Topology of the space of conormal distributions

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 $I'=I'(M,L)=I(M,L;\Omega )'$ of the space of conormal densities. It is shown that I is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and $I'$ 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 $K'$. We construct a locally convex Hausdoff space J and a continuous linear map $I\rightarrow J$ such that the sequence $0\rightarrow K\rightarrow I\rightarrow J\rightarrow 0$ as well as the transpose sequence $0\rightarrow J'\rightarrow I'\rightarrow K'\rightarrow 0$ are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that $I\cap I'=C^\infty (M)$ in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow $\phi =\{\phi ^t\}$ on a compact foliated manifold $(M,\mathcal {F})$. It describes a Lefschetz distribution $L_{\textrm{dis}}(\phi )$ defined by the induced action $\phi ^*=\{\phi ^{t\,*}\}$ on the reduced cohomologies ${\bar{H}}^\bullet I(\mathcal {F})$ and ${\bar{H}}^\bullet I'(\mathcal {F})$ of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $\phi$.

1 Introduction

Given a smooth manifold M and a closed regular submanifold $L\subset M$, the corresponding space $I=I(M,L)$ of conormal distributions was considered in [13, 17, 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. This space was also used 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 ${\text {Diff}}(M,L)\subset {\text {Diff}}(M)$ 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 ${\text {Diff}}(M,L)u\subset H^s(M)$, endowed with the projective topology given by the maps $P: I^{(s)}\rightarrow H^s(M)$ ($P\in {\text {Diff}}(M,L)$).

By definition, $I=\bigcup _sI^{(s)}$ with the corresponding locally convex inductive topology, which is continuously contained in $C^{-\infty }(M)$. There is another expression $I=\bigcup _mI^m$, 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 $I'(M,L)=I(M,L;\Omega )'$, simply denoted by $I'$, is also continuously contained in $C^{-\infty }(M)$. We prove that $I'$ is a complete Montel space and $I'=\varprojlim I^{\prime \,(s)}$, where $I^{\prime \,(s)}=I^{\prime \,(s)}(M,L)=I^{(-s)}(M,L;\Omega )'$ is bornological and barreled (Corollaries 5.1 to 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 $\varvec{M}$ and a projection $\varvec{\pi }:\varvec{M}\rightarrow M$. 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 $\mathcal {A}(\varvec{M})$ (respectively, ${\dot{\mathcal {A}}}(\varvec{M})$) be the locally convex space of extendable (respectively, supported) conormal distributions at the boundary. Then, via the push-down map $\varvec{\pi }_*$, the image of ${\dot{\mathcal {A}}}(\varvec{M})$ is I, and $\mathcal {A}(\varvec{M})$ 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 $I'$, consider also the strong dual spaces $K'(M,L)=K(M,L;\Omega )'$ and $J'(M,L)=J(M,L;\Omega )'$, simply denoted by $K'$ and $J'$. It is proved that K is a limit subspace (Corollary 7.19), the spaces K, J, $K'$ and $J'$ satisfy the properties stated for I and $I'$ (Corollaries 7.21, 7.23 and 8.1, to 8.3), and there are short exact sequences, $0\rightarrow K\rightarrow I\rightarrow J\rightarrow 0$ and its transpose $0\leftarrow K'\leftarrow I'\leftarrow J'\leftarrow 0$, in the category of continuous linear maps between locally convex spaces (Propositions 7.29 and  8.8). These sequences are relevant because J, K, $J'$ and $K'$ have better descriptions than I and $I'$ (Corollaries 6.40 and  6.49 and Proposition 7.26). Finally, it is shown that $I\cap I'=C^\infty (M)$ (Theorem 8.11), extending a result of [15, 25] for the boundary case. Most of these properties are first established in the boundary case (Sect. 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 (Sects. 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 ${\text {Diff}}(M)\equiv K(M^2,\Delta )$, where $\Delta$ is the diagonal of $M^2$. Thus $\Psi (M)$ and ${\text {Diff}}(M)$ become examples of locally convex spaces satisfying the above properties.

The wave front set of any $u\in I(M,L)$ satisfies ${\text {WF}}(u)\subset N^*L{\setminus }0_L$ (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 “Lefschetz distribution” $L_{\textrm{dis}}(\phi )$ 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 $L_{\textrm{dis}}(\phi )$ 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 we 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 $I'(\mathcal {F})$ 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 ${\bar{H}}^\bullet I'(\mathcal {F})$, with actions $\phi ^*$. The spaces $K(\mathcal {F})$, $J(\mathcal {F})$, $K'(\mathcal {F})$ and $J'(\mathcal {F})$ are similarly defined, obtaining short exact sequences, $0\rightarrow {\bar{H}}^\bullet K(\mathcal {F})\rightarrow {\bar{H}}^\bullet I(\mathcal {F})\rightarrow {\bar{H}}^\bullet J(\mathcal {F})\rightarrow 0$ and $0\leftarrow {\bar{H}}^\bullet K'(\mathcal {F})\leftarrow {\bar{H}}^\bullet I'(\mathcal {F})\leftarrow {\bar{H}}^\bullet J'(\mathcal {F})\leftarrow 0$. In this way, the definition of $L_{\textrm{dis}}(\phi )$ for both ${\bar{H}}^\bullet I(\mathcal {F})$ and ${\bar{H}}^\bullet I'(\mathcal {F})$ together can be reduced to the cases of ${\bar{H}}^\bullet K(\mathcal {F})$, ${\bar{H}}^\bullet J(\mathcal {F})$, ${\bar{H}}^\bullet K'(\mathcal {F})$ and ${\bar{H}}^\bullet J'(\mathcal {F})$. This can be done by using the descriptions of $K(\mathcal {F})$, $J(\mathcal {F})$, $K'(\mathcal {F})$ and $J'(\mathcal {F})$, and some additional ingredients. Using these ideas, we define in [4] the Lefschetz distribution $L_{\textrm{dis}}(\phi )$, 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. We hope that this method will provide useful tools in future developments of Deninger’s project.

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, 28, 31], 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. For any LCS X, its (continuous) dual $X'$ is always endowed with the strong topology; i.e., we write $X'=X'_\beta$ 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 $k'\ge k$ such that, for all $k''\ge k'$, the topologies of $X_{k'}$ and $X_{k''}$ coincide on some 0-neighborhood of $X_k$. In this case, $X:=\bigcup _kX_k$ 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 $Y\equiv \bigcup _kY_k$, 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 $X_r\subset X_{r'}$ for $r<r'$ in $\mathbb {R}$ because $\bigcap _rX_r=\bigcap _kX_{r_k}$ and $\bigcup _rX_r=\bigcup _kX_{s_k}$ 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 $0\rightarrow X \rightarrow Y \rightarrow Z\rightarrow 0$ 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

$$\begin{aligned} \Vert u\Vert _{K,C^k}=\sup _{x\in K,\ |\alpha |\le k}|\partial ^\alpha u(x)|\;, \end{aligned}$$

(2.1)

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 $C^\infty _{\textrm{loc}}(U)$ is used for this space, and $C^\infty (U)$ is used for the uniform space denoted by $C^\infty _{\textrm{ub}}(U)$ in this paper.) For any $S\subset U$, 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,

$$\begin{aligned} C^\infty _{\textrm{c}}(U)=\bigcup _KC^\infty _K(U)\;, \end{aligned}$$

(2.2)

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

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

$$\begin{aligned} C^\infty _{{\cdot }/\textrm{c}}(U,\mathbb {K}^l)\equiv C^\infty _{{\cdot }/\textrm{c}}(U)\otimes \mathbb {K}^l\;. \end{aligned}$$

(2.3)

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

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 $\Omega ^aE$ ($a\in \mathbb {R}$) denote the line bundle of a-densities of E, let $\Omega E=\Omega ^1E$, 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 $o(M)=o(TM)$, $\Omega ^aM=\Omega ^aTM$ 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 $\Vert {\cdot }\Vert _{K,C^k}$ 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 $C^\infty (M;\Omega ^a)=C^\infty (M;\Omega ^aM)$. 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 Sect. 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 $C^\infty _{\textrm{c}}(M;E)$ of compactly supported smooth sections is defined like in (2.2), using compact subsets $K\subset M$. There is a continuous inclusion $C^\infty _{\textrm{c}}(M;E)\subset C^\infty (M;E)$. If M is a fiber bundle, the LCHS $C^\infty _{\textrm{cv}}(M;E)$ 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

$$\begin{aligned} C^{-\infty }_{{\cdot }/\textrm{c}}(M;E)=C^\infty _{\textrm{c}/{\cdot }}(M;E^*\otimes \Omega )'\;. \end{aligned}$$

(2.4)

(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 continuous dense inclusion $C^\infty _{{\cdot }/\textrm{c}}(M;E)\subset C^{-\infty }_{{\cdot }/\textrm{c}}(M;E)$. If $U\subset M$ is open, the extension by zero defines a TVS-embedding $C^{\pm \infty }_{\textrm{c}}(U;E)\subset C^{\pm \infty }_{\textrm{c}}(M;E)$.

The above spaces of distributional sections can be also described in terms of the corresponding spaces of distributions as the algebraic tensor product as $C^\infty (M)$-modules

$$\begin{aligned} C^{-\infty }_{{\cdot }/\textrm{c}}(M;E)\equiv C^{-\infty }_{{\cdot }/\textrm{c}}(M)\otimes _{C^\infty (M)}C^\infty (M;E)\;. \end{aligned}$$

(2.5)

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

$$\begin{aligned} C^{-\infty }_{{\cdot }/\textrm{c}}(M;F)&\equiv C^{-\infty }_{{\cdot }/\textrm{c}}(M)\otimes \mathbb {K}^{l'} \equiv C^{-\infty }_{{\cdot }/\textrm{c}}(M)\otimes _{C^\infty (M)}C^\infty (M)\otimes \mathbb {K}^{l'}\\&\equiv C^{-\infty }_{{\cdot }/\textrm{c}}(M)\otimes _{C^\infty (M)}C^\infty (M;F)\;, \end{aligned}$$

and the spaces of  (2.5) clearly correspond by these identities. Expressions like (2.5) hold for most of the LCSs of distributional sections we will consider, which are also $C^\infty (M)$-modules. Thus, from now on, we will mostly define and study those 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 $\Vert {\cdot }\Vert _{K,C^k}$ given like in (2.1), the LF-space $C^k_{\textrm{c}}(M)$ of $C^k$ functions with compact support, defined like in (2.2), and the space $C^{\prime \,-k}_{{\cdot }/\textrm{c}}(M)$ of distributions of order k with arbitrary/compact support, defined like in (2.4). (A prime is added to this notation to distinguish $C^{\prime \,0}_{{\cdot }/\textrm{c}}(M)$ from $C^0_{{\cdot }/\textrm{c}}(M)$.) There are continuous dense inclusions

$$\begin{aligned} C^{k'}_{{\cdot }/\textrm{c}}(M)\subset C^k_{{\cdot }/\textrm{c}}(M)\;,\quad C^{\prime \,-k'}_{\textrm{c}/{\cdot }}(M)\supset C^{\prime \,-k}_{\textrm{c}/{\cdot }}(M)\quad (k<k')\;, \end{aligned}$$

(2.6)

with [28, Exercise 12.108]

$$\begin{aligned} \bigcap _kC^k_{{\cdot }/\textrm{c}}(M)=C^{\infty }_{{\cdot }/\textrm{c}}(M)\;,\quad \bigcup _kC^{\prime \,-k}_{\textrm{c}}(M)=C^{-\infty }_{\textrm{c}}(M)\;. \end{aligned}$$

(2.7)

The space $\bigcup _kC^{\prime \,-k}(M)$ 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], $C^{\infty }_{\textrm{c}}(M)$ and $C^{k}_{\textrm{c}}(M)$ are complete and Hausdorff [16, Examples 2.12.6 and 2.12.8]. $C^{\infty }_{{\cdot }/\textrm{c}}(M)$ and $C^{k}_{{\cdot }/\textrm{c}}(M)$ 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)]. $C^{\pm \infty }_{{\cdot }/\textrm{c}}(M)$ 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]. $C^{\infty }_{{\cdot }/\textrm{c}}(M)$ is a Schwartz space [16, Examples 3.15.2 and 3.15.3], and therefore $C^{-\infty }_{{\cdot }/\textrm{c}}(M)$ is ultrabornological [16, Exercise 3.15.9 (c)]. $C^\infty (M)$ is distinguished [16, Examples 3.16.1]. $C^{\pm \infty }_{{\cdot }/\textrm{c}}(M)$ 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 $A:C^\infty _{\textrm{c}}(M;E)\rightarrow C^\infty (M;F)$ be a continuous linear map. Recall that the transpose of A is the continuous linear map

$$\begin{aligned}{} & {} A^t:C^{-\infty }_{\textrm{c}}(M;F^*\otimes \Omega )\rightarrow C^{-\infty }(M;E^*\otimes \Omega ),\\{} & {} \quad \langle A^tv,u\rangle =\langle v,Au\rangle ,\quad u\in C^\infty _{\textrm{c}}(M;E),\quad v\in C^{-\infty }_{\textrm{c}}(M;F^*\otimes \Omega ). \end{aligned}$$

For instance, the transpose of $C^\infty _{\textrm{c}}(M;E^*\otimes \Omega )\subset C^\infty (M;E^*\otimes \Omega )$ is a continuous dense injection $C^{-\infty }_{\textrm{c}}(M;E)\subset C^{-\infty }(M;E)$. If $A^t$ restricts to a continuous linear map $C^\infty _{\textrm{c}}(M;F^*\otimes \Omega )\rightarrow C^\infty (M;E^*\otimes \Omega )$, then $A^{tt}:C^{-\infty }_{\textrm{c}}(M;E)\rightarrow C^{-\infty }(M;F)$ 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 $C^\infty _{\textrm{c}}(U;E^*\otimes \Omega )\subset C^\infty _{\textrm{c}}(M;E^*\otimes \Omega )$ is the restriction map $C^{-\infty }(M;E)\rightarrow C^{-\infty }(U,E)$, $u\mapsto u|_U$, and the transpose of the restriction map $C^\infty (M;E^*\otimes \Omega )\rightarrow C^\infty (U,E^*\otimes \Omega )$ is the extension by zero $C^{-\infty }_{\textrm{c}}(U;E)\subset C^{-\infty }_{\textrm{c}}(M;E)$. 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 $\phi :M'\rightarrow M$ induces the continuous linear pull-back map

$$\begin{aligned} \phi ^*:C^\infty (M;E)\rightarrow C^\infty (M';\phi ^*E)\;. \end{aligned}$$

(2.8)

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

$$\begin{aligned} \phi _*:C^\infty _{\textrm{c}}(M';\phi ^*E\otimes \Omega _{\textrm{fiber}})\rightarrow C^\infty _{\textrm{c}}(M;E)\;, \end{aligned}$$

(2.9)

where $\Omega _{\textrm{fiber}}=\Omega _{\textrm{fiber}}M'=\Omega \mathcal {V}$ for the vertical subbundle $\mathcal {V}=\ker \phi _*\subset TM'$. Since $\phi ^*\Omega M\equiv \Omega (TM/\mathcal {V})\equiv \Omega ^{-1}_{\textrm{fiber}}\otimes \Omega M'$, the transposes of the versions of (2.8) and (2.9) with $E^*\otimes \Omega M$ are continuous extensions of (2.9) and (2.8) [14, Theorem 6.1.2],

$$\begin{aligned}{} & {} \phi _*:C^{-\infty }_{\textrm{c}}(M';\phi ^*E\otimes \Omega _{\textrm{fiber}})\rightarrow C^{-\infty }_{\textrm{c}}(M;E), \end{aligned}$$

(2.10)

$$\begin{aligned}{} & {} \phi ^*:C^{-\infty }(M;E)\rightarrow C^{-\infty }(M';\phi ^*E), \end{aligned}$$

(2.11)

also called push-forward and pull-back maps. The term integration along the fibers is also used for $\phi _*$.

If $\phi :M'\rightarrow M$ is a proper local diffeomorphism, then we can omit $\Omega _{\textrm{fiber}}$ and the compact support condition in (2.9) and (2.10), and therefore the compositions $\phi _*\phi ^*$ and $\phi ^*\phi _*$ are defined on smooth/distributional sections.

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

$$\begin{aligned} C^{\pm \infty }_{{\cdot }/\textrm{c}}(M';\phi ^*E)=C^{\pm \infty }_{{\cdot }/\textrm{c}}(M')\otimes _{C^\infty (M)}C^\infty (M;E)\;. \end{aligned}$$

(2.12)

Using (2.5) and (2.12), we can describe (2.8)–(2.11) 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 ${\text {Diff}}(M)$ be the filtered algebra and $C^\infty (M)$-module of differential operators, filtered by the order. Every ${\text {Diff}}^m(M)$ ($m\in \mathbb {N}_0$) 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 $C^\infty _{{\cdot }/\textrm{c}}(M)$. In particular, ${\text {Diff}}^0(M)\equiv C^\infty (M)$.

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 ${\text {Diff}}^m(M;E,F)$ (${\text {Diff}}^m(M;E)$ being obtained if $E=F$); here, redundant notation is simplified like in the case of $C^{\pm \infty }(M;E)$ (Sect. 2.4). If E is a line bundle, then

$$\begin{aligned} {\text {Diff}}^m(M;E)&\equiv {\text {Diff}}^m(M)\otimes _{C^\infty (M)}C^\infty (M;E\otimes E^*)\nonumber \\&\equiv {\text {Diff}}^m(M)\otimes _{C^\infty (M)}C^\infty (M)\equiv {\text {Diff}}^m(M)\;. \end{aligned}$$

(2.13)

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

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 $C^\infty _{\textrm{c}}(M;\Omega ^{1/2})$ with the scalar product $\langle u,v\rangle =\int _Mu{\bar{v}}$. The induced norm is denoted by $\Vert {\cdot }\Vert$.

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 $\Vert {\cdot }\Vert$ 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 $\Vert {\cdot }\Vert$ 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 the strict LF-space $L^2_{\textrm{c}}(M;E)$ is defined like in (2.2). On the other hand, let

$$\begin{aligned} L^2_{\textrm{loc}}(M;E)=\{\,u\in C^{-\infty }(M;E)\mid C^\infty _{\textrm{c}}(M)\,u\subset L^2_{\textrm{c}}(M;E)\,\}\;, \end{aligned}$$

(2.14)

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

Any $A\in {\text {Diff}}^m(M;E)$ 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 $A^*\in {\text {Diff}}^m(M;E)$ (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 $\Vert u\Vert _{L^\infty }={\text {ess\,sup}}_{x\in M}|u(x)|$. There is a continuous injection $L^\infty (M;E)\subset L^2_{\textrm{loc}}(M;E)$. If M is compact, then the equivalence class of $\Vert {\cdot }\Vert _{L^\infty }$ 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 $\mathcal {S}(\mathbb {R}^n)$, which extends to a TVS-automorphism of the space $\mathcal {S}(\mathbb {R}^n)'$ of tempered distributions [14, Section 7.1]. In turn, for every $s\in \mathbb {R}$, this automorphism of $\mathcal {S}(\mathbb {R}^n)'$ restricts to unitary isomorphism

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

(2.15)

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 $H^s_{\mathrm{c/loc}}(U)$ are defined like in (2.4) and (2.14), using the spaces $H^s_K(\mathbb {R}^n)$ for compact subsets $K\subset U$. They are continuously included in $C^{-\infty }_{\textrm{c}/{\cdot }}(U)$.

For a manifold M, the definition of the LCHSs $H^s_{\mathrm{c/loc}}(M)$ 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 $C^{-\infty }_{\textrm{c}/{\cdot }}(M)$.

2.10.2 Case of compact manifolds

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

$$\begin{aligned} H^{-s}(M)=H^s(M;\Omega )'\;, \end{aligned}$$

(2.16)

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

$$\begin{aligned} H^s(M)\subset H^{s'}(M), \end{aligned}$$

(2.17)

for $s'<s$, and

$$\begin{aligned}{} & {} H^s(M)\subset C^k(M)\subset H^k(M), \end{aligned}$$

(2.18)

$$\begin{aligned}{} & {} H^{-s}(M)\supset C^{\prime \,-k}(M)\supset H^{-k}(M), \end{aligned}$$

(2.19)

for $s>k+n/2$. The first inclusion of (2.18) is the Sobolev embedding theorem, and (2.19) is the transpose of the version of (2.18) with $\Omega M$. Moreover the inclusions (2.17) are compact (Rellich theorem). So the spaces $H^s(M)$ form a compact spectrum with

$$\begin{aligned} C^\infty (M)=\bigcap _sH^s(M)\;\quad C^{-\infty }(M)=\bigcup _sH^s(M)\;. \end{aligned}$$

(2.20)

Any $A\in {\text {Diff}}^m(M;E)$ defines a bounded operator $A:H^{s+m}(M;E)\rightarrow H^s(M;E)$. 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 ${\bar{A}}^t\in {\text {Diff}}^m(M;{\bar{E}}^*\otimes \Omega )$ via the identity $H^s(M;E)\equiv H^s(M;{\bar{E}})'=H^{-s}(M;{\bar{E}}^*\otimes \Omega )$, where the bar stands for the complex conjugate. In the case $s=0$, the adjoint of A is induced by the formal adjoint $A^*\in {\text {Diff}}^m(M;E)$.

By the elliptic estimate, a scalar product on $H^s(M)$ can be defined by $\langle u,v\rangle _s=\langle (1+P)^su,v\rangle$, for any choice of a nonnegative symmetric elliptic $P\in {\text {Diff}}^2(M)$, where $\langle {\cdot },{\cdot }\rangle$ is defined like in Sect. 2.8 and $(1+P)^s$ is given by the spectral theorem for all $s\in \mathbb {R}$. The corresponding norm $\Vert {\cdot }\Vert _s$ 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 $\omega \in C^\infty (M;\Omega )$ (Sect. 2.8), besides a nonnegative symmetric elliptic $P\in {\text {Diff}}^2(M;E)$. If $E=\Omega ^{1/2}M$, then $\langle {\cdot },{\cdot }\rangle _s$ can be defined independently of $({\cdot },{\cdot })$ and $\omega$ (Sect. 2.8).

