation*}
g_\psi Q_\psi \begin{pmatrix} \partial_z & -1  \\ A & \partial_z
\end{pmatrix}_{\psi}  P_\psi Q_\psi
\begin{pmatrix} U \\ \left(\partial_z\right)_{\psi} U \end{pmatrix}
= g_\psi Q_\psi \begin{pmatrix} 0 \\ F \end{pmatrix} \quad
\text{microlocally at infinity on } I_{\theta_1}
\end{equation*}
where $g_\psi$ is the microlocal cut-off function from the previous section.

Furthermore we will use the following notation: for $l=1,2$ and
 $k \in \mathbb{N}$ we denote by $\OPNerror{k}{l}$ operators of the form
\begin{equation*}
\sum_{k \leq j \leq k+l} R_{2-j,\psi}(y,D_t,D_x) (\partial_z^{j-k})_\psi
\end{equation*}
and the generalized symbol of $R_{2-j,\psi} = R_{2-j,\psi}(y,D_t,D_x)$ is
in $\mathcal{N}_{\mbox{$\mathcal{S}$}^{2-j}}$ for $k \leq j \leq k+l$.

In the following we assume that $R_\psi$ is a $\psi$-pseudodifferential
operator valued 2$\times$2 error matrix with entries in
$\OPNerror{1}{1}$ on $I'_{\theta_1}$. Also, we let
$B_{\pm,\psi}= B_{\pm,\psi}(y,D_t,D_x) \in \operatorname{OP}_\psi \MphgSym[1,0]{1,0}$
 on $I'_{\theta_1}$.

To obtain a diagonalization for the operator in \eqref{EquivSystem} we will
search for operators $P_\psi, Q_\psi, R_\psi$ and $B_{\pm,\psi}$ as above
such that the following adapted formulation of the problem is valid in
the region $I'_{\theta_1}$:
\begin{equation}\label{Ansatz}
Q_{\psi} \Big[ \left(\partial_z\right)_\psi -
\begin{pmatrix} 0 & 1  \\ -A & 0  \end{pmatrix}_{\psi} \Big] P_\psi
= \begin{pmatrix} \partial_z - i B_{+} & 0 \\ 0 & \partial_z - i B_{-}
\end{pmatrix}_{\psi} + R_\psi.
\end{equation}
To show the existence of these operators we will construct them explicitly.
Therefore we presume that \eqref{Ansatz} can be solved for some
$P_\psi, Q_\psi, R_\psi$ and $B_{\pm,\psi}$ with the above conditions.

Multiplying equation \eqref{Ansatz} with
 $Q_\psi \left( \begin{smallmatrix} 1 \\ \partial_z  \end{smallmatrix}\right)_{\psi}$
from the right we obtain for the left hand side an expression of the form
\begin{equation}\label{ExpandAnsatz1}
Q_\psi \begin{pmatrix} \partial_z & -1 \\ A & \partial_z \end{pmatrix}_{\psi}
 P_\psi Q_\psi \begin{pmatrix} 1 \\ \partial_z \end{pmatrix}_{\psi}
= \begin{pmatrix} Q^{12}_\psi ( \partial^2_z + A )^{}_{\psi}
+ S^{(1,+)}_{\psi}  \\ Q^{22}_\psi (\partial^2_z + A )^{}_{\psi}
+ S^{(1,-)}_{\psi}  \end{pmatrix} \quad \mbox{on} \ I'_{\theta_1}
\end{equation}
with $S^{(1,\pm)}_\psi \in \OPNerror{m}{2}$ on $I'_{\theta_1}$.
Similarly we compute for the right hand side
\begin{align} \label{ExpandAnsatz2}
%\begin{split}
& \Big[ \begin{pmatrix} \partial_z \hspace{-2pt}
- \hspace{-1pt} i B_{+} & 0 \\ 0 & \partial_z \hspace{-2pt}
- \hspace{-1pt} i B_{-} \end{pmatrix}_{\psi} + R_\psi \Big]
 Q_\psi \begin{pmatrix} 1 \\ \partial_z  \end{pmatrix}_{\psi} \qquad \nonumber \\
& \qquad = \begin{pmatrix}  (\partial_z \hspace{-1pt}
- \hspace{-1pt} iB_{+})^{}_{\psi} (Q^{12}_\psi (\partial_z)^{}_{\psi}
+ Q^{11}_\psi) + S^{(2,+)}_\psi \enspace \\ (\partial_z \hspace{-2pt}
- \hspace{-1pt} iB_{-})^{}_{\psi} (Q^{22}_\psi (\partial_z)^{}_{\psi}
+ Q^{21}_\psi) + S^{(2,-)}_\psi \end{pmatrix} \quad \mbox{on} \ I'_{\theta_1}
%\end{split}
\end{align}
where $S^{(2,\pm)}_\psi \in \OPNerror{m}{2}$ on $I'_{\theta_1}$.
Combining \eqref{ExpandAnsatz1} and \eqref{ExpandAnsatz2} we get an improved
formulation of \eqref{Ansatz} which reads
\begin{equation}\label{RefinedAnsatz}
\begin{aligned}
Q^{12}_\psi ( \partial^2_z + A )^{}_{\psi}
 &= (\partial_z - iB_{+})^{}_{\psi} \bigl(Q^{12}_\psi (\partial_z)^{}_{\psi}
 + Q^{11}_\psi \bigr) + R^{(+)}_\psi \quad \mbox{on} \ I'_{\theta_1}\\
Q^{22}_\psi ( \partial^2_z + A )^{}_{\psi}
 &= (\partial_z - iB_{-})^{}_{\psi} \bigl(Q^{22}_\psi (\partial_z)^{}_{\psi}
 + Q^{21}_\psi \bigr) + R^{(-)}_\psi \quad \mbox{on} \ I'_{\theta_1},
\end{aligned}
\end{equation}
with $R^{(\pm)}_\psi \in \OPNerror{m}{2}$ on $I'_{\theta_1}$.

