$\displaystyle\lesssim\int_{\begin{subarray}{c}\\{\eta\,;\,(x_{0},\xi_{0}-\eta)\notin
V_{1}\\\ (x_{0},\eta)\in
V_{2}\\}\end{subarray}}\langle{\xi_{0}-\eta}\rangle^{-N-l-n-1}\cdot\langle{\eta}\rangle^{l}\,\mathrm{d}{\eta}$
$\displaystyle\lesssim\langle{\xi_{0}}\rangle^{-N}\int_{{\mathbb{R}^{n}}}\langle{\eta}\rangle^{-l-n-1}\cdot\langle{\eta}\rangle^{l}\,\mathrm{d}{\eta}\quad\big{(}\text{by~{}}\eqref{eq:AngSep2-PM2021}\big{)}$
$\displaystyle\lesssim\langle{\xi_{0}}\rangle^{-N},\quad\forall
N\in\mathbb{N}.$ (8.24)

The estimation of $I_{3}$ is similar to (8.24),

$\displaystyle|I_{3}|$
$\displaystyle\leq\int_{\begin{subarray}{c}\\{\eta\,;\,(x_{0},\xi_{0}-\eta)\in
V_{1}\\\ (x_{0},\eta)\notin V_{2}\\}\end{subarray}}|(\varphi
u)^{\wedge}(\xi_{0}-\eta)|\cdot|(\varphi v)^{\wedge}(\eta)|\,\mathrm{d}{\eta}$
$\displaystyle=\int_{\begin{subarray}{c}\\{\eta\,;\,(x_{0},\gamma)\in V_{1}\\\
(x_{0},\xi_{0}-\gamma)\notin V_{2}\\}\end{subarray}}|(\varphi
u)^{\wedge}(\gamma)|\cdot|(\varphi
v)^{\wedge}(\xi_{0}-\gamma)|\,\mathrm{d}{\gamma}\quad(\gamma=\xi_{0}-\eta)$
$\displaystyle\lesssim\int_{\begin{subarray}{c}\\{\eta\,;\,(x_{0},\gamma)\in
V_{1}\\\ (x_{0},\xi_{0}-\gamma)\notin
V_{2}\\}\end{subarray}}\langle{\gamma}\rangle^{l^{\prime}}\cdot\langle{\xi_{0}-\gamma}\rangle^{-N-l^{\prime}-n-1}\,\mathrm{d}{\gamma}$
$\displaystyle\lesssim\langle{\xi_{0}}\rangle^{-N}\int_{{\mathbb{R}^{n}}}\langle{\gamma}\rangle^{l^{\prime}}\cdot\langle{\gamma}\rangle^{-l^{\prime}-n-1}\,\mathrm{d}{\gamma}\quad\big{(}\text{by~{}}\eqref{eq:AngSep2-PM2021}\big{)}$
$\displaystyle\lesssim\langle{\xi_{0}}\rangle^{-N},\quad\forall
N\in\mathbb{N}.$ (8.25)

Now we work on $I_{4}$. From (8.15), (8.17) and (8.23), we know that
$\xi_{0}\notin V_{1}+V_{2}$, thus the set $\\{\eta\,;\,(x_{0},\xi_{0}-\eta)\in
V_{1},\,(x_{0},\eta)\in V_{2}\\}$ is empty. Therefore $I_{4}=0$. Combining
this fact with (8.18), (8.19), (8.24) and (8.25), we arrive at

$|(\varphi^{2}uv)^{\wedge}(\xi)|\leq
C_{N}\langle{\xi_{0}}\rangle^{-N},\quad\forall N\in\mathbb{N},$

for $(x_{0},\xi_{0})\notin V_{3}\cup V_{1}^{\prime}\cup V_{2}^{\prime}$. This
implies $\operatorname{WF}(u+v)\subset V_{3}\cup V_{1}^{\prime}\cup
V_{2}^{\prime}$. The sets $V_{3}$, $V_{1}^{\prime}$ and $V_{2}^{\prime}$ can
be close to $\operatorname{WF}(u)+\operatorname{WF}(v)$,
$\operatorname{WF}(u)$ and $\operatorname{WF}(v)$, respectively, as close as
possible, so we arrive at (8.16). The proof is complete. ∎

####  8.2.3. Convolution

We define

$\left\\{\begin{aligned}
\operatorname{WF}^{\prime}(K)&:=\\{(x,y;\xi,-\eta)\,;\,(x,y;\xi,\eta)\in\operatorname{WF}(K)\\},\\\
\operatorname{WF}_{x}(K)&:=\\{(x;\xi)\,;\,\exists
y\textrm{~{}s.t.~{}}(x,y;\xi,0)\in\operatorname{WF}(K)\\},\\\ A\circ
B&:=\\{(x,\xi)\,;\,\exists(y,\eta)\in B\textrm{~{}s.t.~{}}(x,y;\xi,\eta)\in
A\\},\\\ O_{x}&:=\\{(x,0)\,;\,x\in\Omega\\}.\end{aligned}\right.$ (8.26)

We need the following lemma.

######  Lemma 8.10.

Assume $f\in\mathcal{D}^{\prime}(\Omega\times\Omega)$, and there is a compact
set $\mathcal{K}\subset\Omega$ such that $\mathop{\rm
supp}f\subset\Omega\times\mathcal{K}$. Then

$\boxed{\operatorname{WF}\big{(}\int
f(x,y)\,\mathrm{d}{y}\big{)}=\operatorname{WF}_{x}(f).}$ (8.27)

###### Proof.

Step 1. ($\supset$). Assume $(x_{0},\xi_{0})\notin\operatorname{WF}\big{(}\int
f(x,y)\,\mathrm{d}{y}\big{)}$, then there exists $\chi_{x_{0}}\in
C_{c}^{\infty}(\Omega)$ such that

$\int
e^{-ix_{0}\cdot\xi_{0}}\chi_{x_{0}}(x)f(x,y)\,\mathrm{d}{y}\,\mathrm{d}{x}=\mathcal{O}(\langle{\xi_{0}}\rangle^{-\infty})=\mathcal{O}(\langle{(\xi_{0},0)}\rangle^{-\infty}).$

which gives

$\forall\bar{y}\in\mathcal{K},\ \int
e^{-i(x_{0},\bar{y})\cdot(\xi_{0},0)}\chi_{x_{0}}(x)\chi(y)f(x,y)\,\mathrm{d}{(}x,y)=\mathcal{O}(\langle{(\xi_{0},0)}\rangle^{-\infty}),$

where $\chi\in C_{c}^{\infty}(\Omega)$ with $\chi\equiv 1$ on $\mathcal{K}$.
This means $(x_{0},\bar{y};\xi_{0},0)\notin\operatorname{WF}(f)$ for
$\forall\bar{y}\in\mathcal{K}$, so
$(x_{0},\xi_{0})\notin\operatorname{WF}_{x}(f)$. Hence,

$\operatorname{WF}\big{(}\int
f(x,y)\,\mathrm{d}{y}\big{)}\supset\operatorname{WF}_{x}(f).$

Step 2. ($\subset$). Assume $(x_{0},\xi_{0})\notin\operatorname{WF}_{x}(f)$,
then for $\forall\bar{y}\in\Omega$ we have
$(x_{0},\bar{y};\xi_{0},0)\notin\operatorname{WF}(f)$. Therefore, for
$\forall\bar{y}\in\Omega$, there is a neighborhood of $\bar{y}$ such that

$\int
e^{-i(x_{0},\bar{y})\cdot(\xi_{0},0)}\chi_{x_{0}}(x)\chi(y)f(x,y)\,\mathrm{d}{(}x,y)=\mathcal{O}(\langle{(\xi_{0},0)}\rangle^{-\infty}),$
(8.28)

for $\chi\equiv 1$ in that neighborhood. Because $\mathcal{K}$ is compact, so
by using partition of unity technique, we can remove the term $\chi(y)$ in
(8.28), and obtain

$\int e^{-ix_{0}\cdot\xi_{0}}\chi_{x_{0}}(x)\big{(}\int
f(x,y)\,\mathrm{d}{y}\big{)}\,\mathrm{d}{x}=\mathcal{O}(\langle{\xi_{0}}\rangle^{-\infty}),$

which gives $(x_{0},\xi_{0})\notin\operatorname{WF}(\int
f(x,y)\,\mathrm{d}{y}).$ Hence,

$\operatorname{WF}\big{(}\int
f(x,y)\,\mathrm{d}{y}\big{)}\subset\operatorname{WF}_{x}(f).$

The proof is done. ∎

######  Theorem 8.11.

Assume $u\in\mathscr{E}^{\prime}(\Omega)$, and
$K\in\mathscr{D}^{\prime}(\Omega\times\Omega)$. When
$\big{(}\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u)\big{)}\cap
O_{x}=\emptyset$, the distribution

$w(x):=\langle{K(x,y),u(y)}\rangle_{y}$

is well-defined in the sense that

$\forall\varphi\in\mathscr{D}(\Omega),\quad
w(\varphi):=\langle{K(x,y),u(y)\otimes\varphi(x)}\rangle,$

and we have the following canonical relation:

$\boxed{\operatorname{WF}(w)\subseteq\big{(}\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u)\big{)}\cup\operatorname{WF}_{x}(K).}$
(8.29)

######  Remark 8.12.

Note that the $\mathop{\rm supp}u$ should be contained in $\Omega$, otherwise
the $w$ may be ill-defined.

###### Proof.

Step 1. Turn into product. Denote $\tilde{u}(x,y)=1(x)\otimes u(y)$ where
$1(x)$ is the constant function. The wavefront set of the function $1(x)$ is
empty, so by Theorem 8.8 we have

$\displaystyle\operatorname{WF}(\tilde{u})$
$\displaystyle\subset\big{(}\operatorname{WF}(1)\times\operatorname{WF}(u)\big{)}\cup\big{(}\operatorname{WF}(1)\times{\rm
supp}_{0}u\big{)}\cup\big{(}{\rm supp}_{0}1\times\operatorname{WF}(u)\big{)}$
$\displaystyle=\emptyset\cup\emptyset\cup\big{(}{\rm
supp}_{0}1\times\operatorname{WF}(u)\big{)}$
$\displaystyle=\\{(x,y;0,\eta)\,;\,x\in\mathop{\rm
supp}\Omega,\,(y,\eta)\in\operatorname{WF}(u)\\}.$ (8.30)

The $w(x)$ can be written as

$w(x)=\langle{K(x,y),u(y)}\rangle_{y}=\int
K(x,y)\cdot\tilde{u}(x,y)\,\mathrm{d}{y}=\int K\tilde{u}(x,y)\,\mathrm{d}{y}.$

where $K\tilde{u}$ stands for the product of $K$ and $\tilde{u}$. By Theorem
8.9, to guarantee the product $K\tilde{u}$ is well-defined, we need to check
if the prerequisite

$\big{(}\operatorname{WF}(K)+\operatorname{WF}(\tilde{u})\big{)}\cap
O_{x,y}=\emptyset$ (8.31)

is true. It can be shown that the condition
$\big{(}\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u)\big{)}\cap
O_{x}=\emptyset$ guarantees (8.31) (see Exercise 8.2), so $K\tilde{u}$ is
well-defined.

Because $u\in\mathscr{E}^{\prime}(\Omega)$, we see that for $\forall
x\in\Omega$, $\mathop{\rm supp}K\tilde{u}(x,\cdot)$ is uniformly compact, so
by Lemma 8.10 we have $\operatorname{WF}\big{(}\int
K\tilde{u}(x,y)\,\mathrm{d}{y}\big{)}=\operatorname{WF}_{x}(K\tilde{u}),$ so,

$\operatorname{WF}(w)=\operatorname{WF}\big{(}\int
K\tilde{u}(x,y)\,\mathrm{d}{y}\big{)}=\operatorname{WF}_{x}(K\tilde{u})=\operatorname{WF}(K\tilde{u})\circ
O_{y},$ (8.32)

where we used the fact that for general distribution
$f\in\mathcal{D}^{\prime}(\Omega\times\Omega)$,

$\operatorname{WF}_{x}(f)=\operatorname{WF}(f)\circ O_{y}.$

Step 2. Use product Theorem. Combining (8.32) with Theorem 8.9, we can have

$\displaystyle\operatorname{WF}(w)$
$\displaystyle=\operatorname{WF}(K\tilde{u})\circ
O_{y}\subset\Big{(}\big{(}\operatorname{WF}(K)+\operatorname{WF}(\tilde{u})\big{)}\cup\operatorname{WF}(K)\cup\operatorname{WF}(\tilde{u})\Big{)}\circ
O_{y}$ $\displaystyle=M_{1}\cup M_{2}\cup M_{3},$ (8.33)

where

$\left\\{\begin{aligned}
M_{1}&:=\big{(}\operatorname{WF}(K)+\operatorname{WF}(\tilde{u})\big{)}\circ
O_{y},\\\ M_{2}&:=\operatorname{WF}(K)\circ O_{y},\\\
M_{3}&:=\operatorname{WF}(\tilde{u})\circ O_{y}.\end{aligned}\right.$

The set $\operatorname{WF}(K)+\operatorname{WF}(\tilde{u})$ can be expressed
as

$\displaystyle\ \operatorname{WF}(K)+\operatorname{WF}(\tilde{u})$
$\displaystyle=$ $\displaystyle\
\\{(x,y;\xi,\eta)\,;\,\xi=\xi_{1}+\xi_{2},\,\eta=\eta_{1}+\eta_{2},\,(x,y;\xi_{1},\eta_{1})\in\operatorname{WF}(K),\,(x,y;\xi_{2},\eta_{2})\in\operatorname{WF}(\tilde{u})\\}$
$\displaystyle=$ $\displaystyle\
\\{(x,y;\xi,\eta)\,;\,\eta=\eta_{1}+\eta_{2},\,(x,y;\xi,\eta_{1})\in\operatorname{WF}(K),\,(y,\eta_{2})\in\operatorname{WF}(u)\\}.\quad\text{(by
\eqref{eq:t1u-PM2021})}$

Thus,

$\displaystyle M_{1}$
$\displaystyle=\big{(}\operatorname{WF}(K)+\operatorname{WF}(\tilde{u})\big{)}\circ
O_{y}=\\{(x,\xi)\,;\,(x,y;\xi,-\eta)\in\operatorname{WF}(K),\,(y,\eta)\in\operatorname{WF}(u)\\}$
$\displaystyle=\\{(x,\xi)\,;\,(x,y;\xi,\eta)\in\operatorname{WF}^{\prime}(K),\,(y,\eta)\in\operatorname{WF}(u)\\}=\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u).$
(8.34)

By (8.30) it can also be checked that

$M_{2}=\operatorname{WF}_{x}(K),\quad M_{3}=\emptyset.$ (8.35)

Combining (8.34), (8.35) with (8.33), we obtain (8.29). The proof is complete.
∎

######  Remark 8.13.

In Theorem 8.11, if we know in advance that

$\operatorname{WF}(K)\subset(T^{*}\Omega_{x}\backslash
0)\times(T^{*}\Omega_{y}\backslash 0),$

then $\operatorname{WF}_{x}(K)=\emptyset$ and (8.29) can be reduced to

$\boxed{\operatorname{WF}(w)\subseteq\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u).}$
(8.36)

The set $\operatorname{WF}^{\prime}(K)$ is called the _twist_ of
$\operatorname{WF}(K)$, and the operation
“$\operatorname{WF}^{\prime}(K)\circ$” is called _canonical relation_ of the
operator:

$u(y)\mapsto w(x):=\langle{K(x,y),u(y)}\rangle$

which takes $K$ as its kernel. These can be generalized to the theory of
_Fourier integral operators_.

###  8.3. The wavefront sets of Fourier integral operators

Recall the notion of phase function given in Definition 3.1.