If $s\in \mathbb {N}_0$, we can also describe

$$\begin{aligned} H^s(M)&=\{\,u\in C^{-\infty }(M)\mid {\text {Diff}}^s(M)\,u\subset L^2(M)\,\}\;, \end{aligned}$$

(2.21)

$$\begin{aligned} H^{-s}(M)&={\text {Diff}}^s(M)\,L^2(M)\;, \end{aligned}$$

(2.22)

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

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 $C^\infty _{\textrm{c}}(M;E)$ 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 $\Vert {\cdot }\Vert _s$ depends on the choices involved. For instance, in (2.15), $H^s(\mathbb {R}^n)$ can be also described with the Laplacian of $\mathbb {R}^n$ and the standard density and Euclidean/Hermitian structure. The version of (2.16) with E can be used to define $H^{-s}(M;E)$. With this generality, the versions of (2.21), (2.22), the right-hand side inclusions of (2.18) and (2.19), and the inclusions “$\subset$” of (2.20) are wrong, but the versions of (2.17), the left-hand side continuous inclusions of (2.18) and (2.19), and the continuous inclusion “$\supset$” of (2.20) are true. Thus the intersection and union of (2.20) define new LCHSs $H^{\pm \infty }(M)$, which are continuously included in $C^{\pm \infty }(M)$. Any $A\in {\text {Diff}}^m(M;E)$ defines continuous linear maps $A:H^s_{\mathrm{c/loc}}(M;E)\rightarrow H^{s-m}_{\mathrm{c/loc}}(M;F)$.

2.11 Weighted spaces

2.11.1 Case of compact manifolds

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 Sect. 2.10. The corresponding norm is denoted by $\Vert {\cdot }\Vert _{hH^s}$. 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.

2.11.2 Extension to non-compact manifolds

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 $\langle u,v\rangle _s$ (Sect. 2.10). We also get the weighted spaces $hH^s_{\mathrm{c/loc}}(M;E)$, and the weighted Banach space $hL^\infty (M;E)$ with the norm $\Vert u\Vert _{hL^\infty }=\Vert h^{-1}u\Vert _{L^\infty }$. There is a continuous injection $hL^\infty (M;E)\subset hL^2_{\textrm{loc}}(M;E)$.

2.12 Bounded geometry

Concerning this topic, we follow [10, 30, 32, 33, 35]; 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 ${\text {inj}}_M\ge 0$ 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 ${\text {inj}}_M>0$. 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 ${\text {inj}}_M>0$ and $\sup |\nabla ^mR|<\infty$ 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 $y_p:V_p\rightarrow B$ 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 $y_p:V_p\rightarrow B$ 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 $|\partial ^\alpha (y_qy_p^{-1})|$ 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 $0<2r\le r_0$, 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 $a_{pq}:V_p\cap V_q\rightarrow {\text {GL}}(l,\mathbb {K})\subset \mathbb {K}^{l\times l}$, 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 $C_m\ge 0$ with $|\nabla ^{m'}u|\le C_m$ on M for all $m'\le m$. These functions form the uniform $C^m$ space $C_{\textrm{ub}}^m(M)$, which is a Banach space with the norm $\Vert {\cdot }\Vert _{C^m_{\textrm{ub}}}$ defined by the best constant $C_m$. Equivalently, we may take the norm $\Vert {\cdot }\Vert '_{C^m_{\textrm{ub}}}$ defined by the best constant $C'_m\ge 0$ such that $|\partial ^\alpha (uy_p^{-1})|\le C'_m$ on B for all $p\in M$ and $|\alpha |\le m$; 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 $C^\infty _{\textrm{ub}}(M)=\bigcap _mC_{\textrm{ub}}^m(M)$. This is a Fréchet space with the semi-norms $\Vert {\cdot }\Vert _{C^m_{\textrm{ub}}}$ or $\Vert {\cdot }\Vert '_{C^m_{\textrm{ub}}}$. 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 $C^\infty _{\textrm{ub}}(M)$.

For a Euclidean/Hermitian vector bundle E of bounded geometry over M, the uniform $C^m$ space $C_{\textrm{ub}}^m(M;E)$, of $C^m$-uniformly bounded sections, can be defined by introducing $\Vert {\cdot }\Vert '_{C^m_{\textrm{ub}}}$ 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 $C^\infty _{\textrm{ub}}(M;E)$ 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 $C^\infty _{\textrm{ub}}(B;\mathbb {C}^l)$. In particular, $\mathfrak {X}_{\textrm{ub}}(M):=C^\infty _{\textrm{ub}}(M;TM)$ is a $C^\infty _{\textrm{ub}}(M)$-submodule and Lie subalgebra of $\mathfrak {X}(M)$.

2.12.3 Differential operators of bounded geometry

Like in Sect. 2.7, by using $\mathfrak {X}_{\textrm{ub}}(M)$ and $C^\infty _{\textrm{ub}}(M)$ instead of $\mathfrak {X}(M)$ and $C^\infty (M)$, we get the filtered subalgebra and $C^\infty _{\textrm{ub}}(M)$-submodule ${\text {Diff}}_{\textrm{ub}}(M)\subset {\text {Diff}}(M)$ of differential operators of bounded geometry. Observe that

$$\begin{aligned} C^m_{\textrm{ub}}(M)=\{\,u\in C^m(M)\mid {\text {Diff}}_{\textrm{ub}}(M)\,u\subset L^\infty (M)\ \forall m'\le m\,\}\;. \end{aligned}$$

(2.23)

For vector bundles of bounded geometry E and F over M, by taking the $C^\infty _{\textrm{ub}}(M)$-tensor product of ${\text {Diff}}_{\textrm{ub}}(M)$ and $C^\infty _{\textrm{ub}}(M;F\otimes E^*)$, we obtain the filtered $C^\infty _{\textrm{ub}}(M)$-submodule ${\text {Diff}}_{\textrm{ub}}(M;E,F)\subset {\text {Diff}}(M;E,F)$ (or ${\text {Diff}}_{\textrm{ub}}(M;E)$ 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, ${\text {Diff}}_{\textrm{ub}}(M;E)$ is a filtered subalgebra of ${\text {Diff}}(M;E)$. Like in (2.13), if E is a line bundle of bounded geometry, then

$$\begin{aligned} {\text {Diff}}_{\textrm{ub}}^m(M;E)\equiv {\text {Diff}}_{\textrm{ub}}^m(M)\;. \end{aligned}$$

(2.24)

Every $A\in {\text {Diff}}_{\textrm{ub}}^m(M;E)$ defines continuous linear maps $A:C^{m+k}_{\textrm{ub}}(M;E)\rightarrow C^k_{\textrm{ub}}(M;E)$ ($k\in \mathbb {N}_0$), which induce a continuous endomorphism A of $C^\infty _{\textrm{ub}}(M;E)$. It is said that A is uniformly elliptic if there is some $C\ge 1$ such that, for all $p\in M$ and $\xi \in T^*_pM$, its leading symbol $\sigma _m(A)$ satisfies

$$\begin{aligned} C^{-1}|\xi |^m\le |\sigma _m(A)(p,\xi )|\le C|\xi |^m. \end{aligned}$$

This condition is independent of the choice of the Hermitian metric of bounded geometry on E. Any $A\in {\text {Diff}}^m_{\textrm{ub}}(M;E,F)$ satisfies the second inequality. The case where $A\in {\text {Diff}}_{\textrm{ub}}^m(M;E,F)$ 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 $P\in {\text {Diff}}^2_{\textrm{ub}}(M;E)$ can be used to define the Sobolev space $H^s(M;E)$ ($s\in \mathbb {R}$) with a scalar product $\langle {\cdot },{\cdot }\rangle _s$ ((Sect. 2.10). Any choice of P defines the same Hilbertian space $H^s(M;E)$, which is a $C^\infty _{\textrm{ub}}(M)$-module. In particular, $L^2(M;E)$ is the $C^\infty _{\textrm{ub}}(M)$-tensor product of $L^2(M;\Omega ^{1/2})$ and $C^\infty _{\textrm{ub}}(M;E\otimes \Omega ^{1/2})$, and $H^s(M;E)$ is the $C^\infty _{\textrm{ub}}(M)$-tensor product of $H^s(M)$ and $C^\infty _{\textrm{ub}}(M;E)$. 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

$$\begin{aligned} \langle u,v\rangle '_s=\sum _k\sum _{|\alpha |\le s}\int _Bf_k^2(x)\cdot \partial ^\alpha (uy_{p_k}^{-1})(x)\cdot \overline{\partial ^\alpha (vy_{p_k}^{-1})(x)}\,dx, \end{aligned}$$

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 $\langle {\cdot },{\cdot }\rangle '_s$ can be defined for $H^s(M;E)$ with the help of local trivializations defining the bounded geometry of E. Every $A\in {\text {Diff}}_{\textrm{ub}}^m(M;E)$ defines bounded operators $A:H^{m+s}(M;E)\rightarrow H^s(M;E)$ ($s\in \mathbb {R}$), 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 $h^{\pm 1}\in C^\infty _{\textrm{ub}}(M)$.

Proposition 2.5

(Roe [30, Proposition 2.8]) If $m'>m+n/2$, then $H^{m'}(M;E)\subset C_{\textrm{ub}}^m(M;E)$, continuously. Thus $H^\infty (M;E)\subset C^\infty _{\textrm{ub}}(M;E)$, 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 $dx=dx^1\wedge \cdots \wedge dx^n$ and $d\xi =d\xi ^1\wedge \cdots \wedge d\xi ^l$. 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 $\beta \in \mathbb {N}_0^l$,

$$\begin{aligned} \Vert a\Vert _{K,\alpha ,\beta ,m}:=\sup _{x\in K,\ \xi \in \mathbb {R}^l}\frac{|\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )|}{(1+|\xi |)^{m-|\beta |}}<\infty \;. \end{aligned}$$

(3.1)

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

$$\begin{aligned} S^m(U\times \mathbb {R}^l)\subset S^{m'}(U\times \mathbb {R}^l)\quad (m<m')\;, \end{aligned}$$

(3.2)

giving rise to the LCSs

$$\begin{aligned} S^\infty (U\times \mathbb {R}^l)=\bigcup _mS^m(U\times \mathbb {R}^l),\quad S^{-\infty }(U\times \mathbb {R}^l)=\bigcap _mS^m(U\times \mathbb {R}^l). \end{aligned}$$

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

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

When $U=\mathbb {R}^0=\{0\}$, the notation $S^m(\mathbb {R}^l)$, $S^{\pm \infty }(\mathbb {R}^l)$ and $S^{(m)}(\mathbb {R}^l)$ is used, and 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 Sect. 2.4).

Proposition 3.1

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

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

$$\begin{aligned} C^\infty _{\textrm{cv}}(U\times \mathbb {R}^l)\subset S^{-\infty }(U\times \mathbb {R}^l)\;,\quad S^\infty (U\times \mathbb {R}^l)\subset C^\infty (U\times \mathbb {R}^l)\;; \end{aligned}$$

(3.3)

in particular, $S^\infty (U\times \mathbb {R}^l)$ is Hausdorff. According to (2.1) and (3.3), we get continuous semi-norms $\Vert {\cdot }\Vert _{Q,C^k}$ 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

$$\begin{aligned} \Vert a\Vert _{Q,C^k}=\sup _{(x,\xi )\in Q,\ |\alpha |+|\beta |\le k}|\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )|\;. \end{aligned}$$

(3.4)

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

$$\begin{aligned} \Vert a\Vert '_{K,\alpha ,\beta ,m}=\sup _{x\in K}\limsup _{|\xi |\rightarrow \infty }\frac{\big |\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )\big |}{|\xi |^{m-|\beta |}}\;. \end{aligned}$$

(3.5)

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 $\phi :{\widehat{S}}^{\prime \,m}(U\times \mathbb {R}^l)\rightarrow C^\infty (U\times \mathbb {R}^l)$ 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 $\Vert \phi (a)\Vert '_{K,\alpha ,\beta ,m}=\Vert a\Vert '_{K,\alpha ,\beta ,m}<\infty$, there are $C,R>0$ so that, if $x\in K$ and $|\xi |\ge R$, then

$$\begin{aligned} \frac{|\partial _x^\alpha \partial _\xi ^\beta \phi (a)(x,\xi )|}{(1+|\xi |)^{m-|\beta |}}\le C. \end{aligned}$$

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 $\Vert \phi (a)\Vert _{Q,C^k}=\Vert a\Vert _{Q,C^k}<\infty$, there is some $C'>0$ such that $|\partial _x^\alpha \partial _\xi ^\beta \phi (a)(x,\xi )|<C'$ for $(x,\xi )\in Q$, yielding

$$\begin{aligned} \frac{|\partial _x^\alpha \partial _\xi ^\beta \phi (a)(x,\xi )|}{(1+|\xi |)^{m-|\beta |}}\le {\left\{ \begin{array}{ll} C' &{} \text {if}\, |\beta |\le m\\ C'(1+R)^{|\beta |-m} &{} \text {if}\, |\beta |\ge m. \end{array}\right. } \end{aligned}$$

This shows that $\Vert \phi (a)\Vert _{K,\alpha ,\beta ,m}<\infty$, 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 $S^m(U\times \mathbb {R}^l)\rightarrow S^{\prime \,m}(U\times \mathbb {R}^l)$ 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]. $\square$

Proposition 3.3

For $m,m'\in \mathbb {N}_0$, $\alpha \in \mathbb {N}_0^n$, $\beta \in \mathbb {N}_0^l$ and any compact $K\subset U$, if $m<m'$, then $\Vert {\cdot }\Vert '_{K,\alpha ,\beta ,m'}=0$ on $S^m(U\times \mathbb {R}^l)$.

Proof

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

$$\begin{aligned} \Vert a\Vert '_{K,\alpha ,\beta ,m'}=\Vert a\Vert '_{K,\alpha ,\beta ,m}\lim _{|\xi |\rightarrow \infty }|\xi |^{m-m'}=0. \end{aligned}$$

$\square$

Corollary 3.4

For $m<m'$, the topologies of $S^{m'}(U\times \mathbb {R}^l)$ 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 $m'>m$, since $O\cap S^{m'}(U\times \mathbb {R}^l)$ is open in $S^{m'}(U\times \mathbb {R}^l)$, it follows from the first assertion that there is some open $P\subset C^\infty (U\times \mathbb {R}^l)$ such that $O\cap S^m(U\times \mathbb {R}^l)=P\cap S^m(U\times \mathbb {R}^l)$. $\square$

Corollary 3.5

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

Proof

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

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

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 (Sect. 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. $\square$

Remark 3.7

Another proof of Corollary 3.5 could be given like in Proposition 6.10.

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 $|\xi ^1|\le m$, and $a_m(x,\xi )=(\xi ^1-m)^m$ if $|\xi ^1|\ge m+1$. Then $a_m\in S^m(U\times \mathbb {R}^l){\setminus } S^{m-1}(U\times \mathbb {R}^l)$ and $a_m\rightarrow 0$ in $C^\infty (U\times \mathbb {R}^l)$ as $m\uparrow \infty$. However $a_m\not \rightarrow 0$ 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 and 3.3 and Corollaries 3.4 to 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

$$\begin{aligned} C^\infty _{\textrm{cv}}(E;\pi ^*F)\subset S^{-\infty }(E;F),\quad S^{\infty }(E;F)\subset C^\infty (E;\pi ^*F), \end{aligned}$$

where $\pi :E\rightarrow M$ 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 $n'$ and dimension $n''$, 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 ${\text {Diff}}(M,L)\subset {\text {Diff}}(M)$ like in Sect. 2.7. We have

$$\begin{aligned} A\in {\text {Diff}}(M,L)\Rightarrow A^t\in {\text {Diff}}(M,L;\Omega )\;. \end{aligned}$$

(4.1)

By the conditions on L, every ${\text {Diff}}^m(M,L)$ ($m\in \mathbb {N}_0$) is locally finitely $C^\infty (M)$-generated, and therefore ${\text {Diff}}(M,L)$ is countably $C^\infty (M)$-generated. The surjective restriction map $\mathfrak {X}(M,L)\rightarrow \mathfrak {X}(L)$, $X\mapsto X|_L$, induces a surjective linear restriction map of filtered algebras and $C^\infty (M)$-modules,

$$\begin{aligned} {\text {Diff}}(M,L)\rightarrow {\text {Diff}}(L)\;,\quad A\mapsto A|_L\;. \end{aligned}$$

(4.2)

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

$$\begin{aligned} x= & {} (x^1,\dots ,x^n)\equiv (x',x''):U\rightarrow U'\times U'',\\ x'= & {} (x'^1,\dots ,x'^{n'}),\quad x''=(x''^{1},\dots ,x''^{n''}),\quad L_0:=L\cap U=\{x'=0\}. \end{aligned}$$

If L is of codimension one, then we will use the notation (x, y) instead of $(x',x'')$. For every $m\in \mathbb {N}_0$, ${\text {Diff}}^m(U,L_0)$ is $C^\infty (U)$-spanned by the operators $x^{\prime \alpha }\partial _{x'}^\beta \partial _{x''}^\gamma$ with $|\beta |+|\gamma |\le m$ and $|\alpha |=|\beta |$; we may use the generators $\partial _{x'}^\beta \partial _{x''}^\gamma x^{\prime \alpha }$ 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

$$\begin{aligned} I^{(s)}(M,L)=\{\,u\in C^{-\infty }(M)\mid {\text {Diff}}(M,L)\,u\subset H^s(M)\,\}\;, \end{aligned}$$

(4.3)

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

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 ${\text {Diff}}(M,L)$, the space $I^{(s)}(M,L)$ has the projective topology given by the maps $P_j:I^{(s)}(M,L)\rightarrow H^s(M)$. Let

$$\begin{aligned} I_k^{(s)}(M,L)=\{\,u\in C^{-\infty }(M)\mid P_ju\subset H^s(M),\ j=0,\dots ,k\,\}, \end{aligned}$$

with the projective topology given by the maps $P_j:I^{(s)}(M,L)\rightarrow H^s(M)$ ($j=0,\dots ,k$). 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

$$\begin{aligned} \langle u,v\rangle _{s,k}=\sum _{j=0}^k\langle P_ju,P_jv\rangle _s, \end{aligned}$$

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

We have continuous inclusions

$$\begin{aligned} I^{(s)}(M,L)\subset I^{(s')}(M,L)\quad (s'<s)\;, \end{aligned}$$

(4.4)

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

$$\begin{aligned} I(M,L)=\bigcup _sI^{(s)}(M,L),\quad I^{(\infty )}(M,L)=\bigcap _sI^{(s)}(M,L). \end{aligned}$$

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,

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

(4.5)

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 $I^{(s)}_{{\cdot }/\textrm{c}}(M,L)$ by using $C^{-\infty }_{{\cdot }/\textrm{c}}(M)$ and $H^s_{\mathrm{loc/c}}(M)$. Every $I^{(s)}(M,L)$ is a Fréchet space, as follows like in the proof of Proposition 4.1, using the Fréchet spaces $H^s_{\textrm{loc}}(M)$. We can describe $I^{(s)}_{\textrm{c}}(M,L)=\bigcup _KI^{(s)}_K(M,L)$ 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 $I_{\textrm{c}}(M,L)=\bigcup _sI_{\textrm{c}}^{(s)}(M,L)$ is an LF-space [28, Exercise 12.108]; moreover $I_{\textrm{c}}(M,L)=\bigcup _KI_K(M,L)$. We also have the LCHS $I^{(\infty )}_{\textrm{c}}(M,L)=\bigcap _sI_{\textrm{c}}^{(s)}(M,L)$. All of these spaces are modules over $C^\infty (M)$; $I_{\textrm{c}}(M,L)$ is a filtered module and $I^{(\infty )}_{\textrm{c}}(M,L)$ a submodule. The extension by zero defines a continuous inclusion $I_{\textrm{c}}(U,L\cap U)\subset I_{\textrm{c}}(M,L)$ 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 $\bigcup _sI^{(s)}(M,L)$, which consists of the conormal distributions with a Sobolev order. But now let (cf. [15, Definition 18.2.6])

$$\begin{aligned} I(M,L)=\{\,u\in C^{-\infty }(M)\mid C^\infty _{\textrm{c}}(M)\,u\subset I_{\textrm{c}}(M,L)\,\}\;, \end{aligned}$$

(4.6)

which is a LCS with the projective topology given by the (multiplication) maps $f_j:I(M,L)\rightarrow I_{\textrm{c}}(M,L)$, for a countable partition of unity $\{f_j\}\subset C^\infty _{\textrm{c}}(M)$. We have $I(M,L)=\bigcup _sI^{(s)}(M,L)$ 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, $I_{{\cdot }/\textrm{c}}(M,L)$ 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 Sect. 4.1 for a chart $(U,x=(x',x''))$ of M adapted to L. We use the identity $U''\times \mathbb {R}^{n'}\equiv N^*U''$, and the symbol spaces $S^m(U''\times \mathbb {R}^{n'})\equiv S^m(N^*U'')$ (Sect. 3). Define

$$\begin{aligned}{} & {} C^\infty _{\textrm{cv}}(N^*U'')\rightarrow C^\infty (U),\quad a\mapsto u, \end{aligned}$$

(4.7)

$$\begin{aligned}{} & {} C^\infty _{\textrm{c}}(U)\rightarrow C^\infty (N^*U''),\quad u\mapsto a, \end{aligned}$$

(4.8)

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

$$\begin{aligned} u(x)&=(2\pi )^{-n'}\int _{\mathbb {R}^{n'}}e^{i\langle x',\xi \rangle }a(x'',\xi )\,d\xi \;,\\ a(x'',\xi )&=\int _{\mathbb {R}^{n'}}e^{-i\langle x',\xi \rangle }u(x',x'')\,dx'\;. \end{aligned}$$

Proposition 4.3

([15, Theorem 18.2.8], [25, Proposition 6.1.1], [27, Lemma 9.33]) If $s<-{\bar{m}}-n'/2$, then (4.7) has a continuous extension $S^{{\bar{m}}}(N^*U'')\rightarrow I^{(s)}(U,L_0)$. If ${\bar{m}}>-s-n'/2$, then (4.8) induces a continuous linear map $I_{\textrm{c}}^{(s)}(U,L_0)\rightarrow S^{{\bar{m}}}(N^*U'')$.

