# A scheme for solving hyperbolic
problems with symbolic structure

Zhuoping Ruan  Department of Mathematics, Nanjing University, Nanjing 210093,
China<EMAIL_ADDRESS>and  Ingo Witt  Mathematical Institute,
University of Göttingen, Bunsenstr. 3-5, D-37073 Göttingen, Germany
<EMAIL_ADDRESS>
###### Abstract.

Hyperbolic problems can at times be solved employing symbolic arguments. This
is especially true for the construction of forward (and backward) fundamental
solutions. We formulate a corresponding abstract scheme and illustrate its
practicality by a number of instructive examples.

###### Key words and phrases:

Symbolic structure, well-posedness, principal symbol map, transport equations.

###### 2010 Mathematics Subject Classification:

Primary: 35L35; Secondary: 35L90

The first author was supported by the NSFC (No. 11771206). Part of the work
was done while the second author was visiting Nanjing University.

###### Contents

  1. 1 Introduction
  2. 2 Verification of results
    1. 2.1 Proof of Theorem 1.1
    2. 2.2 Implications of the assumptions
  3. 3 Various examples
    1. 3.1 Parametrices for elliptic equations
    2. 3.2 A class of ordinary differential operators
    3. 3.3 The strictly hyperbolic Cauchy problem
    4. 3.4 Propagation of conormality
    5. 3.5 Green’s functions
  4. A
    1. A.1 Asymptotic completeness
    2. A.2 Conormal distributions
    3. A.3 The transmission property

##  1\. Introduction

In this article, we present a scheme for solving hyperbolic problems with
symbolic structure. The objective is to promote the idea that using symbolic
arguments is also suitable for hyperbolic problems. In fact, symbolic
arguments are widely used for elliptic equations, but, less known, can
likewise be effectively applied in the hyperbolic realm. The main difference
between both is that one obtains a parametrix (i.e., an inverse up to a
remainder in the residual class) in the elliptic case, whereas one actually
accomplishes to get a genuine inverse for hyperbolic problems.

In the framework we propose the conditions to be imposed are the existence of
a principal symbol map, the unique solvability of the transport equations,
asymptotic completeness, and well-posedness on the level of the residual
class. See below for a precise formulation and a discussion of these
conditions.

Specifically, we are interested in the following situation: Let
${\mathbb{L}}\colon{\mathbb{E}}^{0}\longrightarrow\widetilde{\mathbb{E}}^{0}$
be a linear continuous operator between Fréchet spaces. Suppose that, for a
given $f\in\widetilde{\mathbb{E}}^{0}$, we want to solve the equation

(1.1) $\mathbb{L}u=f$

for $u\in{\mathbb{E}}^{0}$. Suppose, in addition, that Eq. (1.1) comes with a
symbolic structure. By this we mean the following: There are four sequences
$\left\\{{\mathbb{E}}^{j}\right\\}_{j\in\mathbb{N}_{0}}$,
$\left\\{{\mathbb{F}}^{j}\right\\}_{j\in\mathbb{N}_{0}}$,
$\bigl{\\{}\widetilde{{\mathbb{E}}}^{j}\bigr{\\}}_{j\in\mathbb{N}_{0}}$,
$\bigl{\\{}\widetilde{\mathbb{F}}^{j}\bigr{\\}}_{j\in\mathbb{N}_{0}}$ of
Fréchet spaces, where

${\mathbb{E}}^{0}\supseteq{\mathbb{E}}^{1}\supseteq{\mathbb{E}}^{2}\supseteq\dotsc\supseteq{\mathbb{E}}^{\infty},\qquad\widetilde{\mathbb{E}}^{0}\supseteq\widetilde{\mathbb{E}}^{1}\supseteq\widetilde{\mathbb{E}}^{2}\supseteq\dotsc\supseteq\widetilde{\mathbb{E}}^{\infty},$

${\mathbb{E}}^{\infty}=\bigcap_{j}\nolimits{\mathbb{E}}^{j}$,
$\widetilde{\mathbb{E}}^{\infty}=\bigcap_{j}\widetilde{\mathbb{E}}^{j}$, and

${\mathbb{L}}\in\bigcap\nolimits_{j\in\mathbb{N}_{0}}\mathcal{L}\bigl{(}{\mathbb{E}}^{j},\widetilde{\mathbb{E}}^{j}\bigr{)}.$

Moreover, there are linear continuous operators
${\bbsigma}^{j}\colon{\mathbb{E}}^{j}\to{\mathbb{F}}^{j}$,
$\widetilde{\bbsigma}^{j}\colon\widetilde{\mathbb{E}}^{j}\to\widetilde{\mathbb{F}}^{j}$,
and ${\mathbb{T}}^{j}\colon{\mathbb{F}}^{j}\to\widetilde{\mathbb{F}}^{j}$ such
that, for each $j\in\mathbb{N}_{0}$, the diagram

(1.2)
$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>{\mathbb{E}}^{j+1}@>{}>{}>{\mathbb{E}}^{j}@>{{\bbsigma}^{j}}>{}>{\mathbb{F}}^{j}@>{}>{}>0\\\
@V{{\mathbb{L}}}V{}V@V{{\mathbb{L}}}V{}V@V{}V{{\mathbb{T}}^{j}}V\\\
0@>{}>{}>\widetilde{\mathbb{E}}^{j+1}@>{}>{}>\widetilde{\mathbb{E}}^{j}<EMAIL_ADDRESS>
commutes.

We make the following assumptions:

(I) Principal symbol maps. The rows in (1.2) are exact.

(II) Transport equations. The operators
$\mathbb{{\mathbb{T}}}^{j}\colon\mathbb{{\mathbb{F}}}^{j}\to\widetilde{\mathbb{F}}^{j}$
are bijective.

(III) Asymptotic completeness. Given a sequence $\\{u_{j}\\}$ with
$u_{j}\in{\mathbb{E}}^{j}$ for all $j\in\mathbb{N}_{0}$, there is a
$u\in{\mathbb{E}}^{0}$ such that, for all $J\in\mathbb{N}_{0}$,

(1.3) $u-\sum\nolimits_{j<J}u_{j}\in{\mathbb{E}}^{J}.$

(IV) Well-posedness. Given $f\in\widetilde{\mathbb{E}}^{\infty}$, Eq. (1.1)
possesses a unique solution $u\in{\mathbb{E}}^{\infty}$.

The main theorem is the following one:

######  Theorem 1.1.

Suppose that properties (I) through (IV) hold. Then Eq. (1.1) possesses a
unique solution $u\in{\mathbb{E}}^{0}$ for any
$f\in\widetilde{\mathbb{E}}^{0}$.

In applications, it can be fairly hard to provide a framework in which all the
conditions (I) through (IV) are met. This is because to establish such a
framework means nothing less than to reveal the analytic content of the
problem under consideration. We shall discuss five illustrative examples in
Section 3. An even more sophisticated example will appear in [7].

Let us add a few remarks before we proceed:

(a) The open mapping theorem implies that the operator
${\mathbb{L}}\colon{\mathbb{E}}^{0}\to\widetilde{\mathbb{E}}^{0}$ has a
continuous inverse
${\mathbb{L}}^{-1}\colon\widetilde{\mathbb{E}}^{0}\to{\mathbb{E}}^{0}$ if
conditions (I) through (IV) hold. The same is true for
${\mathbb{L}}\colon{\mathbb{E}}^{j}\to\widetilde{\mathbb{E}}^{j}$ and any
$j\in\mathbb{N}_{0}\cup\\{\infty\\}$.

(b) Well-posedness in the residual class, as expressed by (IV), is typical of
hyperbolic problems. For elliptic problems, one usually has conditions (I)
through (III) only. Under these conditions, given
$f\in\widetilde{\mathbb{E}}^{0}$, one finds a $u\in{\mathbb{E}}^{0}$ such
${\mathbb{L}}u-f\in\widetilde{\mathbb{E}}^{\infty}$. This $u$ is unique modulo
${\mathbb{E}}^{\infty}$.

(c) Condition (II) is in effect an ellipticity condition. Indeed, in the
elliptic case, the operator ${\mathbb{T}}^{j}$ is often just multiplication by
the principal symbol of the operator ${\mathbb{L}}$. In the hyperbolic case,
however, by reasons coming from microlocal analysis the principal symbol of
${\mathbb{L}}$ usually vanishes so that the next-order symbol takes control.
(This also relates to the fact that one does not gain as much regularity by
solving Eq. (1.1) as in the elliptic case.) In physics, the ${\mathbb{T}}^{j}$
are usually dubbed transport operators.

(d) Property (1.3) is commonly written as $u\sim\sum_{j}u_{j}$ in
${\mathbb{E}}^{0}$ understanding that, in fact, ${\mathbb{E}}^{0}$ stands for
the filtration $\\{{\mathbb{E}}^{j}\\}$ of ${\mathbb{E}}^{0}$. Notice that
this property determines $u$ uniquely modulo ${\mathbb{E}}^{\infty}$.

When symbolic arguments are exploited in the context of hyperbolic problems,
then this is usually done when working in local coordinates. (The problem
treated in [2, Sec. 5] constitutes a notable exception. It will be reviewed in
Section 3.3.) In such a situation one has symbolic components to all orders
and, accordingly, transport equations are also solved to all orders sort of
instantaneously. One of the fine points about the scheme disclosed here is
that one does not have to care about lower-order symbols, the determination of
them is implicit and runs in the background. Of course, details about lower-
order symbols might be worked out if necessary for the respective application.

The paper is organized as follows: In Section 2, we prove Theorem 1.1 and
discuss a few implications of the assumptions. Section 3 is then devoted to
examples which, as we hope, demonstrate the usefulness of this method. In an
appendix, we describe an abstract scheme for establishing asymptotic
completeness and, for the reader’s convenience, also recall the notion of a
conormal distribution and the transmission property.

Notation. We use standard notation from microlocal analysis, see e.g. [5, 6,
9].

  * •

(Pseudodifferential operators) $\Psi_{\textup{cl}}^{m}(X)$ for
$m\in\mathbb{C}$ is the space of classical pseudodifferential operators of
order $m$ on a ${\mathscr{C}}^{\infty}$ manifold $X$ and
$\sigma_{\psi}^{m}(P)\in S^{(m)}(\dot{T}^{*}X)$ is the principal symbol of an
operator $P\in\Psi_{\textup{cl}}^{m}(X)$.