We note that \eqref{RefinedAnsatz} is a coupled system of two equations each
of which stating a factorization similar to \eqref{eqn:Fact} in
Theorem~\ref{thm:Fact}. In the following we relate \eqref{eqn:Fact}
to \eqref{RefinedAnsatz} which will guarantee existence of the basic
 approach made in \eqref{Ansatz}. Now since Theorem~\ref{thm:Fact} allows
two different factorizations (depending on the sign of the principal symbol)
we will modify both of them to deduce the refined reformulation in
\eqref{RefinedAnsatz}.

Thus, with a view to Theorem~\ref{thm:Fact}, we first write
$L_\psi= ( \partial_z^2  + A )_\psi$ in the  form
\begin{equation}\label{Fact1}
L_\psi = ( \partial_z + A_{11} )_\psi ( \partial_z + A_{12} )_\psi
+ \Gamma_{1,\psi} \quad \mbox{on} \ I'_{\theta_1}
\end{equation}
and for $j= 1,2$, $A_{1j,\psi} = A_{1j,\psi}(y,D_t,D_x)$
is a polyhomogeneous $\psi$-pseudodifferential operator with generalized symbol
$(a_{1j,\varepsilon})_\varepsilon$ as in Theorem~\ref{thm:Fact}.
Moreover we choose $A_{11,\psi}$ and $-A_{12,\psi}$ such that their principal
symbols are equal to $(-i \sqrt{a_\varepsilon})_\varepsilon$.
Furthermore $\Gamma_{1,\psi}$ is in $\OPNerror{0}{1}$ on $I'_{\theta_1}$.
 Likewise we obtain
\begin{equation}\label{Fact2}
L_\psi = ( \partial_z + A_{21} )_\psi ( \partial_z + A_{22} )_\psi
+ \Gamma_{2,\psi} \quad \mbox{on} \ I'_{\theta_1}
\end{equation}
where $A_{2j,\psi} = A_{2j,\psi}(y,D_t,D_x)$, $j=1,2$ are
polyhomogeneous $\psi$-pseudodifferential operators but at this point the
top order symbols of $A_{21,\psi}$ and $-A_{22,\psi}$ equal to
 $(i \sqrt{a_\varepsilon})_\varepsilon$. Again $\Gamma_{2,\psi}$ is in
$\OPNerror{0}{1}$ on $I'_{\theta_1}$.

