# Layer potentials and essentially translation invariant pseudodifferential
operators on manifolds with cylindrical ends

Mirela Kohr  Faculty of Mathematics and Computer Science, Babeş-Bolyai
University, 1 M. Kogălniceanu Str., 400084 Cluj-Napoca, Romania
<EMAIL_ADDRESS>,  Victor Nistor  Université de Lorraine, CNRS, IECL,
F-57000 Metz, France<EMAIL_ADDRESS>and  Wolfgang L. Wendland
Institut für Angewandte Analysis und Numerische Simulation, Universität
Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
<EMAIL_ADDRESS>
###### Abstract.

Motivated by the study of layer potentials on manifolds with straight conical
or cylindrical ends, we introduce and study two classes (or calculi) of
pseudodifferential operators defined on manifolds with cylindrical ends: the
class of pseudodifferential operators that are “translation invariant at
infinity” and the class of “essentially translation invariant operators.”
These are “minimal” classes of pseudodifferential operators containing the
layer potential operators of interest. Both classes are close to the
$b$-calculus considered by Melrose and Schulze and to the $c$-calculus
considered by Melrose and Mazzeo-Melrose. Our calculi, however, are different
and, while some of their properties follow from those of the $b$\- or
$c$-calculi, many of their properties do not. In particular, we prove that the
“essentially translation invariant calculus” is spectrally invariant, a
property not enjoyed by the “translation invariant at infinity” calculus or
the $b$-calculus. For our calculi, we provide easy, intuitive proofs of the
usual properties: stability for products and adjoints, mapping and boundedness
properties for operators acting between Sobolev spaces, regularity properties,
existence of a quantization map, topological properties of our algebras, and
the Fredholm property. Since our applications will be to the Stokes operator,
we systematically work in the setting of Douglis-Nirenberg-elliptic operators.
We also show that our calculi behave well with respect to restrictions to
(suitable) submanifolds, which is crucial for our applications to layer
potential operators.

###### Key words and phrases:

Stokes operator; Sobolev spaces; Pseudodifferential operators; Manifolds with
cylindrical ends.

###### 2000 Mathematics Subject Classification:

Primary 35R01; Secondary 76M.

M.K. has been partially supported by the Babeş-Bolyai University Grant AGC
33656/21.07.2022. V.N. has been partially supported by ANR OpART.

###### Contents

  1. 1 Introduction
    1. 1.1 The framework and some earlier results
    2. 1.2 Main results.
    3. 1.3 Contents of the paper
  2. 2 Background: manifolds with cylindrical ends and Sobolev spaces
    1. 2.1 General notation: duals and connections
    2. 2.2 Manifolds with cylindrical ends
    3. 2.3 Compatible vector bundles and differential operators
    4. 2.4 Sobolev spaces
      1. 2.4.1 Positive order covariant Sobolev spaces
      2. 2.4.2 Negative and non-integer order covariant Sobolev spaces
      3. 2.4.3 Alternative definition
  3. 3 Basics on pseudodifferential operators
    1. 3.1 The “main formula” and pseudodifferential operators on $\mathbb{R}^{n}$
    2. 3.2 Pseudodifferential operators on manifolds
    3. 3.3 Compact manifolds
    4. 3.4 Classical symbols and vector bundles
    5. 3.5 Fredholm operators and ellipticity
  4. 4 Translation invariance in a neighborhood of infinity
    1. 4.1 Definition of the $\operatorname{inv}$-calculus
    2. 4.2 The first properties of the $\operatorname{inv}$-calculus
    3. 4.3 Limit operators
    4. 4.4 Further properties of the $\operatorname{inv}$-calculus
  5. 5 Essentially translation invariant operators
    1. 5.1 Definition of essentially translation invariant operators
    2. 5.2 Properties of essentially translation invariant operators
    3. 5.3 Translation invariance and the Fourier transform
    4. 5.4 Including vector bundles
  6. 6 Douglis-Nirenberg-elliptic operators
    1. 6.1 Definition of Douglis-Nirenberg ellipticity
    2. 6.2 Consequences of regularity
  7. 7 Restriction, Fredholm, and spectral invariance properties
    1. 7.1 Restriction to submanifolds and layer potentials
    2. 7.2 Spectral invariance
    3. 7.3 Definition of layer potential operators
    4. 7.4 Fredholm conditions

##  1\. Introduction

Boundary value problems play a very prominent role in the applications of
mathematics to other areas of science and engineering. The analysis of
boundary value problems is mainly based on _two approaches._ The most commonly
used one relies on energy methods. The second one is based on layer
potentials, which are pseudodifferential operators if the boundary is nice
enough. While less elementary than the energy method approach, the layer
potentials approach to boundary value problems possesses several advantages.

This paper is the second paper in a series of papers in which _we study the
method of layer potentials on manifolds with cylindrical ends._ To this end,
we need to develop two precise calculi of pseudodifferential operators that
would contain our differential operators and their associated layer potential
operators, the “$\operatorname{inv}$-calculus” and the “essentially
translation invariant at infinity calculus”. These two calculi were originally
introduced in [44], with the same motivation of studying the layer potential
operators on manifolds with cylindrical ends. In this paper, we complete and
significantly extend the results of that paper, providing also an alternative
approach to several results in that papers. We thus provide a more
systematical account of the properties of these two calculi. We also more
carefully indicate the relations between our calculi and the closely related
$b$\- and $c$-calculi [40, 41, 42, 49, 50] and to the $c$-calculus of Melrose
and Mazzeo-Melrose. We also work consistently in the framework of Douglis-
Nirenberg-elliptic operators.

###  1.1. The framework and some earlier results

A manifold with _straight cylindrical ends_ is a Riemannian manifold
$({\mathfrak{M}},g)$ isometrically diffeomorphic to one of the form

(1.1)
${\mathfrak{M}}\,=\,{\mathfrak{M}}_{0}\cup(\partial{\mathfrak{M}}_{0}\times(-\infty,0])\,,$

where ${\mathfrak{M}}_{0}$ is a smooth manifold with boundary, the “end”
$\partial{\mathfrak{M}}_{0}\times(-\infty,0]$ is endowed with a product
metric, and we identify $\partial{\mathfrak{M}}_{0}$ with
$\partial{\mathfrak{M}}_{0}\times\\{0\\}$. Most of the time
${\mathfrak{M}}_{0}$ will be compact, but, in order to be able to include
$T^{*}{\mathfrak{M}}$ in our setting, we allow also the case
${\mathfrak{M}}_{0}$ non-compact. See Definition 2.1 and Remark 2.2 for
details. In our applications to layer potentials, ${\mathfrak{M}}$ will be, in
fact, the _boundary_ of a manifold with boundary and straight cylindrical ends
$\Omega$, on which our boundary value problem will be posed. Manifolds with
straight cylindrical ends (for which our calculi were designed) are important
because they can be used to study manifolds with _straight conical_ points via
the Kondratiev transform $r=e^{t}$ ($r$ being the distance to the conical
points and $t$ the coordinate on the cylindrical end). Manifolds with straight
conical points are, of course, less general than manifolds with (plain)
conical points, but they appear often enough in applications to justify an
explicit study, which yields, in particular, stronger results than the ones
available in the general case.

In view of our applications, we develop two pseudodifferential calculi on
${\mathfrak{M}}$ that are specifically tailored to manifolds with straight
cylindrical ends. The two calculi are

${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\ \subset\
{\Psi_{\operatorname{ess}}^{m}}({\mathfrak{M}})\,,$

and are both closely related to the $b$-calculus of Melrose [41, 42] and
Schulze [49, 50] and to the $c$-calculus of Melrose and Mazzeo-Melrose [40].
The first calculus is called the “inv”-calculus and consists of
pseudodifferential operators that are translation invariant in a neighborhood
of infinity on ${\mathfrak{M}}$. The second calculus is called the
“essentially translation invariant”-calculus and is a spectrally invariant
enlargement of the “inv”-calculus.

The motivation for considering these two calculi is the following. The
“inv”-calculus is the smallest calculus that we could imagine and contains the
operators we are interested in on manifolds with straight cylindrical ends.
Saying that an operator belongs to this calculus is a strong statement,
stronger than saying that it belongs to the $b$-calculus (which is larger than
the “inv”-calculus). In general, it is convenient in applications to use the
smallest calculus that does the job. This justifies considering the
“inv”-calculus. The “inv”-calculus is, however, not spectrally invariant,
which then motivated us to consider the slightly larger calculus of
“essentially translation invariant” operators, which we prove to be spectrally
invariant. Again, the resulting algebra is one of the smallest one that is
spectrally invariant and is adapted to the geometry, which again justifies its
study.

As we have mentioned above, our calculi are closely related to the $b$ and
$c$-calculi [40, 41, 42, 49, 50], so we hope that our results will help the
reader better understand those calculi as well. See also [4, 17, 35, 36, 51].
Versions of “inv” and “essentially translation invariant” calculi were
introduced and briefly studied in [44]. Here we provide a more general version
of these calculi and a more systematic account. For our applications, we were
especially interested in the spectral invariance property and the Fredholm
property (Theorem 7.4 and 7.12). There was a lot of work on spectral
invariance properties for pseudodifferential operators. A few more recent
papers include [2, 11, 12, 18, 34, 40, 47]. There was also a lot of work on
Fredholm conditions for pseudodifferential operators. A few more recent
publications include [8, 13, 15, 28, 33, 40, 43, 46, 47, 49, 50]. See the
aforementioned publications for further references.

###  1.2. Main results.

We first develop and study the general properties of two pseudodifferential
calculi mentioned above on ${\mathfrak{M}}$, the “inv” and the “essentially
translation invariant” calculi
(${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ and
${\Psi_{\operatorname{ess}}^{m}}({\mathfrak{M}})$) using symbols in the
classes $S_{1,0}^{m}$ [1, 26, 45, 56, 57, 59]. We prove that these calculi are
stable under products and under adjoints. We show that they induce bounded
operators between suitable Sobolev spaces. We prove that they enjoy the usual
symbolic properties, which leads to regularity results. We introduce and study
natural topologies on these spaces of pseudodifferential operators. We provide
a description of the distribution kernels of the classical, negative order
pseudodifferential operators in our calculi, which is crucial for our
applications to layer potentials. We also extend the concept of normal
operator (which we call “limit operator” for historical reasons) to our
calculi and use it to study the structure of our algebras. These results on
the “inv”-calculus can be found in Section 4 and those about the “essentially
translation invariant”-calculus can be found in Section 5. In these two
sections, we work mostly with operators whose symbols belong to the Hörmander
classes $S_{1,0}^{m}$.

The results of these two sections are important for establishing our three
main results needed in applications: compatiblity with restrictions to
submanifolds (Theorem 7.1), the spectral invariance (Theorem 7.4) and the
Fredholm property (Theorem 7.12). We prove all of the three main results in
the framework of Douglis-Nirenberg-elliptic operators, which is another
novelty of our approach. This is an important feature since our main planned
application is to the Stokes operator. These three main results, however, are
proved in the setting of classical pseudodifferential operators. (The general
case is significantly different and including it would have made this paper
significantly longer.) These results are crucial in our applications to the
study of layer potentials, in particular, these results allow us to introduce
the layer potential operators associated to an invertible classical
pseudodifferential operator of low order. The proof that our calculi behave
well with respect to restrictions to (suitable) submanifolds is the reason for
our approach to the pseudodifferential calculi via distribution kernels.

It can be proved (but it is non-trivial) that the
$\operatorname{inv}$-calculus is a subset of the $b$-calculus and that the
essentially translation invariant calculus is a subset of the $c$-calculus.
(Checking this is an excellent exercise for understanding the $b$-calculus and
the differences between the $b$\- and $c$-calculi). Some of the properties of
our calculi would follow then from the aforementioned inclusions. For our
purposes, however, it was more convenient to include direct proofs. That makes
our paper easier to follow by people with a limited knowledge of
pseudodifferential operators and, hopefully, will make pseudodifferential
operators more accessible. Our paper can also serve as an introduction to the
$b$\- and $c$-calculi. Most importantly, our approach is useful for studying
restrictions to submanifolds (Theorem 7.1), which is needed for the definition
of the layer potential operators.

###  1.3. Contents of the paper

The paper is organized as follows. Section 2 contains definitions and results
on manifolds with _straight_ cylindrical ends and the corresponding Sobolev
spaces. The manifolds with straight cylindrical ends play a central role in
our developments. In Section 3, we recall a few basic results on
pseudodifferential operators (the reader familiar with the topic can skip this
section at a first reading). In Section 4, we recall and study the definition
and the properties of the calculus
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ of pseudodifferential
operators that are invariant in a neighborhood of infinity (the
“inv”-calculus), including a description of the distribution kernel of the
classical operators in this calculus. In Section 5, we introduce and study the
calculus ${\Psi_{\operatorname{ess}}^{\infty}}({\mathfrak{M}})$ of essentially
translation invariant (in a neighborhood of infinity) pseudodifferential
operators. This is a suitable enlargement of
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ that is stable under
inversion of elliptic operators of non-negative order. This stability result
is contained in Theorem 7.4 in Section 7, where we also prove the restriction
to submanifold property (Theorem 7.1) and the Fredholm property for these
operators (Theorem 7.12). A novelty of our approach is that we prove our three
basic results (restriction to submanifolds, the spectral invariance and the
Fredholm property) in the framework of Douglis-Nirenberg-elliptic operators.
The Douglis-Nirenberg-ellipticity and related regularity results are discussed
in Section 6. We expect our methods to extend to other classes of non-compact
manifolds.

We thank Cipriana Anghel, Nadine Große, Massimo Lanza de Cristoforis, and
Sergiu Moroianu for useful discussions. We also thank Joerg Seiler and Elmar
Schrohe for useful discussions and for sending us their papers. The general
introduction to pseudodifferential operators is based on our lectures given at
the “Summer school in Geometric Analysis” at _Université Libre de Bruxelles,_
August-September 2023, school organized by Haydys Andriy and Joel Fine whom we
thank for their kind invitation.

##  2\. Background: manifolds with cylindrical ends and Sobolev spaces

We include in this background section several definitions and results on
manifolds with cylindrical ends and their Sobolev spaces. As explained in the
Introduction, we shall consider only the simplest class of manifolds with
cylindrical ends, the manifolds with “straight cylindrical ends,” because of
its applications and because in this case we can obtain stronger results. Let
us add, nevertheless, that the main difference between the various classes of
manifolds with cylindrical ends is the choice of the relevant algebra of
pseudodifferential operators: choosing a smaller algebra leads to stronger
results on the layer potential operators, but if the algebra is too small, it
will not contain the relevant layer potential operators. In any case, these
algebras will be closely related to the $b$-calculus. There are many further
references on the $b$-calculus, see, for instance [16, 17, 35, 36, 48]. The
general case of manifolds with cylindrical ends requires a different
pseudodifferential calculus and will be treated in a subsequent joint work
with Cipriana Anghel and Sergiu Moroianu, building also on the results on the
$c$-calculus in [4].

###  2.1. General notation: duals and connections

We shall use the following conventions for the dual spaces that we use. First,
$V^{\prime}$ denotes the dual space of a real or complex topological vector
space $V$ and, if $T:V\to W$ is a linear map, then $T^{\prime}:W^{\prime}\to
V^{\prime}$ is its dual. Similarly, if $V$ and $W$ are complex vector spaces
endowed with inner products, then $T^{*}:W\to V$ is the adjoint of $T:V\to W$.
In case of real vector spaces (still endowed with an inner product), instead
of the adjoint we have the transpose $T^{\top}:W\to V$. By $V^{*}$ we shall
denote the complex conjugate of $V^{\prime}$. When there is no danger of
confusion, we use the notation $1$ for the identity operator of any given
space.

Let $M$ be a smooth manifold. The space of smooth sections of a smooth vector
bundle $E\to M$ will be denoted ${\mathcal{C}}^{\infty}(M;E)$ and the space of
smooth sections that in addition have compact support will be denoted
${\mathcal{C}}^{\infty}_{\text{c}}(M;E)$. From now on, $({\mathfrak{M}},g)$
will be a Riemannian manifold. We can use the fixed metric on ${\mathfrak{M}}$
to trivialize the density bundle on ${\mathfrak{M}}$ and hence identify the
dual of the space ${\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})$ with a
space of distributions on ${\mathfrak{M}}$. As usual,
$T{\mathfrak{M}}\to{\mathfrak{M}}$ is the tangent bundle to ${\mathfrak{M}}$
and $T^{*}{\mathfrak{M}}\to{\mathfrak{M}}$ is the cotangent bundle to
${\mathfrak{M}}$ (the dual of $T{\mathfrak{M}}$). On these bundles, as well as
on their tensor products, we shall consider the Levi-Civita connection
$\nabla^{LC}$. All vector bundles $E$ considered in this paper will be finite
dimensional, smooth, Hermitian, and will be endowed with metric compatible
connections $\nabla^{E}$. (We shall write simply $\nabla$ instead of
$\nabla^{E}$ when there is no risk of confusion, that is most of the time,
since there will be typically only one connection on each vector bundle.) A
_closed manifold_ is a smooth, compact manifold without boundary.

###  2.2. Manifolds with cylindrical ends

Let $({\mathfrak{L}},g_{{\mathfrak{L}}})$ be a Riemannian manifold and
$I\subset\mathbb{R}$. Unless stated otherwise, we shall endow
${\mathfrak{L}}\times I$ with the product metric

(2.1) $g_{{\mathfrak{L}}\times I}\,=\,g_{{\mathfrak{L}}}+(dt)^{2}\,,$

where $(dt)^{2}$ is the standard Euclidean metric on $I$. Throughout this
paper, ${\mathfrak{M}}$ will be a Riemannian manifold (in fact, for the most
part, a manifold with straight cylindrical ends). The metric on
${\mathfrak{M}}$ will be denoted simply $g$ and $\operatorname{dist}_{g}$ will
be the distance function on ${\mathfrak{M}}$ defined by $g$. From now on,
${\mathfrak{L}}$ will denote a closed manifold (i.e. smooth, compact, without
boundary) endowed with a metric.

######  Definition 2.1.

We say that ${\mathfrak{M}}$ has _straight cylindrical ends_ if it is
geodesically complete and there exists a Riemannian manifold
${\mathfrak{L}}\neq\emptyset$ and an isometry
$\psi_{{\mathfrak{M}}}:{\mathfrak{L}}\times(-\infty,0)\to{\mathfrak{M}}$ whose
complement is a manifold with boundary ${\mathfrak{M}}_{0}$. (On
${\mathfrak{L}}\times(-\infty,0)$ we consider the product metric, as usual.)
If ${\mathfrak{M}}_{0}$ is compact, we shall say that ${\mathfrak{M}}$ has
compact ends.

Figure 1. A picture of a manifold with straight cylindrical ends.

A manifold with cylindrical ends is thus a non-compact, Riemannian manifold
without boundary. If it has compact ends, it is, moreover, with bounded
geometry. (This can be seen, for instance, from [3, 21].) We let
$r_{{\mathfrak{M}}}$ denote its injectivity radius.

