[columns=3, title=Index of notation]

# Topology of the space of conormal distributions

Jesús A. Álvarez López  Department of Mathematics and CITMAga
University of Santiago de Compostela
15782 Santiago de Compostela
Spain<EMAIL_ADDRESS>,  Yuri A. Kordyukov  Institute of Mathematics
Russian Academy of Sciences
112 Chernyshevsky street
450077 Ufa
Russia<EMAIL_ADDRESS>and  Eric Leichtnam  Institut de Mathématiques
de Jussieu-PRG
CNRS
Batiment Sophie Germain (bureau 740)
Case 7012
75205 Paris Cedex 13, France<EMAIL_ADDRESS>
###### Abstract.

Given a closed manifold $M$ and a closed regular submanifold $L$, consider the
corresponding locally convex space $I=I(M,L)$ of conormal distributions, with
its natural topology, and the strong dual
$I^{\prime}=I^{\prime}(M,L)=I(M,L;\Omega)^{\prime}$ of the space of conormal
densities. It is shown that $I$ is a barreled, ultrabornological, webbed,
Montel, acyclic LF-space, and $I^{\prime}$ is a complete Montel space, which
is a projective limit of bornological barreled spaces. In the case of
codimension one, similar properties and additional descriptions are proved for
the subspace $K\subset I$ of conormal distributions supported in $L$ and for
its strong dual $K^{\prime}$. We construct a locally convex Hausdoff space $J$
and a continuous linear map $I\to J$ such that the sequence $0\to K\to I\to
J\to 0$ as well as the transpose sequence $0\to J^{\prime}\to I^{\prime}\to
K^{\prime}\to 0$ are short exact sequences in the category of continuous
linear maps between locally convex spaces. Finally, it is shown that $I\cap
I^{\prime}=C^{\infty}(M)$ in the space of distributions. These results will be
applied to prove a trace formula for foliated flows, involving the reduced
cohomologies of the complexes of leafwise currents that are conormal and dual-
conormal at the leaves preserved by the flow.

###### Key words and phrases:

Conormal distributions, dual-conormal distributions, barreled,
ultrabornological, webbed, acyclic, Montel space, complete, boundedly
retractive, reflexive

###### 2020 Mathematics Subject Classification:

46F05, 46A13, 46M40

The authors are partially supported by the grants MTM2017-89686-P and
PID2020-114474GB-I00 (AEI/FEDER, UE) and ED431C 2019/10 (Xunta de Galicia,
FEDER)

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Symbols
  4. 4 Conormal distributions
  5. 5 Dual-conormal distributions
  6. 6 Conormal distributions at the boundary
  7. 7 Conormal sequence
  8. 8 Dual-conormal sequence

##  1\. Introduction

Given a manifold $M$ and a closed regular submanifold $L\subset M$, the
corresponding space $I=I(M,L)$ of conormal distributions was considered in
[17, 13], [15, Section 18.2], [36, Chapters 3–5], [25, Chapters 4 and 6], [27,
Chapters 3 and 9]. But its study was mainly oriented to the important role
played in the analysis of pseudodifferential operators. Also as a tool to get
appropriate generalizations of those operators to manifolds with boundary or
corners, stratified spaces, etc. For instance, the existence of asymptotic
expansions of their symbols was well analyzed. However we are not aware of any
publication with a deep study of its natural topology; actually, this project
was begun in [25, Chapters 4 and 6], but that publication remains incomplete.
The main objective of this paper is to fill in that gap of study.

Assume $M$ is compact for the sake of simplicity. Let
$\operatorname{Diff}(M,L)\subset\operatorname{Diff}(M)$ be the subalgebra of
differential operators generated by the vector fields on $M$ tangent to $L$,
and let $H^{s}(M)$ be the Sobolev space of order $s\in\mathbb{R}$. The space
$I^{(s)}=I^{(s)}(M,L)$ of conormal distributions of Sobolev order $s$ consists
of the distributions $u\in C^{-\infty}(M)$ satisfying
$\operatorname{Diff}(M,L)u\subset H^{s}(M)$, endowed with the projective
topology given by the maps $P:I^{(s)}\to H^{s}(M)$
($P\in\operatorname{Diff}(M,L)$).

By definition, $I=\bigcup_{s}I^{(s)}$ with the corresponding locally convex
inductive topology, which is continuously contained in $C^{-\infty}(M)$. There
is another expression $I=\bigcup_{m}I^{m}$, using spaces $I^{m}=I^{m}(M,L)$ of
conormal distributions with symbol order $m$, which can be locally described
in terms of symbol spaces with a partial Fourier transform.

We show that every $I^{(s)}$ is a totally reflexive Fréchet space (Proposition
4.1), and the LF-space $I$ is a barreled, ultrabornological, webbed, acyclic
Montel space, and therefore complete, boundedly retractive and reflexive
(Corollaries 4.2 and 4.7). As a first step, these properties are established
for symbol spaces.

All notions and properties considered here have straightforward extensions for
distributional sections of vector bundles. In particular, for the density
bundle $\Omega=\Omega M$, the strong dual
$I^{\prime}(M,L)=I(M,L;\Omega)^{\prime}$, simply denoted by $I^{\prime}$, is
also continuously contained in $C^{-\infty}(M)$. We prove that $I^{\prime}$ is
a complete Montel space and $I^{\prime}=\varprojlim I^{\prime\,(s)}$, where
$I^{\prime\,(s)}=I^{\prime\,(s)}(M,L)=I^{(-s)}(M,L;\Omega)^{\prime}$ is
bornological and barreled (Corollaries 5.2, 5.1 and 5.3).

Now assume $L$ is of codimension one. For simplicity reasons, consider the
case where $L$ is transversely oriented. Then cut $M$ along $L$ to obtain a
compact manifold with boundary $\boldsymbol{M}$ and a projection
$\boldsymbol{\pi}:\boldsymbol{M}\to M$. In this way, we can take advantage of
the machinery developed in [15, 25] to study conormal distributions at the
boundary; in particular, some notions from small b-calculus are used. For
instance, with the terminology and notation of [15, 25], let
$\mathcal{A}(\boldsymbol{M})$ (respectively, $\mathcal{A}(\boldsymbol{M})$) be
the locally convex space of extendable (respectively, supported) conormal
distributions at the boundary. Then, via the push-down map
$\boldsymbol{\pi}_{*}$, the image of $\dot{\mathcal{A}}(\boldsymbol{M})$ is
$I$, and $\mathcal{A}(\boldsymbol{M})$ becomes isomorphic to another locally
convex space $J=J(M,L)$. Let $K=K(M,L)\subset I$ be the subspace of conormal
distributions supported in $L$. Like in the definition of $I^{\prime}$,
consider also the strong dual spaces $K^{\prime}(M,L)=K(M,L;\Omega)^{\prime}$
and $J^{\prime}(M,L)=J(M,L;\Omega)^{\prime}$, simply denoted by $K^{\prime}$
and $J^{\prime}$. It is proved that $K$ is a limit subspace
(LABEL:c:_K(M_L)_=_bigcup_s_K^{(s)}(M_L)), the spaces $K$, $J$, $K^{\prime}$
and $J^{\prime}$ satisfy the properties stated for $I$ and $I^{\prime}$
(Corollaries 7.19, 7.21, 8.1, 8.2 and 8.3), and there are short exact
sequences, $0\to K\to I\to J\to 0$ and its transpose $0\leftarrow
K^{\prime}\leftarrow I^{\prime}\leftarrow J^{\prime}\leftarrow 0$, in the
category of continuous linear maps between locally convex spaces (Propositions
7.27 and 8.8). These sequences are relevant because $J$, $K$, $J^{\prime}$ and
$K^{\prime}$ have better descriptions than $I$ and $I^{\prime}$ (Corollaries
6.37, 6.46 and 7.24). Finally, it is shown that $I\cap
I^{\prime}=C^{\infty}(M)$ (Theorem 8.11), extending a result of [15, 25] for
the boundary case. Most of these properties are first established in the
boundary case (Section 6).

Besides the extensions for distributional sections of vector bundles, some
results are extended to non-compact manifolds. We also analyze the action of
differential operators on these spaces, as well as the pull-back and push-
forward homomorphisms induced by maps on these spaces (Sections 4, 5, 7 and
8).

Via the Schwartz kernel theorem, the spaces of pseudodifferential and
differential operators can be described as $\Psi(M)\equiv I(M^{2},\Delta)$ and
$\operatorname{Diff}(M)\equiv K(M^{2},\Delta)$, where $\Delta$ is the diagonal
of $M^{2}$. Thus $\Psi(M)$ and $\operatorname{Diff}(M)$ become examples of
locally convex spaces satisfying the above properties.

The wave front set of any $u\in I(M,L)$ satisfies $\operatorname{WF}(u)\subset
N^{*}L\setminus 0_{L}$ (considering $N^{*}L\subset T^{*}M$) [14, Chapter
VIII], [15, Chapter XVIII]; this is the reason of the term “conormal
distribution.” The larger space of all distributions whose wave front set is
contained in any prescribed closed cone of $T^{*}M\setminus 0_{M}$, like
$N^{*}L\setminus 0_{L}$, also has a natural topology which was studied in [6].

Our results for codimension one can be clearly extended to arbitrary
codimension. We only consider codimension one for simplicity reasons. It is
also clear that there are further extensions to manifolds with corners,
stratified spaces, etc.

The case of codimension one is also enough for our application in a trace
formula for simple foliated flows [4]. These are simple flows
$\phi=\\{\phi^{t}\\}$ that preserve the leaves of a foliation $\mathcal{F}$ on
$M$. C. Deninger conjectured the existence of a “Leftchetz distribution”
$L_{\text{\rm dis}}(\phi)$ on $\mathbb{R}$ for the induced pull-back action
$\phi^{*}=\\{\phi^{t\,*}\\}$ on the leafwise reduced cohomology
$\bar{H}^{\bullet}(\mathcal{F})$, and predicted a formula for $L_{\text{\rm
dis}}(\phi)$ involving data from the closed orbits and fixed points [8]. Here,
$\bar{H}^{\bullet}(\mathcal{F})$ is the maximal Hausdorff quotient of the
leafwise cohomology $H^{\bullet}(\mathcal{F})$, defined by the de Rham
derivative of the leaves acting on leafwise differential forms smooth on $M$,
equipped with the $C^{\infty}$ topology. But can not use leafwise forms smooth
on $M$ if there are leaves preserved by $\phi$; they do not work well.
Instead, we consider the spaces $I(\mathcal{F})$ and $I^{\prime}(\mathcal{F})$
of distributional leafwise currents that are conormal and dual-conormal at the
preserved leaves, giving rise to reduced cohomologies,
$\bar{H}^{\bullet}I(\mathcal{F})$ and
$\bar{H}^{\bullet}I^{\prime}(\mathcal{F})$, with actions $\phi^{*}$. The
spaces $K(\mathcal{F})$, $J(\mathcal{F})$, $K^{\prime}(\mathcal{F})$ and
$J^{\prime}(\mathcal{F})$ are similarly defined, obtaining short exact
sequences,
$0\to\bar{H}^{\bullet}K(\mathcal{F})\to\bar{H}(\mathcal{F})^{\bullet}I\to\bar{H}^{\bullet}J(\mathcal{F})\to
0$ and
$0\leftarrow\bar{H}^{\bullet}K^{\prime}(\mathcal{F})\leftarrow\bar{H}^{\bullet}I^{\prime}(\mathcal{F})\leftarrow\bar{H}^{\bullet}J^{\prime}(\mathcal{F})\leftarrow
0$. In this way, the definition of $L_{\text{\rm dis}}(\phi)$ for both
$\bar{H}^{\bullet}I(\mathcal{F})$ and
$\bar{H}^{\bullet}I^{\prime}(\mathcal{F})$ together can be reduced to the
cases of $\bar{H}^{\bullet}K(\mathcal{F})$, $\bar{H}^{\bullet}J(\mathcal{F})$,
$\bar{H}^{\bullet}K^{\prime}(\mathcal{F})$ and
$\bar{H}^{\bullet}J^{\prime}(\mathcal{F})$. This can be done by using the
descriptions of $K(\mathcal{F})$, $J(\mathcal{F})$, $K^{\prime}(\mathcal{F})$
and $J^{\prime}(\mathcal{F})$, and some additional ingredients. In this way we
can define $L_{\text{\rm dis}}(\phi)$, which has the desired expression plus a
zeta invariant produced by the use of the b-trace of R. Melrose [24]. However
the ingredients can be chosen so that the zeta invariant vanishes [3], and the
predicted formula becomes correct.

##  2\. Preliminaries

###  2.1. Topological vector spaces

The field of coefficients is $\mathbb{K}=\mathbb{R},\mathbb{C}$. For the
general theory of topological vector spaces (TVSs), we follow the references
[9, 16, 21, 31, 28], assuming the following conventions. We always consider
locally convex spaces (LCSs), which are not assumed to be Hausdorff (contrary
to the definition of [31]); the abbreviation LCHS is used in the Hausdorff
case. Local convexity is preserved by all operations we use. For any
inductive/projective system (or spectrum) of continuous linear maps between
LCSs, we have its (locally convex) inductive/projective limit; in particular,
when the inductive/projective spectrum consists of a sequence of continuous
inclusions, their union/intersection is endowed with the inductive/projective
limit topology. This applies to the locally convex direct sum and the
topological product of LCSs. LF-spaces are not assumed to be strict. The
(continuous) dual $X^{\prime}$ is always endowed with the strong topology;
i.e., we write $X^{\prime}=X^{\prime}_{\beta}$ with the usual notation.

Some homological theory of LCSs will be used (see [39] and references therein)
For instance, for an inductive spectrum of LCSs of the form
$(X_{k})=(X_{0}\subset X_{1}\subset\cdots)$, the condition of being _acyclic_
can be described as follows [39, Theorem 6.1]: for all $k$, there is some
$k^{\prime}\geq k$ such that, for all $k^{\prime\prime}\geq k^{\prime}$, the
topologies of $X_{k^{\prime}}$ and $X_{k^{\prime\prime}}$ coincide on some
$0$-neighborhood of $X_{k}$. In this case, $X:=\bigcup_{k}X_{k}$ is Hausdorff
if and only if all $X_{k}$ are Hausdorff [39, Proposition 6.3]. It is said
that $(X_{k})$ is _regular_ if any bounded $B\subset X$ is contained and
bounded in some step $X_{k}$. If moreover the topologies of $X$ and $X_{k}$
coincide on $B$, then $(X_{k})$ is said to be _boundedly retractive_. The
conditions of being _compactly retractive_ or _sequentially retractive_ are
similarly defined, using compact sets or convergent sequences.

If the steps $X_{k}$ are Fréchet spaces, the above properties of $(X_{k})$
depend only on the LF-space $X$ [39, Chapter 6, p. 111]; thus it may be said
that they are properties of $X$. In this case, $X$ is acyclic if and only if
it is boundedly/compactly/sequentially retractive [39, Proposition 6.4]. As a
consequence, acyclic LF-spaces are complete and regular [39, Corollary 6.5].

A topological vector subspace $Y\subset X$ is called a _limit subspace_ if
$Y\equiv\bigcup_{k}Y_{k}$, where $Y_{k}=X\cap Y_{k}$. This condition is
satisfied if and only if the spectrum consisting of the spaces $X_{k}/Y_{k}$
is acyclic [39, Chapter 6, p. 110].

Assume the steps $X_{k}$ are LCHSs. It is said that $(X_{k})$ is _compact_ if
the inclusion maps are compact operators. In this case, $(X_{k})$ is clearly
acyclic, and so $X$ is Hausdorff. Moreover $X$ is a complete bornological DF
Montel space [18, Theorem 6’].

The above concepts and properties also apply to an inductive/projective
spectrum consisting of continuous inclusions $X_{r}\subset X_{r^{\prime}}$ for
$r<r^{\prime}$ in $\mathbb{R}$ because $\bigcap_{r}X_{r}=\bigcap_{k}X_{r_{k}}$
and $\bigcup_{r}X_{r}=\bigcup_{k}X_{s_{k}}$ for sequences
$r_{k}\downarrow-\infty$ and $s_{k}\uparrow\infty$.

In the category of continuous linear maps between LCSs, the exactness of a
sequence $0\to X\to Y\to Z\to 0$ means that it is exact as a sequence of
linear maps and consists of topological homomorphisms [39, Sections 2.1 and
2.2].

###  2.2. Smooth functions on open subsets of $\mathbb{R}^{n}$

For an open $U\subset\mathbb{R}^{n}$
($n\in\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$), we will use the Fréchet space
$C^{\infty}(U)$ of smooth ($\mathbb{K}$-valued) functions on $U$ with the
topology of uniform approximation of all partial derivatives on compact
subsets, which is described by the semi-norms

(2.1) $\|u\|_{K,C^{k}}=\sup_{x\in K,\ |\alpha|\leq
k}|\partial^{\alpha}u(x)|\;,$

for any compact $K\subset U$, $k\in\mathbb{N}_{0}$ and
$\alpha\in\mathbb{N}_{0}^{n}$, using standard multi-index notation. (Sometimes
the notation $C^{\infty}_{\text{\rm loc}}(U)$ is used for this space, and
$C^{\infty}(U)$ is used for the uniform space denoted by
$C^{\infty}_{\text{\rm ub}}(U)$ in this paper.) For any $S\subset M$, the
notation $C^{\infty}_{S}(U)$ is used for the subspace of smooth functions
supported in $S$ (with the subspace topology). (The common notation
$C^{\infty}(S)=C^{\infty}_{S}(U)$ would be confusing when extended to other
function spaces.) Recall also the strict LF-space of compactly supported
functions,

(2.2) $C^{\infty}_{\text{\rm c}}(U)=\bigcup_{K}C^{\infty}_{K}(U)\;,$

for compact subsets $K\subset M$ (an exhausting increasing sequence of compact
subsets is enough).

The above definitions have straightforward generalizations to the case of
functions with values in $\mathbb{K}^{l}$, obtaining

(2.3) $C^{\infty}_{{\cdot}/\text{\rm c}}(U,\mathbb{K}^{l})\equiv
C^{\infty}_{{\cdot}/\text{\rm c}}(U)\otimes\mathbb{K}^{l}\;.$

(The notation $C^{\infty}_{{\cdot}/\text{\rm c}}$ or $C^{\infty}_{\text{\rm
c}/{\cdot}}$ refers to both $C^{\infty}$ and $C^{\infty}_{\text{\rm c}}$.)

###  2.3. Vector bundles

The notation $M$ will be used for a smooth manifold of dimension $n$, and $E$
for a ($\mathbb{K}$-) vector bundle over $M$. The fibers of $E$ are denoted by
$E_{x}$ ($x\in M$), the zero in every $E_{x}$ by $0_{x}$, and the image of the
zero section by $0_{M}$. Let $\Omega^{a}E$ ($a\in\mathbb{R}$) denote the line
bundle of $a$-densities of $E$, let $\Omega E=\Omega^{1}E$, and let $o(E)$ be
the flat line bundle of orientations of $E$. We may use the notation
$E_{L}=E|_{L}$ for the restriction of $E$ to a submanifold $L\subset M$. As
particular cases, we have the tangent and cotangent $\mathbb{R}$-vector
bundles, $TM$ and $T^{*}M$, and the associated $\mathbb{K}$-vector bundles
$o(M)=o(TM)$, $\Omega^{a}M=\Omega^{a}TM$ and $\Omega M=\Omega TM$.

###  2.4. Smooth and distributional sections

Our notation for spaces of distributional sections mainly follows [25], with
some minor changes to fit our application in [4]. Some notation from [14, 15]
is also used.

Generalizing $C^{\infty}(U,\mathbb{K}^{l})$, we have the Fréchet space
$C^{\infty}(M;E)$ of smooth sections of $E$, whose topology is described by
semi-norms $\|{\cdot}\|_{K,C^{k}}$ defined as in (2.1) via charts $(U,x)$ of
$M$ and diffeomorphisms of triviality $E_{U}\equiv U\times\mathbb{K}^{l}$,
with $K\subset U$. This procedure is standard and will be used again with
other section spaces.

Redundant notation will be removed as usual. For instance, we write
$C^{\infty}(M)$ (respectively, $C^{\infty}(M,\mathbb{K}^{l})$) in the case of
the trivial vector bundle of rank $1$ (respectively, $l$). We also write
$C^{\infty}(L,E)=C^{\infty}(L,E_{L})$ and
$C^{\infty}(M;\Omega^{a})=C^{\infty}(M;\Omega^{a}M)$. We may write
$C^{\infty}(E)=C^{\infty}(M;E)$ if $M$ is fixed, but this may also mean the
space of smooth functions on $E$. In particular,
$\mathfrak{X}(M)=C^{\infty}(M;TM)$ is the Lie algebra of vector fields. The
subspace $C^{\infty}_{S}(M;E)$ is defined like in Section 2.2. Similar
notation will be used with any LCHS and $C^{\infty}(M)$-module continuously
included in $C^{\infty}(M;E)$, or in the space $C^{-\infty}(M;E)$ defined
below.

The notation $C^{\infty}(M;E)$, or $C^{\infty}(E)$, is also used with any
smooth fiber bundle $E$, obtaining a completely metrizable topological space
with the weak $C^{\infty}$ topology.