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

$$\begin{aligned} a\,|d\xi |\in S^{{\bar{m}}}(N^*U'';\Omega N^*U'')\equiv S^{{\bar{m}}}(N^*L_0;\Omega N^*L_0). \end{aligned}$$

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 $f_ju\in I_{\textrm{c}}(U_j,L_j)$ for all j. According to Proposition 4.3, every $f_ju$ is given by some $a_j\in S^\infty (N^*L_j;\Omega N^*L_j)$. For

$$\begin{aligned} {\bar{m}}=m+n/4-n'/2\;, \end{aligned}$$

(4.9)

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:

$$\begin{aligned} I^m(M,L)\rightarrow C^\infty (M{\setminus } L)\oplus \prod _jS^{{\bar{m}}}(N^*L_j;\Omega N^*L_j)\;,\quad u\mapsto (hu,(a_j))\;. \end{aligned}$$

(4.10)

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

$$\begin{aligned} I^m(M,L)\subset I^{m'}(M,L)\quad (m<m')\;. \end{aligned}$$

(4.11)

The element $\sigma _m(u)\in S^{({\bar{m}})}(N^*L;\Omega N^*L)$ represented by $\sum _ja_j\in S^{{\bar{m}}}(N^*L;\Omega N^*L)$ is called the principal symbol of u. This defines the exact sequence

$$\begin{aligned} 0\rightarrow I^{m-1}(M,L)\hookrightarrow I^m(M,L) \xrightarrow {\sigma _m} S^{({\bar{m}})}(N^*L;\Omega N^*L)\rightarrow 0. \end{aligned}$$

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

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

(4.12)

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

$$\begin{aligned} I(M,L)=\bigcup _mI^m(M,L),\quad I^{(\infty )}(M,L)=I^{-\infty }(M,L):=\bigcap _mI^m(M,L). \end{aligned}$$

The spaces $I^m(M,L)$ form the symbol order filtration of I(M, L). The maps (4.10) induce a TVS-embedding

$$\begin{aligned} I(M,L)\rightarrow C^\infty (M{\setminus } L)\oplus \prod _jS^\infty (N^*L_j;\Omega N^*L_j)\;. \end{aligned}$$

(4.13)

Corollary 4.5

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

Proof

Use Corollary 3.4 and the TVS-embeddings (4.10). $\square$

Corollary 4.6

([25, Eq. (6.2.12)]) For $m<m'$, $C^\infty (M)$ is dense in $I^m(M,L)$ with the topology of $I^{m'}(M,L)$. 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 $f_ju$ by Proposition 4.3, like in (4.10). By Corollary 3.5, there is a sequence $b_{j,k}\in C^\infty _{\textrm{c}}(N^*L_j;\Omega N^*L_j)$ converging to $a_j$ in $S^{{\bar{m}}'}(N^*L_j;\Omega N^*L_j)$ (${\bar{m}}'=m'+n/4-n'/2$). Let $v_{j,k}$ be the sequence in $C^\infty (U)$ that corresponds to $b_{j,k}$ via (4.7); it converges to $f_ju$ in $I^{m'}(U,L)$ as $k\rightarrow \infty$ by Proposition 4.3. Take functions ${\tilde{f}}_j\in C^\infty _{\textrm{c}}(U_j)$ with ${\tilde{f}}_j=1$ on ${\text {supp}}f_j$. Then ${\tilde{f}}_jv_{j,k}\rightarrow f_ju$ in $I^{m'}_{\textrm{c}}(U,L)$, and therefore $hu+\sum _j{\tilde{f}}_jv_{j,k}\in C^\infty (M)$ is convergent to u in $I^{m'}(M,L)$. $\square$

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)]. $\square$

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 $I^m_{\textrm{c}}(M,L)=\bigcup _KI^m_K(M,L)$ like in (2.2). The space of conormal distributions with a symbol order is $\bigcup _mI^m(M,L)$, and let $I^{-\infty }_{{\cdot }/\textrm{c}}(M,L)=\bigcap _mI^m_{{\cdot }/\textrm{c}}(M,L)$. 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 $\bigcup _mI^m(M,L)=\bigcup _sI^{(s)}(M,L)$, $I_{\textrm{c}}(M,L)=\bigcup _mI^m_{\textrm{c}}(M,L)$ and $I^{(\infty )}_{{\cdot }/\textrm{c}}(M,L)=I^{-\infty }_{{\cdot }/\textrm{c}}(M,L)$. Corollary 4.7 has extensions for $\bigcup _mI^m(M,L)$ and $I_{{\cdot }/\textrm{c}}(M,L)$, 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 $\Omega NL\otimes \Omega L\equiv \Omega _LM$. The transpose of the restriction map $C^\infty _{\textrm{c}/{\cdot }}(M;E^*\otimes \Omega M)\rightarrow C^\infty _{\textrm{c}/{\cdot }}(L;E^*\otimes \Omega _LM)$ is a continuous inclusion

$$\begin{aligned}{} & {} C^{-\infty }_{{\cdot }/\textrm{c}}(L;E\otimes \Omega ^{-1}NL)\subset C^{-\infty }_{{\cdot }/\textrm{c}}(M;E), \nonumber \\{} & {} \quad u\mapsto \delta _L^u,\quad \langle \delta _L^u,v\rangle =\langle u,v|_L\rangle ,\quad v\in C^\infty _{\textrm{c}/{\cdot }}(M;E^*\otimes \Omega ). \end{aligned}$$

(4.14)

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

$$\begin{aligned} C^\infty _{{\cdot }/\textrm{c}}(L;E\otimes \Omega ^{-1}NL)\subset C^{-\infty }_{{\cdot }/\textrm{c}}(M;E)\;; \end{aligned}$$

(4.15)

in this case, we can write $\langle \delta _L^u,v\rangle =\int _Lu\,v|_L$. 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

$$\begin{aligned} C^\infty _{{\cdot }/\textrm{c}}(L;E\otimes \Omega ^{-1}NL)\subset H^s_{\mathrm{loc/c}}(M;E)\quad (s<-n'/2) \end{aligned}$$

with

$$\begin{aligned} C^\infty _{{\cdot }/\textrm{c}}(L;E\otimes \Omega ^{-1}NL)\cap H^{-n'/2}_{\mathrm{loc/c}}(M;E)=0. \end{aligned}$$

Proof

First, take $M=\mathbb {R}^n$, $L=\mathbb {R}^{n''}\times \{0\}\equiv \mathbb {R}^{n''}$ and $E=M\times \mathbb {C}$ (the trivial line bundle). Let $\delta _0$ be the Dirac mass at 0 in $\mathbb {R}^{n'}$. For any $\phi \in \mathcal {S}(\mathbb {R}^{n''})$, consider the tensor product distribution $\phi \otimes \delta _0\in \mathcal {S}(\mathbb {R}^n)'$ [14, Section 5.1]. Its Fourier transform is ${\hat{\phi }}\otimes {\hat{\delta }}_0={\hat{\phi }}\otimes 1$. If $\phi \ne 0$, then ${\hat{\phi }}\otimes 1\in L^2(\mathbb {R}^n,(1+|\xi |^2)^s\,d\xi )$ if and only if $1\in L^2(\mathbb {R}^{n'},(1+|\xi |^2)^s\,d\xi )$, which holds just when $s<-n'/2$. Moreover the map

$$\begin{aligned} \mathcal {S}(\mathbb {R}^{n''})\rightarrow L^2(\mathbb {R}^n,(1+|\xi |^2)^s\,d\xi ),\quad \phi \mapsto {\hat{\phi }}\otimes 1, \end{aligned}$$

is continuous if $s<-n'/2$. Thus (2.15) yields versions of the stated properties using $\mathcal {S}(\mathbb {R}^{n''})$ 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. $\square$

For instance, for any $p\in M$ and $u\in E_p\otimes \Omega _p^{-1}M$, we get $\delta _p^u\in H^s_{\textrm{c}}(M;E)$ 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

$$\begin{aligned} M\times C^\infty (M;E\otimes \Omega ^{-1})\rightarrow H^s_{\textrm{c}}(M;E),\quad (p,u)\mapsto \delta _p^{u(p)}. \end{aligned}$$

As a particular case, the Dirac mass at any $p\in \mathbb {R}^n$ is $\delta _p=\delta _p^{1\otimes |dx|^{-1}}\in H^s_{\textrm{c}}(\mathbb {R}^n)$.

4.5 Differential operators on conormal distributional sections

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

$$\begin{aligned} A:I^{(s)}_{{\cdot }/\textrm{c}}(M,L;E)\rightarrow I^{(s-k)}_{{\cdot }/\textrm{c}}(M,L;E)\;, \end{aligned}$$

(4.16)

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

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

$$\begin{aligned} A\delta _L^u=\delta _L^{A'u}\;,\quad A'=((A^t)|_L)^t\in {\text {Diff}}(L;E\otimes \Omega ^{-1}NL)\;, \end{aligned}$$

(4.17)

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

$$\begin{aligned} \langle A\delta _L^u,v\rangle =\langle \delta _L^u,A^tv\rangle =\langle u,(A^tv)|_L\rangle =\langle u,(A^t)|_L(v|_L)\rangle =\langle A'u,v|_L\rangle =\langle \delta _L^{A'u},v\rangle . \end{aligned}$$

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

$$\begin{aligned} C^\infty _{{\cdot }/\textrm{c}}(L;E\otimes \Omega ^{-1}NL)\subset I^{(s)}_{{\cdot }/\textrm{c}}(M,L;E)\quad (s<-n'/2)\;. \end{aligned}$$

(4.18)

4.6 Pull-back of conormal distributions

If a smooth map $\phi :M'\rightarrow M$ is transverse to a regular submanifold $L\subset M$, then $L':=\phi ^{-1}(L)\subset M'$ is a regular submanifold and (the trivial-line-bundle version of) (2.8) has continuous extensions

$$\begin{aligned} \phi ^*:I^m(M,L)\rightarrow I^{m+k/4}(M',L')\quad (m\in \mathbb {R})\;, \end{aligned}$$

(4.19)

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

$$\begin{aligned} \phi ^*:I(M,L)\rightarrow I(M',L')\;. \end{aligned}$$

(4.20)

If $\phi$ is a submersion, then (4.20) is a restriction of (2.11). If $\phi$ is a local diffeomorphism, then (4.20) is compatible with the Sobolev and symbol order filtrations in the sense that it restricts to continuous maps between the spaces defining those filtrations.

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 $\phi :M'\rightarrow M$ be a smooth submersion, and let $L\subset M$ and $L'\subset M'$ be regular submanifolds such that $\phi (L')\subset L$ and the restriction $\phi :L'\rightarrow L$ is also a smooth submersion. Then (2.9) has continuous extensions

$$\begin{aligned} \phi _*:I^m_{\textrm{c}}(M',L';\Omega _{\textrm{fiber}})\rightarrow I^{m+l/2-k/4}_{\textrm{c}}(M,L)\quad (m\in \mathbb {R})\;, \end{aligned}$$

(4.21)

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

$$\begin{aligned} \phi _*:I_{\textrm{c}}(M',L';\Omega _{\textrm{fiber}})\rightarrow I_{\textrm{c}}(M,L)\;, \end{aligned}$$

(4.22)

which is a restriction of (2.10). If $\phi$ is a local diffeomorphism, then (4.22) is compatible with the Sobolev and symbol order filtrations.

4.8 Pseudodifferential operators

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

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 (2.21) and (2.22) hold true using $\Psi ^s(M)$ instead of ${\text {Diff}}^s(M)$ for any $s\in \mathbb {R}$. In this way, $\Psi (M)$ also becomes a LCHS satisfying the properties indicated in Sects. 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 Sect. 2.7, satisfying the analog of (2.13). In this case, we have ${\bar{m}}=m$ in (4.9), and the symbol of any $A\in \Psi ^m(M;E,F)$ can be given by $\sigma _m(A)\equiv \sigma _m(K_A)$; this symbol is used to extend the concept of ellipticity to pseudodifferential operators (see e.g. [24]).

$\Psi (M;E)$ is preserved by taking transposes, and therefore any $A\in \Psi (M;E)$ defines a continuous endomorphism A of $C^{-\infty }(M;E)$ (Sect. 2.5), and ${\text {sing}}{\text {supp}}Au\subset {\text {sing}}{\text {supp}}u$ for all $u\in C^{-\infty }(M;E)$ (pseudolocality).

If $A\in \Psi ^m(M;E)$, it defines a bounded operator $A:H^{s+m}(M;E)\rightarrow H^s(M;E)$. This can be considered as a closable densely defined operator in $H^s(M;E)$, like in the case of differential operators (Sect. 2.10). In the case $s=0$, the adjoint of A is induced by the formal adjoint $A^*\in \Psi ^m(M;E)$. The symbol map on $\Psi (M;E)$ is multiplicative and compatible with transposition and taking adjoints.

The class of pseudodifferential operators is preserved by transposition. So any $A\in \Psi ^m(M)$ defines a continuous endomorphism A of $C^{-\infty }(M)$ (Sect. 2.5), and ${\text {sing}}{\text {supp}}Au\subset {\text {sing}}{\text {supp}}u$ for all $u\in C^{-\infty }_{\textrm{c}}(M)$ (pseudolocality).

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 $A:C^{\pm \infty }_{\textrm{c}}(M)\rightarrow C^{\pm \infty }(M)$ and $A:H^{s+m}_{\textrm{c}}(M)\rightarrow H^s_{\textrm{loc}}(M)$. Thus $\Psi (M)$ is not an algebra in this case. However, if $A\in \Psi ^m(M)$ is properly supported (both factor projections $M^2\rightarrow M$ restrict to proper maps ${\text {supp}}K_A\rightarrow M$), then it defines a continuous endomorphism A of $C^{-\infty }_{\textrm{c}}(M;E)$; 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 Sects. 4.2.2 and 4.3.3.

In the setting of bounded geometry (Sect. 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 Sects. 4.2 to 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 [25, Chapter 6]

$$\begin{aligned} I'(M,L)=I(M,L;\Omega )'\;. \end{aligned}$$

(5.1)

Corollary 5.1

$I'(M,L)$ is a complete Montel space.

Proof

Since $I(M,L;\Omega )$ is bornological (the version Corollary 4.2 with $\Omega M$), $I'(M,L)$ 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 $\Omega M$), $I'(M,L)$ is a Montel space [16, Proposition 3.9.9], [21, 6.27.2 (2)], [31, IV.5.9]. $\square$

Let also

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

(5.2)

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]. $\square$

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

$$\begin{aligned} I^{\prime \,(s)}(M,L)\rightarrow I^{\prime \,(s')}(M,L),\quad I^{\prime \,m}(M,L)\rightarrow I^{\prime \,m'}(M,L). \end{aligned}$$

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

$$\begin{aligned} C^{-\infty }(M)\supset I'(M,L)\supset C^\infty (M)\;, \end{aligned}$$

(5.3)

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

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

(5.4)

Thus

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

(5.5)

Corollary 5.3

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

Proof

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

Alternatively, the following argument can be used. $I'(M,L)$ 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]. $\square$

5.1.2 Extension to non-compact manifolds

If M is not supposed to be compact, we can similarly define the space $I'_K(M,L)$ of dual-conormal distributions supported in any compact $K\subset M$. Then define the LCHSs, $I'_{\textrm{c}}(M,L)=\bigcup _KI'_K(M,L)$ like in (2.2), and $I'(M,L)$ like in (4.6) using $I'_{\textrm{c}}(M,L)$ instead of $I_{\textrm{c}}(M,L)$. 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 $K_j:={\text {supp}}f_j$ is compact, the multiplication by the functions $f_j$ defines closed TVS-embeddings

$$\begin{aligned} I'(M,L)\rightarrow \prod _jI'_{K_j}(M,L)\;,\quad I'_{\textrm{c}}(M,L)\rightarrow \bigoplus _jI'_{K_j}(M,L)\;. \end{aligned}$$

(5.6)

By the extension of Corollary 4.7 for $I_{\textrm{c}}(M,L;\Omega )$, the obvious extension of Corollary 5.2 for every $I'_{K_j}(M,L)$, the indicated extension of (5.1) and the properties of (5.6), we get an extension of Corollary 5.1.

Similarly, we can define the spaces $I^{\prime \,(s)}_{{\cdot }/\textrm{c}}(M,L)$ and $I^{\prime \,m}_{{\cdot }/\textrm{c}}(M,L)$. They satisfy (5.2) interchanging the support condition, and also obvious versions of (5.3)–(5.5). Since $I_{\textrm{c}}(M,L)$ is an acyclic Montel space (Sect. 4.3.3), there is an extension of Corollary 5.3 for $I'(M,L)$.

5.2 Differential operators on dual-conormal distributional sections

For any $A\in {\text {Diff}}(M;E)$, consider $A^t\in {\text {Diff}}(M;E^*\otimes \Omega )$. The transpose of $A^t$ on $I_{\textrm{c}/{\cdot }}(M,L;E^*\otimes \Omega )$ is a continuous endomorphism A of $I'_{{\cdot }/\textrm{c}}(M,L;E)$, which is a restriction of the map A on $C^{-\infty }(M;E)$ (Sect. 2.7). By (4.16), if $A\in {\text {Diff}}^m(M;E)$, we get induced continuous linear maps

$$\begin{aligned} A:I^{\prime \,(s)}_{{\cdot }/\textrm{c}}(M,L;E)\rightarrow I^{\prime \,(s-m)}_{{\cdot }/\textrm{c}}(M,L;E)\;, \end{aligned}$$

(5.7)

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

5.3 Pull-back of dual-conormal distributions

If the conditions of Sect. 4.7 hold, transposing the versions of (4.21) and (4.22) with $\Omega M$ and $-m$, we get continuous linear pull-back maps

$$\begin{aligned}{} & {} \phi ^*:I^{\prime \,m}(M,L)\rightarrow I^{\prime \,m+l/2-k/4}(M',L')\quad (m\in \mathbb {R}), \end{aligned}$$

(5.8)

$$\begin{aligned}{} & {} \phi ^*:I'(M,L)\rightarrow I'(M',L'). \end{aligned}$$

(5.9)

The map (5.9) is an extension of (2.8), a restriction of (2.11) and the projective limit of the maps (5.8). If $\phi$ is a local diffeomorphism, then (5.9) is compatible with the Sobolev and symbol order filtrations.

5.4 Push-forward of dual-conormal distributions

With the notation of Sect. 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

$$\begin{aligned}{} & {} \phi _*:I^{\prime \,m}_{\textrm{c}}(M',L'\otimes \Omega _{\textrm{fiber}})\rightarrow I^{\prime \,m-k/4}_{\textrm{c}}(M,L)\quad (m\in \mathbb {R}), \end{aligned}$$

(5.10)

$$\begin{aligned}{} & {} \phi _*:I'_{\textrm{c}}(M',L';\Omega _{\textrm{fiber}})\rightarrow I'_{\textrm{c}}(M,L). \end{aligned}$$

(5.11)

The map (5.11) is an extension of (2.9), a restriction of (2.10) and the projective limit of the maps (5.10). If $\phi$ is a local diffeomorphism, then (5.11) is compatible with the Sobolev and symbol order filtrations.

6 Conormal distributions at the boundary

For the sake of simplicity, in this section and in Sects. 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 Sects. 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 $x\ge 0$, $\partial M=\{x=0\}$ (i.e., $x^{-1}(0)$) and $dx\ne 0$ 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 $\varpi :T\rightarrow \partial M$ 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, ${}^{\textrm{b}}TM$ and ${}^{\textrm{b}}T^*M$, 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 ${}^{\textrm{b}}\Omega ^sM:=\Omega ^s({}^{\textrm{b}}TM)$ ($s\in \mathbb {R}$) and ${}^{\textrm{b}}\Omega M:={}^{\textrm{b}}\Omega ^1\,M$. Clearly,

$$\begin{aligned} C^\infty (M;\Omega ^s)\equiv x^sC^\infty (M;{}^{\textrm{b}}\Omega ^s)\;. \end{aligned}$$

(6.1)

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

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 ${}^{\textrm{b}}\!N\partial M\subset {}^{\textrm{b}}T_{\partial M}M$ with $T\partial M={}^{\textrm{b}}T_{\partial M}M/{}^{\textrm{b}}\!N\partial M$. So ${}^{\textrm{b}}\Omega ^s_{\partial M}M\equiv \Omega ^s\partial M\otimes \Omega ^s({}^{\textrm{b}}\!N\partial M)$, and a restriction map $C^\infty (M;{}^{\textrm{b}}\Omega ^s)\rightarrow C^\infty (\partial M;\Omega ^s)$ is locally given by

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

A Euclidean structure g on ${}^{\textrm{b}}TM$ is called a b-metric. Locally,

$$\begin{aligned} g=a_0\Big (\frac{dx}{x}\Big )^2+2\sum _{j=1}^{n-1}a_{0j}\,\frac{dx}{x}\,dy^j +\sum _{j,k=1}^{n-1}a_{jk}\,dy^j\,dy^k, \end{aligned}$$

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 of dimension n which contains M as submanifold (for instance, $\breve{M}$ could be the double of M), and let $M'=\breve{M}{\setminus }\mathring{M}$, which is another compact n-submanifold with boundary of $\breve{M}$, with dimension n and $\partial M'=M\cap M'=\partial M$.

The concepts, notation and conventions of Sect. 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

$$\begin{aligned} R:C^\infty (\breve{M})\rightarrow C^\infty (M) \end{aligned}$$

(6.2)

is surjective; in fact, there is a continuous linear extension map $E:C^\infty (M)\rightarrow C^\infty (\breve{M})$ [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 $C^\infty _{M'}(\breve{M})$.

