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\begin{document}

\title[Algebras of pseudodifferential operators]{Algebras of pseudodifferential
operators on complete manifolds}

\author[B. Ammann]{Bernd Ammann} 
\address{Universit\"at Hamburg,
Fachbereich 11--Mathematik, Bundesstrasse 55, D-20146 Hamburg,
Germany} 
\email{ammann@berndammann.de}
\thanks{Ammann was partially supported by the European Contract Human 
Potential Program, Research Training Networks HPRN-CT-2000-00101 
and HPRN-CT-1999-00118;
Nistor was partially supported by NSF Grants DMS 99-1981 and
DMS 02-00808. Manuscripts available from {\tt http://www.math.psu.edu/nistor/}.}

\author[R. Lauter]{Robert Lauter} 
\address{Universit\"at Mainz,
Fachbereich 17--Mathematik, D-55099 Mainz, Germany}
\email{lauter@mathematik.uni-mainz.de, lauterr@web.de}

\author[V. Nistor]{Victor Nistor}
\address{Mathematics Department, Pennsylvania State
University, University Park, PA 16802}
\email{nistor@math.psu.edu}


\commby{Michael E. Taylor}
\date{April 24, 2003}
\subjclass[2000]{Primary 58J40; Secondary 58H05, 65R20}

\keywords{Differential operator, 
pseudodifferential operator, principal symbol,
conormal distribution, Riemannian manifold, Lie
algebra, exponential map}

\begin{abstract}
In several influential works, Melrose has studied examples of
non-compact manifolds $M_0$ whose large scale geometry is described by
a Lie algebra of vector fields $\mathcal V \subset \Gamma(M;TM)$ on a {\em
compactification} of $M_0$ to a manifold with corners $M$. The
geometry of these manifolds---called ``manifolds with a Lie structure
at infinity''---was studied from an axiomatic point of view in a
previous paper of ours. In this paper, we define and study an algebra
$\Psi_{1,0,\mathcal V}^\infty(M_0)$ of pseudodifferential operators
canonically associated to a manifold $M_0$ with a Lie structure at
infinity $\mathcal V \subset \Gamma(M;TM)$.  We show that many of the
properties of the usual algebra of pseudodifferential operators on a
compact manifold extend to the algebras that we introduce. In
particular, the algebra $\Psi_{1,0,\mathcal V}^\infty(M_0)$ is a
``microlocalization'' of the algebra ${\rm Diff}^{*}_{\mathcal V}(M)$ 
of differential
operators with smooth coefficients on $M$ generated by $\mathcal V$ and
$\mathcal{C}^\infty(M)$. This proves a conjecture of Melrose (see his ICM 90
proceedings paper).
\end{abstract}

\maketitle

\section*{Introduction}

In \cite{meicm}, Melrose has formulated a far reaching program to
study the analytic properties of geometric differential operators on
an open manifold $M_0$, provided that its large scale geometry is
controlled by a Lie algebra of vector fields $\VV$ on a
compactification $M \supset M_0$.  Typically, $M$ is a manifold with
corners, $M_0 = M \smallsetminus \pa M$ is obtained by removing all
faces of $M$, and the given Lie algebra of vector fields $\VV \subset
\Gamma(M; TM)$ satisfies a number of axioms (see Section
\ref{sec.LI}).  This structure leads to complete metrics of bounded
curvature on $M_0$, and $\partial M$ is the ``boundary at infinity.''
For example, manifolds with asymptotically Euclidean, asymptotically
hyperbolic, or asymptotically complex hyperbolic ends are obtained in
this way.

An important ingredient in Melrose's program mentioned above is to
define a suitable pseudodifferential calculus on $M_0$ adapted in a
certain sense to $(M,\VV)$. Melrose calls this pseudodifferential
calculus a ``microlocalization of $\DiffV{*}(M )$,'' where
$\DiffV{*}(M )$ is the algebra of differential operators on $M$
generated by $\VV$ and $\CI(M)$. 