  * •

(Fourier integral operators) For $\mu\in\mathbb{C}$, ${\mathscr{C}}^{\infty}$
manifolds $X,\,Y$, and a homogeneous canonical relation $C$ from
$\dot{T}^{*}Y$ to $T^{*}X$, $I_{\textup{cl}}^{\mu}(X,Y;C)$ denotes the
corresponding class of classical Fourier integral operators
$A\colon{\mathscr{C}}_{\textup{c}}^{\infty}(Y)\to{\mathscr{C}}^{\infty}(X)$.
(Especially,
$\Psi_{\textup{cl}}^{m}(X)=I_{\textup{cl}}^{m}(X,X;\Delta_{\dot{T}^{*}X})$,
where $\Delta_{\dot{T}^{*}X}\subset\dot{T}^{*}X\times\dot{T}^{*}X$ is the
diagonal.) We have a principal symbol $\sigma_{\psi}^{\mu}(A)\in
S^{(\bar{\mu})}(C;L)$ of a certain homogeneity $\bar{\mu}$, where $L$ is some
line bundle over $C$. (Both $\bar{\mu}$ and $L$ are of no further concern to
us.) For $P\in\Psi_{\textup{cl}}^{m}(X)$ properly supported, $A\in
I_{\textup{cl}}^{\mu}(X,Y;C)$, and $C\subset\dot{T}^{*}X\times\dot{T}^{*}Y$,
we have $PA\in I_{\textup{cl}}^{m+\mu}(X,Y;C)$ and
$\sigma_{\psi}^{m+\mu}(PA)=\sigma_{\psi}^{m}(P)\bigr{|}_{C}\,\sigma_{\psi}^{\mu}(A)$,
where $\sigma_{\psi}^{m}(P)$ is lifted to a function on
$\dot{T}^{*}X\times\dot{T}^{*}Y$ via the projection onto the left factor.

  * •

(Transport operators) In the previous item, suppose that
$\sigma_{\psi}^{m}(P)$ vanishes on the projection of
$C\subset\dot{T}^{*}X\times\dot{T}^{*}Y$ into $\dot{T}^{*}X$. Then $PA\in
I_{\textup{cl}}^{m+\mu-1}(X,Y;C)$ and
$\sigma_{\psi}^{m+\mu-1}(PA)=T_{C}^{P}\sigma_{\psi}^{\mu}(A)$, where the so-
called transport operator $T_{C}^{P}\in\operatorname{Diff}^{1}(C;L)$ is a
first-order differential operator acting on sections of the bundle $L\to C$.
$T_{C}^{P}$ differs from the Lie derivative
$\left(1/\textup{i}\right)\mathcal{L}_{\sigma_{\psi}^{m}(P)}$ by an operator
of order $0$ and is homogeneous of degree $m-1$, i.e., $T_{C}^{P}$ sends
$S^{(\nu)}(C;L)$ into $S^{(\nu+m-1)}(C;L)$ for each $\nu\in\mathbb{C}$.

##  2\. Verification of results

###  2.1. Proof of Theorem 1.1

We need both existence and uniqueness.

#### Uniqueness

Let $u\in{\mathbb{E}}^{0}$ be a solution to ${\mathbb{L}}u=0$. Then, by
induction, $u\in{\mathbb{E}}^{j}$ for all $j$. Indeed, suppose that
$u\in{\mathbb{E}}^{j}$ for some $j\in\mathbb{N}_{0}$. Then
${\mathbb{T}}^{j}{\bbsigma}^{j}(u)=\widetilde{\bbsigma}^{j}({\mathbb{L}}u)=0$.
Hence, ${\bbsigma}^{j}(u)=0$ and $u\in{\mathbb{E}}^{j+1}$.

Consequently, $u\in\bigcap_{j}{\mathbb{E}}^{j}={\mathbb{E}}^{\infty}$. As
${\mathbb{L}}\colon{\mathbb{E}}^{\infty}\to\widetilde{\mathbb{E}}^{\infty}$ is
bijective, we find that $u=0$ as required.

#### Existence

We proceed in three steps.

##### Step 1

Inductively, we construct $u_{j}\in{\mathbb{E}}_{j}$ such that

$g_{j+1}={\mathbb{L}}(u_{0}+u_{1}+\dots+u_{j})-f\in\widetilde{\mathbb{E}}^{j+1},\quad
j\in\mathbb{N}_{0}.$

Indeed, we first pick $u_{0}\in{\mathbb{E}}^{0}$ with
${\mathbb{T}}^{0}{\bbsigma}^{0}(u_{0})=\widetilde{\bbsigma}^{\,0}(f)$. Then
$g_{1}={\mathbb{L}}u_{0}-f\in\widetilde{\mathbb{E}}^{0}$ and
$\tilde{\bbsigma}^{0}(g_{1})={\mathbb{T}}^{0}{\bbsigma}^{\,0}(u_{0})-\widetilde{\bbsigma}^{\,0}(f)=0$,
hence $g_{1}\in\widetilde{\mathbb{E}}^{1}$.

Now suppose that $u_{0},\dots,u_{j}$ for some $j\in\mathbb{N}_{0}$ have
already been constructed. We pick $u_{j+1}\in{\mathbb{E}}^{j+1}$ with
${\mathbb{T}}^{j+1}{\bbsigma}^{j+1}(u_{j+1})=-\,\widetilde{\bbsigma}^{\,j+1}(g_{j+1})$.
Then

$g_{j+2}={\mathbb{L}}(u_{0}+\dots+u_{j}+u_{j+1})-f=g_{j+1}+{\mathbb{L}}u_{j+1}\in\widetilde{\mathbb{E}}^{j+1}$

and

$\widetilde{\bbsigma}^{\,j+1}(g_{j+2})=\widetilde{\bbsigma}^{\,j+1}(g_{j+1})+{\mathbb{T}}^{j+1}{\bbsigma}^{j+1}(u_{j+1})=0,$

hence $g_{j+2}\in\widetilde{\mathbb{E}}^{j+2}$.

##### Step 2

Asymptotic completeness yields a $u^{\prime}\in{\mathbb{E}}^{0}$ such that
$u^{\prime}-\sum_{j<J}u_{j}\in{\mathbb{E}}^{J}$ for all $J\in\mathbb{N}_{0}$.
Then

${\mathbb{L}}u^{\prime}-f={\mathbb{L}}\left(u^{\prime}-\sum\nolimits_{j<J}u_{j}\right)+g_{J}\in\widetilde{\mathbb{E}}^{J},\quad
J\in\mathbb{N}_{0},$

hence $g^{\prime}={\mathbb{L}}u^{\prime}-f\in\widetilde{\mathbb{E}}^{\infty}$.

##### Step 3

Eventually, let $v\in{\mathbb{E}}^{\infty}$ be the unique solution to
${\mathbb{L}}v=-\,g^{\prime}$. Then $u=u^{\prime}+v$ is the desired solution
to Eq. (1.1).

This finishes the proof. ∎

###  2.2. Implications of the assumptions

For the sake of completeness, we mention two consequences of the assumptions.

######  Lemma 2.1.

Suppose (I) through (IV) hold. Then $\widetilde{\mathbb{E}}^{0}$ is
asymptotically complete for the filtration $\\{\widetilde{\mathbb{E}}^{j}\\}$.

###### Proof.

Let $\\{f_{j}\\}\subset\widetilde{\mathbb{E}}^{0}$ be a sequence with
$f_{j}\in\widetilde{\mathbb{E}}^{j}$ for all $j$. From Theorem 1.1 we infer
that there are (uniquely determined) $u_{j}\in{\mathbb{E}}^{j}$ such that
${\mathbb{L}}u_{j}=f_{j}$. Let $u\in{\mathbb{E}}^{0}$ satisfy (1.3) and
$f=Lu\in\widetilde{\mathbb{E}}^{0}$. Then, for all $J\in\mathbb{N}_{0}$,
$u-\sum\nolimits_{j<J}u_{j}\in{\mathbb{E}}^{J}$ and

$f-\sum\nolimits_{j<J}f_{j}={\mathbb{L}}\left(u-\sum\nolimits_{j<J}u_{j}\right)\in\widetilde{\mathbb{E}}^{J}.$

Hence, $f\sim\sum f_{j}$ in $\widetilde{\mathbb{E}}^{0}$. ∎

Often one works in local coordinates, where one has symbolic components to all
orders. On an abstract level, this corresponds to (the existence of and)
fixing a splitting of the first row of (1.2). So, let us additionally assume
that there are linear continous maps
${\bbtau}^{j}\colon{\mathbb{F}}^{j}\to{\mathbb{E}}^{j}$ such that
${\bbsigma}^{j}\circ{\bbtau}^{j}=\textup{id}_{\,{\mathbb{F}}^{j}}$. Then one
can write

${\mathbb{E}}^{0}\cong{\mathbb{E}}^{j}\oplus{\mathbb{F}}^{0}\oplus\dotsc\oplus{\mathbb{F}}^{j-1}$

(algebraically and topologically), where one passes to the next level using
${\mathbb{E}}^{j}\cong{\mathbb{E}}^{j+1}\oplus{\mathbb{F}}^{j}$ through the
identification
$u\overset{\cong}{\longmapsto}\left(u-{\bbtau}^{j}{\bbsigma}^{j}u,{\bbsigma}^{j}u\right)$.

######  Lemma 2.2.

If ${\bbtau}^{j}$ is a splitting of the first row of (1.2), then so is
${\mathbb{L}}\circ{\bbtau}^{j}\circ({\mathbb{T}}^{j})^{-1}$ for the second
row.

###### Proof.

We have
$\widetilde{\bbsigma}^{\,j}{\mathbb{L}}{\bbtau}^{j}({\mathbb{T}}^{j})^{-1}={\mathbb{T}}^{j}{\bbsigma}^{j}{\bbtau}^{j}({\mathbb{T}}^{j})^{-1}=\textup{id}_{\,\widetilde{\mathbb{F}}^{j}}$.
∎

##  3\. Various examples

We discuss a few examples to demonstrate the practicality of the scheme.

###  3.1. Parametrices for elliptic equations

To illustrate the elliptic case, we recall the standard result about the
existence of a parametrix to an elliptic operator. See e.g. [5, 9]. Notice
that in the elliptic case conditions (I) through (III) hold, but condition
(IV) is usually violated.

######  Theorem 3.1.

Let $X$ be ${\mathscr{C}}^{\infty}$ closed manifold and
$P\in\Psi_{\textup{cl}}^{m}(X)$ be an elliptic classical pseudodifferential
operator of order $m\in\mathbb{C}$. Then there is a pseudodifferential
operator $Q\in\Psi_{\textup{cl}}^{-m}(X)$ such that
$PQ-I,\,QP-I\in\Psi^{-\infty}(X)$. $Q$ is unique modulo $\Psi^{-\infty}(X)$.