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 $\partial M$,

$$\begin{aligned} \dot{C}^\infty (M)=\bigcap _{m\ge 0}x^mC^\infty (M)\subset C^\infty (M)\;. \end{aligned}$$

(6.3)

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

$$\begin{aligned} \dot{C}^\infty (M)\equiv C^\infty _M(\breve{M})\subset C^\infty (\breve{M})\;. \end{aligned}$$

(6.4)

By (6.3),

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

(6.5)

and therefore, by (6.1),

$$\begin{aligned} \dot{C}^\infty (M;{}^{\textrm{b}}\Omega ^s)\equiv \dot{C}^\infty (M;\Omega ^s)\quad (s\in \mathbb {R})\;. \end{aligned}$$

(6.6)

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 $C^\infty (M)=\bigcap _kC^k(M)$ 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

$$\begin{aligned} \dot{C}^{-\infty }(M)=C^\infty (M;\Omega )',\quad C^{-\infty }(M)=\dot{C}^\infty (M;\Omega )'. \end{aligned}$$

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

$$\begin{aligned} \dot{C}^{-\infty }(M)\equiv C^{-\infty }_M(\breve{M})\subset C^{-\infty }(\breve{M})\;. \end{aligned}$$

(6.7)

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

$$\begin{aligned}{} & {} R:C^{-\infty }(\breve{M})\rightarrow C^{-\infty }(M), \end{aligned}$$

(6.8)

$$\begin{aligned}{} & {} R:\dot{C}^{-\infty }(M)\rightarrow C^{-\infty }(M), \end{aligned}$$

(6.9)

which are surjective by the Hahn-Banach theorem. Their null spaces are $C^{-\infty }_{M'}(\breve{M})=\dot{C}^{-\infty }(M')$ 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 continuous dense inclusions [25, Lemma 3.2.1]

$$\begin{aligned} C^\infty _{\textrm{c}}(\mathring{M})\subset \dot{C}^\infty (M)\subset C^\infty (M)\subset \dot{C}^{-\infty }(M)\;, \end{aligned}$$

(6.10)

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 continuous dense inclusion

$$\begin{aligned} C^\infty (M)\subset C^{-\infty }(M)\;. \end{aligned}$$

(6.11)

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

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

$$\begin{aligned} x^mC^{-\infty }(M)&=C^{-\infty }(M)\quad (m\in \mathbb {R})\;, \end{aligned}$$

(6.12)

$$\begin{aligned} C^{-\infty }(M;{}^{\textrm{b}}\Omega ^s)&\equiv C^{-\infty }(M;\Omega ^s)\quad (s\in \mathbb {R})\;. \end{aligned}$$

(6.13)

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 $\bigcup _kC^{\prime -k}(M)=C^{-\infty }(M)$ 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,

$$\begin{aligned} \dot{H}^s(M)=H^s_M(\breve{M})\subset H^s(\breve{M})\;. \end{aligned}$$

(6.14)

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

$$\begin{aligned} R:H^s(\breve{M})\rightarrow H^s(M)\;; \end{aligned}$$

(6.15)

i.e., this is a surjective topological homomorphism. Its null space is $H^s_{M'}(\breve{M})$. The analogs of (2.17)–(2.20) hold true in this setting using $\dot{C}^{\pm \infty }(M)$ and $C^{\pm \infty }(M)$. Furthermore the analogs of (2.17) 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

$$\begin{aligned} \dot{H}^s(M)\equiv H^{-s}(M;\Omega )'\;,\quad H^s(M)\equiv \dot{H}^{-s}(M;\Omega )'\;, \end{aligned}$$

(6.16)

and the map (6.9) has a continuous restriction

$$\begin{aligned} R:\dot{H}^s(M)\rightarrow H^s(M)\;, \end{aligned}$$

(6.17)

which is surjective if $s\le 1/2$, and injective if $s\ge -1/2$. 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]. $\square$

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

Consider the sequences

$$\begin{aligned}{} & {} 0\rightarrow \dot{C}^{-\infty }(M') \xrightarrow {\iota } C^{-\infty }(\breve{M}) \xrightarrow {R} C^{-\infty }(M)\rightarrow 0, \end{aligned}$$

(6.18)

$$\begin{aligned}{} & {} 0\rightarrow \dot{C}^{-\infty }_{\partial M}(M) \xrightarrow {\iota } \dot{C}^{-\infty }(M) \xrightarrow {R} C^{-\infty }(M)\rightarrow 0. \end{aligned}$$

(6.19)

Proposition 6.2 has the following direct consequence.

Corollary 6.3

The sequences (6.18) and (6.19) are exact sequences in the category of continuous linear maps between LCSs.

From (6.7), we get

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

(6.20)

The analogs of the second inclusion of (2.6), (2.17) and (2.19) 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 6.4

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

Proof

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

The following analog of Proposition 6.1 hold true with the same arguments, applying Proposition 6.4 and using that the Hilbertian spaces $\dot{H}^s_{\partial M}(M)$ form a compact spectrum.

Corollary 6.5

$\dot{C}^{-\infty }_{\partial M}(M)$ 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 7.9.

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

The notions of Sect. 2.7 also have straightforward extensions to manifolds with boundary. The action of any $A\in {\text {Diff}}(M)$ 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

(6.21)

However the analogous diagram

(6.22)

may not be commutative. Let us use the notation $u\mapsto u_{\textrm{c}}$ 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 $A(u_{\textrm{c}})-(Au)_{\textrm{c}}\in C^{-\infty }_{\partial M}(M)$ 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

$$\begin{aligned} \partial _x(u_{\textrm{c}})-(\partial _xu)_{\textrm{c}}=u(x_1)\,\delta _{x_1}-u(x_0)\,\delta _{x_0} \end{aligned}$$

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

$$\begin{aligned} {\text {Diff}}(\breve{M})\rightarrow {\text {Diff}}(M)\;,\quad \breve{A}\mapsto \breve{A}|_M\;. \end{aligned}$$

(6.23)

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

(6.24)

where the left-hand side square extends the left-hand side square of (6.21).

If $A\in {\text {Diff}}^m(M)$ ($m\in \mathbb {N}_0$), its actions on $\dot{C}^{-\infty }(M)$ and $C^{-\infty }(M)$ define continuous linear maps,

$$\begin{aligned} A:\dot{H}^s(M)\rightarrow \dot{H}^{s-m}(M)\;,\quad A:H^s(M)\rightarrow H^{s-m}(M)\;. \end{aligned}$$

(6.25)

The maps (6.17) and (6.25) fit into a commutative diagram given by the left-hand side square of (6.21).

6.7 Differential operators tangent to the boundary

The concepts of Sect. 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 $\mathfrak {X}_{\textrm{b}}(M)\subset \mathfrak {X}(M)$ of vector fields tangent to $\partial M$, called b-vector fields. There is a canonical identity $\mathfrak {X}_{\textrm{b}}(M)\equiv C^\infty (M;{}^{\textrm{b}}TM)$. Using $\mathfrak {X}_{\textrm{b}}(M)$ like in Sect. 2.7, we get the filtered $C^\infty (M)$-submodule and filtered subalgebra ${\text {Diff}}_{\textrm{b}}(M)\subset {\text {Diff}}(M)$ of b-differential operators. It consists of the operators $A\in {\text {Diff}}(M)$ such that (6.22) is commutative [25, Exercise 3.4.20]. The extension of ${\text {Diff}}_{\textrm{b}}(M)$ to arbitrary vector bundles is closed by taking transposes and formal adjoints.

Clearly, the restriction map (6.23) satisfies

$$\begin{aligned} {\text {Diff}}(\breve{M},\partial M)|_M={\text {Diff}}_{\textrm{b}}(M)\;. \end{aligned}$$

(6.26)

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

$$\begin{aligned} {\text {Diff}}_{\textrm{b}}^k(M)\,x^a=x^a{\text {Diff}}_{\textrm{b}}^k(M)\;. \end{aligned}$$

(6.27)

Since ${\text {Diff}}(M)$ is spanned by $\partial _x$ and ${\text {Diff}}_{\textrm{b}}(M)$ as algebra, it follows that

$$\begin{aligned} {\text {Diff}}^k(M)\,x^a\subset x^{a-k}{\text {Diff}}^k(M)\;. \end{aligned}$$

(6.28)

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

$$\begin{aligned} {\dot{\mathcal {A}}}^{(s)}(M)&=\{\,u\in \dot{C}^{-\infty }(M)\mid {\text {Diff}}_{\textrm{b}}(M)\,u\subset \dot{H}^s(M)\,\}\;,\\ \mathcal {A}^{(s)}(M)&=\{\,u\in C^{-\infty }(M)\mid {\text {Diff}}_{\textrm{b}}(M)\,u\subset H^s(M)\,\}\;, \end{aligned}$$

with the projective topologies given by the maps $P:{\dot{\mathcal {A}}}^{(s)}(M)\rightarrow H^s(M)$ and $P:\mathcal {A}^{(s)}(M)\rightarrow \dot{H}^s(M)$ ($P\in {\text {Diff}}_{\textrm{b}}(M)$). 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,

$$\begin{aligned} {\dot{\mathcal {A}}}(M)=\bigcup _s{\dot{\mathcal {A}}}^{(s)}(M)\;,\quad \mathcal {A}(M)=\bigcup _s\mathcal {A}^{(s)}(M)\;. \end{aligned}$$

(6.29)

By definition, there are continuous inclusions

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

(6.30)

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

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

Proposition 6.6

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

Corollary 6.7

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

We have

$$\begin{aligned} \bigcap _s{\dot{\mathcal {A}}}^{(s)}(M)=\dot{C}^\infty (M)\;,\quad \bigcap _s\mathcal {A}^{(s)}(M)=C^\infty (M)\;, \end{aligned}$$

(6.31)

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

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

(6.32)

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

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

(6.33)

Using (6.9), (6.15) and the commutativity of the left-hand side square of (6.21), we get the continuous linear restriction maps

$$\begin{aligned} R:{\dot{\mathcal {A}}}^{(s)}(M)\rightarrow \mathcal {A}^{(s)}(M)\;, \end{aligned}$$

(6.34)

which are surjective for $s\le 1/2$ and injective for $s\ge -1/2$ because this is true for the maps (6.17). From (6.3),  (6.32) and (6.34) for $s=0$, we get a continuous inclusion $C^\infty (M)\subset {\dot{\mathcal {A}}}^{(0)}(M)$, yielding a continuous inclusion [25, Proposition 4.1.1]

$$\begin{aligned} C^\infty (M)\subset {\dot{\mathcal {A}}}(M)\;. \end{aligned}$$

(6.35)

The maps (6.34) induce a surjective continuous linear restriction map

$$\begin{aligned} R:{\dot{\mathcal {A}}}(M)\rightarrow \mathcal {A}(M)\;, \end{aligned}$$

(6.36)

with $R=1$ on $C^\infty (M)$ [25, Proposition 4.1.1]. The surjectivity of (6.36) is a consequence of the existence of enough partial extension maps [25, Section 4.4]; the precise statement is recalled in Proposition 6.29 for later use. The maps (6.34) and (6.36) 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 6.7).

Proposition 6.8

The map (6.36) is a surjective topological homomorphism.

Corollary 6.9

The inclusion (6.35) is dense.

6.9 The spaces $x^mL^\infty (M)$

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

$$\begin{aligned} x^mL^\infty (M)\subset C^{-\infty }(M). \end{aligned}$$

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

$$\begin{aligned} x^mL^\infty (M)\subset x^{m'}L^\infty (M)\;. \end{aligned}$$

(6.37)

Proposition 6.10

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

Proof

Given $u\in x^mL^\infty (M)$ and $\epsilon >0$, let B be the ball in $x^{m'}L^\infty (M)$ of center u and radius $\epsilon$. Let $S=\sup _Mx^{m-m'}>0$. 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 $\lambda \ge 0$, $\lambda (r)\le r^m$ if $r>0$, $\lambda (r)=0$ if $r^{m-m'}\le \delta /2$, and $\lambda (r)=r^m$ if $r^{m-m'}\ge \delta$. Let $h=\lambda (x)f\in C^\infty _{\textrm{c}}(\mathring{M})$. If $x^{m-m'}\le \delta$, then, almost everywhere,

$$\begin{aligned} x^{-m'}|h-u|&\le \delta x^{-m}(|h|+|u|)\le \delta (|f|+x^{-m}|u|)\\&\le \delta (|f-x^{-m}u|+2x^{-m}|u|)<\delta \Big (\frac{\epsilon }{2}+\frac{\epsilon }{2\delta }\Big )<\epsilon \;. \end{aligned}$$

If $x^{m-m'}\ge \delta$, then, almost everywhere,

$$\begin{aligned} x^{-m'}|h-u|=x^{m-m'}|x^{-m}\lambda (x)f-x^{-m}u|\le S|f-x^{-m}u|<\epsilon . \end{aligned}$$

Thus $h\in B\cap C^\infty _{\textrm{c}}(\mathring{M})$. $\square$

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

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

$$\begin{aligned} \mathcal {A}^m(M)=\{\,u\in C^{-\infty }(M)\mid {\text {Diff}}_{\textrm{b}}(M)\,u\subset x^mL^\infty (M)\,\}. \end{aligned}$$

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

Example 6.11

([25, Exercises 4.2.23 and 4.2.24]) Via the injection of $\mathbb {R}^l$ into its stereographic compactification $\mathbb {S}^l_+=\{\,x\in \mathbb {S}^l\mid x^{l+1}\ge 0\,\}$, the space $\mathcal {A}^{-m}(\mathbb {S}^l_+)$ corresponds to the symbol space $S^m(\mathbb {R}^l)$ (Sect. 3).

Note that (6.37) yields a continuous inclusion

$$\begin{aligned} \mathcal {A}^m(M)\subset \mathcal {A}^{m'}(M)\quad (m'<m)\;. \end{aligned}$$

(6.38)

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

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

(6.39)

Hence

$$\begin{aligned} \mathcal {A}(M)=\bigcup _m\mathcal {A}^m(M)\;. \end{aligned}$$

(6.40)

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 (6.31),

$$\begin{aligned} \dot{C}^\infty (M)=\bigcap _m\mathcal {A}^m(M). \end{aligned}$$

Let $\{\,P_j\mid j\in \mathbb {N}_0\,\}$ be a countable $C^\infty (M)$-spanning set of ${\text {Diff}}_{\textrm{b}}(M)$. The topology of $\mathcal {A}^m(M)$ can be described by the semi-norms $\Vert {\cdot }\Vert _{k,m}$ ($k\in \mathbb {N}_0$) given by

$$\begin{aligned} \Vert u\Vert _{k,m}=\Vert P_ku\Vert _{x^mL^\infty }=\underset{M}{{\text {ess\,sup}}}\,\big |x^{-m}P_ku\big |=\sup _{\mathring{M}}\big |x^{-m}P_ku\big |\;, \end{aligned}$$

(6.41)

using (6.33) in the last expression. From (2.1) and (6.33), we also get the continuous semi-norms $\Vert {\cdot }\Vert _{K,k,m}$ (for any compact $K\subset \mathring{M}$ and $k\in \mathbb {N}_0$) on $\mathcal {A}^m(M)$ given by

$$\begin{aligned} \Vert u\Vert _{K,k,m}=\sup _K|P_ku|\;. \end{aligned}$$

(6.42)

Other continuous semi-norms $\Vert {\cdot }\Vert '_{k,m}$ ($k\in \mathbb {N}_0$) on $\mathcal {A}^m(M)$ are defined by

$$\begin{aligned} \Vert u\Vert '_{k,m}=\lim _{\epsilon \downarrow 0}\sup _{\{0<x<\epsilon \}}\big |x^{-m}P_ku\big |\;. \end{aligned}$$

(6.43)

The proofs of the following results are similar to the proofs of Propositions 3.2 and 3.3, Corollaries 3.4 to 3.6, using (6.33).

Proposition 6.12

The semi-norms (6.42) and (6.43) together describe the topology of $\mathcal {A}^m(M)$.

Proposition 6.13

For $m,m',k\in \mathbb {N}_0$, if $m'<m$, then $\Vert {\cdot }\Vert '_{k,m'}=0$ on $\mathcal {A}^m(M)$.

Corollary 6.14

If $m'<m$, then the topologies of $\mathcal {A}^{m'}(M)$ 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 6.15

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

Corollary 6.16

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

Remark 6.17

Proposition 6.10 provides an alternative direct proof of Corollary 6.15. Actually, it will be shown that $C^\infty _{\textrm{c}}(\mathring{M})$ is dense in every $\mathcal {A}^m(M)$ with its own topology (Corollary 6.39 and Remark 6.41), reconfirming Corollary 6.9.

The obvious analog of Remark 3.8 makes sense for (6.33) and Corollary 6.14.

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

Using (6.8), (6.15),  (6.26) and the commutativity of the left-hand side square of (6.24), we get continuous linear restriction maps

$$\begin{aligned} R:I^{(s)}(\breve{M},\partial M)\rightarrow \mathcal {A}^{(s)}(M), \end{aligned}$$

which induce a continuous linear restriction map

$$\begin{aligned} R:I(\breve{M},\partial M)\rightarrow \mathcal {A}(M)\;. \end{aligned}$$

(6.44)

By (6.14),  (6.26) and the commutativity of the right-hand side square of (6.24), we get the TVS-identities

$$\begin{aligned} {\dot{\mathcal {A}}}^{(s)}(M)\equiv I^{(s)}_M(\breve{M},\partial M)\;, \end{aligned}$$

(6.45)

inducing a continuous linear isomorphism

$$\begin{aligned} {\dot{\mathcal {A}}}(M) \xrightarrow {\cong } I_M(\breve{M},\partial M)\;. \end{aligned}$$

(6.46)

By (6.46) and Proposition 6.8, the map (6.44) 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 6.7).

Proposition 6.18

The map (6.44) is a surjective topological homomorphism.

The null space of (6.44) is $I_{M'}(\breve{M},\partial M)$. The following analog of Proposition 6.4 follows with formally the same proof, using Proposition 6.18 and Corollary 6.16.

Proposition 6.19

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

Corollary 6.20

The map (6.46) is a TVS-isomorphism.

Proof

Apply (6.29),  (6.45) and Proposition 6.19. $\square$

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

Inspired by (6.45), let

$$\begin{aligned} {\dot{\mathcal {A}}}^m(M)=I^m_M(\breve{M},\partial M)\subset I^m(\breve{M},\partial M)\quad (m\in \mathbb {R})\;, \end{aligned}$$

(6.47)

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

$$\begin{aligned} {\dot{\mathcal {A}}}(M)=\bigcup _m{\dot{\mathcal {A}}}^m(M),\quad \dot{C}^\infty (M)=\bigcap _m{\dot{\mathcal {A}}}^m(M), \end{aligned}$$

and the TVS-isomorphism (6.46) is also compatible with the symbol filtration.

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

Corollary 6.21

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

The following result follows like Corollary 3.6, applying Corollary 6.21 and using that ${\dot{\mathcal {A}}}(M)$ is barreled (Corollary 6.7) and a closed subspace of the Montel space $I(\breve{M},\partial M)$ (Corollary 4.7).

Corollary 6.22

${\dot{\mathcal {A}}}(M)$ 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

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

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 (6.36). They satisfy the analogs of (4.4), (4.11) and (4.12). So

$$\begin{aligned} \bigcup _s\mathcal {K}^{(s)}(M)=\bigcup _m\mathcal {K}^m(M). \end{aligned}$$

The term conormal sequence at the boundary of M will be used for

$$\begin{aligned} 0\rightarrow \mathcal {K}(M) \xrightarrow {\iota } {\dot{\mathcal {A}}}(M) \xrightarrow {R} \mathcal {A}(M)\rightarrow 0\;. \end{aligned}$$

(6.48)

Proposition 6.8 has the following direct consequence.

Corollary 6.23

The conormal sequence at the boundary of M is exact in the category of continuous linear maps between LCSs.

The following analog of Proposition 6.4 holds true with formally the same proof, using Proposition 6.8 and Corollary 6.16.

Proposition 6.24

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

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

$$\begin{aligned} \mathcal {K}^{(s)}(M)=\{\,u\in \dot{C}^{-\infty }_{\partial M}(M)\mid {\text {Diff}}_{\textrm{b}}(M)\,u\subset \dot{H}^s_{\partial M}(M)\,\}, \end{aligned}$$

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

Proposition 6.25

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

Corollary 6.26

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

Now the following analogs of Corollaries 6.21 and 6.22 hold true with formally the same proofs, using Corollaries 6.21, 6.22 and 6.26.

Corollary 6.27

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

Corollary 6.28

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

By Corollary 6.20,

$$\begin{aligned} \mathcal {K}(M)\equiv I_{\partial M}(\breve{M},\partial M)\;, \end{aligned}$$

(6.49)

which restricts to identities between the spaces defining the Sobolev and symbol order filtrations, according to (6.45) and (6.47).

A description of $\mathcal {K}^{(s)}(M)$ and $\mathcal {K}(M)$ will be indicated in Remark 7.28.

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

According to Sect. 4.5, and using (6.26),  (6.45), Proposition 6.18 and locality, any $A\in {\text {Diff}}(M)$ defines continuous endomorphisms A of ${\dot{\mathcal {A}}}(M)$, $\mathcal {A}(M)$ and $\mathcal {K}(M)$. If $A\in {\text {Diff}}^k(M)$, these maps also satisfy the analogs of (4.16). If $A\in {\text {Diff}}_{\textrm{b}}(M)$, 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 Sect. 6.6,  (6.30), (6.32) and (6.33), 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

Given linear subspaces, $X\subset \mathcal {A}(M)$ and $Y\subset {\dot{\mathcal {A}}}(M$), a map $E:X\rightarrow Y$ is called a partial extension map if $R(Y)\subset X$ and $RE=1$ on X.