In \cite{MelroseScattering} and several other papers, Melrose and his
collaborators have completed this program in several important cases
\cite{emm91, zfr, defr, Mazzeo, mame87, MaMeAsian, me81, meaps,
MelroseScattering, mecom, melcor, MelroseMendoza, jaredduke}.  One of
the main points is that the geometric operators on manifolds with a
Lie structure at infinity identify with degenerate differential
operators on the compactification $M$. This type of differential
operators appear naturally, for example, in the study of boundary
value problems on manifolds with singularities. Results in this
direction were obtained also by Schulze and his collaborators, who
typically worked in the framework of the Boutet de Monvel
algebras. See
\cite{ScSc, schwil} and the references therein.  See also \cite{LN1,
nwx, Parenti}.

It is desirable to present all these cases in a unified setting
and to extend the results to a 
larger class of manifolds, namely 
the class of ``manifolds with a Lie structure at infinity.'' These are
open manifolds $M_0$ which are topologically the interior of a compact
manifold $M$ with corners, and the geometry of $M_0$ near $\partial M$
is described by a Lie algebra of vector fields $\VV\subset \Gamma(TM)$
satisfying certain axioms (see Section~\ref{sec.LI} for details).
The geometrical properties of these manifolds were
studied in \cite{aln1}. 
Here we introduce an algebra $\Psi_{1,0,\VV}^{\infty}(M_0)$ of
pseudodifferential operators on $M_0$ that is canonically associated
to the manifold with a Lie structure at infinity $M_0$. 
Then we show that this algebra ``microlocalizes'' $\DiffV{*}(M)$ in
the sense that $\DiffV{*}(M)$ is the algebra of all differential
operators in $\Psi_{1,0,\VV}^{\infty}(M_0)$ and that
$\Psi_{1,0,\VV}^{\infty}(M_0)$ has the usual symbolic properties of
algebras of pseudodifferential operators on a compact manifold.  We
also show that the algebra $\Psi_{1,0,\VV}^{\infty}(M_0)$ is invariant
under the diffeomorphisms of $M_0$ obtained by exponentiating the
vector fields $X \in \VV$ and under conjugation with complex powers of
the functions that define the faces of the compactification $M$ of
$M_0$.

The explicit construction of the algebra
$\Psi_{1,0,\VV}^{\infty}(M_0)$ microlocalizing $\DiffV{*}(M_0)$ in
the sense of \cite{meicm} is, roughly, as follows. First, $\VV$
defines an extension of $TM_0$ to a vector bundle $A \to M$ ($M_0
= M \smallsetminus \pa M$). Denote $V_r := \{d(x,y) < r\} \subset
M_0^2$ and $(A)_r = \{v \in A, \|v\| < r\}$. Let $r>0$ be less
than the injectivity radius of $M_0$ and $V_r \ni (x, y) \mapsto
(x, \tau(x, y)) \in (A)_r$ be a local inverse of the Riemannian
exponential map $TM_0 \ni v \mapsto \exp_x(-v) \in M_0 \times
M_0$. Let $\chi$ be a smooth function on $A$ with support in
$(A)_r$, $\chi = 1$ on $(A)_{r/2}$. For any $a \in
S^m_{1,0}(A^*)$, we define
\begin{equation}
    \big[a_\chi(D)u\big](x) = (2\pi )^{-n} \int_{M_{0}} \left
    (\int_{T^{*}_{x}M_{0}} e^{i \tau(x,y) \cdot \eta} \chi(x,
    \tau(x,y)) a(x, \eta)u(y)\, d \eta \right) dy.
\end{equation}
The algebra $\Psi_{1,0,\VV}^{\infty}(M_0)$ is then generated linearly
by the operators $a_\chi(D)$ and $b_\chi(D)\exp(X_1) \ldots
\exp(X_k)$, $a \in S^\infty(A^*)$, $b \in S^{-\infty}(A^*)$, and $X_j \in
\VV$. (We need to introduce the operators $b_\chi(D)\exp(X_1) \ldots
\exp(X_k)$, where $\exp(X_j)$ is the exponential of the vector field
$X_j$, to make our space of operators closed under products.) The
definition of the operators of order $-\infty$ is in fact the crucial
part of this construction.

In \cite{meicm}, Melrose outlined the construction of a
pseudodifferential calculus on manifolds with Lie structure at
infinity provided certain manifolds with corners (blow-ups) can be
constructed.  (See also \cite{VasyN}.)  Our approach owes a lot to his
approach when it comes to proving that $\Psi_{1,0,\VV}^{\infty}(M_0)$
are algebras, but we replace the blow-ups with groupoids, using also a
deep result of Crainic and Fernandes \cite{CrainicFernandez} on the
integration of Lie algebroids.  (However, to prove the original form
of the conjecture from \cite{meicm}, the earlier results of
\cite{NistorINT} also suffice.)