######  Remark 2.2.

We use the notations of Definition 2.1. By setting ${\mathfrak{M}}_{0}$ to be
the complement of $\psi_{{\mathfrak{M}}}({\mathfrak{L}}\times(-\infty,0))$, we
see that an equivalent way of formulating Definition 2.1 is that the
Riemannian manifold ${\mathfrak{M}}$ contains a smooth submanifold
${\mathfrak{M}}_{0}\subset{\mathfrak{M}}$ with boundary
$\emptyset\neq\partial{\mathfrak{M}}_{0}$ for which there exists a
diffeomorphism

$\phi:\partial{\mathfrak{M}}_{0}\times(-\infty,0]\to{\mathfrak{M}}\smallsetminus\stackrel{{\scriptstyle\circ}}{{{\mathfrak{M}}}}_{0}$

such that $\phi(x,0)=x$ for $x\in\partial{\mathfrak{M}}_{0}$ and the induced
metric on $\partial{\mathfrak{M}}_{0}\times(-\infty,0]$ is the product of the
metric on $\partial{\mathfrak{M}}_{0}$ and the standard metric on
$(-\infty,0]$. We shall use this notation (as in [44]) and assume from now on,
for simplicity, that $\phi$ is the identity map, which recovers the Equation
(1.1) of the Introduction. Formula (1.1) provides a standard decomposition of
${\mathfrak{M}}$.

######  Remark 2.3.

When studying ${\mathfrak{M}}$, it will be necessary to consider also
$\partial{\mathfrak{M}}_{0}\times\mathbb{R}$, which is also a manifold with
straight cylindrical ends.

From now on, ${\mathfrak{M}}$ will always be a manifold with straight
cylindrical ends
(${\mathfrak{M}}={\mathfrak{M}}_{0}\cup\big{(}\partial{\mathfrak{M}}_{0}\times(-\infty,0]\big{)}$)
_or_ a closed manifold. (This is needed for symmetry, because the results that
hold for manifolds with cylindrical ends hold trivially also for compact
manifolds. In fact, if in the definition of a manifold with cylindrical ends
one allows ${\mathfrak{L}}=\partial{\mathfrak{M}}_{0}=\emptyset$ and
${\mathfrak{M}}_{0}$ is compact, the resulting manifold ${\mathfrak{M}}$ is
compact without boundary).

###  2.3. Compatible vector bundles and differential operators

We now introduce the vector bundles and the differential operators that we
will use.

######  Definition 2.4.

Let ${\mathfrak{M}}$ be a manifold with straight cylindrical ends (as in
Remark 2.2, whose notation we continue to use) and let $E\to{\mathfrak{M}}$ be
a Hermitian vector bundle. We shall say that $E$ is _compatible with the
straight cylindrical ends structure on ${\mathfrak{M}}$_ (_compatible_ , for
short) if there exists a constant $R_{E}<0$ such that
$E|_{\partial{\mathfrak{M}}_{0}\times(-\infty,R_{E}]}$ is isometric to the
pull-back to $\partial{\mathfrak{M}}_{0}\times(-\infty,R_{E}]$ of a Hermitian
vector bundle $E_{0}\to\partial{\mathfrak{M}}_{0}$ with connection
$\nabla^{E_{0}}$ and that $E$ has a metric preserving connection which, on
$\partial{\mathfrak{M}}_{0}\times(-\infty,R_{E}]$ coincides with the pull-back
connection of $\nabla^{E_{0}}$. (In the text, we shall often use the same
notation for $E$ and $E_{0}$.)

From now on, $E\to{\mathfrak{M}}$ will always denote a vector bundle that is
compatible with the straight cylindrical ends structure on ${\mathfrak{M}}$.
The following remark justifies our interest in compatible vector bundles.

######  Remark 2.5.

A compatible vector bundle $E\to{\mathfrak{M}}$ as in Definition 2.4 above
will thus also have a product structure on
$\partial{\mathfrak{M}}_{0}\times(-\infty,R_{E}]$. Hence
$\nabla^{E}=\nabla^{E_{0}}+dt\otimes\partial_{t}$ on
$\partial{\mathfrak{M}}_{0}\times(-\infty,R_{E})$. Similarly, the Bochner
Laplacian of $E$ satisfies

$\Delta_{E}\mathrel{\ :=\
}(\nabla^{E})^{*}\nabla^{E}\,=\,(\nabla^{E_{0}})^{*}\nabla^{E_{0}}-\partial_{t}^{2}\,,$

where $\nabla^{E_{0}}$ is the connection on
$E_{0}\to\partial{\mathfrak{M}}_{0}$.

We now introduce an important class of smooth sections.

######  Definition 2.6.

Let $E\to{\mathfrak{M}}$ be as in Remark 2.5 and let
$u\in{\mathcal{C}}^{\infty}({\mathfrak{M}};E)$. We shall say that $u$ is
_translation invariant in a neighborhood of infinity_ (or _translation
invariant at infinity_) if there exists $R_{u}\leq R_{E}$ such that $u(x,t)$
is independent of $t$ for $t\leq R_{u}$. We let
$\mathcal{R}_{\infty}u(x):=u(x,t)$ for $-t$ large. We let
${\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};E)$ denote the set
of sections of $E$ that are translation invariant in a neighborhood of
infinity. We identify
${\mathcal{C}}^{\infty}(\partial{\mathfrak{M}}_{0};E)\simeq{\mathcal{C}}^{\infty}(\partial{\mathfrak{M}}_{0}\times\mathbb{R};E)^{\mathbb{R}}$
and call the induced map

$\begin{gathered}\mathcal{R}_{\infty}:{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};E)\to{\mathcal{C}}^{\infty}(\partial{\mathfrak{M}}_{0}\times\mathbb{R};E)^{\mathbb{R}}\simeq{\mathcal{C}}^{\infty}(\partial{\mathfrak{M}}_{0};E)\\\
u(x,t)\,=:\,\mathcal{R}_{\infty}u(x)\in E\quad\mbox{for }t\leq
R_{u}\,,\end{gathered}$

the _limit at infinity_ map. We let
${\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}}):={\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};\mathbb{C})$.

We now introduce similarly the differential operators we are interested in.

######  Notation 2.7.

Let $F\to{\mathfrak{M}}$ be a second vector bundle on ${\mathfrak{M}}$
compatible with its straight cylindrical ends at infinity. We shall denote by
$\operatorname{Diff}_{\operatorname{inv}}^{m}({\mathfrak{M}};E,F)$ the set of
differential operators $P$ acting on sections of $E$ such that there exist
$a_{j}\in{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};\operatorname{Hom}(T^{*\otimes
j}{\mathfrak{M}}\otimes E,F))$, $j=0,\ldots,m$, with the property that

$Pu\,=\,\sum_{j=0}^{m}a_{j}(\nabla^{E})^{j}u\,.$

In particular,

$\nabla^{LC}\in\operatorname{Diff}^{1}_{\operatorname{inv}}({\mathfrak{M}};T{\mathfrak{M}},T^{*}{\mathfrak{M}}\otimes
T{\mathfrak{M}})\,.$

We let

$\mathcal{R}_{\infty}(P)\mathrel{\ :=\
}\sum_{j=0}^{m}\mathcal{R}_{\infty}(a_{j})(\nabla^{E})^{j}$

be the associated operator on $\partial{\mathfrak{M}}_{0}\times\mathbb{R}$
obtained by “freezing the coefficients” of $P$ near infinity. It will be
called the _limit at infinity_ operator associated to $P$.

We shall need the following simple result.

######  Lemma 2.8.

Let $E,F,G\to{\mathfrak{M}}$ be compatible vector bundles on our manifold with
straight cylindrical ends ${\mathfrak{M}}$. We use the notation introduced in
this subsection.

  1. (1)

The limit at infinity map
$\mathcal{R}_{\infty}:{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};E)\to{\mathcal{C}}^{\infty}(\partial{\mathfrak{M}}_{0}\times\mathbb{R};E)^{\mathbb{R}}$
of Defintion 2.6 is surjective and

$\ker\mathcal{R}_{\infty}\,=\,{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}};E)\,.$

  2. (2)

${\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};\operatorname{Hom}(E,F))={\mathcal{C}}^{\infty}({\mathfrak{M}};\operatorname{Hom}(E,F))\cap\operatorname{Diff}_{\operatorname{inv}}^{m}({\mathfrak{M}};E,F)$.

  3. (3)

$\operatorname{Diff}_{\operatorname{inv}}^{m}({\mathfrak{M}};F,G)\operatorname{Diff}_{\operatorname{inv}}^{m^{\prime}}({\mathfrak{M}};E,F)\subset\operatorname{Diff}_{\operatorname{inv}}^{m+m^{\prime}}({\mathfrak{M}};E,G)$.

  4. (4)

$\operatorname{Diff}_{\operatorname{inv}}^{m}({\mathfrak{M}};E,F){\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};E)\subset{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};F)$.

  5. (5)

$\mathcal{R}_{\infty}:\operatorname{Diff}_{\operatorname{inv}}^{m}({\mathfrak{M}};E,F)\to\operatorname{Diff}_{\operatorname{inv}}^{m}(\partial{\mathfrak{M}}_{0}\times\mathbb{R};E,F)^{\mathbb{R}}$
is well-defined, surjective, and multiplicative.

###### Proof.

Everything follows immediately from the definitions. (For (3) see also [30].)
Let us just add that for (5) we first prove that that
$\mathcal{R}_{\infty}:{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};\operatorname{Hom}(E,F))\to{\mathcal{C}}^{\infty}_{\operatorname{inv}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R};\operatorname{Hom}(E,F))^{\mathbb{R}}$
is well-defined, surjective, and multiplicative and that
$\mathcal{R}_{\infty}(\nabla^{LC})=\nabla^{LC}.$ ∎

We have the following simple consequence (recall that $\nabla$ is the generic
notation for our connections).

######  Corollary 2.9.

Let $u\in{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};E)$ (see
Definition 2.6), then $u$ is bounded and, for all $k\geq 1$,
$\nabla^{k}u\in{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};E\otimes
T^{*\otimes k}{\mathfrak{M}})$. Let
$a\in{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};\operatorname{End}(E))$
with $a\neq 0$ everywhere. Then $a$ is bounded and
$a^{-1}\in{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}};\operatorname{End}(E))$.
In particular, $\mathcal{R}_{\infty}(a)\neq 0$ everywhere.

###### Proof.

This follows from Definition 2.6, Lemma 2.8. We have that
$\mathcal{R}_{\infty}(a)$ is a restriction of $a$, so it doesn’t vanish
either. ∎

Although we shall not really use it, a useful picture when ${\mathfrak{M}}$
has compact ends is that one can “compactify” ${\mathfrak{M}}$ to
$\overline{{\mathfrak{M}}}$ by including the “boundary at infinity”
$\partial{\mathfrak{M}}_{0}\times\\{-\infty\\}$. All sections that are
translation invariant at infinity will extend to sections on this
compactification. In particular, the boundedness of $u$ in Corollary 2.9
follows from the fact that it extends to the compactification
$\overline{{\mathfrak{M}}}$.

###  2.4. Sobolev spaces

In the following, $({\mathfrak{M}},g)$ will be a fixed manifold with straight
cylindrical ends and _compact ends._ Let us recall now the definition of the
Sobolev spaces on ${\mathfrak{M}}$. We agree that
$\mathbb{N}=\\{1,2,\ldots,\\}$ and that
$\mathbb{Z}_{+}=\mathbb{N}\cup\\{0\\}$. For simplicity of the notation, we
introduce vector bundles only later.

####  2.4.1. Positive order covariant Sobolev spaces

By $\,d\operatorname{vol}$ we denote the induced volume form on
${\mathfrak{M}}$ associated to the metric $g$ on ${\mathfrak{M}}$. (Note that
the metric $g$ is fixed everywhere in the paper.) For $p\in[1,\infty)$, we
define the norm

(2.2) $\|u\|_{L^{p}({\mathfrak{M}})}\mathrel{\ :=\
}\left(\int_{{\mathfrak{M}}}|u(x)|^{p}\,d\operatorname{vol}(x)\right)^{1/p}\,.$

As usual, we define the space:

(2.3) $L^{p}({\mathfrak{M}})\mathrel{\ :=\
}\\{u:{\mathfrak{M}}\to\mathbb{C}\mid\|u\|_{L^{p}({\mathfrak{M}})}<+\infty\\}/\ker(\|\cdot\|_{L^{p}({\mathfrak{M}})})\,.$

The space $L^{\infty}({\mathfrak{M}})$ and its norm are defined with the usual
modification, using the essential supremum. By
$(f,g)=(f,g)_{L^{2}({\mathfrak{M}})}$ we shall denote the scalar (inner)
product on $L^{2}({\mathfrak{M}})$. It is linear in the first variable and
conjugate linear in the second one.

There are many ways of introducing Sobolev spaces on a manifold. For open
subsets of compact manifolds, the Sobolev spaces can be defined using
partitions of unity supported in coordinate patches (thus reducing to the
Euclidean case), and the resulting norms are equivalent for different choices
of the partition of unity. In the non-compact case, one cannont use any
partition of unity. One can still use connections, though (see, for instance
[22, 30]).

######  Definition 2.10.

Let $k\in\mathbb{Z}_{+}:=\\{0,1,\ldots\\}$. We let

$W^{k,p}({\mathfrak{M}})\mathrel{\ :=\
}\\{u:{\mathfrak{M}}\to\mathbb{C}\mid\nabla^{j}(u)\in
L^{p}({\mathfrak{M}};T^{*\otimes j}{\mathfrak{M}})\,,\,\mbox{ for all }\,0\leq
j\leq k\\}$

be the order $k$, $L^{p}$–type Sobolev space on ${\mathfrak{M}}$ with the norm

$\|u\|_{W^{k,p}({\mathfrak{M}})}\mathrel{\ :=\ }\ell^{p}\mbox{--norm of
}\\{\|\nabla^{j}(u)\|_{L^{p}({\mathfrak{M}};T^{*\otimes j}{\mathfrak{M}})}\,,\
0\leq j\leq k\\}\,.$

We let $H^{k}:=W^{k,2}$, as usual. Since ${\mathfrak{M}}$ is complete (even
with bounded geometry) the Sobolev space $H^{s}({\mathfrak{M}})$ is the domain
of $(1+\Delta)^{s/2}$, for $s\geq 0$ (we use the convention
$\Delta:=d^{*}d\geq 0$). See [3, 21, 54] for related results (valid also in
the more general setting of manifolds with bounded geometry and, sometimes,
valid even in the setting of complete manifolds).

####  2.4.2. Negative and non-integer order covariant Sobolev spaces

We proceed as usual and we define the negative order Sobolev spaces (just for
$p=2$) by

(2.4) $H^{-s}({\mathfrak{M}})\mathrel{\ :=\ }H^{s}({\mathfrak{M}})^{*}\,,$

where $V^{*}$ is the complex conjugate dual of $V$, as before. We also define
the spaces $H^{r}({\mathfrak{M}})$ for $r\notin\mathbb{Z}$ by complex
interpolation between consecutive integers (see, e.g., [6, 37, 55, 59] for
further details).

####  2.4.3. Alternative definition

Sometimes, the following definition of Sobolev spaces will be useful. All
manifolds $\partial{\mathfrak{M}}_{0}\times(-R-k,-R-k+2)$, $k\in\mathbb{N}$,
are isometric and hence the spaces

$H^{m}(\partial{\mathfrak{M}}_{0}\times(-R-k,-R-k+2))\,,\quad
k\in\mathbb{N}\,,$

are also canonically isometric (they can be defined using a finite partition
of unity on $\partial{\mathfrak{M}}_{0}$.) Let

${\mathfrak{M}}_{R}\mathrel{\ :=\
}{\mathfrak{M}}_{0}\cup(\partial{\mathfrak{M}}_{0}\times(-R,0))\,=\,{\mathfrak{M}}\smallsetminus(\partial{\mathfrak{M}}_{0}\times(-\infty,-R])\,.$

Then an equivalent norm on $H^{m}({\mathfrak{M}})$, $m\in\mathbb{R}$, is

(2.5) $|||u|||_{m}^{2}\mathrel{\ :=\
}\|u\|_{H^{m}({\mathfrak{M}}_{R})}^{2}\,+\,\sum_{k=1}^{\infty}\|u\|_{H^{m}(\partial{\mathfrak{M}}_{0}\times(-R-k,-R-k+2))}^{2}\,.$

There exists a smooth partition of unity $(\phi_{k}^{2})$,
$k\in\mathbb{Z}_{+}$, where $\phi_{k}$ is smooth with support in
$\partial{\mathfrak{M}}_{0}\times(-R-k,-R-k+2)$ and is the translate of
$\phi_{1}$ by $k-1$, for $k\geq 1$, and where $\phi_{0}$ is with support in
${\mathfrak{M}}_{R}$. Then the norm on $H^{m}({\mathfrak{M}})$,
$m\in\mathbb{R}$, is also equivalent to

(2.6) $|||u|||_{m}^{\prime 2}\mathrel{\ :=\
}\|\phi_{0}^{2}u\|_{H^{m}({\mathfrak{M}}_{R})}^{2}\,+\,\sum_{k=1}^{\infty}\|\phi_{k}^{2}u\|_{H^{m}(\partial{\mathfrak{M}}_{0}\times(-R-k,-R-k+2))}^{2}\,.$

These equivalences follows, for instance, from [21].

If ${\mathfrak{M}}={\mathfrak{L}}\times\mathbb{R}$, the same results hold true
if we consider the $\mathbb{Z}$-translation invariant partition of unity
$(\phi_{k}^{2})$, $k\in\mathbb{Z}$ (extending from $k\in\mathbb{N}$).

##  3\. Basics on pseudodifferential operators

We include in this section a rather detailed and comprehensive overview of the
standard theory of pseudodifferential operators on $\mathbb{R}^{n}$, on
general manifolds, and on compact manifolds. This serves to fix the notation,
to help the reader, but also to provide a couple of new results that are
needed in the sequel. The material of this section is based on (and expands)
our talks at the “Summer school in Geometric Analysis” at the Free University
of Bruxelles, in 2023. In order to avoid complications, we shall assume that
our manifolds are Hausdorff and paracompact. The results of this section are,
for the most part, not new, but rather classical. A quick, but lucid
introduction to pseudodifferential operators can be found in [56]. Two more
recent accounts that contain also more detail as well as some needed non-
standard results are [1] and [23]. We thank Rafe Mazzeo for brining HintzLN’
book [23] to our attention. The classical reference monograph on
pseudodifferential operators containing also historical material and many
extensions is [26]. Otherwise, there are many other very good textbooks
containing material on pseudodifferential operators, among which we mention
[27, 45, 57, 59]. We include some proofs for the benefit of the reader and as
complementary material (not because any of these results in this section are
really new).