######  Theorem 8.14.

Assume $\varphi\in C^{\infty}({\mathbb{R}^{n}}\times\mathbb{R}^{N})$ is a
phase function of order $1$, and $a\in S^{m}$ is a symbol. Define $A(x)$ as

$A(x):=\int e^{i\varphi(x,\theta)}a(x,\theta)\,\mathrm{d}{\theta},$ (8.37)

where the integral is understood as an oscillatory integral. Then $A$ induces
a distribution (also denoted as $A$) $A\in\mathscr{D}^{\prime}(\Omega)$ for
any domain $\Omega\subset{\mathbb{R}^{n}}$, i.e. $A\colon
u\in\mathscr{D}(\Omega)\mapsto I_{\varphi}(au)$ by

$A(u):=I_{\varphi}(au)=\langle{A,u}\rangle=\int
e^{i\varphi(x,\theta)}a(x,\theta)u(x)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$

in oscillatory integral sense. The wavefront set of $A$ satisfies

$\boxed{\operatorname{WF}(A)\subset\\{(x,\varphi_{x}(x,\theta))\,;\,\varphi_{\theta}(x,\theta)=0,\,(x,\theta)\notin\operatorname{Smo}(a)\\}.}$
(8.38)

######  Remark 8.15.

When the following conditions are satisfied, the inclusion “$\subset$” in
(8.38) can be improved to “$=$” (see contexts preceding [ho202Xmicro, Theorem
3.9], [chen1997fio, Theorem 3.9]):

  1. (1)

the phase function $\varphi$ is non-degenerate on
$C_{\phi}:=\\{(x,\theta)\,;\,\varphi_{\theta}(x,\theta)=0,\,(x,\theta)\notin\operatorname{Smo}(a)\\}$,
i.e. the $N$-$(n+N)$ matrix $\mathrm{d}\phi_{\theta}$ is full rank on
$C_{\phi}$, here

$\mathrm{d}\phi_{\theta}(x,\theta)=\begin{pmatrix}\phi_{\theta
x}(x,\theta)&\phi_{\theta\theta}(x,\theta)\end{pmatrix};$

  2. (2)

the map $(x,\theta)\mapsto(x,\varphi_{x}(x,\theta))$ is injective when
restricted to $C_{\phi}$.

Readers may distinguish the $A$ appeared in Theorem 8.14 with the operator $B$
defined as

$Bu(x):=\int
e^{i\varphi(x,y,\theta)}a(x,y,\theta)u(y)\,\mathrm{d}{y}\,\mathrm{d}{\theta}.$

The $A$ is a distribution while the $B$ just defined is an operator, namely,
$A$ maps a function to a scalar while $B$ maps a function to another function.

However, $A$ is a generalization of $B$, because

$\displaystyle\langle{Bu(x),v(x)}\rangle$ $\displaystyle=\int
e^{i\varphi(x,y,\theta)}a(x,y,\theta)u(y)v(x)\,\mathrm{d}{x}\,\mathrm{d}{y}\,\mathrm{d}{\theta}$
$\displaystyle=\langle{\int
e^{i\varphi(x,y,\theta)}a(x,y,\theta)\,\mathrm{d}{\theta},(v\otimes
u)(x,y)}\rangle$ $\displaystyle=\langle{K_{B},v\otimes u}\rangle.$

where $\tilde{B}$ is defined as

$K_{B}(x,y):=\int e^{i\varphi(x,y,\theta)}a(x,y,\theta)\,\mathrm{d}{\theta}.$

Hence the operator $B$ can be turned into a form of (8.37).

Moreover, we have $Bu(x)=\langle{K_{B}(x,y),u(y)}\rangle$, so by combining
Theorems 8.14 and 8.11, hopefully we can obtain $\operatorname{WF}(Bu)$.

###### Short proof of Theorem 8.14.

This short proof is for summarizing the key idea of proving this theorem and
thus the details may not be rigorously correct. After this short proof, we
also present a formal proof of Theorem 8.14.

According to the Definition 8.4 ,we fix a cutoff function $\phi$ with
$\phi(x_{0})\neq 0$ and compute

$\displaystyle\widehat{\phi A}(\xi)$ $\displaystyle=A(\phi
e^{-ix\cdot\xi})=(2\pi)^{-n/2}\int
e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta},$

and the basic idea is to use $N$ times (with $N$ large enough) the operator
$L:=\frac{(\varphi_{x}(x,\theta)-\xi)\cdot\nabla_{x}}{i|\varphi_{x}(x,\theta)-\xi|^{2}}$
acting on $e^{i(\varphi(x,\theta)-x\cdot\xi)}$ and the fact (8.3) to get the
desired estimate. But in order to do so, one needs to first address some
singularities in the oscillatory integral. We have

$\displaystyle\widehat{\phi A}(\xi)$
$\displaystyle\simeq\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\mathbb{R}_{\theta}^{n}}e^{i\varphi(x,\theta)}a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\simeq\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\mathbb{R}_{\theta}^{n}}(\frac{\varphi_{\theta}(x,\theta)\cdot\nabla_{\theta}}{|\varphi_{\theta}(x,\theta)|^{2}})^{N}e^{i\varphi(x,\theta)}a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\sim\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\mathbb{R}_{\theta}^{n}}e^{i\varphi(x,\theta)}|\varphi_{\theta}(x,\theta)|^{-N}\partial_{\theta}^{N}a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}.$
(8.39)

Then as $N$ be large enough, the $\partial_{\theta}^{N}a(x,\theta)$ will be
integrable w.r.t. $\theta$. But we notice that
$|\varphi_{\theta}(x,\theta)|^{-1}$ has singularity at $\theta=0$, so we first
exclude the neighborhood of the origin of $\theta$ by using a cutoff function
$\chi$ with $\chi(0)\neq 0$ as follows

$\displaystyle\widehat{\phi A}(\xi)$ $\displaystyle\simeq\int
e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)\chi(\theta)a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}+\int
e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\mathbb{R}_{\theta}^{n}}e^{i\varphi(x,\theta)}\chi(\theta)a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\quad+\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\Gamma_{x}}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\quad+\int_{\theta\notin\Gamma_{x}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=:I_{1}(\xi)+I_{2}(\xi)+I_{3}(\xi),$

where

$\Gamma_{x}:=\\{\theta\,;\,a(x,\theta)=\mathcal{O}(|\theta|^{-\infty})\\}.$
(8.40)

The $I_{1}(\xi)$ and $I_{2}(\xi)$ are $O(|\xi|^{-\infty})$ as
$|\xi|\to+\infty$, because these terms
$\int_{\mathbb{R}_{\theta}^{n}}e^{i\varphi(x,\theta)}\chi(\theta)a(x,\theta)\,\mathrm{d}{\theta}$
and
$\int_{\Gamma_{x}}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}$
are smooth in terms of $x$ (for the first term, it is because the actual
integral domain is compact, i.e. is contained in $\mathop{\rm supp}\theta$;
for the second term, it is because the integrand decays at infinity order).
Then we can compute $I_{3}$ as follows,

$\displaystyle I_{3}(\xi)$
$\displaystyle\simeq\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\varphi_{\theta}(x,\theta)\neq
0\text{~{}and~{}}\theta\notin\Gamma_{x}}(\frac{\varphi_{\theta}(x,\theta)\cdot\nabla_{\theta}}{|\varphi_{\theta}(x,\theta)|^{2}})^{N}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\quad+\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\varphi_{\theta}(x,\theta)=0\text{~{}and~{}}\theta\notin\Gamma_{x}}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle=:I_{4}(\xi)+I_{5}(\xi).$

Now here comes the key point: to obtain $\langle{\xi}\rangle^{-N}$, for
$I_{4}$ we differentiate $e^{-ix\cdot\xi}$ w.r.t. $x$, and for $I_{5}$ we
differentiate $e^{i(\varphi-x\cdot\xi)}$ w.r.t. $x$.

We can estimate $I_{4}$ by using the computation as in (8.39),

$\displaystyle I_{4}(\xi)$
$\displaystyle\simeq\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\varphi_{\theta}(x,\theta)\neq
0\text{~{}and~{}}\theta\notin\Gamma_{x}}(\frac{\varphi_{\theta}(x,\theta)\cdot\nabla_{\theta}}{|\varphi_{\theta}(x,\theta)|^{2}})^{N}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\sim\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\varphi_{\theta}(x,\theta)\neq
0\text{~{}and~{}}\theta\notin\Gamma_{x}}e^{i\varphi(x,\theta)}|\varphi_{\theta}(x,\theta)|^{-N}\partial_{\theta}^{N}((1-\chi)a)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x},$

where the integer $N$ can be arbitrary. And hence we have
$I_{4}(\xi)=\mathcal{O}(|\xi|^{-\infty})$ for the same reason as $I_{1}$.

It is the $I_{5}$ which finally decides $\operatorname{WF}(A)$. For
$\xi\neq\varphi_{x}(x,\theta)$, we can have $I_{3}$ as follows,

$\displaystyle I_{5}(\xi)$
$\displaystyle=\int_{\begin{subarray}{c}\varphi_{\theta}(x,\theta)=0,\\\
(x,\theta)\notin\operatorname{Smo}(a)\end{subarray}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
(8.41)
$\displaystyle=\int_{\begin{subarray}{c}\varphi_{\theta}(x,\theta)=0,\\\
(x,\theta)\notin\operatorname{Smo}(a)\end{subarray}}\big{[}\big{(}\frac{(\varphi_{x}(x,\theta)-\xi)\cdot\nabla_{x}}{i|\varphi_{x}(x,\theta)-\xi|^{2}}\big{)}^{N_{1}+N_{2}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\big{]}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
(8.42)
$\displaystyle\lesssim\int_{\begin{subarray}{c}\varphi_{\theta}(x,\theta)=0,\\\
(x,\theta)\notin\operatorname{Smo}(a)\end{subarray}}\langle{\varphi_{x}(x,\theta)-\xi}\rangle^{-N_{1}-N_{2}}\phi(x)|a(x,\theta)|\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle\lesssim\int_{\begin{subarray}{c}\varphi_{\theta}(x,\theta)=0,\\\
(x,\theta)\notin\operatorname{Smo}(a)\end{subarray}}\langle{\xi}\rangle^{-N_{1}}\langle{\varphi_{x}(x,\theta)}\rangle^{-N_{2}}\phi(x)\langle{\theta}\rangle^{m}\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle\simeq\langle{\xi}\rangle^{-N_{1}}\int_{\begin{subarray}{c}\varphi_{\theta}(x,\theta)=0,\\\
(x,\theta)\notin\operatorname{Smo}(a)\end{subarray}}\phi(x)\langle{\theta}\rangle^{m-N_{2}}\,\mathrm{d}{x}\,\mathrm{d}{\theta}\simeq\langle{\xi}\rangle^{-N_{1}},$

To guarantee the derivation from (8.41) to (8.42), we need
$\xi\neq\varphi_{x}(x,\theta)$ for these $(x,\theta)$ which satisfy
$\varphi_{x}(x,\theta)=0$ and $(x,\theta)\notin\operatorname{Smo}(a)$. We
finished the proof. ∎

###### Formal proof of Theorem 8.14.

We do some preparation first. Define $\mathcal{A}$ as the collection of
subsets $\Omega$ in
$\mathbb{R}_{x}^{n}\times(\mathbb{R}_{\xi}^{n}\backslash\\{0\\})$ where
$(x,\xi)\in\Omega\Rightarrow(x,t\xi)\in\Omega$ for any $t>0$, and
$\mathcal{B}$ as the collection of subsets in
$\mathbb{R}_{x}^{n}\times\mathbb{S}_{\xi}^{n-1}$. Then there is a one-to-one
correspondence between $\mathcal{A}$ and $\mathcal{B}$, and we denote the one-
to-one mapping as $S$,

$S\colon\Omega\in\mathcal{A}\ \mapsto\
S\Omega=\\{(x,\eta)\,;\,\exists\xi\in{\mathbb{R}^{n}}\textrm{~{}s.t.~{}}\eta=\xi/|\xi|\text{~{}and~{}}(x,\xi)\in\Omega\\}\in\mathcal{B}.$

Let $T\colon(x,\theta)\mapsto(x,\varphi_{x}(x,\theta))$. Note that $ST=TS$.
For any positive integer $k$, denote

$V_{k}:=\\{(x,\theta)\in\mathbb{R}_{x}^{n}\times\mathbb{R}_{\xi}^{n}\,;\,|\varphi_{\theta}(x,\theta)|\leq
1/k\\}.$ (8.43)

It can be checked that

  * •

$\\{V_{k}\\}_{k}$ and $\\{TV_{k}\\}_{k}$ are decreasing in terms of $k$,

  * •

$V_{k}$ and $TV_{k}$ are closed in
$\mathbb{R}_{x}^{n}\times(\mathbb{R}_{\xi}^{n}\backslash\\{0\\})$,

  * •

$V_{k}$, $TV_{k}\in\mathcal{A}$,

  * •

$SV_{k}$ and $STV_{k}$ ($=TSV_{k}$) are also closed in
$\mathbb{R}_{x}^{n}\times\mathbb{S}_{\xi}^{n-1}$.

Now let’s assume

$(x_{0},\xi_{0})\notin TV_{k},$ (8.44)

then $(x_{0},\xi_{0}/|\xi_{0}|)$ is not in $STV_{k}$, which is a closed set.
Therefore, there exists $\epsilon_{1}>0$ such that

$\\{(x,\xi/|\xi|)\,;\,|x-x_{0}|\leq\epsilon_{1},\,|\xi/|\xi|-\xi_{0}/|\xi_{0}||\leq\epsilon_{1}\\}\cap
STV_{k}=\emptyset.$ (8.45)

Because $\nabla_{(x,\theta)}\varphi(x,\theta)$ is always assumed to be
nonzero, the number

$\inf_{|x-x_{0}|\leq\epsilon_{1},\,\theta\in\mathbb{S}^{n-1}}\big{(}|\varphi_{x}(x,\theta)|+|\varphi_{\theta}(x,\theta)|\big{)}$

exists and is positive and we denote it as $\epsilon_{2}$,

$\epsilon_{2}:=\inf_{|x-x_{0}|\leq\epsilon_{1},\,\theta\in\mathbb{S}^{n-1}}\big{(}|\varphi_{x}(x,\theta)|+|\varphi_{\theta}(x,\theta)|\big{)}>0.$
(8.46)

Let $k_{0}$ be any positive integer such that

$k_{0}>2\lceil 1/\epsilon_{2}\rceil.$ (8.47)

Now, for any $(x,\theta)\in SV_{k_{0}}$, we know $(x,\theta)\in V_{k_{0}}$, so
(8.43) gives $|\varphi_{\theta}(x,\theta)|\leq 1/k_{0}<\epsilon_{2}/2$, so
from (8.46) we can conclude that

$|\varphi_{x}(x,\theta)|>\epsilon_{2}/2>0\quad\text{in}\quad
W:=\\{(x,\theta)\in SV_{k_{0}}\,;\,|x-x_{0}|\leq\epsilon_{1}\\}.$ (8.48)

Fix some $\phi\in C_{c}^{\infty}(B(x_{0},\epsilon_{1}))$. And $\chi\in
C_{c}^{\infty}({\mathbb{R}^{n}})$ is a cut-off function with support
containing the origin. Now we estimate $\widehat{\phi A}(\xi)$. We have

$\displaystyle\widehat{\phi A}(\xi)$ $\displaystyle=A(\phi
e^{-ix\cdot\xi})=(2\pi)^{-n/2}\int
e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle\simeq\int
e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)\chi(\theta)a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}+\int
e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\mathbb{R}_{\theta}^{n}}e^{i\varphi(x,\theta)}\chi(\theta)a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\quad+\int_{(x,\theta)\in\operatorname{Smo}^{\prime}(a)}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\,\mathrm{d}{x}$
$\displaystyle\quad+\int_{(x,\theta)\notin\operatorname{Smo}^{\prime}(a)}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=:I_{1}(\xi)+I_{2}(\xi)+I_{3}(\xi),$ (8.49)

where $\operatorname{Smo}^{\prime}(a)$ is an arbitrary subset of
$\operatorname{Smo}(a)$ such that for every fixed $x$, the projection of the
intersect $S((x,\mathbb{R}_{\xi}^{n})\cap\operatorname{Smo}(a))$ is a compact
subset of the sphere $\mathbb{S}^{n-1}$. The $(x,\mathbb{R}_{\xi}^{n})$ means
$\\{(x,\xi)\in\mathbb{R}_{x}^{n}\times\mathbb{R}_{\xi}^{n}\,;\,\xi\in{\mathbb{R}^{n}}\\}$.
The term $I_{1}$ is easy to estimate. The
$\int_{\mathbb{R}_{\theta}^{n}}e^{i\varphi(x,\theta)}\chi(\theta)a(x,\theta)\,\mathrm{d}{\theta}$
in $I_{1}$ is $C^{\infty}$-smooth in terms of $x$, so by using integration by
parts we can have

$|I_{1}(\xi)|=|\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\mathcal{C}(x)\,\mathrm{d}{x}|\leq
C_{\alpha}\xi^{-\alpha},\quad\forall\xi,\,\forall\,\text{multi-
index~{}}\alpha.$ (8.50)

where $\mathcal{C}$ is some function in $C_{c}^{\infty}({\mathbb{R}^{n}})$.
The estimation (8.50) gives