This paper is an announcement. 
Complete proofs will be published in \cite{aln2}.


\subsubsection*{Acknowledgements:}\ V. N. would like to thank the
Institute Erwin Schr\"odinger in Vienna, where part of this work was
completed.  We also thank A.~Vasy who has contributed in several ways to
the results in this paper.



\section{Manifolds with Lie structure at infinity}
\label{sec.LI}


For the convenience of the reader, let us recall the definition of
Riemannian manifolds with a Lie structure at infinity and some of
their basic properties \cite{aln1}.


In the sequel, by a {\em manifold} we shall always understand a
$C^\infty$-manifold {\em with corners}, whereas a {\em smooth
manifold} is a $C^\infty$-manifold {\em without corners}.  By
definition, every point $p$ in a manifold with corners $M$ has a
coordinate neighborhood diffeomorphic to $[0,\infty)^k \times
\RR^{n-k}$ such that the transition functions are smooth up to the
boundary. We then call $p$ a point of {\em boundary depth at most $k$}
and write $\codim(p) \leq k$.  Points $p$ with $\codim(p) \leq k$ but
not $\codim(p) \leq k-1$ are said to be of {\em boundary depth
$k$}. (This terminology is in agreement with the terminology for
stratified spaces, if we stratify a manifold with corners by its open
subfaces; see \cite{mather}.) We denote by $\pa M$ the union of all
non-trivial faces of $M$.  Usually, we write $M_0$ for the interior of
$M$, i.e., $M_0:= M \smallsetminus \pa M$.

A Lie subalgebra $\VV\subseteq \Gamma(M; TM)$ of the Lie algebra of
all smooth vector fields on $M$ is said to be {\em a structural Lie
algebra of vector fields} provided it is a finitely generated,
projective $\CI(M)$-module and each $V\in\VV $ is tangent to all
hyperfaces of $M$. (We shall denote the sections of a vector bundle $V
\to X$ by $\Gamma(X; V)$, unless $X$ is understood, in which case we
shall write simply $\Gamma(V)$.) By the Serre-Swan theorem
\cite{Karoubi}, there exists a smooth vector bundle
$A=A_\VV\rightarrow M$ together with a natural map $\varrho =
\varrho_\VV : A \to TM$ making the following diagram commutative
\begin{equation}\label{eq.anchor}
    \begin{array}{rcccl}
    A_\VV \kern-3mm & & \stackrel{\varrho}{\longrightarrow}
    & & \kern-3mm TM \\ & \searrow\kern-3mm && \kern-3mm
    \swarrow & \\ && M
    \end{array}
\end{equation}
and such that $\VV = \varrho (\Gamma(A_\VV))$.


\begin{definition}\label{def.unif.str}\
A {\em Lie structure at infinity} on a smooth manifold $M_0$ is a
triple $(M_0, M, \VV)$, where $M$ is a compact manifold, possibly
with corners, and $\VV \subset \Gamma(M; TM)$ is a structural Lie
algebra of vector fields on $M$ with the following properties:
\begin{enumerate}
\item\ $M_0$ is diffeomorphic to the interior $M \smallsetminus \pa M$
of $M$;
\item\ If $\varrho : A \to TM$ is the anchor map defined by diagram
\eqref{eq.anchor}, then $\varrho$ restricts to an isomorphism
$A\vert_{M\smallsetminus\partial M} \rightarrow
TM\vert_{M\smallsetminus\partial M}$.
\end{enumerate}
\end{definition}


Note that for a given manifold $M_{0}$ in general there can exist
many Lie structures at infinity. Examples of Lie structures at
infinity were discussed in \cite{aln1}.

From now on, we identify  $M_0$ with $M \smallsetminus \pa M$ and
$A\vert_{M_0}$ with $TM_0$. Because $A$ and $\VV$ determine each
other up to isomorphism, we sometimes specify a Lie structure at
infinity on $M$ by the pair $(M, A)$. Also, it follows that
$\varrho : \Gamma(M; A) \to \Gamma(M; TM)$ is injective, so we
shall identify $\Gamma(M; A)$ with $\VV = \varrho(\Gamma(M; A))$.