Proposition 6.29

(Cf. [25, Section 4.4]) For all $m\in \mathbb {R}$, there is a continuous linear partial extension map $E_m:\mathcal {A}^m(M)\rightarrow {\dot{\mathcal {A}}}^{(s)}(M)$, where $s=0$ if $m\ge 0$, and $m>s\in \mathbb {Z}^-$ if $m<0$. For $m\ge 0$, $E_m:\mathcal {A}^m(M)\rightarrow {\dot{\mathcal {A}}}^{(0)}(M)$ is a continuous inclusion map.

Remark 6.30

By (6.39) and Proposition 6.29, for any $A\in {\text {Diff}}^k(M)$, the endomorphism A of $\mathcal {A}(M)$ (Sect. 6.14) is induced by the continuous linear compositions

$$\begin{aligned} \mathcal {A}^m(M) \xrightarrow {E_m} {\dot{\mathcal {A}}}^{(s)}(M) \xrightarrow {A} {\dot{\mathcal {A}}}^{(s-k)}(M) \xrightarrow {R} \mathcal {A}^{(s-k)}(M) \subset \mathcal {A}^{m'-k}(M), \end{aligned}$$

where $m'=s-n/2-1$ for m and s satisfying the conditions of Proposition 6.29.

6.16 $L^2$ half-b-densities

By (6.1),

$$\begin{aligned} C^\infty \left( M;\Omega ^{-\frac{1}{2}}\otimes {}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right)&\equiv C^\infty \left( M;\Omega ^{-\frac{1}{2}}\right) \otimes _{C^\infty (M)}C^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \\&\equiv C^\infty \left( M;\Omega ^{-\frac{1}{2}}\right) \otimes _{C^\infty (M)}x^{-\frac{1}{2}}C^\infty \left( M;\Omega ^{\frac{1}{2}}\right) \\&\equiv x^{-\frac{1}{2}}C^\infty \left( M;\Omega ^{-\frac{1}{2}}\otimes \Omega ^{\frac{1}{2}}\right) \equiv x^{-\frac{1}{2}}C^\infty (M)\;. \end{aligned}$$

So

$$\begin{aligned} L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right)&\equiv L^2\left( M;\Omega ^{\frac{1}{2}}\right) \otimes _{C^\infty (M)}C^\infty \left( M;\Omega ^{-\frac{1}{2}}\otimes {}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \nonumber \\&\equiv L^2\left( M;\Omega ^{\frac{1}{2}}\right) \otimes _{C^\infty (M)}x^{-\frac{1}{2}}C^\infty (M) \equiv x^{-\frac{1}{2}}L^2\left( M;\Omega ^{\frac{1}{2}}\right) \;. \end{aligned}$$

(6.50)

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

$$\begin{aligned} L^2\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \equiv L^2(M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}})\;. \end{aligned}$$

(6.51)

6.17 $L^\infty$ half-b-densities

Like in (6.50), we get

$$\begin{aligned} L^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \equiv x^{-\frac{1}{2}}L^\infty \left( M;\Omega ^{\frac{1}{2}}\right) \;, \end{aligned}$$

(6.52)

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

Equip M with a b-metric g (Sect. 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 ${}^{\textrm{b}}\Omega ^{1/2}M$, we have the identity of Banach spaces

$$\begin{aligned} L^\infty \left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \equiv L^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \;. \end{aligned}$$

(6.53)

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 (2.16), (2.21) and (2.22):

$$\begin{aligned}{} & {} H_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) =\left\{ \,u\in L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \mid {\text {Diff}}_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \,u\subset L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \,\right\} ,\\{} & {} {\text {Diff}}_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \,L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) =H_{\textrm{b}}^{-m} \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) = H_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) '. \end{aligned}$$

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 ${\text {Diff}}_{\textrm{b}}^m(M;{}^{\textrm{b}}\Omega ^{1/2})$ to define a scalar product on $H_{\textrm{b}}^{\pm m}(M;{}^{\textrm{b}}\Omega ^{1/2})$. The intersections and unions of the b-Sobolev spaces are denoted by $H_{\textrm{b}}^{\pm \infty }(M;{}^{\textrm{b}}\Omega ^{1/2})$. In particular, $H_{\textrm{b}}^\infty (M;{}^{\textrm{b}}\Omega ^{1/2})=\mathcal {A}^{(0)}(M;{}^{\textrm{b}}\Omega ^{1/2})$.

6.19 Weighted b-Sobolev spaces

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

$$\begin{aligned} \bigcap _{a,m}x^aH^m_{\textrm{b}}\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) =\dot{C}^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

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

Like in (2.13),

$$\begin{aligned} {\text {Diff}}_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \equiv {\text {Diff}}_{\textrm{b}}^m(M)\equiv {\text {Diff}}_{\textrm{b}}^m\left( M;\Omega ^{\frac{1}{2}}\right) \;. \end{aligned}$$

(6.54)

By (6.27), for all $k\in \mathbb {N}_0$, $m\in \mathbb {Z}$ and $a\in \mathbb {R}$, any $A\in {\text {Diff}}_{\textrm{b}}^k(M;{}^{\textrm{b}}\Omega ^{1/2})$ defines a continuous linear map [24, Lemma 5.14]

$$\begin{aligned} A:x^aH^m_{\textrm{b}}\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \rightarrow x^aH^{m-k}_{\textrm{b}}\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

Thus it induces a continuous endomorphism A of $x^aH^{\pm \infty }_{\textrm{b}}(M;{}^{\textrm{b}}\Omega ^{1/2})$.

6.21 A description of $\mathcal {A}(M)$

In this subsection, unless the contrary is indicated, assume the following properties:

• (A) $\mathring{M}$ is of bounded geometry with g.

• (B) The collar neighborhood T of $\partial M$ can be chosen so that:

  • (a) every $A\in \mathfrak {X}(\partial M)$ has an extension $A'\in \mathfrak {X}_{\textrm{b}}(T)$ such that $A'$ is $\varpi$-projectable to A, and $A'|_{\mathring{T}}$ is orthogonal to the $\varpi$-fibers; and

  • (b) $\mathfrak {X}_{\textrm{ub}}(\mathring{M})|_{\mathring{T}}$ is $C^\infty _{\textrm{ub}}(\mathring{M})|_{\mathring{T}}$-generated by $x\partial _x$ and the restrictions $A'|_{\mathring{T}}$ of the vector fields $A'$ of (a), for $A\in \mathfrak {X}(\partial M)$.

For instance, these properties hold if $\mathring{T}$ is cylindrical with g (Sect. 6.1).

Lemma 6.31

$\mathfrak {X}_{\textrm{b}}(M)|_T$ is $C^\infty (M)|_T$-generated by $x\partial _x$ and the vector fields $A'$ of (B), for $A\in \mathfrak {X}(\partial M)$.

Proof

For every $A\in \mathfrak {X}(\partial M)$, there is a unique $A''\in \mathfrak {X}(T)$ such that $A''$ is $\varpi$-projectable to A and $dx(A'')=0$. Since $A'-A''$ is tangent to the $\varpi$-fibers and vanishes on $\partial M$, we have $A'-A''=fx\partial _x$ for some $f\in C^\infty (T)$. Then the result follows because $\mathfrak {X}_{\textrm{b}}(M)|_T$ is $C^\infty (M)|_T$-spanned by $x\partial _x$ and the vector fields $A''$. $\square$

Consider the notation of Sects. 2.12.2 to 2.12.4 for $\mathring{M}$ with g.

Corollary 6.32

The restriction to $\mathring{M}$ defines a continuous injection $C^\infty (M)\subset C^\infty _{\textrm{ub}}(\mathring{M})$; in particular, $C^\infty _{\textrm{ub}}(\mathring{M})$ becomes a $C^\infty (M)$-module.

Proof

It is enough to work on a collar neighborhood T of the boundary satisfying (A) and (B). But, by (2.23),  (b) and Lemma 6.31, the restriction to $\mathring{T}$ defines an injection of $C^\infty (M)|_T$ into $C^\infty _{\textrm{ub}}(\mathring{M})|_{\mathring{T}}$. $\square$

Proposition 6.33

There is a canonical identity of $C^\infty _{\textrm{ub}}(\mathring{M})$-modules,

$$\begin{aligned} {\text {Diff}}_{\textrm{ub}}^m(\mathring{M})\equiv {\text {Diff}}_{\textrm{b}}^m(M)\otimes _{C^\infty (M)}C^\infty _{\textrm{ub}}(\mathring{M}). \end{aligned}$$

Proof

We have to prove that ${\text {Diff}}_{\textrm{ub}}^m(\mathring{M})$ is $C^\infty _{\textrm{ub}}(\mathring{M})$-spanned by ${\text {Diff}}_{\textrm{b}}^m(M)$. It is enough to consider the case $m=1$ because the filtered algebra ${\text {Diff}}_{\textrm{ub}}(\mathring{M})$ (respectively, ${\text {Diff}}_{\textrm{b}}(M)$) is spanned by ${\text {Diff}}_{\textrm{ub}}^1(\mathring{M})$ (respectively, ${\text {Diff}}_{\textrm{b}}^1(M)$). Moreover it is clearly enough to work on a collar neighborhood T of the boundary satisfying (A) and (B). By (b), Lemma 6.31 and Corollary 6.32, the restriction to $\mathring{T}$ defines an injection of ${\text {Diff}}_{\textrm{b}}^1(M)|_T$ as a $C^\infty _{\textrm{ub}}(\mathring{M})|_{\mathring{T}}$-spanning subset of ${\text {Diff}}_{\textrm{ub}}^1(\mathring{M})|_{\mathring{T}}$. $\square$

Corollary 6.34

There is a canonical identity of $C^\infty _{\textrm{ub}}(\mathring{M})$-modules,

$$\begin{aligned} {\text {Diff}}_{\textrm{ub}}^m\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \equiv {\text {Diff}}_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \otimes _{C^\infty (M)}C^\infty _{\textrm{ub}}(\mathring{M}). \end{aligned}$$

Proof

This follows from (2.24) for $\mathring{M}$, (6.54) and Proposition 6.33. $\square$

Corollary 6.35

$H^m(\mathring{M};\Omega ^{1/2})\equiv H_{\textrm{b}}^m(M;{}^{\textrm{b}}\Omega ^{1/2})$ $(m\in \mathbb {Z})$ as $C^\infty (M)$-modules and Hilbertian spaces, and therefore $H^{\pm \infty }(\mathring{M};\Omega ^{1/2})\equiv H_{\textrm{b}}^{\pm \infty }(M;{}^{\textrm{b}}\Omega ^{1/2})$.

Proof

We show the case where $m\ge 0$, and the case where $m<0$ follows by taking dual spaces. For any $m\in \mathbb {N}_0$, let $\{P_k\}$ be a finite $C^\infty (M)$-spanning set of ${\text {Diff}}_{\textrm{b}}^m(M;{}^{\textrm{b}}\Omega ^{1/2})$, which is also a $C^\infty _{\textrm{ub}}(\mathring{M})$-spanning set of ${\text {Diff}}_{\textrm{ub}}^m(\mathring{M};\Omega ^{1/2})$ by Corollary 6.34. Then, by (6.51),

$$\begin{aligned} H^m\left( \mathring{M};\Omega ^{\frac{1}{2}}\right)&=\left\{ \,u\in L^2\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \mid P_ku\in L^2\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \ \forall k\,\right\} \\&\equiv \{\,u\in L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \mid P_ku\in L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \ \forall k\,\} =H_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

Moreover $\{P_k\}$ can be used to define scalar products on both $H^m(\mathring{M};\Omega ^{1/2})$ and $H_{\textrm{b}}^m(M;{}^{\textrm{b}}\Omega ^{1/2})$, obtaining that the above identity is unitary. $\square$

Proposition 6.36

$\mathcal {A}^m(M;\Omega ^{1/2})\equiv x^{m+1/2}H_{\textrm{b}}^\infty (M;{}^{\textrm{b}}\Omega ^{1/2})$ $(m\in \mathbb {R})$.

Proof

By (6.52),  (6.53), Proposition 2.5 and Corollary 6.35, we get the following identities and continuous inclusions:

$$\begin{aligned} H_{\textrm{b}}^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right)&\equiv H^\infty \left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \subset C^\infty _{\textrm{ub}}\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \nonumber \\&\subset L^\infty \left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \equiv L^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \equiv x^{-\frac{1}{2}}L^\infty \left( M;\Omega ^{\frac{1}{2}}\right) \;. \end{aligned}$$

So, according to Sect. 6.20, every $A\in {\text {Diff}}_{\textrm{b}}(M;{}^{\textrm{b}}\Omega ^{1/2})$ induces a continuous linear map

$$\begin{aligned} x^{m+\frac{1}{2}}H_{\textrm{b}}^\infty (M;{}^{\textrm{b}}\Omega ^{1/2}) \xrightarrow {A} x^{m+\frac{1}{2}}H_{\textrm{b}}^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \subset x^mL^\infty \left( M;\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

Hence there is a continuous inclusion

$$\begin{aligned} x^{m+\frac{1}{2}}H_{\textrm{b}}^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \subset \mathcal {A}^m\left( M;\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

On the other hand, by (6.13) and the version of (6.12) with $\Omega ^{1/2}M$, for all $a\in \mathbb {R}$,

$$\begin{aligned} x^a\mathcal {A}^m\left( M;\Omega ^{\frac{1}{2}}\right) \subset x^aC^{-\infty }\left( M;\Omega ^{\frac{1}{2}}\right) =C^{-\infty }\left( M;\Omega ^{\frac{1}{2}}\right) \equiv C^{-\infty }\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

Moreover, by (6.50) and (6.54), every $A\in {\text {Diff}}_{\textrm{b}}(M;{}^{\textrm{b}}\Omega ^{1/2})$ induces a continuous linear map

$$\begin{aligned} \mathcal {A}^m\left( M;\Omega ^{\frac{1}{2}}\right) \xrightarrow {A} x^mL^\infty \left( M;\Omega ^{\frac{1}{2}}\right) \subset x^mL^2\left( M;\Omega ^{\frac{1}{2}}\right) \equiv x^{m+\frac{1}{2}}L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

Hence, by (6.27) and (6.54), A induces a continuous linear map

$$\begin{aligned} A:x^{-m-\frac{1}{2}}\mathcal {A}^m\left( M;\Omega ^{\frac{1}{2}}\right) \rightarrow L^2\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

It follows that there is a continuous inclusion

$$\begin{aligned} x^{-m-\frac{1}{2}}\mathcal {A}^m\left( M;\Omega ^{\frac{1}{2}}\right) \subset H_{\textrm{b}}^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) , \end{aligned}$$

or, equivalently, there is a continuous inclusion

$$\begin{aligned} \mathcal {A}^m\left( M;\Omega ^{\frac{1}{2}}\right) \subset x^{m+\frac{1}{2}}H_{\textrm{b}}^\infty \left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) . \end{aligned}$$

$\square$

Corollary 6.37

$H^m(\mathring{M})=x^{-1/2}H_{\textrm{b}}^m(M)$ $(m\in \mathbb {Z})$ as $C^\infty (M)$-modules and Hilbertian spaces, and therefore $H^{\pm \infty }(\mathring{M})=x^{-1/2}H_{\textrm{b}}^{\pm \infty }(M)$.

Proof

By (6.1) and Corollary 6.35,

$$\begin{aligned} H^m(\mathring{M})&\equiv H^m\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \otimes _{C^\infty _{\textrm{ub}}(\mathring{M})} C^\infty _{\textrm{ub}}\left( \mathring{M};\Omega ^{-\frac{1}{2}}\right) \\&\equiv H^m\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \otimes _{C^\infty _{\textrm{ub}}(\mathring{M})} \left( C^\infty \left( M;\Omega ^{-\frac{1}{2}}\right) \otimes _{C^\infty (M)}C^\infty _{\textrm{ub}}(\mathring{M})\right) \\&\equiv H^m\left( \mathring{M};\Omega ^{\frac{1}{2}}\right) \otimes _{C^\infty (M)}C^\infty (M;\Omega ^{-\frac{1}{2}})\\&\equiv H_{\textrm{b}}^m\left( M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}}\right) \otimes _{C^\infty (M)}x^{-\frac{1}{2}}C^\infty \left( M;{}^{\textrm{b}}\Omega ^{-\frac{1}{2}}\right) \equiv x^{-\frac{1}{2}}H_{\textrm{b}}^m(M)\;. \end{aligned}$$

$\square$

Corollary 6.38

$\mathcal {A}^m(M)\equiv x^mH_{\textrm{b}}^\infty (M)\equiv x^{m+1/2}H^\infty (\mathring{M})$ $(m\in \mathbb {R})$.

Proof

The second identity is given by Corollary 6.37. By Proposition 6.36 and (6.1),

$$\begin{aligned} x^mH_{\textrm{b}}^\infty (M)&\equiv x^{m+\frac{1}{2}}H_{\textrm{b}}^\infty (M;{}^{\textrm{b}}\Omega ^{\frac{1}{2}})\otimes _{C^\infty (M)} x^{-\frac{1}{2}}C^\infty (M;{}^{\textrm{b}}\Omega ^{-\frac{1}{2}})\\&\equiv \mathcal {A}^m(M;\Omega ^{\frac{1}{2}})\otimes _{C^\infty (M)}C^\infty (M;\Omega ^{-\frac{1}{2}}) \equiv \mathcal {A}^m(M)\;. \end{aligned}$$

$\square$

By (6.39) and (6.40), we get the following consequences of Corollary 6.38.

Corollary 6.39

$C^\infty _{\textrm{c}}(\mathring{M})$ is dense in every $\mathcal {A}^m(M)$ and $\mathcal {A}^{(s)}(M)$.

Corollary 6.40

$\mathcal {A}(M)\equiv \bigcup _mx^mH_{\textrm{b}}^\infty (M)=\bigcup _mx^mH^\infty (\mathring{M})$.

Remark 6.41

Corollary 6.39 and the first identities of Corollaries 6.38 and 6.40 are independent of g. So they hold true without the assumptions (A) and (B). Observe that Corollary 6.39 is stronger than Corollary 6.15.

6.22 Dual-conormal distributions at the boundary

Consider the LCHSs [15, Section 18.3], [25, Chapter 4]

$$\begin{aligned} \mathcal {K}'(M)=\mathcal {K}(M;\Omega )',\quad \mathcal {A}'(M)={\dot{\mathcal {A}}}(M;\Omega )',\quad {\dot{\mathcal {A}}}'(M)=\mathcal {A}(M;\Omega )'. \end{aligned}$$

The elements of $\mathcal {A}'(M)$ (respectively, ${\dot{\mathcal {A}}}'(M)$) will be called extendible (respectively, supported) dual-conormal distributions at the boundary. The following analog of Corollary 5.1 holds true with formally the same proof, using the versions with $\Omega M$ of Corollaries 6.7, 6.22, 6.16, 6.26 and 6.28

Proposition 6.42

$\mathcal {K}'(M)$, $\mathcal {A}'(M)$ and ${\dot{\mathcal {A}}}'(M)$ are complete Montel spaces.

We also define the LCHSs

$$\begin{aligned} \mathcal {K}^{\prime \,(s)}(M)&=\mathcal {K}^{(-s)}(M;\Omega )'\;,&\quad \mathcal {K}^{\prime \,m}(M)&=\mathcal {K}^{-m}(M;\Omega )'\;,\\ \mathcal {A}^{\prime \,(s)}(M)&={\dot{\mathcal {A}}}^{(-s)}(M;\Omega )'\;,&\quad \mathcal {A}^{\prime \,m}(M)&={\dot{\mathcal {A}}}^{-m}(M;\Omega )'\;,\\ {\dot{\mathcal {A}}}^{\prime \,(s)}(M)&=\mathcal {A}^{(-s)}(M;\Omega )'\;,&\quad {\dot{\mathcal {A}}}^{\prime \,m}(M)&=\mathcal {A}^{-m}(M;\Omega )'\;. \end{aligned}$$

Transposing the analogs of (4.4) and (4.11) for the spaces $\mathcal {K}^{(s)}(M;\Omega )$, $\mathcal {K}^m(M;\Omega )$, ${\dot{\mathcal {A}}}^{(s)}(M;\Omega )$ and ${\dot{\mathcal {A}}}^m(M;\Omega )$, we get continuous linear restriction maps

$$\begin{aligned}{} & {} \mathcal {K}^{\prime \,(s')}(M)\rightarrow \mathcal {K}^{\prime \,(s)}(M),\quad \mathcal {K}^{\prime \,m}(M)\rightarrow \mathcal {K}^{\prime \,m'}(M),\\{} & {} \quad \mathcal {A}^{\prime \,(s')}(M)\rightarrow \mathcal {A}^{\prime \,(s)}(M),\quad \mathcal {A}^{\prime \,m}(M)\rightarrow \mathcal {A}^{\prime \,m'}(M), \end{aligned}$$

for $s<s'$ and $m<m'$. These maps form projective spectra, giving rise to projective limits. The spaces $\mathcal {K}^{\prime \,(s)}(M)$, $\mathcal {K}^{\prime \,m}(M)$, $\mathcal {A}^{\prime \,(s)}(M)$ and $\mathcal {A}^{\prime \,m}(M)$ satisfy the analogs of (5.4). So the corresponding projective limits satisfy the analogs of (5.5).

Similarly, transposing the analog of (4.4) for the spaces $\mathcal {A}^{(s)}(M;\Omega )$ and the version of (6.38) with $\Omega M$, by Corollary 6.39 and Remark 6.41, we get continuous inclusions

$$\begin{aligned} {\dot{\mathcal {A}}}^{\prime \,(s')}(M)\subset {\dot{\mathcal {A}}}^{\prime \,(s)}(M), \quad {\dot{\mathcal {A}}}^{\prime \,m'}(M)\subset {\dot{\mathcal {A}}}^{\prime \,m}(M), \end{aligned}$$

for $s<s'$ and $m<m'$. The version of (6.39) with $\Omega M$ yields continuous inclusions

$$\begin{aligned} {\dot{\mathcal {A}}}^{\prime \,(s)}(M)\supset {\dot{\mathcal {A}}}^{\prime \,m}(M)\supset {\dot{\mathcal {A}}}^{\prime \,(\max \{m,0\})}(M)\quad (m>s+n/2+1)\;. \end{aligned}$$

(6.55)

Therefore

$$\begin{aligned} \bigcap _s{\dot{\mathcal {A}}}^{\prime \,(s)}(M)=\bigcap _m{\dot{\mathcal {A}}}^{\prime \,m}(M)\;. \end{aligned}$$

(6.56)

The following analogs of Corollaries 5.2 and 5.3 hold true with formally the same proofs, using the versions with $\Omega M$ of Propositions 6.6 and 6.25 and Corollaries 6.22, 6.16 and 6.28; or use Proposition 6.42 for an alternative proof.