$|I_{1}(\xi)|\leq C_{N}\langle{\xi}\rangle^{-N},\quad\forall\xi,\,\forall
N\in\mathbb{N}.$ (8.51)

And $I_{2}$ can be estimated as follows,

$\displaystyle|I_{2}(\xi)|$
$\displaystyle=|\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\phi(x)\big{(}\int_{\\{\theta\,;\,(x,\theta)\in\operatorname{Smo}^{\prime}(a)\\}}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}|$
$\displaystyle=|\int_{\mathbb{R}_{x}^{n}}\big{[}\big{(}i^{|\alpha|}\xi^{-\alpha}\partial_{x}^{\alpha}\big{)}e^{-ix\cdot\xi}\big{]}\phi(x)\big{(}\int_{\\{\theta\,;\,(x,\theta)\in\operatorname{Smo}^{\prime}(a)\\}}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}|$
$\displaystyle=\xi^{-\alpha}|\int_{\mathbb{R}_{x}^{n}}e^{-ix\cdot\xi}\partial_{x}^{\alpha}\Big{[}\phi(x)\big{(}\int_{\\{\theta\,;\,(x,\theta)\in\operatorname{Smo}^{\prime}(a)\\}}e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\Big{]}\,\mathrm{d}{x}|.$

It is easy to check that the term in $\big{[}\cdots\big{]}$ are
$C^{\infty}$-smooth and compactly supported w.r.t. $x$, thus it is integrable.
Therefore,

$|I_{2}(\xi)|\leq C_{N}\langle{\xi}\rangle^{-N},\quad\forall\xi,\,\forall
N\in\mathbb{N}.$ (8.52)

Then we move on to $I_{3}$,

$\displaystyle I_{3}(\xi)$
$\displaystyle=\int_{(\operatorname{Smo}^{\prime}(a))^{c}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=\int_{(\operatorname{Smo}^{\prime}(a))^{c}\cap(V_{k_{0}})^{c}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle\quad+\int_{(\operatorname{Smo}^{\prime}(a))^{c}\cap
V_{k_{0}}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=:I_{4}(\xi)+I_{5}(\xi),$ (8.53)

where $(\operatorname{Smo}^{\prime}(a))^{c}$ signifies the complementary set
of $\operatorname{Smo}^{\prime}(a)$. Note that in $(V_{k_{0}})^{c}$, the
$|\varphi_{\theta}(x,\theta)|$ is no less that $1/k_{0}$ (c.f. (8.43)), thus
no singularity will accrue when $|\varphi_{\theta}(x,\theta)|$ appears in the
denominator. Hence, for $I_{4}$ we have

$\displaystyle I_{4}(\xi)$
$\displaystyle=\int_{(\operatorname{Smo}^{\prime}(a))^{c}\cap(V_{k_{0}})^{c}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\,\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=\int\hskip 32.0pte^{-ix\cdot\xi}\hskip 32.0pt\phi(x)\big{(}\int
e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle=\int\big{[}(i^{|\alpha|}\xi^{-\alpha}\partial_{x}^{\alpha})e^{-ix\cdot\xi}\big{]}\,\phi(x)\big{(}\int
e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\simeq\xi^{-\alpha}\int
e^{-ix\cdot\xi}\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\partial_{x}^{\alpha-\beta}\phi(x)\cdot\partial_{x}^{\beta}\big{(}\int
e^{i\varphi(x,\theta)}(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\simeq\xi^{-\alpha}\int
e^{-ix\cdot\xi}\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\partial_{x}^{\alpha-\beta}\phi(x)$
$\displaystyle\quad\cdot\partial_{x}^{\beta}\big{(}\int(\frac{-i\varphi_{\theta}(x,\theta)\cdot\nabla_{\theta}}{|\varphi_{\theta}(x,\theta)|^{2}})^{N}(e^{i\varphi(x,\theta)})(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\lesssim\xi^{-\alpha}\sum_{\beta\leq\alpha}\int
e^{-ix\cdot\xi}C_{\alpha,\beta}\partial_{x}^{\alpha-\beta}\phi(x)\big{(}\int\langle{\theta}\rangle^{m+|\beta|-N}\,\mathrm{d}{\theta}\big{)}\,\mathrm{d}{x}$
$\displaystyle\leq
C_{\alpha}\xi^{-\alpha},\quad\forall\xi,\,\forall\,\text{multi-
index~{}}\alpha.$ (8.54)

Now for the estimation of $I_{5}$, we need some constraints on the direction
of $\xi$. It is this term that determines $\operatorname{WF}(A)$. Because
$(x_{0},\xi_{0})\notin TV_{k_{0}}$ (see (8.44)), according to (8.45), there is
a cone $W\subset\mathbb{R}_{\xi}^{n}\backslash\\{0\\}$ such that

$\xi_{0}\in W\quad\text{and}\quad|\nabla_{x}\varphi(x,\theta)-\xi|\geq
C(|\nabla_{x}\varphi(x,\theta)|+|\xi|),\forall\xi\in W.$ (8.55)

Define
$L:=\frac{-i(\partial_{x}\varphi(x,\theta)-\xi)\cdot\nabla_{x}}{|\nabla_{x}\varphi(x,\theta)-\xi|^{2}}$.
By using the fact that

$|\partial^{\alpha}Lf(x)|\lesssim|\nabla_{x}\varphi(x,\theta)-\xi|^{-1}\sum_{\beta}|\partial^{\beta}f(x)|,$

we can have, for all $\xi\in W$,

$\displaystyle I_{5}(\xi)$
$\displaystyle=\int_{(\operatorname{Smo}^{\prime}(a))^{c}\cap
V_{k_{0}}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=\int_{(\operatorname{Smo}^{\prime}(a))^{c}\cap
V_{k_{0}}}L^{N}(e^{i(\varphi(x,\theta)-x\cdot\xi)})\phi(x)(1-\chi(\theta))a(x,\theta)\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle=\int_{(\operatorname{Smo}^{\prime}(a))^{c}\cap
V_{k_{0}}}e^{i(\varphi(x,\theta)-x\cdot\xi)}\,{}^{t}\\!L^{N}\big{(}\phi(x)(1-\chi(\theta))a(x,\theta)\big{)}\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle\lesssim\int_{\mathop{\rm
supp}\phi\times\mathbb{R}_{\theta}^{n}}(1+|\varphi_{x}(x,\theta)-\xi|)^{-N}\langle{\theta}\rangle^{m}\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle(N_{1}+N_{2}=N)\quad$
$\displaystyle\lesssim\langle{\xi}\rangle^{-N_{1}}\int_{\mathop{\rm
supp}\phi\times\mathbb{R}_{\theta}^{n}}\langle{|\theta|\cdot\varphi_{x}(x,\theta/|\theta|)}\rangle^{-N_{2}}\langle{\theta}\rangle^{m}\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle\big{(}\text{by~{}}\eqref{eq:gredx-PM2021}\big{)}\quad$
$\displaystyle\lesssim\langle{\xi}\rangle^{-N_{1}}\int_{\mathop{\rm
supp}\phi\times\mathbb{R}_{\theta}^{n}}\langle{\theta}\rangle^{-N_{2}}\langle{\theta}\rangle^{m}\,\mathrm{d}{x}\,\mathrm{d}{\theta}$
$\displaystyle\lesssim\langle{\xi}\rangle^{-N_{1}}\quad\forall
N_{1}\in\mathbb{N}.$ (8.56)

Combining (8.44), (8.47), (8.49), (8.51), (8.52), (8.53), (8.54) and (8.56),
we arrive at

$|\widehat{\phi A}(\xi)|\lesssim\langle{\xi}\rangle^{-N},\quad\xi\in
W,~{}\forall N\in\mathbb{N},$ (8.57)

where the $W$ is a cone containing $\xi_{0}$, and
$(x_{0},\xi_{0})\neq(x,\varphi_{x}(x,\theta))$ for these $(x,\theta)$
satisfying

$(x,\theta)\in(\operatorname{Smo}^{\prime}(a))^{c}\cap V_{k_{0}}.$

Therefore, for any $k_{0}>2\lceil 1/\epsilon_{2}\rceil$, there holds

$\Big{(}\\{(x,\varphi_{x}(x,\theta))\,;\,|\varphi_{\theta}(x,\theta)|\leq
1/k_{0},\,(x,\theta)\in(\operatorname{Smo}^{\prime}(a))^{c}\\}\Big{)}^{c}\subset\Big{(}\operatorname{WF}(A)\Big{)}^{c},$

thus

$\operatorname{WF}(A)\subset\\{(x,\varphi_{x}(x,\theta))\,;\,|\varphi_{\theta}(x,\theta)|\leq
1/k_{0},\,(x,\theta)\notin\operatorname{Smo}^{\prime}(a)\\}.$

Finally, let $k_{0}$ goes to zero and choose $\operatorname{Smo}^{\prime}(a)$
to be arbitrarily close to $\operatorname{Smo}(a)$, we arrive at the
conclusion (8.38). ∎

###  8.4. Applications

Now we are ready to apply those results.

####  8.4.1. Microlocality of $\Psi$DOs

######  Proposition 8.16.

Assume $a\in
S^{+\infty}(\mathbb{R}_{x}^{n}\times\mathbb{R}_{y}^{n}\times\mathbb{R}_{\xi}^{n})$
is symbol and $K$ is the kernel of the corresponding $\Psi$DO of $a$, then

$\boxed{\operatorname{WF}(K)\subset\\{(x,x;\xi,-\xi)\,;\,(x,x,\xi)\notin\operatorname{Smo}(a)\\}.}$
(8.58)

###### Proof.

Denote the corresponding $\Psi$DO as $A$, then

$\displaystyle\langle{K(x,y),u\otimes v(x,y)}\rangle$
$\displaystyle=\langle{K(x,y),u(x)\otimes v(y)}\rangle=\langle\langle
K(x,y),u(x)\rangle_{x},v(y)\rangle_{y}$
$\displaystyle=\langle{Au(y),v(y)}\rangle=(2\pi)^{-n}\int
e^{i(x-y)\cdot\xi}a(x,y,\xi)u(x)v(y)\,\mathrm{d}{x}\,\mathrm{d}{y}\,\mathrm{d}{\xi}$
$\displaystyle=(2\pi)^{-n}\int e^{i(x-y)\cdot\xi}a(x,y,\xi)u\otimes
v(x,y)\,\mathrm{d}{x}\,\mathrm{d}{y}\,\mathrm{d}{\xi}.$

Therefore, in the oscillatory integral sense,

$K(x,y)=(2\pi)^{-n}\int e^{i(x-y)\cdot\xi}a(x,y,\xi)\,\mathrm{d}{\xi}.$

According to Theorem 8.14, we have

$\operatorname{WF}(K)\subset\\{(x,y;\varphi_{x,y}(x,y,\xi))\,;\,\varphi_{\xi}(x,y,\xi)=0,\,(x,y,\xi)\notin\operatorname{Smo}(a)\\},$

where $\varphi(x,y,\xi)=(x-y)\cdot\xi$. Hence,

$\displaystyle\operatorname{WF}(K)$
$\displaystyle\subset\\{(x,y;\xi,-\xi)\,;\,x-y=0,\,(x,y,\xi)\notin\operatorname{Smo}(a)\\}$
$\displaystyle=\\{(x,x;\xi,-\xi)\,;\,(x,x,\xi)\notin\operatorname{Smo}(a)\\}.$

The proof is complete. ∎

######  Theorem 8.17.

Assume that $A$ is a $\Psi$DO, then for $u\in\mathscr{E}^{\prime}$ we have

$\text{microlocality:}\quad\boxed{\operatorname{WF}(Au)\subseteq\operatorname{WF}(u)\backslash\operatorname{Smo}(A).}$
(8.59)

Moreover, if $A$ is elliptic in the sense of Definition 6.4, then
$\operatorname{Smo}(A)=\emptyset$ and

$\boxed{\operatorname{WF}(Au)=\operatorname{WF}(u).}$ (8.60)

###### Proof.

We have $Au(x)=\langle{K(x,y),u(y)}\rangle$ where $K$ is its kernel, so
according to Theorem 8.11 and Proposition 8.16, we can conclude

$\displaystyle\operatorname{WF}(Au)$
$\displaystyle\subseteq\big{(}\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u)\big{)}\cup\operatorname{WF}_{x}(K)$
$\displaystyle=\big{(}\\{(x,x;\xi,\xi)\,;\,(x,\xi)\notin\operatorname{Smo}(a)\\}\circ\operatorname{WF}(u)\big{)}\cup\emptyset$
$\displaystyle=\operatorname{WF}(u)\backslash\operatorname{Smo}(a)=\operatorname{WF}(u)\backslash\operatorname{Smo}(A).$

If $A$ is elliptic, then according to the definition, we have

$|a(x,\xi)|\geq
C\langle{\xi}\rangle^{m},\quad\text{when~{}}x\in{\mathbb{R}^{n}},\ |\xi|\geq
R,$

for some constants $m\in\mathbb{R}$, $C>0$ and $R>0$, so it is obvious that
$\operatorname{Smo}(A)=\emptyset$, thus by (8.59),

$\operatorname{WF}(Au)\subseteq\operatorname{WF}u.$

Also, because $A$ is elliptic, then by Theorem 6.6 we know $A$ has a
parametrix $B$ such that $R:=BA-I\in\Psi^{-\infty}$, so

$\operatorname{WF}(u)=\operatorname{WF}(BAu-
Ru)\subset\operatorname{WF}(BAu)\cup\operatorname{WF}(Ru)\subset\operatorname{WF}(Au).$

In total, $\operatorname{WF}(Au)=\operatorname{WF}u$. The proof is complete. ∎

######  Lemma 8.18.

Assume $a$ is a symbol and $u\in\mathscr{E}^{\prime}(\Omega)$. Denote the
corresponding $\Psi$DO of $a$ as $A$, then

$\boxed{\operatorname{WF}(Au)\cap\operatorname{Smo}(a)=\emptyset.}$ (8.61)

###### Proof.

This is a straight forward outcome of (8.59). ∎

####  8.4.2. Pull-back of distributions

######  Theorem 8.19.

Let $\Omega_{1}$ and $\Omega_{2}$ be two domain in ${\mathbb{R}^{n}}$, and
$\psi\colon\Omega_{1}\to\Omega_{2}$ is an diffeomorphism. Then for any
$u\in\mathscr{D}^{\prime}(\Omega_{2})$, we have
$\psi^{*}u\in\mathscr{D}^{\prime}(\Omega_{1})$, and

$\boxed{\operatorname{WF}(\psi^{*}u)=\\{(x,{}^{t}\psi^{\prime}|_{x}\eta)\,;\,(\psi(x),\eta)\in\operatorname{WF}(u)\\},}$
(8.62)

where $\psi^{\prime}$ signifies the matrix whose ($i$-row, $j$-column) element
is $\partial_{x_{i}}\psi^{j}$, and $({}^{t}\psi^{\prime})^{-1}|_{x}$ is the
inverse of transpose of the matrix $\psi^{\prime}$ evaluated at $x$, and
$({}^{t}\psi^{\prime})^{-1}|_{x}\eta$ stands for the matrix multiplication of
the matrix $({}^{t}\psi^{\prime})^{-1}|_{x}$ and the vertical vector $\eta$.

The mapping in (8.62),

$(x,{}^{t}\psi^{\prime}|_{x}\eta)\ \mapsto\ (\psi(x),\eta)$

is invariant on the cotangent bundle (see [maLSNH2020, for Theorem 18.1.17]).

###### First proof of Theorem 8.19.

It is obvious that $\psi^{*}u\in\mathscr{D}^{\prime}(\Omega_{1})$.