###  3.1. The “main formula” and pseudodifferential operators on
$\mathbb{R}^{n}$

Let $\langle x,\xi\rangle:=x_{1}\xi_{1}+x_{2}\xi_{2}+\ldots x_{n}\xi_{n}$ for
$x,\xi\in\mathbb{R}^{n}$, $\imath:=\sqrt{-1}$, and let

${\hat{u}(\xi)}\,=\,\mathcal{F}u(\xi)\mathrel{\ :=\
}\int_{\mathbb{R}}e^{{-}\imath\langle{y},\xi\rangle}\,{u(y)\,dy}$

be the _Fourier transform_ of $u.$ The following formula

$a(x,D)u(x)\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\,e^{\imath\langle
x,\xi\rangle}\,a(x,\xi){\hat{u}(\xi)}\,d\xi\,,$

will be referred to as “the main formula” (of pseudodifferential theory) and,
as we will see, it gives the “local form” of pseudodifferential operators on
manifolds. In this formula,
$a:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{C}$ is a suitable “symbol”
that will depend on the requirements of the problem. (The terms in quotation
marks will be defined below.) Let us see what the main formula becomes in a
few particular cases of symbols $a:\mathbb{R}^{2n}\to\mathbb{C}$.

######  Example 3.1.

Let $a(x,\xi)=a(x)$ be a smooth function independent of $\xi$. Then

$a(x,D)u(x)\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\,e^{\imath\langle
x,\xi\rangle}\,a(x)\hat{u}(\xi)\,d\xi\,=\,a(x)u(x)\,,$

by the _Fourier inversion formula,_

$(\mathcal{F}^{-1}w)(x)\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}}e^{\imath\langle
x,\xi\rangle}w(\xi)d\xi\,.$

Recall that the _convolution_ of two function
$f,g:\mathbb{R}^{n}\to\mathbb{C}$ is (when defined)

(3.1) $f*g(x)\,=\,\int_{\mathbb{R}^{n}}f(y)g(x-y)dy\,.$

######  Example 3.2.

Let next $a(x,\xi)=a(\xi)$ be a function independent of $\xi$, and integrable
in $\xi$. Then

$\displaystyle a(D)u(x)\,=\,a(x,D)u(x)\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\,e^{\imath\langle
x,\xi\rangle}\,a(\xi)\,\hat{u}(\xi)\,d\xi\,=\,(\mathcal{F}^{-1}a)*u(x)$

is the _convolution_ operator with $\mathcal{F}^{-1}a$. This remains true if
$a\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ (that is, if the function $a$
defines a tempered distribution on $\mathbb{R}^{n}$, in that case “oscillatory
integrals” may be useful; see [58] for a pedagogical introduction and [1] for
a thorough account). In particular, if $a(x,\xi):=a_{j}(\xi):=\imath\xi_{j},$
then

$\displaystyle a_{j}(D)u(x)\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\,e^{\imath\langle
x,\xi\rangle}\,\imath\xi_{j}\,\hat{u}(\xi)\,d\xi\,=\,\frac{\partial
u(x)}{\partial x_{j}}\,=:\,\partial_{j}u(x)\,,$

because the Fourier transform interchanges multiplication by $\imath\xi_{j}$
with $\partial_{j}.$ Morover, if we denote by $a$ also the multiplication
operator with $a$, then $a(D)=\mathcal{F}^{-1}a\mathcal{F}$. Thus, if both $a$
and $b:=a^{-1}$ are in $\mathcal{S}^{\prime}(\mathbb{R}^{n})$, then

$a(D)b(D)\,=\,(\mathcal{F}^{-1}a\mathcal{F})(\mathcal{F}^{-1}b\mathcal{F})\,=\,\mathcal{F}^{-1}ab\mathcal{F}\,=\,1\,.$

In particular, if $a(\xi)=1+|\xi|^{2}$, then $a(D)=1-\Delta$. Let
$b(\xi)=(1+|\xi|^{2})^{-1}$, then $a(D)^{-1}=b(D)$, which is the convolution
operator with a true function.

The next example combines the first two examples.

######  Example 3.3.

Combinging the first two examples and using for any multi-index
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in\mathbb{Z}_{+}^{n}$,
$|\alpha|:=\sum_{j=1}^{n}\alpha_{j},$ and

$\partial^{\alpha}u\mathrel{\ :=\
}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}\ldots\partial_{n}^{\alpha_{n}}u\,=\,{\mathcal{F}^{-1}}\big{(}{(\imath\xi)^{\alpha}}\hat{u}\big{)}\,,\quad(\imath\xi)^{\alpha}\mathrel{\
:=\
}\imath^{|\alpha|}\xi_{1}^{\alpha_{1}}\xi_{2}^{\alpha_{2}}\ldots\xi_{n}^{\alpha_{n}}\,,$

we obtain for $a(x,\xi):=\sum_{|\alpha|\leq
m}\,a_{\alpha}(x)\,(\imath\xi)^{\alpha},$ (a _polynomial in_ $\xi$):

$\displaystyle a(x,D)u(x)$ $\displaystyle\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{\imath\langle
x,\xi\rangle}\underbrace{\sum_{|\alpha|\leq
m}\,a_{\alpha}(x)\,{(\imath\xi)^{\alpha}}}_{a(x,\xi)}\,\hat{u}(\xi)d\xi$
$\displaystyle\,=\,\sum_{{{|\alpha|\leq
m}}}\,a_{\alpha}(x)\,{\partial^{\alpha}}u(x)\,,$

a _differential operator_.

Let us specify now our choice of symbols.

######  Notation 3.4.

We shall denote an element of $\mathbb{R}^{2n}$ by $(x,\xi)$, with
$x,\xi\in\mathbb{R}^{n}$. For $a:U\times\mathbb{R}^{n}\to\mathbb{C}$,
$U\subset\mathbb{R}^{n}$ open, $\alpha,\beta\in\mathbb{Z}_{+}^{n}$, and
$s\in\mathbb{R}$, we let

  1. (1)

$p_{\alpha,\beta}^{[q]}(a)\mathrel{\ :=\ }\sup_{(x,\xi)\in
U\times\mathbb{R}^{n}}\left|\frac{\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)}{(1+|\xi|)^{q-|\beta|}}\right|\quad\mbox{and}$

  2. (2)

$p_{N}^{[q]}(a):=\max_{|\alpha|+|\beta|\leq N}p_{\alpha,\beta}^{[q]}(a).$

  3. (3)

We then let

$S_{unif}^{q}(U\times\mathbb{R}^{n})\mathrel{\ :=\
}\\{a:U\times\mathbb{R}^{n}\to\mathbb{C}\mid\ p_{N}^{[q]}(a)<\infty\mbox{ for
all }N\in\mathbb{Z}_{+}\\}\,.$

This class of symbols was introduced by Kohn and Nirenberg in [29]. See [26]
and [56] for more historical information and for other classes of symbols.

######  Remark 3.5.

We think of $U\times\mathbb{R}^{n}$ as $T^{*}U$. Similar definitions can be
given for any vector bundle $V\to M$ over a manifold $M$, replacing
$\partial_{x}^{\alpha}$ with a suitable covariant derivative and replacing
$\partial_{\xi}^{\beta}$ with derivatives with respect to vertical vector
fields that are translation invariant in the fibers to obtan the spaces
$S_{unif}^{m}(V)$. Notice that in this notation we consider $V$ as a vector
bundle over $M$, not simply as a manifold, and hence this space should better
be denoted $S_{unif}(M;V)$ to indicated the additionals structure on which it
depends. Notice also that we consider estimates that are _uniform_ in $x$, as
in [1, 23, 26, 56, 58]. These definitions can be localized on compact subsets
of $U$ as in [27]. Also, a more general class of symbols is
$S_{\rho,\delta}^{m}(\mathbb{R}^{2n})$ considered by Hörmander, but we do not
need it in our applications.

Let $M$ be a smooth manifold. Recall that a linear map
$T:{\mathcal{C}}^{\infty}_{\text{c}}(M)\to{\mathcal{C}}^{\infty}_{\text{c}}(M)^{\prime}$
is called _continuous_ if $\langle T\phi,\psi\rangle$ is continuous with
respect to $\phi\in{\mathcal{C}}^{\infty}_{\text{c}}(M),$ for any
$\psi\in{\mathcal{C}}^{\infty}_{\text{c}}(M).$ It is not difficult to see
that, if $a\in S_{unif}^{q}(\mathbb{R}^{2n})$, then
$a(x,D):{\mathcal{C}}^{\infty}_{\text{c}}(\mathbb{R}^{n})\to{\mathcal{C}}^{\infty}(\mathbb{R}^{n})$
is well-defined and continuous. (A stronger statement is included below.) This
proposition shows that $a(x,D)$ satisfies the hypotheses of the Schwartz’
kernel theorem, which we recall next. Let $\langle\ ,\
\rangle:{\mathcal{C}}^{\infty}_{\text{c}}(M)^{\prime}\times{\mathcal{C}}^{\infty}_{\text{c}}(M)\to\mathbb{C}$
be the _paring_ between distributions and test functions.

######  Theorem 3.6 (Schwartz’ kernel theorem).

Let $M$ be a smooth manifold. Let
$T:{\mathcal{C}}^{\infty}_{\text{c}}(M)\to{\mathcal{C}}^{\infty}_{\text{c}}(M)^{\prime}$
be linear and continuous. Then there exists a unique
$k_{T}\in{\mathcal{C}}^{\infty}_{\text{c}}(M\times M)^{\prime}$ such that

$\langle T\phi,\psi\rangle\,=\,\langle k_{T},\psi\boxtimes\phi\rangle\,,$

where $\psi\boxtimes\phi\in{\mathcal{C}}^{\infty}_{\text{c}}(M^{2})$ is
defined by $(\psi\boxtimes\phi)(x,y)=\psi(x)\phi(y)\,.$ The distribution
$k_{T}$ is called the _distribution kernel_ of $T.$

The converse is also true, in the sense that every
$K\in{\mathcal{C}}^{\infty}_{\text{c}}(M^{2})^{\prime}$ corresponds to a
continuous linear map
$T:{\mathcal{C}}^{\infty}_{\text{c}}(M)\to{\mathcal{C}}^{\infty}_{\text{c}}(M)^{\prime}$,
namely, $K=k_{T}.$ This is the “easy part” of the theorem but will not be
necessary to us in what follows. A natural question that arises then is to
find the kernel $k_{a(x,D)}$ of the operator $a(x,D)$.

######  Remark 3.7.

If in the _main formula_

$a(x,D)u(x)\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\,e^{\imath\langle
x,\xi\rangle}\,a(x,\xi){\hat{u}(\xi)}\,d\xi$

we substitute the Fourier transform
$\hat{u}(\xi)\,=\,\mathcal{F}u(\xi)=\int_{\mathbb{R}}e^{-i\langle{y},\xi\rangle}\,u(y)\,dy,$
we obtain the _second main formula_ of pseudodifferential operator theory:

$a(x,D)u(x)\,=\,\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}^{n}}\,e^{\imath\langle
x-y,\xi\rangle}\,a(x,\xi)\,{u(y)\,dy}\right)d\xi\,.$

This second main formula allows us to determine the distribution kernel of
$a(x,D)$.

######  Remark 3.8.

Assume first that $a\in S_{unif}^{-n-\epsilon}(\mathbb{R}^{2n})$,
$\epsilon>0$, and let

$K(x,y):=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{\imath\langle
x-y,\xi\rangle}a(x,\xi)d\xi,$

which is a continuous function of $(x,y)$ by the continuity of integrals
depending on parameters by bounding with the integrable function
$(1+|\xi|)^{-n-\epsilon}$. The second main formula (see Remark 3.7) gives

$\displaystyle\langle a(x,D)u,v\rangle$
$\displaystyle=\int_{\mathbb{R}^{n}}\left(\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{\imath\langle
x-y,\xi\rangle}a(x,\xi)\,u(y)dy{d\xi}\right)\,v(x)dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\,K(x,y)v(x)u(y)\,dydx$
$\displaystyle=\langle K,v\boxtimes u\rangle\,,$

using an integration with respect to $\xi.$ Therefore

(3.2) $k_{a(x,D)}(x,y)\,=\,K(x,y)\mathrel{\ :=\
}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{\imath\langle
x-y,\xi\rangle}a(x,\xi)d\xi\,.$

In general, this formula for $k_{a(x,D)}(x,y)$ must be interpreted in
distribution sense, as the Fourier transform of a temperate distribution (this
can be done using oscillatory integrals, as in [1, 58], for instance).

We obtain the following.

######  Proposition 3.9.

Let $m\in\mathbb{R}$ and $a\in S_{unif}^{m}(\mathbb{R}^{2n}).$

  1. (1)

The distribution $k_{a(x,D)}$ is defined by a smooth function on
$\\{(x,y)\in\mathbb{R}^{2n}\mid x\neq y\\}$ (that is, away from the diagonal).

  2. (2)

If $k_{a(x,D)}=0$, then $a=0$.

  3. (3)

Consequently, $k_{a(x,D)}$ determines $a$ and hence the map
$a(x,D):{\mathcal{C}}^{\infty}_{\text{c}}(\mathbb{R}^{n})\to{\mathcal{C}}^{\infty}(\mathbb{R}^{n})$
also determines $a$.

###### Proof.

The stronger statement that
$(x-y)^{\alpha}k_{a(x,D)}(x,y)\in\mathcal{C}^{j}(\mathbb{R}^{2n})$ for
$|\alpha|$ large is contained in the proof of Proposition 2.1 in [56] (this
result holds true also for suitable symbols in the Hörmander class
$S_{\rho,\delta}^{m}$). The second point is obtained from Equation (3.2) by
the Fourier inversion formula. ∎

The following remark explains “what is lost” by looking only at the
distribution kernel on $\\{(x,y)\mid x\neq y\\}.$ It will not be needed for
any proof.

######  Remark 3.10.

If $a,b\in S_{unif}^{q}(\mathbb{R}^{2n})$ and
$k_{a(x,D)}(x,y)=k_{b(x,D)}(x,y)$ for $x\neq y$, then $a(x,D)-b(x,D)$ has
distribution kernel supported on the diagonal $(x,x)\subset\mathbb{R}^{2n}$
and hence it is a differential operator, by Peetre’s theorem. In particular,
the function $k_{a(x,D)}$ away from the diagonal determines any operator of
negative order.

Let us now discuss the mapping properties of our operators. Recall that

$H^{s}(\mathbb{R}^{n})\mathrel{\ :=\ }\Big{\\{}u\,\Big{|}\
\int_{\mathbb{R}^{n}}(1+|\xi|)^{2s}|\hat{u}(\xi)|^{2}\,d\xi<+\infty\,\Big{\\}}\,,\quad
s\in\mathbb{R}\,.$

In particular, if $m\in\mathbb{Z}_{+}$,

$H^{m}(\mathbb{R}^{n})\mathrel{\ :=\ }\big{\\{}u\,\big{|}\
\partial^{\alpha}u\in L^{2}(\mathbb{R}^{n})\ \mbox{ for all }\
|\alpha|\leq{m}\,\big{\\}}\,.$

These definitions coincide, of course, with our previous definitions of
Subsection 2.4. We let

(3.3) $\Psi_{unif}^{m}(\mathbb{R}^{n})\mathrel{\ :=\ }\\{a(x,D)\mid a\in
S_{unif}^{m}(\mathbb{R}^{2n})\\}$

and (for some fixed $p\in[1,\infty]$), we let

$\mathcal{S}(\mathbb{R}^{n}):=\\{u:\mathbb{R}^{n}\to\mathbb{C}\mid
x^{\alpha}\partial^{\beta}u\in
L^{p}(\mathbb{R}^{n}),\,\forall\,\alpha,\beta\in\mathbb{Z}_{+}^{n}\\},$

be the usual _Schwartz space_ on $\mathbb{R}^{n}$ with the induced norms (the
resulting space does not depend on $p\in[1,\infty]$, but its semi-norms do).

######  Theorem 3.11.

Let $s,m\in\mathbb{R}$ and $\Psi_{unif}^{m}(\mathbb{R}^{n})$ be as in Equation
(3.3).

  1. (1)

The map
$S_{unif}^{m}(\mathbb{R}^{2n})\times\mathcal{S}(\mathbb{R}^{n})\ni(a,u)\to
a(x,D)u\in\mathcal{S}(\mathbb{R}^{n})$ is well defined and continuous.

  2. (2)

If $a\in S_{unif}^{m}(\mathbb{R}^{2n})$ and $b\in
S_{unif}^{m^{\prime}}(\mathbb{R}^{2n}),$ then $ab\in
S_{unif}^{m+m^{\prime}}(\mathbb{R}^{2n})$ and

$b(x,D)a(x,D)-(ab)(x,D)=c(x,D)\in\Psi_{unif}^{m+m^{\prime}-1}(\mathbb{R}^{n})\,.$

Moreover, for any $N\in\mathbb{Z}_{+}$, there is $C_{N}>0$ such that (see 3.4)

$p_{N}^{[m+m^{\prime}-1]}(c)\leq
C_{N}\sum_{j=1}^{n}\big{[}p_{N}^{[m]}(a)p_{N}^{[m^{\prime}-1]}(\partial_{\xi_{j}}b)+p_{N}^{[m-1]}(\partial_{\xi_{j}}a)p_{N}^{[m^{\prime}]}(b)\big{]}\,.$

In particular, if $P\in\Psi_{unif}^{m}(\mathbb{R}^{n})$ and
$Q\in\Psi_{unif}^{m^{\prime}}(\mathbb{R}^{n})$, then
$PQ\in\Psi_{unif}^{m+m^{\prime}}(\mathbb{R}^{n})$.

  3. (3)

$P=a(x,D):H^{s}(\mathbb{R}^{n})\to H^{s-m}(\mathbb{R}^{n})$ is bounded and its
norm depends continuously on $a\in S_{unif}^{m}(\mathbb{R}^{2n})$.

  4. (4)

Similarly, if $a(x,D)\in\Psi_{unif}^{m}(U)$, then
$a(x,D)^{*}-\overline{a}(x,D)\in\Psi_{unif}^{m-1}(U)$. In particular, if
$P\in\Psi_{unif}^{m}(\mathbb{R}^{n})$, then
$P^{*}\in\Psi_{unif}^{m}(\mathbb{R}^{n})$.