Elements in the enveloping algebra $\DiffV{*}(M)$ of $\VV$ are called
{\em $\VV$-differential operators on $M$}. By the injectivity of the
induced structural map $\varrho_{\VV} :
\Gamma(A_{\VV})\rightarrow\Gamma(TM)$, the algebra of
$\VV$-differential operators can be realized as a subalgebra of all
differential operators on $M$; in particular, they act continuously on
the space $\CI(M)$.  Moreover, the order of differential operators
induces a filtration $\DiffV{m}(M )$, $m \in \ZZ_{+}$, on the algebra
$\DiffV{*}(M )$. Since $\DiffV{*}(M)$ is a $\CI(M)$-module, we can
introduce $\VV$-differential operators acting between sections of
smooth vector bundles $E, F \rightarrow M$, $E, F \subset M \times
\CC^N$ by
\begin{equation}\label{vectorvalued}
    \DiffV{*}(M ;E,F) := e_F M_N(\DiffV{*}(M )) e_E\,,
\end{equation}
where $e_E, e_F \in M_N(\CI(M))$ are the projections onto $E$ and,
respectively, $F$. It follows that $\DiffV{*}(M ;E,E) = :
\DiffV{*}(M ;E)$ is an algebra that is closed under adjoints and
contains all geometric operators on $M_0$ that are associated to a
metric on $M_0$ that comes from a metric on $A$. (See
\cite{aln1}.)

Since any metric on $A$ induces a natural metric on $TM_0 =
A\vert_{M_0}$, we obtain the following definition.


\begin{definition}
\label{def.R.usi}\ A manifold $M_0$ with a Lie structure at
infinity $(M, A)$ and with metric $g_{0}$ on $M_0$ obtained by
restricting a metric $g$ from $A$ to $TM_0$ is called a {\em
Riemannian manifold with a Lie structure at infinity}.
\end{definition}


The geometry of Riemannian manifolds $(M_{0},g_{0})$ with a Lie
structure at infinity has been studied in \cite{aln1}. For instance,
$(M_{0},g_{0})$ is necessarily of infinite volume and complete.
Moreover, all the covariant derivatives of the Riemannian curvature
tensor are bounded. Under additional mild assumptions, we also know
that the injectivity radius ${\rm injrad}(p)$, viewed as a function
depending on $p\in M$, is bounded below by a positive constant,
or, equivalently, $(M_{0},g_{0})$ is of bounded geometry in the sense
of \cite{Shubin} and references therein.  We shall denote by $r_0 :=
\inf_p {\rm injrad}(p)$ the {\em injectivity radius} of $M_0$.

On a Riemannian manifold $M_0$ with a uniform structure at
infinity $(M, A)$, the exponential map $\exp_p : TM_0 \to M_0$ is
well defined for all $p \in M_0$ and extends to a differentiable
map $\exp_p : A_p \to M$ depending smoothly on $p \in M$. A
convenient way to introduce the exponential map is via the
geodesic spray, as done in \cite{aln1}. Similarly, any vector
field $X \in \Gamma(A)$ is integrable and will map any (connected)
face to itself. The resulting diffeomorphism of $M_0$ will be
denoted $\psi_X$.

{\em We shall also assume from now on that $r_0$, the injectivity
radius of $(M_{0},g_{0})$, is positive.}


\section{Kohn-Nirenberg quantization and pseudodifferential
operators} \label{sec.KN}


We now introduce the algebras $\Psi_{cl, \VV}(M_0)$ and $\Psi_{1,
0, \VV}(M_0)$ of pseudodifferential operators on $M_0$ adapted to
the Lie structure at infinity $(M, \VV)$. We also state some of
their main properties.


\subsection{Riemann-Weyl fibration}
Fix now a Riemannian metric $g$ on the bundle $A$, and let $g_{0}
= g|_{M_{0}}$ be its restriction to the interior $M_{0}$ of $M$,
defined in view of the identification $A\vert_{M_0} = TM_0$. We
shall use this metric to trivialize all density bundles on $M$.
Denote by $\pi : TM_0 \to M_0$ the natural projection.