For $(y_{0},\xi_{0})$, because $\psi$ is a diffeomorphism, we can find
$(x_{0},\eta_{0})$ such that $x_{0}=\psi^{-1}(y_{0})$ and
$\xi_{0}=(\psi^{-1})^{\prime}|_{y_{0}}\cdot\eta_{0}$. Assume that
$(y_{0},\xi_{0})\notin\operatorname{WF}(u)$. For a smooth cutoff function
$\varphi_{x_{0}}$ satisfying $\varphi_{x_{0}}(x_{0})\neq 1$, we have

$\displaystyle(\varphi_{x_{0}}\psi^{*}u)^{\wedge}(\eta)\simeq\int
e^{-ix\cdot\eta}\varphi_{x_{0}}(x)\psi^{*}u(x)\,\mathrm{d}{x}=\int
e^{-ix\cdot\eta_{0}}\varphi_{x_{0}}(x)u(\psi(x))\,\mathrm{d}{x}$
$\displaystyle=$ $\displaystyle\int
e^{-i\psi^{-1}(y)\cdot\eta}\varphi_{x_{0}}(\psi^{-1}(y))u(y)\,\mathrm{d}{\psi}^{-1}(y)\quad\big{(}y=\psi(x)\big{)}$
$\displaystyle=$ $\displaystyle\int
e^{-i{}^{t}\psi^{-1}(y)\eta}\cdot\varphi_{x_{0}}(\psi^{-1}(y))|\frac{\partial\psi^{-1}}{\partial
y}(y)|\phi^{-1}(y)\cdot(\phi u)(y)\,\mathrm{d}{y}$ $\displaystyle=$
$\displaystyle\int
e^{-i{}^{t}\psi^{-1}(y)\eta}\cdot\tilde{\varphi}_{y_{0}}(y)\cdot(\phi
u)(y)\,\mathrm{d}{y}=\int
e^{-i({}^{t}\psi^{-1}(y)\eta-y\cdot\xi)}\cdot\tilde{\varphi}_{y_{0}}(y)\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$ $\displaystyle=$
$\displaystyle\int_{|\hat{\xi}_{0}-\hat{\xi}|>1}e^{-i({}^{t}\psi^{-1}(y)\eta-y\cdot\xi)}\cdot\tilde{\varphi}_{y_{0}}(y)\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$
$\displaystyle+\int_{|\hat{\xi}_{0}-\hat{\xi}|\leq
1}e^{-i{}^{t}\psi^{-1}(y)\eta}e^{iy\cdot\xi}\cdot\tilde{\varphi}_{y_{0}}(y)\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$ $\displaystyle=$
$\displaystyle\\!\\!:I_{1}+I_{2},$

where $\hat{\xi}:=\xi/|\xi|$ and the same for $\hat{\xi}_{0}$. For convenience
we have written $\psi^{-1}(y)\cdot\eta$ as ${}^{t}\psi^{-1}(y)\eta$, where
${}^{t}M$ signifies the transpose operation for any matrix $M$. By doing so it
will be more straightforward when we make derivatives. Here
$\tilde{\varphi}_{y_{0}}(y)$ is a generic function which is
$C^{\infty}$-smooth and is compactly supported w.r.t. $y$ and whose precise
definition may varies from line to line.

For $I_{1}$, because $|\hat{\xi}_{0}-\hat{\xi}|>1$ and
$(y_{0},\xi_{0})\notin\operatorname{WF}(u)$, we have
$\big{|}({}^{t}\psi^{-1})^{\prime}|_{y}\eta-\xi\big{|}\geq\frac{1}{2}$ and
$|\widehat{\phi u}(\xi)|\lesssim\langle{\xi}\rangle^{-N_{0}}$ for some integer
$N_{0}$. The number $N_{0}$ comes from the fact that $\phi u$ is a compactly
supported distribution so its Fourier transform has (at most) polynomial
growth. Hence,

$\displaystyle I_{1}$
$\displaystyle=\int_{|\hat{\xi}_{0}-\hat{\xi}|>1}\Big{(}\frac{1+(({}^{t}\psi^{-1})^{\prime}|_{y}\eta-\xi)\cdot\nabla_{y}}{\langle{({}^{t}\psi^{-1})^{\prime}|_{y}\eta-\xi}\rangle^{2}}\Big{)}^{N}\big{(}e^{-i(\psi^{-1}(y)\cdot\eta-y\cdot\xi)}\big{)}\cdot\tilde{\varphi}_{y_{0}}(y)\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$
$\displaystyle\simeq\int_{|\hat{\xi}_{0}-\hat{\xi}|>1}\langle{({}^{t}\psi^{-1})^{\prime}|_{y}\eta-\xi}\rangle^{-N}e^{-i(\psi^{-1}(y)\cdot\eta-y\cdot\xi)}\cdot\tilde{\varphi}_{y_{0}}(y)\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$
$\displaystyle\lesssim\int_{|\hat{\xi}_{0}-\hat{\xi}|>1}\langle{({}^{t}\psi^{-1})^{\prime}|_{y}\eta}\rangle^{-N_{1}}\langle{\xi}\rangle^{-N_{2}}\cdot|\tilde{\varphi}_{y_{0}}(y)|\cdot|\widehat{\phi
u}(\xi)|\,\mathrm{d}{\xi}\,\mathrm{d}{y}\quad(N=N_{1}+N_{2})$
$\displaystyle\lesssim\langle{\eta}\rangle^{-N_{1}}\int_{|\hat{\xi}_{0}-\hat{\xi}|>1}\langle{\xi}\rangle^{-N_{2}}\cdot(\int|\tilde{\varphi}_{y_{0}}(y)|\,\mathrm{d}{y})\cdot|\widehat{\phi
u}(\xi)|\,\mathrm{d}{\xi}$
$\displaystyle\lesssim\langle{\eta}\rangle^{-N_{1}}\int_{|\hat{\xi}_{0}-\hat{\xi}|>1}\langle{\xi}\rangle^{-N_{2}}\langle{\xi}\rangle^{N_{0}}\,\mathrm{d}{\xi}\lesssim\langle{\eta}\rangle^{-N_{1}},$

provided that $N_{2}-N_{0}>$ the dimension of $\xi$.

For $I_{2}$, we have

$\displaystyle I_{2}$ $\displaystyle=\int_{|\hat{\xi}_{0}-\hat{\xi}|\leq
1}\Big{(}\frac{1+(({}^{t}\psi^{-1})^{\prime}|_{y}\eta)\cdot\nabla_{y}}{\langle{({}^{t}\psi^{-1})^{\prime}|_{y}\eta}\rangle^{2}}\Big{)}^{N}\big{(}e^{-i\psi^{-1}(y)\cdot\eta}\big{)}e^{iy\cdot\xi}\cdot\tilde{\varphi}_{y_{0}}(y)\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$
$\displaystyle\simeq\int_{|\hat{\xi}_{0}-\hat{\xi}|\leq
1}\langle{({}^{t}\psi^{-1})^{\prime}|_{y}\eta}\rangle^{-N}e^{-i\psi^{-1}(y)\cdot\eta}\cdot\big{(}\nabla_{y}\big{)}^{N}\big{(}e^{iy\cdot\xi}\cdot\tilde{\varphi}_{y_{0}}(y)\big{)}\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$
$\displaystyle\simeq\int_{|\hat{\xi}_{0}-\hat{\xi}|\leq
1}\langle{\eta}\rangle^{-N}e^{-i\psi^{-1}(y)\cdot\eta}\cdot\langle{\xi}\rangle^{N}\tilde{\varphi}_{y_{0}}(y)\cdot\widehat{\phi
u}(\xi)\,\mathrm{d}{\xi}\,\mathrm{d}{y}$
$\displaystyle\lesssim\langle{\eta}\rangle^{-N}\int_{|\hat{\xi}_{0}-\hat{\xi}|\leq
1}\langle{\xi}\rangle^{N}\cdot(\int|\tilde{\varphi}_{y_{0}}(y)|\,\mathrm{d}{y})\cdot|\widehat{\phi
u}(\xi)|\,\mathrm{d}{\xi}$
$\displaystyle\lesssim\langle{\eta}\rangle^{-N}\int_{|\hat{\xi}_{0}-\hat{\xi}|\leq
1}\langle{\xi}\rangle^{N}\langle{\xi}\rangle^{-N-n-1}\,\mathrm{d}{\xi}\lesssim\langle{\eta}\rangle^{-N}.$

In total, we have
$|(\varphi_{x_{0}}\psi^{*}u)^{\wedge}(\eta)|\lesssim\langle{\eta}\rangle^{-N}$
for any integer $N$ if $\hat{\eta}$ is in a small neighborhood of
$\hat{\eta}_{0}$ where
$(\psi(x_{0}),({}^{t}\psi^{-1})^{\prime}|_{y_{0}}\eta_{0})\notin\operatorname{WF}(u)$,
namely,

$(\psi(x_{0}),({}^{t}\psi^{-1})^{\prime}|_{\psi(x_{0})}\eta_{0})\notin\operatorname{WF}(u)\
\Rightarrow\ (x_{0},\eta_{0})\notin\operatorname{WF}(\psi^{*}u).$

Therefore,

$(x_{0},\eta_{0})\in\operatorname{WF}(\psi^{*}u)\ \Rightarrow\
(\psi(x_{0}),({}^{t}\psi^{-1})^{\prime}|_{\psi(x_{0})}\eta_{0})\in\operatorname{WF}(u),$

so

$\operatorname{WF}(\psi^{*}u)\subset\\{(x,\eta)\,;\,(\psi(x),({}^{t}\psi^{-1})^{\prime}|_{\psi(x)}\eta)\in\operatorname{WF}(u)\\}.$

Because $\psi$ is invertible, we can obtain the opposite inclusion by looking
at $\psi^{-1*}(\psi^{*}u)$.

It can be shown that
$(\psi^{-1})^{\prime}|_{\psi(x)}=(\psi^{\prime}(x))^{-1}$. Indeed, by
differentiating $x=\psi^{-1}(\psi(x))$ w.r.t. $x$ we obtain
$I=(\psi^{-1})^{\prime}|_{\psi(x)}\cdot\psi^{\prime}(x),$ so
$(\psi^{-1})^{\prime}|_{\psi(x)}=(\psi^{\prime}(x))^{-1},$ and by taking
transpose we obtain

$({}^{t}\psi^{-1})^{\prime}|_{\psi(x)}=({}^{t}\psi^{\prime}(x))^{-1},$

so

$\displaystyle\operatorname{WF}(\psi^{*}u)$
$\displaystyle=\\{(x,\eta)\,;\,(\psi(x),({}^{t}\psi^{\prime})^{-1}|_{x}\eta)\in\operatorname{WF}(u)\\}$
$\displaystyle=\\{(x,{}^{t}\psi^{\prime}|_{x}\xi)\,;\,(\psi(x),\xi)\in\operatorname{WF}(u)\\}.$

The proof is complete. ∎

There is also another proof for Theorem 8.19. As in Remark 8.15, we can use
Theorems 8.14 and 8.11 to obtain $\operatorname{WF}(\psi^{*}u)$.

###### second proof of Theorem 8.19.

The pull-back $\psi^{*}$ has a kernel: for $f\in C^{\infty}(\Omega_{2})$ and
$g\in C^{\infty}(\Omega_{1})$, we have

$\displaystyle\langle{\psi^{*}f,g}\rangle$
$\displaystyle=\int\psi^{*}f(x)g(x)\,\mathrm{d}{x}=(2\pi)^{-n}\int
e^{i(\psi(x)-y)\cdot\eta}f(y)g(x)\,\mathrm{d}{x}\,\mathrm{d}{y}\,\mathrm{d}{\eta}$
$\displaystyle\simeq\langle{\int
e^{i(\psi(x)-y)\cdot\eta}\,\mathrm{d}{\eta},(g\otimes f)(x,y)}\rangle$
$\displaystyle=\langle{K,g\otimes f}\rangle,\quad\text{where}\quad K(x,y)=\int
e^{i({}^{t}\psi(x)-{}^{t}y)\eta}\,\mathrm{d}{\eta}.$

and $\psi^{*}u(x)=\langle{K(x,y),u(y)}\rangle.$ By Theorem 8.14 we have

$\operatorname{WF}(K)\subset\\{(x,\psi(x);{}^{t}\psi^{\prime}|_{x}\eta,-\eta)\\},$

and then by Theorem 8.11 we have

$\displaystyle\operatorname{WF}(\psi^{*}u)$
$\displaystyle\subseteq\big{(}\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u)\big{)}\cup\operatorname{WF}_{x}(K)=\\{(x,{}^{t}\psi^{\prime}|_{x}\eta)\,;\,(\psi(x),\eta)\in\operatorname{WF}(u)\\}.$

The opposite inclusion can be obtained by looking at $\psi^{-1*}(\psi^{*}u)$.
We obtain (8.62). ∎

### Exercise

######  Exercise 8.1.

Prove Lemme 8.3.

######  Exercise 8.2.

Prove that the condition
$\big{(}\operatorname{WF}^{\prime}(K)\circ\operatorname{WF}(u)\big{)}\cap
O_{x}=\emptyset$ guarantees (8.31).

######  Exercise 8.3.

Show the details in the computations (8.34) and (8.35).

##  Chapter 9 Propagation of the singularities

###  9.1. Microlocal parametrix

To study the microlocal parametrix, we recall notion of conic sets and the
smooth direction “$\operatorname{Smo}$” given in Definitions 8.1 & 8.2, and
$T^{*}{\mathbb{R}^{n}}\backslash 0$ stands for the cotangent bundle with the
zero section excluded. Now we generalize Definitions 6.1 & 6.4, microlocally,
as follows.

######  Definition 9.1 (Microlocal parametrix).

Assume $m\in\mathbb{R}$ and $T\in\Psi^{m}$. We call a $\Psi$DO $S$ a _left
(resp. right) microlocal parametrix_ of $T$ if there exists a nonempty open
conic set $\Gamma\subset T^{*}{\mathbb{R}^{n}}\backslash 0$ such that

$\operatorname{Smo}(ST-I)=\Gamma\quad\text{(resp.~{}}\operatorname{Smo}(TS-I)=\Gamma\text{)}.$

We call $S$ a _microlocal parametrix_ of $T$ if it is both a left and a right
microlocal parametrix under the same set $\Gamma$.

######  Definition 9.2 (Microlocal ellipticity).

Assume $m\in\mathbb{R}$ and $a\in S^{m}$, and $A$ is the $\Psi$DO of $a$. Let
$\Gamma\subset T^{*}{\mathbb{R}^{n}}\backslash 0$ be a open conic set. We say
$a$ (and $A$) is _microlocally elliptic in $\Gamma$_ if for some constants
$C_{\Gamma}>0$, $R>0$,

$\boxed{|a(x,\xi)|\geq
C_{\Gamma}\langle{\xi}\rangle^{m},\quad\forall(x,\xi)\in\Gamma,\ |\xi|\geq
R.}$

We write $\boxed{\operatorname{Char}A:=(\bigcup\mathscr{F})^{c}}$, where
$\mathscr{F}=\\{\Gamma\,;\,A\text{~{}is microlocally elliptic
in~{}}\Gamma\\}$, and the notation $\Omega^{c}$ stands for the complement of
$\Omega$ in $T^{*}{\mathbb{R}^{n}}\backslash 0$.

From Definition 9.2, it is obvious that $\operatorname{Char}A$ is always
closed. The following claim is trivial.

######  Lemma 9.3.

$\operatorname{Char}A=\emptyset$ if and only if $A$ is elliptic in the sense
of Definition 6.4.

######  Lemma 9.4.

Assume $P$ is the $\Psi$DO with principal symbol $p_{m}(x,\xi)$ homogeneous in
$\xi$, then $\operatorname{Char}P=p_{m}^{-1}(0)$, where $p_{m}^{-1}(0)$
signifies the set $\\{(x,\xi)\in T^{*}{\mathbb{R}^{n}}\backslash
0\,;\,p_{m}(x,\xi)=0\\}$.

The proof is left as an exercise. The $\operatorname{Char}A$ and
$\operatorname{Smo}(A)$ is closely related. Results in §6.1 can be modified to
a microlocal version.

######  Theorem 9.5 (Microlocal ellipticity $\Leftrightarrow$ Microlocal
parametrix).

Let $m\in\mathbb{R}$ and $A\in\Psi^{m}$. Assume $A$ is microlocally elliptic
in $\Gamma$, where $\Gamma=(\operatorname{Char}A)^{c}$ is non-empty. Then $A$
has a microlocal parametrix $B$. Moreover, they satisfies

$\boxed{(\operatorname{Char}A)^{c}\subset\operatorname{Smo}(I-BA).}$ (9.1)

Conversely, if $A$ has either a right or left microlocal parametrix, then $A$
is microlocally elliptic.