###### Proof.

Most of these results are standard. We include only some references to the
less standard material. The point (1) is a particular case of Theorem 3.6. in
[1]. See also Proposition 4.4 in [23]. The norm estimate in (2) follows from
Proposition 3.5 and Theorems 3.15 and 3.16 in [1]. (Proposition 3.5 and
Theorem 4.8 and 4.16 from [23] are also relevant in this regard.) The
continuity between the Sobolev spaces is of course very well-known, the
slightly stronger statement provided here is a consequence of the proof of
continuity on $L^{2}(\mathbb{R}^{n})$ using Hörmander’s trick [26] in [1, 23],
using also point (2). ∎

Let
$S_{unif}^{m}/S_{unif}^{m-1}(\mathbb{R}^{2n}):=S_{unif}^{m}(\mathbb{R}^{2n})/S_{unif}^{m-1}(\mathbb{R}^{2n})$
and

(3.4) $\sigma_{m}:\Psi_{unif}^{m}(\mathbb{R}^{n})\to
S_{unif}^{m}/S_{unif}^{m-1}(\mathbb{R}^{2n})\,,\quad\sigma_{m}(a(x,D))\mathrel{\
:=\ }a+S_{unif}^{m-1}(\mathbb{R}^{2n})\,,$

which is well-defined by Proposition 3.9. Theorem 3.11(2) then gives right
away that $\sigma_{m+m^{\prime}}(QP)=\sigma_{m^{\prime}}(Q)\sigma_{m}(P).$

We shall need also a “local form” of the last theorem. Let
$U\subset\mathbb{R}^{n}$ be open and let

$\begin{gathered}\Psi_{comp}^{m}(U)\mathrel{\ :=\
}\\{a(x,D)\mid\,\operatorname{supp}{k_{a(x,D)}}\Subset U\times
U\\}\quad\mbox{and}\\\\[2.84526pt] S_{comp}^{m}(T^{*}U)\mathrel{\ :=\ }\\{a\in
S_{unif}^{m}(\mathbb{R}^{2n})\mid\operatorname{supp}a\subset
K\times\mathbb{R}^{n}\,,\ K\Subset U\\}\,,\end{gathered}$

where $A\Subset B$ means that the closure of $A$ is a compact subset of the
interior of $B$. Remark 3.8 shows that $a(x,D)\in\Psi_{comp}^{m}(U)\Rightarrow
a\in S_{comp}^{m}(T^{*}U)\subset S_{unif}^{m}(\mathbb{R}^{2n})$ and that every
$P\in\Psi_{comp}^{m}(U)$ maps ${\mathcal{C}}^{\infty}_{\text{c}}(U)$ to
itself. In particular, we can compose these actions.

######  Theorem 3.12.

For $U\subset\mathbb{R}^{n}$ an open subset,
$\Psi_{comp}^{\infty}(U):=\cup_{m\in\mathbb{Z}}\Psi_{comp}^{m}(U)$ is an
algebra invariant under diffeomorphisms and under adjoints.

The adjoints are taken with respect to the usual metric on $\mathbb{R}^{n}$.
Let us make the diffeomorphism invariance more precise.

######  Remark 3.13.

Let us consider in Theorem 3.12 a diffeomorphism $\phi:U\to W$ of two open
subsets of $\mathbb{R}^{n}$. Then $\phi$ induces:

  1. (1)

a vector bundle isomorphism $\phi^{*}:T^{*}W\to T^{*}U,$ where

$\phi^{*}(x,\xi)=(\phi^{-1}(x),(d\phi)^{\top}\xi)$

is the action of $\phi$ between the cotangent bundles (with $L^{\top}$ being
the _transpose_ , or dual, of $L$);

  2. (2)

a linear isomorphism
$\phi_{*}:{\mathcal{C}}^{\infty}_{\text{c}}(U)\to{\mathcal{C}}^{\infty}_{\text{c}}(W),$
where $\phi_{*}u:=u\circ\phi^{-1}$; and

  3. (3)

a linear isomorphism $\phi_{*}:S_{comp}^{m}(T^{*}U)\to S_{comp}^{m}(T^{*}W),$
where $\phi_{*}a=a\circ\phi^{*}$.

Let $a(x,D)\in\Psi_{comp}^{m}(U)$, then $\phi_{*}a=a\circ\phi^{*}\in
S_{comp}^{m}(T^{*}W)$ and a very important result [25, 29] is that

$\phi_{*}\circ
a(x,D)\circ\phi_{*}^{-1}-(\phi_{*}a)(x,D)\in\Psi_{comp}^{m-1}(W)\,.$

Consequently, $\phi_{*}\circ a(x,D)\circ\phi_{*}^{-1}\in\Psi_{comp}^{m}(W)$
and this gives the diffeomorphism invariance of the spaces
$\Psi_{comp}^{m}(U)$ with $U\subset\mathbb{R}^{n}$ open. (See [26] for
historical comments and [56] for an inventive proof of the diffeomorphism
invariance using Egorov’s theorem).

Let $\phi:U\to W$ be a diffeomorphism, as before. Then a consequence of the
above discussion is that

$\textstyle{S_{comp}^{m}(T^{*}U)\ni a}$$\textstyle{\phi_{*}(a)\in
S_{comp}^{m}(T^{*}W)}$$\textstyle{{\Psi_{comp}^{m}(U)}\ni
a(x,D)}$$\textstyle{\
\phi_{*}a(x,D)\phi_{*}^{-1}\in\Psi_{comp}^{m}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{*}}$

commutes up to _lower order symbols._ This allows to obtain the local
definition of the principal symbol on $\mathbb{R}^{n}$.

######  Theorem 3.14.

If $a\in S_{unif}^{m}(\mathbb{R}^{2n})$ is such that
$a(x,D)\in\Psi_{comp}^{m}(U)$, then the _principal symbol_

$\sigma_{m}(a(x,D))\mathrel{\ :=\ }a+S_{unif}^{m-1}(T^{*}U)\in
S_{unif}^{m}(T^{*}U)/S_{unif}^{m-1}(T^{*}U)$

is well-defined and diffeomorphism invariant in the sense that

$\sigma_{m}(\phi_{*}a(x,D)\phi_{*}^{-1})\,=\,\phi_{*}(\sigma_{m}(a(x,D)))\mathrel{\
:=\ }\sigma_{m}(a(x,D))\circ\phi^{*}\,.$

One can then obtain the definition of the principal symbol on
$\Psi_{unif}^{m}(\mathbb{R}^{n})$, but we prefer to do it directly in the case
of non-compact manifolds. For $\eta\in\mathbb{R}^{n}$, we let
$e_{\eta}:\mathbb{R}^{n}\to\mathbb{C}$, $e_{\eta}(x):=e^{\imath\langle
x,\eta\rangle}$. The behavior of the principal symbol can be obtained from the
formula

(3.5) $[a(x,D)e^{\eta}](x)\,=\,e^{\imath\langle x,\eta\rangle}a(x,\eta)\,,$

for suitable $a$, for instance, for $a\in S_{unif}^{m}(\mathbb{R}^{2n})$ an
order $m$ symbol such that $a(x,D)$ with compactly supported distribution
kernel (this formula is stated in the proof of [26] and in Equation (3.32) in
[1], but see also Equation (2.53) in [23]).

###  3.2. Pseudodifferential operators on manifolds

The material of this section is completely standard, so we do not include
proofs or references, but, in addition to the textbooks mentioned above, see
also Seeley’s paper [52]. Let $M$ be a _smooth manifold,_ assumed Hausdorff
and paracompact, for simplicity, as before. Let also $U\subset M$ be open and
let $\phi:U\simeq W\subset\mathbb{R}^{n}$ be a diffeomorphism. We use the
notation of Remark 3.13. We then define

(3.6) $\Psi_{comp}^{m}(U)\mathrel{\ :=\
}\phi_{*}^{-1}\Psi_{comp}^{m}(W)\phi_{*}\mathrel{\ :=\
}\\{\phi_{*}^{-1}a(x,D)\phi_{*}\mid a\in S_{1,0}^{m}(\mathbb{R}^{2n})\\\
\mbox{ and }k_{a(x,D)}\mbox{ compactly supported in }W\times W\\}\,.$

Theorem 3.12 shows that the definition of $\Psi_{comp}^{m}(U)$, Equation
(3.6), does not depend on the choice of the diffeomorphism $\phi$.

######  Definition 3.15.

A linear map
$P:{\mathcal{C}}^{\infty}_{\text{c}}(M)\to{\mathcal{C}}^{\infty}(M)$ is a
_pseudodifferential operator_ of order $\leq m$ on $M$ with symbol of class
$S_{1,0}^{m}$ if, for any $U\simeq W\subset\mathbb{R}^{n}$ and
$\eta\in{\mathcal{C}}^{\infty}_{\text{c}}(U),$ we have $\eta
P\eta\in\Psi_{comp}^{m}(U),$ where $\Psi_{comp}^{m}(U)$ is as in Equation
(3.6). We let $\Psi_{1,0}^{m}(M)$ denote the set of all _pseudodifferential
operators_ on $M$ with symbols in the class $S_{1,0}^{m}$.

Similarly, we define

(3.7) $S_{1,0}^{m}(T^{*}M)\mathrel{\ :=\
}\\{b:T^{*}M\to\mathbb{C}\mid\phi_{*}(b|_{U})\mathrel{\ :=\
}(b|_{U})\circ\phi^{*-1}\in S_{unif}^{m}(T^{*}W)\\\ \mbox{ for }\phi:U\simeq
W\subset\mathbb{R}^{n}\,,\ U\Subset M\,,\ W\Subset\mathbb{R}^{n}\\}$

and

(3.8) $\sigma_{m}:\Psi_{comp}^{m}(U)\to
S_{unif}^{m}(T^{*}U)/S_{unif}^{m-1}(T^{*}U)$

by the formula

(3.9) $\sigma_{m}(\phi_{*}^{-1}a(x,D)\phi_{*})\mathrel{\ :=\
}(\phi_{*})^{-1}(\sigma_{m}(a(x,D)))\mathrel{\ :=\
}\sigma_{m}(a(x,D))\circ\phi^{*-1}\,.$

Theorem 3.14 shows that the definition of $\Psi_{comp}^{m}(U)$, Equation
(3.8), does not depend on the choice of the diffeomorphism $\phi:U\to W$.
Moreover, for each $m\in\mathbb{R}$, the map of Equation (3.8) extends to a
_surjective map_

(3.10) $\sigma_{m}:\Psi_{1,0}^{m}(M)\to
S_{1,0}^{m}/S_{1,0}^{m-1}(T^{*}M)\mathrel{\ :=\
}S_{1,0}^{m}(T^{*}M)/S_{1,0}^{m-1}(T^{*}M)\,.$

######  Theorem 3.16.

Let $M$ be a smooth manifold, as before.

  1. (1)

$\Psi_{1,0}^{m}(M)\subset\Psi_{1,0}^{m^{\prime}}(M)$ if $m<m^{\prime}.$

  2. (2)

If $P\in\Psi_{1,0}^{m}(M)$, then
$P:{\mathcal{C}}^{\infty}_{c}(M)\to{\mathcal{C}}^{\infty}(M)$ _continuously_
and hence $P$ has a distribution kernel $k_{P}$.

  3. (3)

$\Psi_{1,0}^{m}(M)$ and $\sigma_{m}:\Psi_{1,0}^{m}(M)\to
S_{1,0}^{m}/S_{1,0}^{m-1}(T^{*}M)$ are _diffeomorphism invariant._

  4. (4)

$\Psi_{1,0}^{m}(M)$ contains all the _differential operators_ of order $m$
with smooth coefficients. In particular,
${\mathcal{C}}^{\infty}(M)\subset\Psi_{1,0}^{0}(M).$

  5. (5)

$\Psi_{1,0}^{m}(M)$ contains all operators with _smooth distribution kernel._

  6. (6)

If $\phi$ and $\psi$ are smooth functions on $M$, then

$k_{\phi P\psi}\,=\,\phi k_{P}\psi\,,$

Consequently, $k_{P}$ is _smooth_ _away from the diagonal_ and, _at the
diagonal,_ $k_{P}$ has the _same behavior as the kernels of the operators in_
$\Psi_{unif}(\mathbb{R}^{n})$.

If $P\in\Psi_{1,0}^{m}(M)$ is such that
$P{\mathcal{C}}^{\infty}_{\text{c}}(M)\subset{\mathcal{C}}^{\infty}_{\text{c}}(M)$
and this action extends by continuity to a linear map
$P:{\mathcal{C}}^{\infty}(M)\subset{\mathcal{C}}^{\infty}(M)$, then we say
that $P$ is _properly supported._

######  Theorem 3.17.

Let us assume that a smooth measure was chosen on $M$.

  1. (1)

Let $P\in\Psi_{1,0}^{m}(M)$ and let $P^{*}$ be defined with respect to the
inner product on $M$. Then $P^{*}\in\Psi_{1,0}^{m}(M)$. We have that $P$ is
properly supported if, and only if,
$P{\mathcal{C}}^{\infty}_{\text{c}}(M)\subset{\mathcal{C}}^{\infty}_{\text{c}}(M)$
and
$P^{*}{\mathcal{C}}^{\infty}_{\text{c}}(M)\subset{\mathcal{C}}^{\infty}_{\text{c}}(M)$.
Thus $P$ is properly supported if, and only if, $P^{*}$ is properly supported.

  2. (2)

If $P\in\Psi_{1,0}^{m}(M)$ and $Q\in\Psi_{1,0}^{m^{\prime}}(M)$ and at least
one is properly supported, then $PQ$ is defined, belongs to
$\Psi_{1,0}^{m+m^{\prime}}(M)$, and

$\sigma_{m+m^{\prime}}(PQ)=\sigma_{m}(P)\sigma_{m^{\prime}}(Q).$

If both $P$ and $Q$ are properly supported, then $PQ$ is also properly
supported.

We also notice that, if $M=\mathbb{R}^{n}$, then
$\Psi_{unif}^{m}(\mathbb{R}^{n})\subset\Psi_{1,0}^{m}(\mathbb{R}^{n})$, but we
do not have equality (if $n>0$). Similarly,
$S_{unif}^{m}(\mathbb{R}^{n})\subset S_{0,1}^{m}(\mathbb{R}^{n})$, but we
again fail to have equality (in general).

###  3.3. Compact manifolds

We can say more if our manifold $M$ is compact. Assume this is the case in
this subsection. To start with, all operators in $\Psi_{1,0}^{m}(M)$ are
properly supported and hence we can compose any two pseudodifferential
operators on $M$. This simplifies matters, but the main result we are looking
for is the Fredholm property (whose definition is reminded below). Recall
that, for $M$ compact, the Sobolev spaces $H^{s}(M)$ defined in Subsection 2.4
are independent of choices (they can be defined either using a finite
partition of unity or a connection).

######  Theorem 3.18.

Let us assume $M$ to be compact.

  1. (1)

$\Psi_{1,0}^{\infty}(M):=\cup_{k\in\mathbb{Z}}\Psi_{1,0}^{k}(M)$ is a filtered
$*$-algebra.

  2. (2)

If $P\in\Psi_{1,0}^{m}(M)$, then $P:H^{s}(M)\to H^{s-m}(M)$ is _bounded_ for
all $s\in\mathbb{R}$. If this map is invertible for some $s\in\mathbb{R}$,
then $P^{-1}\in\Psi_{1,0}^{-m}(M).$

  3. (3)

$H^{s}(M)\to H^{s^{\prime}}(M)$ is compact if $s>s^{\prime}$, so any
$P\in\Psi_{1,0}^{q}(M),$ with $q<0,$ induces a _compact operator_
$P:H^{s}(M)\to H^{s}(M).$

“Filtered” in point (1) above means that the order of the product of two
operators does not exceed the sum of their orders. The point (2) goes back to
Beals [5]. The point (3) is, of course, nothing but Rellich’s lemma. We notice
that
$\cup_{k\in\mathbb{Z}}\Psi_{1,0}^{k}(M)=\cup_{m\in\mathbb{R}}\Psi_{1,0}^{m}(M),$
but this is not true for the classical pseudodifferential operators discussed
below. The Fredholm property is related to ellipticity, which we now discuss.

######  Proposition 3.19.

Let $a\in S_{1,0}^{m}(T^{*}M)$. The following are equivalent

  1. (1)

There are $C,R>0$ such that $|a(\xi)|\geq C|\xi|^{m}$ for $\xi\in T^{*}M\,,\
|\xi|\geq R.$

  2. (2)

There exist $b\in S_{1,0}^{-m}(T^{*}M)$ such that $ab-1\in
S_{1,0}^{-1}(T^{*}M).$

If one of these properties is satisfied, then $a$ is called _elliptic_ and
this property is a property of its class in
$S_{1,0}^{m}/S_{1,0}^{m-1}(T^{*}M).$

(If $M$ is non-compact, the two properties of the above proposition are no
longer equivalent, for some trivial reasons, and we retain the second one as
the definition of ellipticity. This will not be a problem in our paper.) This
leads to the following definition.

######  Definition 3.20.

$P\in\Psi_{1,0}^{m}(M)$ is _elliptic_ if $\sigma_{m}(P)\in
S_{1,0}^{m}/S_{1,0}^{m-1}(T^{*}M)$ is elliptic.

Below, we shall also need the notion of “injectively elliptic” operators,
which is completely analogous (Definition 3.29).

###  3.4. Classical symbols and vector bundles

We now recall the definition of a smaller class of symbols that is better
suited for the study of the parametrices of elliptic differential operators,
namely, the class of “classical symbols.” It is also better suited for the
study of the method of layer potentials. We continue to assume that $M$ is a
smooth (Hausdorff, paracompact) manifold, but we no longer assume $M$ to be
compact.

######  Definition 3.21.

We shall say that $a\in S_{1,0}^{m}(T^{*}M)$ is homogeneous of order $m,$ if
$a_{k}(t\xi)=t^{k}a_{k}(\xi)\ \mbox{ for }\ \xi\in T^{*}M,\ t,|\xi|\geq 1.$ We
shall say that $a\in S_{1,0}^{m}(\mathbb{R}^{n})$ is _classical_ of order $m$
if there exist $a_{m-j}\in S_{1,0}^{m-j}(M)$ homogeneous of order $m-j,$
$j\in\mathbb{Z}_{+}$ such that $a_{m}$ does not vanish identically and
$a-\sum_{j=0}^{N}a_{m-j}\in S_{1,0}^{m-N-1}(M)\,.$

In other words, we have the asymptotic expansion
$a\sim\sum_{j=0}^{\infty}a_{m-j}.$ If we allow only classical symbols, then
all the results of this section so far remain true, except 3.16(1).

Let $S_{cl}^{m}(T^{*}M)$ be the set of classical symbols of order
$\in\mathbb{R}$. Similarly, if $U\subset\mathbb{R}^{n}$, we let
$\Psi_{c,cl}^{m}(U):=\\{a(x,D)\in\Psi_{comp}^{m}(U)\mid a\in
S_{cl}^{m}(T^{*}M)\\}$ and, finally, we let $\Psi_{cl}^{m}(M)$ be defined
using the sets $\Psi_{c,cl}^{m}(U)$ with $U\Subset M$ open. Whenever using
classical pseudodifferential operators we shall include the index “cl” (or no
index at all, beginning with the next section). We let
$S_{cl}^{m}/S_{cl}^{m-1}(T^{*}M):=S_{cl}^{m}(T^{*}M)/S_{cl}^{m-1}(T^{*}M)$. In
addition to the property proved so far, the classical pseudodifferential
operators enjoy some additional properties.