Corollary 6.43

$\mathcal {K}^{\prime \,(s)}(M)$, $\mathcal {A}^{\prime \,(s)}(M)$ and ${\dot{\mathcal {A}}}^{\prime \,(s)}(M)$ are bornological and barreled.

Corollary 6.44

We have

$$\begin{aligned} \mathcal {K}'(M)\equiv \varprojlim \mathcal {K}^{\prime \,(s)}(M),\quad \mathcal {A}'(M)\equiv \varprojlim \mathcal {A}^{\prime \,(s)}(M),\quad {\dot{\mathcal {A}}}'(M)=\bigcap _s{\dot{\mathcal {A}}}^{\prime \,(s)}(M). \end{aligned}$$

Transposing the versions of (6.3),  (6.30), (6.32),  (6.35) and Corollary 6.9 with $\Omega M$, we get continuous inclusions [25, Section 4.6]

$$\begin{aligned}&C^\infty (M)\subset \mathcal {A}'(M)\subset C^{-\infty }(M),\dot{C}^{-\infty }(M)\;, \end{aligned}$$

(6.57)

$$\begin{aligned}&\dot{C}^\infty (M)\subset {\dot{\mathcal {A}}}'(M)\subset \dot{C}^{-\infty }(M),C^{-\infty }(M)\;, \end{aligned}$$

(6.58)

and $R:\dot{C}^{-\infty }(M)\rightarrow C^{-\infty }(M)$ restricts to the identity map on $\mathcal {A}'(M)$ and ${\dot{\mathcal {A}}}'(M)$.

6.23 Dual-conormal sequence at the boundary

Transposing maps in the version of (6.48) with $\Omega M$, we get the sequence

$$\begin{aligned} 0\leftarrow \mathcal {K}'(M) \xleftarrow {\iota ^t} \mathcal {A}'(M) \xleftarrow {R^t} {\dot{\mathcal {A}}}'(M)\leftarrow 0, \end{aligned}$$

which will called the dual-conormal sequence at the boundary of M.

Proposition 6.45

The dual-conormal sequence at the boundary of M is exact in the category of continuous linear maps between LCSs.

Proof

By Corollary 6.8 and [39, Lemma 7.6], it is enough to prove that the map (6.36) satisfies the following condition of “topological lifting of bounded sets.”

Claim 6.46

For all bounded subset $A\subset \mathcal {A}(M)$, there is some bounded subset $B\subset {\dot{\mathcal {A}}}(M)$ such that, for all 0-neighborhood $U\subset {\dot{\mathcal {A}}}(M)$, there is a 0-neighborhood $V\subset \mathcal {A}(M)$ so that $A\cap V\subset R(B\cap U)$.

Since $\mathcal {A}(M)$ is boundedly retractive (Corollary 6.16), A is contained and bounded in some step $\mathcal {A}^m(M)$. For any $m'>m$, let $E_{m'}:\mathcal {A}^m(M)\rightarrow {\dot{\mathcal {A}}}^{(s)}(M)$ be the partial extension map given by Proposition 6.29. Then $B:=E_{m'}(A)$ is bounded in ${\dot{\mathcal {A}}}^{(s)}(M)$, and therefore in ${\dot{\mathcal {A}}}(M)$. Moreover, given any 0-neighborhood $U\subset {\dot{\mathcal {A}}}(M)$, there is some 0-neighborhood $W\subset \mathcal {A}^{m'}(M)$ so that $E_{m'}(W)\subset U\cap {\dot{\mathcal {A}}}^{(s)}(M)$. By Corollary 6.14, there is some 0-neighborhood $V\subset \mathcal {A}(M)$ such that $V\cap \mathcal {A}^m(M)=W\cap \mathcal {A}^m(M)$. Hence $E_{m'}(V\cap \mathcal {A}^m(M))\subset U\cap {\dot{\mathcal {A}}}^{(s)}(M)$, yielding

$$\begin{aligned} A\cap V=R(E_{m'}(A\cap V))\subset R(E_{m'}(A)\cap E_{m'}(V\cap \mathcal {A}^m(M)))\subset R(B\cap U). \end{aligned}$$

$\square$

Remark 6.47

Proposition 6.45 does not agree with [25, Proposition 4.6.2], which seems to be a minor error of that book project.

6.24 ${\dot{\mathcal {A}}}(M)$ and $\mathcal {A}(M)$ versus $\mathcal {A}'(M)$

Using (6.30), (6.32) and (6.57), we have [15, Proposition 18.3.24], [25, Theorem 4.6.1]

$$\begin{aligned} {\dot{\mathcal {A}}}(M)\cap \mathcal {A}'(M)=C^\infty (M)\;. \end{aligned}$$

(6.59)

(Actually, the a priori weaker equality $\mathcal {A}(M)\cap \mathcal {A}'(M)=C^\infty (M)$ is proved in [25, Theorem 4.6.1], but it is equivalent to (6.59) because $R=1$ on $\mathcal {A}'(M)$.)

6.25 A description of ${\dot{\mathcal {A}}}'(M)$

In this subsection, assume again (A) and (B).

Corollary 6.48

${\dot{\mathcal {A}}}^{\prime \,m}(M)\equiv x^mH_{\textrm{b}}^{-\infty }(M)=x^{m-\frac{1}{2}}H^{-\infty }(\mathring{M})$ $(m\in \mathbb {R})$.

Proof

Apply the version of Corollary 6.38 with $\Omega M$. $\square$

Corollary 6.49

${\dot{\mathcal {A}}}'(M)\equiv \bigcap _mx^mH_{\textrm{b}}^{-\infty }(M)=\bigcap _mx^mH^{-\infty }(\mathring{M})$.

Proof

Apply (6.56) and Corollaries 6.44 and 6.48. $\square$

Corollary 6.50

$C^\infty _{\textrm{c}}(\mathring{M})$ is dense in every ${\dot{\mathcal {A}}}^{\prime \,m}(M)$ and in ${\dot{\mathcal {A}}}'(M)$. Therefore the first inclusion of (6.58) is also dense.

Proof

Since $C^\infty _{\textrm{c}}(\mathring{M})$ is dense in $H^{-\infty }(\mathring{M})$, we get that $C^\infty _{\textrm{c}}(\mathring{M})=x^mC^\infty _{\textrm{c}}(\mathring{M})$ is dense in every $x^mH^{-\infty }(\mathring{M})\equiv {\dot{\mathcal {A}}}^{\prime \,m}(M)$ (Corollary 6.48), and therefore in ${\dot{\mathcal {A}}}'(M)$ (Corollary 6.49). $\square$

Remark 6.51

Like in Remark 6.41, Corollary 6.50 and the first identities of Corollary 6.49 are independent of g, and hold true without the assumptions (A) and (B).

6.26 Action of ${\text {Diff}}(M)$ on $\mathcal {A}'(M)$, ${\dot{\mathcal {A}}}'(M)$ and $\mathcal {K}'(M)$

According to Sects. 2.5 and 6.14, any $A\in {\text {Diff}}(M)$ induces continuous linear endomorphisms A of $\mathcal {A}'(M)$, ${\dot{\mathcal {A}}}'(M)$ and $\mathcal {K}'(M)$ [25, Proposition 4.6.1], which are the transposes of $A^t$ on ${\dot{\mathcal {A}}}(M;\Omega )$, $\mathcal {A}(M;\Omega )$ and $\mathcal {K}(M;\Omega )$. If $A\in {\text {Diff}}^k(M)$, these maps satisfy the analogs of (5.7). If $A\in {\text {Diff}}_{\textrm{b}}(M)$, it induces continuous endomorphisms of $\mathcal {A}^{\prime \,(s)}(M)$, $\mathcal {A}^{\prime \,m}(M)$, ${\dot{\mathcal {A}}}^{\prime \,(s)}(M)$ and $\mathcal {K}^{\prime \,(s)}(M)$.

7 Conormal sequence

7.1 Cutting along a submanifold

Let M be a closed connected manifold, and $L\subset M$ be a regular closed submanifold of codimension one. L may not be connected, and therefore $M{\setminus } L$ may have several connected components. First assume also that L is transversely oriented. Then, like in the boundary case Sect. 6.1, there is some real-valued smooth function x on some tubular neighborhood T of L in M, with projection $\varpi :T\rightarrow L$, so that $L=\{x=0\}$ and $dx\ne 0$ on L. Any function x satisfying these conditions is called a defining function of L on T. We can suppose $T\equiv (-\epsilon ,\epsilon )_x\times L$, for some $\epsilon >0$, so that $\varpi :T\rightarrow L$ is the second factor projection. For any atlas $\{V_j,y_j\}$ of L, we get an atlas of T of the form $\{U_j\equiv (-\epsilon ,\epsilon )_x\times V_j,(x,y)\}$, whose charts are adapted to L. The corresponding local vector fields $\partial _x\in \mathfrak {X}(U_j)$ can be combined to define a vector field $\partial _x\in \mathfrak {X}(T)$; we can consider $\partial _x$ as the derivative operator on $C^\infty (T)\equiv C^\infty ((-\epsilon ,\epsilon ),C^\infty (L))$. For every j, ${\text {Diff}}(U_j,L\cap U_j)$ is spanned by $x\partial _x,\partial _j^1,\dots ,\partial _j^{n-1}$ using the operations of $C^\infty (U_j)$-module and algebra, where $\partial _j^\alpha =\partial /\partial y_j^\alpha$. Using $T\equiv (-\epsilon ,\epsilon )_x\times L$, any $A\in {\text {Diff}}(L)$ induces an operator $1\otimes A\in {\text {Diff}}(T,L)$, such that $(1\otimes A)(u(x)v(y))=u(x)\,(Av)(y)$ for $u\in C^\infty (-\epsilon ,\epsilon )$ and $v\in C^\infty (L)$. This defines a canonical injection ${\text {Diff}}(L)\equiv 1\otimes {\text {Diff}}(L)\subset {\text {Diff}}(T,L)$ so that $(1\otimes A)|_L=A$. (This also shows the surjectivity of (4.2) in this case.) Moreover ${\text {Diff}}(T)$ (respectively, ${\text {Diff}}(T,L)$) is spanned by $\partial _x$ (respectively, $x\partial _x$) and $1\otimes {\text {Diff}}(L)$ using the operations of $C^\infty (T)$-module and algebra. Clearly,

$$\begin{aligned}{}[\partial _x,1\otimes {\text {Diff}}(L)]=0\;,\quad [\partial _x, x\partial _x]=\partial _x\;, \end{aligned}$$

(7.1)

yielding

$$\begin{aligned}{}[\partial _x,{\text {Diff}}^k(T,L)]\subset {\text {Diff}}^k(T,L)+{\text {Diff}}^{k-1}(T,L)\,\partial _x\;. \end{aligned}$$

(7.2)

${\text {Diff}}^k(T,L)$ and ${\text {Diff}}^k(T)$ satisfy the obvious versions of (6.27) and (6.28).

For a vector bundle E over M, there is an identity $E_T\equiv (-\epsilon ,\epsilon )\times E_L$ over $T\equiv (-\epsilon ,\epsilon )\times L$, which can be used to define $\partial _x\in {\text {Diff}}^1(T;E)$ using the above charts. With this interpretation of $\partial _x$ and using tensor products like in (2.5), the vector bundle versions of the properties and spaces of distributions of this section are straightforward.

Let $\varvec{M}$ be the smooth manifold with boundary defined by “cutting” M along L; i.e., modifying M only on the tubular neighborhood $T\equiv (-\epsilon ,\epsilon )\times L$, which is replaced with $\varvec{T}=((-\epsilon ,0]\sqcup [0,\epsilon ))\times L$ in the obvious way. ($\varvec{M}$ is the blowing-up [M, L] of M along L [25, Chapter 5].) Thus $\partial \varvec{M}\equiv L\sqcup L$ because L is transversely oriented, and $\mathring{\varvec{M}}\equiv M{\setminus } L$. A canonical projection $\varvec{\pi }:\varvec{M}\rightarrow M$ is defined as the combination of the identity map $\mathring{\varvec{M}}\rightarrow M{\setminus } L$ and the map $\varvec{T}\rightarrow T$ given by the product of the canonical projection $(-\epsilon ,0]\sqcup [0,\epsilon )\rightarrow (-\epsilon ,\epsilon )$ and ${\text {id}}_L$. This projection realizes M as a quotient space of $\varvec{M}$ by the equivalence relation defined by the homeomorphism $h\equiv h_0\times {\text {id}}$ of $\partial \varvec{M}\equiv \partial \varvec{T}=(\{0\}\sqcup \{0\})\times L$, where $h_0$ switches the two points of $\{0\}\sqcup \{0\}$. Moreover $\varvec{\pi }:\varvec{M}\rightarrow M$ is a local embedding of a compact manifold with boundary to a closed manifold of the same dimension.

Like in Sect. 2.6, we have the continuous linear pull-back map

$$\begin{aligned} \varvec{\pi }^*:C^\infty (M)\rightarrow C^\infty (\varvec{M})\;, \end{aligned}$$

(7.3)

which is clearly injective. Then the transpose of the version of (7.3) with $\Omega M$ and $\Omega \varvec{M}\equiv \varvec{\pi }^*\Omega M$ is the continuous linear push-forward map

$$\begin{aligned} \varvec{\pi }_*:\dot{C}^{-\infty }(\varvec{M})\rightarrow C^{-\infty }(M)\;, \end{aligned}$$

(7.4)

which is surjective by a consequence of the Hahn-Banach theorem [31, Theorem II.4.2].

After distinguishing a connected component $L_0$ of L, let ${\widetilde{M}}$ and ${\widetilde{L}}$ be the quotients of $\varvec{M}\sqcup \varvec{M}\equiv \varvec{M}\times \mathbb {Z}_2$ and $\partial \varvec{M}\sqcup \partial \varvec{M}\equiv \partial \varvec{M}\times \mathbb {Z}_2$ by the equivalence relation generated by $(p,a)\sim (h(p),a)$ if $\varvec{\pi }(p)\in L{\setminus } L_0$ and $(p,a)\sim (h(p),a+1)$ if $\varvec{\pi }(p)\in L_0$ ($p\in \varvec{\pi }^{-1}(L)=\partial \varvec{M}$ in both cases). Let us remark that ${\widetilde{M}}$ may not be homeomorphic to the double of $\varvec{M}$, which is the quotient of $\varvec{M}\times \mathbb {Z}_2$ by the equivalence relation generated by $(p,0)\sim (p,1)$, for $p\in \partial \varvec{M}$. Note that ${\widetilde{M}}$ is a closed connected manifold and ${\widetilde{L}}$ is a closed regular submanifold. Moreover the quotient $\widetilde{T}$ of $\textbf{T}\sqcup \textbf{T}$ becomes a tubular neighborhood of ${\widetilde{L}}$ in ${\widetilde{M}}$. The combination $\varvec{\pi }\sqcup \varvec{\pi }:\varvec{M}\sqcup \varvec{M}\rightarrow M$ induces a two-fold covering map ${\tilde{\pi }}:{\widetilde{M}}\rightarrow M$, whose restrictions to ${\widetilde{L}}$ and ${\widetilde{T}}$ are trivial two-fold coverings of L and T, respectively; i.e., ${\widetilde{L}}\equiv L\sqcup L$ and ${\widetilde{T}}\equiv T\sqcup T$. The group of deck transformations of ${\tilde{\pi }}:{\widetilde{M}}\rightarrow M$ is $\{{\text {id}},\sigma \}$, where $\sigma :{\widetilde{M}}\rightarrow {\widetilde{M}}$ is induced by the map $\sigma _0:\varvec{M}\times \mathbb {Z}_2\rightarrow \varvec{M}\times \mathbb {Z}_2$ defined by switching the elements of $\mathbb {Z}_2$. The composition of the injection $\varvec{M}\rightarrow \varvec{M}\times \mathbb {Z}_2$, $p\mapsto (p,0)$, with the quotient map $\varvec{M}\sqcup \varvec{M}\rightarrow {\widetilde{M}}$ is a smooth embedding $\varvec{M}\rightarrow {\widetilde{M}}$. This will be considered as an inclusion map of a regular submanifold with boundary, obtaining $\partial \varvec{M}\equiv {\widetilde{L}}$.

Since ${\tilde{\pi }}$ is a two-fold covering map, we have continuous linear maps (Sect. 2.6)

$$\begin{aligned}{} & {} {\tilde{\pi }}_*:C^\infty ({\widetilde{M}}) \rightarrow C^\infty (M),\quad {\tilde{\pi }}^*:C^\infty (M) \rightarrow C^\infty ({\widetilde{M}}),\nonumber \\{} & {} \quad {\tilde{\pi }}^*:C^{-\infty }(M) \rightarrow C^{-\infty }({\widetilde{M}}),\quad {\tilde{\pi }}_*:C^{-\infty }({\widetilde{M}}) \rightarrow C^{-\infty }(M), \end{aligned}$$

(7.5)

both pairs of maps satisfying

$$\begin{aligned} {\tilde{\pi }}_*{\tilde{\pi }}^*=2{\text {id}},\quad {\tilde{\pi }}^*{\tilde{\pi }}_*=A_\sigma \;, \end{aligned}$$

(7.6)

where $A_\sigma :C^{\pm \infty }({\widetilde{M}})\rightarrow C^{\pm \infty }({\widetilde{M}})$ is given by $A_\sigma u=u+\sigma _*u$. Using the continuous linear restriction and inclusion maps given by (6.2) and (6.7), we get the commutative diagrams

(7.7)

the second one being the transpose of the density-bundles version of the first one.

7.2 Lift of differential operators from M to ${\widetilde{M}}$

For any $A\in {\text {Diff}}(M)$, let ${\widetilde{A}}\in {\text {Diff}}({\widetilde{M}})$ denote its lift via the covering map ${\tilde{\pi }}:{\widetilde{M}}\rightarrow M$. The action of ${\widetilde{A}}$ on $C^{\pm \infty }({\widetilde{M}})$ corresponds to the action of A on $C^{\pm \infty }(M)$ via ${\tilde{\pi }}^*:C^{\pm \infty }(M)\rightarrow C^{\pm \infty }({\widetilde{M}})$ and ${\tilde{\pi }}_*:C^{\pm \infty }({\widetilde{M}})\rightarrow C^{\pm \infty }(M)$. According to (6.23), ${\widetilde{A}}|_{\varvec{M}}\in {\text {Diff}}(\varvec{M})$ is the lift of A via the local embedding $\varvec{\pi }:\varvec{M}\rightarrow M$, sometimes also denoted by ${\widetilde{A}}$. The action of ${\widetilde{A}}$ on $C^\infty (\varvec{M})$ (respectively, $C^{-\infty }(\varvec{M})$) corresponds to the action of A on $C^\infty (M)$ (respectively, $C^{-\infty }(M)$) via $\varvec{\pi }^*:C^\infty (M)\rightarrow C^\infty (\varvec{M})$ (respectively, $\varvec{\pi }_*:C^{-\infty }(\varvec{M})\rightarrow C^{-\infty }(M)$). If $A\in {\text {Diff}}(M,L)$, then ${\widetilde{A}}\in {\text {Diff}}({\widetilde{M}},{\widetilde{L}})$ and ${\widetilde{A}}|_{\varvec{M}}\in {\text {Diff}}_{\textrm{b}}(\varvec{M})$ by (6.26).

7.3 The spaces $C^{\pm \infty }(M,L)$

Consider the closed subspaces,

$$\begin{aligned} C^\infty (M,L)\subset C^\infty (M)\;,\quad C^k(M,L)\subset C^k(M)\quad (k\in \mathbb {N}_0)\;, \end{aligned}$$

(7.8)

consisting of functions that vanish to all orders at the points of L in the first case, and that vanish up to order k at the points of L in the second case. Then let

$$\begin{aligned} C^{-\infty }(M,L)=C^\infty (M,L;\Omega )',\quad C^{\prime \,-k}(M,L)=C^k(M,L;\Omega )'. \end{aligned}$$

Note that (7.3) restricts to TVS-isomorphisms

$$\begin{aligned} \varvec{\pi }^*:C^\infty (M,L)\xrightarrow {\cong }\dot{C}^\infty (\varvec{M})\;,\quad \varvec{\pi }^*:C^k(M,L)\xrightarrow {\cong }\dot{C}^k(\varvec{M})\;. \end{aligned}$$

(7.9)

Taking the transposes of its versions with density bundles, it follows that (7.4) restricts to TVS-isomorphisms

$$\begin{aligned} \varvec{\pi }_*:C^{-\infty }(\varvec{M})\xrightarrow {\cong }C^{-\infty }(M,L)\;,\quad \varvec{\pi }_*:C^{\prime \,-k}(\varvec{M})\xrightarrow {\cong }C^{\prime \,-k}(M,L)\;. \end{aligned}$$

(7.10)

So the spaces $C^\infty (M,L)$, $C^k(M,L)$, $C^{-\infty }(M,L)$ and $C^{\prime \,-k}(M,L)$ satisfy the analogs of (2.6) and (2.7).