The strict LF-space $C^{\infty}_{\text{\rm c}}(M;E)$ of compactly supported
smooth sections is defined like in (2.2), using compact subsets $K\subset M$.
There is a continuous inclusion $C^{\infty}_{\text{\rm c}}(M;E)\subset
C^{\infty}(M;E)$. If $M$ is a fiber bundle, the LCHS $C^{\infty}_{\text{\rm
cv}}(M;E)$ of smooth sections with compact support in the vertical direction
is similarly defined using (2.1) and (2.2) with closed subsets $K\subset M$
whose intersection with the fibers is compact (now an exhaustive increasing
sequence of such subsets $K$ is not enough).

The space of distributional sections with arbitrary/compact support is

(2.4) $C^{-\infty}_{{\cdot}/\text{\rm c}}(M;E)=C^{\infty}_{\text{\rm
c}/{\cdot}}(M;E^{*}\otimes\Omega)^{\prime}\;.$

(In [14], these dual spaces are endowed with the weak topology, contrary to
our convention.) Integration of smooth densities on $M$ and the canonical
pairing of $E$ and $E^{*}$ define a dense continuous inclusion
$C^{\infty}_{{\cdot}/\text{\rm c}}(M;E)\subset C^{-\infty}_{{\cdot}/\text{\rm
c}}(M;E)$. If $U\subset M$ is open, the extension by zero defines a TVS-
embedding $C^{\pm\infty}_{\text{\rm c}}(U;E)\subset C^{\pm\infty}_{\text{\rm
c}}(M;E)$.

The above spaces of distributional sections are locally finitely
$C^{\infty}(M)$-generated, and can be also described in terms of the
corresponding spaces of distributions as the algebraic tensor product as
$C^{\infty}(M)$-modules

(2.5) $C^{-\infty}_{{\cdot}/\text{\rm c}}(M;E)\equiv
C^{-\infty}_{{\cdot}/\text{\rm
c}}(M)\otimes_{C^{\infty}(M)}C^{\infty}(M;E)\;.$

To show this identity, $E$ can be realized as a vector subbundle of a trivial
vector bundle $F=M\times\mathbb{K}^{l^{\prime}}$ [12, Theorem 4.3.1]. Then,
like in (2.3),

$\displaystyle C^{-\infty}_{{\cdot}/\text{\rm c}}(M;F)$ $\displaystyle\equiv
C^{-\infty}_{{\cdot}/\text{\rm c}}(M)\otimes\mathbb{K}^{l^{\prime}}\equiv
C^{-\infty}_{{\cdot}/\text{\rm
c}}(M)\otimes_{C^{\infty}(M)}C^{\infty}(M)\otimes\mathbb{K}^{l^{\prime}}$
$\displaystyle\equiv C^{-\infty}_{{\cdot}/\text{\rm
c}}(M)\otimes_{C^{\infty}(M)}C^{\infty}(M;F)\;,$

and the spaces of (2.5) clearly correspond by these identities. Expressions
like (2.5) hold for most of the spaces of distributional sections we will
consider. Thus, from now on, we will mostly define and study spaces for the
trivial line bundle or density bundles, and then the notation for arbitrary
vector bundles will be used without further comment, and the properties have
straightforward extensions.

Consider also the Fréchet space $C^{k}(M)$ ($k\in\mathbb{N}_{0}$) of $C^{k}$
functions, with the semi-norms $\|{\cdot}\|_{K,C^{k}}$ given by (2.1), the LF-
space $C^{k}_{\text{\rm c}}(M)$ of $C^{k}$ functions with compact support,
defined like in (2.2), and the space $C^{\prime\,-k}_{{\cdot}/\text{\rm
c}}(M)$ of distributions of order $k$ with arbitrary/compact support, defined
like in (2.4). (A prime is added to this notation to distinguish
$C^{\prime\,0}_{{\cdot}/\text{\rm c}}(M)$ from $C^{0}_{{\cdot}/\text{\rm
c}}(M)$.) There are dense continuous inclusions

(2.6) $C^{k^{\prime}}_{{\cdot}/\text{\rm c}}(M)\subset
C^{k}_{{\cdot}/\text{\rm c}}(M)\;,\quad C^{\prime\,-k^{\prime}}_{\text{\rm
c}/{\cdot}}(M)\supset C^{\prime\,-k}_{\text{\rm
c}/{\cdot}}(M)\quad(k<k^{\prime})\;,$

with $\bigcap_{k}C^{k}_{{\cdot}/\text{\rm c}}(M)=C^{\infty}_{{\cdot}/\text{\rm
c}}(M)$ and $\bigcup_{k}C^{\prime\,-k}_{\text{\rm
c}}(M)=C^{-\infty}_{\text{\rm c}}(M)$ [28, Exercise 12.108]. The space
$\bigcup_{k}C^{\prime\,-k}(M)$ consists of the distributions with some order;
it is $C^{-\infty}(M)$ just when $M$ is compact.

Let us recall some properties of the spaces we have seen. In addition of the
fact that $C^{\infty}(M)$ and $C^{k}(M)$ are Fréchet spaces [16, Example
2.9.3], $C^{\infty}_{\text{\rm c}}(M)$ and $C^{k}_{\text{\rm c}}(M)$ are
complete and Hausdorff [16, Examples 2.12.6 and 2.12.8].
$C^{\infty}_{{\cdot}/\text{\rm c}}(M)$ and $C^{k}_{{\cdot}/\text{\rm c}}(M)$
are ultrabornological because this property is satisfied by Fréchet spaces and
preserved by inductive limits [28, Example 13.2.8 (d) and Theorem 13.2.11],
and therefore they are barreled [29, Observation 6.1.2 (b)].
$C^{\pm\infty}_{{\cdot}/\text{\rm c}}(M)$ is a Montel space (in particular,
barreled) [16, Examples 3.9.3, 3.9.4 and 3.9.6 and Proposition 3.9.9], [9,
Section 8.4.7, Theorem 8.4.11 and Application 8.4.12], [31, the paragraph
before IV.5.9], and therefore reflexive [9, Section 8.4.7], [21, 6.27.2 (1)],
[31, IV.5.8]. $C^{\infty}_{{\cdot}/\text{\rm c}}(M)$ is a Schwartz space [16,
Examples 3.15.2 and 3.15.3], and therefore $C^{-\infty}_{{\cdot}/\text{\rm
c}}(M)$ is ultrabornological [16, Exercise 3.15.9 (c)]. $C^{\infty}(M)$ is
distinguished [16, Examples 3.16.1]. $C^{\pm\infty}_{{\cdot}/\text{\rm c}}(M)$
is webbed because this property is satisfied by LF-spaces and strong duals of
strict inductive limits of sequences of metrizable LCSs [7, Proposition
IV.4.6], [22, 7.35.1 (4) and 7.35.4 (8)], [28, Theorem 14.6.5].

###  2.5. Linear operators on section spaces

Let $E$ and $F$ be vector bundles over $M$, and let $A:C^{\infty}_{\text{\rm
c}}(M;E)\to C^{\infty}(M;F)$ be a continuous linear map. Recall that the
_transpose_ of $A$ is the continuous linear map

$\displaystyle A^{t}:C^{-\infty}_{\text{\rm c}}(M;F^{*}\otimes\Omega)\to
C^{-\infty}(M;E^{*}\otimes\Omega)\;,$ $\displaystyle\langle
A^{t}v,u\rangle=\langle v,Au\rangle\;,\quad u\in C^{\infty}_{\text{\rm
c}}(M;E)\;,\quad v\in C^{-\infty}_{\text{\rm c}}(M;F^{*}\otimes\Omega)\;.$

For instance, the transpose of $C^{\infty}_{\text{\rm
c}}(M;E^{*}\otimes\Omega)\subset C^{\infty}(M;E^{*}\otimes\Omega)$ is a
continuous dense injection $C^{-\infty}_{\text{\rm c}}(M;E)\subset
C^{-\infty}(M;E)$. If $A^{t}$ restricts to a continuous linear map
$C^{\infty}_{\text{\rm c}}(M;F^{*}\otimes\Omega)\to
C^{\infty}(M;E^{*}\otimes\Omega)$, then $A^{tt}:C^{-\infty}_{\text{\rm
c}}(M;E)\to C^{-\infty}(M;F)$ is a continuous extension of $A$, also denoted
by $A$.

There are versions of the construction of $A^{t}$ and $A^{tt}$ when both the
domain and codomain of $A$ have compact support, or no support restriction.
For example, for any open $U\subset M$, the transpose of the extension by zero
$C^{\infty}_{\text{\rm c}}(U;E^{*}\otimes\Omega)\subset C^{\infty}_{\text{\rm
c}}(M;E^{*}\otimes\Omega)$ is the restriction map $C^{-\infty}(M;E)\to
C^{-\infty}(U,E)$, $u\mapsto u|_{U}$, and the transpose of the restriction map
$C^{\infty}(M;E^{*}\otimes\Omega)\to C^{\infty}(U,E^{*}\otimes\Omega)$ is the
extension by zero $C^{-\infty}_{\text{\rm c}}(U;E)\subset
C^{-\infty}_{\text{\rm c}}(M;E)$. In the whole paper, inclusion maps may be
denoted by $\iota$ and restriction maps by $R$, without further comment.

Other related concepts and results, like singular support, Schwartz kernel and
the Schwartz kernel theorem, can be seen e.g. in [24].

###  2.6. Pull-back and push-forward of distributional sections

Recall that any smooth map $\phi:M^{\prime}\to M$ induces the continuous
linear pull-back map

(2.7) $\phi^{*}:C^{\infty}(M;E)\to C^{\infty}(M^{\prime};\phi^{*}E)\;.$

Suppose that moreover $\phi$ is a submersion. Then it also induces the
continuous linear push-forward map

(2.8) $\phi_{*}:C^{\infty}_{\text{\rm
c}}(M^{\prime};\phi^{*}E\otimes\Omega_{\text{\rm fiber}})\to
C^{\infty}_{\text{\rm c}}(M;E)\;,$

where $\Omega_{\text{\rm fiber}}=\Omega_{\text{\rm
fiber}}M^{\prime}=\Omega\mathcal{V}$ for the vertical subbundle
$\mathcal{V}=\ker\phi_{*}\subset TM^{\prime}$. Since $\phi^{*}\Omega
M\equiv\Omega(TM/\mathcal{V})\equiv\Omega^{-1}_{\text{\rm fiber}}\otimes\Omega
M^{\prime}$, the transposes of the versions of (2.7) and (2.8) with
$E^{*}\otimes\Omega M$ are continuous extensions of (2.8) and (2.7) [14,
Theorem 6.1.2],

(2.9) $\displaystyle\phi_{*}:C^{-\infty}_{\text{\rm
c}}(M^{\prime};\phi^{*}E\otimes\Omega_{\text{\rm fiber}})\to
C^{-\infty}_{\text{\rm c}}(M;E)\;,$ (2.10)
$\displaystyle\phi^{*}:C^{-\infty}(M;E)\to
C^{-\infty}(M^{\prime};\phi^{*}E)\;,$

also called push-forward and pull-back linear maps. The term integration along
the fibers is also used for $\phi_{*}$.

If $\phi:M^{\prime}\to M$ is a proper local diffeomorphism, then we can omit
$\Omega_{\text{\rm fiber}}$ and the compact support condition in (2.8) and
(2.9), and therefore the compositions $\phi_{*}\phi^{*}$ and
$\phi^{*}\phi_{*}$ are defined on smooth/distributional sections.

The space $C^{\infty}(M^{\prime};\phi^{*}E)$ becomes a $C^{\infty}(M)$-module
via the algebra homomorphism $\phi^{*}:C^{\infty}(M)\to
C^{\infty}(M^{\prime})$, and we have

(2.11) $C^{\pm\infty}_{{\cdot}/\text{\rm
c}}(M^{\prime};\phi^{*}E)=C^{\pm\infty}_{{\cdot}/\text{\rm
c}}(M^{\prime})\otimes_{C^{\infty}(M)}C^{\infty}(M;E)\;.$

Using (2.5) and (2.11), we can describe (2.7)–(2.10) as the
$C^{\infty}(M)$-tensor products of their trivial-line-bundle versions with the
identity map on $C^{\infty}(M;E)$. Thus, from now on, only pull-back and push-
forward of distributions will be considered.

###  2.7. Differential operators

Let $\operatorname{Diff}(M)$ be the filtered algebra and
$C^{\infty}(M)$-module of differential operators, filtered by the order. Every
$\operatorname{Diff}^{m}(M)$ ($m\in\mathbb{N}_{0}$) is spanned as
$C^{\infty}(M)$-module by all compositions of up to $m$ elements of
$\mathfrak{X}(M)$, considered as the Lie algebra of derivations of
$C^{\infty}_{{\cdot}/\text{\rm c}}(M)$. In particular,
$\operatorname{Diff}^{0}(M)\equiv C^{\infty}(M)$.

For vector bundles $E$ and $F$ over $M$, the above concepts can be extended by
taking the $C^{\infty}(M)$-tensor product with $C^{\infty}(M;F\otimes E^{*})$,
obtaining $\operatorname{Diff}^{m}(M;E,F)$ ($\operatorname{Diff}^{m}(M;E)$
being obtained if $E=F$); here, redundant notation is simplified like in the
case of $C^{\pm\infty}(M;E)$ (Section 2.4). If $E$ is a line bundle, then

$\displaystyle\operatorname{Diff}^{m}(M;E)$
$\displaystyle\equiv\operatorname{Diff}^{m}(M)\otimes_{C^{\infty}(M)}C^{\infty}(M;E\otimes
E^{*})$ (2.12)
$\displaystyle\equiv\operatorname{Diff}^{m}(M)\otimes_{C^{\infty}(M)}C^{\infty}(M)\equiv\operatorname{Diff}^{m}(M)\;.$

Any $A\in\operatorname{Diff}^{m}(M;E)$ defines a continuous linear
endomorphism $A$ of $C^{\infty}_{{\cdot}/\text{\rm c}}(M;E)$. We get
$A^{t}\in\operatorname{Diff}^{m}(M;E^{*}\otimes\Omega)$ using integration by
parts. So $A$ has continuous extensions to a continuous endomorphism $A$ of
$C^{-\infty}_{{\cdot}/\text{\rm c}}(M;E)$ (Section 2.5). A similar map is
defined when $A\in\operatorname{Diff}^{m}(M;E,F)$.

Other related concepts like symbols and ellipticity can be seen e.g. in [24].

###  2.8. $L^{2}$ sections

Recall that the Hilbert space $L^{2}(M;\Omega^{1/2})$ of square-integrable
half-densities is the completion of $C^{\infty}_{\text{\rm
c}}(M;\Omega^{1/2})$ with the scalar product $\langle
u,v\rangle=\int_{M}u\bar{v}$. The induced norm is denoted by $\|{\cdot}\|$.

If $M$ is compact, the space $L^{2}(M;E)$ of square-integrable sections of $E$
can be described as the $C^{\infty}(M)$-tensor product of
$L^{2}(M;\Omega^{1/2})$ and $C^{\infty}(M;\Omega^{-1/2}\otimes E)$. It becomes
a Hilbert space with the scalar product $\langle
u,v\rangle=\int_{M}(u,v)\,\omega$ determined by the choice of a
Euclidean/Hermitian structure $({\cdot},{\cdot})$ on $E$ and a non-vanishing
$\omega\in C^{\infty}(M;\Omega)$. The equivalence class of its norm
$\|{\cdot}\|$ is independent of those choices; in this sense, $L^{2}(M;E)$ is
called a _Hilbertian space_ if no norm is distinguished.

When $M$ is not assumed to be compact, any choice of $({\cdot},{\cdot})$ and
$\omega$ can be used to define $L^{2}(M;E)$ and
$\langle{\cdot},{\cdot}\rangle$. Now $L^{2}(M;E)$ and the equivalence class of
$\|{\cdot}\|$ depends on the choices involved. The independence still holds
for sections supported in any compact $K\subset M$, obtaining the Hilbertian
space $L^{2}_{K}(M;E)$. Then we get the strict LF-space $L^{2}_{\text{\rm
c}}(M;E)$ is defined like in (2.2). On the other hand, let

(2.13) $L^{2}_{\text{\rm loc}}(M;E)=\\{\,u\in C^{-\infty}(M;E)\mid
C^{\infty}_{\text{\rm c}}(M)\,u\subset L^{2}_{\text{\rm c}}(M;E)\,\\}\;,$

which is a Fréchet space with the semi-norms $u\mapsto\|f_{k}u\|$, for a
countable partition of unity $\\{f_{k}\\}\subset C^{\infty}_{\text{\rm
c}}(M)$. If $M$ is compact, then $L^{2}_{\text{\rm loc/c}}(M;E)\equiv
L^{2}(M;E)$ as TVSs. The spaces $L^{2}_{\text{\rm loc/c}}(M;E)$ satisfy the
obvious version of (2.4).

Any $A\in\operatorname{Diff}^{m}(M;E)$ can be considered as a densely defined
operator in $L^{2}(M;E)$. Integration by parts shows that the adjoint $A^{*}$
is defined by an element $A^{*}\in\operatorname{Diff}^{m}(M;E)$ (the _formal
adjoint_ of $A$).

###  2.9. $L^{\infty}$ sections

A Euclidean/Hermitian structure can be also used to define the Banach space
$L^{\infty}(M;E)$ of its essentially bounded sections, with the norm
$\|u\|_{L^{\infty}}=\operatorname{ess\,sup}_{x\in M}|u(x)|$. There is a
continuous injection $L^{\infty}(M;E)\subset L^{2}_{\text{\rm loc}}(M;E)$. If
$M$ is compact, then the equivalence class of $\|{\cdot}\|_{L^{\infty}}$ is
independent of $({\cdot},{\cdot})$.

###  2.10. Sobolev spaces

####  2.10.1. Local and compactly supported versions

Recall that the Fourier transform, $f\mapsto\hat{f}$, defines a TVS-
automorphism of the Schwartz space $\SS(\mathbb{R}^{n})$, which extends to a
TVS-automorphism of the space $\SS(\mathbb{R}^{n})^{\prime}$ of tempered
distributions [14, Section 7.1]. In turn, for every $s\in\mathbb{R}$, this
automorphism of $\SS(\mathbb{R}^{n})^{\prime}$ restricts to unitary
isomorphism

(2.14)
$H^{s}(\mathbb{R}^{n})\xrightarrow{\cong}L^{2}(\mathbb{R}^{n},(1+|\xi|^{2})^{s}\,d\xi)\;,\quad
f\mapsto\hat{f}\;,$

for some Hilbert space $H^{s}(\mathbb{R}^{n})$, called the Sobolev space of
order $s$ of $\mathbb{R}^{n}$. There is a canonical continuous inclusion
$H^{s}(\mathbb{R}^{n})\subset C^{-\infty}(\mathbb{R}^{n})$.

For any compact $K\subset\mathbb{R}^{n}$, we have the Hilbert subspace
$H^{s}_{K}(\mathbb{R}^{n})\subset H^{s}(\mathbb{R}^{n})$ of elements supported
in $K$. Then the LCHSs $H^{s}_{\text{\rm c/loc}}(U)$ are defined like in (2.4)
and (2.13), using the spaces $H^{s}_{K}(\mathbb{R}^{n})$ for compact subsets
$K\subset U$. They are continuously included in $C^{-\infty}_{\text{\rm
c}/{\cdot}}(U)$.

For a manifold $M$, the definition of the LCHSs $H^{s}_{\text{\rm c/loc}}(M)$
can be extended in a standard way, by using a locally finite atlas and a
partition of unity consisting of compactly supported smooth functions. These
are the compactly supported and local versions of the Sobolev space of order
$s$ of $M$. They are continuously included in $C^{-\infty}_{\text{\rm
c}/{\cdot}}(M)$.

####  2.10.2. Case of compact manifolds

Suppose for a while that $M$ is compact. Then $H^{s}(M):=H_{\text{\rm
loc}}^{s}(M)=H^{s}_{\text{\rm c}}(M)$ is a Hilbertian space called the
_Sobolev space_ of order $s$ of $M$. We have

(2.15) $H^{-s}(M)=H^{s}(M;\Omega)^{\prime}\;,$

given by (2.4). Moreover there are dense continuous inclusions,

(2.16) $\displaystyle H^{s}(M)\subset H^{s^{\prime}}(M)\;,$ for
$s^{\prime}<s$, and (2.17) $\displaystyle H^{s}(M)\subset C^{k}(M)\subset
H^{k}(M)\;,$ (2.18) $\displaystyle H^{-s}(M)\supset C^{\prime\,-k}(M)\supset
H^{-k}(M)\;,$

for $s>k+n/2$. The first inclusion of (2.17) is the Sobolev embedding theorem,
and (2.18) is the transpose of the version of (2.17) with $\Omega M$. Moreover
the inclusions (2.16) are compact (Rellich theorem). So the spaces
$H^{s}_{\text{\rm loc/c}}(M)$ form a compact spectrum with

(2.19) $C^{\infty}(M)=\bigcap_{s}H^{s}(M)\;\quad
C^{-\infty}(M)=\bigcup_{s}H^{s}(M)\;.$

Any $A\in\operatorname{Diff}^{m}(M;E)$ defines a bounded operator
$A:H^{s+m}(M;E)\to H^{s}(M;E)$. It can be considered as a densely defined
operator in $H^{s}(M;E)$, which is closable because, after fixing a scalar
product in $H^{s}(M;E)$, the adjoint of $A$ in $H^{s}(M;E)$ is densely defined
since it is induced by
$\bar{A}^{t}\in\operatorname{Diff}^{m}(M;\bar{E}^{*}\otimes\Omega)$ via the
identity $H^{s}(M;E)\equiv
H^{s}(M;\bar{E})^{\prime}=H^{-s}(M;\bar{E}^{*}\otimes\Omega)$, where the bar
stands for the complex conjugate. In the case $s=0$, the adjoint of $A$ is
induced by the formal adjoint $A^{*}\in\operatorname{Diff}^{m}(M;E)$.