###### Proof.

To illustrate the applicability of the abstract scheme, we show how to
construct a right parametrix. A left parametrix is constructed in the same
manner, the proof is then concluded in the usual way.

We identify all the players in the scheme: ${\mathbb{L}}$ is multiplication by
$P$ from the left (and then $u=Q$, $f=I$),
${\mathbb{E}}^{j}=\Psi_{\textup{cl}}^{-m-j}(X)$,
$\widetilde{\mathbb{E}}^{j}=\Psi_{\textup{cl}}^{-j}(X)$,
${\mathbb{F}}^{j}=S^{(-m-j)}(\dot{T}^{*}X)$,
$\widetilde{\mathbb{F}}^{j}=S^{(-j)}(\dot{T}^{*}X)$,
${\bbsigma}^{j}=\sigma_{\psi}^{-m-j}$,
$\widetilde{\bbsigma}^{\,j}=\sigma_{\psi}^{-j}$, and ${\mathbb{T}}^{j}$ is
multiplication by $\sigma_{\psi}^{m}(P)$.

It is well-known that (I) through (III) hold in this situation. Hence, there
is a $Q\in\Psi_{\textup{cl}}^{-m}(X)$ such that $PQ-I\in\Psi^{-\infty}(X)$. ∎

######  Remark.

The Neumann series argument usually invoked in the parametrix construction is
not required here.

###  3.2. A class of ordinary differential operators

We construct a specific fundamental system for a class of ordinary
differential operators $L$ on the half-line
$\overline{\mathbb{R}}_{+}=[0,\infty)$. The result presented here, when it
applies, is more precise than known results, see e.g. [3, 10, 11]. Beyond
that, the proof is considerably shorter.

Let ${\ell_{*}}>-1$. The ordinary differential operators $L$ under
consideration are of the form

$L=D_{t}^{m}+\sum_{r=1}^{m}a_{r}(t)D_{t}^{m-r},$

where $a_{r}\in
S_{{\textup{cl}},{\ell_{*}}+1}^{r{\ell_{*}}}(\overline{\mathbb{R}}_{+})$ for
$1\leq r\leq m$. Here, $b\in
S_{{\textup{cl}},{\ell_{*}}+1}^{\mu}(\overline{\mathbb{R}}_{+})$ for
$\mu\in\mathbb{C}$ means that
$b\in{\mathscr{C}}^{\infty}(\overline{\mathbb{R}}_{+})$ and there are
coefficients $b_{j}\in\mathbb{C}$ such that, for all $J,\,k\in\mathbb{N}_{0}$,

$\left|\partial_{t}^{k}\left(b(t)-\chi(t)\sum\nolimits_{j<J}b_{j}\,t^{\mu-j({\ell_{*}}+1)}\right)\right|\lesssim\langle
t\rangle^{\Re\mu-J({\ell_{*}}+1)-k}.$

Here, $\chi\in{\mathscr{C}}^{\infty}(\overline{\mathbb{R}}_{+})$ is an
excision function, i.e., $\chi(t)=0$ for $t\leq 1$ and $\chi(t)=1$ for $t\geq
2$. Writing
$a_{r}(t)=a_{r0}t^{r{\ell_{*}}}+a_{r1}t^{r{\ell_{*}}-({\ell_{*}}+1)}+O(t^{r{\ell_{*}}-2({\ell_{*}}+1)})$
as $t\to\infty$, we introduce

(3.1) $l_{0}(\tau)=\tau^{m}+\sum_{r=1}^{m}a_{r0}\tau^{m-r},\quad
l_{1}(\tau)=\sum_{r=1}^{m}a_{r1}\tau^{m-r}.$

######  Proposition 3.2.

Let $\mu\in\mathbb{R}$ be a simple root of the polynomial $l_{0}$ and

$\Delta=\Delta(\mu)=-\,\frac{{\ell_{*}}\mu\,l_{0}^{\prime\prime}(\mu)/2+\textup{i}\,l_{1}(\mu)}{l_{0}^{\prime}(\mu)}\in\mathbb{C}.$

Then the equation $Lu=0$ has a unique solution of the form

$u(t)=\textup{e}^{\textup{i}\mu\,\frac{t^{{\ell_{*}}+1}}{{\ell_{*}}+1}}c(t),$

where $c\in
S_{{\textup{cl}},{\ell_{*}}+1}^{\Delta}(\overline{\mathbb{R}}_{+})$ and
$c(t)=t^{\Delta}+O(t^{\Delta-({\ell_{*}}+1)})$ as $t\to\infty$.

###### Proof.

For $\delta\in\mathbb{C}$, we set

$\mathcal{Z}^{\delta}=\bigl{\\{}\textup{e}^{\textup{i}\mu\,\frac{t^{{\ell_{*}}+1}}{{\ell_{*}}+1}}c(t)\bigm{|}c\in
S_{{\textup{cl}},{\ell_{*}}+1}^{\delta}(\overline{\mathbb{R}}_{+})\bigr{\\}}.$

We further set $\Gamma_{0}^{\delta}u=c_{0}$, $\Gamma_{1}^{\delta}u=c_{1}$ for
$u=\textup{e}^{\textup{i}\mu\,\frac{t^{{\ell_{*}}+1}}{{\ell_{*}}+1}}c(t)\in\mathcal{Z}^{\delta}$
if
$c(t)=c_{0}t^{\delta}+c_{1}t^{\delta-({\ell_{*}}+1)}+O(t^{\delta-2({\ell_{*}}+1)})$
as $t\to\infty$. Notice that
$\Gamma_{0}^{\delta}u=\Gamma_{1}^{\delta+({\ell_{*}}+1)}u$ if
$u\in\mathcal{Z}^{\delta}\subset\mathcal{Z}^{\delta+({\ell_{*}}+1)}$.

A direct computation yields that
$L\colon\mathcal{Z}^{\delta}\to\mathcal{Z}^{\delta+m{\ell_{*}}}$,
$\Gamma_{0}^{\delta+m{\ell_{*}}}(Lu)=l_{0}(\mu)\,\Gamma_{0}^{\delta}u=0$, and

$\Gamma_{1}^{\delta+m{\ell_{*}}}(Lu)=l_{0}(\mu)\,\Gamma_{1}^{\delta}u-\textup{i}\left[\delta\,l_{0}^{\prime}(\mu)+{\ell_{*}}\mu\,l_{0}^{\prime\prime}(\mu)/2+\textup{i}\,l_{1}(\mu)\right]\Gamma_{0}^{\delta}u.$

So,
$L\colon\mathcal{Z}^{\delta}\to\mathcal{Z}^{\delta+m{\ell_{*}}-({\ell_{*}}+1)}$
in view of $l_{0}(\mu)=0$ and

$\Gamma_{0}^{\delta+m{\ell_{*}}-({\ell_{*}}+1)}(Lu)=-\textup{i}\left[\delta\,l_{0}^{\prime}(\mu)+{\ell_{*}}\mu\,l_{0}^{\prime\prime}(\mu)/2+\textup{i}\,l_{1}(\mu)\right]\Gamma_{0}^{\delta}u.$

Note that $\Gamma_{0}^{\Delta+m{\ell_{*}}-({\ell_{*}}+1)}(Lu)=0$ for all
$u\in\mathcal{Z}^{\Delta}$ by construction.

We now apply the abstract scheme with ${\mathbb{L}}=L$,

${\mathbb{E}}^{j}=\mathcal{Z}^{\Delta-j({\ell_{*}}+1)},\quad\widetilde{\mathbb{E}}^{j}=\mathcal{Z}^{\Delta+m{\ell_{*}}-(j+1)({\ell_{*}}+1)},\quad{\mathbb{F}}^{j}=\widetilde{\mathbb{F}}^{j}=\mathbb{C},$

${\bbsigma}^{j}=\Gamma_{0}^{\Delta-j({\ell_{*}}+1)}$,
$\widetilde{\bbsigma}^{\,j}=\Gamma_{0}^{\Delta+m{\ell_{*}}-(j+1)({\ell_{*}}+1)}$,
and ${\mathbb{T}}^{j}$ is multiplication by $\textup{i}j({\ell_{*}}+1)$. Then
${\mathbb{E}}^{\infty}=\widetilde{\mathbb{E}}^{\infty}={\mathscr{S}}(\overline{\mathbb{R}}_{+})$
in view of $\mu\in\mathbb{R}$. We observe that properties (I) through (IV) are
fulfilled for $j\geq 1$.

Accordingly, we choose a $u_{0}\in\mathcal{Z}^{\Delta}$ with
$\Gamma_{0}^{\Delta}u=1$. There is a unique solution
$v\in\mathcal{Z}^{\Delta-({\ell_{*}}+1)}$ to the equation $L(u_{0}+v)=0$. (To
see this, write
$Lv=-\,Lu_{0}\in\mathcal{Z}^{\Delta+m{\ell_{*}}-2({\ell_{*}}+1)}$.) Then
$u=u_{0}+v$ is the sought solution. ∎

######  Corollary 3.3.

Suppose that the polynomial $l_{0}$ has $m$ simple real roots
$\mu_{1}<\dotsc<\mu_{m}$,

$l_{0}(\tau)=\prod_{h=1}^{m}(\tau-\mu_{h}).$

Then the ordinary differential operator $L$ possesses a fundamental system

$\textup{e}^{\textup{i}\mu_{1}\,\frac{t^{{\ell_{*}}+1}}{{\ell_{*}}+1}}c_{1}(t),\dotsc,\textup{e}^{\textup{i}\mu_{m}\,\frac{t^{{\ell_{*}}+1}}{{\ell_{*}}+1}}c_{m}(t),$

where $c_{h}\in
S_{{\textup{cl}},{\ell_{*}}+1}^{\Delta_{h}}(\overline{\mathbb{R}}_{+})$ with
$\Delta_{h}=\Delta(\mu_{h})$ for $1\leq h\leq m$.

###  3.3. The strictly hyperbolic Cauchy problem

The following example is briefly touched upon. This example is taken from [2,
Sec. 5] and revised to fit into the abstract scheme. More mathematical detail
will be provided for the example studied in Section 3.4, which is similar in
spirit. For the present example, the assumptions needed for the scheme to work
follow from the theory of Fourier integral operators. See e.g. [2, 6] as well
as the notational paragraph at the end of Section 1.