An expansion in \eqref{Fact1} and \eqref{Fact2} then gives the following for $j=1,2$
\begin{equation*}
(\partial^2_z + A)_\psi = (\partial^2_z)_\psi
+ (A_{j1}+A_{j2})_\psi(\partial_z)_\psi + \operatorname{OP}_\psi\bigl(
\bigl(\varepsilon \partial_z a_{j2}\bigr)_\varepsilon \bigr) + A_{j1,\psi}A_{j2,\psi}
+ \Gamma_{j,\psi}
\end{equation*}
on $I'_{\theta_1}$ where we have used the Leibniz rule.
By construction of the generalized symbols of $A_{j1,\psi}$ and $A_{j2,\psi}$
we observe that $(a_{j1,\varepsilon})_\varepsilon = (-a_{j2,\varepsilon})_\varepsilon$
modulo $\mathcal{N}_{\mbox{$\mathcal{S}$}^1}$ on $I'_{\theta_1}$, $j=1,2$.
Using this \eqref{Fact1} and \eqref{Fact2} then read
\begin{equation}\label{TwoFact}
\begin{split}
(\partial^2_z + A)_\psi
&= (\partial_z + A_{11})_\psi (\partial_z - A_{11})_\psi
 \quad \mod{} \OPNerror{0}{1} \ \mbox{on} \ I'_{\theta_1} \\
&= (\partial_z + A_{21})_\psi (\partial_z - A_{21})_\psi
 \quad  \mod{} \OPNerror{0}{1} \ \mbox{on} \ I'_{\theta_1}.
\end{split}
\end{equation}
With $m$ being the fixed real number from the beginning of this section we
now choose $\widetilde{Q}^{(\pm)}_\psi \in \operatorname{OP}_\psi \MphgSym[m,0]{1,0}$
and so that the requirements of Lemma~\ref{lem:psiparametrix} are fulfilled.
Using the same result we obtain the existence of two $\psi$-pseudodifferential
operators $Q^{(\pm)}_\psi$ with generalized symbols in $\MphgSym[-m,0]{1,0}$
satisfying \eqref{AE:parametrix} and
\begin{equation*}
\widetilde{Q}^{(\pm)}_\psi Q^{(\pm)}_\psi = Q^{(\pm)}_\psi \widetilde{Q}^{(\pm)}_\psi
 = I_\psi \quad \mod{} \operatorname{OP}_\psi \mathcal{N}_{\mbox{$\mathcal{S}$}^0} \
\text{on } I'_{\theta_1}.
\end{equation*}
Inserting $\widetilde{Q}^{(+)}_\psi Q^{(+)}_\psi$ into the first line of
\eqref{TwoFact} and $\widetilde{Q}^{(-)}_\psi Q^{(-)}_\psi$ into the second
line yields
\begin{align*}
(\partial^2_z + A)_\psi
&= (\partial_z + A_{11})^{}_{\psi} \widetilde{Q}^{(+)}_\psi
 \bigl(Q^{(+)}_\psi (\partial_z)^{}_{\psi} - Q^{(+)}_\psi A_{11,\psi}^{} \bigr)
 \quad \mod{} \OPNerror{0}{2} \text{ on } I'_{\theta_1}\ \\
&= (\partial_z + A_{21})^{}_{\psi} \widetilde{Q}^{(-)}_\psi
 \bigl(Q^{(-)}_\psi (\partial_z)^{}_{\psi} - Q^{(-)}_\psi A_{21,\psi}^{} \bigr)
 \quad \mod{} \OPNerror{0}{2} \text{ on } I'_{\theta_1}.
\end{align*}
Here we define
\begin{equation}\label{SolutionsB}
\begin{aligned}
-iB_{+,\psi}^{}  &:=  Q^{(+)}_\psi A_{11,\psi}^{} \widetilde{Q}^{(+)}_\psi