By the elliptic estimate, a scalar product on $H^{s}(M)$ can be defined by
$\langle u,v\rangle_{s}=\langle(1+P)^{s}u,v\rangle$, for any choice of a
nonnegative symmetric elliptic $P\in\operatorname{Diff}^{2}(M)$, where
$\langle{\cdot},{\cdot}\rangle$ is defined like in Section 2.8 and $(1+P)^{s}$
is given by the spectral theorem for all $s\in\mathbb{R}$. The corresponding
norm $\|{\cdot}\|_{s}$ is independent of the choice of $P$. For a vector
bundle $E$, a precise scalar product on $H^{s}(M;E)$ can be defined as above,
using any choice of a Euclidean/Hermitian structure $({\cdot},{\cdot})$ on $E$
and a non-vanishing $\omega\in C^{\infty}(M;\Omega)$ (Section 2.8), besides a
nonnegative symmetric elliptic $P\in\operatorname{Diff}^{2}(M;E)$. If
$E=\Omega^{1/2}M$, then $\langle{\cdot},{\cdot}\rangle_{s}$ can be defined
independently of $({\cdot},{\cdot})$ and $\omega$ (Section 2.8).

If $s\in\mathbb{N}_{0}$, we can also describe

(2.20) $\displaystyle H^{s}(M)$ $\displaystyle=\\{\,u\in
C^{-\infty}(M)\mid\operatorname{Diff}^{s}(M)\,u\subset L^{2}(M)\,\\}\;,$
(2.21) $\displaystyle H^{-s}(M)$
$\displaystyle=\operatorname{Diff}^{s}(M)\,L^{2}(M)\;,$

with the respective projective and injective topologies given by the maps
$A:H^{s}(M)\to L^{2}(M)$ and $A:L^{2}(M)\to H^{-s}(M)$
($A\in\operatorname{Diff}^{s}(M)$).

####  2.10.3. Extension to non-compact manifolds

If $M$ is not assumed to be compact, then $H^{s}(M;E)$ can be defined as the
completion of $C^{\infty}_{\text{\rm c}}(M;E)$ with respect to the scalar
product $\langle{\cdot},{\cdot}\rangle_{s}$ defined by the above choices of
$({\cdot},{\cdot})$, $\omega$ and $P$; in this case, $H^{s}(M;E)$ and the
equivalence class of $\|{\cdot}\|_{s}$ depends on the choices involved. For
instance, in (2.14), $H^{s}(\mathbb{R}^{n})$ can be also described with the
Laplacian of $\mathbb{R}^{n}$ and the standard density and Euclidean/Hermitian
structure. The version of (2.15) with $E$ can be used to define $H^{-s}(M;E)$.
With this generality, the versions of (2.20), (2.21), the right-hand side
inclusions of (2.17) and (2.18), and the inclusions “$\subset$” of (2.19) are
wrong, but the versions of (2.16), the left-hand side continuous inclusions of
(2.17) and (2.18), and the continuous inclusion “$\supset$” of (2.19) are
true. Thus the intersection and union of (2.19) define new LCHSs
$H^{\pm\infty}(M)$, which are continuously included in $C^{\pm\infty}(M)$. Any
$A\in\operatorname{Diff}^{m}(M;E)$ defines continuous linear maps
$A:H^{s}_{\text{\rm c/loc}}(M;E)\to H^{s-m}_{\text{\rm c/loc}}(M;F)$.

###  2.11. Weighted spaces

Assume first that $M$ is compact. Take any $h\in C^{\infty}(M)$ which is
positive almost everywhere; for instance, $\\{h=0\\}$ could be any countable
union of submanifolds of positive codimension. Then the _weighted Sobolev
space_ $hH^{s}(M;E)$ is a Hilbertian space; a scalar product
$\langle{\cdot},{\cdot}\rangle_{hH^{s}}$ is given by $\langle
u,v\rangle_{hH^{s}}=\langle h^{-1}u,h^{-1}v\rangle_{s}$, depending on the
choice of a scalar product $\langle{\cdot},{\cdot}\rangle_{s}$ on $H^{s}(M;E)$
like in Section 2.10. The corresponding norm is denoted by
$\|{\cdot}\|_{hH^{s}}$. In particular, we get the _weighted $L^{2}$ space_
$hL^{2}(M;E)$. We have $h>0$ just when $hH^{m}(M;E)=H^{m}(M;E)$; in this case,
$\langle{\cdot},{\cdot}\rangle_{hH^{s}}$ can be described like
$\langle{\cdot},{\cdot}\rangle_{s}$ using $h^{-2}\omega$ instead of $\omega$.
Thus the notation $hH^{m}(M;E)$ for $h>0$ is used when changing the density;
e.g., if it is different from a distinguished choice, say a Riemanian volume.

If $M$ is not compact, $hH^{s}(M;E)$ and $\langle u,v\rangle_{hH^{s}}$ depend
on $h$ and the chosen definitions of $H^{s}(M;E)$ and $\langle u,v\rangle_{s}$
(Section 2.10). We also get the weighted spaces $hH^{s}_{\text{\rm
c/loc}}(M;E)$, and the weighted Banach space $hL^{\infty}(M;E)$ with the norm
$\|u\|_{hL^{\infty}}=\|h^{-1}u\|_{L^{\infty}}$. There is a continuous
injection $hL^{\infty}(M;E)\subset hL^{2}_{\text{\rm loc}}(M;E)$.

###  2.12. Bounded geometry

Concerning this topic, we follow [10, 30, 35, 32, 33]; see also [2] for the
way we present it and examples.

####  2.12.1. Manifolds and vector bundles of bounded geometry

The concepts recalled here become relevant when $M$ is not compact. Equip $M$
with a Riemannian metric $g$, and let $\nabla$ denote its Levi-Civita
connection, $R$ its curvature and $\operatorname{inj}_{M}\geq 0$ its
injectivity radius (the infimum of the injectivity radius at all points). If
$M$ is connected, we have an induced distance function $d$. If $M$ is not
connected, we can also define $d$ taking $d(p,q)=\infty$ if $p$ and $q$ belong
to different connected components. Observe that $M$ is complete if
$\operatorname{inj}_{M}>0$. For $r>0$ and $p\in M$, let $B(p,r)$ and
$\overline{B}(p,r)$ denote the open and closed $r$-balls centered at $p$.

Recall that $M$ is said to be of _bounded geometry_ if
$\operatorname{inj}_{M}>0$ and $\sup|\nabla^{m}R|<\infty$ for every
$m\in\mathbb{N}_{0}$. This concept has the following chart description.

######  Theorem 2.1 (Eichhorn [10]; see also [30, 32, 33]).

$M$ is of bounded geometry if and only if, for some open ball
$B\subset\mathbb{R}^{n}$ centered at $0$, there are normal coordinates at
every $p\in M$ defining a diffeomorphism $y_{p}:V_{p}\to B$ such that the
corresponding Christoffel symbols $\Gamma^{i}_{jk}$, as a family of functions
on $B$ parametrized by $i$, $j$, $k$ and $p$, lie in a bounded set of the
Fréchet space $C^{\infty}(B)$. This equivalence holds as well replacing the
Cristoffel symbols with the metric coefficients $g_{ij}$.

From now on in this subsection, assume $M$ is of bounded geometry and consider
the charts $y_{p}:V_{p}\to B$ given by Theorem 2.1. The radius of $B$ is
denoted by $r_{0}$.

######  Proposition 2.2 (Schick [32, Theorem A.22], [33, Proposition 3.3]).

For every $\alpha\in\mathbb{N}_{0}^{n}$, the function
$|\partial^{\alpha}(y_{q}y_{p}^{-1})|$ is bounded on $y_{p}(V_{p}\cap V_{q})$,
uniformly on $p,q\in M$.

######  Proposition 2.3 (Shubin [35, Appendix A1.1, Lemma 1.2]).

For any $0<2r\leq r_{0}$, there is a subset $\\{p_{k}\\}\subset M$ and some
$N\in\mathbb{N}$ such that the balls $B(p_{k},r)$ cover $M$, and every
intersection of $N+1$ sets $B(p_{k},2r)$ is empty.

A vector bundle $E$ of rank $l$ over $M$ is said to be of _bounded geometry_
when it is equipped with a family of local trivializations over the charts
$(V_{p},y_{p})$, for small enough $r_{0}$, with corresponding defining cocycle
$a_{pq}:V_{p}\cap
V_{q}\to\operatorname{GL}(l,\mathbb{K})\subset\mathbb{K}^{l\times l}$, such
that, for all $\alpha\in\mathbb{N}_{0}^{n}$, the function
$|\partial^{\alpha}(a_{pq}y_{p}^{-1})|$ is bounded on $y_{p}(V_{p}\cap
V_{q})$, uniformly on $p,q\in M$. When referring to local trivializations of a
vector bundle of bounded geometry, we always mean that they satisfy this
condition. If the corresponding defining cocycle is valued in the
orthogonal/unitary group, then $E$ is said to be of _bounded geometry_ as a
Euclidean/Hermitian vector bundle.

####  2.12.2. Uniform spaces

For every $m\in\mathbb{N}_{0}$, a function $u\in C^{m}(M)$ is said to be
_$C^{m}$ -uniformy bounded_ if there is some $C_{m}\geq 0$ with
$|\nabla^{m^{\prime}}u|\leq C_{m}$ on $M$ for all $m^{\prime}\leq m$. These
functions form the _uniform $C^{m}$ space_ $C_{\text{\rm ub}}^{m}(M)$, which
is a Banach space with the norm $\|{\cdot}\|_{C^{m}_{\text{\rm ub}}}$ defined
by the best constant $C_{m}$. Equivalently, we may take the norm
$\|{\cdot}\|^{\prime}_{C^{m}_{\text{\rm ub}}}$ defined by the best constant
$C^{\prime}_{m}\geq 0$ such that $|\partial^{\alpha}(uy_{p}^{-1})|\leq
C^{\prime}_{m}$ on $B$ for all $p\in M$ and $|\alpha|\leq m$; in fact, it is
enough to consider any subset of points $p$ so that $\\{V_{p}\\}$ covers $M$
[32, Theorem A.22], [33, Proposition 3.3]. The _uniform $C^{\infty}$ space_ is
$C^{\infty}_{\text{\rm ub}}(M)=\bigcap_{m}C_{\text{\rm ub}}^{m}(M)$. This is a
Fréchet space with the semi-norms $\|{\cdot}\|_{C^{m}_{\text{\rm ub}}}$ or
$\|{\cdot}\|^{\prime}_{C^{m}_{\text{\rm ub}}}$. It consists of the functions
$u\in C^{\infty}(M)$ such that all functions $uy_{p}^{-1}$ lie in a bounded
set of $C^{\infty}(B)$, which are said to be _$C^{\infty}$ -uniformy bounded_.

The same definitions apply to functions with values in $\mathbb{C}^{l}$.
Moreover the definition of uniform spaces with covariant derivative can be
also considered for non-complete Riemannian manifolds.

######  Proposition 2.4 (Shubin [35, Appendix A1.1, Lemma 1.3]; see also [33,
Proposition 3.2]).

Given $r$, $\\{p_{k}\\}$ and $N$ like in Proposition 2.3, there is a partition
of unity $\\{f_{k}\\}$ subordinated to the open covering $\\{B(p_{k},r)\\}$,
which is bounded in the Fréchet space $C^{\infty}_{\text{\rm ub}}(M)$.

For a Euclidean/Hermitian vector bundle $E$ of bounded geometry over $M$, the
_uniform $C^{m}$ space_ $C_{\text{\rm ub}}^{m}(M;E)$, of _$C^{m}$ -uniformly
bounded_ sections, can be defined by introducing
$\|{\cdot}\|^{\prime}_{C^{m}_{\text{\rm ub}}}$ like the case of functions,
using local trivializations of $E$ to consider every $uy_{p}^{-1}$ in
$C^{m}(B,\mathbb{C}^{l})$ for all $u\in C^{m}(M;E)$. Then, as above, we get
the _uniform $C^{\infty}$ space_ $C^{\infty}_{\text{\rm ub}}(M;E)$ of
_$C^{\infty}$ -uniformly bounded_ sections, which are the sections $u\in
C^{\infty}(M;E)$ such that all functions $uy_{p}^{-1}$ define a bounded set of
$C^{\infty}_{\text{\rm ub}}(B;\mathbb{C}^{l})$. In particular,
$\mathfrak{X}_{\text{\rm ub}}(M):=C^{\infty}_{\text{\rm ub}}(M;TM)$ is a
$C^{\infty}_{\text{\rm ub}}(M)$-submodule and Lie subalgebra of
$\mathfrak{X}(M)$. Observe that

(2.22) $C^{m}_{\text{\rm ub}}(M)=\\{\,u\in C^{m}(M)\mid\mathfrak{X}_{\text{\rm
ub}}(M)\,\overset{(m^{\prime})}{\cdots}\,\mathfrak{X}_{\text{\rm
ub}}(M)\,u\subset L^{\infty}(M)\ \forall m^{\prime}\leq m\,\\}\;.$

####  2.12.3. Differential operators of bounded geometry

Like in Section 2.7, by using $\mathfrak{X}_{\text{\rm ub}}(M)$ and
$C^{\infty}_{\text{\rm ub}}(M)$ instead of $\mathfrak{X}(M)$ and
$C^{\infty}(M)$, we get the filtered subalgebra and $C^{\infty}_{\text{\rm
ub}}(M)$-submodule $\operatorname{Diff}_{\text{\rm
ub}}(M)\subset\operatorname{Diff}(M)$ of differential operators of _bounded
geometry_. Moreover this concept can be extended to vector bundles of bounded
geometry $E$ and $F$ over $M$ by taking the $C^{\infty}_{\text{\rm
ub}}(M)$-tensor product of $\operatorname{Diff}_{\text{\rm ub}}(M)$ and
$C^{\infty}_{\text{\rm ub}}(M;F\otimes E^{*})$, obtaining the filtered
$C^{\infty}_{\text{\rm ub}}(M)$-submodule $\operatorname{Diff}_{\text{\rm
ub}}(M;E,F)\subset\operatorname{Diff}(M;E,F)$ (or
$\operatorname{Diff}_{\text{\rm ub}}(M;E)$ if $E=F$). Bounded geometry of
differential operators is preserved by compositions and by taking transposes,
and by taking formal adjoints in the case of Hermitian vector bundles of
bounded geometry; in particular, $\operatorname{Diff}_{\text{\rm ub}}(M;E)$ is
a filtered subalgebra of $\operatorname{Diff}(M;E)$. Like in (2.12), we have

(2.23) $\operatorname{Diff}_{\text{\rm
ub}}^{m}(M;\Omega^{\frac{1}{2}})\equiv\operatorname{Diff}_{\text{\rm
ub}}^{m}(M)\;.$

Every $A\in\operatorname{Diff}_{\text{\rm ub}}^{m}(M;E)$ defines continuous
linear maps $A:C^{m+k}_{\text{\rm ub}}(M;E)\to C^{k}_{\text{\rm ub}}(M;E)$
($k\in\mathbb{N}_{0}$), which induce a continuous endomorphism $A$ of
$C^{\infty}_{\text{\rm ub}}(M;E)$. It is said that $A$ is _uniformly elliptic_
if there is some $C\geq 1$ such that, for all $p\in M$ and $\xi\in
T^{*}_{p}M$, its leading symbol $\sigma_{m}(A)$ satisfies

$C^{-1}|\xi|^{m}\leq|\sigma_{m}(A)(p,\xi)|\leq C|\xi|^{m}\;.$

This condition is independent of the choice of the Hermitian metric of bounded
geometry on $E$. Any $A\in\operatorname{Diff}^{m}_{\text{\rm ub}}(M;E,F)$
satisfies the second inequality. The case where
$A\in\operatorname{Diff}_{\text{\rm ub}}^{m}(M;E,F)$ is similar.

####  2.12.4. Sobolev spaces of manifolds of bounded geometry

For any Hermitian vector bundle $E$ of bounded geometry over $M$, any
nonnegative symmetric uniformly elliptic
$P\in\operatorname{Diff}^{2}_{\text{\rm ub}}(M;E)$ can be used to define the
Sobolev space $H^{s}(M;E)$ ($s\in\mathbb{R}$) with a scalar product
$\langle{\cdot},{\cdot}\rangle_{s}$ (Section 2.10). Any choice of $P$ defines
the same Hilbertian space $H^{s}(M;E)$, which is a $C^{\infty}_{\text{\rm
ub}}(M)$-module. In particular, $L^{2}(M;E)$ is the $C^{\infty}_{\text{\rm
ub}}(M)$-tensor product of $L^{2}(M;\Omega^{1/2})$ and $C^{\infty}_{\text{\rm
ub}}(M;E\otimes\Omega^{1/2})$, and $H^{s}(M;E)$ is the $C^{\infty}_{\text{\rm
ub}}(M)$-tensor product of $H^{s}(M)$ and $C^{\infty}_{\text{\rm ub}}(M;E)$.
For instance, we may take $P=\nabla^{*}\nabla$ for any unitary connection
$\nabla$ of bounded geometry on $E$. For $s\in\mathbb{N}_{0}$, the Sobolev
space $H^{s}(M)$ can be also described with the scalar product

$\langle u,v\rangle^{\prime}_{s}=\sum_{k}\sum_{|\alpha|\leq
s}\int_{B}f_{k}^{2}(x)\cdot\partial^{\alpha}(uy_{p_{k}}^{-1})(x)\cdot\overline{\partial^{\alpha}(vy_{p_{k}}^{-1})(x)}\,dx\;,$

using the partition of unity $\\{f_{k}\\}$ given by Proposition 2.4 [32,
Theorem A.22], [33, Propositions 3.2 and 3.3], [35, Appendices A1.2 and A1.3].
A similar scalar product $\langle{\cdot},{\cdot}\rangle^{\prime}_{s}$ can be
defined for $H^{s}(M;E)$ with the help of local trivializations defining the
bounded geometry of $E$. Every $A\in\operatorname{Diff}_{\text{\rm
ub}}^{m}(M;E)$ defines bounded operators $A:H^{m+s}(M;E)\to H^{s}(M;E)$
($s\in\mathbb{R}$), which induce a continuous endomorphism $A$ of
$H^{\pm\infty}(M;E)$. For any almost everywhere positive $h\in C^{\infty}(M)$,
we have $hH^{m}(M;E)=H^{m}(M;E)$ if and only if $h>0$ and $h^{\pm 1}\in
C^{\infty}_{\text{\rm ub}}(M)$.

######  Proposition 2.5 (Roe [30, Proposition 2.8]).

If $m^{\prime}>m+n/2$, then $H^{m^{\prime}}(M;E)\subset C_{\text{\rm
ub}}^{m}(M;E)$, continuously. Thus $H^{\infty}(M;E)\subset
C^{\infty}_{\text{\rm ub}}(M;E)$, continuously.

##  3\. Symbols

The canonical coordinates of $\mathbb{R}^{n}\times\mathbb{R}^{l}$
($n,l\in\mathbb{N}_{0}$) are denoted by
$(x,\xi)=(x^{1},\dots,x^{n},\xi^{1},\dots,\xi^{l})$, and let
$dx=dx^{1}\wedge\dots\wedge dx^{n}$ and $d\xi=d\xi^{1}\wedge\dots\wedge
d\xi^{l}$. Recall that a _symbol_ of _order_ at most $m\in\mathbb{R}$ on
$U\times\mathbb{R}^{l}$, or simply on $U$, is a function $a\in
C^{\infty}(U\times\mathbb{R}^{l})$ such that, for any compact $K\subset U$,
and multi-indices $\alpha\in\mathbb{N}_{0}^{n}$ and
$\beta\in\mathbb{N}_{0}^{l}$,

(3.1) $\|a\|_{K,\alpha,\beta,m}:=\sup_{x\in K,\
\xi\in\mathbb{R}^{l}}\frac{|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|}{(1+|\xi|)^{m-|\beta|}}<\infty\;.$

The set of symbols of order at most $m$, $S^{m}(U\times\mathbb{R}^{l})$,
becomes a Fréchet space with the semi-norms $\|{\cdot}\|_{K,\alpha,\beta,m}$
given by (3.1). There are continuous inclusions

(3.2) $S^{m}(U\times\mathbb{R}^{l})\subset
S^{m^{\prime}}(U\times\mathbb{R}^{l})\quad(m<m^{\prime})\;,$

giving rise to the LCSs

$S^{\infty}(U\times\mathbb{R}^{l})=\bigcup_{m}S^{m}(U\times\mathbb{R}^{l})\;,\quad
S^{-\infty}(U\times\mathbb{R}^{l})=\bigcap_{m}S^{m}(U\times\mathbb{R}^{l})\;.$

The LF-space $S^{\infty}(U\times\mathbb{R}^{l})$ is a filtered algebra and
$C^{\infty}(U)$-module with the pointwise multiplication. The Fréchet space
$S^{-\infty}(U\times\mathbb{R}^{l})$ is a filtered ideal and
$C^{\infty}(U)$-submodule of $S^{\infty}(U\times\mathbb{R}^{l})$. The
homogeneous components of the corresponding graded algebra are

$S^{(m)}(U\times\mathbb{R}^{l})=S^{m}(U\times\mathbb{R}^{l})/S^{m-1}(U\times\mathbb{R}^{l})\;.$

The notation $S^{m}(\mathbb{R}^{l})$, $S^{\pm\infty}(\mathbb{R}^{l})$, etc. is
used when $U=\mathbb{R}^{0}=\\{0\\}$; in this case, the subscripts $K$ and
$\alpha$ are omitted from the notation of the semi-norms in (3.1).