Let $X$ be ${\mathscr{C}}^{\infty}$ manifold, $P\in\Psi_{\textup{cl}}^{m}(X)$,
$Q_{r}\in\Psi_{\textup{cl}}^{m_{r}}(X)$ for $1\leq r\leq\mu$, and $Y\subset X$
be a closed ${\mathscr{C}}^{\infty}$ hypersurface. We assume the following:

  * •

The principal symbol $p\in S^{(m)}(\dot{T}^{*}X)$ of $P$ is real-valued and
characteristic curves of $P$ meet $Y$ transversally.

  * •

The algebraic problem $p(y,\xi)=0$ and $\xi\bigr{|}_{T_{y}Y}=\eta$ has, for
any $(y,\eta)\in\dot{T}^{*}Y$, exactly $\mu$ solutions
$\xi_{1}(y,\eta),\dots,\xi_{\mu}(y,\eta)\in T_{y}^{*}X$.

  * •

The matrix $\bigl{(}q_{r}(y,\xi_{j}(y,\eta))\bigr{)}_{j,r=1,\dots,\mu}$ is
non-singular for all $(y,\eta)\in\dot{T}^{*}Y$. Here, $q_{r}\in
S^{(m_{r})}(\dot{T}^{*}X)$ is the principal symbol of $Q_{r}$.

Let $\gamma_{Y}\colon{\mathscr{C}}^{\infty}(X)\to{\mathscr{C}}^{\infty}(Y)$,
$v\mapsto v\bigr{|}_{Y}$ be restriction to $Y$.

######  Theorem 3.4.

Under the assumptions above, there is a neighborhood $X_{0}\subseteq X$ of $Y$
in which a parametrix $E$ to the problem

$\left\\{\begin{aligned} Pu&=0&&\textup{in $X_{0}$,}\\\
\gamma_{Y}Q_{r}u&=g_{r}&&\textup{on $Y$,}&1\leq r\leq\mu,\end{aligned}\right.$

exists. This parametrix
$E\colon\mathscr{E}^{\prime}(Y;\mathbb{C}^{\mu})\to\mathscr{D}^{\prime}(X_{0})$
is of the form

$\bigl{(}g_{1},\dotsc,g_{\mu}\bigr{)}\mapsto\sum\nolimits_{r=1}^{\mu}E_{r}g_{r},$

where $E_{r}\in I_{\textup{cl}}^{-m_{r}-1/4}(X_{0},Y;C)$ for $1\leq r\leq\mu$
with the canonical relation $C$ being the flow-out from
$\bigl{\\{}(x,\xi,y,\eta)\in\dot{T}^{*}X_{0}\times\dot{T}^{*}Y\bigm{|}x=y,\,\xi\bigr{|}_{T_{y}Y}=\eta,\,p(x,\xi)=0\bigr{\\}}$
under the Hamiltonian vector field $H_{p}$.

###### Proof.

First observe that $C\subset(T^{*}X_{0}\times T^{*}Y)\setminus 0$ is an
embedded submanifold if the neighborhood $X_{0}$ of $Y$ is chosen small
enough. (It has actually $\mu$ components corresponding to the $\mu$
characteristic roots $\xi_{j}$.) So, the class
$I_{\textup{cl}}^{\nu}(X_{0},Y;C)$ for $\nu\in\mathbb{C}$ of Fourier integral
operators is well-defined.

Then we set up the scheme as follows: ${\mathbb{L}}$ is the linear operator

$\bigl{\\{}E_{s}\bigr{\\}}_{s=1}^{\mu}\mapsto\biggl{(}\bigl{\\{}PE_{s}\bigr{\\}}_{s=1}^{\mu},\bigl{\\{}\gamma_{Y}Q_{r}E_{s}\bigr{\\}}_{r,s=1}^{\mu}\biggr{)}.$

Moreover,

$\displaystyle{\mathbb{E}}^{j}$
$\displaystyle=\bigoplus\nolimits_{s=1}^{\mu}I_{\textup{cl}}^{-m_{s}-j-1/4}(X_{0},Y;C),$
$\displaystyle\widetilde{\mathbb{E}}^{j}$
$\displaystyle=\bigoplus\nolimits_{s=1}^{\mu}I_{\textup{cl}}^{m-m_{s}-j-5/4}(X_{0},Y;C)\oplus\bigoplus\nolimits_{r,s=1}^{\mu}\Psi_{\textup{cl}}^{m_{r}-m_{s}-j}(Y),$
$\displaystyle{\mathbb{F}}^{j}$
$\displaystyle=\bigoplus\nolimits_{s=1}^{\mu}S^{(-\bar{m}_{s}-j-1/4)}(C;L),$
$\displaystyle\widetilde{\mathbb{F}}^{j}$
$\displaystyle=\bigoplus\nolimits_{s=1}^{\mu}S^{(m-\bar{m}_{s}-j-5/4)}(C;L)\oplus\bigoplus\nolimits_{r,s=1}^{\mu}S^{(m_{r}-m_{s}-j)}(\dot{T}^{*}Y),$

${\bbsigma}^{j}$ and $\widetilde{\bbsigma}^{\,j}$ are composed of the
respective principal symbol maps, and the transport operators are given as

${\mathbb{T}}^{j}=\Bigl{(}\\{T_{C}^{P}\\}_{1\leq s\leq\mu},H^{j}\Bigr{)}$

where the linear operator
$H^{j}\colon{\mathbb{F}}^{j}\to\bigoplus\nolimits_{r,s=1}^{\mu}S^{(m_{r}-m_{s}-j)}(\dot{T}^{*}Y)$
is composition with the principal symbol of $\gamma_{Y}\in
I_{\textup{cl}}^{1/4}(Y,X_{0};R)$ (here $R=(\dot{N}_{Y\times
X_{0}}\Delta_{Y})^{\prime}$) followed by pointwise multiplication from the
left by the matrix
$\bigl{(}q_{r}(y,\xi_{j}(y,\eta))\bigr{)}_{j,r=1,\dots,\mu}$. This provides
initial conditions for solving the transport equations.

By construction, conditions (I) through (III) are fulfilled. ∎

The (microlocal) constructions above can be globalized upon making additional
global assumptions on the characteristic flow. Moreover, for the genuine
strictly hyperbolic Cauchy problem and $P$ being a differential operator,
${\mathscr{C}}^{\infty}$ well-posedness turns the parametrix $E$ into a
fundamental solution. For details, see again [2, 6].

###  3.4. Propagation of conormality

Propagation of conormality (see Appendix A.2) is much related to the well-
posedness of the hyperbolic Cauchy problem. It is a more precise statement
than the better known statement about the mere propagation of singularities
(which, in turn, is a microlocalized version of the basic energy
inequalities), insofar as the kind of singularities being propagated gets
specified.

We consider the strictly hyperbolic Cauchy problem. Coordinate invariance
being known, we can work in local coordinates
$(t,x)\in(0,T)\times\mathbb{R}^{n}$. So, let

$L=D_{t}^{m}+\sum_{\begin{subarray}{c}j+|\alpha|\leq m,\\\
j<m\end{subarray}}a_{j\alpha}(t,x)D_{t}^{j}D_{x}^{\alpha}$

with coefficients
$a_{j\alpha}\in{\mathscr{C}}_{\textup{b}}^{\infty}([0,T]\times\mathbb{R}^{n})$.
We assume $L$ to be strictly hyperbolic, i.e.,

$\sigma_{\psi}^{m}(L)(t,x,\tau,\xi)=\prod_{h=1}^{m}(\tau-\mu_{h}(t,x,\xi))$

for real-valued $\mu_{h}\in
S^{(1)}([0,T]\times\mathbb{R}^{n}\times\dot{\mathbb{R}}^{n})$, where, in
addition,

$\left|\mu_{h}(t,x,\xi)-\mu_{h^{\prime}}(t,x,\xi)\right|\gtrsim|\xi|,\quad
h\neq h^{\prime}.$

We are interested in the inhomogeneous Cauchy problem

(3.2) $\left\\{\enspace\begin{aligned}
&Lu=f(t,x),&&(t,x)\in(0,T)\times\mathbb{R}^{n},\\\
&D_{t}^{p}u\bigr{|}_{t=0}=g_{p}(x),&&0\leq p\leq m-1.\end{aligned}\right.$

Let $\Sigma\subset\mathbb{R}^{n}$ by a ${\mathscr{C}}^{\infty}$ hypersurface,
say of bounded geometry. Let $T>0$ be small enough so that we have $m$
characteristic hypersurfaces
$\Sigma_{1},\dots,\Sigma_{m}\subset[0,T]\times\mathbb{R}^{n}$ of $L$ emanating
from $\\{0\\}\times\Sigma$, with $\Sigma_{h}$ belonging to the $h$th
characteristic root $\mu_{h}$.

######  Theorem 3.5.

Let $\mu\in\mathbb{C}$, $f=\sum_{h=1}^{m}f_{h}$, where $f_{h}\in
I_{\textup{cl}}^{m+\mu-1}([0,T]\times\mathbb{R}^{n},\Sigma_{h})$ for $1\leq
h\leq m$, and $g_{p}\in I_{\textup{cl}}^{\mu+p+1/4}(\mathbb{R}^{n},\Sigma)$
for $0\leq p\leq m-1$. Then the unique solution $u$ to (3.2) is of the form
$u=\sum_{h=1}^{m}u_{h}$, where $u_{h}\in
I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\Sigma_{h})$ for $1\leq h\leq
m$.

###### Proof.

The bundles $\Omega_{\dot{N}^{*}\Sigma_{h}}^{1,\textup{v}}$ (see Appendix A.2)
are trivialized as we work in local coordinates. Hence,
$\sigma_{\psi}^{\nu}(v)\in S^{(\bar{\nu})}(\dot{N}^{*}\Sigma_{h})$ for $v\in
I_{\textup{cl}}^{\nu}([0,T]\times\mathbb{R}^{n},\Sigma_{h})$, where now
$\bar{\nu}=\nu+(n-1)/4$. Similarly for $g\in
I_{\textup{cl}}^{\nu+1/4}(\mathbb{R}^{n};\Sigma)$ and
$\sigma_{\psi}^{\nu}(g)\in S^{(\bar{\nu})}(\dot{N}^{*}\Sigma)$.