- \operatorname{OP}^{}_{\psi} \bigl( (\varepsilon \partial_z
  q_\varepsilon^{(+)} )^{}_{\varepsilon} \bigr) \widetilde{Q}^{(+)}_\psi  \\
-iB_{-,\psi}^{}  &:=  Q^{(-)}_\psi A_{21,\psi}^{} \widetilde{Q}^{(-)}_\psi
- \operatorname{OP}^{}_{\psi} \bigl( (\varepsilon
 \partial_z q_\varepsilon^{(-)} )^{}_{\varepsilon} \bigr) \widetilde{Q}^{(-)}_\psi.
\end{aligned}
\end{equation}
Consequently, $B_{\pm,\psi}$ are polyhomogeneous $\psi$-pseudodifferential
operators with real-valued top order symbol in $\MhgSym[1,0]{1,0}$ on
$I'_{\theta_1}$. Furthermore a straightforward calculation shows that with
this choices for $B_{\pm,\psi}$ we have
\begin{align*}
\bigl( \partial_z + A_{11} \bigr)_\psi \widetilde{Q}^{(+)}_\psi
 &= \widetilde{Q}^{(+)}_\psi \bigl( \partial_z - i B_{+} \bigr)_\psi
 \quad \mod{} \OPNerror{1-m}{0} \ \mbox{on} \ I'_{\theta_1}\\
\bigl( \partial_z + A_{21} \bigr)_\psi \widetilde{Q}^{(-)}_\psi
 &= \widetilde{Q}^{(-)}_\psi \bigl( \partial_z - i B_{-} \bigr)_\psi
 \quad \mod{} \OPNerror{1-m}{0} \ \mbox{on} \ I'_{\theta_1}.
\end{align*}
leading to the following decomposition on $I'_{\theta_1}$,
\begin{align*}
\bigl( \partial^2_z  + A \bigr)^{}_{\psi}
 & = \widetilde{Q}^{(+)}_\psi \big( \partial_z - iB_{+} \big)^{}_{\psi}
  \big( Q^{(+)}_\psi  (\partial_z)^{}_\psi - Q^{(+)}_\psi A^{}_{11,\psi} \big)
  \quad \mod{} \OPNerror{0}{2}\phantom{.} \\
 & = \widetilde{Q}^{(-)}_\psi \big( \partial_z - iB_{-} \big)^{}_{\psi}
  \big( Q^{(-)}_\psi  (\partial_z)^{}_\psi - Q^{(-)}_\psi A^{}_{21,\psi} \big)
  \quad \mod{} \OPNerror{0}{2}.
\end{align*}
Comparing this with the refined approach \eqref{RefinedAnsatz} we make the
following choice for the matrix $Q_\psi$ from the beginning
\begin{gather*}
    Q^{11}_\psi := - Q^{(+)}_\psi A^{}_{11,\psi}, \quad
  Q^{12}_\psi := Q^{(+)}_\psi, \\
    Q^{21}_\psi := - Q^{(-)}_\psi A^{}_{21,\psi}, \quad
  Q^{22}_\psi := Q^{(-)}_\psi.
\end{gather*}
Hence
\begin{equation}\label{TransferMatrix}
Q_\psi = \begin{pmatrix} -Q^{(+)}_\psi A^{}_{11,\psi} & Q^{(+)}_\psi \\
 -Q^{(-)}_\psi A^{}_{21,\psi} & Q^{(-)}_\psi \end{pmatrix}
= \begin{pmatrix} Q^{(+)} & 0 \\ 0 & Q^{(-)} \end{pmatrix}_{\psi} \begin{pmatrix}
 -A_{11} & 1 \\ -A_{21} & 1 \end{pmatrix}_{\psi}
\end{equation}
and $Q^{(\pm)}_\psi$ are polyhomogeneous $\psi$-pseudodifferential operators
of order $-m$ on $I'_{\theta_1}$ and elliptic in the sense of \eqref{StrongEll}.