Since $S^{\infty}(U\times\mathbb{R}^{l})$ is an LF-space, we get the following
(see Section 2.4).

######  Proposition 3.1.

$S^{\infty}(U\times\mathbb{R}^{l})$ is barreled, ultrabornological and webbed.

There are continuous inclusions (see Section 2.4 for the definition of
$C^{\infty}_{\text{\rm cv}}(U\times\mathbb{R}^{l})$)

(3.3) $C^{\infty}_{\text{\rm cv}}(U\times\mathbb{R}^{l})\subset
S^{-\infty}(U\times\mathbb{R}^{l})\;,\quad
S^{\infty}(U\times\mathbb{R}^{l})\subset C^{\infty}(U\times\mathbb{R}^{l})\;;$

in particular, $S^{\infty}(U\times\mathbb{R}^{l})$ is Hausdorff. According to
(2.1) and (3.3), we get continuous semi-norms $\|{\cdot}\|_{Q,C^{k}}$ on
$S^{\infty}(U\times\mathbb{R}^{l})$, for any compact $Q\subset
U\times\mathbb{R}^{l}$ and $k\in\mathbb{N}_{0}$, given by

(3.4) $\|a\|_{Q,C^{k}}=\sup_{(x,\xi)\in Q,\ |\alpha|+|\beta|\leq
k}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\;.$

With the notation of (3.1), consider also the continuous semi-norms
$\|{\cdot}\|^{\prime}_{K,\alpha,\beta,m}$ on $S^{m}(U\times\mathbb{R}^{l})$
given by

(3.5) $\|a\|^{\prime}_{K,\alpha,\beta,m}=\sup_{x\in
K}\limsup_{|\xi|\to\infty}\frac{\big{|}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)\big{|}}{|\xi|^{m-|\beta|}}\;.$

In the case of $S^{m}(\mathbb{R}^{l})$, the subscripts $K$ and $\alpha$ are
omitted from the notation of the semi-norms (3.5).

######  Proposition 3.2.

The semi-norms (3.4) and (3.5) together describe the topology of
$S^{m}(U\times\mathbb{R}^{l})$.

###### Proof.

Let $S^{\prime\,m}(U\times\mathbb{R}^{l})$ denote the LCHS defined by endowing
the vector space $S^{m}(U\times\mathbb{R}^{l})$ with the topology induced by
the semi-norms (3.4) and (3.5) together; in fact, countably many semi-norms of
these types are enough to describe its topology (taking exhausting increasing
sequences of compact sets), and therefore
$S^{\prime\,m}(U\times\mathbb{R}^{l})$ is metrizable. Let
$\widehat{S}^{\prime\,m}(U\times\mathbb{R}^{l})$ denote its completion, where
the stated semi-norms have continuous extensions. There is a continuous
inclusion $S^{\prime\,m}(U\times\mathbb{R}^{l})\subset
C^{\infty}(U\times\mathbb{R}^{l})$, which can be extended to a continuous map
$\phi:\widehat{S}^{\prime\,m}(U\times\mathbb{R}^{l})\to
C^{\infty}(U\times\mathbb{R}^{l})$ because $C^{\infty}(U\times\mathbb{R}^{l})$
is complete. For any $a\in\widehat{S}^{\prime\,m}(U\times\mathbb{R}^{l})$, and
$K$, $\alpha$ and $\beta$ like in (3.5), since
$\|\phi(a)\|^{\prime}_{K,\alpha,\beta,m}=\|a\|^{\prime}_{K,\alpha,\beta,m}<\infty$,
there are $C,R>0$ so that, if $x\in K$ and $|\xi|\geq R$, then

$\frac{|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\phi(a)(x,\xi)|}{(1+|\xi|)^{m-|\beta|}}\leq
C\;.$

Let $B_{R}\subset\mathbb{R}^{l}$ denote the open ball of center $0$ and radius
$R$. For $Q=K\times\overline{B_{R}}\subset U\times\mathbb{R}^{l}$ and
$k=|\alpha|+|\beta|$, since $\|\phi(a)\|_{Q,C^{k}}=\|a\|_{Q,C^{k}}<\infty$,
there is some $C^{\prime}>0$ such that
$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\phi(a)(x,\xi)|<C^{\prime}$ for
$(x,\xi)\in Q$, yielding

$\frac{|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\phi(a)(x,\xi)|}{(1+|\xi|)^{m-|\beta|}}\leq\begin{cases}C^{\prime}&\text{if
$|\beta|\leq m$}\\\ C^{\prime}(1+R)^{|\beta|-m}&\text{if $|\beta|\geq
m$}\;.\end{cases}$

This shows that $\|\phi(a)\|_{K,\alpha,\beta,m}<\infty$, obtaining that
$a\equiv\phi(a)\in S^{m}(U\times\mathbb{R}^{l})$. Hence
$S^{\prime\,m}(U\times\mathbb{R}^{l})$ is complete, and therefore it is a
Fréchet space. Thus the identity map $S^{m}(U\times\mathbb{R}^{l})\to
S^{\prime\,m}(U\times\mathbb{R}^{l})$ is a continuous linear isomorphism
between Fréchet spaces, obtaining that it is indeed a homeomorphism by a
version of the open mapping theorem [21, Section 15.12], [31, Theorem II.2.1],
[28, Theorem 14.4.6]. ∎

######  Proposition 3.3.

For $m,m^{\prime}\in\mathbb{N}_{0}$, $\alpha\in\mathbb{N}_{0}^{n}$,
$\beta\in\mathbb{N}_{0}^{l}$ and any compact $K\subset U$, if $m<m^{\prime}$,
then $\|{\cdot}\|^{\prime}_{K,\alpha,\beta,m^{\prime}}=0$ on
$S^{m}(U\times\mathbb{R}^{l})$.

###### Proof.

According to (3.5), for all $a\in S^{m}(U\times\mathbb{R}^{l})$,

$\|a\|^{\prime}_{K,\alpha,\beta,m^{\prime}}=\|a\|^{\prime}_{K,\alpha,\beta,m}\lim_{|\xi|\to\infty}|\xi|^{m-m^{\prime}}=0\;.\qed$

######  Corollary 3.4.

For $m<m^{\prime}$, the topologies of $S^{m^{\prime}}(U\times\mathbb{R}^{l})$
and $C^{\infty}(U\times\mathbb{R}^{l})$ coincide on
$S^{m}(U\times\mathbb{R}^{l})$. Therefore the topologies of
$S^{\infty}(U\times\mathbb{R}^{l})$ and $C^{\infty}(U\times\mathbb{R}^{l})$
coincide on $S^{m}(U\times\mathbb{R}^{l})$.

###### Proof.

The first assertion is a consequence of Propositions 3.2 and 3.3.

To prove the second assertion, by (3.3), it is enough to show that the
topology of $S^{\infty}(U\times\mathbb{R}^{l})$ is finer or equal than the
topology of $C^{\infty}(U\times\mathbb{R}^{l})$ on
$S^{m}(U\times\mathbb{R}^{l})$. For every open $O\subset
S^{\infty}(U\times\mathbb{R}^{l})$ and $m^{\prime}>m$, since $O\cap
S^{m^{\prime}}(U\times\mathbb{R}^{l})$ is open in
$S^{m^{\prime}}(U\times\mathbb{R}^{l})$, it follows from the first assertion
that there is some open $P\subset C^{\infty}(U\times\mathbb{R}^{l})$ such that
$O\cap S^{m}(U\times\mathbb{R}^{l})=P\cap S^{m}(U\times\mathbb{R}^{l})$. ∎

######  Corollary 3.5.

For $m<m^{\prime}$, $C^{\infty}_{\text{\rm c}}(U\times\mathbb{R}^{l})$ is
dense in $S^{m}(U\times\mathbb{R}^{l})$ with the topology of
$S^{m^{\prime}}(U\times\mathbb{R}^{l})$. Therefore $C^{\infty}_{\text{\rm
c}}(U\times\mathbb{R}^{l})$ is dense in $S^{\infty}(U\times\mathbb{R}^{l})$.

###### Proof.

The first assertion is given by Corollary 3.4 and the density of
$C^{\infty}_{\text{\rm c}}(U\times\mathbb{R}^{l})$ in
$C^{\infty}(U\times\mathbb{R}^{l})$.

To prove the second assertion, take any open $O\neq\emptyset$ in
$S^{\infty}(U\times\mathbb{R}^{l})$. We have $O\cap
S^{m}(U\times\mathbb{R}^{l})\neq\emptyset$ for some $m$. This intersection is
open in $S^{m}(U\times\mathbb{R}^{l})$ with the topology of any
$S^{m^{\prime}}(U\times\mathbb{R}^{l})$ for all $m^{\prime}\geq m$. So $O\cap
C^{\infty}_{\text{\rm c}}(U\times\mathbb{R}^{l})\neq\emptyset$ by the first
assertion. ∎

######  Corollary 3.6.

$S^{\infty}(U\times\mathbb{R}^{l})$ is an acyclic Montel space, and therefore
complete, boundedly retractive and reflexive.

###### Proof.

Corollary 3.4 gives the property of being acyclic, and therefore complete and
boundedly retractive (Section 2.1). Since $S^{\infty}(U\times\mathbb{R}^{l})$
is barreled (Proposition 3.1) and every Montel space is reflexive [21, 6.27.2
(1)], [9, Section 8.4.7], [31, after the examples of IV.5.8], it only remains
to prove that $S^{\infty}(U\times\mathbb{R}^{l})$ is semi-Montel.

Take any closed bounded subset $B\subset S^{\infty}(U\times\mathbb{R}^{l})$;
in particular, $B$ is complete because $S^{\infty}(U\times\mathbb{R}^{l})$ is
complete. Since $S^{\infty}(U\times\mathbb{R}^{l})$ is boundedly retractive,
$B$ is contained and bounded in some $S^{m}(U\times\mathbb{R}^{l})$, and the
topologies of $S^{\infty}(U\times\mathbb{R}^{l})$ and
$S^{m}(U\times\mathbb{R}^{l})$ coincide on $B$. By Corollary 3.4, it follows
that $B$ is a complete bounded subspace of
$C^{\infty}(U\times\mathbb{R}^{l})$, and therefore closed because
$C^{\infty}(U\times\mathbb{R}^{l})$ is complete. So $B$ is compact because
$C^{\infty}(U\times\mathbb{R}^{l})$ is a Montel space. ∎

######  Remark 3.7.

Another proof of Corollary 3.5 could be given like in Proposition 6.8.

######  Remark 3.8.

Despite of Corollary 3.4, the following argument shows that the second
inclusion of (3.3) is not a TVS-embedding. Let $a_{m}\in
S^{\infty}(U\times\mathbb{R}^{l})$ ($m\in\mathbb{N}_{0}$) such that
$a_{m}(x,\xi)=0$ if $|\xi^{1}|\leq m$, and $a_{m}(x,\xi)=(\xi^{1}-m)^{m}$ if
$|\xi^{1}|\geq m+1$. Then $a_{m}\in S^{m}(U\times\mathbb{R}^{l})\setminus
S^{m-1}(U\times\mathbb{R}^{l})$ and $a_{m}\to 0$ in
$C^{\infty}(U\times\mathbb{R}^{l})$ as $m\uparrow\infty$. However
$a_{m}\not\to 0$ in $S^{\infty}(U\times\mathbb{R}^{l})$; otherwise, since
$S^{\infty}(U\times\mathbb{R}^{l})$ is sequentially retractive (Corollary
3.6), all $a_{m}$ would lie in some step $S^{m_{0}}(U\times\mathbb{R}^{l})$, a
contradiction.

With more generality, a symbol of order $m$ on a vector bundle $E$ over $M$ is
a smooth function on $E$ satisfying (3.1) via charts of $M$ and local
trivializations of $E$, with $K$ contained in the domains of charts where $E$
is trivial. As above, they form a Fréchet space $S^{m}(E)$ with the topology
described by the semi-norms given by this version of (3.1). The version of
(3.2) in this setting is true, obtaining the corresponding inductive and
projective limits $S^{\pm\infty}(E)$, and quotient spaces $S^{(m)}(E)$. We can
similarly define the norms (3.4) and (3.5) on $S^{m}(E)$, and Propositions
3.2, 3.3, 3.4, 3.5 and 3.6 can be directly extended to this setting.

Given another vector bundle $F$ over $M$, we can further take the
$C^{\infty}(M)$-tensor product of these spaces with $C^{\infty}(M;F)$,
obtaining the spaces $S^{m}(E;F)$, $S^{\pm\infty}(E;F)$ and $S^{(m)}(E;F)$,
satisfying analogous properties and results. Now (3.3) becomes

$C^{\infty}_{\text{\rm cv}}(E;\pi^{*}F)\subset S^{-\infty}(E;F)\;,\quad
S^{\infty}(E;F)\subset C^{\infty}(E;\pi^{*}F)\;,$

where $\pi:E\to M$ is the vector bundle projection.

##  4\. Conormal distributions

###  4.1. Differential operators tangent to a submanifold

Let $L$ is a regular submanifold of $M$ of codimension $n^{\prime}$ and
dimension $n^{\prime\prime}$, which is a closed subset. Let
$\mathfrak{X}(M,L)\subset\mathfrak{X}(M)$ be the Lie subalgebra and
$C^{\infty}(M)$-submodule of vector fields tangent to $L$. Using
$\mathfrak{X}(M,L)$ instead of $\mathfrak{X}(M)$, we can define the filtered
subalgebra and $C^{\infty}(M)$-submodule
$\operatorname{Diff}(M,L)\subset\operatorname{Diff}(M)$ like in Section 2.7.
We have

(4.1) $A\in\operatorname{Diff}(M,L)\Rightarrow
A^{t}\in\operatorname{Diff}(M,L;\Omega)\;.$

By the conditions on $L$, every $\operatorname{Diff}^{m}(M,L)$
($m\in\mathbb{N}_{0}$) is locally finitely $C^{\infty}(M)$-generated, and
therefore $\operatorname{Diff}(M,L)$ is countably $C^{\infty}(M)$-generated.
The surjective restriction map $\mathfrak{X}(M,L)\to\mathfrak{X}(L)$,
$X\mapsto X|_{L}$, induces a surjective linear restriction map of filtered
algebras and $C^{\infty}(M)$-modules,

(4.2) $\operatorname{Diff}(M,L)\to\operatorname{Diff}(L)\;,\quad A\mapsto
A|_{L}\;.$

Let $(U,x)$ be a chart of $M$ adapted to $L$; i.e., for open subsets
$U^{\prime}\subset\mathbb{R}^{n^{\prime}}$ and
$U^{\prime\prime}\subset\mathbb{R}^{n^{\prime\prime}}$,

$\displaystyle x=(x^{1},\dots,x^{n})\equiv(x^{\prime},x^{\prime\prime}):U\to
U^{\prime}\times U^{\prime\prime}\;,$ $\displaystyle x^{\prime}=(x^{\prime
1},\dots,x^{\prime n^{\prime}})\;,\quad x^{\prime\prime}=(x^{\prime\prime
1},\dots,x^{\prime\prime n^{\prime\prime}})\;,\quad L_{0}:=L\cap
U=\\{x^{\prime}=0\\}\;.$

If $L$ is of codimension one, then we will use the notation $(x,y)$ instead of
$(x^{\prime},x^{\prime\prime})$. For every $m\in\mathbb{N}_{0}$,
$\operatorname{Diff}^{m}(U,L_{0})$ is $C^{\infty}(U)$-spanned by the operators
$x^{\prime\alpha}\partial_{x^{\prime}}^{\beta}\partial_{x^{\prime\prime}}^{\gamma}$
with $|\beta|+|\gamma|\leq m$ and $|\alpha|=|\beta|$; we may use the
generators
$\partial_{x^{\prime}}^{\beta}\partial_{x^{\prime\prime}}^{\gamma}x^{\prime\alpha}$
as well, with the same conditions on the multi-indices.

###  4.2. Conormal distributions filtered by Sobolev order

####  4.2.1. Case of compact manifolds

Suppose first that $M$ is compact. Then the space of _conormal distributions_
at $L$ of _Sobolev order_ at most $s\in\mathbb{R}$ is the LCS and
$C^{\infty}(M)$-module

(4.3) $I^{(s)}(M,L)=\\{\,u\in
C^{-\infty}(M)\mid\operatorname{Diff}(M,L)\,u\subset H^{s}(M)\,\\}\;,$

with the projective topology given by the maps $P:I^{(s)}(M,L)\to H^{s}(M)$
($P\in\operatorname{Diff}(M,L)$).

######  Proposition 4.1.

$I^{(s)}(M,L)$ is a totally reflexive Fréchet space.

###### Proof.

For any countable $C^{\infty}(M)$-spanning set $\\{P_{j}\mid
j\in\mathbb{N}_{0}\\}$ of $\operatorname{Diff}(M,L)$, the space $I^{(s)}(M,L)$
has the projective topology given by the maps $P_{j}:I^{(s)}(M,L)\to
H^{s}(M)$. Let

$I_{k}^{(s)}(M,L)=\\{\,u\in C^{-\infty}(M)\mid P_{j}u\subset H^{s}(M),\
j=0,\dots,k\,\\}\;,$

with the projective topology given by the maps $P_{j}:I^{(s)}(M,L)\to
H^{s}(M)$ ($j=0,\dots,k$). We can assume $P_{0}=1$, and therefore
$I_{0}^{(s)}(M,L)=H^{s}(M)$. Every $I_{k}^{(s)}(M,L)$ is a Hilbert space with
the scalar product

$\langle u,v\rangle_{s,k}=\sum_{j=0}^{k}\langle P_{j}u,P_{j}v\rangle_{s}\;,$

there are continuous inclusions $I_{k^{\prime}}^{(s)}(M,L)\subset
I_{k}^{(s)}(M,L)$ ($k<k^{\prime}$), and
$I^{(s)}(M,L)=\bigcap_{k}I_{k}^{(s)}(M,L)$. So $I^{(s)}(M,L)$ is a totally
reflexive Fréchet space [38, Theorem 4]. ∎

We have continuous inclusions

(4.4) $I^{(s)}(M,L)\subset I^{(s^{\prime})}(M,L)\quad(s^{\prime}<s)\;,$

and consider the LCSs and $C^{\infty}(M)$-modules

$I(M,L)=\bigcup_{s}I^{(s)}(M,L)\;,\quad
I^{(\infty)}(M,L)=\bigcap_{s}I^{(s)}(M,L)\;.$

$I(M,L)$ is an LF-space, and $I^{(\infty)}(M,L)$ is a Fréchet space and
submodule of $I(M,L)$. The elements of $I(M,L)$ are called _conormal
distributions_ of $M$ at $L$ (or of $(M,L)$). The spaces $I^{(s)}(M,L)$ form
the _Sobolev order filtration_ of $I(M,L)$. From (4.3), it follows that there
are canonical continuous inclusions,

(4.5) $C^{\infty}(M)\subset I^{(\infty)}(M,L)\;,\quad I(M,L)\subset
C^{-\infty}(M)\;;$

in particular, $I(M,L)$ is Hausdorff.

Since every $I^{(s)}(M,L)$ is a Fréchet space (Proposition 4.1), the following
analog of Proposition 3.1 holds true by the same reason.

######  Corollary 4.2.

$I(M,L)$ is barreled, ultrabornological and webbed.

####  4.2.2. Extension to non-compact manifolds

If $M$ is not assumed to be compact, we can similarly define the LCHS
$I^{(s)}_{{\cdot}/\text{\rm c}}(M,L)$ by using $C^{-\infty}_{{\cdot}/\text{\rm
c}}(M)$ and $H^{s}_{\text{\rm loc/c}}(M)$. Every $I^{(s)}(M,L)$ is a Fréchet
space, as follows like in the proof of Proposition 4.1, using the Fréchet
spaces $H^{s}_{\text{\rm loc}}(M)$. We can describe $I^{(s)}_{\text{\rm
c}}(M,L)=\bigcup_{K}I^{(s)}_{K}(M,L)$ like in (2.2), which is a strict LF-
space because every $I^{(s)}_{K}(M,L)$ satisfies an analog of Proposition 4.1.
Therefore $I_{\text{\rm c}}(M,L)=\bigcup_{s}I_{\text{\rm c}}^{(s)}(M,L)$ is an
LF-space [28, Exercise 12.108]; moreover $I_{\text{\rm
c}}(M,L)=\bigcup_{K}I_{K}(M,L)$. We also have the Fréchet space
$I^{(\infty)}_{\text{\rm c}}(M,L)=\bigcap_{s}I_{\text{\rm c}}^{(s)}(M,L)$. All
of these spaces are modules over $C^{\infty}(M)$; $I_{\text{\rm c}}(M,L)$ is a
filtered module and $I^{(\infty)}_{\text{\rm c}}(M,L)$ a submodule. The
extension by zero defines a continuous inclusion $I_{\text{\rm c}}(U,L\cap
U)\subset I_{\text{\rm c}}(M,L)$ for any open $U\subset M$. We also define the
space $I^{(\infty)}(M,L)$ like in the compact case, as well as the space
$\bigcup_{s}I^{(s)}(M,L)$, which consists of the conormal distributions with a
Sobolev order. But now let (cf. [15, Definition 18.2.6])

(4.6) $I(M,L)=\\{\,u\in C^{-\infty}(M)\mid C^{\infty}_{\text{\rm
c}}(M)\,u\subset I_{\text{\rm c}}(M,L)\,\\}\;,$

which is a LCS with the projective topology given by the (multiplication) maps
$f_{j}:I(M,L)\to I_{\text{\rm c}}(M,L)$, for a countable partition of unity
$\\{f_{j}\\}\subset C^{\infty}_{\text{\rm c}}(M)$. We have
$I(M,L)=\bigcup_{s}I^{(s)}(M,L)$ if and only if $L$ is compact; thus the
spaces $I^{(s)}(M,L)$ form a filtration of $I(M,L)$ just when $L$ is compact.
There is an extension of (4.5) for non-compact $M$, taking arbitrary/compact
support; in particular, $I_{{\cdot}/\text{\rm c}}(M,L)$ is Hausdorff.