We apply the abstract scheme employing the following setup:
${\mathbb{L}}u=\left(Lu,u\bigr{|}_{t=0},D_{t}u\bigr{|}_{t=0},\dotsc,\right.$
$\left.D_{t}^{m-1}u\bigr{|}_{t=0}\right)$,

$\displaystyle{\mathbb{E}}^{j}$
$\displaystyle=\biggl{\\{}\sum\nolimits_{h=1}^{m}u_{h}\biggm{|}\text{$u_{h}\in
I_{\textup{cl}}^{\mu-j}([0,T]\times\mathbb{R}^{n},\Sigma_{h})$ for $1\leq
h\leq m$}\biggr{\\}},$ $\displaystyle\widetilde{\mathbb{E}}^{j}$
$\displaystyle=\biggl{\\{}\sum\nolimits_{h=1}^{m}f_{h}\biggm{|}\text{$f_{h}\in
I_{\textup{cl}}^{m+\mu-j-1}([0,T]\times\mathbb{R}^{n},\Sigma_{h})$ for $1\leq
h\leq
m$}\biggr{\\}}\oplus\bigoplus_{p=0}^{m-1}I_{\textup{cl}}^{\mu+p-j+1/4}(\mathbb{R}^{n},\Sigma).$

To justify this, note that $\sigma_{\psi}^{m+\mu-j}(Lu)=0$ for $u\in
I_{\textup{cl}}^{\mu-j}([0,T]\times\mathbb{R}^{n},\Sigma_{h})$ due to the fact
that $\sigma_{\psi}^{m}(L)\bigr{|}_{\dot{N}^{*}\Sigma_{h}}=0$. We further set

${\mathbb{F}}^{j}=\bigoplus_{h=1}^{m}S^{\bar{\mu}-j}(\dot{N}^{*}\Sigma_{h}),\quad\widetilde{\mathbb{F}}^{j}=\bigoplus_{h=1}^{m}S^{m+\bar{\mu}-j-1}(\dot{N}^{*}\Sigma_{h})\oplus\bigoplus_{p=0}^{m-1}S^{(\bar{\mu}+p-j)}(\dot{N}^{*}\Sigma)$

and

$\displaystyle{\bbsigma}^{j}(u)$
$\displaystyle=\Bigl{(}\\{\sigma_{\psi}^{\mu-j}(u_{h})\\}_{1\leq h\leq
m}\Bigr{)},$
$\displaystyle\widetilde{\bbsigma}^{\,j}\bigl{(}f,\\{g_{p}\\}_{0\leq p\leq
m-1}\bigr{)}$
$\displaystyle=\Bigl{(}\\{\sigma_{\psi}^{m+\mu-j-1}(f_{h})\\}_{1\leq h\leq
m},\\{\sigma_{\psi}^{\mu+p-j+1/4}(g_{p})\\}_{0\leq p\leq m-1}\Bigr{)}$

writing $u=\sum_{h}u_{h}$ and $f=\sum_{h}f_{h}$ as above. The transport
operators are

${\mathbb{T}}^{j}=\Bigl{(}\\{T_{\Sigma_{h}}^{p}\\}_{1\leq h\leq
m},H^{j}\Bigr{)}$

where
$H^{j}\colon{\mathbb{F}}^{j}\to\bigoplus_{p=0}^{m-1}S^{(\bar{\mu}+p-j)}(\dot{N}^{*}\Sigma)$
is (up to a multiplicative factor of $(2\pi)^{-1}$) the linear operator of
restriction to $\\{t=0\\}$ followed by pointwise multiplication from the left
by the matrix $\bigl{(}\mu_{h}(0,x,\xi)^{p}\bigr{)}_{\begin{subarray}{c}1\leq
h\leq m,\\\ 0\leq p\leq m-1\end{subarray}}$ for $(x,\xi)\in\dot{N}^{*}\Sigma$.
To see this note that
$\sigma_{\psi}^{\nu+p}(D_{t}^{p}v)=\tau^{p}\sigma_{\psi}^{\nu}(v)$ for $v\in
I_{\textup{cl}}^{\nu}([0,T]\times\mathbb{R}^{n};\Sigma_{h})$ and
$\tau=\mu_{h}(0,x,\xi)$ for $(0,x,\tau,\xi)\in\dot{N}^{*}\Sigma_{h}$. Further
note that the matrix
$\bigl{(}\mu_{h}(0,x,\xi)^{p}\bigr{)}_{\begin{subarray}{c}1\leq h\leq m,\\\
0\leq p\leq m-1\end{subarray}}$ is non-singular as a Vandermonde matrix,
considering that the characteristic roots are pairwise distinct.

Hence, the ${\mathbb{T}}^{j}$ are indeed bijective (as implied by the method
of characteristics) and conditions (I) through (IV) of the abstract scheme are
fulfilled. Notice that condition (IV) is nothing else than the
${\mathscr{C}}^{\infty}$ well-posedness of the Cauchy problem (3.2). ∎

###  3.5. Green’s functions

It is a well-known fact that the future light cone is the singular support of
the forward Green’s function (henceforth called Green’s function) of the wave
operator and that, in odd space dimensions, the Green’s function is smooth
from both sides up to the future light cone.111Indeed, it is zero outside the
future light cone by finite propagation speed. This property holds in greater
generality as we are going to show now. As we shall see, only certain
algebraic properties of the amplitude functions, which arise when writing the
conormal distributions under consideration as oscillatory integrals, are
indeed responsable for that result to hold, and these algebraic properties are
propagated. General references are [1, 4].

######  Remark.

A similar discussion is possible for even space dimensions. In this case, the
result is that the Green’s function of the wave operator is zero outside the
future light cone, in particular, smooth from the outside of the future light
cone up the future light cone, and it is what one calls diffuse from the
inside.

We consider a second-order strictly hyperbolic differential operator $P$ with
coefficients in
${\mathscr{C}}_{\textup{b}}^{\infty}([0,T]\times\mathbb{R}^{n})$, which we
write as

(3.3) $P=D_{t}^{2}+p_{1}(t,x,D_{x})D_{t}+p_{2}(t,x,D_{x}).$

We denote by $q_{l}(t,x,\xi)$ for $l=1,2$ the principal symbol of
$p_{l}(t,x,D_{x})$. We assume that $q_{1}$, $q_{2}$ are real-valued and that

$q_{2}(t,x,\xi)\lesssim-|\xi|^{2}.$

######  Example.

An example is the wave operator $\Box=D_{t}^{2}+c^{2}(t,x)\Delta_{x}$ with
variable propagation speed, where
$c\in{\mathscr{C}}_{\textup{b}}^{\infty}([0,T]\times\mathbb{R}^{n})$ and
$c(t,x)\gtrsim 1$.

Let $\mu^{\pm}\in
S^{(1)}([0,T]\times\mathbb{R}^{n}\times\dot{\mathbb{R}}^{n})$ with

$\mu^{\pm}(t,x,\xi)=-\,\frac{1}{2}\,q_{1}(t,x,\xi)\pm\frac{1}{2}\sqrt{q_{1}^{2}(t,x,\xi)-4q_{2}(t,x,\xi)}$

denote the characteristic roots of $P$. Notice that
$\pm\,\mu^{\pm}(t,x,\xi)\gtrsim|\xi|$ and that

(3.4) $\mu^{\pm}(t,x,-\xi)=-\,\mu^{\mp}(t,x,\xi).$

Further let $[0,T]\ni
t\mapsto\gamma^{\pm}(t;x^{0},\xi^{0})=\left(t,x(t),\tau(t),\xi(t)\right)$ for
$(x^{0},\xi^{0})\in\mathbb{R}^{n}\times\dot{\mathbb{R}}^{n}$ denote the null
bicharacteristic emanating from
$\left(0,x^{0},\mu^{\pm}(0,x^{0},\xi^{0}),\xi^{0}\right)$. It is given as the
solution to the system

$\left\\{\enspace\begin{aligned}
&x^{\prime}(t)=-\,\mu_{\xi}^{\pm}(t,x(t),\xi(t)),\quad\xi^{\prime}(t)=\mu_{x}^{\pm}(t,x(t),\xi(t)),\\\
&x(0)=x^{0},\quad\xi(0)=\xi^{0},\end{aligned}\right.$

and $\tau(t)=\mu^{\pm}(t,x(t),\xi(t))$ follows. Notice that in view of (3.4)
we have that

(3.5)
$x^{\pm}(t;x^{0};\xi^{0})=x^{\mp}(t;x^{0};-\xi^{0}),\quad\xi^{\pm}(t;x^{0};\xi^{0})=-\,\xi^{\mp}(t;x^{0};-\xi^{0}),$

whereas $\pm\tau^{\pm}(t;x^{0},\xi^{0})>0$.

We also introduce the Lagrangian manifolds with boundary

$\Lambda^{\pm}=\\{\gamma^{\pm}(t;0,\xi^{0})\mid
t\in[0,T],\,\xi^{0}\in\dot{\mathbb{R}}^{n}\\}\subset\dot{T}^{\ast}([0,T]\times\mathbb{R}^{n}).$

Notice that $\Lambda^{+}\cap\Lambda^{-}=\emptyset$. $\Lambda^{\pm}$ are
parametrized by the non-degenerate phase functions
$\phi^{\pm}=\phi^{\pm}(t,x,\xi)$ which are found as solutions to the eikonal
equations

(3.6) $\left\\{\enspace\begin{aligned}
\partial_{t}\phi^{\pm}&=\mu^{\pm}(t,x,\nabla\phi^{\pm}),&&(t,x)\in(0,T)\times\mathbb{R}^{n},\\\
\phi^{\pm}\bigr{|}_{t=0}&=x\cdot\xi.\end{aligned}\right.$

Here, $\xi\in\dot{\mathbb{R}}^{n}$ acts as a parameter. These are Hamilton-
Jacobi equation which are solved utilizing the method of characteristics.
Notice that, in view of our assumptions, Eq. (3.6) can be solved up to time
$t=T$.

######  Example.

For the wave operator $P=D_{t}^{2}+c^{2}(t)\Delta_{x}$ with $c$ independent of
$x$, we have

$\Lambda^{\pm}=\\{(t,x,\tau,\xi)\mid
x=\mp\,C(t)\xi/|\xi|,\,\tau=\pm\,c(t)|\xi|,\,t\in[0,T],\,\xi\in\dot{\mathbb{R}}^{n}\\}.$

and $\phi^{\pm}(t,x,\xi)=x\cdot\xi\pm C(t)|\xi|$, where
$C(t)=\int_{0}^{t}c(s)\,\textup{d}s$.