Define
\begin{equation}\label{eq.Phi}
    \Phi:TM_{0}\longrightarrow M_{0}\times M_{0}, \quad
    \Phi(v) := (x, \exp_{x}(-v)), \; x = \pi(v).
\end{equation}

Recall that, for $v \in T_{x}M$, we have $\exp_{x}(v)=\gamma_{v}(1)$,
where $\gamma_{v}$ is the unique geodesic with
$\gamma_{v}(0)=\pi(v)=x$ and $\gamma_{v}'(0)=v$.  It is known that
there is an open neighborhood $U$ of the zero-section $M_{0}$ in
$TM_{0}$ such that $\Phi|_{U}$ is a diffeomorphism onto an open
neighborhood $V$ of the diagonal $\Delta_{M_{0}} \subseteq M_{0}\times
M_{0}$.

To fix notation, let $E$ be a vector bundle with a norm
$\|\;\cdot\;\|$. We shall denote by $(E)_r$ the set of all vectors $v$
of $E$ with norm $\|v\|< r$. Our assumption that the injectivity
radius $r_0$ of $M_0$ is positive gives that, for $0 < r \le r_{0}$,
the restriction $\Phi|_{(TM_{0})_{r}}$ is a diffeomorphism onto an
open neighborhood $V_{r}$ of the diagonal $\Delta_{M_{0}}$.  We
continue, by slight abuse of notation, to write $\Phi$ for that
restriction.  Following~\cite{emmhei}, we shall call $\Phi$ a {\em
Riemann-Weyl fibration}. However, note that the Riemann-Weyl
fibrations are defined in a slightly different way in~\cite{emmhei};
the difference will be of no importance for us.  The inverse of $\Phi$
is given by
\begin{equation*}
        M_{0}\times M_{0}\supseteq V_{r}\ni(x,y)\longmapsto
        (x,\tau(x,y))\in (TM_{0})_{r}\,,
\end{equation*}
where $\tau(x,y)\in T_{x}M_{0}$ is the tangent vector at $x$ to
the shortest geodesic $\gamma : [0,1] \to M$ such that $\gamma(0)
= x$ and $\gamma(1) = y$.

We shall denote by $S^m_{1,0}(E)$ the space of symbols of order $m$
and type $(1,0)$ on $E$ (in H\"ormander's sense) and by $S^m_{cl}(E)$
the space of classical symbols of order $m$ on $E$ \cite{hor3,
Taylor1, Taylor2}. These spaces are reviewed in \cite{aln2} in our
framework.

Let $\chi \in \CI(A^*)$ be a smooth function that is equal to $1$ on
$(A^*)_{r}$ and is equal to $0$ outside $(A^*)_{2r}$, for some $r <
r_0/3$. Then, following \cite{aln2}, we define
\begin{equation*}
        a_\chi(D)u(x) =  (2\pi )^{-n}
        \int_{T^{*}M_{0}} e^{i\tau(x,y)\cdot\eta}
        \chi(x,\tau(x,y)) a(x,\eta)u(y)\, d \eta \, d y \,.
\end{equation*}
This integral is an oscillatory integral with respect to the
symplectic measure on $T^*M_0$ \cite{hor3}. Alternatively, we can
consider the measures on $M_0$ and on $T^*_xM_0$ defined by some
choice of a metric on $A$ and then we integrate first along the fibers
$T^*_x M_0$ and then along $M_0$.


\begin{proposition}\label{cor.comp.symb}\
The map $\sigma_{tot} : S_{1,0}^{m}(A^{*}) \to
\Psi^{m}(M_{0})/\Psi^{-\infty}(M_{0})$,
\[
        \sigma_{tot}(a) := a_\chi(D) + \Psi^{-\infty}(M_{0}),
\]
is independent of the choice of the function $\chi\in
\CIc((A)_{r})$ used to define $a_\chi(D)$.
\end{proposition}

The space of all operators of the form $a_\chi(D)$, with $a\in
S_{1,0}^{*}(A^{*})$, is not closed under composition. In order to make
it closed under composition, we are going to include more operators of
order $-\infty$ in our calculus.