###  4.3. Filtration of $I(M,L)$ by the symbol order

####  4.3.1. Local description of conormal distributions with symbols

Consider the notation of Section 4.1 for a chart
$(U,x=(x^{\prime},x^{\prime\prime}))$ of $M$ adapted to $L$. We use the
identity $U^{\prime\prime}\times\mathbb{R}^{n^{\prime}}\equiv
N^{*}U^{\prime\prime}$, and the symbol spaces
$S^{m}(U^{\prime\prime}\times\mathbb{R}^{n^{\prime}})\equiv
S^{m}(N^{*}U^{\prime\prime})$ (Section 3). Define

(4.7) $\displaystyle C^{\infty}_{\text{\rm cv}}(N^{*}U^{\prime\prime})\to
C^{\infty}(U)\;,\quad a\mapsto u\;,$ (4.8) $\displaystyle
C^{\infty}_{\text{\rm c}}(U)\to C^{\infty}(N^{*}U^{\prime\prime})\;,\quad
u\mapsto a\;,$

by the following partial inverse Fourier transform and partial Fourier
transform:

$\displaystyle u(x)$
$\displaystyle=(2\pi)^{-n^{\prime}}\int_{\mathbb{R}^{n^{\prime}}}e^{i\langle
x^{\prime},\xi\rangle}a(x^{\prime\prime},\xi)\,d\xi\;,$ $\displaystyle
a(x^{\prime\prime},\xi)$
$\displaystyle=\int_{\mathbb{R}^{n^{\prime}}}e^{-i\langle
x^{\prime},\xi\rangle}u(x^{\prime},x^{\prime\prime})\,dx^{\prime}\;.$

######  Proposition 4.3 ([15, Theorem 18.2.8], [25, Proposition 6.1.1], [27,
Lemma 9.33]).

If $s<-\bar{m}-n^{\prime}/2$, then (4.7) has a continuous extension
$S^{\bar{m}}(N^{*}U^{\prime\prime})\to I^{(s)}(U,L_{0})$. If
$\bar{m}>-s-n^{\prime}/2$, then (4.8) induces a continuous linear map
$I_{\text{\rm c}}^{(s)}(U,L_{0})\to S^{\bar{m}}(N^{*}U^{\prime\prime})$.

######  Remark 4.4.

The continuity of the maps of Proposition 4.3 is not stated in [15, Theorem
18.2.8], [25, Proposition 6.1.1], [27, Lemma 9.33], but it follows easily from
their proofs.

When applying Proposition 4.3 to $M$ via $(U,x)$, it will be convenient to use

$a\,|d\xi|\in S^{\bar{m}}(N^{*}U^{\prime\prime};\Omega
N^{*}U^{\prime\prime})\equiv S^{\bar{m}}(N^{*}L_{0};\Omega N^{*}L_{0})\;.$

####  4.3.2. Case of compact manifolds

Assume first that $M$ is compact. Take a finite cover of $L$ by relatively
compact charts $(U_{j},x_{j})$ of $M$ adapted to $L$, and write $L_{j}=L\cap
U_{j}$. Let $\\{h,f_{j}\\}$ be a $C^{\infty}$ partition of unity of $M$
subordinated to the open covering $\\{M\setminus L,U_{j}\\}$. Then $I(M,L)$
consists of the distributions $u\in C^{-\infty}(M)$ such that $hu\in
C^{\infty}(M\setminus L)$ and $f_{j}u\in I_{\text{\rm c}}(U_{j},L_{j})$ for
all $j$. Then, according to Proposition 4.3, every $f_{j}u$ is given by some
$a_{j}\in S^{\infty}(N^{*}L_{j};\Omega N^{*}L_{j})$. For

(4.9) $\bar{m}=m+n/4-n^{\prime}/2\;,$

the condition $a_{j}\in S^{\bar{m}}(N^{*}L_{j};\Omega N^{*}L_{j})$ describes
the elements $u$ of a $C^{\infty}(M)$-submodule $I^{m}(M,L)\subset I(M,L)$,
which is independent of the choices involved [27, Proposition 9.33] (see also
[25, Definition 6.2.19] and [36, Definition 4.3.9]). Moreover, applying the
versions of semi-norms (2.1) on $C^{\infty}(M\setminus L)$ to $hu$ and
versions of semi-norms (3.1) on $S^{\bar{m}}(N^{*}L_{j};\Omega N^{*}L_{j})$ to
every $a_{j}$, we get semi-norms on $I^{m}(M,L)$, which becomes a Fréchet
space [25, Sections 6.2 and 6.10]. In other words, the following map is
required to be a TVS-embedding:

(4.10) $I^{m}(M,L)\to C^{\infty}(M\setminus
L)\oplus\prod_{j}S^{\bar{m}}(N^{*}L_{j};\Omega N^{*}L_{j})\;,\quad
u\mapsto(hu,(a_{j}))\;.$

The version of (3.2) for the spaces $S^{\bar{m}}(N^{*}L_{j};\Omega
N^{*}L_{j})$ gives continuous inclusions

(4.11) $I^{m}(M,L)\subset I^{m^{\prime}}(M,L)\quad(m<m^{\prime})\;.$

The element $\sigma_{m}(u)\in S^{(\bar{m})}(N^{*}L;\Omega N^{*}L)$ represented
by $\sum_{j}a_{j}\in S^{\bar{m}}(N^{*}L;\Omega N^{*}L)$ is called the
_principal symbol_ of $u$. This defines the exact sequence

$0\to I^{m-1}(M,L)\hookrightarrow
I^{m}(M,L)\xrightarrow{\sigma_{m}}S^{(\bar{m})}(N^{*}L;\Omega N^{*}L)\to 0\;.$

From Proposition 4.3 and (4.9), we also get continuous inclusions

(4.12) $I^{(-m-n/4+\epsilon)}(M,L)\subset I^{m}(M,L)\subset
I^{(-m-n/4-\epsilon)}(M,L)\;,$

for all $m\in\mathbb{R}$ and $\epsilon>0$ (cf. [25, Eq. (6.2.5)], [27, Eq.
(9.35)]). So

$I(M,L)=\bigcup_{m}I^{m}(M,L)\;,\quad
I^{(\infty)}(M,L)=I^{-\infty}(M,L):=\bigcap_{m}I^{m}(M,L)\;.$

The spaces $I^{m}(M,L)$ form the _symbol order filtration_ of $I(M,L)$. The
maps (4.10) induce a TVS-embedding

(4.13) $I(M,L)\to C^{\infty}(M\setminus
L)\oplus\prod_{j}S^{\infty}(N^{*}L_{j};\Omega N^{*}L_{j})\;.$

######  Corollary 4.5.

For $m<m^{\prime},m^{\prime\prime}$, the topologies of $I^{m^{\prime}}(M,L)$
and $I^{m^{\prime\prime}}(M,L)$ coincide on $I^{m}(M,L)$.

###### Proof.

Use Corollary 3.4 and the TVS-embeddings (4.10). ∎

######  Corollary 4.6 ([25, Eq. (6.2.12)]).

For $m<m^{\prime}$, $C^{\infty}(M)$ is dense in $I^{m}(M,L)$ with the topology
of $I^{m^{\prime}}(M,L)$. Therefore $C^{\infty}(M)$ is dense in $I(M,L)$.

###### Proof.

$C^{\infty}(M)$ is contained in the stated spaces by (4.5).

Let us prove the first density, and the second one follows like in Corollary
3.5. Given $u\in I^{m}(M,L)$, let $a_{j}\in S^{\bar{m}}(N^{*}L_{j};\Omega
N^{*}L_{j})$ be the symbol corresponding to $f_{j}u$ by Proposition 4.3, like
in (4.10). By Corollary 3.5, there is a sequence $b_{j,k}\in
C^{\infty}_{\text{\rm c}}(N^{*}L_{j};\Omega N^{*}L_{j})$ converging to $a_{j}$
in $S^{\bar{m}^{\prime}}(N^{*}L_{j};\Omega N^{*}L_{j})$
($\bar{m}^{\prime}=m^{\prime}+n/4-n^{\prime}/2$). Let $v_{j,k}$ be the
sequence in $C^{\infty}(U)$ that corresponds to $b_{j,k}$ via (4.7); it
converges to $f_{j}u$ in $I^{m^{\prime}}(U,L)$ as $k\to\infty$ by Proposition
4.3. Take functions $\tilde{f}_{j}\in C^{\infty}_{\text{\rm c}}(U_{j})$ with
$\tilde{f}_{j}=1$ on $\operatorname{supp}f_{j}$. Then $\tilde{f}_{j}v_{j,k}\to
f_{j}u$ in $I^{m^{\prime}}_{\text{\rm c}}(U,L)$, and therefore
$hu+\sum_{j}\tilde{f}_{j}v_{j,k}\in C^{\infty}(M)$ is convergent to $u$ in
$I^{m^{\prime}}(M,L)$. ∎

######  Corollary 4.7.

$I(M,L)$ is an acyclic Montel space, and therefore complete, boundedly
retractive and reflexive.

###### Proof.

Like in Corollary 3.6, by Corollaries 4.2 and 4.5, it is enough to prove that
$I(M,L)$ is semi-Montel. The TVS-embedding (4.13) is closed because $I(M,L)$
is complete. Then $I(M,L)$ is semi-Montel because $C^{\infty}(M\setminus L)$
and $S^{\infty}(N^{*}L_{j};\Omega N^{*}L_{j})$ are Montel spaces (Corollary
3.6), and this property is inherited by closed subspaces and products [16,
Propositions 3.9.3 and 3.9.4], [28, Exercise 12.203 (c)]. ∎

######  Remark 4.8.

The reflexivity of $I(M,L)$ is also a consequence of the reflexivity of
$I^{(s)}(M,L)$ (Proposition 4.1) and the regularity of $I(M,L)$ (Corollary
4.7) [23].

####  4.3.3. Extension to non-compact manifolds

When $M$ is not assumed to be compact, the definition of $I^{m}(M,L)$ can be
immediately extended assuming $\\{U_{j}\\}$ is a locally finite cover of $L$,
obtaining an analog of (4.10). We can similarly define $I^{m}_{K}(M,L)$ for
all compact $K\subset M$, and take $I^{m}_{\text{\rm
c}}(M,L)=\bigcup_{K}I^{m}_{K}(M,L)$ like in (2.2). The space of conormal
distributions with a symbol order is $\bigcup_{m}I^{m}(M,L)$, and let
$I^{-\infty}_{{\cdot}/\text{\rm c}}(M,L)=\bigcap_{m}I^{m}_{{\cdot}/\text{\rm
c}}(M,L)$. There are extensions of (4.10)–(4.13) and Corollaries 4.5 and 4.6,
with arbitrary/compact support (using direct sums instead of products in the
case of compact support). So $\bigcup_{m}I^{m}(M,L)=\bigcup_{s}I^{(s)}(M,L)$,
$I_{\text{\rm c}}(M,L)=\bigcup_{m}I^{m}_{\text{\rm c}}(M,L)$ and
$I^{(\infty)}_{{\cdot}/\text{\rm c}}(M,L)=I^{-\infty}_{{\cdot}/\text{\rm
c}}(M,L)$. Corollary 4.7 can be extended with $\bigcup_{m}I^{m}(M,L)$ and
$I_{{\cdot}/\text{\rm c}}(M,L)$, except acyclicity in the case of $I(M,L)$.

###  4.4. Dirac sections at submanifolds

Let $NL$ and $N^{*}L$ denote the normal and conormal bundles of $L$. We have
identity $\Omega NL\otimes\Omega L\equiv\Omega_{L}M$. The transpose of the
restriction map $C^{\infty}_{\text{\rm c}/{\cdot}}(M;E^{*}\otimes\Omega M)\to
C^{\infty}_{\text{\rm c}/{\cdot}}(L;E^{*}\otimes\Omega_{L}M)$ is a continuous
inclusion

(4.14) $\displaystyle C^{-\infty}_{{\cdot}/\text{\rm
c}}(L;E\otimes\Omega^{-1}NL)\subset C^{-\infty}_{{\cdot}/\text{\rm
c}}(M;E)\;,$ $\displaystyle
u\mapsto\delta_{L}^{u}\;,\quad\langle\delta_{L}^{u},v\rangle=\langle
u,v|_{L}\rangle\;,\quad v\in C^{\infty}_{\text{\rm
c}/{\cdot}}(M;E^{*}\otimes\Omega)\;.$

By restriction of (4.14), we get a continuous inclusion [11, p. 310],

(4.15) $C^{\infty}_{{\cdot}/\text{\rm c}}(L;E\otimes\Omega^{-1}NL)\subset
C^{-\infty}_{{\cdot}/\text{\rm c}}(M;E)\;;$

in this case, we can write $\langle\delta_{L}^{u},v\rangle=\int_{L}u\,v|_{L}$.
This is the subspace of _$\delta$ -sections_ or _Dirac sections_ at $L$.
Actually, the following sharpening of (4.15) is true.

######  Proposition 4.9.

The inclusion (4.15) induces a continuous injection

$C^{\infty}_{{\cdot}/\text{\rm c}}(L;E\otimes\Omega^{-1}NL)\subset
H^{s}_{\text{\rm loc/c}}(M;E)\quad(s<-n^{\prime}/2)$

with

$C^{\infty}_{{\cdot}/\text{\rm c}}(L;E\otimes\Omega^{-1}NL)\cap
H^{-n^{\prime}/2}_{\text{\rm loc/c}}(M;E)=0\;.$

###### Proof.

First, take $M=\mathbb{R}^{n}$,
$L=\mathbb{R}^{n^{\prime\prime}}\times\\{0\\}\equiv\mathbb{R}^{n^{\prime\prime}}$
and $E=M\times\mathbb{C}$ (the trivial line bundle). Let $\delta_{0}$ be the
Dirac mass at $0$ in $\mathbb{R}^{n^{\prime}}$. For any
$\phi\in\SS(\mathbb{R}^{n^{\prime\prime}})$, consider the tensor product
distribution $\phi\otimes\delta_{0}\in\SS(\mathbb{R}^{n})^{\prime}$ [14,
Section 5.1]. Its Fourier transform is
$\hat{\phi}\otimes\hat{\delta}_{0}=\hat{\phi}\otimes 1$. If $\phi\neq 0$, then
$\hat{\phi}\otimes 1\in L^{2}(\mathbb{R}^{n},(1+|\xi|^{2})^{s}\,d\xi)$ if and
only if $1\in L^{2}(\mathbb{R}^{n^{\prime}},(1+|\xi|^{2})^{s}\,d\xi)$, which
holds just when $s<-n^{\prime}/2$. Moreover the map

$\SS(\mathbb{R}^{n^{\prime\prime}})\to
L^{2}(\mathbb{R}^{n},(1+|\xi|^{2})^{s}\,d\xi)\;,\quad\phi\mapsto\hat{\phi}\otimes
1\;,$

is continuous if $s<-n^{\prime}/2$. Thus (2.14) yields versions of the stated
properties using $\SS(\mathbb{R}^{n^{\prime\prime}})$ and $H^{s}(R^{n})$.

For arbitrary $M$, $L$ and $E$, the result follows from the previous case by
using a locally finite atlas, a subordinated partition of unity, and
diffeomorphisms of triviality of $E$. ∎

For instance, for any $p\in M$ and $u\in E_{p}\otimes\Omega_{p}^{-1}M$, we get
$\delta_{p}^{u}\in H^{s}_{\text{\rm c}}(M;E)$ if $s<-n/2$, with
$\langle\delta_{p}^{u},v\rangle=u\cdot v(p)$ for $v\in
C^{\infty}(M;E^{*}\otimes\Omega)$, obtaining a continuous map

$M\times C^{\infty}(M;E\otimes\Omega^{-1})\to H^{s}_{\text{\rm
c}}(M;E)\;,\quad(p,u)\mapsto\delta_{p}^{u(p)}\;.$

As a particular case, the Dirac mass at any $p\in\mathbb{R}^{n}$ is
$\delta_{p}=\delta_{p}^{1\otimes|dx|^{-1}}\in H^{s}_{\text{\rm
c}}(\mathbb{R}^{n})$.

###  4.5. Differential operators on conormal distributional sections

Any $A\in\operatorname{Diff}^{k}(M;E)$ induces continuous linear maps [25,
Lemma 6.1.1]

(4.16) $A:I^{(s)}_{{\cdot}/\text{\rm c}}(M,L;E)\to
I^{(s-k)}_{{\cdot}/\text{\rm c}}(M,L;E)\;,$

which induce a continuous endomorphism $A$ of $I_{{\cdot}/\text{\rm
c}}(M,L;E)$. If $A\in\operatorname{Diff}(M,L;E)$, then it clearly induces a
continuous endomorphism $A$ of every $I^{(s)}_{{\cdot}/\text{\rm c}}(M,L;E)$.

According to (4.14), for $A\in\operatorname{Diff}(M,L;E)$ and $u\in
C^{\infty}_{{\cdot}/\text{\rm c}}(L;E\otimes\Omega^{-1}NL)$,

(4.17) $A\delta_{L}^{u}=\delta_{L}^{A^{\prime}u}\;,\quad
A^{\prime}=((A^{t})|_{L})^{t}\in\operatorname{Diff}(L;E\otimes\Omega^{-1}NL)\;,$

where $A^{t}\in\operatorname{Diff}(M,L;E^{*}\otimes\Omega)$ and
$(A^{t})|_{L}\in\operatorname{Diff}(L,E^{*}\otimes\Omega_{L}M)$ using the
vector bundle versions of (4.1) and (4.2). In fact, for $v\in
C^{\infty}_{\text{\rm c}/{\cdot}}(M;E^{*}\otimes\Omega)$,

$\langle A\delta_{L}^{u},v\rangle=\langle\delta_{L}^{u},A^{t}v\rangle=\langle
u,(A^{t}v)|_{L}\rangle=\langle u,(A^{t})|_{L}(v|_{L})\rangle=\langle
A^{\prime}u,v|_{L}\rangle=\langle\delta_{L}^{A^{\prime}u},v\rangle\;.$

By (4.17), $\operatorname{Diff}(M,L;E)$ preserves the subspace of Dirac
sections given by (4.15). Thus the continuous inclusion of Proposition 4.9
induces a continuous inclusion

(4.18) $C^{\infty}_{{\cdot}/\text{\rm c}}(L;E\otimes\Omega^{-1}NL)\subset
I^{(s)}_{{\cdot}/\text{\rm c}}(M,L;E)\quad(s<-n^{\prime}/2)\;.$

###  4.6. Pull-back of conormal distributions

If a smooth map $\phi:M^{\prime}\to M$ is transverse to a regular submanifold
$L\subset M$, then $L^{\prime}:=\phi^{-1}(L)\subset M^{\prime}$ is a regular
submanifold and (the trivial-line-bundle version of) (2.7) has continuous
extensions

(4.19) $\phi^{*}:I^{m}(M,L)\to
I^{m+k/4}(M^{\prime},L^{\prime})\quad(m\in\mathbb{R})\;,$

where $k=\dim M-\dim M^{\prime}$ [36, Theorem 5.3.8], [25, Proposition 6.6.1].
Taking inductive limits and using (4.12), we get a continuous linear map

(4.20) $\phi^{*}:I(M,L)\to I(M^{\prime},L^{\prime})\;.$

If $\phi$ is a submersion, this is a restriction of (2.10).

If $\phi$ is a local diffeomorphism, then (2.7) also has continuous extensions

(4.21) $\phi^{*}:I^{(s)}(M,L)\to
I^{(s)}(M^{\prime},L^{\prime})\quad(s\in\mathbb{R})\;.$

A more general pull-back of distributional sections can be defined under
conditions on the wave front set [14, Theorem 8.2.4], but we will not use it.

###  4.7. Push-forward of conormal distributional sections

Now let $\phi:M^{\prime}\to M$ be a smooth submersion, and let $L\subset M$
and $L^{\prime}\subset M^{\prime}$ be regular submanifolds such that
$\phi(L^{\prime})\subset L$ and the restriction $\phi:L^{\prime}\to L$ is also
a smooth submersion. Then (2.8) has continuous extensions

(4.22) $\phi_{*}:I^{m}_{\text{\rm c}}(M^{\prime},L^{\prime};\Omega_{\text{\rm
fiber}})\to I^{m+l/2-k/4}_{\text{\rm c}}(M,L)\quad(m\in\mathbb{R})\;,$

where $k=\dim M^{\prime}-\dim M$ and $l=\dim L^{\prime}-\dim L$ [36, Theorem
5.3.6], [25, Proposition 6.7.2]. Taking inductive limits, we get a continuous
linear map

(4.23) $\phi_{*}:I_{\text{\rm c}}(M^{\prime},L^{\prime};\Omega_{\text{\rm
fiber}})\to I_{\text{\rm c}}(M,L)\;,$

which is a restriction of (2.9).