######  Proposition 3.22.

Let $m\in\mathbb{R}$ and $S^{*}M$ be the set of vectors of length 1 in
$T^{*}M.$ Then restriction to $S^{*}M$ defines a bijection

$S_{cl}^{m}/S_{cl}^{m-1}(T^{*}M)\mathrel{\ :=\
}S_{cl}^{m}(T^{*}M)/S_{cl}^{m-1}(T^{*}M)\simeq{\mathcal{C}}^{\infty}(S^{*}M)\,.$

In particular, $\sigma_{m}:\Psi_{cl}^{m}(M)\to{\mathcal{C}}^{\infty}(S^{*}M)$
has kernel $\Psi_{cl}^{m-1}(M)$.

For a general statement of the Fredholm result, we now explain how we can
include _vector bundles._

######  Remark 3.23.

Let ${E}\to M$ be a complex vector bundle. Without loss of generality, we can
assume that a metric has been chosen on $E$.

  1. (1)

Let $E\subset\mathbb{C}^{N}$ be a smooth embedding and let
$e\in{\mathcal{C}}^{\infty}(M;M_{N}(\mathbb{C}))$ be the orthogonal projection
$\mathbb{C}^{N}\to E.$

  2. (2)

Then $\Psi_{1,0}^{m}(M;E)\mathrel{\ :=\ }eM_{N}(\Psi_{1,0}^{m}(M))e$ acts on
$H^{s}(M;E)\simeq eH^{s}(M).$

  3. (3)

The same constructions applies to classical pseudodifferential operators, but
recall that $\Psi_{cl}^{\infty}(M;E):=\cup_{m\in\mathbb{Z}}\Psi_{cl}^{m}(M;E)$
and that this space no longer contains $\Psi_{cl}^{s}(M;E)$ for
$s\notin\mathbb{Z}$.

  4. (4)

The principal symbols then become

$\begin{gathered}\sigma_{m}:\Psi_{1,0}^{m}(M;E)\to
S_{1,0}^{m}/S_{1,0}^{m-1}(T^{*}M;\operatorname{End}(E))\quad\mbox{and}\\\
\sigma_{m}:\Psi_{cl}^{m}(M;E)\to
S_{cl}^{m}/S_{cl}^{m-1}(T^{*}M;\operatorname{End}(E))\simeq{\mathcal{C}}^{\infty}(S^{*}M;\operatorname{End}(E))\,.\end{gathered}$

The distribution kernels of operators acting on vector bundles change in an
evident way in the following way:

######  Remark 3.24.

Let $V_{i}\to M_{i}$ be vector bundles, $i=1,2$. Then $V_{1}\boxtimes V_{2}\to
M_{1}\times M_{2}$ will denote their external tensor product. Thus
$(V_{1}\boxtimes
V_{2})_{(x_{1},x_{2})}=(V_{1})_{x_{1}}\otimes(V_{2})_{x_{2}}$. We are
interested in the external tensor product because the distribution kernels of
the operators in ${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}};E;F)$ will be
distributions with values in $F\boxtimes E^{\prime}$. (Thus, if
$E=F=\mathbb{C}$, then $E\boxtimes E^{\prime}=\mathbb{C}$, which simplifies
the notation and recovers the old definition of the distribution kernels.)

###  3.5. Fredholm operators and ellipticity

Let us recall next a few more needed general definions and results.

######  Definition 3.25.

Let $X,Y$ be normed spaces. The space of continuous linear maps $T:X\to Y$
will be denoted by $\mathcal{B}(X;Y)$. If $X$ and $Y$ are complete (i.e.
Banach), then $T\in\mathcal{B}(X;Y)$ is _Fredholm_ if

$\dim\ker(P)\,,\ \dim(Y/PX)<\infty.$

The following theorem [27, 58] will be important for us.

######  Theorem 3.26 (Atkinson).

Let $X$ and $Y$ be Hilbert spaces; $T\in\mathcal{B}(X;Y)$ is _Fredholm_ if,
and only if, there exists $Q\in\mathcal{B}(Y;X)$ such that both $TQ-1_{Y}$ and
$QT-1_{X}$ are compact.

An important property of compact manifolds and of their elliptic operators is
the Fredholm property. We include a proof for convenience.

######  Theorem 3.27.

Let $E,F\to M$ be vector bundles over a complete Riemannian manifold $M$ and
let $P\in\Psi_{1,0}^{m}(M;E,F)$, $m\in\mathbb{R}.$

  1. (1)

Assume that $M$ is compact and that $P$ is _elliptic._ Then, for any
$s\in\mathbb{R},$ $P:H^{s}(M;E)\to H^{s-m}(M;F)$ is Fredholm.

  2. (2)

Conversely, if $P:H^{s}(M;E)\to H^{s-m}(M;F)$ is Fredholm for some
$s\in\mathbb{R}$, then $P$ is elliptic.

Note that for the converse implication we no longer assume $M$ to be compact.

###### Proof.

There is $b\in S_{1,0}^{-m}(M;\operatorname{End}(E))$ such that
$\sigma_{m}(P)b-1\in S_{1,0}^{-1}(M;\operatorname{End}(E))$ because $P$ is
elliptic. The surjectivity of $\sigma_{-m}$ and the multiplicativity of the
principal symbol give that there exists $Q\in\Psi_{1,0}^{-m}(M;E)$ such that
$PQ-1,QP-1\in\Psi_{1,0}^{-1}(M;E)\,.$ Consequently, $PQ-1$ and $QP-1$ are
compact operators. Atkinson’s theorem 3.26 then gives that $P$ is Fredholm.

The proof of the converse statement follows from the following sequence of
lemmas of independent interest from [20]. ∎

For the following lemmas (obtained in discussions with Nadine Große [20]), we
keep the assumptions of Theorem 3.27, unless explicitly stated otherwise (some
of these assumptions are reminded occasionally). We follow Section 19.5 in
[26].

######  Lemma 3.28.

Let $M$ be a complete Riemannian manifold, $m,s\in\mathbb{R}$, and let
$P:H^{s}(M;E)\to H^{s-m}(M;F)$ be a continuous linear map that has finite
dimensional kernel and closed range. Then, for every $t\leq s$ there exists
$C_{t}>0$ such that, for every $u\in H^{s}(M;E)$, we have the _$L^{2}$
-coercive regularity estimate_

(3.11) $\|u\|_{H^{s}(M;E)}\leq
C_{t}\big{(}\|Pu\|_{H^{s-m}(M;F)}+\|u\|_{H^{t}(M;E)}\big{)}\,.$

For operators acting on vector bundles, the injectivity of the principal
symbol does not imply its invertibility. We thus introduce the following
definition.

######  Definition 3.29.

A symbol $a\in S_{1,0}^{m}(T^{*}M;\operatorname{Hom}(E;F))$ is _injectively
elliptic_ if there exists $b\in S_{1,0}^{-m}(T^{*}M;\operatorname{Hom}(E;F))$
such that $ba-1\in S_{1,0}^{-1}(T^{*}M;\operatorname{End}(E))$ (i.e. $a$ is
left invertible modulo lower order symbols). If $P\in\Psi_{1,0}^{m}(M;E,F)$,
we say that $P$ is _injectively elliptic_ if $\sigma_{m}(P)$ is injectively
elliptic.

As usual, we obtain the following alternative description of injectively
elliptic symbols.

######  Lemma 3.30.

We have that $a\in S_{1,0}^{m}(T^{*}M;\operatorname{Hom}(E;F))$ is injectively
elliptic if, and only if, for every $x\in M$ and any compact neighborhood $V$
of $x$ in $M$, there exist $R_{V},C_{V}>0$ such that, for all $w\in E|_{V}$
with $\|w\|\geq R_{V}$, we have

$\|a(\xi)w\|\geq C_{V}\|w\|\,.$

It is easy to see then that $a\in S_{1,0}^{m}{T^{*}M;\operatorname{Hom}(E;F)}$
is injectively elliptic if, and only if, $a^{*}a$ is elliptic (in
$S_{1,0}^{2m}(T^{*}M;\operatorname{End}(E))$). (This is used, implicitly, in
the proof of Theorem 19.5.1 in [26].) Also, it follows from definitions (since
$E$ and $F$ are finite dimensional vector bundles), that $a\in
S_{1,0}^{m}(T^{*}M;\operatorname{Hom}(E,F))$ is elliptic if, and only if, both
$a$ and $a^{*}$ are injectively elliptic.

######  Lemma 3.31.

Let $M$ be a complete Riemannian manifold and $m,s,t\in\mathbb{R}$, $t<s$. Let
$U\subset M$ be open and let $P\in\Psi_{1,0}^{m}(M;E,F)$ satisfy an _$L^{2}$
-coercive regularity estimate_ in $U$, that is, there exists $C_{t}>0$ such
that, for all $u\in{\mathcal{C}}^{\infty}_{\text{c}}(U;E)$, we have the
estimate of Equation (3.11). Then $P$ is injectively elliptic in $U$.

We can now complete the proof of Theorem 3.27. Recall that we have proved the
direct part and it remains to prove the converse, that is point (2) of that
Theorem.

###### End of the proof of Theorem 3.27.

We use twice the last two lemmas, first for $P$, and then for
$P^{*}\in\Psi^{m}(M;F,E)$. Since $P:H^{s}(M;E)\to H^{s-m}(M;F)$ is Fredholm,
it has finite dimensional kernel and closed range. Lemma 3.28 implies that $P$
satisfies the $L^{2}$-coercive regularity estimate of Equation (3.11) with
$t=s-1$. In turn, Lemma 3.31 then gives that $P$ is injectively elliptic. The
same argument applied to the Fredholm operator $P^{*}$ gives that $P^{*}$ is
also injectively elliptic. Consequently, $P$ is elliptic. ∎

The same ideas as the ones used in proof of Lemma 3.31 in [20] give the
following result. (It is also a direct consequence of the analogous results in
the case of compact manifolds [26, 56].)

######  Proposition 3.32.

Let $M$ be a complete Riemannian manifold, $m,s\in\mathbb{R}$, and
$P\in\Psi_{cl}^{m}(M;E,F)$ be such that $P:H^{s}(M;E)\to H^{s-m}(M;F)$ is
compact. Then $\sigma_{m}(P)=0$.

##  4\. Translation invariance in a neighborhood of infinity

Pseudodifferential operators have long played an important role in the study
of layer potential operators [27, 56]. In this paper, we consider mostly two
classes of pseudodifferential operators associated to our manifolds with
straight cylindrical ends. These two classes differ only in the regularizing
operators in a sense that will be made clear below. In this section, we
introduce the smallest of these two classes of operators and study their
properties. The operators in this class are “$b$-pseudodifferential” operators
(in the sense of Melrose [41, 42] and Schulze [50, 49]; see the nice paper by
Lauter and Seiler [35] for a comparison of the constructions of Melrose and
Schulze). A few of the results in the following two subsections are
consequences of the properties of the $b$-calculus and many are similar to
some analogous results that are valid for the $b$-calculus. We nevertheless
include the proofs, for the convenience of the reader and because our approach
is different from the one used for the $b$-calculus. We let $n$ denote the
dimension of our manifold with cylindrical ends ${\mathfrak{M}}$.

###  4.1. Definition of the $\operatorname{inv}$-calculus

Let
${\mathfrak{M}}={\mathfrak{M}}_{0}\cup\left(\partial{\mathfrak{M}}_{0}\times(-\infty,0]\right)$
be our fixed manifold with straight cylindrical ends, as in Remark 2.2. Recall
that we assume that ${\mathfrak{M}}$ has compact ends, that is, that
${\mathfrak{M}}_{0}$ is compact. This assumption will remain in place until
the end of the paper.

For $R,s\geq 0$, we let

(4.1)
$\Phi_{s}:\partial{\mathfrak{M}}_{0}\times(-\infty,-R]\to\partial{\mathfrak{M}}_{0}\times(-\infty,-R-s]\,,\quad\Phi_{s}(x,t)=(x,t-s)\,,$

denote the bijective isometry given by translation with $-s$ in the
$t$-direction. If $s<0$, then $\Phi_{s}$ is defined as the inverse of
$\Phi_{-s}$. Let also $\Phi_{s}(f):=f\circ\Phi_{s}$, whenever the compositions
are defined. If $\varepsilon>0$, the set

(4.2)
$\\{(x,y)\in{\mathfrak{M}}\times{\mathfrak{M}}\mid\operatorname{dist}_{g}(x,y)<\varepsilon\\}$

will be called _the $\varepsilon$–neighborhood of the diagonal_
$\delta_{\mathfrak{M}}:=\\{(x,x)\mid x\in{\mathfrak{M}}\\}$ of
${\mathfrak{M}}$.

######  Definition 4.1.

A linear and continuous operator
$P:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})\to{\mathcal{C}}^{\infty}({\mathfrak{M}})^{\prime}$
is called translation invariant in a neighborhood of infinity if:

  1. (1)

there exists $\varepsilon_{P}>0$ such that its Schwartz (or distribution)
kernel is supported in the $\varepsilon_{P}$–neighborhood of the diagonal
(Equation (4.2)).

  2. (2)

there exists $R_{P}>\varepsilon_{P}$ such that, for all
$u\in{\mathcal{C}}^{\infty}_{\text{c}}\big{(}\partial{\mathfrak{M}}_{0}\times(-\infty,-R_{P})\big{)}$
and all $s>0$, we have $P\Phi_{s}(u)=\Phi_{s}P(u).$

We let ${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ denote the space of
all pseudodifferential operators $P\in\Psi_{1,0}^{m}({\mathfrak{M}})$ of order
$\leq m\in\mathbb{R}\cup\\{-\infty\\}$ that are translation invariant in a
neighborhood of infinity. We let
${\Psi_{\operatorname{inv}}^{\infty}}({\mathfrak{M}}):=\cup_{m\in\mathbb{Z}}{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$.

######  Remark 4.2.

Let us notice that, if $u$ has support in
$\partial{\mathfrak{M}}_{0}\times(-\infty,-R_{P})$, then $\Phi_{s}(u)$ has
support in $\partial{\mathfrak{M}}_{0}\times(-\infty,-R_{P}+\varepsilon_{P})$,
so $P\Phi_{s}(u)$ and $\Phi_{s}(Pu)$ are both defined since
$R_{P}>\varepsilon_{P}$. Consequently, the condition
$P\Phi_{s}(u)=\Phi_{s}P(u)$ makes sense.

The calculus ${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ will be called
the _$\operatorname{inv}$ -calculus._ Our $\operatorname{inv}$-calculus is
smaller than the “$b$-calculus” in the following sense.

######  Remark 4.3.

Let us compactify ${\mathfrak{M}}$ to $\overline{{\mathfrak{M}}}$ using the
function $r:=e^{t}$ (the Kondratiev transform), so that
${\mathfrak{M}}=\\{r>0\\}$ and $\partial\overline{{\mathfrak{M}}}=\\{r=0\\}$.
Then it is not difficult to show that

${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\subset\Psi_{b}^{m}(\overline{{\mathfrak{M}}})\,,$

where $\Psi_{b}^{m}(\overline{{\mathfrak{M}}})$ is the “$b$-calculus” of
Melrose [41, 42] and Schulze [50, 49]. (Schulze calls it the “cone-calculus,”
see the nice paper by Lauter and Seiler [35] comparing the two approaches for
further references. See also [10, 17, 51, 53].) Some of the properties of
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ proved in this paper would
follow from this inclusion. The space $\Psi_{b}^{m}({\mathfrak{M}})$ is,
nevertheless, not too far from our space
${\Psi_{\operatorname{inv}}^{m}}(\overline{{\mathfrak{M}}})$, since the former
consists of operators whose kernels are “exponentially close” to being
translation invariant, whereas ours are exactly translation invariant (in a
neighborhood of infinity). Our proofs can thus be regarded as a different
approach to the $b$-calculus.

We have already encountered operators in the $\operatorname{inv}$-calculus
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$, as explained in the next
remark.

######  Remark 4.4.

The endomorphisms and, more generally, the differential operators in
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ have already been used.
Indeed, it follows from definitions that

$\begin{gathered}{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}})\,=\,{\mathcal{C}}^{\infty}({\mathfrak{M}})\cap{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\quad\mbox{
and}\\\
\operatorname{Diff}_{\operatorname{inv}}^{m}({\mathfrak{M}})\,=\,\\{P\mbox{ a
differential operator on
}{\mathfrak{M}}\\}\cap{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\,.\end{gathered}$

###  4.2. The first properties of the $\operatorname{inv}$-calculus

In this subsection, we recall the definition of the
$\operatorname{inv}$-calculus
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ and prove its first
properties. These properties are direct analogues of the corresponding
properties for the classical pseudodifferential operators on compact manifolds
(or of the $b$-calculus). However, none of the results of this subsection
seems to follow from the inclusion of our $\operatorname{inv}$-calculus in the
$b$-calculus. We begin with a characterization of the distribution kernels of
the negative order operators.

We obtain more examples of pseudodifferential operators that are translation
invariant in a neighborhood of infinity by describing their distribution
kernels. Recall that
$k_{P}\in{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}}\times{\mathfrak{M}})^{\prime}$
denotes the distribution kernel of a continuous, linear operator
$P:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})\to{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})^{\prime}$,
Theorem 3.6. Also, recall that ${\mathfrak{M}}$ has dimension $n$, that
$r_{{\mathfrak{M}}}$ denotes its injectivity radius, and that
$\mathbb{N}=\\{1,2,\ldots\\}$. We let $\exp:T{\mathfrak{M}}\to{\mathfrak{M}}$
be the exponential map (which is defined everywhere since ${\mathfrak{M}}$ is
geodesically complete).

######  Proposition 4.5.

Let ${\mathfrak{M}}$ be a manifold with straight cylindrical ends, as before,
and $m<0$, $m\notin\mathbb{Z}$. A continuous, linear operator
$P:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})\to{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})^{\prime}$
is _a classical pseudodifferential_ operator in
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ if, and only if, it
satisfies the following conditions:

  1. (1)

Its distribution kernel $k_{P}$ is a function with support in some
$\epsilon$-neighborhood of the diagonal (Equation 4.2) and is smooth away from
the diagonal.

  2. (2)

There exists $R_{P}>0$ such that, if $x,y\in\partial{\mathfrak{M}}_{0}$,
$t,s\leq-R_{P}$, $(x,t)\neq(y,s)$, and $\lambda\geq 0$, then

$k_{P}(x,t-\lambda,y,s-\lambda)\,=\,k_{P}(x,t,y,s)\,.$

  3. (3)

Let $S{\mathfrak{M}}\subset T{\mathfrak{M}}$ be the subset of vectors of
length one and $p:S{\mathfrak{M}}\to{\mathfrak{M}}$ be the canonical
projection. Then there exist functions
$a_{j}\in{\mathcal{C}}^{\infty}(S{\mathfrak{M}})$ and $C_{j}\geq 0$,
$j\in\mathbb{Z}_{+}$, such that, for all $\xi\in S{\mathfrak{M}}$, all
$t\in(0,r_{\mathfrak{M}}/2]$, and all $N\in\mathbb{N}$, we have

$|k_{P}(p(\xi),\exp(t\xi))-\sum_{j=0}^{N}a_{j}(\xi)t^{-n-m+j}|\leq
C_{N}t^{-n-m+N+1}\,.$

If that is the case, then
$a_{j}\in{\mathcal{C}}^{\infty}_{inv}(S{\mathfrak{M}})$, more precisely,
$a_{j}(\eta,t)=a_{j}(\eta,t-s)$ for all $s\geq 0$ and $t<R_{P}$ (independent
of $j$!), where $(\eta,t)$ are the canonical coordinates on the cylindrical
end of $S{\mathfrak{M}}$.