On the other hand, there are Hilbertian spaces $H^r(M,L)$ ($r>n/2$) and $H^{\prime \,s}(M,L)$ ($s\in \mathbb {R}$), continuously included in $C^0(M,L)$ and $C^{-\infty }(M,L)$, respectively, such that the second map of (7.9) for $k=0$ and the first map of (7.10) restrict to a TVS-isomorphisms

$$\begin{aligned} \varvec{\pi }^*:H^r(M,L)\xrightarrow {\cong }\dot{H}^r(\varvec{M})\;,\quad \varvec{\pi }_*:H^s(\varvec{M})\xrightarrow {\cong }H^{\prime \,s}(M,L)\;. \end{aligned}$$

(7.11)

By (6.16),

$$\begin{aligned} H^{\prime \,-r}(M,L)\equiv H^r(M,L;\Omega )'\;,\quad H^r(M,L)\equiv H^{\prime \,-r}(M,L;\Omega )'\;. \end{aligned}$$

(7.12)

Now, the second identity of (7.12) can be used to extend the definition of $H^r(M,L)$ for all $r\in \mathbb {R}$.

Alternatively, we may also use trace theorems [1, Theorem 7.53 and 7.58] to define $H^m(M,L)$ for $m\in \mathbb {Z}^+$, and then use the first identity of (7.12) to define $H^{\prime \,-m}(M,L)$.

From (7.3),  (7.4),  (7.11) and the analogs of (2.18)–(2.20) mentioned in Sect. 6.4, we get

$$\begin{aligned} C^\infty (M,L)&=\bigcap _rH^r(M,L)\;,&\quad C^{-\infty }(M,L)&=\bigcup _sH^{\prime \,s}(M,L)\;, \end{aligned}$$

(7.13)

as well as a continuous inclusion and a continuous linear surjection,

$$\begin{aligned} C^\infty (M)&\subset \bigcap _sH^{\prime \,s}(M,L)\;,&\quad C^{-\infty }(M)&\leftarrow \bigcup _rH^r(M,L)\;. \end{aligned}$$

(7.14)

By (7.12) and (7.13),

$$\begin{aligned} C^\infty (M,L)=C^{-\infty }(M,L;\Omega )'\;. \end{aligned}$$

(7.15)

Proposition 6.1 and (7.10) have the following consequence.

Corollary 7.1

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

The transpose of the version of the first inclusion of (7.8) with $\Omega M$ is a continuous linear restriction map

$$\begin{aligned} R:C^{-\infty }(M)\rightarrow C^{-\infty }(M,L)\;, \end{aligned}$$

(7.16)

whose restriction to $C^\infty (M)$ is the identity. This map can be also described as the composition

$$\begin{aligned} C^{-\infty }(M) \xrightarrow {{\tilde{\pi }}^*} C^{-\infty }({\widetilde{M}}) \xrightarrow {R} C^{-\infty }(\varvec{M}) \xrightarrow {\varvec{\pi }_*} C^{-\infty }(M,L). \end{aligned}$$

The canonical pairing between $C^\infty (M)$ and $C^\infty (M,L;\Omega )$ defines a continuous inclusion

$$\begin{aligned} C^\infty (M)\subset C^{-\infty }(M,L) \end{aligned}$$

(7.17)

such that (7.16) is the identity on $C^\infty (M)$. We also get commutative diagrams

(7.18)

the second one being the transpose of the density-bundles version of the first one.

7.4 The space $C^{-\infty }_L(M)$

The closed subspaces of elements supported in L,

$$\begin{aligned} C^{-\infty }_L(M)\subset C^{-\infty }(M),\quad C^{\prime \,-k}_L(M)\subset C^{\prime \,-k}(M),\quad H^s_L(M)\subset H^s(M), \end{aligned}$$

are the null spaces of restrictions of (7.16). These spaces satisfy continuous inclusions analogous to (2.6), (2.17) and (2.19).

According to (6.20) and Sect. 7.1,

$$\begin{aligned} \dot{C}^{-\infty }_{\partial \varvec{M}}(\varvec{M})&\equiv C^{-\infty }_{{\widetilde{L}}}({\widetilde{M}}) \equiv C^{-\infty }_{{\widetilde{L}}}({\widetilde{T}})\equiv C^{-\infty }_L(T)\oplus C^{-\infty }_L(T)\nonumber \\&\equiv C^{-\infty }_L(M)\oplus C^{-\infty }_L(M)\;. \end{aligned}$$

(7.19)

The maps (7.4) and (7.5) have restrictions

$$\begin{aligned} \varvec{\pi }_*={\tilde{\pi }}_*:\dot{C}^{-\infty }_{\partial \varvec{M}}(\varvec{M})\rightarrow C^{-\infty }_L(M)\;,\quad {\tilde{\pi }}^*:C^{-\infty }_L(M)\rightarrow \dot{C}^{-\infty }_{\partial \varvec{M}}(\varvec{M})\;. \end{aligned}$$

(7.20)

Using (7.19), these maps are given by $\varvec{\pi }_*(u,v)=u+v$ and ${\tilde{\pi }}^*u=(u,u)$.

From (7.19), Proposition 6.4 and Corollary 6.5, we get the following.

Corollary 7.2

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

Corollary 7.3

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

Moreover the right-hand side diagram of (7.18) can be completed to get the commutative diagram

(7.21)

Proposition 7.4

The maps (7.4) and (7.16) are surjective topological homomorphisms.

Proof

In (7.21), the top row is exact in the category of continuous linear maps between LCSs by Corollary 6.3, the left-hand side vertical map is onto by (7.6), and the right-hand side vertical map is a TVS-isomorphism. Then, by the commutativity of its right-hand side square, the map (7.16) is surjective, and therefore the bottom row of (7.21) is exact in the category of linear maps between vector spaces.

By the above properties, chasing (7.21), we get that (7.4) is surjective. Since $\dot{C}^{-\infty }(\varvec{M})$ is webbed (Proposition 6.1) and $C^{-\infty }(M)$ ultrabornological, by the open mapping theorem, it also follows that (7.4) is a topological homomorphism.

To get that (7.16) is another surjective topological homomorphism, apply the commutativity of the right-hand side square of (7.21) and the above properties. $\square$

Corollary 7.5

The bottom row of (7.21) is exact in the category of continuous linear maps between LCSs.

Corollary 7.6

The inclusion (7.17) is dense.

Proof

Apply Proposition 7.4 and the density of $C^\infty (M)$ in $C^{-\infty }(M)$. $\square$

7.5 A description of $C^{-\infty }_L(M)$

According to Sects. 2.7 and 7.1 and (4.14), we have the subspaces

$$\begin{aligned} \partial _x^mC^{-\infty }(L;\Omega ^{-1}NL)\subset C^{-\infty }_L(M) \end{aligned}$$

(7.22)

for $m\in \mathbb {N}_0$, and continuous linear isomorphisms

$$\begin{aligned} \partial _x^m:C^{-\infty }(L;\Omega ^{-1}NL)\xrightarrow {\cong }\partial _x^mC^{-\infty }(L;\Omega ^{-1}NL)\;. \end{aligned}$$

(7.23)

They induce a continuous linear injection

$$\begin{aligned} \bigoplus _{m=0}^\infty C^0_m\rightarrow C^{-\infty }_L(M)\;, \end{aligned}$$

(7.24)

where $C^0_m=C^{-\infty }(L;\Omega ^{-1}NL)$ for all m.

Proposition 7.7

The map (7.24) is a TVS-isomorphism, which restricts to TVS-isomorphisms

$$\begin{aligned} \bigoplus _{m=0}^kC^{m-k}(L;\Omega ^{-1}NL)\xrightarrow {\cong }C^{\prime \,-k}_L(M)\quad (k\in \mathbb {N}_0)\;. \end{aligned}$$

(7.25)

Proof

In the case where $M=\mathbb {R}^n$ and L is a linear subspace, it is known that (7.25) is a linear isomorphism [14, Theorem 2.3.5 and Example 5.1.2], which is easily seen to be continuous. This can be easily extended to arbitrary M by using charts of M adapted to L. Then we get the continuous linear isomorphism (7.24) by taking the locally convex inductive limit of (7.25) as $k\uparrow \infty$. Since $\bigoplus _mC^0_m$ is webbed and $C^{-\infty }_L(M)$ ultrabornological (Corollary 7.3), the map (7.24) is a TVS-isomorphism by the open mapping theorem. $\square$

Remark 7.8

Proposition 7.7 reconfirms Corollary 7.2.

Remark 7.9

(See [25, Exercise 3.3.18]) In Sect. 6.5, for any compact manifold with boundary M, the analog of Proposition 7.7 for $\dot{C}^{-\infty }_{\partial M}(M)$ follows from the application of Proposition 7.7 to $C^{-\infty }_{\partial M}(\breve{M})$.

Corollary 7.10

Every map (7.23) is a TVS-isomorphism.

7.6 Action of ${\text {Diff}}(M)$ on $C^{-\infty }(M,L)$ and $C^{-\infty }_L(M)$

For every $A\in {\text {Diff}}(M)$, $A^t$ preserves $C^\infty (M,L;\Omega )$, and therefore A induces a continuous linear map $A=A^{tt}$ on $C^{-\infty }(M,L)$. By locality, it restricts to a continuous endomorphism A of $C^{-\infty }_L(M)$.

7.7 The space J(M, L)

According to Sects. 6.8 and 7.3, there is a LCHS J(M, L), continuously included in $C^{-\infty }(M,L)$, so that (7.10) restricts to a TVS-isomorphism

$$\begin{aligned} \varvec{\pi }_*:\mathcal {A}(\varvec{M})\xrightarrow {\cong }J(M,L)\;, \end{aligned}$$

(7.26)

where $\mathcal {A}(\varvec{M})$ is defined in (6.29). By (6.33), there is a continuous inclusion

$$\begin{aligned} J(M,L)\subset C^\infty (M{\setminus } L). \end{aligned}$$

We also get spaces $J^{(s)}(M,L)$ and $J^m(M,L)$ ($s,m\in \mathbb {R}$) corresponding to $\mathcal {A}^{(s)}(\varvec{M})$ and $\mathcal {A}^m(\varvec{M})$ via (7.26). Extend |x| to a function $\varvec{x}$ on M that is positive and smooth on $M{\setminus } L$. Its lift $\varvec{\pi }^*\varvec{x}$ is a boundary defining function of $\varvec{M}$, also denoted by $\varvec{x}$. Using the first map of (7.10) and second map of (7.11), and according to Sect. 7.2, we can also describe

$$\begin{aligned} J^{(s)}(M,L)&=\{\,u\in C^{-\infty }(M,L)\mid {\text {Diff}}(M,L)\,u\subset H^{\prime \,s}(M,L)\,\}\;,\\ J^m(M,L)&=\{\,u\in C^{-\infty }(M,L)\mid {\text {Diff}}(M,L)\,u\subset \varvec{x}^mL^\infty (M)\,\}\;,\nonumber \end{aligned}$$

(7.27)

equipped with topologies like in Sects. 6.8 and 6.10. These spaces satisfy the analogs of (4.4), (6.29) and (6.38)–(6.40). By (7.14) and (7.27), and by the definition of J(M, L), there are continuous inclusions,

$$\begin{aligned} C^\infty (M)\subset J^{(\infty )}(M,L):=\bigcap _sJ^{(s)}(M,L)\;,\quad J(M,L)\subset C^{-\infty }(M,L)\;. \end{aligned}$$

(7.28)

Moreover the following analogs of Proposition 6.6 and Corollaries 6.7 and 6.14 to 6.16 hold true.

Corollary 7.11

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

Corollary 7.12

J(M, L) is barreled, ultrabornological and webbed.

Corollary 7.13

If $m'<m$, then the topologies of $J^{m'}(M,L)$ and $C^\infty (M{\setminus } L)$ coincide on $J^m(M,L)$. Therefore the topologies of J(M, L) and $C^\infty (M{\setminus } L)$ coincide on $J^m(M,L)$.

Corollary 7.14

For $m'<m$, $C^\infty _{\textrm{c}}(M{\setminus } L)$ is dense in $J^m(M,L)$ with the topology of $J^{m'}(M,L)$. Therefore $C^\infty _{\textrm{c}}(M{\setminus } L)$ is dense in J(M, L).

Corollary 7.15

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

The analog of Remark 6.17 makes sense for J(M, L).

7.8 A description of J(M, L)

Take a b-metric $\varvec{g}$ on $\varvec{M}$ satisfying (A) and (B), and consider its restriction to $\mathring{\varvec{M}}$. Consider also the boundary defining function $\varvec{x}$ of $\varvec{M}$ (Sect. 7.7). Corollaries 6.38 to 6.40 and (7.26) have the following direct consequences.

Corollary 7.16

$J^m(M,L)\cong \varvec{x}^mH_{\textrm{b}}^\infty (\varvec{M})\equiv \varvec{x}^{m+1/2}H^{\infty }(\mathring{\varvec{M}})$ $(m\in \mathbb {R})$.

Corollary 7.17

$C^\infty _{\textrm{c}}(M{\setminus } L)$ is dense in every $J^m(M,L)$ and $J^{(s)}(M,L)$.

Corollary 7.18

$J(M,L)\cong \bigcup _m\varvec{x}^mH_{\textrm{b}}^\infty (\varvec{M})=\bigcup _m\varvec{x}^mH^\infty (\mathring{\varvec{M}})$.

The analog of Remark 6.41 makes sense for J(M, L).

7.9 I(M, L) versus ${\dot{\mathcal {A}}}(\varvec{M})$ and J(M, L)

According to Sects. 4.6 and  4.7, we have the continuous linear maps

$$\begin{aligned} {\tilde{\pi }}^*:I(M,L)\rightarrow I({\widetilde{M}},{\widetilde{L}})\;,\quad {\tilde{\pi }}_*:I({\widetilde{M}},{\widetilde{L}})\rightarrow I(M,L)\;, \end{aligned}$$

(7.29)

which are restrictions of the maps (7.5), and therefore they satisfy (7.6). These maps are compatible with the Sobolev and symbol order filtrations because ${\tilde{\pi }}:{\widetilde{M}}\rightarrow M$ is a covering map (Sects. 4.6 and 4.7).

Using the TVS-embedding ${\dot{\mathcal {A}}}(\varvec{M})\subset I({\widetilde{M}},{\widetilde{L}})$ (Corollary 6.20), which is compatible with the Sobolev and symbol order filtrations, the map ${\tilde{\pi }}_*$ of (7.29) has the restriction

$$\begin{aligned} \varvec{\pi }_*:{\dot{\mathcal {A}}}(\varvec{M})\rightarrow I(M,L)\;. \end{aligned}$$

(7.30)

This map is compatible with the Sobolev and symbol order filtration by the above properties.

On the other hand, the map (7.16) restricts to a continuous linear map

$$\begin{aligned} R:I(M,L)\rightarrow J(M,L)\;, \end{aligned}$$

(7.31)

which can be also described as the composition

$$\begin{aligned} I(M,L) \xrightarrow {{\tilde{\pi }}^*} I({\widetilde{M}},{\widetilde{L}}) \xrightarrow {R} \mathcal {A}(\varvec{M}) \xrightarrow {\varvec{\pi }_*} J(M,L). \end{aligned}$$

From the properties of (6.44),  (7.26) and (7.29), it follows that (7.31) is compatible with the Sobolev order filtration. According to (4.5) and (7.28), the map (7.31) is the identity on $C^\infty (M)$.

7.10 The space K(M, L)

Like in Sect. 6.13, the condition of being supported in L defines the LCHSs and $C^\infty (M)$-modules

$$\begin{aligned} K^{(s)}(M,L)=I^{(s)}_L(M,L),\quad K^m(M,L)=I^m_L(M,L),\quad K(M,L)=I_L(M,L). \end{aligned}$$

These are closed subspaces of $I^{(s)}(M,L)$, $I^m_L(M,L)$ and I(M, L), respectively; more precisely, they are the null spaces of the corresponding restrictions of the map (7.31). According to Corollary 6.20, the identity (7.19) restricts to a TVS-identity

$$\begin{aligned} \mathcal {K}(\varvec{M})\equiv K(M,L)\oplus K(M,L)\;. \end{aligned}$$

(7.32)

Furthermore the maps (7.20) induce continuous linear maps

$$\begin{aligned} \varvec{\pi }_*:\mathcal {K}(\varvec{M})\rightarrow K(M,L)\;,\quad {\tilde{\pi }}^*:K(M,L)\rightarrow \mathcal {K}(\varvec{M})\;. \end{aligned}$$

(7.33)

Using (7.32), these maps are given by $\varvec{\pi }_*(u,v)=u+v$ and ${\tilde{\pi }}^*u=(u,u)$.

By (6.45) and (6.47), $K^{(s)}(M,L)$ and $K^m(M,L)$ satisfy analogs of (7.32), using $\mathcal {K}^{(s)}(\varvec{M})$ and $\mathcal {K}^m(\varvec{M})$. Thus we get the following consequences of Propositions 6.24 and 6.25 and Corollaries 6.26 to 6.28.

Corollary 7.19

K(M, L) is a limit subspace of the LF-space I(M, L).

Corollary 7.20

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

Corollary 7.21

$K^{(s)}(M,L)$ is barreled, ultrabornological and webbed, and therefore so is K(M, L).

Corollary 7.22

For $m<m',m''$, the topologies of $K^{m'}(M,L)$ and $K^{m''}(M,L)$ coincide on $K^m(M,L)$.

Corollary 7.23

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

Example 7.24

With the notation of Sect. 4.8, ${\text {Diff}}(M)\equiv K(M^2,\Delta )$ becomes a filtered $C^\infty (M^2)$-submodule of $\Psi (M)$, with the order filtration corresponding to the symbol order filtration. In this way, ${\text {Diff}}(M)$ also becomes a LCHS satisfying the above properties. If M is compact, it is also a filtered subalgebra of $\Psi (M)$.

7.11 A description of K(M, L)

By (4.18) and (4.16),

$$\begin{aligned} \partial _x^mC^\infty (L;\Omega ^{-1}NL)\subset K^{(s-m)}(M,L)\quad (s<-1/2)\;, \end{aligned}$$

(7.34)

and (7.23) restricts to a continuous linear isomorphism

$$\begin{aligned} \partial _x^m:C^\infty (L;\Omega ^{-1}NL)\xrightarrow {\cong }\partial _x^mC^\infty (L;\Omega ^{-1}NL)\;. \end{aligned}$$

(7.35)

Lemma 7.25

For all $m\in \mathbb {N}_0$,

$$\begin{aligned} \partial _x^mC^\infty (L;\Omega ^{-1}NL)\cap K^{(-\frac{1}{2}-m)}(M,L)=0. \end{aligned}$$

Proof

We proceed by induction on m. The case $m=0$ is given by Proposition 4.9. Now assume $m\ge 1$, and let $v\in C^\infty (L;\Omega ^{-1}NL)$ with $u=\partial _x^m\delta _L^v\in K^{(-\frac{1}{2}-m)}(M,L)$. Take any $A\in {\text {Diff}}^2(L;\Omega NL)$ such that $-\partial _x^2+B\in {\text {Diff}}^2(T)$ is elliptic, where $B=(1\otimes A^t)^t\in {\text {Diff}}^2(T,L)$; for instance, given a Riemannian metric on M, A can be the Laplacian of the flat line bundle $\Omega NL$. By (7.34), $u_0:=\partial _x^{m-1}\delta _L^v\in K^{(\frac{1}{2}-m-\epsilon )}(M,L)$ for $0<\epsilon <1$. By (7.2), given any $B_0\in {\text {Diff}}(M,L)$, there is some $B_1,B_2,B_3\in {\text {Diff}}(M,L)$ such that $[\partial _x^2,B_0]=B_1+B_2\partial _x+B_3\partial _x^2$. So, according to Sect. 4.5, (4.17) and (7.2), for all $B_0\in {\text {Diff}}(M,L)$,

$$\begin{aligned} (-\partial _x^2+B)B_0u_0&=-B_0\partial _xu-B_1u_0-B_2u-B_3\partial _xu +\partial _x^{m-1}\delta _L^{B_0'Av}+[B,B_0]u_0\\&\quad \in K^{\left( -\frac{3}{2}-m\right) }(M,L)+K^{\left( \frac{1}{2}-m-\epsilon \right) }(M,L)=K^{\left( -\frac{3}{2}-m\right) }(M,L)\;. \end{aligned}$$

Hence $B_0u_0\in H^{\frac{1}{2}-m}(M)$ by elliptic regularity. Since $B_0$ is arbitrary, we get $u_0\in K^{(\frac{1}{2}-m)}(M,L)$. So $u_0=0$ by the induction hypothesis, yielding $u=\partial _xu_0=0$. $\square$

By Proposition 7.7, the TVS-isomorphism (7.24) restricts to a linear injection

$$\begin{aligned} \bigoplus _{m=0}^\infty C^1_m\rightarrow K(M,L)\;, \end{aligned}$$

(7.36)

where $C^1_m=C^\infty (L;\Omega ^{-1}NL)$ for all $m\in \mathbb {N}_0$, which is easily seen to be continuous.

Proposition 7.26

The map (7.36) is a TVS-isomorphism, which induces TVS-isomorphisms

$$\begin{aligned} \bigoplus _{m<-s-\frac{1}{2}}C^1_m\xrightarrow {\cong }K^{(s)}(M,L)\quad (s<-1/2)\;. \end{aligned}$$

(7.37)

Proof

To prove that (7.36) is surjective, take any $u\in K(M,L)$. By Proposition 7.7, we can assume $u\in \partial _x^mC^{-\infty }(L;\Omega NL)$ for some m. For any $A\in {\text {Diff}}(L;\Omega NL)$, let $B=(1\otimes A^t)^t\in {\text {Diff}}(T,L)$. Since $u\in K(M,L)$ and B is local, it follows from the definition of I(M, L) and (2.19) that $Bu\in H^{-k}_L(T)\subset C^{\prime \,-k}_L(T)$ for some $k\ge m$. On the other hand, $u=\partial _x^m\delta _L^v$ for some $v\in C^{-\infty }(L;\Omega NL)$. Then (4.17) and (7.1) yield

$$\begin{aligned} Bu=B\partial _x^m\delta _L^v=\partial _x^mB\delta _L^v=\partial _x^m\delta _L^{B'v} =\partial _x^m\delta _L^{Av}. \end{aligned}$$

Therefore, by Proposition 7.7,

$$\begin{aligned} Bu\in C^{\prime \,-k}_L(M)\cap \partial _x^mC^{-\infty }(L;\Omega NL)=\partial _x^mC^{\prime \,m-k}(L;\Omega NL). \end{aligned}$$

This means that $Av\in C^{\prime \,m-k}(L;\Omega NL)$. So $v\in C^\infty (L;\Omega NL)$ because A is arbitrary, and therefore $u\in \partial _x^mC^\infty (L;\Omega NL)$.