If $\phi$ is a local diffeomorphism, then (2.8) also has continuous extensions

(4.24) $\phi_{*}:I^{(s)}_{\text{\rm c}}(M^{\prime},L^{\prime})\to
I^{(s)}_{\text{\rm c}}(M,L)\quad(s\in\mathbb{R})\;.$

###  4.8. Pseudodifferential operators

This type of operators is the main application of conormal distributions (see
e.g. [37, 15, 26, 36]).

####  4.8.1. Case of compact manifolds

Suppose first that $M$ is compact. The filtered algebra and
$C^{\infty}(M^{2})$-module of pseudodifferential operators, $\Psi(M)$,
consists of the continuous endomorphisms $A$ of $C^{\infty}(M)$ with Schwartz
kernel $K_{A}\in I(M^{2},\Delta)$, where $\Delta$ is the diagonal. In fact, by
the Schwartz kernel theorem, we may consider $\Psi(M)\equiv I(M^{2},\Delta)$.
It is filtered by the symbol order, $\Psi^{m}(M)\equiv I^{m}(M^{2},\Delta)$
($m\in\mathbb{R}$), and $\Psi^{-\infty}(M)\equiv I^{-\infty}(M^{2},\Delta)$
consists of the smoothing operators. The analogs of (2.20) and (2.21) hold
true using $\Psi^{s}(M)$ instead of $\operatorname{Diff}^{s}(M)$ for any
$s\in\mathbb{R}$. In this way, $\Psi(M)$ also becomes a LCHS satisfying the
properties indicated in Sections 4.2.1 and 4.3.2.

Taking the $C^{\infty}(M^{2})$-tensor product of $\Psi(M)$ with
$C^{\infty}(M;F\boxtimes E^{*})$, we get $\Psi(M;E,F)$ (or $\Psi(M;E)$ if
$E=F$) as in Section 2.7, satisfying the analog of (2.12). $\Psi(M;E)$ is
preserved by taking transposes, and therefore any $A\in\Psi(M;E)$ defines a
continuous endomorphism $A$ of $C^{-\infty}(M;E)$ (Section 2.5), and
$\operatorname{sing}\operatorname{supp}Au\subset\operatorname{sing}\operatorname{supp}u$
for all $u\in C^{-\infty}(M;E)$ (_pseudolocality_). The concepts of symbols
and ellipticity can be extended to pseudodifferential operators via the
Schwartz kernels (see e.g. [24]), and the symbol map is multiplicative.

If $A\in\Psi^{m}(M;E)$, it defines a bounded operator $A:H^{s+m}(M;E)\to
H^{s}(M;E)$. This can be considered as a closable densely defined operator in
$H^{s}(M;E)$, like in the case of differential operators (Section 2.10). In
the case $s=0$, the adjoint of $A$ is induced by the formal adjoint
$A^{*}\in\Psi^{m}(M;E)$.

The class of pseudodifferential operators is preserved by transposition. So
any $A\in\Psi^{m}(M)$ defines a continuous endomorphism $A$ of
$C^{-\infty}(M)$ (Section 2.5), and
$\operatorname{sing}\operatorname{supp}Au\subset\operatorname{sing}\operatorname{supp}u$
for all $u\in C^{-\infty}_{\text{\rm c}}(M)$ (_pseudolocality_). The concepts
of symbols and ellipticity can be extended to pseudodifferential operators via
the Schwartz kernels (see e.g. [24]), and the symbol map is multiplicative.

####  4.8.2. Extension to non-compact manifolds

If $M$ is not assumed to be compact, $\Psi(M)$ is similarly defined with the
change that any $A\in\Psi^{m}(M)$ defines continuous linear maps
$A:C^{\pm\infty}_{\text{\rm c}}(M)\to C^{\pm\infty}(M)$ and
$A:H^{s+m}_{\text{\rm c}}(M)\to H^{s}_{\text{\rm loc}}(M)$. Thus $\Psi(M)$ is
not an algebra in this case. However, if $A\in\Psi^{m}(M)$ is properly
supported (both factor projections $M^{2}\to M$ restrict to proper maps
$\operatorname{supp}K_{A}\to M$), then it defines a continuous endomorphism
$A$ of $C^{-\infty}_{\text{\rm c}}(M;E)$; in this sense, properly supported
pseudodifferential operators can be composed. Pseudodifferential operators are
properly supported modulo $\Psi^{-\infty}(M)$. Like in the compact case,
$\Psi(M)\equiv I(M^{2},\Delta)$ becomes a filtered $C^{\infty}(M^{2})$-module
and LCHS satisfying the properties indicated in Sections 4.2.2 and 4.3.3.

In the setting of bounded geometry (Section 2.12.3), properly supported
pseudodifferential operators with uniformly bounded symbols, and their uniform
ellipticity, were studied in [19, 20].

##  5\. Dual-conormal distributions

###  5.1. Dual-conormal distributions

Consider the notation of Sections 4.2 and 4.3.

####  5.1.1. Case of compact manifolds

Assume first that $M$ is compact. The space of _dual-conormal distributions_
of $M$ at $L$ (or of $(M,L)$) is the strong dual space [25, Chapter 6]

(5.1) $I^{\prime}(M,L)=I(M,L;\Omega)^{\prime}\;.$

######  Corollary 5.1.

$I^{\prime}(M,L)$ is a complete Montel space.

###### Proof.

Since $I(M,L;\Omega)$ is bornological (the version Corollary 4.2 with $\Omega
M$), $I^{\prime}(M,L)$ is complete [31, IV.6.1], [29, Corollary 6.1.18], [28,
Theorem 13.2.13].

Since $I(M,L;\Omega)$ is a Montel space (the version Corollary 4.7 with
$\Omega M$), $I^{\prime}(M,L)$ is a Montel space [16, Proposition 3.9.9], [21,
6.27.2 (2)], [31, IV.5.9]. ∎

Let also

(5.2) $I^{\prime\,(s)}(M,L)=I^{(-s)}(M,L;\Omega)^{\prime}\;,\quad
I^{\prime\,m}(M,L)=I^{-m}(M,L;\Omega)^{\prime}\;.$

######  Corollary 5.2.

$I^{\prime\,(s)}(M,L)$ is bornological and barreled.

###### Proof.

Since $I^{(-s)}(M,L;\Omega)$ is a reflexive Fréchet space (the version of
Proposition 4.1 with $\Omega M$), $I^{\prime\,(s)}(M,L)$ is bornological [31,
Corollary 1 of IV.6.6], and therefore barreled [31, IV.6.6]. ∎

Transposing the versions of (4.4) and (4.11) with $\Omega M$, we get
continuous restrictions, for $s^{\prime}<s$ and $m<m^{\prime}$,

$I^{\prime\,(s)}(M,L)\to I^{\prime\,(s^{\prime})}(M,L)\;,\quad
I^{\prime\,m}(M,L)\to I^{\prime\,m^{\prime}}(M,L)\;.$

These maps form projective systems, giving rise to $\varprojlim
I^{\prime\,(s)}(M,L)$ as $s\uparrow\infty$ and $\varprojlim
I^{\prime\,m}(M,L)$ as $m\downarrow-\infty$. Transposing the versions of (4.5)
and (4.12) with $\Omega M$, we get continuous inclusions

(5.3) $C^{-\infty}(M)\supset I^{\prime}(M,L)\supset C^{\infty}(M)\;,$

and, for all $m\in\mathbb{R}$ and $\epsilon>0$, continuous restrictions

(5.4) $I^{\prime\,(-m+n/4-\epsilon)}(M,L)\leftarrow
I^{\prime\,m}(M,L)\leftarrow I^{\prime\,(-m+n/4+\epsilon)}(M,L)\;.$

Thus

(5.5) $\varprojlim I^{\prime\,(s)}(M,L)\equiv\varprojlim
I^{\prime\,m}(M,L)\;.$

######  Corollary 5.3.

$I^{\prime}(M,L)\equiv\varprojlim I^{\prime\,(s)}(M,L)$.

###### Proof.

This holds because $I(M,L)$ is regular (Corollary 4.7) [23, Lemma 1].

Alternatively, the following argument can be used. $I^{\prime}(M,L)$ is a
Montel space (Corollary 5.1); in particular, it is barreled, and therefore a
Mackey space [31, IV.3.4]. On the other hand, every $I^{\prime\,(s)}(M,L)$ is
bornological (Corollary 5.2), and therefore a Mackey space [16, Proposition
3.7.2], [31, IV.3.4], [28, Theorem 13.2.10]. So the result follows applying
[31, Remark of IV.4.5]. ∎

####  5.1.2. Extension to non-compact manifolds

If $M$ is not supposed to be compact, we can similarly define the space
$I^{\prime}_{K}(M,L)$ of dual-conormal distributions supported in any compact
$K\subset M$. Then define the LCHSs, $I^{\prime}_{\text{\rm
c}}(M,L)=\bigcup_{K}I^{\prime}_{K}(M,L)$ like in (2.2), and $I^{\prime}(M,L)$
like in (4.6) using $I^{\prime}_{\text{\rm c}}(M,L)$ instead of $I_{\text{\rm
c}}(M,L)$. These spaces satisfy a version of (5.1), interchanging
arbitrary/compact support like in (2.4). Given a smooth partition of unity
$\\{f_{j}\\}$ so that every $K_{j}:=\operatorname{supp}f_{j}$ is compact, the
multiplication by the functions $f_{j}$ defines closed TVS-embeddings

(5.6) $I^{\prime}(M,L)\to\prod_{j}I^{\prime}_{K_{j}}(M,L)\;,\quad
I^{\prime}_{\text{\rm c}}(M,L)\to\bigoplus_{j}I^{\prime}_{K_{j}}(M,L)\;.$

Then, by the extension of Corollary 4.7 for $I_{\text{\rm c}}(M,L;\Omega)$,
the obvious extension of Corollary 5.2 for every $I^{\prime}_{K_{j}}(M,L)$,
(5.1) and (5.6), we get an extension of Corollary 5.1. Similarly, we can
define the spaces $I^{\prime\,(s)}_{{\cdot}/\text{\rm c}}(M,L)$ and
$I^{\prime\,m}_{{\cdot}/\text{\rm c}}(M,L)$, which satisfy (5.2) interchanging
the support condition. Since $I_{\text{\rm c}}(M,L)$ is an acyclic Montel
space (Section 4.3.3), there are extensions of Corollaries 5.1 and 5.3 for
$I^{\prime}(M,L)$.

###  5.2. Differential operators on dual-conormal distributional sections

For any $A\in\operatorname{Diff}(M;E)$, consider
$A^{t}\in\operatorname{Diff}(M;E^{*}\otimes\Omega)$. The transpose of $A^{t}$
on $I_{\text{\rm c}/{\cdot}}(M,L;E^{*}\otimes\Omega)$ is a continuous
endomorphism $A$ of $I^{\prime}_{{\cdot}/\text{\rm c}}(M,L;E)$, which is a
restriction of the map $A$ on $C^{-\infty}(M;E)$ (Section 2.7). By (4.16), if
$A\in\operatorname{Diff}^{m}(M;E)$, we get induced continuous linear maps

(5.7) $A:I^{\prime\,(s)}_{{\cdot}/\text{\rm c}}(M,L;E)\to
I^{\prime\,(s-m)}_{{\cdot}/\text{\rm c}}(M,L;E)\;,$

If $A\in\operatorname{Diff}(M,L;E)$, the transpose of $A^{t}$ of
$I^{(-s)}_{\text{\rm c}/{\cdot}}(M,L;E^{*}\otimes\Omega)$ is a continuous
endomorphism $A$ of $I^{\prime\,(s)}_{{\cdot}/\text{\rm c}}(M,L;E)$.

###  5.3. Pull-back of dual-conormal distributions

If the conditions of Section 4.7 hold, transposing the versions of (4.22) and
(4.23) with $E^{*}\otimes\Omega M$ and $-m$, we get continuous linear pull-
back maps

(5.8) $\displaystyle\phi^{*}:I^{\prime\,m}(M,L)\to
I^{\prime\,m+l/2-k/4}(M^{\prime},L^{\prime})\quad(m\in\mathbb{R})\;,$ (5.9)
$\displaystyle\phi^{*}:I^{\prime}(M,L)\to
I^{\prime}(M^{\prime},L^{\prime})\;.$

The map (5.9) is an extension of (2.7), a restriction of (2.10) and the
projective limit of the maps (5.8).

If $\phi$ is a local diffeomorphism, we similarly get from (4.24) continuous
pull-back linear maps

(5.10) $\phi^{*}:I^{\prime\,(s)}(M,L)\to
I^{\prime\,(s)}(M^{\prime},L^{\prime})\quad(s\in\mathbb{R})\;.$

###  5.4. Push-forward of dual-conormal distributions

With the notation of Section 4.6, if $\phi$ is a submersion, transposing the
versions of (4.19) and (4.20) with $\Omega M$ and $-m$, we get continuous
linear push-forward maps

(5.11) $\displaystyle\phi_{*}:I^{\prime\,m}_{\text{\rm
c}}(M^{\prime},L^{\prime}\otimes\Omega_{\text{\rm fiber}})\to
I^{\prime\,m-k/4}_{\text{\rm c}}(M,L)\quad(m\in\mathbb{R})\;,$ (5.12)
$\displaystyle\phi_{*}:I^{\prime}_{\text{\rm
c}}(M^{\prime},L^{\prime};\Omega_{\text{\rm fiber}})\to I^{\prime}_{\text{\rm
c}}(M,L)\;.$

The map (5.12) is an extension of (2.8), a restriction of (2.9) and the
projective limit of the maps (5.11).

If $\phi$ is a local diffeomorphism, we similarly get from (4.21) continuous
linear push-forward maps

(5.13) $\phi_{*}:I^{\prime\,(s)}(M^{\prime},L^{\prime})\to
I^{\prime\,(s)}(M,L)\quad(s\in\mathbb{R})\;.$

##  6\. Conormal distributions at the boundary

For the sake of simplicity, in this section and in Sections 7 and 8, we only
consider the case of compact manifolds unless otherwise stated. But the
concepts, notation and some of the results, can be extended to the non-compact
case like in Sections 4.2.2, 4.3.3 and 5.1.2, using arbitrary/compact support
conditions. Such extensions to non-compact manifolds may be used without
further comment.

###  6.1. Some notions of b-geometry

R. Melrose introduced b-calculus, a way to extend calculus to manifolds with
boundary [24, 25]. We will only use a part of it called small b-calculus. Let
$M$ be a compact (smooth) $n$-manifold with boundary; its interior is denoted
by $\mathring{M}$. There exists a function $x\in C^{\infty}(M)$ so that $x\geq
0$, $\partial M=\\{x=0\\}$ (i.e., $x^{-1}(0)$) and $dx\neq 0$ on $\partial M$,
which is called a _boundary defining function_. Let ${}_{+}N\partial M\subset
N\partial M$ be the inward-pointing subbundle of the normal bundle to the
boundary. There is a unique trivialization $\nu\in C^{\infty}(\partial
M;{}_{+}N\partial M)$ of ${}_{+}N\partial M$ so that $dx(\nu)=1$. Take a
collar neighborhood $T\equiv[0,\epsilon_{0})_{x}\times\partial M$ of $\partial
M$, whose projection $\varpi:T\to\partial M$ is the second factor projection.
(In a product expression, every factor projection may be indicated as
subscript of the corresponding factor.) Given coordinates
$y=(y^{1},\dots,y^{n-1})$ on some open $V\subset\partial M$, we get via
$\varpi$ coordinates $(x,y)=(x,y^{1},\dots,y^{n-1})$ adapted (to $\partial M$)
on the open subset $U\equiv[0,\epsilon_{0})\times V\subset M$. There are
vector bundles over $M$, ${}^{\text{\rm b}}TM$ and ${}^{\text{\rm b}}T^{*}M$,
called _b-tangent_ and _b-cotangent_ bundles, which have the same restrictions
as $TM$ and $T^{*}M$ to $\mathring{M}$, and such that
$x\partial_{x},\partial_{y^{1}},\dots,\partial_{y^{n-1}}$ and
$x^{-1}dx,dy^{1},\dots,dy^{n-1}$ extend to smooth local frames around boundary
points. This gives rise to versions of induced vector bundles, like
${}^{\text{\rm b}}\Omega^{s}M:=\Omega^{s}({}^{\text{\rm b}}TM)$
($s\in\mathbb{R}$) and ${}^{\text{\rm b}}\Omega M:={}^{\text{\rm
b}}\Omega^{1}M$. Clearly,

(6.1) $C^{\infty}(M;\Omega^{s})\equiv x^{s}C^{\infty}(M;{}^{\text{\rm
b}}\Omega^{s})\;.$

Thus the integration operator $\int_{M}$ is defined on
$xC^{\infty}(M;{}^{\text{\rm b}}\Omega)$, and induces a pairing between
$C^{\infty}(M)$ and $xC^{\infty}(M;{}^{\text{\rm b}}\Omega)$.

At the points of $\partial M$, the local section $x\partial_{x}$ is
independent of the choice of adapted local coordinates, spanning a trivial
line subbundle ${}^{\text{\rm b}}\\!N\partial M\subset{}^{\text{\rm
b}}T_{\partial M}M$ with $T\partial M={}^{\text{\rm b}}T_{\partial
M}M/{}^{\text{\rm b}}\\!N\partial M$. So ${}^{\text{\rm
b}}\Omega^{s}_{\partial M}M\equiv\Omega^{s}\partial
M\otimes\Omega^{s}({}^{\text{\rm b}}\\!N\partial M)$, and a restriction map
$C^{\infty}(M;{}^{\text{\rm b}}\Omega^{s})\to C^{\infty}(\partial
M;\Omega^{s})$ is locally given by

$u=a(x,y)\,\Big{|}\frac{dx}{x}dy\Big{|}^{s}\mapsto u|_{\partial
M}=a(0,y)\,|dy|^{s}\;.$

A Euclidean structure $g$ on ${}^{\text{\rm b}}TM$ is called a _b-metric_.
Locally,

$g=a_{0}\Big{(}\frac{dx}{x}\Big{)}^{2}+2\sum_{j=1}^{n-1}a_{0j}\,\frac{dx}{x}\,dy^{j}+\sum_{j,k=1}^{n-1}a_{jk}\,dy^{j}\,dy^{k}\;,$

where $a_{0}$, $a_{0j}$ and $a_{jk}$ are $C^{\infty}$ functions, on condition
that $g$ is positive definite. If moreover $a_{0}=1+O(x^{2})$ and
$a_{0j}=O(x)$ as $x\downarrow 0$, then $g$ is called _exact_. In this case,
the restriction of $g$ to $\mathring{T}\equiv(0,\epsilon_{0})\times\partial M$
is asymptotically cylindrical, and therefore $g|_{\mathring{M}}$ is complete.
This restriction is of bounded geometry if it is cylindrical around the
boundary; i.e., $g=(\frac{dx}{x})^{2}+h$ on $\mathring{T}$ for (the pull-back
via $\varpi$ of) some Riemannian metric $h$ on $\partial M$, taking
$\epsilon_{0}$ small enough; i.e., $a_{0}=1$ and $a_{0j}=0$ using adapted
local coordinates.

###  6.2. Supported and extendible functions

Let $\breve{M}$ be any closed manifold containing $M$ as submanifold of
dimension $n$ (for instance, $\breve{M}$ could be the double of $M$), and let
$M^{\prime}=\breve{M}\setminus\mathring{M}$, which is another compact
$n$-submanifold with boundary of $\breve{M}$, with dimension $n$ and $\partial
M^{\prime}=M\cap M^{\prime}=\partial M$.

The concepts, notation and conventions of Section 2.4 have straightforward
extensions to manifolds with boundary, like the Fréchet space $C^{\infty}(M)$.
Its elements are called _extendible functions_ because the continuous linear
restriction map

(6.2) $R:C^{\infty}(\breve{M})\to C^{\infty}(M)$

is surjective; in fact, there is a continuous linear extension map
$E:C^{\infty}(M)\to C^{\infty}(\breve{M})$ [34]. Since $C^{\infty}(\breve{M})$
and $C^{\infty}(M)$ are Fréchet spaces, the map (6.2) is open by the open
mapping theorem, and therefore it is a surjective topological homomorphism.
Its null space is $C^{\infty}_{M^{\prime}}(\breve{M})$.

The Fréchet space of _supported_ functions is the closed subspace of the
smooth functions on $M$ that vanish to all orders at the points of $\partial
M$,

(6.3) $\dot{C}^{\infty}(M)=\bigcap_{m\geq 0}x^{m}C^{\infty}(M)\subset
C^{\infty}(M)\;,$

The extension by zero realizes $\dot{C}^{\infty}(M)$ as the closed subspace of
functions on $\breve{M}$ supported in $M$,

(6.4) $\dot{C}^{\infty}(M)\equiv C^{\infty}_{M}(\breve{M})\subset
C^{\infty}(\breve{M})\;.$

By (6.3),

(6.5) $x^{m}\dot{C}^{\infty}(M)=\dot{C}^{\infty}(M)\quad(m\in\mathbb{R})\;,$

and therefore, by (6.1),

(6.6) $\dot{C}^{\infty}(M;{}^{\text{\rm
b}}\Omega^{s})\equiv\dot{C}^{\infty}(M;\Omega^{s})\quad(s\in\mathbb{R})\;.$

We can similarly define Banach spaces $C^{k}(M)$ and $\dot{C}^{k}(M)$
($k\in\mathbb{N}_{0}$) satisfying the analogs of (6.2)–(6.4), which in turn
yield analogs of the first inclusions of (2.6), obtaining
$C^{\infty}(M)=\bigcap_{k}C^{k}(M)$ and
$\dot{C}^{\infty}(M)=\bigcap_{k}\dot{C}^{k}(M)$.