Any vector field $X \in \Gamma(A)$ generates a global flow $\Psi_{X} :
\RR\times M\rightarrow M$ because $X$ is tangent to all boundary faces
of $M$ and $M$ is compact.  Evaluating at $t=1$ yields a
diffeomorphism
\begin{equation}\label{eq.def.psiX}
        \psi_X := \Psi_{X}(1,\cdot) : M \rightarrow M.
\end{equation}

We continue to assume that the injectivity radius $r_0$ of our fixed
manifold with a Lie structure at infinity $(M, M_0, A)$ is strictly
positive.

\begin{definition}\label{def.psi.MA}\ Fix $0< r < r_0$ and
$\chi \in \CIc((A)_{r})$ such that $\chi = 1$ in a neighborhood of $M$
in $A$.  For $m \in \RR$, the space $\Psi_{1,0,\VV}^{m}(M_{0})$ of
{\em pseudodifferential operators generated by the Lie structure at
infinity $(M,A)$} is the linear space of operators $\CIc(M_0)
\rightarrow \CIc(M_{0})$ generated by $a_\chi(D)$, $a \in
S_{1,0}^m(A^*)$, and $b_{\chi}(D)\psi_{X_1}\ldots \psi_{X_k}$, $b \in
S^{-\infty}(A^*)$ and $X_j \in \Gamma(A)$, for all $j$.

Similarly, the space $\Psi_{cl,\VV}^{m}(M_{0})$ of {\em classical
pseudodifferential operators generated by the Lie structure at
infinity $(M,A)$} is obtained by using classical symbols $a$ in the
construction above.
\end{definition}


We also obtain that $\Psi_{1, 0, \VV}^m(M_{0})$ and $\Psi_{cl,\VV}^m(M_{0})$
are algebras independent of the choices made in
their definition.

\begin{theorem}\label{theorem.indep}\
The spaces $\Psi_{1, 0, \VV}^m(M_{0})$ and $\Psi_{cl,\VV}^m(M_{0})$
are filtered algebras. They do not depend on the choice of the metric
on $A$ and the function $\chi$ used to define it, but depend, in
general, on the Lie structure at infinity $(M,A)$ on $M_0$.
\end{theorem}

The fact that $\Psi_{1, 0, \VV}^m(M_{0})$ and
$\Psi_{cl,\VV}^m(M_{0})$ are filtered algebras means that
\begin{equation*}
    \Psi_{1,0, \VV}^m(M_{0}) \Psi_{1, 0, \VV}^{m'}(M_{0}) \subseteq
    \Psi_{1, 0, \VV}^{m + m'}(M_{0})\,, \qquad \Psi_{cl,
    \VV}^m(M_{0}) \Psi_{cl, \VV}^{m'}(M_{0}) \subseteq \Psi_{cl,
    \VV}^{m + m'}(M_{0})\,,
\end{equation*}
for all $m,m' \in \CC \cup \{-\infty\}.$

The proof of Theorem \ref{theorem.indep} is obtained by realizing
$\Psi_{1,0, \VV}^{\infty}(M_{0})$ as the homomorphic image of the
algebra $\Psi^{\infty}_{1,0}(\GR)$ of pseudodifferential operators on
a groupoid $\GR$ integrating the Lie algebroid $A$ (that is, with
$\Gamma(A) = \VV$). This is possible due to the results of
\cite{CrainicFernandez, NistorINT, nwx}. 
Note that this proof provides also an alternative definition of the
algebras $\Psi_{1,0, \VV}^{\infty}(M_{0})$ and $\Psi_{cl,
\VV}^{\infty}(M_{0})$. The advantage of our original definition,
however, is that it is intrinsically formulated in terms of the
geometry of $M$.

As for the usual algebras of pseudodifferential operators, we have the
following basic property of the principal symbol.


\begin{proposition}\label{prop.princ.symb}\
The principal symbol establishes isomorphisms
\begin{equation}\label{eq.def.princ}
    \sigma^{(m)} : \Psi_{1,0,\VV}^{m}(M_{0}) /
    \Psi_{1,0,\VV}^{m-1}(M_{0})
    \to S^m_{1,0}(A^*) / S^{m-1}_{1,0}(A^*)
\end{equation}
and
\begin{equation}\label{eq.def.princ'}
    \sigma^{(m)} : \Psi_{cl,\VV}^{m}(M_{0}) /
    \Psi_{cl,\VV}^{m-1}(M_{0})
    \to S^m_{cl}(A^*) / S^{m-1}_{cl}(A^*) \simeq 
	\CI(S^*A),
\end{equation}
where $S^*A$ is the set of unit vectors in $A^*$.
\end{proposition}


We have the following boundedness result.