If $-m\in\mathbb{N}$, the result is true if, for each $j\geq m+n$, we replace
the term $a_{j}(\xi)t^{-n-m+j}$ with $a_{j}(\xi)t^{-n-m+j}+p_{j}(t\xi)\ln t$,
where $p_{j}$ is a polynomial in $\xi$ of degree $-n-m+j$ with smooth
coefficients.

###### Proof of Proposition 4.5.

This follows almost directly from the definitions and classical results, see,
for instance, Proposition 2.8 of [56]. The support condition, Condition (1) is
part of the definition of ${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$.
Condition (2) is equivalent to the invariance of the operators in a
neighborhood of infinity. Finally, the last condition, Condition (3), states
that $P$ is an order $m<0$ classical pseudodifferential operator, see [26, 56,
58]. This proves the desired equivalence. The last statement (including
$a_{j}\in{\mathcal{C}}^{\infty}_{inv}(S{\mathfrak{M}})$) follows from the
uniqueness of the coefficients $a_{j}$ and the translation invariance at
infinity of the operator $P$ (with $R_{P}$ the same in our statement and in
the definition of translation invariance at infinity for $P$). ∎

Recall the definition of the principal symbol, Equation (3.10). We notice that
the principal symbol of a pseudodifferential operator that is translation
invariant in a neighborhood of infinity is also translation invariant at
infinity. We spell this and some related needed facts in the following remark.

######  Remark 4.6.

The manifold $T^{*}{\mathfrak{M}}$ also has cylindrical ends (but does not
have compact ends). The translation invariance at infinity of a
pseudodifferential operator implies the same property for its principal symbol
(by the behavior of the principal symbol with respect to diffeomorphisms). Let

$S_{inv}^{m}(T^{*}{\mathfrak{M}})\mathrel{\ :=\
}S_{1,0}^{m}(T^{*}{\mathfrak{M}})\cap{\mathcal{C}}^{\infty}_{inv}(T^{*}{\mathfrak{M}}\,.)$

We obtain that $\sigma_{m}$ defines a linear map

$\sigma_{m}:{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\to
S_{inv}^{m}(T^{*}{\mathfrak{M}})/S_{inv}^{m-1}(T^{*}{\mathfrak{M}})\,.$

We mention the following property that is not satisfied by the $b$\- or
$c$-calculi.

######  Remark 4.7.

Corollary 2.9 implies that every elliptic operator
$P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ is also _uniformly
elliptic,_ in fact, $\sigma_{m}(P)^{-1}$ is a principal symbol of order $-m$
by Corollary 2.9. Again, this is one of the reasons which shows that the
$\operatorname{inv}$-calculus is simpler than the $b$-calculus and why we
relegate the use of the $b$-calculus and of other related calculi, such as the
$c$-calculus, to another paper.

The following remark introduces the quantization map.

######  Remark 4.8.

Let $S_{\operatorname{inv}}^{m}(T^{*}{\mathfrak{M}})$ be as in Remark 4.6.
That is, it is the set of (scalar) symbols of order $\leq m$ that are
translation invariant in a neighborhood of infinity in the usual sense, that
is that $a\in S_{\operatorname{inv}}^{m}(T^{*}{\mathfrak{M}})$ if, and only
if, there exists $R_{a}>0$ such that $a(x,t,v)=a(x,t-s,v)$ for all $s>0$ and
$(x,t,v)\in T^{*}{\mathfrak{M}}$ with $(x,t)\in{\mathfrak{M}}$ cylindrical
coordinates with $x\in\partial{\mathfrak{M}}_{0}$ and $t<-R_{a}$ and $v\in
T_{(x,t)}^{*}{\mathfrak{M}}$. Let
$\chi:{\mathfrak{M}}\times{\mathfrak{M}}\to[0,1]$ be a smooth cut-off function
that is supported in an $\varepsilon$-neighborhood of the diagonal, is equal
to 1 in a $\varepsilon^{\prime}$-neighborhood of the diagonal, and is
translation invariant in a neighborhood of infinity, in the sense that it
satisfies the condition of Proposition 4.5(2). We shall also need the usual
quantization map

$\begin{gathered}q:S^{m}(T^{*}{\mathfrak{M}})\to\Psi_{1,0}^{m}({\mathfrak{M}})\,,\\\
\big{[}q(a)u\big{]}(x)\mathrel{\ :=\
}(2\pi)^{-n}\int_{T_{x}^{*}{\mathfrak{M}}}a(w)\left(\int_{T_{x}{\mathfrak{M}}}e^{-\imath\langle
w,z\rangle}\chi(x,\exp(z))u(\exp(z))dz\right)dw\,.\end{gathered}$

(To see that this recovers the usual formula, one should notice that the value
at $x$ is computed in coordinates centered at $x$, so it is natural to use the
substitution $z=y-x$, which leads us to the usual formula, except maybe for
the cut-off factor $\chi$.) The volume forms (measures) on
$T_{x}^{*}{\mathfrak{M}}$ and $T_{x}{\mathfrak{M}}$ are those induced by the
metric. Then
$q(S_{inv}^{m}(T^{*}{\mathfrak{M}}))\subset\Psi_{inv}^{m}({\mathfrak{M}})$

We have the following notion of asymptotic completeness.

######  Remark 4.9.

Let $m\in\mathbb{R}$ and $a_{j}\in
S_{\operatorname{inv}}^{m-j}(T^{*}{\mathfrak{M}})$, $j\in\mathbb{Z}_{+}$ be
such that one can choose _the same $R_{a_{j}}$ for all $j$_ (i.e. there exists
$R>0$ such that $a_{j}(x,t-s,v)=a_{j}(x,t,v)$ for all $j\in\mathbb{Z}_{+}$,
$t<-R$, and $s>0$). Then there exists $a\in
S_{\operatorname{inv}}^{m}(T^{*}{\mathfrak{M}})$ such that

$a-\sum_{j=0}^{N}a_{j}\,\in\,S_{\operatorname{inv}}^{m-N-1}(T^{*}{\mathfrak{M}})\,,\quad\forall\,N\in\mathbb{N}\,.$

This follows from the asymptotic completeness of the usual symbols. (Here we
have not included the most general result, but rather the form of the result
that we will need.) See Proposition 3.14 in [23] for a useful general
extension of this result.

We have the following result extending one of the usual properties of
pseudodifferential operators.

######  Proposition 4.10.

The quantization map
$q:S_{\operatorname{inv}}^{m}(T^{*}{\mathfrak{M}})\to{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$,
$m\in\mathbb{Z}$, of Remark 4.8 satisfies

$\sigma_{m}(q(a))\,=\,a+S_{\operatorname{inv}}^{m-1}(T^{*}{\mathfrak{M}})\,.$

Consequently, the induced map

$q:S_{\operatorname{inv}}^{m}(T^{*}{\mathfrak{M}})/S_{\operatorname{inv}}^{-\infty}(T^{*}{\mathfrak{M}})\to{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})/{\Psi_{\operatorname{inv}}^{-\infty}}({\mathfrak{M}})$

is a linear bijection. The maps $q$ for all $m$ are compatible and they
combine to a map
$q:S_{\operatorname{inv}}^{\infty}(T^{*}{\mathfrak{M}})\to{\Psi_{\operatorname{inv}}^{\infty}}({\mathfrak{M}})$.

###### Proof.

The translation invariance of the exponential map shows that
$q(a)\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ whenever $a\in
S_{\operatorname{inv}}^{m}(T^{*}{\mathfrak{M}})$. The rest is as in the usual
case of pseudodifferential operators (it is a consequence of the corresponding
properties for pseudodifferential operators and of Remark 4.9). ∎

######  Remark 4.11.

Proposition 4.10 gives, in particular, that the map

$\overline{q}:S_{\operatorname{inv}}^{m}(T^{*}{\mathfrak{M}})\oplus{\Psi_{\operatorname{inv}}^{-\infty}}({\mathfrak{M}})\ni(a,R)\to
q(a)+R\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$

is onto, which allows us to endow the later space with the quotient topology
(of an inductive limit of Fréchet spaces). he kernel of this map is

$\ker\overline{q}\,=\,\\{(a,-q(a))\mid a\in
S_{\operatorname{inv}}^{-\infty}(T^{*}{\mathfrak{M}})\\}\,.$

We summarize now some of the easy properties of the
$\operatorname{inv}$-calculus ${\Psi_{\operatorname{inv}}^{m}}$ (these
properties can be found in many of the works on the $b$-calculus, once one
takes into account the simpler definition of ellipticity for the operators in
the $\operatorname{inv}$-calculus, see Theorem 3.14 and Proposition 3.19, they
are also implicit in [44]). Operators that are (almost) translation invariant
were also studied recently in [24]. We begin with the properties that are
direct analogues of the corresponding properties for the usual
pseudodifferential calculus on compact manifolds.

######  Theorem 4.12.

Let
${\mathfrak{M}}={\mathfrak{M}}_{0}\cup\big{(}\partial{\mathfrak{M}}_{0}\times(-\infty,0]\big{)}$
be a manifold with straight cylindrical ends.

  1. (1)

${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ is stable under adjoints
and, for $m,q\in\mathbb{R}$,

${\Psi_{\operatorname{inv}}^{q}}({\mathfrak{M}}){\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\subset{\Psi_{\operatorname{inv}}^{q+m}}({\mathfrak{M}})\,.$

  2. (2)

The principal symbol
$\sigma_{m}:{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\to
S_{inv}^{m}(T^{*}{\mathfrak{M}})/S_{inv}^{m-1}(T^{*}{\mathfrak{M}})$ of Remark
4.6 is onto and its kernel is
${\Psi_{\operatorname{inv}}^{m-1}}({\mathfrak{M}})$. It is $*$-multiplicative,
in the sense that

$\sigma_{q+m}(QP)\,=\,\sigma_{q}(Q)\sigma_{m}(P)\ \mbox{ and }\
\sigma_{m}(P^{*})\,=\,\sigma_{m}(P)^{*}\,.$

###### Proof.

To prove (1), let $P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ and
$Q\in{\Psi_{\operatorname{inv}}^{q}}({\mathfrak{M}})$. Both operators are
properly supported, by assumption, so the composition $QP$ is defined and
$QP\in\Psi^{m+q}({\mathfrak{M}})$. Moreover, the support of the distribution
kernel of $QP$ is contained in the
$(\varepsilon_{P}+\varepsilon_{Q})$–neighbhorhood of the diagonal. The
invariance in a neighborhood of infinity follows from
$QP\Phi_{s}(u)=Q\Phi_{s}Pu=\Phi_{s}QPu$ whenever
$u\in{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})$ has support in
$\partial{\mathfrak{M}}_{0}\times(-\infty,-R)$ for $R$ large enough (say
$R>R_{P}+R_{Q}$). Thus
$QP\in{\Psi_{\operatorname{inv}}^{q+m}}({\mathfrak{M}})$. The relation
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})^{*}={\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$
is similar.

The surjectivity in (2) is a consequence of Proposition 4.10. The
multiplicativity follows from the usual multiplicativity of the principal
symbols. The kernel of the usual principal symbol is
$\Psi^{m-1}({\mathfrak{M}})$. The result then follows from
$\Psi^{m-1}({\mathfrak{M}})\cap{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})={\Psi_{\operatorname{inv}}^{m-1}}({\mathfrak{M}})$.
∎

###  4.3. Limit operators

An important concept associated to the operators in the $b$-calculus is the
“normal operator” (in Melrose’s terminology) or “conormal principal symbol”
(in Schulze’s terminology, who uses the term “cone-calculus” instead of the
$b$-calculus). This concept is especially easy to define for our translation
invariant at infinity operators. We shall call these operators “limit at
infinity operators,” (or “limit operators,” for short) since this is the
terminology that is used for their generalizations by many other authors. A
few results of this section follow from the properties of the $b$-calculus,
but we include their simple proof for completeness (and since the proofs are
different than the ones for the $b$-calculus). Recall that we assume that
${\mathfrak{M}}$ has compact ends (that is, that $\partial{\mathfrak{M}}_{0}$
is compact). It is, of course, a manifold with straight cylindrical ends.

To simplify the notation, we shall set as in Melrose’s work

(4.3) ${\psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}\mathrel{\ :=\
}{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}\,.$

We begin with some simple observations on the supports of our operators that
will help us to define the normal (or limit) operator associated to an
operator that is translation invariant in a neighborhood of infinity.

######  Remark 4.13.

First, it follows from the definition of translation invariant
pseudodifferential operators in a neighborhood of infinity that every
$P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ is properly supported,
that is,
$P({\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}}))\subset{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})$.
We use this observation for ${\mathfrak{M}}={\mathfrak{L}}\times\mathbb{R}$ to
conclude that
${\psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}:={\Psi_{\operatorname{inv}}^{m}}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$
consists of the set of translation invariant, properly supported,
pseudodifferential operators on ${\mathfrak{L}}\times\mathbb{R}$ (invariant
with respect to the translations by $t\in\mathbb{R}$), that is

${\psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}\,=\,\\{P\in\Psi^{m}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}\mid\
P\mbox{ properly supported}\\}.$

This type of operators appear in the following result that will provide the
definition of limit operators. Let
$\Phi_{s}:\partial{\mathfrak{M}}_{0}\times\mathbb{R}\to\partial{\mathfrak{M}}_{0}\times\mathbb{R}$
be the translation by $-s$, $\Phi_{s}(f)(x)=f(x-s)$, that is, we allow
$R=-\infty$ in (4.1).

######  Lemma 4.14.

Let $P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ and
$u\in{\mathcal{C}}^{\infty}_{\text{c}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})$.
Then, there exists $R_{P,u}>0$ such that $P\Phi_{s}(u)$ is defined and
independent of $s>R_{P,u}$. Let

$\widetilde{P}(u)\mathrel{\ :=\ }\Phi_{-s}P\Phi_{s}(u)\,,$

for any $s>R_{P,u}$.

  1. (1)

If
$a\in{\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}})\subset{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$,
then $\tilde{a}=\mathcal{R}_{\infty}(a)$ (Definition 2.6).

  2. (2)

In general,
${\mathcal{R}_{\infty}}(P):=\widetilde{P}\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}:={\Psi_{\operatorname{inv}}^{m}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})^{\mathbb{R}}$.

###### Proof.

For $s$ large, $\Phi_{s}(u)$ has support in
$\partial{\mathfrak{M}}_{0}\times(-\infty,0)\subset{\mathfrak{M}}$, and hence
$P\Phi_{s}(u)$ is defined. The independence of $s$ follows from the definition
of ${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ (see Definition 4.1).
Indeed, we have independence of $s$ as soon as $s$ is large enough for
$\Phi_{s}(u)$ to have support in
$\partial{\mathfrak{M}}_{0}\times(-\infty,-R_{P})$. The relation
$\tilde{a}=\mathcal{R}_{\infty}(a)$ follows from the definition of
${\mathcal{C}}^{\infty}_{\operatorname{inv}}({\mathfrak{M}})$ (Definition
2.6). The resulting operator ${\mathcal{R}_{\infty}}(P)$ is clearly invariant
by translations. This completes the proof. ∎

In analogy with Proposition 4.5, we have the following proposition, whose
proof is completely similar. We do not include its proof since we will not use
this proposition, for the same reason, we state it explicitly only in the case
$E=\mathbb{C}$. Recall that our manifold ${\mathfrak{M}}$ has dimension $n$.
The following results, while not used in this paper, is meant to be used for
${\mathfrak{L}}=\partial{\mathfrak{M}}_{0}$, which thus has dimension $n-1$.
(The asymptotic of the distribution kernel is, however, on
${\mathfrak{L}}\times\mathbb{R}$, thus on a manifold of dimension $n$.)

######  Proposition 4.15.

Let ${\mathfrak{L}}$ be a closed manifold of dimension $n-1$ and $m<0$,
$m\notin\mathbb{Z}$. A continuous, linear map
$T:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{L}}\times\mathbb{R})\to{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{L}}\times\mathbb{R})^{\prime}$
is a _classical pseudodifferential_ operator in
${\psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}:={\Psi_{\operatorname{inv}}^{m}}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$
if, and only if, it satisfies the following conditions.

  1. (1)

Its distribution kernel $k_{T}$ is a function with support in some
$\epsilon$-neighborhood of the diagonal (Equation 4.2) and is smooth away from
the diagonal.

  2. (2)

For all $x,y\in{\mathfrak{L}}$, $t,s,\lambda\in\mathbb{R}$, $(x,t)\neq(y,s)$,
we have

$k_{T}(x,t-\lambda,y,s-\lambda)\,=\,k_{T}(x,t,y,s)\,.$

  3. (3)

Let $S({\mathfrak{L}}\times\mathbb{R})\subset
T({\mathfrak{L}}\times\mathbb{R})$ be the subset of vectors of length one, as
usual, and
$p:S({\mathfrak{L}}\times\mathbb{R})\to{\mathfrak{L}}\times\mathbb{R}$ be the
canonical projection. Then there exist functions
$a_{j}\in{\mathcal{C}}^{\infty}\big{(}S({\mathfrak{L}}\times\mathbb{R})\big{)}^{\mathbb{R}}$
and $C_{j}\geq 0$, $j\in\mathbb{Z}_{+}$, such that, for all $N\in\mathbb{N}$,
$\xi\in S({\mathfrak{L}}\times\mathbb{R})$, and $t\in(0,r_{\mathfrak{L}}/2]$

$|k_{T}(p(\xi),\exp(t\xi))-\sum_{j=0}^{N}a_{j}(\xi)t^{-n-m+j}|\leq
C_{N}t^{-n-m+N+1}\,.$

We also have the following property of the distribution kernel of
$\tilde{P}={\mathcal{R}_{\infty}}(P)$, which is a defining property for $m<0$
(by Peetre’s theorem).

######  Proposition 4.16.

Let $P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$. Then

$k_{\widetilde{P}}(x,t,y,s)\,=\,k_{P}(x,t-\lambda,y,s-\lambda)$

for $\lambda$ large enough and
$(x,t)\neq(y,s)\in\partial{\mathfrak{M}}_{0}\times(-\infty,0)$.

###### Proof.

This follows from the translation invariance at infinity, knowing that the
distribution kernel of a pseudodifferential operator is a smooth function
outside the diagonal. ∎

We now introduce an important concept without analog in the compact case.