At last, we introduce the future light cone with vertex at the origin
$\mathscr{O}=(0,0)$,

$\mathcal{C}=\\{(t,x)\in[0,T]\times\mathbb{R}^{n}\mid\text{$x=x^{\pm}(t;0,\xi^{0})$
for some $\pm\,\xi^{0}\in\dot{\mathbb{R}}^{n}$}\\},$

see (3.5)

######  Lemma 3.6.

For $T>0$ small,
$\mathcal{C}\setminus\mathscr{O}\subset(0,T]\times\mathbb{R}^{n}$ is a
hypersurface222The fact that $\mathcal{C}\setminus\mathscr{O}$ has a boundary
at $t=T$ plays no role here.. Furthermore,

$\dot{N}^{*}(\mathcal{C}\setminus\mathscr{O})=(\Lambda^{+}\sqcup\Lambda^{-})\cap\\{0<t\leq
T\\}.$

For $(t,x,\tau,\xi)\in\dot{N}^{*}(\mathcal{C}\setminus\mathscr{O})$, we have
that $(t,x,\tau,\xi)\in\Lambda^{\pm}$ if and only if $\pm\tau>0$.

In view of Lemma (3.6), we set

$I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})=I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\Lambda^{+})+I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\Lambda^{-})$

for $\mu\in\mathbb{C}$ and say that distributions in
$I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})$ are classically
conormal, of order $\mu$, with respect to $\mathcal{C}$. Notice that
$I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\Lambda^{+})\cap
I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\Lambda^{-})={\mathscr{C}}_{\textup{b}}^{\infty}([0,T]\times\mathbb{R}^{n})$.

######  Remark.

It holds

$I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})\subset{\mathscr{C}}^{\infty}([0,T];{\mathscr{S}}^{\prime}(\mathbb{R}^{n})).$

More precisely, for $u\in
I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})$, we have that
$u\bigr{|}_{\\{t\\}\times\mathbb{R}^{n}}\in
I_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\mathcal{C}\cap(\\{t\\}\times\mathbb{R}^{n}))$
for $0<t\leq T$, while $u\bigr{|}_{\\{0\\}\times\mathbb{R}^{n}}\in
I_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\\{0\\})$.

We now consider the inhomogeneous Cauchy problem

(3.7) $\left\\{\enspace\begin{aligned}
&Pu=f(t,x),\qquad(t,x)\in(0,T)\times\mathbb{R}^{n},\\\
&u\bigr{|}_{t=0}=g_{0}(x),\quad
D_{t}u\bigr{|}_{t=0}=g_{1}(x).\end{aligned}\right.$

We first show that the solution $u$ is conormal with respect to $\mathcal{C}$
under the condition that the initial data $(g_{0},g_{1})$ is conormal with
respect to the origin and that the right-hand side $f$ is also conormal with
respect to $\mathcal{C}$.

######  Theorem 3.7.

Let $\mu\in\mathbb{C}$, $g_{0}\in
I_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\\{0\\})$, $g_{1}\in
I_{\textup{cl}}^{\mu+5/4}(\mathbb{R}^{n},\\{0\\})$, and $f\in
I_{\textup{cl}}^{\mu+1}([0,T]\times\mathbb{R}^{n},\mathcal{C})$. Then the
unique solution $u$ to Eq. (3.7) belongs to
$I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})$.

###### Proof.

We apply the abstract scheme with
${\mathbb{L}}u=\bigl{(}Pu,u\bigr{|}_{t=0},D_{t}u\bigr{|}_{t=0}\bigr{)}$ and

$\displaystyle{\mathbb{E}}_{j}$
$\displaystyle=I_{\textup{cl}}^{\mu-j}([0,T]\times\mathbb{R}^{n},\mathcal{C}),$
$\displaystyle\widetilde{\mathbb{E}}_{j}$
$\displaystyle=I_{\textup{cl}}^{\mu-j+1}([0,T]\times\mathbb{R}^{n},\mathcal{C})\oplus
I_{\textup{cl}}^{\mu-j+1/4}(\mathbb{R}^{n},\\{0\\})\oplus
I_{\textup{cl}}^{\mu-j+5/4}(\mathbb{R}^{n},\\{0\\}),$
$\displaystyle{\mathbb{F}}_{j}$
$\displaystyle=S^{(\bar{\mu}-j)}(\Lambda^{+}\sqcup\Lambda^{-}),$
$\displaystyle\widetilde{\mathbb{F}}_{j}$
$\displaystyle=S^{(\bar{\mu}-j+1)}(\Lambda^{+}\sqcup\Lambda^{-})\oplus
S^{(\bar{\mu}-j)}(\dot{\mathbb{R}}^{n})\oplus
S^{(\bar{\mu}-j+1)}(\dot{\mathbb{R}}^{n}),$

where $\bar{\mu}=\mu-\frac{n-1}{4}$. The maps
${\bbsigma}^{j}\colon{\mathbb{E}}^{j}\to{\mathbb{F}}^{j}$ and
$\widetilde{\bbsigma}^{j}\colon\widetilde{\mathbb{E}}^{j}\to{\mathbb{F}}^{j}$
are the principal symbol maps, while the operator
${\mathbb{T}}^{j}\colon{\mathbb{F}}^{j}\to\widetilde{\mathbb{F}}^{j}$ is given
as ${\mathbb{T}}^{j}=\left(T_{\Lambda^{+}}^{P}\oplus
T_{\Lambda^{-}}^{P},\gamma_{0},\gamma_{1}\right)$, where

(3.8) $\left\\{\enspace\begin{aligned}
&\gamma_{0}\colon\bigl{(}a_{(\bar{\mu}-j)}^{+},a_{(\bar{\mu}-j)}^{-}\bigr{)}\mapsto
a_{(\bar{\mu}-j)}^{+}\bigr{|}_{t=0}+a_{(\bar{\mu}-j)}^{-}\bigr{|}_{t=0},\\\
&\gamma_{1}\colon\bigl{(}a_{(\bar{\mu}-j)}^{+},a_{(\bar{\mu}-j)}^{-}\bigr{)}\mapsto\mu^{+}a_{(\bar{\mu}-j)}^{+}\bigr{|}_{t=0}+\mu^{-}a_{(\bar{\mu}-j)}^{-}\bigr{|}_{t=0}.\end{aligned}\right.$

As before, we readily establish that properties (I) through (IV) hold. ∎

######  Remark.

The solution $u$ to (3.7) is of the form

(3.9)
$u(t,x)=\int_{\mathbb{R}^{n}}\left(\textup{e}^{\textup{i}\phi^{+}(t,x,\xi)}a^{+}(t,\xi)+\textup{e}^{\textup{i}\phi^{-}(t,x,\xi)}a^{-}(t,\xi)\right){\mathchar
22\relax\mkern-11.0mu\textup{d}}\xi\mod{{\mathscr{C}}_{\textup{b}}^{\infty}}$

for suitable $a^{\pm}\in
S_{\textup{cl}}^{\bar{\mu}}([0,T]\times\mathbb{R}^{n})$.

As announced at the beginning of this section, we now strengthen the statement
of Theorem 3.7. To this end, we need to discuss the transmission property, see
Appendix A.3. From now on, the space dimension $n$ is odd.

######  Proposition 3.8.

Let $\bar{\mu}=\mu-(n-1)/4\in\mathbb{Z}$ and $n$ be odd. Then $u=u(t,x)$ as
given in (3.9) possesses the two-sided transmission property with respect to
$\mathcal{C}$ if and only if

(3.10)
$a_{(\bar{\mu}-j)}^{-}(t,-\xi)=(-1)^{\bar{\mu}-j}a_{(\bar{\mu}-j)}^{+}(x,\xi),\quad
j\in\mathbb{N}_{0}.$

###### Proof.

This follows by writing, for $0<t\leq T$,

$u(t,x)=\int_{-\infty}^{\infty}\textup{e}^{\textup{i}(t-\psi(x))\tau}b(x,\tau)\,{\mathchar
22\relax\mkern-11.0mu\textup{d}}\tau\mod{\mathscr{C}}^{\infty}$

for a suitable333$b=b(x,\tau)$ grows like $|x|^{-(n-1)/2}$ as $x\to 0$, but
this is irrelevant here. $b\in
S_{\textup{cl}}^{\bar{\mu}+(n-1)/2}((\mathbb{R}^{n}\setminus\\{0\\})\times\mathbb{R})$,
where $t-\psi(x)=0$ is a defining equation for
$\mathcal{C}\setminus\mathscr{O}$, and, using the stationary phase method, by
expressing $b_{(\bar{\mu}-j+(n-1)/2)}(x,\tau)$ for $j\in\mathbb{N}_{0}$ in
terms of $a_{(\bar{\mu}-k)}^{+}(t,\xi)$ for $k\leq j$ when $\tau>0$ and in
terms of $a_{(\bar{\mu}-k)}^{-}(t,\xi)$ for $k\leq j$ when $\tau<0$. The
condition

$b_{(\bar{\mu}-j+(n-1)/2)}(x,-\tau)=(-1)^{(\bar{\mu}-j+(n-1)/2)}b_{(\bar{\mu}-j+(n-1)/2)}(x,-\tau),\quad
j\in\mathbb{N}_{0},$

then translates into (3.10). See also [1, 4]. ∎

For the proof of the next theorem, we need to work out suitable subspaces of
the function spaces ${\mathbb{E}}^{j}$, $\widetilde{\mathbb{E}}^{j}$,
${\mathbb{F}}^{j}$, and $\widetilde{\mathbb{F}}^{j}$ used in the proof of
Theorem 3.7. For $\bar{\mu}\in\mathbb{Z}$ and $n$ odd, we define the space
$I_{{\textup{cl}},\textup{tr}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})$
to consist of all $u\in
I_{\textup{cl}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})$ that possess the
two-sided transmission property with respect to the future light cone
$\mathcal{C}$. Further we introduce

$S_{\textup{tr}}^{(\bar{\mu})}(\Lambda^{+}\sqcup\Lambda^{-})=\\{(a_{(\bar{\mu})}^{+},a_{(\bar{\mu})}^{-})\in
S^{(\bar{\mu})}(\Lambda^{+}\sqcup\Lambda^{-})\mid
a_{(\bar{\mu})}^{-}(t,-\xi)=(-1)^{\bar{\mu}}a_{(\bar{\mu})}^{+}(t,\xi)\\}.$

We also need to impose certain symmetry conditions on the initial data
$(g_{0},g_{1})$. For $\bar{\mu}\in\mathbb{Z}$, let

$\tilde{S}^{(\bar{\mu})}(\dot{\mathbb{R}}^{n})=\left\\{b_{(\bar{\mu})}\in
S^{(\bar{\mu})}(\dot{\mathbb{R}}^{n})\mid
b_{(\bar{\mu})}(-\xi)=(-1)^{\bar{\mu}}b_{(\bar{\mu})}(\xi)\right\\}$

and let $\tilde{I}_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\\{0\\})$ consist of
all $g\in I_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\\{0\\})$ such that the
Fourier transform $\widehat{g}\in S_{\textup{cl}}^{\bar{\mu}}(\mathbb{R}^{n})$
admits an asymptotic expansion $\widehat{g}(\xi)\sim\sum_{j\geq
0}\chi(\xi)b_{(\bar{\mu}-j)}(\xi)$ with
$b_{(\bar{\mu}-j)}\in\tilde{S}^{(\bar{\mu}-j)}(\dot{\mathbb{R}}^{n})$ for all
$j$.