###  6.3. Supported and extendible distributions

The spaces of _supported_ and _extendible_ distributions on $M$ are

$\dot{C}^{-\infty}(M)=C^{\infty}(M;\Omega)^{\prime}\;,\quad
C^{-\infty}(M)=\dot{C}^{\infty}(M;\Omega)^{\prime}\;.$

Transposing the version of (6.2) with $\Omega M$, we get [25, Proposition
3.2.1]

(6.7) $\dot{C}^{-\infty}(M)\equiv C^{-\infty}_{M}(\breve{M})\subset
C^{-\infty}(\breve{M})\;.$

Similarly, (6.4) and (6.3) give rise to continuous linear restriction maps

(6.8) $\displaystyle R:C^{-\infty}(\breve{M})\to C^{-\infty}(M)\;,$ (6.9)
$\displaystyle R:\dot{C}^{-\infty}(M)\to C^{-\infty}(M)\;,$

which are surjective by the Hahn-Banach theorem. Their null spaces are
$C^{-\infty}_{M^{\prime}}(\breve{M})=\dot{C}^{-\infty}(M^{\prime})$ and
$\dot{C}^{-\infty}_{\partial M}(M)$ [25, Proposition 3.3.1], respectively.
According to (6.7), the map (6.9) is a restriction of (6.8). As a consequence
of (6.7), there are dense continuous inclusions [25, Lemma 3.2.1]

(6.10) $C^{\infty}_{\text{\rm
c}}(\mathring{M})\subset\dot{C}^{\infty}(M)\subset
C^{\infty}(M)\subset\dot{C}^{-\infty}(M)\;,$

the last one given by the integration pairing between $C^{\infty}(M)$ and
$C^{\infty}(M;\Omega)$. The restriction of this pairing to
$\dot{C}^{\infty}(M;\Omega)$ induces a dense continuous inclusion

(6.11) $C^{\infty}(M)\subset C^{-\infty}(M)\;.$

Moreover (6.9) is the identity map on $C^{\infty}(M)$.

As before, from (6.5) and (6.6), we get

(6.12) $\displaystyle x^{m}C^{-\infty}(M)$
$\displaystyle=C^{-\infty}(M)\quad(m\in\mathbb{R})\;,$ (6.13) $\displaystyle
C^{-\infty}(M;{}^{\text{\rm b}}\Omega^{s})$ $\displaystyle\equiv
C^{-\infty}(M;\Omega^{s})\quad(s\in\mathbb{R})\;.$

The Banach spaces $C^{\prime-k}(M)$ and $\dot{C}^{\prime-k}(M)$
($k\in\mathbb{N}_{0}$) are similarly defined and satisfy the analogs of
(6.7)–(6.13). These spaces satisfy the analogs of the second inclusions of
(2.6), obtaining $\bigcup_{k}C^{\prime-k}(M)=C^{-\infty}(M)$ and
$\bigcup_{k}\dot{C}^{\prime-k}(M)=\dot{C}^{-\infty}(M)$.

###  6.4. Supported and extendible Sobolev spaces

The _supported_ Sobolev space of order $s\in\mathbb{R}$ is the closed subspace
of the elements supported in $M$,

(6.14) $\dot{H}^{s}(M)=H^{s}_{M}(\breve{M})\subset H^{s}(\breve{M})\;.$

On the other hand, using the map (6.9), the _extendible_ Sobolev space of
order $s$ is $H^{s}(M)=R(H^{s}(\breve{M}))$ with the inductive topology given
by

(6.15) $R:H^{s}(\breve{M})\to H^{s}(M)\;;$

i.e., this is a surjective topological homomorphism. Its null space is
$H^{s}_{M^{\prime}}(\breve{M})$. The analogs of (2.16)–(2.19) hold true in
this setting using $\dot{C}^{\pm\infty}(M)$ and $C^{\pm\infty}(M)$.
Furthermore the analogs of (2.16) are also compact operators because (6.14) is
a closed embedding and (6.15) a surjective topological homomorphism.

The following properties are satisfied [25, Proposition 3.5.1].
$C^{\infty}(M)$ is dense in $H^{s}(M)$, we have

(6.16) $\dot{H}^{s}(M)\equiv H^{-s}(M;\Omega)^{\prime}\;,\quad
H^{s}(M)\equiv\dot{H}^{-s}(M;\Omega)^{\prime}\;,$

and the map (6.9) has a continuous restriction

(6.17) $R:\dot{H}^{s}(M)\to H^{s}(M)\;,$

which is surjective if $s\leq 1/2$, and injective if $s\geq-1/2$. In
particular, $\dot{H}^{0}(M)\equiv H^{0}(M)\equiv L^{2}(M)$. The null space of
(6.17) is $\dot{H}^{s}_{\partial M}(M)$.

Since $\dot{H}^{s}(M)$ and $H^{s}(M)$ form compact spectra of Hilbertian
spaces, we get the following result.

######  Proposition 6.1.

$\dot{C}^{-\infty}(M)$ and $C^{-\infty}(M)$ are barreled, ultrabornological,
webbed, acyclic DF Montel spaces, and therefore complete, boundedly retractive
and reflexive.

######  Proposition 6.2.

The maps (6.8) and (6.9) are surjective topological homomorphisms.

###### Proof.

We already know that these maps are linear, continuous and surjective. Since
$C^{-\infty}(\breve{M})$ is webbed, and $\dot{C}^{-\infty}(M)$ and
$C^{-\infty}(M)$ are webbed and ultrabornological (Proposition 6.1), the
stated maps are also open by the open mapping theorem [22, 7.35.3 (1)], [28,
Exercise 14.202 (a)], [7, Section IV.5], [5]. ∎

###  6.5. The space $\dot{C}^{-\infty}_{\partial M}(M)$

Proposition 6.2 means that the following are short exact sequences in the
category of continuous linear maps between LCSs:

$\displaystyle
0\to\dot{C}^{-\infty}(M^{\prime})\xrightarrow{\iota}C^{-\infty}(\breve{M})\xrightarrow{R}C^{-\infty}(M)\to
0\;,$ (6.18) $\displaystyle 0\to\dot{C}^{-\infty}_{\partial
M}(M)\xrightarrow{\iota}\dot{C}^{-\infty}(M)\xrightarrow{R}C^{-\infty}(M)\to
0\;.$

From (6.7), we get

(6.19) $\dot{C}^{-\infty}_{\partial M}(M)\equiv C^{-\infty}_{\partial
M}(\breve{M})\subset C^{-\infty}(\breve{M})\;.$

The analogs of the second inclusion of (2.6), (2.16) and (2.18) for the spaces
$\dot{C}^{\prime\,-k}(M)$ and $\dot{H}^{s}(M)$ yield corresponding analogs for
the spaces $\dot{C}^{\prime\,-k}_{\partial M}(M)$ and $\dot{H}^{s}_{\partial
M}(M)$. Thus the spaces $\dot{C}^{\prime\,-k}_{\partial M}(M)$ and
$\dot{H}^{s}_{\partial M}(M)$ form spectra with the same union; the spectrum
of spaces $\dot{H}^{s}_{\partial M}(M)$ is compact.

######  Proposition 6.3.

$\dot{C}^{-\infty}_{\partial M}(M)$ is a limit subspace of the LF-space
$\dot{C}^{-\infty}(M)$.

###### Proof.

By Propositions 6.2 and 6.1, $\dot{C}^{-\infty}(M)/\dot{C}^{-\infty}_{\partial
M}(M)\equiv C^{-\infty}(M)$ is acyclic, which is equivalent to the statement.
∎

The following analog of Proposition 6.1 hold true with the same arguments,
applying Proposition 6.3 and using that the Hilbertian spaces
$\dot{H}^{s}_{\partial M}(M)$ form a compact spectrum.

######  Corollary 6.4.

$\dot{C}^{-\infty}_{\partial M}(M)$ is barreled, ultrabornological, webbed
acyclic DF Montel space, and therefore complete, boundedly retractive and
reflexive.

A description of $\dot{C}^{-\infty}_{\partial M}(M)$ will be indicated in
Remark 7.7.

###  6.6. Differential operators acting on $C^{-\infty}(M)$ and
$\dot{C}^{-\infty}(M)$

The notions of Section 2.7 also have straightforward extensions to manifolds
with boundary. The action of any $A\in\operatorname{Diff}(M)$ on
$C^{\infty}(M)$ preserves $\dot{C}^{\infty}(M)$. Taking the version of this
property with $\Omega M$, we get that $A^{t}$ acts on
$\dot{C}^{\infty}(M;\Omega)$ and $C^{\infty}(M;\Omega)$. Using the transpose
again, we get extended continuous actions of $A$ on $C^{-\infty}(M)$ and
$\dot{C}^{-\infty}(M)$. They fit into commutative diagrams

(6.20) $\begin{CD}\dot{C}^{-\infty}(M)@>{A}>{}>\dot{C}^{-\infty}(M)\\\
@V{R}V{}V@V{}V{R}V\\\
C^{-\infty}(M)@>{A}>{}>C^{-\infty}(M)\end{CD}\qquad\begin{CD}C^{-\infty}(M)@>{A}>{}>C^{-\infty}(M)\\\
@A{\iota}A{}A@A{}A{\iota}A\\\ C^{\infty}(M)@>{A}>{}>C^{\infty}(M)\;.\end{CD}$

However the analogous diagram

(6.21) $\begin{CD}\dot{C}^{-\infty}(M)@>{A}>{}>\dot{C}^{-\infty}(M)\\\
@A{\iota}A{}A@A{}A{\iota}A\\\ C^{\infty}(M)@>{A}>{}>C^{\infty}(M)\end{CD}$

may not be commutative. Let us use the notation $u\mapsto u_{\text{\rm c}}$
for the injection $C^{\infty}(M)\subset\dot{C}^{-\infty}(M)$ (see (6.10)).
(Following Melrose, the subscript “c” stands for “cutoff at the boundary.”) We
have $A(u_{\text{\rm c}})-(Au)_{\text{\rm c}}\in C^{-\infty}_{\partial M}(M)$
for all $u\in C^{\infty}(M)$ [25, Eq. (3.4.8)]. For instance, if
$M=[x_{0},x_{1}]$, where $x_{0}<x_{1}$ in $\mathbb{R}$, and $A=\partial_{x}$,
integration by parts gives

$\partial_{x}(u_{\text{\rm c}})-(\partial_{x}u)_{\text{\rm
c}}=u(x_{1})\,\delta_{x_{1}}-u(x_{0})\,\delta_{x_{0}}$

for all $u\in C^{\infty}([x_{0},x_{1}])$, using the Dirac mass at $x_{j}$
($j=0,1$).

Using (6.2) and its version for vector fields, we get a surjective restriction
map

(6.22)
$\operatorname{Diff}(\breve{M})\to\operatorname{Diff}(M)\;,\quad\breve{A}\mapsto\breve{A}|_{M}\;.$

For any $\breve{A}\in\operatorname{Diff}(\breve{M})$ with $\breve{A}|_{M}=A$,
we have the commutative diagrams

(6.23)
$\begin{CD}C^{-\infty}(\breve{M})@>{\breve{A}}>{}>C^{-\infty}(\breve{M})\\\
@V{R}V{}V@V{}V{R}V\\\
C^{-\infty}(M)@>{A}>{}>C^{-\infty}(M)\;,\end{CD}\qquad\begin{CD}C^{-\infty}(\breve{M})@>{\breve{A}}>{}>C^{-\infty}(\breve{M})\\\
@A{\iota}A{}A@A{}A{\iota}A\\\
\dot{C}^{-\infty}(M)@>{A}>{}>\dot{C}^{-\infty}(M)\;,\end{CD}$

where the left-hand side square extends the left-hand side square of (6.20).

If $A\in\operatorname{Diff}^{m}(M)$ ($m\in\mathbb{N}_{0}$), its actions on
$\dot{C}^{-\infty}(M)$ and $C^{-\infty}(M)$ define continuous linear maps,

(6.24) $A:\dot{H}^{s}(M)\to\dot{H}^{s-m}(M)\;,\quad A:H^{s}(M)\to
H^{s-m}(M)\;.$

The maps (6.17) and (6.24) fit into a commutative diagram given by the left-
hand side square of (6.20).

###  6.7. Differential operators tangent to the boundary

The concepts of Section 4 can be generalized to the case with boundary when
$L=\partial M$ [25, Chapter 6] (see also [24, Section 4.9]), giving rise to
the Lie subalgebra and $C^{\infty}(M)$-submodule $\mathfrak{X}_{\text{\rm
b}}(M)\subset\mathfrak{X}(M)$ of vector fields tangent to $\partial M$, called
_b-vector fields_. There is a canonical identity $\mathfrak{X}_{\text{\rm
b}}(M)\equiv C^{\infty}(M;{}^{\text{\rm b}}TM)$. Using
$\mathfrak{X}_{\text{\rm b}}(M)$ like in Section 2.7, we get the filtered
$C^{\infty}(M)$-submodule and filtered subalgebra
$\operatorname{Diff}_{\text{\rm b}}(M)\subset\operatorname{Diff}(M)$ of _b-
differential operators_. It consists of the operators
$A\in\operatorname{Diff}(M)$ such that (6.21) is commutative [25, Exercise
3.4.20]. The extension of $\operatorname{Diff}_{\text{\rm b}}(M)$ to arbitrary
vector bundles is closed by taking transposes and formal adjoints.

Clearly, the restriction map (6.22) satisfies

(6.25) $\operatorname{Diff}(\breve{M},\partial
M)|_{M}=\operatorname{Diff}_{\text{\rm b}}(M)\;.$

For all $a\in\mathbb{R}$ and $k\in\mathbb{Z}$, we have [25, Eqs. (4.2.7) and
(4.2.8)]

(6.26) $\operatorname{Diff}_{\text{\rm
b}}^{k}(M)\,x^{a}=x^{a}\operatorname{Diff}_{\text{\rm b}}^{k}(M)\;.$

Since $\operatorname{Diff}(M)$ is spanned by $\partial_{x}$ and
$\operatorname{Diff}_{\text{\rm b}}(M)$ as algebra, it follows that

(6.27) $\operatorname{Diff}^{k}(M)\,x^{a}\subset
x^{a-k}\operatorname{Diff}^{k}(M)\;.$

###  6.8. Conormal distributions at the boundary

The spaces of _supported_ and _extendible_ conormal distributions at the
boundary of Sobolev order $s\in\mathbb{R}$ are the $C^{\infty}(M)$-modules and
LCSs

$\displaystyle\dot{\mathcal{A}}^{(s)}(M)$
$\displaystyle=\\{\,u\in\dot{C}^{-\infty}(M)\mid\operatorname{Diff}_{\text{\rm
b}}(M)\,u\subset\dot{H}^{s}(M)\,\\}\;,$ $\displaystyle\mathcal{A}^{(s)}(M)$
$\displaystyle=\\{\,u\in C^{-\infty}(M)\mid\operatorname{Diff}_{\text{\rm
b}}(M)\,u\subset H^{s}(M)\,\\}\;,$

with the projective topologies given by the maps
$P:\dot{\mathcal{A}}^{(s)}(M)\to H^{s}(M)$ and
$P:\mathcal{A}^{(s)}(M)\to\dot{H}^{s}(M)$ ($P\in\operatorname{Diff}_{\text{\rm
b}}(M)$). They satisfy the analogs of the continuous inclusions (4.4), giving
rise to the filtered $C^{\infty}(M)$-modules and LF-spaces of _supported_ and
_extendible_ conormal distributions at the boundary,

(6.28)
$\dot{\mathcal{A}}(M)=\bigcup_{s}\dot{\mathcal{A}}^{(s)}(M)\;,\quad\mathcal{A}(M)=\bigcup_{s}\mathcal{A}^{(s)}(M)\;.$

By definition, there are continuous inclusions

(6.29)
$\dot{\mathcal{A}}(M)\subset\dot{C}^{-\infty}(M)\;,\quad\mathcal{A}(M)\subset
C^{-\infty}(M)\;.$

Thus $\dot{\mathcal{A}}(M)$ and $\mathcal{A}(M)$ are Hausdorff.

The following analogs of Propositions 4.1 and 3.1 hold true with formally the
same proofs.

######  Proposition 6.5.

$\dot{\mathcal{A}}^{(s)}(M)$ and $\mathcal{A}^{(s)}(M)$ are totally reflexive
Fréchet spaces.

######  Corollary 6.6.

$\dot{\mathcal{A}}(M)$ and $\mathcal{A}(M)$ are barreled, ultrabornological
and webbed.

We have

(6.30)
$\bigcap_{s}\dot{\mathcal{A}}^{(s)}(M)=\dot{C}^{\infty}(M)\;,\quad\bigcap_{s}\mathcal{A}^{(s)}(M)=C^{\infty}(M)\;,$

obtaining dense continuous inclusions [25, Lemma 4.6.1 and the proof of
Proposition 4.1.1]

(6.31) $\dot{C}^{\infty}(M)\subset\dot{\mathcal{A}}(M)\;,\quad
C^{\infty}(M)\subset\mathcal{A}(M)\;.$

By elliptic regularity, we also get continuous inclusions [25, Eq. (4.1.4)]

(6.32) $\dot{\mathcal{A}}(M)|_{\mathring{M}},\mathcal{A}(M)\subset
C^{\infty}(\mathring{M})\;.$

Using (6.9), (6.15) and the commutativity of the left-hand side square of
(6.20), we get the continuous linear restriction maps

(6.33) $R:\dot{\mathcal{A}}^{(s)}(M)\to\mathcal{A}^{(s)}(M)\;,$

which are surjective for $s\leq 1/2$ and injective for $s\geq-1/2$ because
this is true for the maps (6.17). In particular, there is a dense continuous
inclusion

(6.34) $C^{\infty}(M)\subset\dot{\mathcal{A}}^{(0)}(M)\;.$

The maps (6.33) induce a continuous linear restriction map

(6.35) $R:\dot{\mathcal{A}}(M)\to\mathcal{A}(M)\;.$

This map is surjective [25, Proposition 4.1.1], which is a consequence of the
existence of corresponding partial extension maps [25, Section 4.4]; the
precise statement is recalled in Proposition 6.26 for later use in this work.
The maps (6.33) and (6.35) are restrictions of (6.9).

The following analog of Proposition 6.2 holds true with formally the same
proof, using that $\dot{\mathcal{A}}(M)$ is webbed and $\mathcal{A}(M)$
ultrabornological (Corollary 6.6).

######  Proposition 6.7.

The map (6.35) is a surjective topological homomorphism.

###  6.9. The spaces $x^{m}L^{\infty}(M)$

For $m\in\mathbb{R}$, consider the weighted space $x^{m}L^{\infty}(M)$
(Section 2.11). From (6.12) and since $L^{\infty}(M)\subset C^{-\infty}(M)$,
it follows that there is a continuous inclusion

$x^{m}L^{\infty}(M)\subset C^{-\infty}(M)\;.$

Moreover, for $m^{\prime}<m$, from $x^{m-m^{\prime}}\in L^{\infty}(M)$, we
easily get a continuous inclusion

(6.36) $x^{m}L^{\infty}(M)\subset x^{m^{\prime}}L^{\infty}(M)\;,$

######  Proposition 6.8.

For $m^{\prime}<m$, $C^{\infty}_{\text{\rm c}}(\mathring{M})$ is dense in
$x^{m}L^{\infty}(M)$ with the topology of $x^{m^{\prime}}L^{\infty}(M)$.

###### Proof.

Given $u\in x^{m}L^{\infty}(M)$ and $\epsilon>0$, let $B$ be the ball in
$x^{m^{\prime}}L^{\infty}(M)$ of center $u$ and radius $\epsilon$. Let
$S=\sup_{M}x^{m-m^{\prime}}>0$. Since $C^{\infty}(M)$ is dense in
$L^{\infty}(M)$, there is some $f\in C^{\infty}(M)$ so that
$|f-x^{-m}u|<\min\\{\epsilon/2,\epsilon/S\\}$ (Lebesgue-) almost everywhere.
(Recall that the sets of Lebesgue measure zero are well-defined in any $C^{1}$
manifold [12, Lemma 3.1.1].) There is some $0<\delta<1$ so that $\delta
x^{-m}|u|<\epsilon/4$ almost everywhere. Take some $\lambda\in
C^{\infty}(\mathbb{R})$ so that $\lambda\geq 0$, $\lambda(r)\leq r^{m}$ if
$r>0$, $\lambda(r)=0$ if $r^{m-m^{\prime}}\leq\delta/2$, and
$\lambda(r)=r^{m}$ if $r^{m-m^{\prime}}\geq\delta$. Let $h=\lambda(x)f\in
C^{\infty}_{\text{\rm c}}(\mathring{M})$. If $x^{m-m^{\prime}}\leq\delta$,
then, almost everywhere,

$\displaystyle x^{-m^{\prime}}|h-u|$ $\displaystyle\leq\delta
x^{-m}(|h|+|u|)\leq\delta(|f|+x^{-m}|u|)$
$\displaystyle\leq\delta(|f-x^{-m}u|+2x^{-m}|u|)<\delta\Big{(}\frac{\epsilon}{2}+\frac{\epsilon}{2\delta}\Big{)}<\epsilon\;.$

If $x^{m-m^{\prime}}\geq\delta$, then, almost everywhere,

$x^{-m^{\prime}}|h-u|=x^{m-m^{\prime}}|x^{-m}\lambda(x)f-x^{-m}u|\leq
S|f-x^{-m}u|<\epsilon\;.$

Thus $h\in B\cap C^{\infty}_{\text{\rm c}}(\mathring{M})$. ∎

###  6.10. Filtration of $\mathcal{A}(M)$ by bounds

For every $m\in\mathbb{R}$, let

$\mathcal{A}^{m}(M)=\\{\,u\in C^{-\infty}(M)\mid\operatorname{Diff}_{\text{\rm
b}}(M)\,u\subset x^{m}L^{\infty}(M)\,\\}\;.$

This is another $C^{\infty}(M)$-module and LCS, with the projective topology
given by the maps $P:\mathcal{A}^{m}(M)\to x^{m}L^{\infty}(M)$
($P\in\operatorname{Diff}_{\text{\rm b}}(M)$).