######  Definition 4.17.

Let $P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ and

$\widetilde{P}\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}\mathrel{\
:=\ }\Psi_{inv}^{m}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})^{\mathbb{R}}$

be as in Lemma 4.14. Then $\widetilde{P}$ is called the _limit operator_
associated to $P$. The resulting map

${\mathcal{R}_{\infty}}:{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\ni
P\longmapsto\widetilde{P}\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}$

will be called the _limit operator map._

For the proof and for further use, we shall need the following cut-off
function $\eta$ that will be fixed throughout the paper:

######  Notation 4.18.

We let $\eta:\partial{\mathfrak{M}}_{0}\times\mathbb{R}\to[0,1]$ be a fixed
smooth “cut-off” function

$\eta(x,t)\,=\,\begin{cases}\ 1\ \mbox{ for }\ t\leq-2\\\ \ 0\ \mbox{ for }\
t\geq-1\,.\end{cases}$

We also regard $\eta$ as a function on
${\mathfrak{M}}={\mathfrak{M}}_{0}\cup(\partial{\mathfrak{M}}_{0}\times(-\infty,0])$
by setting it to be $=0$ on ${\mathfrak{M}}_{0}$.

The cut-off function $\eta$ of 4.18 will be used to show that the limit-
operator map ${\mathcal{R}_{\infty}}$ is surjective by constructing a cross-
section for it.

######  Definition 4.19.

For
$T\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}:={\Psi_{\operatorname{inv}}^{m}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})^{\mathbb{R}}$,
we let

$s_{0}(T)\mathrel{\ :=\ }\eta
T\eta:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})\to{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})\,.$

It is known that the map $s_{0}$ is well defined from the properties of the
$b$-calculus. We nevertheless prove directly some of its needed properties,
for completeness and for the benefit of the reader.

######  Lemma 4.20.

Let $T\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}$ and
$s_{0}(T):=\eta T\eta$ (Definition 4.19).

  1. (1)

$s_{0}(T)$ is well defined.

  2. (2)

$s_{0}(T)\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$.

  3. (3)

We have ${\mathcal{R}_{\infty}}\circ s_{0}=id$, where ${\mathcal{R}_{\infty}}$
is as in Definition 4.17.

###### Proof.

(1) Let $u\in{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})$. Then $\eta u$
has support in $\partial{\mathfrak{M}}_{0}\times(-\infty,0)$, and hence it can
be identified with a function in
${\mathcal{C}}^{\infty}_{\text{c}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})$,
so $T(\eta
u)\in{\mathcal{C}}^{\infty}_{\text{c}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})$
is defined. Then $\eta T(\eta u)$ in turn has support in
$\partial{\mathfrak{M}}_{0}\times(-\infty,0)\subset{\mathfrak{M}}$, and hence
can be identified with a function in
${\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})$. Thus $s_{0}(T)u:=\eta
T(\eta u)\in{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})$ is well defined
and we have (1).

(2) Clearly $s_{0}(T)$ is an order $m$ pseudodifferential operator, by the
general properties of pseudodifferential operators. Let
$u\in{\mathcal{C}}^{\infty}_{\text{c}}(\partial{\mathfrak{M}}_{0}\times(-\infty,-R))$
for some large $R>0$. Then $\eta\Phi_{s}(u)=\Phi_{s}(u)$, for $s\geq 0$. The
same is true for $u$ replaced with $T\Phi_{s}(u)$, $s\geq 0$, since $T$ is
invariant and properly supported (so it increases the support of $\Phi_{s}(u)$
by at most $\varepsilon_{T}$). We thus have

$s_{0}(T)\Phi_{s}(u)=\eta
T(\eta\Phi_{s}(u))=T(\Phi_{s}(u))=\Phi_{s}T(u)=\Phi_{s}\eta(T(\eta
u))=\Phi_{s}s_{0}(T)u\,.$

Hence $s_{0}(T)\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$, as
claimed. To prove (3) the same reasoning gives

$\Phi_{-s}s_{0}(T)\Phi_{s}(u)=\Phi_{-s}\eta
T(\eta\Phi_{s}(u))=\Phi_{-s}T(\Phi_{s}(u))=T(u)$

for $s$ large enough. Thus ${\mathcal{R}_{\infty}}(s_{0}(T))=T$. ∎

We shall need to take a closer look at the principal symbols of operators in
${\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}$. To this end, we
specialize Theorem 3.14 and Proposition 3.19 to these operators.

######  Remark 4.21.

Let us replace ${\mathfrak{M}}$ with
$\partial{\mathfrak{M}}_{0}\times\mathbb{R}$ in Remark 4.6 and restrict to
translation invariant operators. Recall that
${\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}\mathrel{\ :=\
}{\Psi_{\operatorname{inv}}^{m}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})^{\mathbb{R}}$.
Then the principal symbol yields a map

$\sigma_{m}:{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}\to
S_{\operatorname{inv}}^{m}(T^{*}(\partial{\mathfrak{M}}_{0}\times\mathbb{R}))^{\mathbb{R}}/S_{\operatorname{inv}}^{m-1}(T^{*}(\partial{\mathfrak{M}}_{0}\times\mathbb{R}))^{\mathbb{R}}\,.$

Next, we can identify

$S^{m}(T^{*}(\partial{\mathfrak{M}}_{0}\times\mathbb{R}))^{\mathbb{R}}\simeq
S^{m}(T^{*}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})/\mathbb{R})$

and, furthermore, we can also identify

$T^{*}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})/\mathbb{R}\,\simeq\,T^{*}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})|_{t=0}\,\simeq\,(T^{*}{\mathfrak{M}}_{0})|_{\partial{\mathfrak{M}}_{0}}\,\simeq\,\partial
T^{*}{\mathfrak{M}}_{0}\,,$

where
$T^{*}{\mathfrak{M}}_{0}|_{\partial{\mathfrak{M}}_{0}}:=p^{-1}(\partial{\mathfrak{M}}_{0})$
is the restriction of the bundle
$p:T^{*}{\mathfrak{M}}_{0}\to{\mathfrak{M}}_{0}$ to the boundary of
${\mathfrak{M}}_{0}$. (All these identifications are obtained using that the
standard decomposition of $T^{*}{\mathfrak{M}}$ as a manifold with straight
cylindrical ends is
$T^{*}{\mathfrak{M}}\,=\,T^{*}{\mathfrak{M}}_{0}\cup\big{[}\partial
T^{*}{\mathfrak{M}}_{0}\times(-\infty,0]\big{]}\,.$) Let
$\mathcal{R}_{\infty}$ be as in Definition 2.6. By taking into account all
these identifications, we obtain the map

$\sigma_{m}^{\mathbb{R}}\,=\,\mathcal{R}_{\infty}\circ\sigma_{m}:{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}\to
S_{inv}^{m}(\partial T^{*}{\mathfrak{M}}_{0})/S_{inv}^{m-1}(\partial
T^{*}{\mathfrak{M}}_{0})\,.$

Let $S^{*}{\mathfrak{M}}_{0}$ be the set of unit vectors in
$T^{*}{\mathfrak{M}}_{0}$. Then, in the case of classical symbols, the last
quotient identifies, as usual with ${\mathcal{C}}^{\infty}(\partial
S^{*}{\mathfrak{M}}_{0})$.

We obtain the following proposition.

######  Proposition 4.22.

We use the notation of Remark 4.21.

  1. (1)

For any $T\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$, we have

$\sigma_{m}^{\mathbb{R}}({\mathcal{R}_{\infty}}(T))\,=\,\mathcal{R}_{\infty}(\sigma_{m}(T))\,.$

  2. (2)

The principal symbol

$\sigma_{m}^{\mathbb{R}}:{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}\
\longrightarrow\ S_{inv}^{m}(\partial
T^{*}{\mathfrak{M}}_{0})/S_{inv}^{m-1}(\partial T^{*}{\mathfrak{M}}_{0})$

is surjective and $*$-multiplicative with kernel
${\psi_{\operatorname{sus}}^{m-1}(\partial{\mathfrak{M}}_{0})}$.

###### Proof.

By definition, we have ${\mathcal{R}_{\infty}}(T)=T$ on
$\partial{\mathfrak{M}}_{0}\times(-\infty,-R)$, for some large $R$. Hence

$\sigma_{m}^{\mathbb{R}}({\mathcal{R}_{\infty}}(T))\,=\,\sigma_{m}^{\mathbb{R}}(T)\,=\,\mathcal{R}_{\infty}\circ\sigma_{m}(T)\,,$

see Remark 4.21, thus (1). The surjectivity in (2) follows from the fact that
the quantization map of Remark 4.8 is translation invariant and from
Proposition 4.10, all applied to ${\mathfrak{M}}_{0}\times\mathbb{R}$ instead
of ${\mathfrak{M}}$. The other properties of $\sigma_{m}^{\mathbb{R}}$ follow
from Theorem 4.12(2), again applied to ${\mathfrak{M}}_{0}\times\mathbb{R}$
instead of ${\mathfrak{M}}$. ∎

###  4.4. Further properties of the $\operatorname{inv}$-calculus

We now include some properties of our $\operatorname{inv}$-calculus that are
no longer direct analogues of the classical calculus on compact manifolds, but
they still have counterparts in the $b$-calculus. The mapping properties of
this subsection are a consequence of the corresponding properties for the
$b$-calculus, but the proof presented here are new. Recall the limit operator
map
${\mathcal{R}_{\infty}}:{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\to{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}:={\Psi_{\operatorname{inv}}^{\infty}}(\partial{\mathfrak{M}}_{0}\times{\mathbb{R}})^{\mathbb{R}}$
of Definition 4.17.

######  Definition 4.23.

We let $\Psi_{comp}^{m}({\mathfrak{M}})$ denote the space of
pseudodifferential operators on ${\mathfrak{M}}$ with compactly supported
distribution kernel.

######  Theorem 4.24.

Let
${\mathfrak{M}}={\mathfrak{M}}_{0}\cup\big{(}\partial{\mathfrak{M}}_{0}\times(-\infty,0]\big{)}$
be a manifold with straight cylindrical ends.

  1. (1)

${\mathcal{R}_{\infty}}:{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\to{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}$
is surjective and its kernel consists of $\Psi_{comp}^{m}({\mathfrak{M}})$,
the set of operators $T\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$
whose distribution (or Schwartz) kernel is compactly supported.

  2. (2)

We have a direct sum decomposition

$\begin{gathered}{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\,=\,s_{0}({\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})})+\Psi_{comp}^{m}({\mathfrak{M}})\\\
P\to(s_{0}({\mathcal{R}_{\infty}}(P)),P-s_{0}({\mathcal{R}_{\infty}}(P)))\,.\end{gathered}$

###### Proof.

The surjectivity in (1) follows from Lemma 4.20(3). The form of its kernel
follows from the definition of operators that are translation invariant in a
neighborhood of infinity. Indeed, let
$P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ with
${\mathcal{R}_{\infty}}(P)=0$. Let $R_{P}>0$ be as in Definition 4.1, that is
$P\Phi_{s}(u)=\Phi_{s}P(u)$ for all
$u\in{\mathcal{C}}^{\infty}_{\text{c}}\big{(}\partial{\mathfrak{M}}_{0}\times(-\infty,-R_{P})\big{)}$
and all $s>0$. By the definition of
${\mathcal{R}_{\infty}}(P)v=\Phi_{-s}P\Phi_{s}(v)$ for $s$ large, we obtain
that $Pu=0$ if $u$ has support in
$\partial{\mathfrak{M}}_{0}\times(-\infty,-R_{P})=:{\mathfrak{M}}_{R_{P}}^{c}:={\mathfrak{M}}\smallsetminus{\mathfrak{M}}_{R_{P}}$.
This shows that the distribution kernel of $P$ is contained in
${\mathfrak{M}}\times{\mathfrak{M}}_{R_{P}}$. The same argument applied to
$P^{*}\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ shows that the
distribution kernel of $P$ is contained in
${\mathfrak{M}}_{R_{P^{*}}}\times{\mathfrak{M}}$. Therefore, the distribution
kernel of $P$ is contained in
${\mathfrak{M}}_{R_{P^{*}}}\times{\mathfrak{M}}_{R_{P}}$, which is a compact
set.

(2) follows from (1) and the relation ${\mathcal{R}_{\infty}}\circ s_{0}=id$
of Lemma 4.20. Indeed,

${\mathcal{R}_{\infty}}(P-s_{0}({\mathcal{R}_{\infty}}(P)))\,=\,{\mathcal{R}_{\infty}}(P)-{\mathcal{R}_{\infty}}(s_{0}({\mathcal{R}_{\infty}}(P)))\,=\,{\mathcal{R}_{\infty}}(P)-{\mathcal{R}_{\infty}}(P)\,=\,0\,.$

Therefore
$P-s_{0}({\mathcal{R}_{\infty}}(P))\in\Psi_{comp}^{m}({\mathfrak{M}})$ by (1).
∎

We now discuss a property of the $\operatorname{inv}$-calculus that has no
analogue in the classical case of pseudodifferential operators on a compact
manifold or for the $b$-calculus. We shall denote by $t$ the cylindrical
variable in
$(x,t)\in\partial{\mathfrak{M}}_{0}\times(-\infty,0)\subset{\mathfrak{M}}$ and
with the same letter the corresponding multiplication operator.

######  Proposition 4.25.

Let $t\in\mathbb{R}$ be the second coordinate function on
$\partial{\mathfrak{M}}_{0}\times\mathbb{R}$ and let
$\rho\in{\mathcal{C}}^{\infty}({\mathfrak{M}})$ be such that $\rho(x,t)=t$ for
all $(x,t)\in\partial{\mathfrak{M}}_{0}\times(-\infty,-1]$.

  1. (1)

Let $T\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}$. Then
$[t,T]\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}$.

  2. (2)

If $T\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}$, then
$[\rho,s_{0}(T)]=s_{0}([t,T])$.

  3. (3)

Let $P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$, we have
$[\rho,P]\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ and
${\mathcal{R}_{\infty}}([\rho,P])=[t,{\mathcal{R}_{\infty}}(P)]$.

###### Proof.

(1) is well known from the $b$-calculus. We include the argument in [44].
First, the operator $[t,T]$ is pseudodifferential of order $m-1$. It is also
properly supported. It is translation invariant since

$\Phi_{-s}[t,T]\Phi_{s}\,=\,[\Phi_{-s}t\Phi_{s},\Phi_{-s}T\Phi_{s}]\,=\,[t-s,T]\,=\,[t,T]\,,$

because $s$ is a constant. This completes the proof of (1).

The operators $t$, $\rho$, and $\eta$ are multiplication operators, and hence
they commute. Moreover, the assumption gives that $\rho\eta=t\eta$. The
relation (2) is thus immediate from

$[\rho,s_{0}(T)]\mathrel{\ :=\ }\rho\eta T\eta-\eta
T\eta\rho\,=\,\eta[t,T]\eta\,=:\,s_{0}([t,T])\,.$

To prove (3), let us write first $P=s_{0}({\mathcal{R}_{\infty}}(P))+P_{c}$,
where $P_{c}\in\Psi_{comp}^{m}({\mathfrak{M}})$. Then

$[\rho,P]\,=\,[\rho,s_{0}({\mathcal{R}_{\infty}}(P))]+[\rho,P_{c}]\,=\,s_{0}([t,{\mathcal{R}_{\infty}}(P)])+[\rho,P_{c}]\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\,,$

by (1) and (2) already proved and by Theorem 4.24(2), using also the fact that
$[\rho,P_{c}]\in\Psi_{comp}^{m}({\mathfrak{M}})$. Finally, let
$u\in{\mathcal{C}}^{\infty}_{\text{c}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})$
and $s\in\mathbb{R}$ be large enough. Then

${\mathcal{R}_{\infty}}([\rho,P])u\,=\,\Phi_{-s}([\rho,P])\Phi_{s}u\,=\,[t-s,{\mathcal{R}_{\infty}}(P)]u\,=\,[t,{\mathcal{R}_{\infty}}(P)]u\,.$

This completes the proof. ∎

We shall need also the following boundedness result. In a certain sense, it
should have been part of Theorem 4.12, but it is easier to prove now. Also, we
formulate it as a lemma, not as theorem, since stronger results will be proved
later on. Given normed spaces $X$ and $Y$, we let $\mathcal{B}(X;Y)$ denote
the space of bounded, linear operators $T:X\to Y$. We let
$\mathcal{B}(X):=\mathcal{B}(X;X)$, as usual.

######  Lemma 4.26.

If $P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ and $s\in\mathbb{R}$,
then the map
$P:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})\to{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})$
extends by continuity to a map $H^{s}({\mathfrak{M}})\to
H^{s-m}({\mathfrak{M}}).$ Moreover, the map

${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})\to\mathcal{B}(H^{s}({\mathfrak{M}}),H^{s-m}({\mathfrak{M}}))$

is continuous for the natural topology on
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ (see Remark 4.11).

###### Proof.

Let us write
$P=P_{c}+s_{0}(P_{i})\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$,
where $P_{c}$ has compactly supported distribution kernel and
$P_{i}:={\mathcal{R}_{\infty}}(P)$, as in Theorem 4.24(2). Then
$P_{c}:H^{s}({\mathfrak{M}})\to H^{s-m}({\mathfrak{M}})$ is bounded by the
mapping properties of pseudodifferential operators on compact manifolds. Let
us prove next that $s_{0}(P_{i}):H^{s}({\mathfrak{M}})\to
H^{s-m}({\mathfrak{M}})$ is bounded. Since $s_{0}(P_{i})=\eta P_{i}\eta$ and
$\eta$ is bounded on all $H^{s}$ spaces, it is enough to prove that
$P_{i}:H^{s}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})\to
H^{s-m}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})$ is bounded. Let us
consider a $\mathbb{Z}$-translation invariant partition of unity $(\phi_{k})$,
$k\in\mathbb{Z}$, as in Section 2.4.3. Then
$\phi_{k}P\phi_{j}:H^{s}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})\to
H^{s-m}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})$ is bounded, by the case
of pseudodifferential on compact manifolds. Moreover, $\phi_{k}P\phi_{j}=0$
for $|j-k|$ large and the operators $\phi_{k}^{2}P\phi_{j}^{2}$ and
$\phi_{k+\ell}^{2}P\phi_{j+\ell}^{2}$ are unitary equivalent for
$\ell\in\mathbb{N}$ and $k$ (equivalently $j$) large enough. It follows then
by standard arguments (a lemma of Cotlar, Knapp, Stein, Calderon,
Vaillancourt, Bony, and Lerner, Lemma 18.6.5 in [26]) that
$P=\sum_{k,j\in\mathbb{Z}_{+}}\phi_{k}^{2}P\phi_{j}^{2}$ is bounded. The
continuity for our algebras follows also from the compact case by the same
arguments. ∎

We shall need to following “order reduction operators” (a stronger version
will be obtained later).

######  Corollary 4.27.

Let $s\in\mathbb{R}$. There exists an elliptic operator
$\Lambda_{s}\in{\Psi_{\operatorname{inv}}^{s}}({\mathfrak{M}})$ such that the
induced map $\Lambda_{s}:H^{s}({\mathfrak{M}})\to L^{2}({\mathfrak{M}})$ is an
isomorphism.