###### Proof.

($\Rightarrow$) Fix $(x_{0},\xi_{0})\in\Gamma$. In the proof of Theorem 6.6,
we modify the function $\chi(\xi)$ to $\chi(x,\xi)$ which is given as
$\chi(x,\xi):=\chi(x-x_{0})\chi(\xi/|\xi|-\xi_{0}/|\xi_{0}|).$ Define
$b_{j}(x,\xi):=(1-\chi(x,\xi))/a(x,\xi)\cdot r_{j}(x,\xi)$ ($j\geq 0$) the
same way, with $r_{0}\equiv 1$, and follow the same steps as in the proof of
Theorem 6.6 we can obtain $\forall N\in\mathbb{N}$,

$\sigma(AB)=1-(1+r_{1}+\cdots+r_{N})\chi-r_{N+1}+S^{-N-1}=1+S^{-N-1}\
\text{in}\subset\Gamma,$

as in (6.3), so

$\sigma(I-AB)\in S^{-\infty}\text{~{}in~{}}\subset\Gamma.$

Due to the arbitrary of $\chi$, we conclude

$\sigma(I-AB)\in S^{-\infty}\text{~{}in~{}}\Gamma.$

Hence, by Definition 8.2 we obtain $\Gamma\subset\operatorname{Smo}(I-AB).$

($\Leftarrow$) Assume $B$ is the right parametrix of $A$, then there exists a
nonempty conic open set $\Gamma\subset T^{*}{\mathbb{R}^{n}}\backslash 0$ such
that $\operatorname{Smo}(AB-I)\subset\Gamma,$ which means
$a\\#b=1+S^{-\infty}\ \text{in}\ \Gamma.$ where $a$ and $b$ are symbols of $A$
and $B$, respectively. Similar to the proof of Theorem 6.7, we can prove that

$|a(x,\xi)|\geq\langle{\xi}\rangle^{m}/2,\quad\text{when}\quad(x,\xi)\in\Gamma,\
\langle{\xi}\rangle\geq C/2.$

Therefore, $a$ is microlocally elliptic in $\Gamma$. The proof for the left-
case is similar.

The proof is complete. ∎

######  Corollary 9.6.

For any $A\in\Psi^{+\infty}$ and any $u\in\mathscr{S}^{\prime}$, there holds

$\boxed{\operatorname{WF}(Au)\subset\operatorname{WF}u\subset\operatorname{WF}(Au)\cup\operatorname{Char}A.}$

###### Proof.

The “$\operatorname{WF}(Au)\subset\operatorname{WF}u$” is from Theorem Theorem
8.17.

When $\operatorname{Char}A=T^{*}{\mathbb{R}^{n}}\backslash 0$, the claim is
trivial. When $\operatorname{Char}A\subsetneqq T^{*}{\mathbb{R}^{n}}\backslash
0$, $(\operatorname{Char}A)^{c}$ is non-empty, so by Theorem 9.5, there exists
a microlocal parametrix $B$ of $A$, so we can have

$\displaystyle\operatorname{WF}u$
$\displaystyle=\operatorname{WF}(BAu+(I-BA)u)$
$\displaystyle\subset\operatorname{WF}(BAu)\cup\operatorname{WF}((I-BA)u)$
$\displaystyle\subset\operatorname{WF}(BAu)\cup\big{(}\operatorname{WF}u\backslash\operatorname{Smo}(I-BA)\big{)}\qquad(\text{by~{}}\eqref{eq:MLpr-
PM2021})$
$\displaystyle=\operatorname{WF}(BAu)\cup\big{(}\operatorname{WF}u\cap(\operatorname{Smo}(I-BA))^{c}\big{)}$
$\displaystyle\subset\operatorname{WF}(BAu)\cup\big{(}\operatorname{WF}u\cap\operatorname{Char}A\big{)}\qquad(\text{by
Theorem~{}}\ref{thm:mpa1-PM2021})$
$\displaystyle=\big{(}\operatorname{WF}(BAu)\cup\operatorname{WF}u\big{)}\cap\big{(}\operatorname{WF}(BAu)\cup\operatorname{Char}A\big{)}$
$\displaystyle\subset\operatorname{WF}u\cap\big{(}\operatorname{WF}(BAu)\cup\operatorname{Char}A\big{)},\qquad(\text{by~{}}\eqref{eq:MLpr-
PM2021})$

which gives

$\operatorname{WF}u\subset\operatorname{WF}(BAu)\cup\operatorname{Char}A\subset\operatorname{WF}(Au)\cup\operatorname{Char}A.$

The proof is complete. ∎

######  Remark 9.7.

Combining Corollary 9.6 and Lemma 9.3, we have

$\left\\{\begin{aligned} A\text{~{}is microlocally
elliptic:~{}}&\operatorname{WF}(Au)\subset\operatorname{WF}u\subset\operatorname{WF}(Au)\cup\operatorname{Char}A,\\\
A\text{~{}is
elliptic:~{}}&\operatorname{WF}(Au)\subset\operatorname{WF}u\subset\operatorname{WF}(Au)\
\Leftrightarrow\
\operatorname{WF}(Au)=\operatorname{WF}u.\end{aligned}\right.$

Hence, Corollary 9.6 can be viewed as a generalization of (8.60).

The following result is important.

######  Theorem 9.8.

Assume $u\in\mathscr{S}^{\prime}$, then

$\boxed{\operatorname{WF}(u)=\bigcap_{A\in\Psi^{+\infty},\,Au\in
C^{\infty}}\operatorname{Char}A.}$

###### Proof.

By Corollary 9.6, we have
$\operatorname{WF}u\subset\operatorname{WF}(Au)\cup\operatorname{Char}A$, so

$\operatorname{WF}(u)\subset\bigcap_{Au\in C^{\infty}}\operatorname{Char}A.$

For the another direction, assume $(x_{0},\xi_{0})\notin\operatorname{WF}(u)$,
then we shall construct a suitable $\Psi$DO $A$ such that

$Au\in
C^{\infty},\quad\text{and}\quad(x_{0},\xi_{0})\notin\operatorname{Char}(A),$
(9.2)

which gives $(x_{0},\xi_{0})\notin\bigcap_{Au\in
C^{\infty}}\operatorname{Char}A$, and so

$(\operatorname{WF}(u))^{c}\subset(\bigcap_{Au\in
C^{\infty}}\operatorname{Char}A)^{c}\ \Rightarrow\ \bigcap_{Au\in
C^{\infty}}\operatorname{Char}A\subset\operatorname{WF}(u),$

and the proof will be finished.

It remains to construct such an operator $A$, and we present two ways to do
it.

Method 1. Because $\operatorname{WF}(u)$ is closed and
$(x_{0},\xi_{0})\notin\operatorname{WF}(u)$, so there exist bounded open
neighborhoods $\omega$, $\omega^{\prime}$ of $x_{0}$ and conic open
neighborhoods $V$, $V^{\prime}$ of $\xi_{0}$ such that

$\left\\{\begin{aligned} &\omega\subsetneqq\omega^{\prime},\quad V\subsetneqq
V^{\prime},\\\ &\omega^{\prime}\times
V^{\prime}\cap\operatorname{WF}(u)=\emptyset.\end{aligned}\right.$

We denote $\Gamma:=\omega\times V$ and $\Gamma^{\prime}:=\omega^{\prime}\times
V^{\prime}$, then $\Gamma\subset\Gamma^{\prime}$ and
$\Gamma^{\prime}\cap\operatorname{WF}(u)=\emptyset$, so

$\operatorname{WF}(u)\subset\Gamma^{c}.$ (9.3)

Choose $a\in C^{\infty}(\mathbb{R}^{2n})$ such that

$\left\\{\begin{aligned} &\mathop{\rm supp}a\subset\Gamma,\quad\text{and}\quad
a(x_{0},\xi_{0})=1,\\\ &a(x,\xi)=a(x,\xi/|\xi|)\ \text{when}\ |\xi|\geq
1.\end{aligned}\right.$ (9.4)

It can be shown that $a\in S^{0}$ (see Exercise 9.2) and
$\underline{(x_{0},\xi_{0})\notin\operatorname{Char}(T_{a})}$. Moreover,
because $a\equiv 0$ in $\Gamma^{c}$ and $\Gamma^{c}$ is a conic set, we can
conclude

$\Gamma^{c}\subset\operatorname{Smo}(T_{a})\quad\Rightarrow\quad(\operatorname{Smo}(T_{a}))^{c}\subset\Gamma.$
(9.5)

Hence by Theorem 8.17 we have

$\displaystyle\operatorname{WF}(T_{a}u)$
$\displaystyle\subset\operatorname{WF}(u)\backslash\operatorname{Smo}(T_{a})=\operatorname{WF}(u)\cap(\operatorname{Smo}(T_{a}))^{c}\subset\operatorname{WF}(u)\cap\Gamma\quad(\text{by~{}}\eqref{eq:GmA-
PM2021})$
$\displaystyle=\operatorname{WF}(u)\backslash\Gamma^{c}=\emptyset,\quad(\text{by~{}}\eqref{eq:wfG-
PM2021})$

which implies $\underline{T_{a}u\in C^{\infty}}$. Condition (9.2) is
satisfied.

Method 2. Because $(x_{0},\xi_{0})\notin\operatorname{WF}(u)$, there exists
$\phi\in C_{c}^{\infty}({\mathbb{R}^{n}})$ such that $\widehat{\phi u}(\xi)$
is rapidly decaying when $\xi/|\xi|$ and $\xi_{0}/|\xi_{0}|$ are close enough,
say, $\big{|}\xi/|\xi|-\xi_{0}/|\xi_{0}|\big{|}\leq\epsilon$ for certain
$\epsilon>0$. Hence we choose $\psi\in C^{\infty}(\mathbb{S}^{n-1})$ such that
$\psi(\xi_{0}/|\xi_{0}|)=1$ and $\psi(\eta)=0$ when
$\big{|}\eta-\xi_{0}/|\xi_{0}|\big{|}>\epsilon$ where
$\eta\in\mathbb{S}^{n-1}$. Choose $\chi\in C_{c}^{\infty}({\mathbb{R}^{n}})$
such that $\chi(\xi)=1$ when $|\xi|\geq 1/10$ and $\chi(\xi)=0$ when
$|\xi|\geq 1/5$. Now we define a operator $A$ as follows

$A\varphi(x):=(2\pi)^{-n}\int_{\mathbb{R}^{2n}}e^{i(x-y)\cdot\xi}\phi(y)(1-\chi(\xi))\psi(\xi/|\xi|)\varphi(y)\,\mathrm{d}{y}\,\mathrm{d}{\xi},\quad\varphi\in\mathscr{S}({\mathbb{R}^{n}}).$

The purpose of the term “$1-\chi(\xi)$” is to cutoff the singularity near
$\xi=0$. By Theorem 5.6 we see $A$ is a $\Psi$DO of order $0$ with symbol

$a(x,\xi)=\phi(x)(1-\chi(\xi))\psi(\xi/|\xi|)+S^{-1},$ (9.6)

By (9.6) we can show $\underline{(x_{0},\xi_{0})\notin\operatorname{Char}A}$,
see Exercise 9.3. We can extend $A$ from $\mathscr{S}$ to
$\mathscr{S}^{\prime}$, and we have

$\widehat{Au}(\xi)=(1-\chi(\xi))\psi(\xi/|\xi|)\widehat{\phi u}(\xi),$

so $\widehat{Au}$ is rapidly decaying, which means $\underline{Au\in
C^{\infty}}$. Condition (9.2) is satisfied.

The proof is complete. ∎

###  9.2. Bicharacteristics

To prove a main result, we first introduce the notion of bicharacteristics.
The Hamiltonian $H_{p}$ of $p$ is defined as:

$H_{p}:=\nabla_{\xi}p\cdot\nabla_{x}-\nabla_{x}p\cdot\nabla_{\xi}.$ (9.7)

By Theorem 5.3 we can see

$\displaystyle\sigma([P,Q])$
$\displaystyle=\nabla_{\xi}p_{m}\cdot\nabla_{x}q_{m}-\nabla_{x}p_{m}\cdot\nabla_{\xi}q_{m}+S^{m_{1}+m_{2}-2}$
$\displaystyle=H_{p_{m}}q_{m}+S^{m_{1}+m_{2}-2},$

where $m_{1}$, $m_{2}$ are the order of $P$ and $Q$, and $p_{m}$, $q_{m}$ are
principal symbols of $P$ and $Q$, respectively.

In what follows we use the notation $T^{*}{\mathbb{R}^{n}}\backslash
0:=\\{(x,\xi)\in T^{*}{\mathbb{R}^{n}}\,;\,\xi\neq 0\\}$. We introduce the
notion of bicharacteristic. For more details on the Hamiltonian flows, see
[salo2006sta, §2].

######  Definition 9.9 (Null bicharacteristic).

Assume $p\in C^{1}(T^{*}{\mathbb{R}^{n}}\backslash 0;\mathbb{R})$, and
$I~{}(\ni 0)$ is an open connected subset of $\mathbb{R}$. Let a curve
$\gamma_{x_{0},\xi_{0}}\colon s\in I\mapsto(x(s),\xi(s))\in
T^{*}{\mathbb{R}^{n}}\backslash 0$ satisfies

$\left\\{\begin{aligned} &\dot{x}(s)=\nabla_{\xi}p(x(s),\xi(s)),\
\dot{\xi}(s)=-\nabla_{x}p(x(s),\xi(s)),\\\ &(x(0),\xi(0))=(x_{0},\xi_{0})\in
T^{*}{\mathbb{R}^{n}}\backslash 0.\end{aligned}\right.$ (9.8)

We call $\boxed{\gamma_{x_{0},\xi_{0}}(t)}$ a _bicharacteristic_ of $p$.
Furthermore, if $p(x_{0},\xi_{0})=0$, then we shall have $p(x(s),\xi(s))=0$
for $\forall s$ and call $\gamma_{x_{0},\xi_{0}}(t):=(x(t),\xi(t))$ a _null
bicharacteristic_ of $p$.

Note that in Definition 9.9, the function $p$ is assumed to be real-valued.
Without this assumption, we cannot guarantee $(x(s),\xi(s))$ are coordinates.

######  Lemma 9.10.

For small enough $\epsilon>0$, there exists a unique solution
$\gamma\colon(-\epsilon,\epsilon)\mapsto(x(s),\xi(s))\in
T^{*}{\mathbb{R}^{n}}\backslash 0$ for the Hamiltonian equation (9.8).
Moreover, assume either

  1. (1)

$\nabla_{(x,\xi)}p$ is uniformly bounded in
$\\{(x(s),\xi(s)/|\xi(s)|)\,;\,s\in(-\epsilon,\epsilon)\\}$, $p$ is
homogeneous of order $1$;

  2. (2)

or $\nabla_{x}p$ is uniformly bounded in
$\\{(x(s),\xi(s))\,;\,s\in(-\epsilon,\epsilon)\\}$, $p$ is homogeneous of
order $\mu\geq 1$, and $\nabla_{\xi}p$ is bounded in uniformly bounded in
$\\{(x(s),\xi(s)/|\xi(s)|)\,;\,s\in(-\epsilon,\epsilon)\\}$;

then the domain of definition of $\gamma$ can be extended from
$(-\epsilon,\epsilon)$ to $\mathbb{R}$.

###### Proof.

Part 1: local solution. We use the Banach fixed-point theorem to show the
existence of local solution. For simplicity denote $\eta_{0}:=(x_{0},\xi_{0})$
and $\eta(s):=(x(s),\xi(s))$ and
$F(\eta(s)):=(\nabla_{\xi}p(\eta(s)),-\nabla_{x}p(\eta(s)))$, and we define a
mapping $\mathscr{F}$:

$\mathscr{F}\colon\eta\in C(I,T^{*}{\mathbb{R}^{n}}\backslash 0)\ \mapsto\
\eta_{0}+\int_{0}^{s}F(\eta(\tau))\,\mathrm{d}{\tau}\in
C(I,T^{*}{\mathbb{R}^{n}}\backslash 0).$

Fix $\epsilon\leq(2\|{\nabla F}\|)^{-1}$, and let $I=(-\epsilon,\epsilon)$.
Then for any $\eta_{1}$, $\eta_{2}\in C(I,T^{*}{\mathbb{R}^{n}}\backslash 0)$,
we have

$\displaystyle\|{\mathscr{F}\eta_{1}-\mathscr{F}\eta_{2}}\|_{C(I)}$
$\displaystyle=\|{\int_{0}^{s}[F(\eta_{1}(\tau))-F(\eta_{2}(\tau))]\,\mathrm{d}{\tau}}\|_{C(I)}$
$\displaystyle\leq\epsilon\|{F(\eta_{1})-F(\eta_{2})}\|_{C(I)}\leq\epsilon\|{\nabla
F}\|\|{\eta_{1}-\eta_{2}}\|_{C(I)}$
$\displaystyle\leq\frac{1}{2}\|{\eta_{1}-\eta_{2}}\|_{C(I)}.$

The Banach fixed-point theorem can be applied, and we can find a fixed point
$\eta$ of $\mathscr{F}$ such that

$\eta(s)=\eta_{0}+\int_{0}^{s}F(\eta(\tau))\,\mathrm{d}{\tau},\ \forall s\in
I\quad\Rightarrow\quad\eta\text{~{}satisfies~{}}\eqref{eq:Has-PM2021}.$

We proved the existence.