######  Lemma 3.9.

Let $\bar{\mu}\in\mathbb{Z}$ and $n$ be odd. Then, for $u\in
I_{{\textup{cl}},\textup{tr}}^{\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})$,

$u\bigr{|}_{t=0}\in\tilde{I}_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\\{0\\}),\quad
D_{t}u\bigr{|}_{t=0}\in\tilde{I}_{\textup{cl}}^{\mu+5/4}(\mathbb{R}^{n},\\{0\\}).$

###### Proof.

A direct verification that employs (3.8). ∎

######  Theorem 3.10.

Let $\bar{\mu}\in\mathbb{Z}$ and $n$ be odd. Suppose that
$g_{0}\in\tilde{I}_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\\{0\\})$,
$g_{1}\in\tilde{I}_{\textup{cl}}^{\mu+5/4}(\mathbb{R}^{n},\\{0\\})$, and $f\in
I_{{\textup{cl}},\textup{tr}}^{\mu+1}([0,T]\times\mathbb{R}^{n},\mathcal{C})$.
Then the solution $u$ as described in the previous theorem belongs to the
space
$I_{{\textup{cl}},\textup{tr}}^{\,\mu}([0,T]\times\mathbb{R}^{n},\mathcal{C})$,
i.e., it possesses the two-sided transmission property with respect to the
future light cone $\mathcal{C}$.

###### Proof.

We change the setup in the proof of the previous theorem as follows:

$\displaystyle{\mathbb{E}}_{j}$
$\displaystyle=I_{{\textup{cl}},\textup{tr}}^{\mu-j}([0,T]\times\mathbb{R}^{n},\mathcal{C}),$
$\displaystyle\widetilde{\mathbb{E}}_{j}$
$\displaystyle=I_{{\textup{cl}},\textup{tr}}^{\mu-j+1}([0,T]\times\mathbb{R}^{n},\mathcal{C})\oplus\tilde{I}_{\textup{cl}}^{\mu+1/4}(\mathbb{R}^{n},\\{0\\})\oplus\tilde{I}_{\textup{cl}}^{\mu+5/4}(\mathbb{R}^{n},\\{0\\}),$
$\displaystyle{\mathbb{F}}_{j}$
$\displaystyle=S_{\textup{tr}}^{(\bar{\mu}-j)}(\Lambda^{+}\sqcup\Lambda^{-}),$
$\displaystyle\widetilde{\mathbb{F}}_{j}$
$\displaystyle=S_{\textup{tr}}^{(\bar{\mu}-j+1)}(\Lambda^{+}\sqcup\Lambda^{-})\oplus\tilde{S}^{(\bar{\mu}-j)}(\dot{\mathbb{R}}^{n})\oplus\tilde{S}^{(\bar{\mu}-j+1)}(\dot{\mathbb{R}}^{n}).$

The operators ${\bbsigma}^{j}$, $\widetilde{\bbsigma}^{j}$, and
${\mathbb{T}}^{j}$ are as before, but acting on the indicated subspaces. Now,
given $b_{(\bar{\mu}-j)}\in\tilde{S}^{(\bar{\mu}-j)}(\dot{\mathbb{R}}^{n})$,
$c_{(\bar{\mu}-j+1)}\in\tilde{S}^{(\bar{\mu}-j+1)}(\dot{\mathbb{R}}^{n})$ and
solving the linear system

$\gamma_{0}\bigl{(}a_{(\bar{\mu}-j)}^{+}\bigr{|}_{t=0},a_{(\bar{\mu}-j)}^{-}\bigr{|}_{t=0}\bigr{)}=b_{(\bar{\mu}-j)},\quad\gamma_{1}\bigl{(}a_{(\bar{\mu}-j)}^{+}\bigr{|}_{t=0},a_{(\bar{\mu}-j)}^{-}\bigr{|}_{t=0}\bigr{)}=c_{(\bar{\mu}-j+1)}$

for $a_{(\bar{\mu}-j)}^{+}\bigr{|}_{t=0}$,
$a_{(\bar{\mu}-j)}^{-}\bigr{|}_{t=0}$, we find that
$a_{(\bar{\mu}-j)}^{+}\bigr{|}_{t=0},a_{(\bar{\mu}-j)}^{-}\bigr{|}_{t=0}\in\tilde{S}^{(\bar{\mu}-j)}(\dot{\mathbb{R}}^{n})$
and

(3.11)
$a_{(\bar{\mu}-j)}^{-}\bigr{|}_{t=0}(-\xi)=(-1)^{\bar{\mu}-j}a_{(\bar{\mu}-j)}^{+}\bigr{|}_{t=0}(\xi).$

Consequently, in view of the symmetry properties of the transport operators
$T_{\Lambda^{+}}^{P}$, $T_{\Lambda^{-}}^{P}$, the relation in (3.11) continues
to hold also for $0<t\leq T$ and we readily obtain that the operator

${\mathbb{T}}^{j}\colon
S_{\textup{tr}}^{(\bar{\mu}-j)}(\Lambda^{+}\sqcup\Lambda^{-})\to
S_{\textup{tr}}^{(\bar{\mu}-j+1)}(\Lambda^{+}\sqcup\Lambda^{-})\oplus\tilde{S}^{(\bar{\mu}-j)}(\dot{\mathbb{R}}^{n})\oplus\tilde{S}^{(\bar{\mu}-j+1)}(\dot{\mathbb{R}}^{n})$

is an isomorphism. Again, properties (I) through (IV) are satisfied. ∎

######  Example.

Let $n$ be odd. Then the Green’s function of the operator $P$ in (3.3) is
found when $g_{0}=0$,
$g_{1}=\textup{i}\delta(x)\in\tilde{I}_{\textup{cl}}^{n/4}(\mathbb{R}^{n},\\{0\\})$,
and $f=0$. We have $\bar{\mu}=-1$, and all the conditions of Theorem 3.10 are
fulfilled. Consequently, we recover the known result that the forward Green’s
function of $P$ is smooth from both sides up the future light cone.

## Appendix A

###  A.1. Asymptotic completeness

Here, we report an abstract scheme for establishing asymptotic completeness.
It is in the spirit of this article. Details can be found in [8, Prop.
1.1.17].

As before, let $\bigl{\\{}{\mathbb{E}}^{j}\bigr{\\}}_{j\in\mathbb{N}_{0}}$ be
a sequence of Fréchet spaces such that

${\mathbb{E}}^{0}\supseteq{\mathbb{E}}^{1}\supseteq{\mathbb{E}}^{2}\supseteq\dotsc\supseteq{\mathbb{E}}^{\infty},$

where ${\mathbb{E}}^{\infty}=\bigcap_{j}{\mathbb{E}}^{j}$. Suppose that there
is family $\\{\bbchi(c)\mid c\geq 1\\}$ of linear operators such that

  * •

$\bbchi(c)\in\bigcap_{j\in\mathbb{N}_{0}}\mathcal{L}({\mathbb{E}}^{j},{\mathbb{E}}^{j})$,

  * •

$1-\bbchi(c)\colon{\mathbb{E}}^{0}\to{\mathbb{E}}^{\infty}$,

  * •

$\bbchi(c)u\to 0$ as $c\to\infty$ in ${\mathbb{E}}^{j}$ for all
$u\in{\mathbb{E}}^{j+1}$.

######  Theorem A.1.

Under the conditions above, for each sequence
$\\{u_{j}\\}\subset{\mathbb{E}}^{0}$ with $u_{j}\in{\mathbb{E}}^{j}$ for all
$j$, there is a sequence $\\{c_{j}\\}\subset[1,\infty)$ (with $c_{j}\to\infty$
as $j\to\infty$ sufficiently fast depending on $\\{u_{j}\\}$) such that, for
any $J\in\mathbb{N}_{0}$, the series

$\sum\nolimits_{j\geq J}\bbchi(c_{j})u_{j}$ converges in ${\mathbb{E}}^{J}$.

Especially, setting $u=\sum\nolimits_{j\geq 0}\bbchi(c_{j})u_{j}$, we have
that $u\sim\sum\nolimits_{j\geq 0}u_{j}$ in ${\mathbb{E}}^{0}$. Indeed, for
$J\in\mathbb{N}_{0}$,

$u-\sum\nolimits_{j<J}u_{j}=-\,\sum\nolimits_{j<J}\left(1-\bbchi(c_{j})\right)u_{j}+\sum\nolimits_{j\geq
J}\bbchi(c_{j})u_{j}\in{\mathbb{E}}^{J}.$

###  A.2. Conormal distributions

We remind the reader of the concept of a conormal distribution and list some
of the basic properties of those distributions. For details, see e.g. [5, Sec.
18.2].

Let $X$ be a ${\mathscr{C}}^{\infty}$ manifold, of dimension $n$, and
$Y\subset X$ be a closed ${\mathscr{C}}^{\infty}$ submanifold, of codimension
$k\leq n$. A distribution $u\in{\mathscr{D}}^{\prime}(X)$ is said to be
conormal with respect to $Y$, of order $\nu\in\mathbb{R}$, if $T_{1}\dotsc
T_{l}u\in B_{2,\infty}^{-\nu-n/4}(X)$ for any number $l$ of vector fields
$T_{j}$ on $X$ tangent to $Y$ (where $B_{p,q}^{s}(X)$ is to designate the
local Besov spaces on $X$). This class of conormal distributions is denoted by
$I^{\nu}(X,Y)$. We have that $\operatorname{WF}(u)\subseteq\dot{N}^{*}Y$ for
$u\in I^{\nu}(X,Y)$, where $\dot{N}^{*}Y$ is the conormal bundle of $Y$ in $X$
with the zero section removed. In particular, $u\bigr{|}_{X\setminus
Y}\in{\mathscr{C}}^{\infty}(X\setminus Y)$.