###### Proof.

Let $a_{s,t}(\xi):=(t^{2}+\|\xi\|^{2})^{s/2}$, where $t>0$ is a parameter and
$\xi\in T^{*}M$. We have that $a_{s,t}\in S^{s}(T^{*}{\mathfrak{M}})$ and
$a_{s,t}a_{-s,t}=1$. Then

$\displaystyle q(a_{s,t})q(a_{-s,t})-1\to 0\ \mbox{ for }t\to\infty$

in the natural topology on ${\Psi_{\operatorname{inv}}^{0}}({\mathfrak{M}})$
given by Remark 4.11 by the standard formula for the product of
pseudodifferential operators in local coordinates, see Theorem 3.11(2). See
[47] for some similar results. Hence $q(a_{s,t})q(a_{-s,t})$ will be
invertible on each $H^{m}({\mathfrak{M}})$, as long as $t$ is large enough. We
use this for $m=0$ and $m=s$. Let $\Lambda_{s}:=q(a_{s,t})$. Then
$\Lambda_{s}:H^{s}({\mathfrak{M}})\to L^{2}({\mathfrak{M}})$ is continuous, by
Lemma 4.26. The same argument gives that $Q:=q(a_{-s,t})$ maps
$L^{2}({\mathfrak{M}})$ to $H^{s}({\mathfrak{M}})$ continuously. Since
$\Lambda_{s}Q$ is invertible on $L^{2}({\mathfrak{M}})$ (for $t$ large), it
follows that $\Lambda_{s}$ is surjective. Similarly (for $t$ large), we may
assume that $Q\Lambda_{s}$ is invertible on $H^{s}({\mathfrak{M}})$. If
follows that $\Lambda_{s}$ is also injective. Thus we can choose
$\Lambda_{s}:=q(a_{s,t})$, for $t$ large. ∎

We shall use the function $\rho$ of Proposition 4.25, thus

$\rho\in{\mathcal{C}}^{\infty}({\mathfrak{M}})\,,\ \rho<0\mbox{ everywhere,
and }\ \rho(x,t)\,=\,t\ \mbox{ for }\ t\leq-1\,.$

######  Corollary 4.28.

Let $\rho$ be as above, and $m,s\in\mathbb{R}$. Let $k\in\mathbb{Z}_{+}$ and
$\rho^{k}H^{s}({\mathfrak{M}})$ be the space of those functions $u$ such that
$\rho^{-k}u\in H^{s}({\mathfrak{M}})$. On $\rho^{k}H^{s}({\mathfrak{M}})$ we
consider the induced topology. Then every
$P\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$ defines a continuous map

$P:\rho^{k}H^{s}({\mathfrak{M}})\to\rho^{k}H^{s-m}({\mathfrak{M}})\,.$

###### Proof.

The result is known for $k=0$, by Lemma 4.26. Let us prove it by induction on
$k\in\mathbb{Z}_{+}$. Let us assume the result is true for some
$k\in\mathbb{Z}_{+}$ and prove it for $k+1$. We know that
$[\rho,P]\in{\Psi_{\operatorname{inv}}^{m-1}}({\mathfrak{M}})$. Then

$\displaystyle\|\rho^{-k}P(\rho^{k}u)\|_{H^{s-m}}$
$\displaystyle\leq\|\rho^{-k+1}P(\rho^{k-1}u)\|_{H^{s-m}}+\|\rho^{-1}\rho^{-k+1}[P,\rho](\rho^{k-1}u)\|_{H^{s-m}}$
$\displaystyle\leq
C\|u\|_{H^{s}}+C\|\rho^{-k+1}[P,\rho](\rho^{k-1}u)\|_{H^{s-m}}$
$\displaystyle\,=\,C\|u\|_{H^{s-m}}\,,$

where $C>0$ is a generic constant (different at each occurrence) and in the
last two steps we have used the induction properties and the fact that
$\rho^{-1}\in W^{\infty,\infty}({\mathfrak{M}})$. ∎

We stress that the weight used in the last lemma is not the one typically used
in the $b$-calculus (where the weights $e^{\lambda\rho}$ are used instead). We
now spell out some simple consequences of the results in this subsection, for
later use. First, by replacing ${\mathfrak{M}}$ with
${\mathfrak{L}}\times\mathbb{R}$ in Lemma 4.26, we obtain

######  Corollary 4.29.

Let
$P\in{\psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}:={\Psi_{\operatorname{inv}}^{m}}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$.
Then

$P:H^{s}({\mathfrak{L}}\times\mathbb{R})\to
H^{s-m}({\mathfrak{L}}\times\mathbb{R})$

is well-defined and continuous for all $s\in\mathbb{R}$. Moreover, the map
${\psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}\to\mathcal{B}(H^{s}({\mathfrak{L}}\times\mathbb{R});H^{s-m}({\mathfrak{L}}\times\mathbb{R}))$
is continuous.

By introducing an extra factor, we see that the action on the first component
yields a continuous action on the product:

######  Corollary 4.30.

Let $M$ be a Riemannian manifold. Then
${\Psi_{\operatorname{inv}}^{0}}({\mathfrak{M}})$ acts continuously on the
product $L^{2}({\mathfrak{M}}\times M)$.

###### Proof.

We know that $P$ acts continuously on $L^{2}({\mathfrak{M}})$. The corollary
then follows from the fact that $L^{2}({\mathfrak{M}}\times M)\simeq
L^{2}({\mathfrak{M}})\overline{\otimes}L^{2}(M)$, where $\overline{\otimes}$
is the Hilbert space tensor product and the action of $P$ on the tensor
product is $P\overline{\otimes}1$ with respect to this identification. ∎

We also have the following characterization of Sobolev spaces.

######  Corollary 4.31.

Let $u\in H^{t}({\mathfrak{M}})$, for some $t\in\mathbb{R}$. We have $u\in
H^{s}({\mathfrak{M}})$ if, and only if, $Pu\in L^{2}({\mathfrak{M}})$ for all
$P\in{\Psi_{\operatorname{inv}}^{s}}({\mathfrak{M}})$.

###### Proof.

If $u\in H^{s}({\mathfrak{M}})$, then $Pu\in L^{2}({\mathfrak{M}})$ for all
$P\in{\Psi_{\operatorname{inv}}^{s}}({\mathfrak{M}})$ by Lemma 4.26. Let $P$
be as in Corollary 4.27 and $Q$ as in its proof. By the same proof, we may
assume $QP$ to be an isomorphism of $H^{s}({\mathfrak{M}})$. Let us assume
$Pu\in L^{2}({\mathfrak{M}})$. Then $QPu\in H^{s}({\mathfrak{M}})$ and hence
$u\in H^{s}({\mathfrak{M}})$, because we have chosen $P$ and $Q$ such that
$QP$ is an isomorphism of $H^{s}({\mathfrak{M}})$ and $u=(QP)^{-1}QPu$. ∎

An important further property of the $\operatorname{inv}$-calculus is the
characterization of the Fredholm operators in
${\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$. Of course, this can be
obtained from the corresponding result in the (bigger) $b$-calculus. In this
paper, however, we shall obtain this result in the (also bigger) class of
“essentially translation invariant in a neighborhood of infinity operators,”
which we discuss in the next section (see Theorem 7.12).

##  5\. Essentially translation invariant operators

The algebra ${\Psi_{\operatorname{inv}}^{\infty}}({\mathfrak{M}})$ is _not
stable under inversion_ in the sense that if
$T\in{\Psi_{\operatorname{inv}}^{\infty}}({\mathfrak{M}})$ is an invertible
operator on the space $L^{2}({\mathfrak{M}})$, then its inverse does not
necessarily belong to ${\Psi_{\operatorname{inv}}^{\infty}}({\mathfrak{M}})$.
As in Mitrea and Nistor [44, Section 2.1], we now introduce a suitable
enlargement ${\Psi_{\operatorname{ess}}^{\infty}}({\mathfrak{M}})$ of
${\Psi_{\operatorname{inv}}^{\infty}}({\mathfrak{M}})$ that will be stable
under the inversion of elliptic operators of non-negative order. The operators
in ${\Psi_{\operatorname{ess}}^{\infty}}({\mathfrak{M}})$ are called
_essentially translation invariant in a neighborhood of infinity_ (or
_essentially translation invariant at infinity_ or, simply, _essentially
translation invariant_ ; they were called _almost invariant_ in [44]).

Recall that
${\mathfrak{M}}={\mathfrak{M}}_{0}\cup(\partial{\mathfrak{M}}_{0}\times(-\infty,0])$
is a manifold with _straight, compact cylindrical ends._

###  5.1. Definition of essentially translation invariant operators

Let us first look at the distribution kernels of the operators in
${\Psi_{\operatorname{inv}}^{-\infty}}({\mathfrak{M}})$. Recall the discussion
in Remark 3.24. Recall that a continuous linear map
$P:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})\to{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}})^{\prime}$
is in $\Psi^{-\infty}({\mathfrak{M}})$ if, and only if, its distribution
kernel $k_{P}$ is smooth. In particular,

######  Remark 5.1.

Recall that $\Psi_{comp}^{m}({\mathfrak{M}})$ denotes the space of
pseudodifferential operators on ${\mathfrak{M}}$ with compactly supported
distribution kernel. We then have $P\in\Psi_{comp}^{-\infty}({\mathfrak{M}})$
if, and only if,
$k_{P}\in{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{M}}\times{\mathfrak{M}})$.

Recall that ${\mathfrak{L}}$ denotes a closed Riemannian manifold. Also,
similarly to Proposition 4.5, we have the following proposition.

######  Proposition 5.2.

Let ${\mathfrak{L}}$ be a closed Riemannian manifold, as usual. A continuous,
linear operator
$T:{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{L}}\times\mathbb{R})\to{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{L}}\times\mathbb{R})^{\prime}$
is in $\Psi^{-\infty}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$ if, and
only if, there exists
$C_{T}\in{\mathcal{C}}^{\infty}({\mathfrak{L}}\times{\mathfrak{L}}\times\mathbb{R})$
such that

$k_{T}(x,t,y,s)\,=\,C_{T}(x,y,t-s)\in\mathbb{C}\,.$

The kernel $C_{T}$ is called the _convolution kernel_ of $T$. We have
$T\in{\Psi_{\operatorname{inv}}^{-\infty}}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$
if, and only if,
$C_{T}\in{\mathcal{C}}^{\infty}_{\text{c}}({\mathfrak{L}}\times{\mathfrak{L}}\times\mathbb{R})$.

The main point of the last proposition is that the _convolution_ kernel
$C_{T}$ of $T$ is compactly supported, unlike its _distribution_ kernel
$k_{T}$. Explicitly, we have

(5.1)
$Tu(x,t)\,=\,\int_{{\mathfrak{L}}\times\mathbb{R}}C_{T}(x,y,t-s)u(y,s)\,d\operatorname{vol}_{{\mathfrak{L}}}(y)ds\,.$

One of the main points of the upcoming definition of
${\Psi_{\operatorname{ess}}^{m}}({\mathfrak{M}})$ is that this formula
(Equation 5.1) makes sense also for _Schwartz functions,_ whose standard
definition we recall next using a function $\rho$ as in Proposition 4.25:

######  Definition 5.3.

Let $0>\rho\in{\mathcal{C}}^{\infty}({\mathfrak{M}})$, $\rho(x,t)=t$ for
$t\leq-1$.

  1. (1)

We let $\mathcal{S}({\mathfrak{M}}\times{\mathfrak{M}})$ be the space of
functions $u:{\mathfrak{M}}\times{\mathfrak{M}}\to\mathbb{C}$ such that

$\int_{{\mathfrak{M}}^{2}}|\rho(x)^{i}\rho(y)^{j}\nabla_{x}^{k}\nabla_{y}^{l}u(x,y)|^{2}\,d\operatorname{vol}_{{\mathfrak{M}}^{2}}(x,y)<\infty\,.$

  2. (2)

Similarly, let
$\mathcal{S}({\mathfrak{L}}\times{\mathfrak{L}}\times\mathbb{R})$ be the space
of functions
$u:{\mathfrak{L}}\times{\mathfrak{L}}\times\mathbb{R}\to\mathbb{C}$ such that

$\int_{{\mathfrak{L}}^{2}\times\mathbb{R}}|t^{i}\partial_{t}^{j}\nabla_{x}^{k}\nabla_{y}^{l}u(x,y,t)|^{2}\,d\operatorname{vol}_{{\mathfrak{L}}^{2}\times\mathbb{R}}(x,y,t)<\infty\,.$

In these two formulas, $i,j,k,l\in\mathbb{Z}_{+}$ are arbitrary.

######  Remark 5.4.

Of course, to obtain the Schwartz spaces of Definition 5.3, we could have used
the $L^{\infty}$-norm (or any other $L^{p}$-norm, $1\leq p\leq\infty$) instead
of the $L^{2}$-norm appearing in the definition. In particular, we have

$\begin{gathered}\mathcal{S}({\mathfrak{M}}\times{\mathfrak{M}})\,=\,\bigcap_{k\in\mathbb{Z}}\,\rho(x)^{-k}\rho(y)^{-k}W^{\infty,p}({\mathfrak{M}}\times{\mathfrak{M}})\
\mbox{ and}\\\
\mathcal{S}({\mathfrak{L}}\times{\mathfrak{L}}\times\mathbb{R})\,=\,\bigcap_{k\in\mathbb{Z}}\rho(t)^{-k}\,W^{\infty,p}({\mathfrak{L}}\times{\mathfrak{L}}\times\mathbb{R})\,.\end{gathered}$

######  Definition 5.5.

Let ${\mathfrak{L}}$ be a closed Riemannian manifold.

  1. (1)

We let

${\Psi_{\operatorname{sus}}^{-\infty}({\mathfrak{L}})}\subset\Psi^{-\infty}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$

be the set of linear operators
$T\in\Psi^{-\infty}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$ whose
convolution kernel satisfies
$C_{T}\in\mathcal{S}({\mathfrak{L}}\times{\mathfrak{L}}\times\mathbb{R})$ (see
Proposition 5.2 and Equation 5.1).

  2. (2)

Then we let ${\Psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}\mathrel{\ :=\
}{\psi_{\operatorname{sus}}^{m}({\mathfrak{L}})}+{\Psi_{\operatorname{sus}}^{-\infty}({\mathfrak{L}})}\subset\Psi^{m}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$
(see 4.3).

######  Remark 5.6.

Definition 5.5 gives right away that

${\psi_{\operatorname{sus}}^{-\infty}({\mathfrak{L}})}\mathrel{\ :=\
}{\Psi_{\operatorname{inv}}^{-\infty}}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}\subset{\Psi_{\operatorname{sus}}^{-\infty}({\mathfrak{L}})}\,.$

Indeed, the first space consists of operators with _compactly_ supported
convolution kernels, whereas the second one consists of operators with
_Schwartz_ convolution kernels.

Recall the cut-off function $\eta:{\mathfrak{M}}\to[0,1]$ of 4.18 (defined by
$\eta$ smooth with support in $\partial{\mathfrak{M}}_{0}\times(-\infty,-1)$
and $\eta=1$ on $\partial{\mathfrak{M}}_{0}\times(-\infty,-2]$). Also, recall
that, for any
$T\in{\psi_{\operatorname{sus}}^{m}(\partial{\mathfrak{M}}_{0})}:={\Psi_{\operatorname{inv}}^{m}}(\partial{\mathfrak{M}}_{0}\times\mathbb{R})^{\mathbb{R}}$,
we have $s_{0}(T):=\eta
T\eta\in{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})$, see Lemma 4.20 and
its proof. Recall the following definition from [44].

######  Definition 5.7.

Let ${\mathfrak{M}}$ be a manifold with straight cylindrical ends
${\mathfrak{M}}={\mathfrak{M}}_{0}\cup(\partial{\mathfrak{M}}_{0}\times(-\infty,0])$.

  1. (1)

We let $\Psi_{\mathcal{S}}^{-\infty}({\mathfrak{M}})$ be the set of
pseudodifferential operators in $\Psi^{-\infty}({\mathfrak{M}})$ with
distribution kernel in $\mathcal{S}({\mathfrak{M}}\times{\mathfrak{M}})$.

  2. (2)

Let ${\Psi_{\operatorname{sus}}^{-\infty}(\partial{\mathfrak{M}}_{0})}$ be as
in Definition 5.5. We let

${\Psi_{\operatorname{ess}}^{-\infty}}({\mathfrak{M}})\mathrel{\ :=\
}s_{0}({\Psi_{\operatorname{sus}}^{-\infty}(\partial{\mathfrak{M}}_{0})})+\Psi_{\mathcal{S}}^{-\infty}({\mathfrak{M}})\,.$

  3. (3)

Finally, we define the space of essentially translation invariant operators in
a neighborhood of infinity of order $m$ as

${\Psi_{\operatorname{ess}}^{m}}({\mathfrak{M}})\mathrel{\ :=\
}{\Psi_{\operatorname{inv}}^{m}}({\mathfrak{M}})+{\Psi_{\operatorname{ess}}^{-\infty}}({\mathfrak{M}})\,.$

In particular,
${\Psi_{\operatorname{sus}}^{-\infty}({\mathfrak{L}})}=:{\Psi_{\operatorname{ess}}^{-\infty}}({\mathfrak{L}}\times\mathbb{R})^{\mathbb{R}}$.

######  Remark 5.8.

Recall that $\Psi_{comp}^{m}({\mathfrak{M}})$ denotes the space of
pseudodifferential operators on ${\mathfrak{M}}$ with compactly supported
distribution kernel. The last two definitions provide the following simple
inclusions. First,

$\Psi_{comp}^{-\infty}({\mathfrak{M}})\subset\Psi_{\mathcal{S}}^{-\infty}({\mathfrak{M}})\,,$

because the space of Schwartz sections contains the space of compactly
supported sections. Let $s_{0}$ be as in the last definition (it was first
introduced in Definition 4.19). Using also the inclusion of Remark 5.6, we
obtain

${\Psi_{\operatorname{inv}}^{-\infty}}({\mathfrak{M}})\,=\,s_{0}({\psi_{\operatorname{sus}}^{-\infty}(\partial{\mathfrak{M}}_{0})})+\Psi_{comp}^{-\infty}({\mathfrak{M}})\\\
\subset
s_{0}({\Psi_{\operatorname{sus}}^{-\infty}(\partial{\mathfrak{M}}_{0})})+\Psi_{\mathcal{S}}^{-\infty}({\mathfrak{M}})\,=:\,{\Psi_{\operatorname{ess}}^{-\infty}}({\mathfrak{M}})\,.$

###  5.2. Properties of essentially translation invariant operators

We now proceed to prove some of the basic properties of
${\Psi_{\operatorname{ess}}^{m}}({\mathfrak{M}})$ (many of which were implicit
in [44]; their proof in that paper were based on suitable norms on
${\Psi_{\operatorname{inv}}^{-\infty}}({\mathfrak{M}})$). Here we propose a