For the uniqueness, assume $\eta_{1}$, $\eta_{2}$ solve (9.8). Because
$\eta_{1}(0)=\eta_{2}(0)$, if there are not equal, their derivatives must be
differ at a point, but this violates the first two equations in (9.8). The
first part of the claim is proven.

Part 2: global solution (cf [salo2006sta, §2]). To obtain the global solution,
we can extend the local solution from $(-\epsilon,\epsilon)$ to
$[-\epsilon,\epsilon]$, and then just paste local solutions on
$[-\epsilon,\epsilon]$,
$[\epsilon-\epsilon^{\prime},\epsilon+\epsilon^{\prime}]$,
$[\epsilon+\epsilon^{\prime}-\epsilon^{\prime\prime},\epsilon+\epsilon^{\prime}+\epsilon^{\prime\prime}]$,
etc. Now we show the endpoints extensions can be done. Assume $\gamma$ is a
local solution on $I=(-\epsilon,\epsilon)$ as given in Part 1.

Assume $\nabla_{(x,\xi)}p$ is uniformly bounded in
$\\{(x(s),\xi(s)/|\xi(s)|)\,;\,s\in(-\epsilon,\epsilon)\\}$, and $p$ is
homogeneous of order $1$. From (9.8) we can have

$\frac{\mathrm{d}}{\mathrm{d}s}\big{(}|\xi(s)|^{2}\big{)}=2\xi(s)\cdot\dot{\xi}(s)=-2\xi(s)\cdot\nabla_{x}p(x(s),\xi(s))=m(s)|\xi(s)|^{2},$
(9.9)

where $m(s):=-2\hat{\xi}(s)\cdot\nabla_{x}p(x(s),\hat{\xi}(s))$ with
$\hat{\xi}(s):=\xi(s)/|\xi(s)|$. Here because $\xi(0)=\xi_{0}\neq 0$, and
$\xi(s)$ is continuous on $s$, so we can choose the interval
$I=(-\epsilon,\epsilon)$ to be small enough such that $\xi(s)\neq 0$ for
$\forall s\in I$, and this can make $\hat{\xi}(s)$ always well-defined. Solve
(9.9) we obtain

$|\xi(s)|=e^{-\int_{0}^{s}\hat{\xi}(\tau)\cdot\nabla_{x}p(x(\tau),\hat{\xi}(\tau))\,\mathrm{d}{\tau}}|\xi(0)|,$

so

$e^{-\epsilon M_{1}}|\xi(0)|\leq|\xi(s)|\leq e^{\epsilon M_{1}}|\xi(s)|$

where
$M_{1}=\sup_{\\{(x(s),\xi(s)/|\xi(s)|)\,;\,s\in(-\epsilon,\epsilon)\\}}|\nabla_{x}p|$,
so $\\{\xi(s)\,;\,s\in(-\epsilon,\epsilon)\\}$ is contained in a bounded
domain. Similarly, for $x(s)$ we have

$|\dot{x}(s)|=|\nabla_{\xi}p(x(s),\xi(s))|=|\nabla_{\xi}p(x(s),\hat{\xi}(s))|\leq\sup_{\mathbb{R}_{x}^{n}\times\mathbb{S}^{n-1}}|\nabla_{\xi}p|$

where
$M_{2}=\sup_{\\{(x(s),\xi(s)/|\xi(s)|)\,;\,s\in(-\epsilon,\epsilon)\\}}|\nabla_{\xi}p|$.
Note that we used the homogeneity of $p$ again. This gives

$|x(s)-x(0)|\leq sn^{1/2}M_{2}.$

Or, if $\nabla_{x}p$ is uniformly bounded in
$\\{(x(s),\xi(s))\,;\,s\in(-\epsilon,\epsilon)\\}$, $p$ is homogeneous of
order $\mu\geq 1$, and $\nabla_{\xi}p$ is uniformly bounded in
$\\{(x(s),\xi(s)/|\xi(s)|)\,;\,s\in(-\epsilon,\epsilon)\\}$, by
$|\dot{\xi}(s)|=|\nabla_{x}p(x(s),\xi(s))|$ we can have

$|\xi(s)|=|\xi(0)+\int_{0}^{s}\dot{\xi}(\tau)\,\mathrm{d}{\tau}|\leq|\xi(0)|+\int_{0}^{s}|\nabla_{x}p|\,\mathrm{d}{\tau}\leq|\xi(0)|+sM\leq|\xi(0)|+\epsilon
M_{1}^{\prime}.$

where
$M_{1}^{\prime}=\sup_{\\{(x(s),\xi(s))\,;\,s\in(-\epsilon,\epsilon)\\}}|\nabla_{x}p|$.
And similarly, for $x(s)$ we have

$|\dot{x}(s)|=|\nabla_{\xi}p(x(s),\xi(s))|=|\xi(s)|^{\mu-1}|\nabla_{\xi}p(x(s),\hat{\xi}(s))|\leq(|\xi(0)|+\epsilon
M_{1}^{\prime})M_{2}^{\prime},$

where
$M_{2}^{\prime}=\sup_{\\{(x(s),\xi(s)/|\xi(s)|)\,;\,s\in(-\epsilon,\epsilon)\\}}|\nabla_{\xi}p|$.

Therefore, in both two cases the $(x(s),\xi(s))$ lives in a bounded domain
when $s\in(-\epsilon,\epsilon)$, thus due to the continuity of $x(s)$ and
$\xi(s)$ we can extend the domain of definition of $\gamma$ from
$(-\epsilon,\epsilon)$ to $[-\epsilon,\epsilon]$.

After extension, we set new initial value $(x_{0},\xi_{0})$ to be
$(x(\epsilon),\xi(\epsilon))$ and by Part 1 we can get a local solution on
$(\epsilon-\epsilon^{\prime},\epsilon+\epsilon^{\prime})$ for some small
enough $\epsilon^{\prime}$. By doing this repeatedly, we can obtain a solution
defined in $\mathbb{R}$. The proof is complete. ∎

######  Lemma 9.11.

Let a symbol $p$ be homogeneous, i.e. $p(x,\lambda\xi)=\lambda p(x,\xi)$ for
$\lambda\in\mathbb{R}_{+}$. Then we have
$\boxed{\gamma_{x_{0},\lambda\xi_{0}}(t)=(x(t),\lambda\xi(t))}$ for
$\forall\lambda\in\mathbb{R}_{+}$.

###### Proof.

Because $p$ is homogeneous, from (9.8) we have

$\left\\{\begin{aligned}
&\dot{x}(s)=\nabla_{\xi}p(x(s),\xi(s))=\nabla_{\xi}p(x(s),\lambda\xi(s)),\\\
&\lambda\dot{\xi}(s)=-\lambda\nabla_{x}p(x(s),\xi(s))=-\nabla_{x}p(x(s),\lambda\xi(s)),\\\
&p(x_{0},\lambda\xi_{0})=\lambda p(x_{0},\xi_{0}),\end{aligned}\right.$

so $(x(t),\lambda\xi(t))$ is also a solution of (9.8), with
$(x(0),\lambda\xi(0))=(x_{0},\lambda\xi_{0})$. The proof is done. ∎

######  Lemma 9.12.

Let $T>0$. Assume a real-valued symbol $p\in S^{1}$ is homogeneous of order 1,
and $F\in C^{\infty}([0,T]\times(T^{*}{\mathbb{R}^{n}}\backslash 0))$ and
$\phi\in C^{\infty}(T^{*}{\mathbb{R}^{n}}\backslash 0)$. Then there exists a
unique solution $q\in
C^{\infty}(\mathbb{R}\times(T^{*}{\mathbb{R}^{n}}\backslash 0))$ satisfying

$\left\\{\begin{aligned} (\partial_{t}+H_{p})q(t,x,\xi)&=F(t,x,\xi),\\\
q(0,x,\xi)&=\phi(x,\xi),\end{aligned}\right.$

where $H_{p}$ is the Hamiltonian of $p$. The solution is given by

$\forall(x_{0},\xi_{0})\in T^{*}{\mathbb{R}^{n}},\quad
q(t,\gamma_{x_{0},\xi_{0}}(t))=\phi(x_{0},\xi_{0})+\int_{0}^{t}F(\tau,\gamma_{x_{0},\xi_{0}}(\tau))\,\mathrm{d}{\tau}.$

More, when $F$ and $\phi$ are homogeneous (with $\xi$) of order
$m\in\mathbb{R}$, then $q$ is also homogeneous (with $\xi$) of order
$m\in\mathbb{R}$.

###### Proof.

Let $\gamma_{x_{0},\xi_{0}}(t)=(x(t),\xi(t))$ be the bicharacteristic of $p$
starting from $(x_{0},\xi_{0})$. The existence of $\gamma_{x_{0},\xi_{0}}$ is
guaranteed by Lemma 9.10. Then we have

$\displaystyle(\partial_{t}+H_{p})q(t,x(t),\xi(t))$
$\displaystyle=(\partial_{t}+\nabla_{\xi}p\cdot\nabla_{x}-\nabla_{x}p\cdot\nabla_{\xi})q(t,x(t),\xi(t))$
$\displaystyle=(\partial_{t}+\dot{x}(t)\cdot\nabla_{x}+\dot{\xi}(t)\cdot\nabla_{\xi})q(t,x(t),\xi(t))$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\big{(}q(t,x(t),\xi(t))\big{)},$

so
$\frac{\mathrm{d}}{\mathrm{d}t}\big{(}q(t,x(t),\xi(t))\big{)}=F(t,x(t),\xi(t)),$
which gives

$\displaystyle q(t,x(t),\xi(t))$
$\displaystyle=q(0,x(0),\xi(0))+\int_{0}^{t}F(\tau,x(\tau),\xi(\tau))\,\mathrm{d}{\tau}$
$\displaystyle=\phi(x_{0},\xi_{0})+\int_{0}^{t}F(\tau,x(\tau),\xi(\tau))\,\mathrm{d}{\tau}.$

For the homogeneity, fix $(x,\xi)\in T^{*}{\mathbb{R}^{n}}\backslash 0$, we
solve the Hamiltonian equation with initial point $(x,\xi)$ and we can obtain
a bicharacteristic $\gamma_{x,\xi}$. Fix $t\in\mathbb{R}$, we set
$(x_{0},\xi_{0}):=\gamma_{x,\xi}(-t)$, so reversely we represent $(x,\xi)$ as
$\gamma_{x_{0},\xi_{0}}(t)=(x(t),\xi(t))$. Because $p$ is homogeneous of order
1, by Lemma 9.11 we have
$(x(t),\lambda\xi(t))=\gamma_{x_{0},\lambda\xi_{0}}(t)$, so

$\displaystyle q(t,x,\lambda\xi)$
$\displaystyle=q(t,x(t),\lambda\xi(t))=q(t,\gamma_{x_{0},\lambda\xi_{0}}(t))=\phi(x_{0},\lambda\xi_{0})+\int_{0}^{t}F(\tau,\gamma_{x_{0},\lambda\xi_{0}}(\tau))\,\mathrm{d}{\tau}$
$\displaystyle=\phi(x_{0},\lambda\xi_{0})+\int_{0}^{t}F(\tau,(x(t),\lambda\xi(\tau)))\,\mathrm{d}{\tau}$
$\displaystyle=\lambda^{m}[\phi(x_{0},\xi_{0})+\int_{0}^{t}F(\tau,x(\tau),\xi(\tau))\,\mathrm{d}{\tau}]=\lambda^{m}q(t,x(t),\xi(t)).$

The proof is done. ∎

###  9.3. Propagation of singularities

For other literature on the topic, [jos99int, §10] is a good reference for
this section. See [shu2001pse, A.1.3], [grigis94mic, §8] for different proofs.
Now we are ready for the main result.

######  Theorem 9.13.

Assume $m\in\mathbb{R}$ and $P\in\Psi^{m}$ is classical $\Psi$DO of real
principal type, and denote its principal symbol as $p_{m}(x,\xi)$. We assume
either

  * •

$u\in\mathscr{S}^{\prime}({\mathbb{R}^{n}})$, or,

  * •

$P$ is properly supported and $u\in\mathscr{D}^{\prime}({\mathbb{R}^{n}})$.

Let $Pu\in C^{\infty}$ and $p_{m}(x_{0},\xi_{0})=0$. If
$(x_{0},\xi_{0})\notin\operatorname{WF}(u)$, then
$\gamma_{x_{0},\xi_{0}}\cap\operatorname{WF}(u)=\emptyset$ where the
$\gamma_{x_{0},\xi_{0}}$ is a null bicharacteristic of $p_{m}$ defined in
Definition 9.9. In other words, for a null bicharacteristic $\gamma$, it holds
either $\gamma\subset\operatorname{WF}(u)$ or
$\gamma\cap\operatorname{WF}(u)=\emptyset$.

###### Proof.

Step 1: change to $\Psi^{1}$. Choose an elliptic $T_{a}\in\Psi^{1-m}$ with
$a(x,\xi)>0$ and $a(x,\xi)$ be real-valued, then

$\operatorname{WF}(T_{a}Pu)=\operatorname{WF}(Pu)\quad\Rightarrow\quad
T_{a}Pu\in C^{\infty}\ \text{if and only if}\ Pu\in C^{\infty},$

namely, $T_{a}$ doesn’t change the wavefront set.

Also, we can show $T_{a}$ doesn’t change null bicharacteristics of the
principal symbols as follows. Assume $(x(s),\xi(s))$ solves (9.8) with
$p_{m}(x(0),\xi(0))=0$. Let’s assume we can find a function $f(s)$ such that

$f(s):=\int_{0}^{s}a(x(f(r)),\xi(f(r)))\,\mathrm{d}{r}.$

This is possible because it amounts to find a fix point of the transform
$\mathcal{I}\circ a\circ\gamma\colon C^{\infty}(\mathbb{R};\mathbb{R})\to
C^{\infty}(\mathbb{R};\mathbb{R})$ where
$\mathcal{I}g(s):=\int_{0}^{s}g(r)\,\mathrm{d}{r}$ and
$\gamma(s):=(x(s),\xi(s))$.

After obtained such an $f$, we can see $f$ is a bijection because
$f^{\prime}=a>0$. Denote

$\tilde{x}(s):=x(f(s)),\quad\tilde{\xi}(s):=\xi(f(s)).$

If $(x(s),\xi(s))$ is defined on a interval $I$, then we say
$(\tilde{x}(s),\tilde{\xi}(s))$ is defined on a interval
$I^{\prime}:=f^{-1}(I)$, so $p_{m}(\tilde{x}(s),\tilde{\xi}(s))=0$ for $s\in
I^{\prime}$, and we can have

$\displaystyle\dot{\tilde{x}}(s)$
$\displaystyle=f^{\prime}(s)\dot{x}(f(s))=a(x(f(s)),\xi(f(s)))\nabla_{\xi}p_{m}(x(f(s)),\xi(f(s)))$
$\displaystyle=a(\tilde{x}(s),\tilde{\xi}(s))\nabla_{\xi}p_{m}(\tilde{x}(s),\tilde{\xi}(s))$
$\displaystyle=a(\tilde{x}(s),\tilde{\xi}(s))\nabla_{\xi}p_{m}(\tilde{x}(s),\tilde{\xi}(s))+p_{m}(\tilde{x}(s),\tilde{\xi}(s))\nabla_{\xi}a(\tilde{x}(s),\tilde{\xi}(s))$
$\displaystyle=\nabla_{\xi}(ap_{m})(\tilde{x}(s),\tilde{\xi}(s)).$

Similarly, we have

$\dot{\tilde{\xi}}(s)=-\nabla_{x}(ap_{m})(\tilde{x}(s),\tilde{\xi}(s)).$

These mean the null bicharacteristic $(x(s),\xi(s))$ of $p_{m}$, after a
reparametrization, is also a null bicharacteristic of $ap_{m}$. Note that
$ap_{m}$ is the principal symbol of $T_{a}P$. Hence, to prove the claim for
$P\in\Psi^{m}$ is equivalent to prove the claim for $P\in\Psi^{1}$, so, in the
rest of the proof we assume $P\in\Psi^{1}$ of real principal type.