The surjectivity of (7.37) follows from Lemma 7.25 and the surjectivity of (7.36).

Finally, (7.36) is open like in Proposition 7.7, using that $C^\infty (L;\Omega ^{-1}NL)$ is webbed and K(M, L) ultrabornological (Corollary 7.21). So (7.37) is also open. $\square$

Remark 7.27

Proposition 7.26 reconfirms Corollary 7.19. It also follows from Proposition 7.26 that (7.35) is a TVS-isomorphism.

Remark 7.28

In Sect. 6.13, for any compact manifold with boundary M, the analog of Proposition 7.7 for $\mathcal {K}(M)$ follows from (6.49) and the application of Proposition 7.7 to $K(\breve{M},\partial M)$.

7.12 The conormal sequence

The diagram (7.21) has the restriction

(7.38)

The bottom row of (7.38) will be called the conormal sequence of M at L (or of (M, L)). The following analog of Proposition 7.4 holds true with formally the same proof, using (7.38).

Proposition 7.29

The maps (7.30) and (7.31) are surjective topological homomorphisms.

Corollary 7.30

The conormal sequence of M at L is exact in the category of continuous linear maps between LCSs.

The surjectivity of (7.31) can be realized with partial extension maps, as stated in the following analog of Proposition 6.29.

Corollary 7.31

For all $m\in \mathbb {R}$, there is a continuous linear partial extension map $E_m:J^m(M,L)\rightarrow I^{(s)}(M,L)$, where $s=0$ if $m\ge 0$, and $m>s\in \mathbb {Z}^-$ if $m<0$.

Proof

By the commutativity of (7.38), we can take $E_m$ equal to the composition

$$\begin{aligned} J^m(M,L) \xrightarrow {\varvec{\pi }_*^{-1}} \mathcal {A}^m(\varvec{M}) \xrightarrow {E_m} {\dot{\mathcal {A}}}^{(s)}(\varvec{M}) \xrightarrow {\varvec{\pi }_*} I^{(s)}(M,L), \end{aligned}$$

where this map $E_m$ is given by Proposition 6.29. $\square$

Corollary 7.32

$C^\infty (M)$ is dense in J(M, L).

Proof

Apply (7.28), Corollary 4.6 and Proposition 7.29. $\square$

7.13 Action of ${\text {Diff}}(M)$ on the conormal sequence

According to Sect. 4.5, every $A\in {\text {Diff}}(M)$ defines a continuous linear map A on I(M, L), which preserves K(M, L) because A is local. Therefore it induces a linear map A on J(M, L), which is continuous by Proposition 7.29. This map satisfies the analog of (4.16).

The map A on J(M, L) can be also described as a restriction of A on $C^{-\infty }(M,L)$ (Sect. 7.6). On the other hand, according to Sect. 6.14, the lift ${\widetilde{A}}\in {\text {Diff}}(\varvec{M})$ defines continuous linear maps on the top spaces of (7.38) which correspond to the operators defined by A on the bottom spaces via the maps $\varvec{\pi }_*$. If $A\in {\text {Diff}}(M,L)$, then it defines continuous endomorphisms A of $J^{(s)}(M,L)$ and $J^m(M,L)$.

7.14 Pull-back of elements of the conormal sequence

Consider the notation and conditions of Sect. 4.6. By locality, the map (4.20) has a restriction $\phi ^*:K(M,L)\rightarrow K(M',L')$. So it also induces a linear map $\phi ^*:J(M,L)\rightarrow J(M',L')$, which is continuous by Proposition 7.29.

7.15 Push-forward of elements of the conormal sequence

Consider the notation and conditions of Sect. 4.7. As above, the map (4.22) has a restriction $\phi _*:K_{\textrm{c}}(M',L';\Omega _{\textrm{fiber}})\rightarrow K_{\textrm{c}}(M,L)$. Thus it induces a linear map $\phi _*:J_{\textrm{c}}(M',L';\Omega _{\textrm{fiber}})\rightarrow J_{\textrm{c}}(M,L)$, which is continuous by Proposition 7.29.

7.16 Case where L is not transversely orientable

If L is not transversely orientable, we still have a tubular neighborhood T of L in M, but there is no defining function x of L in T trivializing the projection $\varpi :T\rightarrow L$. We can cut M along L as well to produce a bounded compact manifold, $\varvec{M}$, with a projection $\varvec{\pi }:\varvec{M}\rightarrow M$ and a boundary collar $\varvec{T}$ over T.

By using a boundary defining function $\varvec{x}$ of $\varvec{M}$, we get the same definitions, properties and descriptions of $C^{\pm \infty }(M,L)$ and J(M, L) (Sects. 7.3,  7.7 and 7.8).

$C^{-\infty }_L(M)$ and K(M, L) also have the same definitions (Sects. 7.4 and 7.10). However (7.19) and (7.32) are not true because the covering map $\varvec{\pi }:\partial \varvec{M}\rightarrow L$ is not trivial, and the descriptions given in Propositions 7.7 and 7.26 need a slight modification. This problem can be solved as follows.

Let ${\check{\pi }}:\check{L}\rightarrow L$ denote the two-fold covering of transverse orientations of L, and let ${\check{\sigma }}$ denote its deck transformation different from the identity. Since the lift of NL to $\check{L}$ is trivial, ${\check{\pi }}$ on $\check{L}\equiv \{0\}\times \check{L}$ can be extended to a two-fold covering ${\check{\pi }}:\check{T}:=(-\epsilon ,\epsilon )_x\times \check{L}\rightarrow T$, for some $\epsilon >0$. Its deck transformation different from the identity is an extension of ${\check{\sigma }}$ on $\check{L}\equiv \{0\}\times \check{L}$, also denoted by ${\check{\sigma }}$. Then $\check{L}$ is transversely oriented in $\check{T}$; i.e., its normal bundle $N\check{L}$ is trivial. Thus $C^{-\infty }_{\check{L}}(\check{T})$ and $K(\check{T},\check{L})$ satisfy (7.19),  (7.32) and Propositions 7.7 and Proposition 7.26. Since $N\check{L}\equiv {\check{\pi }}^*NL$, the map ${\check{\sigma }}$ lifts to a homomorphism of $N\check{L}$, which induces a homomorphism of $\Omega ^{-1}NL$ also denoted by ${\check{\sigma }}$. Let $L_{-1}$ be the union of non-transversely oriented connected components of L, and $L_1$ the union of its transversely oriented components. Correspondingly, let $\check{L}_{\pm 1}={\check{\pi }}^{-1}(L_{\pm 1})$ and $\check{T}_{\pm 1}=(-\epsilon ,\epsilon )\times \check{L}_{\pm 1}$. Since ${\check{\sigma }}^*x=\pm x$ on $T_{\pm 1}$, Propositions 7.7 and 7.26 become true in this case by replacing $C^r(L;\Omega ^{-1}NL)$ ($r\in \mathbb {Z}\cup \{\pm \infty \}$) with the direct sum of the spaces

$$\begin{aligned} \{\,u\in C^r(\check{L}_{\pm 1};\Omega ^{-1}N\check{L}_{\pm 1})\mid {\check{\sigma }}^*u=\pm u\,\}. \end{aligned}$$

Now the other results about $C^{-\infty }_L(M)$ and K(M, L), indicated in Sects. 7.4, 7.5, 7.10 and 7.11, can be obtained by using these extensions of Propositions 7.7 and 7.26 instead of (7.19) and (7.32). Sections 7.12 to 7.15 also have straightforward extensions.

8 Dual-conormal sequence

8.1 The spaces $K'(M,L)$ and $J'(M,L)$

Consider the notation of Sect. 7 assuming that L is transversely oriented; the extension to the non-transversely orientable case can be made like in Sect. 7.16. Like in Sects. 5.1 and 6.22, let

$$\begin{aligned} K'(M,L)=K(M,L;\Omega )',\quad J'(M,L)=J(M,L;\Omega )'. \end{aligned}$$

By (7.26) and (7.32),

$$\begin{aligned} \mathcal {K}'(\varvec{M})\equiv K'(M,L)\oplus K'(M,L)\;,\quad {\dot{\mathcal {A}}}'(\varvec{M})\equiv J'(M,L)\;. \end{aligned}$$

(8.1)

Let also

$$\begin{aligned} \left\{ \begin{aligned} K^{\prime \,(s)}(M,L)=K^{(-s)}(M,L;\Omega )'\;,\quad K^{\prime \,m}(M,L)=K^{-m}(M,L;\Omega )'\;,\\ J^{\prime \,(s)}(M,L)=J^{(-s)}(M,L;\Omega )'\;,\quad J^{\prime \,m}(M,L)=J^{-m}(M,L;\Omega )'\;, \end{aligned}\right. \end{aligned}$$

(8.2)

which satisfy the analogs of (8.1). Like in Sect. 6.22, for $s<s'$ and $m<m'$, we get continuous linear restriction maps

$$\begin{aligned} K^{\prime \,(s')}(M,L)\rightarrow K^{\prime \,(s)}(M,L),\quad K^{\prime \,m}(M,L)\rightarrow K^{\prime \,m'}(M,L), \end{aligned}$$

and continuous injections

$$\begin{aligned} J^{\prime \,(s')}(M,L)\subset J^{\prime \,(s)}(M,L),\quad J^{\prime \,m'}(M,L)\subset J^{\prime \,m}(M,L), \end{aligned}$$

forming projective spectra. By (8.1), its analogs for the spaces (8.2) and according to Sect. 6.22, the spaces $K^{\prime \,(s)}(M,L)$ and $K^{\prime \,m}(M,L)$ satisfy the analogs of (5.4) and (5.5), and the spaces $J^{\prime \,(s)}(M,L)$ and $J^{\prime \,m}(M,L)$ satisfy the analogs of (6.55) and (6.56). Using (8.1), we get the following consequences of Proposition 6.42 and Corollaries 6.43 and 6.44.

Corollary 8.1

$K'(M,L)$ and $J'(M,L)$ are complete Montel spaces.

Corollary 8.2

$K^{\prime \,(s)}(M,L)$ and $J^{\prime \,(s)}(M,L)$ are bornological and barreled.

Corollary 8.3

$K'(M,L)\equiv \varprojlim K^{\prime \,(s)}(M,L)$ and $J'(M,L)=\bigcap _sJ^{\prime \,(s)}(M,L)$.

Like in Sect. 6.22, the versions of (7.15),  (7.28) and Corollary 7.14 with $\Omega M$ induce continuous inclusions

$$\begin{aligned} C^{-\infty }(M)\supset J'(M,L)\supset C^\infty (M,L)\;. \end{aligned}$$

(8.3)

8.2 A description of $J'(M,L)$

With the notation and conditions of Sect. 7.8, the identity (8.1) and Corollaries 6.48 to  6.50 have the following consequences.

Corollary 8.4

$J^{\prime \,m}(M,L)\cong \varvec{x}^mH_{\textrm{b}}^{-\infty }(\varvec{M})=\varvec{x}^{m-\frac{1}{2}}H^{-\infty }(\mathring{\varvec{M}})$ $(m\in \mathbb {R})$.

Corollary 8.5

$J'(M,L)\cong \bigcap _m\varvec{x}^mH_{\textrm{b}}^{-\infty }(\varvec{M})=\bigcap _m\varvec{x}^mH^{-\infty }(\mathring{\varvec{M}})$.

Corollary 8.6

$C^\infty _{\textrm{c}}(M{\setminus } L)$ is dense in every $J^{\prime \,m}(M,L)$ and in $J'(M,L)$. Therefore the right-hand side inclusion of (8.3) is also dense.

The analog of Remark 6.51 makes sense for $J'(M,L)$.

8.3 Description of $K'(M,L)$

The version of Proposition 7.26 with $\Omega M$ has the following direct consequence, where we set

$$\begin{aligned} C^2_m=C^{\infty }(L;\Omega ^{-1}NL\otimes \Omega M)'=C^{\infty }(L;\Omega )'=C^{-\infty }(L) \end{aligned}$$

for every $m\in \mathbb {N}_0$.

Corollary 8.7

The transposes of the versions of (7.36) and (7.37) with $\Omega M$ are TVS-isomorphisms,

$$\begin{aligned} K'(M,L)\xrightarrow {\cong }\prod _{m=0}^\infty C^2_m,\quad K^{\prime \,(s)}(M,L)\xrightarrow {\cong }\prod _{m<s-1/2}C^2_m\quad (s>1/2). \end{aligned}$$

8.4 Dual-conormal sequence

Transposing the density-bundles version of (7.38), we get the commutative diagram

(8.4)

Its maps are compatible with the Sobolev and symbol order filtrations. Its bottom row will be called the dual-conormal sequence of M at L (or of (M, L)). The following analog of Proposition 6.45 holds true with formally the same proof, using Proposition 7.29 and Corollaries 7.13, 7.15 and 7.31.

Proposition 8.8

The dual-conormal sequence of M at L is exact in the category of continuous linear maps between LCSs.

8.5 Action of ${\text {Diff}}(M)$ on the dual-conormal sequence

With the notation of Sect. 7.13, consider the actions of $A^t$ and ${\widetilde{A}}^t$ on the bottom and top spaces of the version of (7.38) with $\Omega M$ and $\Omega \varvec{M}$. Taking transposes again, we get induced actions of A and ${\widetilde{A}}$ on the bottom and top spaces of (8.4), which correspond one another via the linear maps $\varvec{\pi }^*$. These maps satisfy the analogs of (5.7).

8.6 Pull-back of elements of the dual-conormal sequence

With the notation and conditions of Sect. 5.3, besides (5.9), we get continuous linear pull-back maps $\phi ^*:K'(M,L)\rightarrow K'(M',L')$ and $\phi ^*:J'(M,L)\rightarrow J'(M',L')$.

8.7 Push-forward of elements of the dual-conormal sequence

With the notation and conditions of Sect. 5.4, besides (5.11), we get continuous linear push-forward maps $\phi _*:K'_{\textrm{c}}(M',L';\Omega _{\textrm{fiber}})\rightarrow K'_{\textrm{c}}(M,L)$ and $\phi _*:J'_{\textrm{c}}(M',L';\Omega _{\textrm{fiber}})\rightarrow J'_{\textrm{c}}(M,L)$.

8.8 I(M, L) versus $I'(M,L)$

Lemma 8.9

For all $m\in \mathbb {N}_0$, $\varvec{\pi }^*\big (H^{-m}(M)\cap I'(M,L)\big )\subset \dot{H}^{-m}(\varvec{M})\cap \mathcal {A}'(\varvec{M})$.

Proof

Using a volume form on M and its lift to $\varvec{M}$ to define a scalar product of $L^2(M)$ and $L^2(\varvec{M})$, it follows that $\varvec{\pi }^*:C^\infty (M)\rightarrow C^\infty (\varvec{M})$ induces a unitary isomorphism $\varvec{\pi }^*:L^2(M)\rightarrow L^2(\varvec{M})$. Hence the statement is true for $m=0$ because $L^2(\varvec{M})\equiv \dot{H}^0(\varvec{M})$ (Sect. 6.4). Then, for arbitrary m, by (2.22) and (6.25), and according to Sect. 7.1,

$$\begin{aligned}{} & {} \varvec{\pi }^*\big (H^{-m}(M)\cap I'(M,L)\big )\\{} & {} \quad \begin{aligned}&=\varvec{\pi }^*\big ({\text {Diff}}^m(M)\,L^2(M)\cap I'(M,L)\big ) \subset {\text {Diff}}^m(\varvec{M})\,\varvec{\pi }^*L^2(M)\cap \mathcal {A}'(\varvec{M})\\&={\text {Diff}}^m(\varvec{M})\,\dot{H}^0(\varvec{M})\cap \mathcal {A}'(\varvec{M}) \subset \dot{H}^{-m}(\varvec{M})\cap \mathcal {A}'(\varvec{M}). \end{aligned} \end{aligned}$$

$\square$

Lemma 8.10

$\varvec{\pi }^*\big (I(M,L)\cap I'(M,L)\big )\subset C^\infty (\varvec{M})$.

Proof

For every $u\in I(M,L)\cap I'(M,L)$, there is some $m\in \mathbb {N}_0$ such that $u\in I^{(-m)}(M,L)$. Then, by Lemma 8.9, for any $B\in {\text {Diff}}(M,L)$,

$$\begin{aligned} {\widetilde{B}}\varvec{\pi }^*u=\varvec{\pi }^*Bu\in \varvec{\pi }^*\big (H^{-m}(M)\cap I'(M,L)\big )\subset \dot{H}^{-m}(\varvec{M})\cap \mathcal {A}'(\varvec{M}). \end{aligned}$$

Since the operators ${\widetilde{B}}$ ($B\in {\text {Diff}}(M,L)$) generate ${\text {Diff}}({\widetilde{M}},{\widetilde{L}})$ as $C^\infty ({\widetilde{M}})$-module, it follows that $u\in {\dot{\mathcal {A}}}(\varvec{M})\cap \mathcal {A}'(\varvec{M})=C^\infty (\varvec{M})$ by (6.59). $\square$

Theorem 8.11

$I(M,L)\cap I'(M,L)=C^\infty (M)$.

Proof

Suppose there is some non-smooth $u\in I(M,L)\cap I'(M,L)$. However $\varvec{\pi }^*u\in C^\infty (\varvec{M})$ by Lemma 8.10. Then there is a chart (V, y) of L such that, for the induced chart $(U\equiv (-\epsilon ,\epsilon )\times V,(x,y))$ of M, the function u is smooth on $((-\epsilon ,0)\cup (0,\epsilon ))\times V$, and has smooth extensions to $(-\epsilon ,0]\times V$ and $[0,\epsilon )\times V$, but $\partial _x^mu(0^-,y_0)\ne \partial _x^m(0^+,y_0)$ for some $m\in \mathbb {N}_0$ and $y_0\in V$. After multiplying u by a smooth function supported in U whose value at $y_0$ is nonzero, we can assume u is supported in $(-\epsilon /2,\epsilon /2)\times V$. Then there is some $v\in C^\infty (L;\Omega )$ such that ${\text {supp}}v\subset V$ and

$$\begin{aligned} \int _{y\in V}(u(0^-,y)-u(0^+,y))\,v(y)\ne 0\;. \end{aligned}$$

(8.5)

On the other hand, there is a sequence $\phi _k\in C^\infty _{\textrm{c}}(-\epsilon ,\epsilon )$ so that the restrictions of mth derivatives $\phi _k^{(m)}$ to $(-\epsilon /2,\epsilon /2)$ are compactly supported and converge to $\delta _0$ in $C^{-\infty }(-\epsilon /2,\epsilon /2)$ as $k\rightarrow \infty$. For instance, we may take

$$\begin{aligned} \phi _k(t)=h(t)\int _0^t\int _0^{t_{m-1}}\cdots \int _0^{t_1}f_k(t_0)\,dt_0\cdots dt_{m-1}, \end{aligned}$$

where $h,f_k\in C^\infty _{\textrm{c}}(-\epsilon ,\epsilon )$ with $h=1$ on $(-\epsilon /2,\epsilon /2)$, ${\text {supp}}f_k\subset (-\epsilon /2,\epsilon /2)$, $f_k$ is even, and $f_k\rightarrow \delta _0$ in $C^{-\infty }_{\textrm{c}}(-\epsilon /2,\epsilon /2)$ and $f_k(0)\rightarrow \infty$ as $k\rightarrow \infty$. Thus

$$\begin{aligned}{} & {} \phi _k^{(m)}(0)=f_k(0)\rightarrow \infty , \end{aligned}$$

(8.6)

$$\begin{aligned}{} & {} \int _{-\infty }^0a(t)\phi _k^{(m)}(t)\,dt\rightarrow \frac{a(0)}{2},\quad \int _0^\infty b(t)\phi _k^{(m)}(t)\,dt\rightarrow \frac{b(0)}{2}, \end{aligned}$$

(8.7)

for all $a\in C^\infty _{\textrm{c}}(-\infty ,0]$ and $b\in C^\infty _{\textrm{c}}[0,\infty )$.

The sequence $w_k\equiv \phi _k(x)\,v(y)\otimes |dx|\in C^\infty _{\textrm{c}}(T;\Omega )\subset C^\infty (M;\Omega )$ satisfies

$$\begin{aligned} \partial _x^mw_k\equiv \phi _k^{(m)}(x)\,v(y)\otimes |dx|\rightarrow \delta _0(x)\,v(y)\otimes |dx|\equiv \delta _L^v \end{aligned}$$

in $I(M,L;\Omega )$ as $k\rightarrow \infty$. Since $u\in I'(M,L)$ and $\partial _x^{m+1}w_k\in I(M,L;\Omega )$, it follows that $\langle u,\partial _x^{m+1}w_k\rangle \rightarrow \langle u,\partial _x\delta _L^v\rangle$ as $k\rightarrow \infty$. But

$$\begin{aligned} \langle u,\partial _x^{m+1}w_k\rangle&=\int _{y\in V}\int _{-\epsilon /2}^0u(x,y)\,\phi _k^{(m+1)}(x)\,v(y)\,dx\\&\quad +\int _{y\in V}\int _0^{\epsilon /2}u(x,y)\,\phi _k^{(m+1)}(x)\,v(y)\,dx\\&=\phi _k^{(m)}(0)\int _{y\in V}\big (u(0^-,y)-u(0^+,y)\big )\,v(y)\\&\quad -\int _{y\in V}\int _{-\epsilon /2}^0\partial _xu(x,y)\,\phi _k^{(m)}(x)\,v(y)\,dx\\&\quad -\int _{y\in V}\int _0^{\epsilon /2}\partial _xu(x,y)\,\phi _k^{(m)}(x)\,v(y)\,dx\;, \end{aligned}$$

which is divergent by (8.5)–(8.7). $\square$