Since the top order symbol of $A_{j1,\psi}$ is given by
$(\pm i \sqrt{a_\varepsilon})_\varepsilon$, $j=1,2$ it follows from
\eqref{TransferMatrix} that $Q_\psi$ is elliptic in the sense of \eqref{StrongEll}.
 Furthermore, the approximative inverse matrix $P_\psi$ of $Q_\psi$ is given by
\begin{equation*}
P_\psi = \begin{pmatrix} -C_\psi & C_\psi \\ -A_{21,\psi} C_\psi & 1_\psi
+ A_{21,\psi} C_\psi \end{pmatrix} \begin{pmatrix} \widetilde{Q}^{(+)} & 0 \\ 0 &
 \widetilde{Q}^{(-)} \end{pmatrix}_{\psi} \quad \text{on } I'_{\theta_1}
\end{equation*}
where $C_\psi$ is the generalized parametrix of $(A_{11} - A_{21})_\psi$ in the
sense of Lemma~\ref{lem:psiparametrix}. We have therefore found an appropriate
operator-valued matrix $P_\psi,Q_\psi,R_\psi$ and operators $B_{\pm,\psi}$
solving \eqref{Ansatz}.
We have therefore proved the following theorem.

\begin{theorem} \label{thm6.2}
Let $L_\psi$ as in \eqref{GovOp} and $U, F \in \mathcal{G}_{2,2}(\mathbb{R}^{n+1})$.
Then there are operators $Q_\psi$ as in \eqref{TransferMatrix} and
$B_{\pm,\psi}$ as in \eqref{SolutionsB} such that the equation
\begin{equation}\label{eqn:IMLGov}
L_\psi U = F \quad \mbox{microlocally at infinity on} \ I_{\theta_1}
\end{equation}
holds if and only if
\begin{equation}\label{eqn:IMLSystem}
\begin{aligned}
\bigl( \partial_z - iB_{+}(y,D_t,D_x) \bigr)_\psi u_{+} &= f_{+} \quad \mbox{microlocally at infinity on } I_{\theta_1} \mbox{ and}\\
\bigl( \partial_z - iB_{-}(y,D_t,D_x) \bigr)_\psi u_{-} &= f_{-} \quad \mbox{microlocally at infinity on } I_{\theta_1}.
\end{aligned}
\end{equation}
Furthermore, the coupling effect is computed as follows
\begin{equation}
\begin{pmatrix} u_{+} \\ u_{-} \end{pmatrix}
:= Q_\psi \begin{pmatrix} U \\ (\partial_z)_\psi U \end{pmatrix} ,\quad
\begin{pmatrix} f_{+} \\ f_{-} \end{pmatrix} := Q_\psi \begin{pmatrix} 0 \\ F
\end{pmatrix}.
\end{equation}
\end{theorem}

\subsection*{Closing remarks}
Let us briefly summarize the above. First we explained a factorization procedure
for the semiclassical operator $L_\psi$ in the non-trivial case of generalized
coefficients of log-type when acting on $\mathcal{G}_{2,2}$. To overcome the
error made in this factorization we further introduced an adapted notion of
microlocal regularity.

Concerning the motivating part of this paper we want to make the following remarks.
Because of the lack of an adequate description of propagation of singularities
in this setting it is not clear so far if and how one can derive approximated
solutions to \eqref{eqn:IMLGov} from solutions of a perturbation of the
problem \eqref{eqn:IMLSystem} as we have seen in the smooth case in
Subsection~\ref{subsec:PreviousResults}. Even if we allow the operator
$L_\varepsilon$ given in \eqref{eqn:Gov} to have logarithmic slow scale regular
coefficients this problem remains unsolved. Once again, we want to mention
that in the case of logarithmic slow scale case no semiclassical interpretation
of the situation is necessary and microlocal regularity is based on local
$\mathcal{G}_{2,2}^\infty$-regularity.

Moreover we note that apart from the microlocal restrictions in the equations
of \eqref{eqn:IMLSystem} the operators $L_{j,\psi}$ meet the conditions of
Theorem 3.1 of \cite{Hoermann:04}, $j=1,2$, if the coefficients in
\eqref{eqn:Gov} from the beginning are of log-type with an appropriately
chosen exponent $r \in \mathbb{N}$ which depends only on the dimension $n$.
More precisely $r$ plays the same role as $k$ does in
\cite[Remark 3.2]{Hoermann:04}.
Therefore if it is possible to associate to \eqref{eqn:IMLSystem}
a global description of the same by only slight manipulations of the
operators $L_{j,\psi} = \bigl( \partial_z - iB_{\pm}(y,D_t,D_x) \bigr)_\psi$,
 $j=1,2$ in a microlocal sense (as in the smooth setting) one can derive
well-posedness to the corresponding Cauchy problems which in turn
approximate \eqref{eqn:IMLGov} on $I_{\theta_1}$.

However, as already pointed out in Subsection~\ref{subsec:ScColombeau}
we so far only handled the case where the semiclassical asymptotic regime
 was restricted to $\hbar(\varepsilon)=\varepsilon$ as $\varepsilon \to 0$.
This suggests itself to ask for general criteria of other possible choices
for the semiclassical scale $\hbar(\varepsilon)$. Under the viewpoint of
parameter-dependent representation theory for generalized pseudodifferential
operators this will hopefully also give more insight in the notion of microlocal
regularity at infinity. Here a future aim is to obtain a refined notion
of microlocalization which is capable to give a global characterization
and hence has to include regularity results for Colombeau generalized
objects also for finite points.

\subsection*{Acknowlegdments}
The author is very grateful to G\"{u}nther H\"{o}rmann for many helpful discussions.

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\end{document}