Step 2: find a $t$-dependent $Q=Q(t,x,D)$ such that

$Qu|_{t=0}\in C^{\infty}.$ (9.10)

Our plan is to construct a sequence of $t$-dependent $\Psi$DOs
$Q_{j}=Q_{j}(t,x,D)\in\Psi^{-j}$ ($j\geq 0$) having classical symbol $q_{j}$,
and set $Q\sim\sum_{j}Q_{j}$. Here $Q_{j}(t,x,D)\in\Psi^{-j}$ means its symbol
$q_{j}(t,x,\xi)$ is in
$S^{-j}([0,T]\times\mathbb{R}_{x}^{n}\times\mathbb{R}_{\xi}^{n})$, i.e.,

$|\partial_{t}^{\alpha^{\prime}}\partial_{x}^{\alpha^{\prime\prime}}\partial_{\xi}^{\beta}q_{j}(t,x,\xi)|\lesssim\langle{\xi}\rangle^{-j-|\beta|},$

see Definition 2.4.

Because $(x_{0},\xi_{0})\notin\operatorname{WF}(u)$, we have
$(x_{0},t\xi_{0})\notin\operatorname{WF}(u)$ for $\forall t>0$, and we can
find a open conic neighborhood $\omega$ of $(x_{0},\xi_{0})$ such that
$\omega\cap\operatorname{WF}(u)=\emptyset$. Choose a function $\chi(x,\xi)\in
C^{\infty}$ satisfying

$\left\\{\begin{aligned} &\chi(x,\xi)\in C^{\infty},\ \mathop{\rm
supp}\chi\subset\omega,\
\chi(x,\lambda\xi)=\chi(x,\xi)~{}(\forall\lambda>0),\\\ &\chi\equiv
1\text{~{}in a sufficiently small open conic
neighborhood~{}}\tilde{\omega}\text{~{}of~{}}(x_{0},\xi_{0}).\end{aligned}\right.$
(9.11)

Set $q_{0}(0,x,\xi):=\chi(x,\xi)$, then
$\omega^{c}\subset\operatorname{Smo}(Q_{0}|_{t=0})$ where $\omega^{c}$
signifies the complement of the set $\omega$ in $T^{*}{\mathbb{R}^{n}}$, so by
Theorem 8.17,

$\displaystyle\operatorname{WF}(Q_{0}u|_{t=0})$
$\displaystyle=\operatorname{WF}((Q_{0}|_{t=0})u)\subset\operatorname{WF}(u)\backslash\operatorname{Smo}(Q_{0}|_{t=0})$
$\displaystyle=\operatorname{WF}(u)\cap\big{(}\operatorname{Smo}(Q_{0}|_{t=0})\big{)}^{c}$
$\displaystyle\subset\operatorname{WF}(u)\cap\omega=\emptyset,$

so $Q_{0}u|_{t=0}\in C^{\infty}$. For $Q_{j}~{}(j\geq 1)$, we set their symbol
at $t=0$ as zero, i.e.,

$q_{0}(0,x,\xi):=\chi(x,\xi),\quad q_{j}(0,\cdot,\cdot):=0~{}(j\geq 1),$
(9.12)

then $Q_{j}|_{t=0}\in\Psi^{-\infty}$, so $Q_{j}u|_{t=0}\in C^{\infty}$ for
$j\geq 1$. By Theorem 2.6 we can find a $Q$ satisfying $Q\sim\sum_{j}Q_{j}$
(thus $Q$ is also $t$-dependent)111Note that such $Q$ is not unique.. $Q$ is
of order $0$. We can conclude (9.10).

Step 3: to make $Q$ satisfy

$(D_{t}+P)(Qu)\in C^{\infty}.$ (9.13)

To achieve (9.13) is equivalent to achieve

$[D_{t}+P,Q]\in\Psi^{-\infty}$ (9.14)

because

$\displaystyle(D_{t}+P)(Qu)$ $\displaystyle=[D_{t}+P,Q]u+Q(D_{t}+P)u$
$\displaystyle=[D_{t}+P,Q]u+QPu=[P,Q]u+C^{\infty}.$

Here we used $\operatorname{WF}(QPu)\subset\operatorname{WF}(Pu)=\emptyset$ so
$QPu\in C^{\infty}$ The fact $QD_{t}u=0$ is because $u$ is independent of $t$.
Readers may note that in Step 2 we only determined $q_{j}$ on $\\{t=0\\}$,
while $q_{j}$ on $\\{t>0\\}$ hasn’t been fixed yet. Here we design
$q_{j}|_{t>0}$ to achieve (9.14).

We use the notation $\sigma(A)$ to signify the symbol of $A$. Because $P$ is
classical, we can expand $\sigma(P)$ as $\sum_{k}p_{k}$ for some homogeneous
symbols $p_{k}\in S^{1-k}$. Recall Step 1, we see the integral curve of
$H_{p_{m}}$ is the same as $H_{p_{1}}$. Then by Theorem 5.3 and Remark 5.5, we
have

$\displaystyle\sigma([D_{t},Q])$
$\displaystyle\sim\sum_{\alpha}\frac{(-i)^{|\alpha|}}{\alpha!}(\partial_{\tau}^{\alpha}\tau)\partial_{t}^{\alpha}(\sum_{j\geq
0}q_{j})-\sum_{\alpha}\frac{(-i)^{|\alpha|}}{\alpha!}(\partial_{t}^{\alpha}\tau)\partial_{\tau}^{\alpha}(\sum_{j\geq
0}q_{j})$ $\displaystyle=\tau\sum_{j\geq 0}q_{j}+(-i)\partial_{t}\sum_{j\geq
0}q_{j}-\tau\sum_{j\geq 0}q_{j}=\frac{1}{i}\sum_{j\geq 0}\partial_{t}q_{j},$
(9.15)

and

$\displaystyle\sigma([P,Q])$
$\displaystyle\sim\sum_{\alpha}\frac{(-i)^{|\alpha|}}{\alpha!}\partial_{\xi}^{\alpha}(\sum_{k\geq
0}p_{k})\partial_{x}^{\alpha}(\sum_{j\geq
0}q_{j})-\sum_{\alpha}\frac{(-i)^{|\alpha|}}{\alpha!}\partial_{x}^{\alpha}(\sum_{k\geq
0}p_{k})\partial_{\xi}^{\alpha}(\sum_{j\geq 0}q_{j})$
$\displaystyle=\sum_{k\geq 0}\sum_{j\geq
0}\sum_{\alpha}\frac{(-i)^{|\alpha|}}{\alpha!}\big{(}(\partial_{\xi}^{\alpha}p_{k})\partial_{x}^{\alpha}-(\partial_{x}^{\alpha}p_{k})\partial_{\xi}^{\alpha}\big{)}q_{j}$
$\displaystyle=\sum_{\ell\geq
0}\sum_{j+k+|\alpha|=\ell}\frac{(-i)^{|\alpha|}}{\alpha!}\big{(}(\partial_{\xi}^{\alpha}p_{k})\partial_{x}^{\alpha}-(\partial_{x}^{\alpha}p_{k})\partial_{\xi}^{\alpha}\big{)}q_{j}$
$\displaystyle=\sum_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ell\geq
1}\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}|\alpha|\geq
1}\end{subarray}}L_{j,k,\alpha}q_{j},\quad(\text{it can be checked
that~{}}L_{j,k,\alpha}q_{j}\in S^{1-\ell})$

where the linear differential operator
$L_{j,k,\alpha}:=\frac{(-i)^{|\alpha|}}{\alpha!}(\partial_{\xi}^{\alpha}p_{k})\partial_{x}^{\alpha}-(\partial_{x}^{\alpha}p_{k})\partial_{\xi}^{\alpha}$.
Note that $p_{0}$ is the principal symbol of $P$ so $p_{0}$ is real-valued.
Also note that the restriction $\ell\geq 1$ and $|\alpha|\geq 1$ come from the
fact that when $|\alpha|=0$, $L_{j,k,\alpha}=p_{k}-p_{k}=0$. It can be checked

$|\alpha|=1\ \Rightarrow\
\sum_{|\alpha|=1}L_{j,k,\alpha}q_{j}=\frac{1}{i}H_{p_{k}}q_{j}.$ (9.16)

We can further compute $\sigma([P,Q])$ as

$\displaystyle\sigma([P,Q])$
$\displaystyle\sim\frac{1}{i}H_{p_{0}}q_{0}+\sum_{\ell\geq
2}\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell\\\ |\alpha|\geq
1\end{subarray}}L_{j,k,\alpha}q_{j}$
$\displaystyle=\frac{1}{i}H_{p_{0}}q_{0}+\sum_{\ell\geq
2}\big{(}\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell\\\ |\alpha|\geq
1,\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}j=\ell-1}\end{subarray}}L_{j,k,\alpha}q_{j}+\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell\\\
|\alpha|\geq
1,\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}j<\ell-1}\end{subarray}}L_{j,k,\alpha}q_{j}\big{)}$
$\displaystyle=\frac{1}{i}H_{p_{0}}q_{0}+\sum_{\ell\geq
2}\big{(}\sum_{|\alpha|=1}L_{j=\ell-1,k=0,\alpha}q_{j}+\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell\\\
|\alpha|\geq 1,\,j<\ell-1\end{subarray}}L_{j,k,\alpha}q_{j}\big{)}$
$\displaystyle=\frac{1}{i}H_{p_{0}}q_{0}+\sum_{\ell\geq
2}\big{(}\frac{1}{i}H_{p_{0}}q_{\ell-1}+\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell\\\
|\alpha|\geq
1,\,j<\ell-1\end{subarray}}L_{j,k,\alpha}q_{j}\big{)}\qquad(\text{by~{}}\eqref{eq:aLH-
PM2021})$ $\displaystyle=\frac{1}{i}H_{p_{0}}q_{0}+\sum_{\ell\geq
1}\big{(}\frac{1}{i}H_{p_{0}}q_{\ell}+\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell+1\\\
|\alpha|\geq
1,\,j<\ell\end{subarray}}L_{j,k,\alpha}q_{j}\big{)}\qquad(\ell\to\ell-1).$
(9.17)

Combining (9.15) with (9.17), we obtain

$\displaystyle\sigma([D_{t}+P,Q])$
$\displaystyle\sim\frac{1}{i}(\partial_{t}+H_{p_{0}})q_{0}+\sum_{\ell\geq
1}\big{(}\frac{1}{i}(\partial_{t}+H_{p_{0}})q_{\ell}+\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell+1\\\
|\alpha|\geq 1,\,j<\ell\end{subarray}}L_{j,k,\alpha}q_{j}\big{)}.$ (9.18)

The requirement (9.14) thus amounts to require $\sigma([D_{t}+P,Q])\in
S^{-\infty}$, namely,

$\left\\{\begin{aligned} (\partial_{t}+H_{p_{0}})q_{0}&=0,\\\
\frac{1}{i}(\partial_{t}+H_{p_{0}})q_{\ell}&=-\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell+1\\\
|\alpha|\geq 1,\,j<\ell\end{subarray}}L_{j,k,\alpha}q_{j},\quad\ell\geq
1.\end{aligned}\right.$ (9.19)

Combining (9.19) with initial condition (9.12), these $q_{j}~{}(j\geq 0)$ can
be solved iteratively in $[0,T]\times T^{*}{\mathbb{R}^{n}}$ by using see
Lemma 9.12 (recall that $p_{0}$ is real-valued), and gives, $\forall(x,\xi)\in
T^{*}{\mathbb{R}^{n}}\backslash 0$,

$\left\\{\begin{aligned} q_{0}(t,\gamma_{x,\xi}(t))&=\chi(x,\xi),\\\
q_{\ell}(t,\gamma_{x,\xi}(t))&=-i\int_{0}^{t}\sum_{\begin{subarray}{c}j+k+|\alpha|=\ell+1\\\
|\alpha|\geq
1,\,j<\ell\end{subarray}}L_{j,k,\alpha}q_{j}(\tau,\gamma_{x,\xi}(\tau))\,\mathrm{d}{\tau},\quad\ell\geq
1.\end{aligned}\right.$ (9.20)

And they guarantee $\sigma([D_{t}+P,Q])\in S^{-\infty}$, so (9.14) is
achieved, thus (9.13) is satisfied.

By iteration we can show the RHS of (9.20) is of order $-\ell$, so the second
conclusion in Lemma 9.12 implies $q_{\ell}$ is homogeneous of order $-\ell$,
so they are all classical symbols.

Step 4: apply a hyperbolic PDE result. Combining (9.10) and (9.13), we can
conclude

$\left\\{\begin{aligned} (D_{t}+P)(Qu)&=F\ \text{in}\
\mathbb{R}_{+}\times{\mathbb{R}^{n}},\\\ Qu|_{t=0}&=\varphi\ \text{on}\
{\mathbb{R}^{n}},\end{aligned}\right.$ (9.21)

for some $F\in C^{\infty}(\mathbb{R}_{+}\times{\mathbb{R}^{n}})$ and
$\varphi\in C^{\infty}({\mathbb{R}^{n}})$. Now we use Lemma 9.17 in advance to
conclude $Qu\in C([0,T],C^{\infty}({\mathbb{R}^{n}}))$.

Step 5: conclusion. From $Qu\in C([0,T],C^{\infty}({\mathbb{R}^{n}}))$ we see
$\underline{Q|_{t}u\in C^{\infty}}$ for each $t\in[0,T]$, where $Q|_{t}$ is an
abbreviation of $Q(t,x,D)$. From (9.20) we see
$q_{0}(t,\gamma_{x_{0},\xi_{0}}(t))=\chi(x_{0},\xi_{0}).$ By Lemma 9.11, the
homogeneity of $p_{0}$ gives
$(x(t),\lambda\xi(t))=\gamma_{x_{0},\lambda\xi_{0}}(t)$, so

$\displaystyle\forall\lambda>0,\ q_{0}(t,x(t),\lambda\xi(t))$
$\displaystyle=q_{0}(t,\gamma_{x_{0},\lambda\xi_{0}}(t))\qquad\text{(by Lemma
\ref{lem:pHom-PM2021})}$ (9.22)
$\displaystyle=\chi(x_{0},\lambda\xi_{0})=\chi(x_{0},\xi_{0})\qquad\text{(by
\eqref{eq:HQjs-PM2021}, \eqref{eq:chho-PM2021})}$ $\displaystyle\neq
0.\qquad\text{(by \eqref{eq:chho-PM2021})}$

This means $Q$ is elliptic at $(x(t),\xi(t))=\gamma_{x_{0},\xi_{0}}(t)$, i.e.
$\gamma_{x_{0},\xi_{0}}(t)\notin\operatorname{Char}(Q|_{t}).$ Therefore, by
Corollary 9.6,

$\displaystyle\operatorname{WF}(u)$
$\displaystyle\subset\operatorname{WF}(Q|_{t}u)\cup\operatorname{Char}(Q|_{t})=\operatorname{Char}(Q|_{t}),$

so

$\gamma_{x_{0},\xi_{0}}(t)\notin\operatorname{WF}(u),\text{~{}for
any~{}}t\in[0,T].$

which means $\gamma_{x_{0},\xi_{0}}\cap\operatorname{WF}(u)=\emptyset$. The
proof is complete. ∎

######  Remark 9.14.

The condition that $P$ is of real principal type is used in the following
ways:

  * •

real-valued: in Step 3, in order to use Lemma 9.12, $p_{m}$ has to be real-
valued; Also, when $p_{m}$ is real-valued, then $R:=iP+(iP)^{*}$ is of order
$0$. This is used in Step 4 which calls for Lemma 9.17;

  * •

$|\nabla_{\xi}p_{m}(x,\xi)|\neq 0$ when $p(x,\xi)=0$: related to the
solvability of (9.20)? Every $(x,\xi)\in T^{*}{\mathbb{R}^{n}}\backslash 0$
shall be reachable;

  * •

homogeneity: the condition “$p_{m}(x,\lambda\xi)=\lambda p_{m}(x,\xi)$” is
used at (9.22) to guarantee
$(x(t),\lambda\xi(t))=\gamma_{x_{0},\lambda\xi_{0}}(t)$.

Theorem 9.13 can be interpreted by the following claim.

######  Corollary 9.15.

Assume $m\in\mathbb{R}$ and $P\in\Psi^{m}$ is classical $\Psi$DO of real
principal type, and denote its symbol as $p(x,\xi)$. Assume $Pu$ is well-
defined and $Pu\in C^{\infty}$. Then $\operatorname{WF}(u)$ is made of null
bicharacteristic curves $\gamma_{x,\xi}$ for some $(x,\xi)\in p_{m}^{-1}(0)$.