\begin{proposition}\label{cor.bdd}\
Any operator $P \in \Psi^{m}_{1, 0, \VV}(M_{0})$ defines a
continuous linear operator on $\CIc(M_{0})$ and $\CI(M)$. If $m =
0$, it also defines a bounded operator on $L^2(M_0)$.
\end{proposition}

Part (i) of the following result is an analog of a standard result
about the $b$-calculus \cite{meaps}, whereas the second formula is the
independence of diffeomorphisms of the algebras $\Psi_{cl,
\VV}^\infty(M_{0})$, in the framework of manifolds with a Lie
structure at infinity. Recall that if $X\in\Gamma(A)$, we have denoted
by $\psi_{X} := \Psi_{X}(1,\cdot):M \rightarrow M$ the diffeomorphism
defined by integrating $X$ (and specializing at $t = 1$).


\begin{proposition} \label{prop.auto}\ (i) Let $x$ be a  defining
function of some hyperface of $M$. Then
\begin{equation*}
        x^{s} \Psi_{1,0, \VV}^m(M_{0}) x^{-s} = \Psi_{1, 0,
        \VV}^m(M_{0}) \quad \text{and} \quad x^{s} \Psi_{cl,
        \VV}^m(M_{0}) x^{-s} = \Psi_{cl, \VV}^m(M_{0})
\end{equation*}
for any $s \in \CC$.

(ii) Similarly,
\begin{equation*}
        \psi_X \Psi_{1,0, \VV}^m(M_{0}) \psi_X^{-1} = \Psi_{1,0,
        \VV}^m(M_{0}) \quad \text{and}\quad \psi_X \Psi_{cl, \VV}^m(M_{0})
        \psi_X^{-1} = \Psi_{cl, \VV}^m(M_{0}),
\end{equation*}
for any $X \in \Gamma(A)$.
\end{proposition}


Let us notice that (ii) remains true for any diffeomorphism
of $M_0$ that extends to an automorphism of $(M,A)$. Recall that
an autormorphism of the Lie algebroid $(M,A)$ is a morphism of
vector bundles $(\phi,\psi)$, $\phi : M \to M$, $\psi: A \to A$,
such that $\phi$ and $\psi$ are diffeomorphisms and
\begin{equation*}
        \varrho_\Gamma \circ \psi_\Gamma
        = \phi_* \circ \varrho_\Gamma,
\end{equation*}
where $\varrho_\Gamma : \Gamma(A) \to \Gamma(TM)$ and $\psi_\Gamma
: \Gamma(A) \to \Gamma(A)$ are the maps defined by the anchor map
$\varrho$ and $\psi$, respectively,  and $\phi_* : \Gamma(TM) \to
\Gamma(TM)$ is given by the differential of $\phi$.

A proof of the above proposition can be obtained using the
corresponding results for the algebras of pseudodifferential
operators on $\GR$.

The proof of the following proposition relies on the
Campbell-Hausdorff formula.


\begin{proposition}\label{prop.quant}\
Let $X \in \Gamma(A)$ and denote by $a_X (\xi) = \xi(X)$ the
associated linear function on $A^*$. Then $a_X \in S^1(A^*)$ and
$a_X(D) = -\imath X$. Moreover,
\begin{equation*}
    \{a_\chi(D), \, a =\mbox{ \rm polynomial in each fiber}\,\} =
    \DiffV{*}(M_0).
\end{equation*}
\end{proposition}

From this we obtain the following result.

\begin{theorem}\label{cor.quant}\
Let $\Diff{M_0}$ be the algebra of all differential operators on
$M_0$. Then
\begin{equation*}
        \Psi_{1,0,\VV}^\infty(M_0) \cap \Diff{M_0} = \DiffV{*}(M_0).
\end{equation*}
\end{theorem}


Since $\DiffV{*}(M_0) \subset \Psi_{cl,\VV}^\infty(M_0)$, it also
follows that
\begin{equation*}
        \Psi_{cl,\VV}^\infty(M_0) \cap \Diff{M_0} = \DiffV{*}(M_0).
\end{equation*}
Thus the algebras $\Psi_{1,0,\VV}^\infty(M_0)$ and
$\Psi_{1,0,\VV}^\infty(M_0)$ are microlocalizations of
$\DiffV{*}(M_0)$, which, together with the other properties of
these algebras stated above, shows that our constructions solve a
conjecture from \cite{meicm}.


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\end{document}