There is an alternative description of conormal distributions through
oscillatory integrals. Specifically, in local coordinates
$x=(x^{\prime},x^{\prime\prime})\in\mathbb{R}^{k}\times\mathbb{R}^{n-k}$ such
that $Y=\\{x^{\prime}=0\\}$,

$u(x)=\int_{\mathbb{R}^{k}}\textup{e}^{\textup{i}x^{\prime}\xi^{\prime}}a(x^{\prime\prime},\xi^{\prime})\,{\mathchar
22\relax\mkern-11.0mu\textup{d}}\xi^{\prime}\mod{{\mathscr{C}}^{\infty}(\mathbb{R}^{n})},$

where $a\in S^{\nu+(n-2k)/4}(\mathbb{R}^{n-k}\times\mathbb{R}^{k})$ and
${\mathchar
22\relax\mkern-11.0mu\textup{d}}\xi^{\prime}=(2\pi)^{-(n+2k)/4}\,\textup{d}\xi^{\prime}$.
The subclass $I_{\textup{cl}}^{\mu}(X,Y)\subset I^{\Re\mu}(X,Y)$ for
$\mu\in\mathbb{C}$ of classical conormal distributions results upon requiring
that $a\in
S_{\textup{cl}}^{\mu+(n-2k)/4}(\mathbb{R}^{n-k}\times\mathbb{R}^{k})$. In this
case,
$a(x^{\prime\prime},\xi^{\prime})=\chi(\xi^{\prime})a_{(\bar{\mu}-k)}(x^{\prime\prime},\xi^{\prime})+a_{r}(x^{\prime\prime},\xi^{\prime})$,
where $a_{(\bar{\mu}-k)}\in
S^{(\bar{\mu}-k)}(\mathbb{R}^{n-k}\times\dot{\mathbb{R}}^{k})$,
$\bar{\mu}=\mu+(n+2k)/4$, $\chi\in{\mathscr{C}}^{\infty}(\mathbb{R}^{k})$,
$\chi(\xi^{\prime})=0$ for $|\xi^{\prime}|\leq 1$, $\chi(\xi^{\prime})=1$ for
$|\xi^{\prime}|\geq 2$, and $a_{r}\in
S_{\textup{cl}}^{\bar{\mu}-k-1}(\mathbb{R}^{n-k}\times\mathbb{R}^{j})$. Then

$\sigma_{\psi}^{\mu}(u)=a_{(\bar{\mu}-k)}(x^{\prime\prime},\xi^{\prime})\left|\textup{d}\xi^{\prime}\right|$

is the local coordinate expression for the principal symbol
$\sigma_{\psi}^{\mu}(u)\in
S^{(\bar{\mu})}(\dot{N}^{*}Y;\Omega_{\dot{N}^{*}Y}^{1,\textup{v}})$ with
$\Omega_{\dot{N}^{*}Y}^{1,\textup{v}}=\Omega_{\dot{N}^{*}Y}^{1/2}\otimes\bigl{(}\pi\bigr{|}_{\dot{N}^{*}Y}\bigr{)}^{*}\Omega_{X}^{-1/2}$
being the vertical $1$-density bundle over $\dot{N}^{*}Y$. Here,
$\Omega_{Z}^{\alpha}$ is the $\alpha$-density bundle over a manifold $Z$ and
$\pi\colon\dot{T}^{*}X\to X$ denotes the canonical projection.

In Section 3.4, we make use of the following properties:

(a) (Principal symbol map) The principal symbol map fits into a short exact
sequence

$0\longrightarrow I_{\textup{cl}}^{\mu-1}(X,Y)\longrightarrow
I_{\textup{cl}}^{\mu}(X,Y)\overset{\sigma_{\psi}^{\mu}}{\longrightarrow}S^{(\bar{\mu})}(\dot{N}^{*}Y;\Omega_{\dot{N}^{*}Y}^{1,\textup{v}})\longrightarrow
0,$

which splits. As a consequence, $I_{\textup{cl}}^{\mu}(X,Y)$ is a nuclear
Fréchet space equipped with the projective limit topology of
$\prod_{j<J}S^{(\bar{\mu}-j)}(\dot{N}^{*}Y;\Omega_{\dot{N}^{*}Y}^{1,\textup{v}})\times
I^{\Re\mu-J}(X,Y)$ as $J\to\infty$, with each factor carrying its natural
Fréchet topology.

(b) (Application of pseudodifferential operators) For
$P\in\Psi_{\textup{cl}}^{m}(X)$ and $u\in I_{\textup{cl}}^{\mu}(X,Y)$, we have
$Pu\in I_{\textup{cl}}^{m+\mu}(X,Y)$ and
$\sigma_{\psi}^{m+\mu}(Pu)=\sigma_{\psi}^{m}(P)\bigr{|}_{\dot{N}^{*}Y}\,\sigma_{\psi}^{\mu}(u)$.
If $Y$ is characteristic for $P$ (i.e.,
$\sigma_{\psi}^{m}(P)\bigr{|}_{\dot{N}^{*}Y}=0$), then $Pu\in
I_{\textup{cl}}^{m+\mu-1}(X,Y)$ and
$\sigma_{\psi}^{m+\mu-1}(Pu)=T_{Y}^{P}\sigma_{\psi}^{m}u(u)$, where
$T_{Y}^{P}\in\operatorname{Diff}^{1}(\dot{N}^{*}Y;\Omega_{\dot{N}^{*}Y}^{1,\textup{v}})$
is a first-order differential operator acting on sections of the bundle
$\Omega_{\dot{N}^{*}Y}^{1,\textup{v}}$ (similar to the transport operators
$T_{C}^{P}$ introduced at the end of Section 1).

(c) (Restriction to submanifolds) Let $u\in I_{\textup{cl}}^{\mu}(X,Y)$ and
$Z\subset X$ be a closed ${\mathscr{C}}^{\infty}$ submanifold that meets $Y$
transversally. Then $u\bigr{|}_{Z}\in I_{\textup{cl}}^{\mu+l/4}(Z,Y\cap Z)$,
where $l$ is the codimension of $Z$ in $X$, and
$\sigma_{\psi}^{\mu+l/4}\bigl{(}u\bigr{|}_{Z}\bigr{)}=(2\pi)^{-l/4}\sigma_{\psi}^{\mu}(u)\bigr{|}_{\dot{N}_{Z}^{*}(Y\cap
Z)}$.

The restriction to $\dot{N}_{Z}^{*}(Y\cap Z)$ is to be understand as follows:
$T_{p}Y+T_{p}Z=T_{p}X$ for $p\in Y\cap Z$ implies that $T_{p}X/T_{p}Y\cong
T_{p}Z/T_{p}(Y\cap Z)$ and, therefore, $N_{X}Y\bigr{|}_{Y\cap Z}\cong
N_{Z}(Y\cap Z)$ canonically for the normal bundles. Passing to the dual
bundles, $N_{Z}^{*}(Y\cap Z)\cong N_{X}^{*}Y\bigr{|}_{Y\cap Z}\hookrightarrow
N_{X}^{*}Y$.

(d) (Asymptotic completeness) Let $u_{j}\in I_{\textup{cl}}^{\mu-j}(X,Y)$ for
$j\in\mathbb{N}_{0}$. Then there exists a $u\in I_{\textup{cl}}^{\mu}(X,Y)$
with the property that $u-\sum_{j<J}u_{j}\in I_{\textup{cl}}^{\mu-J}(X,Y)$ for
all $J\in\mathbb{N}_{0}$. This $u$ is uniquely determined modulo
${\mathscr{C}}^{\infty}(X)$.

###  A.3. The transmission property

Let $X$ be a ${\mathscr{C}}^{\infty}$ manifold and $Y\subset X$ be a closed
${\mathscr{C}}^{\infty}$ hypersurface. We discuss here the two-sided
transmission property for distributions $u\in{\mathscr{D}}^{\prime}(X)$ with
respect to $Y$. For a more detailed discussion, see [1, 4, 5].

######  Definition A.2.

$u\in I_{\textup{cl}}^{\mu}(X,Y)$ is said to have the two-sided transmission
property with respect to $Y$ if $u\bigr{|}_{X\setminus
Y}\in{\mathscr{C}}^{\infty}(X\setminus Y)$ extends smoothly (locally from both
sides) to $Y$.

A necessary condition is that
$\bar{\mu}=\mu+\frac{n}{4}-\frac{1}{2}\in\mathbb{Z}$, where $n=\dim X$.

The two-sided transmission property is a local property. In local coordinates
$x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}$, let
$Y=\\{x_{n}=0\\}$. Then $u\in I_{\textup{cl}}^{\mu}(X,Y)$ if and only if

(A.1)
$u(x)=\int_{-\infty}^{\infty}\textup{e}^{\textup{i}x_{n}\xi}a(x^{\prime},\xi)\,{\mathchar
22\relax\mkern-11.0mu\textup{d}}\xi\bmod{{\mathscr{C}}_{\textup{b}}^{\infty}},$

where $a\in S_{\textup{cl}}^{\bar{\mu}}(\mathbb{R}^{n-1}\times\mathbb{R})$.
Let

(A.2) $a(x^{\prime},\xi)\sim\sum_{k\geq
0}\chi(\xi)a_{(\bar{\mu}-k)}(x^{\prime},\xi),$

where $a_{(\bar{\mu}-k)}\in
S^{(\bar{\mu}-k)}(\mathbb{R}^{n-1}\times\dot{\mathbb{R}})$.

######  Proposition A.3.

For $\bar{\mu}\in\mathbb{Z}$, the distribution $u\in
I_{\textup{cl}}^{\mu}(X,Y)$ given in (A.1) with (A.2) possesses the two-sided
transmission property if and only if

$a_{(\bar{\mu}-k)}(x^{\prime},-\xi)=(-1)^{\bar{\mu}-k}a_{(\bar{\mu}-k)}(x^{\prime},\xi),\quad
k\in\mathbb{N}_{0}.$

######  Remark.

Let $\bar{\mu}\in\mathbb{Z}$ and $u\in I_{\textup{cl}}^{\mu}(X,Y)$ have the
two-sided transmission property. Then, locally,

$u=u\bigr{|}_{\\{x_{n}\neq 0\\}}+\sum_{0\leq
j\leq\bar{\mu}}c_{j}(x^{\prime})\otimes\delta^{(j)}(x_{n}),$

where $c_{j}\in{\mathscr{C}}_{\textup{b}}^{\infty}(\mathbb{R}^{n-1})$. In
particular, $u$ is a regular distribution if $\bar{\mu}<0$.

######  Remark.

For general $\mu\in\mathbb{C}$, one similarly defines a one-sided transmission
property (locally with respect to either side of $Y$). All three conditions
are equivalent if $\bar{\mu}\in\mathbb{Z}$.

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