###### Proof.

We see that $P$ is a $\Psi$DO with principal symbol $p_{m}(x,\xi)$ homogeneous
in $\xi$, so we can apply Lemma 9.4 to conclude
$\operatorname{Char}P=p_{m}^{-1}(0)$. Also, when $Pu\in C^{\infty}$, by
Corollary 9.6 we have $\operatorname{WF}(u)\subset\operatorname{Char}P$, so

$\operatorname{WF}(u)\subset p_{m}^{-1}(0).$ (9.23)

For any $(x,\xi)\in\operatorname{WF}(u)$, by (9.23) we know $(x,\xi)\in
p_{m}^{-1}(0)$. Denote as $\gamma_{x,\xi}$ the null bicharacteristic of
$p_{m}$ passing through $(x,\xi)$, then $\gamma_{x,\xi}\subset p_{m}^{-1}(0)$
because the value of $p_{m}$ is constant in bicharacteristics. According to
Theorem 9.13, we can conclude $\gamma_{x,\xi}\subset\operatorname{WF}(u)$. In
summary, for every $(x,\xi)\in\operatorname{WF}(u)$ we have
$\gamma_{x,\xi}\subset\operatorname{WF}(u)$ and $\gamma_{x,\xi}$ is a null
bicharacteristic, so $\operatorname{WF}(u)$ is made of null bicharacteristic
curves. ∎

###  9.4. Cauchy problems of hyperbolic PDEs

######  Lemma 9.16.

Assume $T>0$ and $s\in\mathbb{R}$, $P\in\Psi^{1}$ has a real-valued principal
symbol. Denote $L=D_{t}+P$. There exists a constant $\lambda_{0}>0$ such that
for any

$u\in C^{1}([0,T],H^{s}({\mathbb{R}^{n}}))\cap
C([0,T],H^{s+1}({\mathbb{R}^{n}})),$

we have

$\sup_{t\in[0,T]}e^{-\lambda
t}\|{u(t,\cdot)}\|_{H^{s}}\leq\|{u(0,\cdot)}\|_{H^{s}}+2\int_{0}^{T}e^{-\lambda
t}\|{Lu(t,\cdot)}\|_{H^{s}}\,\mathrm{d}{t}.$ (9.24)

###### Proof.

Denote $Q=iP$ and $L^{\prime}=iL=\partial_{t}+Q$. Then

$\displaystyle\sigma(Q+Q^{*})$
$\displaystyle=\sigma(iP+(iP)^{*})=i\sigma(P-P^{*})=i[\sigma(P)+S^{0}-\overline{\sigma(P)}-S^{0}]\in
S^{0},$

because the principal symbol of $P$ is real-valued. We denote $R=Q+Q^{*}$,
then $R\in\Psi^{0}$ and thus is bounded in $L^{2}$.

We prove the case $s=0$ first. Denote $f(t):=\|{e^{-\lambda
t}u(t,\cdot)}\|_{L^{2}}^{2}$, then

$\displaystyle f^{\prime}(t)$ $\displaystyle=2e^{-2\lambda
t}\Re(\partial_{t}u,u)-2\lambda f(t)=2e^{-2\lambda
t}\Re((L^{\prime}-Q)u,u)-2\lambda f(t)$ $\displaystyle=2e^{-2\lambda
t}\Re(L^{\prime}u,u)+e^{-2\lambda t}(-Ru,u)-2\lambda f(t)$ $\displaystyle\leq
2e^{-2\lambda t}\|{Lu(t,\cdot)}\|_{L^{2}}\|{u(t,\cdot)}\|_{L^{2}}+e^{-2\lambda
t}\|{Ru(t,\cdot)}\|_{L^{2}}\|{u(t,\cdot)}\|_{L^{2}}-2\lambda f(t)$
$\displaystyle\leq 2e^{-2\lambda
t}\|{Lu(t,\cdot)}\|_{L^{2}}\|{u(t,\cdot)}\|_{L^{2}}-(2\lambda-\|{R}\|)\|{e^{-\lambda
t}u(t,\cdot)}\|_{L^{2}}^{2}$ $\displaystyle\leq 2e^{-2\lambda
t}\|{Lu(t,\cdot)}\|_{L^{2}}\|{u(t,\cdot)}\|_{L^{2}},\qquad(\text{when~{}}\lambda>\|{R}\|/2),$

where $\|{R}\|$ is the $L^{2}$ operator norm. Hence, for any $t\in[0,T]$,

$\displaystyle e^{-2\lambda t}\|{u(t,\cdot)}\|_{L^{2}}^{2}$
$\displaystyle\leq\|{u(0,\cdot)}\|_{L^{2}}^{2}+2\int_{0}^{t}e^{-2\lambda
s}\|{Lu(s,\cdot)}\|_{L^{2}}\|{u(s,\cdot)}\|_{L^{2}}\,\mathrm{d}{s}$
$\displaystyle\leq\|{u(0,\cdot)}\|_{L^{2}}^{2}+2\int_{0}^{T}e^{-2\lambda
t}\|{Lu(t,\cdot)}\|_{L^{2}}\|{u(t,\cdot)}\|_{L^{2}}\,\mathrm{d}{t}.$

By denoting $M:=\sup_{t\in[0,T]}e^{-\lambda t}\|{u(t,\cdot)}\|_{H^{s}}$, we
can continue

$\displaystyle M^{2}$
$\displaystyle\leq\|{u(0,\cdot)}\|_{L^{2}}^{2}+2\int_{0}^{T}e^{-2\lambda
t}\|{Lu(t,\cdot)}\|_{L^{2}}\|{u(t,\cdot)}\|_{L^{2}}\,\mathrm{d}{t}$
$\displaystyle\leq M\|{u(0,\cdot)}\|_{L^{2}}+2\int_{0}^{T}e^{-\lambda
t}\|{Lu(t,\cdot)}\|_{L^{2}}M\,\mathrm{d}{t}$ $\displaystyle\leq
M(\|{u(0,\cdot)}\|_{L^{2}}+2\int_{0}^{T}e^{-\lambda
t}\|{Lu(t,\cdot)}\|_{L^{2}}\,\mathrm{d}{t}).$

We arrive at the conclusion for $s=0$.

For $s\neq 0$, we can do something similar as in Step 2 of the proof of
Theorem 6.18. This completes the proof. ∎

Based on the energy estimate in Lemma 9.16, we can obtain the following
result.

######  Lemma 9.17.

Assume $T>0$ and $s\in\mathbb{R}$, $P\in\Psi^{1}$ has a real-valued principal
symbol. Let $f\in L^{1}((0,T),H^{s}({\mathbb{R}^{n}}))$ and $\phi\in
H^{s}({\mathbb{R}^{n}})$. Then there is a unique solution $u\in
C([0,T],H^{s}({\mathbb{R}^{n}}))$ of the PDE

$\left\\{\begin{aligned} (D_{t}+P)u&=f\ \text{in}\
(0,T)\times{\mathbb{R}^{n}},\\\ u|_{t=0}&=\phi\ \text{on}\
{\mathbb{R}^{n}},\end{aligned}\right.$ (9.25)

###### Proof.

Step 1: variational formulation. Denote

$\left\\{\begin{aligned} &X:=\\{\varphi\in
C^{\infty}([0,T]\times{\mathbb{R}^{n}})\,;\,\varphi(T,\cdot)\equiv 0\\}\\\
&\ell(\varphi):=\int_{0}^{T}(f,\varphi)\,\mathrm{d}{t}+\frac{1}{i}(\phi,\varphi).\end{aligned}\right.$

We say $u\in\mathscr{S}^{\prime}([0,T]\times{\mathbb{R}^{n}})$ is a weak
solution of (9.25) if $u$ satisfies

$\int_{0}^{T}(u,(D_{t}+P^{*})\varphi)\,\mathrm{d}{t}=\ell(\varphi),\quad\forall\varphi\in
X.$ (9.26)

To find a $u\in L^{\infty}((0,T),H^{s})$ satisfying (9.26), we are to show
$|\ell(\varphi)|\leq\|{(D_{t}+P^{*})\varphi}\|_{L^{1}H^{-s}}$, and the call
for the Hahn-Banach theorem. Here $\|{f}\|_{L^{1}H^{-s}}$ is a shorthand for
$\int_{0}^{T}\|{f}\|_{H^{-s}}\,\mathrm{d}{t}$.

Step 2: energy estimate. Because $P\in\Psi^{1}$ has a real-valued principal
symbol, we see $-P^{*}\in\Psi^{1}$ and $-P^{*}$ also has a real-valued
principal symbol. Apply Lemma 9.16 to $D_{t}+(-P^{*})$ and $\varphi(T-t,x)$ we
obtain

$\sup_{t\in[0,T]}e^{-\lambda
t}\|{\varphi(T-t,\cdot)}\|_{H^{-s}}\leq\|{\varphi(T,\cdot)}\|_{H^{-s}}+2\int_{0}^{T}e^{-\lambda
t}\|{(D_{t}+(-P^{*}))(\varphi(T-t,\cdot))}\|_{H^{-s}}\,\mathrm{d}{t},$

which gives

$\sup_{t\in[0,T]}e^{\lambda t}\|{\varphi(t,\cdot)}\|_{H^{-s}}\leq
2\int_{0}^{T}e^{\lambda
t}\|{(D_{t}+P^{*})\varphi(t,\cdot)}\|_{H^{-s}}\,\mathrm{d}{t}.$

so

$\forall
s\in\mathbb{R},\quad\sup_{t\in[0,T]}\|{\varphi(t,\cdot)}\|_{H^{-s}}\leq
2e^{\lambda T}\|{(D_{t}+P^{*})\varphi}\|_{L^{1}H^{-s}}.$ (9.27)

This means the map $\varphi\mapsto(D_{t}+P^{*})\varphi$ is injective. (9.27)
can be understood as a coercive condition.

Step 3: Hahn-Banach theorem. By using (9.27), we can estimate $\ell$ as
follows,

$\displaystyle|\ell(\varphi)|$
$\displaystyle\leq\int_{0}^{T}|(f,\varphi)|\,\mathrm{d}{t}+|(\phi,\varphi)|\leq\int_{0}^{T}\|{f}\|_{H^{s}}\|{\varphi}\|_{H^{-s}}\,\mathrm{d}{t}+\|{\phi}\|_{H^{s}}\|{\varphi}\|_{H^{-s}}$
$\displaystyle\leq\big{(}\int_{0}^{T}\|{f}\|_{H^{s}}\,\mathrm{d}{t}+\|{\phi}\|_{H^{s}}\big{)}\sup_{t\in[0,T]}\|{\varphi(t,\cdot)}\|_{H^{-s}}$
$\displaystyle\leq
C\big{(}\int_{0}^{T}\|{f}\|_{H^{s}}\,\mathrm{d}{t}+\|{\phi}\|_{H^{s}}\big{)}\|{(D_{t}+P^{*})\varphi}\|_{L^{1}H^{-s}}.$

Therefore, the linear functional $\ell(\varphi)$ is also a linear functional
for $(D_{t}+P^{*})\varphi\in X$ under the norm $L^{1}((0,T),H^{-s})$. Because
the dual space of $L^{1}((0,T),H^{-s})$ is $L^{\infty}((0,T),H^{s})$, by the
Hahn-Banach theorem, there exists a $u\in L^{\infty}((0,T),H^{s})$ such that

$\ell(\varphi)=(u,(D_{t}+P^{*})\varphi)_{t,x},\quad\forall\varphi\in X,$

which is (9.26). This $u$ is a weak solution.

Step 4: weak to strong solution. Because $u$ is a distribution, on $(0,T)$ we
have

$D_{t}u+Pu=f.$

Because $u\in L^{\infty}((0,T),H^{s})$, $Pu\in L^{\infty}((0,T),H^{s-1})$.

Let $f$, $\phi$ be Schwartz, then $f\in L^{\infty}([0,T],H^{s})$, so
$D_{t}u=f-Pu\in L^{\infty}((0,T),H^{s-1})$, which implies

$u\in C([0,T],H^{s-1}).$

Again, $f\in C([0,T],H^{s-2})$ and $Pu\in C([0,T],H^{s-2})$, so
$D_{t}u=f-Pu\in C((0,T),H^{s-2})$, which implies

$u\in C^{1}([0,T],H^{s-2})\cap C([0,T],H^{s-1})\ \text{with}\ u(0)=\phi.$

Due to the arbitrary of $s$, we can conclude

$u\in C^{1}([0,T],H^{s})\cap C([0,T],H^{s+1})\ \text{with}\ u(0)=\phi.$ (9.28)

Therefore, $(u,(D_{t}+P^{*})\varphi)_{t,x}$ can be legally write as
$((D_{t}+P)u,\varphi)_{t,x}$, which implies $u$ is a strong solution of
(9.25).

Step 5: density arguments for $f$, $\phi$. (9.28) is true when $f$ and $\phi$
are Schwartz. For general $f\in L^{1}((0,T),H^{s}({\mathbb{R}^{n}}))$ and
$\phi\in H^{s}({\mathbb{R}^{n}})$, due to the density, we can find
$\\{f_{k}\\}\subset\mathscr{S}$ and $\\{\phi_{k}\\}\subset\mathscr{S}$ such
that

$f_{k}\to f\ \text{in}\ L^{1}((0,T),H^{s}({\mathbb{R}^{n}})),\
\phi_{k}\to\phi\ \text{in}\ H^{s}({\mathbb{R}^{n}}),\quad\text{and}\quad$
$(D_{t}+P)u_{k}=f_{k},\ u|_{t=0}=\phi_{k},\ u_{k}\in C^{1}([0,T],H^{s})\cap
C([0,T],H^{s+1}).$ (9.29)

From (9.29) and Lemma 9.16 we can obtain

$e^{-\lambda
T}\|{u_{k}-u_{k^{\prime}}}\|_{C([0,T{]},H^{s})}\leq\|{\phi_{k}-\phi_{k^{\prime}}}\|_{H^{s}}+2\|{(f_{k}-f_{k^{\prime}})(t,\cdot)}\|_{L^{1}((0,T),H^{s})},$

so $\\{u_{k}\\}$ is Cauchy in $C([0,T{]},H^{s})$ and the limit $u\in
C([0,T{]},H^{s})$ is a desired solution.

Step 6: uniqueness. By the energy estimate (9.24) it is easy to show the
uniqueness of $u$.

The proof is complete. ∎

### Exercise

######  Exercise 9.1.

Proof Lemma 9.4.

######  Exercise 9.2.

Prove the function $a$ constructed in (9.4) is in $S^{0}$.

######  Exercise 9.3.

Show that $a$ defined in (9.6) gives
$(x_{0},\xi_{0})\notin\operatorname{Char}T_{a}$. Hint: to borrow ideas from
Lemma 6.17.

## Bibliography

## Index

  * adjoint 5.15, 5.15
  * adjoint of quantizations Theorem 7.14
  * asymptotics Definition 2.5
  * bicharacteristic Definition 9.9
  * canonical relation Theorem 8.11, Remark 8.13
  * Carleman estimates Theorem 7.24
  * commutator Exercise 5.2
  * composition of quantizations Theorem 7.11, Theorem 7.17
  * composition of $\Psi$DOs Theorem 5.3
  * conic set Definition 8.1
  * direct product §8.2.1
  * ellipticity Definition 6.4
  * Fourier integral operators Remark 8.13
  * Fourier Transform Definition 1.15
  * Gårding’s inequality Theorem 6.18
  * kernel §2.3, Definition 3.8
  * microlocal ellipticity Definition 9.2
  * microlocal parametrix Definition 9.1
  * null bicharacteristic Definition 9.9
  * oscillatory integral I Definition 3.6
  * oscillatory integral II Definition 3.14
  * parametrix Definition 6.1
  * Parseval’s Relation item 3
  * phase function Definition 3.1
  * Plancherel Theorem Theorem 1.19
  * product of distributions §8.2.2
  * propagation of the singularities Theorem 9.13
  * properly supported §2.3
  * pseudodifferential operators Definition 2.7
  * pseudolocal property Theorem 2.24
  * Schur estimate Lemma 6.10
  * Schwartz function Definition 1.4
  * Schwartz Space Definition 1.4
  * sharp Gårding’s Inequality Theorem 6.21
  * singular support Definition 2.23
  * smooth direction, $\operatorname{Smo}$ Definition 8.2
  * Sobolev spaces Definition 2.15
  * stationary phase lemma §4.2
  * strong ellipticity Definition 6.16
  * symbols Definition 2.1
  * symplectic 2-form Remark 7.16
  * symplectic product Definition 7.15
  * Taylor’s expansion §4.1.1
  * tempered Distributions Definition 1.11
  * tempered functions Definition 1.9
  * transpose 5.15, 5.15
  * twist Remark 8.13
  * wavefront set Definition 8.4