######  Example 6.9 ([25, Exercises 4.2.23 and 4.2.24]).

Via the injection of $\mathbb{R}^{l}$ into its stereographic compactification
$\mathbb{S}^{l}_{+}=\\{\,x\in\mathbb{S}^{l}\mid x^{l+1}\geq 0\,\\}$, the space
$\mathcal{A}^{-m}(\mathbb{S}^{l}_{+})$ corresponds to the symbol space
$S^{m}(\mathbb{R}^{l})$ (Section 3).

Note that (6.36) yields a continuous inclusion

(6.37)
$\mathcal{A}^{m}(M)\subset\mathcal{A}^{m^{\prime}}(M)\quad(m^{\prime}<m)\;.$

Moreover there are continuous inclusions [25, Proof of Proposition 4.2.1]

(6.38)
$\mathcal{A}^{(s)}(M)\subset\mathcal{A}^{m}(M)\subset\mathcal{A}^{(\min\\{m,0\\})}(M)\quad(m<s-n/2-1)\;.$

Hence

(6.39) $\mathcal{A}(M)=\bigcup_{m}\mathcal{A}^{m}(M)\;.$

Despite of defining the same LF-space, the filtrations of $\mathcal{A}(M)$
defined by the spaces $\mathcal{A}^{(s)}(M)$ and $\mathcal{A}^{m}(M)$ are not
equivalent because, in contrast with (6.30),

$\dot{C}^{\infty}(M)=\bigcap_{m}\mathcal{A}^{m}(M)\;.$

Let $\\{\,P_{j}\mid j\in\mathbb{N}_{0}\,\\}$ be a countable
$C^{\infty}(M)$-spanning set of $\operatorname{Diff}_{\text{\rm b}}(M)$. The
topology of $\mathcal{A}^{m}(M)$ can be described by the semi-norms
$\|{\cdot}\|_{k,m}$ ($k\in\mathbb{N}_{0}$) given by

(6.40)
$\|u\|_{k,m}=\|P_{k}u\|_{x^{m}L^{\infty}}=\underset{M}{\operatorname{ess\,sup}}\,\big{|}x^{-m}P_{k}u\big{|}=\sup_{\mathring{M}}\big{|}x^{-m}P_{k}u\big{|}\;,$

using (6.32) in the last expression. From (2.1) and (6.32), we also get the
continuous semi-norms $\|{\cdot}\|_{K,k,m}$ (for any compact
$K\subset\mathring{M}$ and $k\in\mathbb{N}_{0}$) on $\mathcal{A}^{m}(M)$ given
by

(6.41) $\|u\|_{K,k,m}=\sup_{K}|P_{k}u|\;.$

Other continuous semi-norms $\|{\cdot}\|^{\prime}_{k,m}$
($k\in\mathbb{N}_{0}$) on $\mathcal{A}^{m}(M)$ are defined by

(6.42) $\|u\|^{\prime}_{k,m}=\lim_{\epsilon\downarrow
0}\sup_{\\{0<x<\epsilon\\}}\big{|}x^{-m}P_{k}u\big{|}\;.$

The proofs of the following results are similar to the proofs of Propositions
3.2, 3.3, 3.4, 3.5 and 3.6, using (6.32).

######  Proposition 6.10.

The semi-norms (6.41) and (6.42) together describe the topology of
$\mathcal{A}^{m}(M)$.

######  Proposition 6.11.

For $m,m^{\prime},k\in\mathbb{N}_{0}$, if $m^{\prime}<m$, then
$\|{\cdot}\|^{\prime}_{k,m^{\prime}}=0$ on $\mathcal{A}^{m}(M)$.

######  Corollary 6.12.

If $m^{\prime}<m$, then the topologies of $\mathcal{A}^{m^{\prime}}(M)$ and
$C^{\infty}(\mathring{M})$ coincide on $\mathcal{A}^{m}(M)$. Therefore the
topologies of $\mathcal{A}(M)$ and $C^{\infty}(\mathring{M})$ coincide on
$\mathcal{A}^{m}(M)$.

######  Corollary 6.13.

For $m^{\prime}<m$, $C^{\infty}_{\text{\rm c}}(\mathring{M})$ is dense in
$\mathcal{A}^{m}(M)$ with the topology of $\mathcal{A}^{m^{\prime}}(M)$.
Therefore $C^{\infty}_{\text{\rm c}}(\mathring{M})$ is dense in
$\mathcal{A}(M)$.

######  Corollary 6.14.

$\mathcal{A}(M)$ is an acyclic Montel space, and therefore complete, boundedly
retractive and reflexive.

######  Remark 6.15.

Proposition 6.8 provides an alternative direct proof of Corollary 6.13.
Actually, it will be shown that $C^{\infty}_{\text{\rm c}}(\mathring{M})$ is
dense in every $\mathcal{A}^{m}(M)$ with its own topology (Corollaries 6.36
and 6.38).

The obvious analog of Remark 3.8 makes sense for (6.32) and Corollary 6.12.

###  6.11. $\dot{\mathcal{A}}(M)$ and $\mathcal{A}(M)$ vs
$I(\breve{M},\partial M)$

Using (6.8), (6.15), (6.25) and the commutativity of the left-hand side square
of (6.23), we get continuous linear restriction maps

$R:I^{(s)}(\breve{M},\partial M)\to\mathcal{A}^{(s)}(M)\;,$

which induce a continuous linear restriction map

(6.43) $R:I(\breve{M},\partial M)\to\mathcal{A}(M)\;.$

By (6.14), (6.25) and the commutativity of the right-hand side square of
(6.23), we get the TVS-identities

(6.44) $\dot{\mathcal{A}}^{(s)}(M)\equiv I^{(s)}_{M}(\breve{M},\partial M)\;,$

inducing a continuous linear isomorphism

(6.45) $\dot{\mathcal{A}}(M)\xrightarrow{\cong}I_{M}(\breve{M},\partial M)\;.$

By (6.45) and Proposition 6.7, the map (6.43) is also surjective. Then the
following analog of Proposition 6.2 follows with formally the same proof,
using that $I(\breve{M},\partial M)$ is webbed (Corollary 4.2) and
$\mathcal{A}(M)$ ultrabornological (Corollary 6.6).

######  Proposition 6.16.

The map (6.43) is a surjective topological homomorphism.

The null space of (6.43) is $I_{M^{\prime}}(\breve{M},\partial M)$. The
following analog of Proposition 6.3 follows with formally the same proof,
using Propositions 6.16 and 6.14.

######  Proposition 6.17.

$I_{M}(\breve{M},\partial M)$ is a limit subspace of the LF-space
$I(\breve{M},\partial M)$.

######  Corollary 6.18.

The map (6.45) is a TVS-isomorphism.

###### Proof.

Apply (6.28), (6.44) and Proposition 6.17. ∎

###  6.12. Filtration of $\dot{\mathcal{A}}(M)$ by the symbol order

Inspired by (6.44), let

(6.46) $\dot{\mathcal{A}}^{m}(M)=I^{m}_{M}(\breve{M},\partial M)\subset
I^{m}(\breve{M},\partial M)\quad(m\in\mathbb{R})\;,$

which are closed subspaces satisfying the analogs of (4.11) and (4.12). Thus

$\dot{\mathcal{A}}(M)=\bigcup_{m}\dot{\mathcal{A}}^{m}(M)\;,\quad\dot{C}^{\infty}(M)=\bigcap_{m}\dot{\mathcal{A}}^{m}(M)\;,$

and the TVS-isomorphism (6.45) is also compatible with the symbol filtration.

The following is a consequence of Corollary 4.5 applied to
$(\breve{M},\partial M)$.

######  Corollary 6.19.

For $m<m^{\prime},m^{\prime\prime}$, the topologies of
$\dot{\mathcal{A}}^{m^{\prime}}(M)$ and
$\dot{\mathcal{A}}^{m^{\prime\prime}}(M)$ coincide on
$\dot{\mathcal{A}}^{m}(M)$.

The following result follows like Corollary 3.6, applying Corollary 6.19 and
using that $\dot{\mathcal{A}}(M)$ is barreled (Corollary 6.6) and a closed
subspace of the Montel space $I(\breve{M},\partial M)$ (Corollary 4.7).

######  Corollary 6.20.

$\dot{\mathcal{A}}(M)$ is an acyclic Montel space, and therefore complete,
boundedly retractive and reflexive.

###  6.13. The space $\mathcal{K}(M)$

Using the condition of being supported in $\partial M$, define the LCHSs and
$C^{\infty}(M)$-modules

$\mathcal{K}^{(s)}(M)=\dot{\mathcal{A}}^{(s)}_{\partial
M}(M)\;,\quad\mathcal{K}^{m}(M)=\dot{\mathcal{A}}^{m}_{\partial
M}(M)\;,\quad\mathcal{K}(M)=\dot{\mathcal{A}}_{\partial M}(M)\;.$

These are closed subspaces of $\dot{\mathcal{A}}^{(s)}(M)$,
$\dot{\mathcal{A}}^{m}(M)$ and $\dot{\mathcal{A}}(M)$, respectively; more
precisely, they are the null spaces of the corresponding restrictions of the
map (6.35). They satisfy the analogs of (4.4), (4.11) and (4.12). So

$\bigcup_{s}\mathcal{K}^{(s)}(M)=\bigcup_{m}\mathcal{K}^{m}(M)\;.$

Proposition 6.7 means that the following sequence is exact in the category of
continuous linear maps between LCSs:

(6.47)
$0\to\mathcal{K}(M)\xrightarrow{\iota}\dot{\mathcal{A}}(M)\xrightarrow{R}\mathcal{A}(M)\to
0\;.$

The following analog of Proposition 6.3 holds true with formally the same
proof, using Propositions 6.7 and 6.14.

######  Proposition 6.21.

$\mathcal{K}(M)$ is a limit subspace of the LF-space $\dot{\mathcal{A}}(M)$.

From the definition of $\dot{\mathcal{A}}^{(s)}(M)$ (Section 6.8), we get

$\mathcal{K}^{(s)}(M)=\\{\,u\in\dot{C}^{-\infty}_{\partial
M}(M)\mid\operatorname{Diff}_{\text{\rm b}}(M)\,u\subset\dot{H}^{s}_{\partial
M}(M)\,\\}\;,$

with the projective topology given by the maps
$P:\mathcal{K}^{(s)}(M)\to\dot{H}^{s}_{\partial M}(M)$
($P\in\operatorname{Diff}_{\text{\rm b}}(M)$). Hence the following analogs of
Propositions 4.1 and 3.1 hold true with formally the same proofs.

######  Proposition 6.22.

$\mathcal{K}^{(s)}(M)$ is a totally reflexive Fréchet space.

######  Corollary 6.23.

$\mathcal{K}(M)$ is barreled, ultrabornological and webbed.

Now the following analogs of Corollaries 6.19 and 6.20 hold true with formally
the same proofs, using Corollaries 6.19, 6.20 and 6.23.

######  Corollary 6.24.

For $m<m^{\prime},m^{\prime\prime}$, the topologies of
$\mathcal{K}^{m^{\prime}}(M)$ and $\mathcal{K}^{m^{\prime\prime}}(M)$ coincide
on $\mathcal{K}^{m}(M)$.

######  Corollary 6.25.

$\mathcal{K}(M)$ is an acyclic Montel space, and therefore complete, boundedly
retractive and reflexive.

By Corollary 6.18,

(6.48) $\mathcal{K}(M)\equiv I_{\partial M}(\breve{M},\partial M)\;,$

which restricts to identities between the spaces defining the Sobolev and
symbol order filtrations, according to (6.44) and (6.46).

A description of $\mathcal{K}^{(s)}(M)$ and $\mathcal{K}(M)$ will be indicated
in Remark 7.26.

###  6.14. Action of $\operatorname{Diff}(M)$ on $\dot{\mathcal{A}}(M)$,
$\mathcal{A}(M)$ and $\mathcal{K}(M)$

According to Section 4.5, and using (6.25), (6.44), Proposition 6.16 and
locality, any $A\in\operatorname{Diff}(M)$ defines continuous endomorphisms
$A$ of $\dot{\mathcal{A}}(M)$, $\mathcal{A}(M)$ and $\mathcal{K}(M)$. If
$A\in\operatorname{Diff}^{k}(M)$, these maps also satisfy the analogs of
(4.16). If $A\in\operatorname{Diff}_{\text{\rm b}}(M)$, it clearly defines
continuous endomorphisms of $\dot{\mathcal{A}}^{(s)}(M)$,
$\mathcal{A}^{(s)}(M)$, $\mathcal{A}^{m}(M)$ and $\mathcal{K}^{(s)}(M)$.

According to Section 6.6, (6.29), (6.31) and (6.32), the maps of this
subsection are restrictions of the endomorphisms $A$ of
$\dot{C}^{-\infty}(M)$, $C^{-\infty}(M)$ and $C^{\infty}(\mathring{M})$, and
extensions of the endomorphisms $A$ of $\dot{C}^{\infty}(M)$ and
$C^{\infty}(M)$.

###  6.15. Partial extension maps

######  Proposition 6.26 (Cf. [25, Section 4.4]).

For all $m\in\mathbb{R}$, there is a continuous linear partial extension map
$E_{m}:\mathcal{A}^{m}(M)\to\dot{\mathcal{A}}^{(s)}(M)$, where $s=0$ if $m\geq
0$, and $m>s\in\mathbb{Z}^{-}$ if $m<0$. For $m\geq 0$,
$E_{m}:\mathcal{A}^{m}(M)\to\dot{\mathcal{A}}^{(0)}(M)$ is a continuous
inclusion map.

######  Remark 6.27.

By (6.38) and Proposition 6.26, for any $A\in\operatorname{Diff}^{k}(M)$, the
endomorphism $A$ of $\mathcal{A}(M)$ is induced by the continuous linear
compositions

$\mathcal{A}^{m}(M)\xrightarrow{E_{m}}\dot{\mathcal{A}}^{(s)}(M)\xrightarrow{A}\dot{\mathcal{A}}^{(s-k)}(M)\xrightarrow{R}\mathcal{A}^{(s-k)}(M)\subset\mathcal{A}^{m^{\prime}-k}(M)\;,$

where $m^{\prime}=s-n/2-1$ for $m$ and $s$ satisfying the conditions of
Proposition 6.26.

###  6.16. $L^{2}$ half-b-densities

By (6.1),

$\displaystyle C^{\infty}(M;\Omega^{-\frac{1}{2}}\otimes{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})$ $\displaystyle\equiv
C^{\infty}(M;\Omega^{-\frac{1}{2}})\otimes_{C^{\infty}(M)}C^{\infty}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})$ $\displaystyle\equiv
C^{\infty}(M;\Omega^{-\frac{1}{2}})\otimes_{C^{\infty}(M)}x^{-\frac{1}{2}}C^{\infty}(M;\Omega^{\frac{1}{2}})$
$\displaystyle\equiv
x^{-\frac{1}{2}}C^{\infty}(M;\Omega^{-\frac{1}{2}}\otimes\Omega^{\frac{1}{2}})\equiv
x^{-\frac{1}{2}}C^{\infty}(M)\;.$

So

$\displaystyle L^{2}(M;{}^{\text{\rm b}}\Omega^{\frac{1}{2}})$
$\displaystyle\equiv
L^{2}(M;\Omega^{\frac{1}{2}})\otimes_{C^{\infty}(M)}C^{\infty}(M;\Omega^{-\frac{1}{2}}\otimes{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})$ (6.49) $\displaystyle\equiv
L^{2}(M;\Omega^{\frac{1}{2}})\otimes_{C^{\infty}(M)}x^{-\frac{1}{2}}C^{\infty}(M)\equiv
x^{-\frac{1}{2}}L^{2}(M;\Omega^{\frac{1}{2}})\;.$

This is an identity of Hilbert spaces, using the weighted $L^{2}$ space
structure of $x^{-1/2}L^{2}(M;\Omega^{1/2})$ (Section 2.11) and the Hilbert
space structure on $L^{2}(M;{}^{\text{\rm b}}\Omega^{1/2})$ induced by the
canonical identity

(6.50) $L^{2}(\mathring{M};\Omega^{\frac{1}{2}})\equiv L^{2}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})\;.$

###  6.17. $L^{\infty}$ half-b-densities

Like in (6.49), we get

(6.51) $L^{\infty}(M;{}^{\text{\rm b}}\Omega^{\frac{1}{2}})\equiv
x^{-\frac{1}{2}}L^{\infty}(M;\Omega^{\frac{1}{2}})\;,$

as LCSs endowed with a family of equivalent Banach space norms.

Equip $M$ with a b-metric $g$ (Section 6.1), and endow $\mathring{M}$ with the
restriction of $g$, also denoted by $g$. With the corresponding
Euclidean/Hermitean structures on $\Omega^{1/2}\mathring{M}$ and
${}^{\text{\rm b}}\Omega^{1/2}M$, we have the identity of Banach spaces

(6.52) $L^{\infty}(\mathring{M};\Omega^{\frac{1}{2}})\equiv
L^{\infty}(M;{}^{\text{\rm b}}\Omega^{\frac{1}{2}})\;.$

###  6.18. b-Sobolev spaces

For $m\in\mathbb{N}_{0}$, the _b-Sobolev spaces_ of _order_ $\pm m$ are
defined by the following analogs of (2.15), (2.20) and (2.21):

$\displaystyle H_{\text{\rm b}}^{m}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})=\\{\,u\in L^{2}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})\mid\operatorname{Diff}_{\text{\rm
b}}^{m}(M;{}^{\text{\rm b}}\Omega^{\frac{1}{2}})\,u\subset
L^{2}(M;{}^{\text{\rm b}}\Omega^{\frac{1}{2}})\,\\}\;,$
$\displaystyle\operatorname{Diff}_{\text{\rm b}}^{m}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})\,L^{2}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})=H_{\text{\rm b}}^{-m}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})=H_{\text{\rm b}}^{m}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})^{\prime}\;.$

These are $C^{\infty}(M)$-modules and Hilbertian spaces with no canonical
choice of a scalar product in general; we can use any finite set of
$C^{\infty}(M)$-generators of $\operatorname{Diff}_{\text{\rm
b}}^{m}(M;{}^{\text{\rm b}}\Omega^{1/2})$ to define a scalar product on
$H_{\text{\rm b}}^{\pm m}(M;{}^{\text{\rm b}}\Omega^{1/2})$. The intersections
and unions of the b-Sobolev spaces are denoted by $H_{\text{\rm
b}}^{\pm\infty}(M;{}^{\text{\rm b}}\Omega^{1/2})$. In particular,
$H_{\text{\rm b}}^{\infty}(M;{}^{\text{\rm
b}}\Omega^{1/2})=\mathcal{A}^{(0)}(M;{}^{\text{\rm b}}\Omega^{1/2})$.

###  6.19. Weighted b-Sobolev spaces

We will also use the _weighted b-Sobolev space_ $x^{a}H_{\text{\rm
b}}^{m}(M;{}^{\text{\rm b}}\Omega^{1/2})$ ($a\in\mathbb{R}$), which is another
Hilbertian space with no canonical choice of a scalar product; given a scalar
product on $H_{\text{\rm b}}^{m}(M;{}^{\text{\rm b}}\Omega^{1/2})$ with norm
$\|{\cdot}\|_{H_{\text{\rm b}}^{m}}$, we get a scalar product on
$x^{a}H_{\text{\rm b}}^{m}(M;{}^{\text{\rm b}}\Omega^{1/2})$ with norm
$\|{\cdot}\|_{x^{a}H_{\text{\rm b}}^{m}}$, like in Section 2.11. Observe that

(6.53) $\bigcap_{a,m}x^{a}H^{m}_{\text{\rm b}}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})=\dot{C}^{\infty}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})\;.$

###  6.20. Action of $\operatorname{Diff}_{\text{\rm b}}^{m}(M)$ on weighted
b-Sobolev spaces

Like in (2.12),

(6.54) $\operatorname{Diff}_{\text{\rm b}}^{m}(M;{}^{\text{\rm
b}}\Omega^{\frac{1}{2}})\equiv\operatorname{Diff}_{\text{\rm
b}}^{m}(M)\equiv\operatorname{Diff}_{\text{\rm
b}}^{m}(M;\Omega^{\frac{1}{2}})\;.$

By (6.26), for all $k\in\mathbb{N}_{0}$, $m\in\mathbb{Z}$ and
$a\in\mathbb{R}$, any $A\in\operatorname{Diff}_{\text{\rm
b}}^{k}(M;{}^{\text{\rm b}}\Omega^{1/2})$ defines a continuous linear map [24,
Lemma 5.14]

$A:x^{a}H^{m}_{\text{\rm b}}(M;{}^{\text{\rm b}}\Omega^{\frac{1}{2}})\to