op}({r}){\rm op}({\kappa^{1/2}\lambda_{1}^{1/2}})W_{1},{\rm
op}({\kappa^{1/2}\lambda_{3}^{1/2}})W_{3})|$
$\displaystyle\leq\big{(}\sqrt{3}\,{\bar{\varepsilon}}\|{\rm
op}({b})\|+CM^{-1/2}\big{)}\|{\rm op}({\kappa^{1/2}\varLambda^{1/2}})W\|^{2}.$

Next consider $({\rm op}({\lambda_{2}}){\rm op}({b})W_{1},W_{2})$ for $b\in
S(\sigma^{-1/2},g)=S(\lambda_{2}^{-1/2},g)$. Denote

$r=(1+k)\\#(\kappa^{-1/2}\lambda_{2}^{-1/2})\\#(\lambda_{2}\\#b)\\#(\lambda_{1}^{-1/2}\kappa^{-1/2})\\#(1+{\tilde{k}})\in
S(1,g)$

such that
$(\kappa^{1/2}\lambda_{2}^{1/2})\\#r\\#(\lambda_{1}^{1/2}\kappa^{1/2})=\lambda_{2}\\#b$.
Write $r=(\kappa^{-1}\lambda_{1}^{-1/2})\\#(\lambda_{2}^{1/2}b)+{\tilde{r}}$
with ${\tilde{r}}\in S(M^{-1},g)$. Thus repeating the same arguments as above
one conclude the last assertion. ∎

In particular this lemma implies

######  Corollary 10.1.

Let $B=(b_{ij})\in S(1,g)$. Then with $W={\rm op}({w^{-n}})V$

$\displaystyle\big{|}({\rm op}({\varLambda}){\rm
op}({B})W,W)\big{|}\leq(\sqrt{3}\,{\bar{\varepsilon}}\|{\rm
op}({b_{31}})\|+CM^{-1/2}){\mathcal{E}}_{1}(V)+C{\mathcal{E}}_{2}(V).$

From Proposition 9.1 it results
$\phi^{-n}\\#{\mathcal{B}}_{1}-{\mathcal{B}}_{1}\\#\phi^{-n}\in
S(M^{-1}\phi^{-n},g)$ then one concludes by Corollary 10.1 that

(10.5) $\big{|}({\rm op}({\varLambda})[{\rm op}({w^{-n}}),{\rm
op}({{\mathcal{B}}_{1}})]V,W)\big{|}\leq
CM^{-1}{\mathcal{E}}_{1}(V)+C{\mathcal{E}}_{2}(V)$

where $W={\rm op}({w^{-n}})V$ again. Write
$T^{-1}\partial_{t}T=({\tilde{t}}_{ij})$ and recall (5.12) and note that
${\tilde{t}}_{12}=-{\tilde{t}}_{21}\in{\mathcal{C}}(\sigma^{-1/2})$ and
${\tilde{t}}_{31}\in S(1,g)$. Then thanks to Lemma 6.5 one sees that
$\lambda_{j}\\#(\phi^{-n}\\#{\tilde{t}}_{j1}-{\tilde{t}}_{j1}\\#\phi^{-n})\\#\phi^{n}$
is in

$\displaystyle
S(\sigma^{1-j/2}\omega^{-1}\rho^{1/2}\langle{\xi}\rangle_{\gamma}^{-1},g)\subset
S(M^{-1}\sqrt{\kappa\lambda_{1}}\,\sqrt{\kappa\lambda_{j}},g),\quad j=2,3$

because $C\lambda_{1}\geq M\rho\langle{\xi}\rangle_{\gamma}^{-1}$,
$C\lambda_{2}\geq M\langle{\xi}\rangle_{\gamma}^{-1}$ and
$\omega^{-1}\leq\kappa$. Therefore repeating similar arguments one concludes

(10.6) $\big{|}({\rm op}({\varLambda})[{\rm op}({w^{-n}}),{\rm
op}({T^{-1}\partial_{t}T})]V,W)\big{|}\leq CM^{-1}{\mathcal{E}}_{1}(V).$

Recalling ${\mathcal{B}}={\mathcal{B}}_{1}-T^{-1}D_{t}T$ it follows from
(10.5) and (10.6) that

(10.7) $\big{|}({\rm op}({\varLambda})[{\rm op}({w^{-n}}),{\rm
op}({{\mathcal{B}}})]V,W)\big{|}\leq
CM^{-1}{\mathcal{E}}_{1}(V)+C{\mathcal{E}}_{2}(V).$

With ${\mathcal{B}}=(q_{ij})$ we see that $q_{ij}\in S(\sigma^{-1/2},g)$ for
$j\geq i$ and

$q_{21}=i\partial_{t}(3b/2a_{M})+S(1),\;q_{31}={\bar{b}}_{3}+i{\bar{e}}/3+S(M^{-1}),\;q_{32}=-2M{\bar{e}}+S(M^{-1})$

by Proposition 9.1. Applying Lemma 10.2 we have from (5.13), recalling
Proposition 5.1 and ${\bar{\varepsilon}}=\sqrt{3}/{\bar{e}}{\bar{c}}^{1/2}$,
that

(10.8) $\begin{split}&\big{|}({\rm op}({\varLambda}){\rm
op}({{\mathcal{B}}}){\rm op}({w^{-n}})V,{\rm op}({w^{-n}})V)\big{|}\\\
&\leq\big{(}|3{\bar{b}}_{3}+i{\bar{e}}|/{\bar{e}}{\bar{c}}^{1/2}+(6+\sqrt{2})/{\bar{c}}^{1/2}+CM^{-1/2}\big{)}{\mathcal{E}}_{1}(V)+CM^{4}{\mathcal{E}}_{2}(V).\end{split}$

Combining the estimates (10.8) and (10.7) we obtain

######  Lemma 10.3.

The term $\big{|}({\rm op}({\varLambda}){\rm op}({w^{-n}}){\rm op}({{\cal
B}})V,{\rm op}({w^{-n}})V)\big{|}$ is bounded by the right-hand side of
(10.8).

###  10.3 Term $({\rm op}({\varLambda}){\rm op}({w^{-n}}){\rm
op}({i{\mathcal{A}}})V,{\rm op}({w^{-n}})V)$

Note that $\phi^{-n}\\#([\xi]r)-([\xi]r)\\#\phi^{-n}\in
S(\phi^{-n}\sigma^{s-1/2}\omega^{-1}\rho^{1/2},g)$ for
$r\in{\mathcal{C}}(\sigma^{s})$ by Lemma 6.5. Recalling Corollary 5.3 and
${\mathcal{A}}=A^{T}[\xi]$ then denoting
$\phi^{-n}\\#{\mathcal{A}}-{\mathcal{A}}\\#\phi^{-n}=(r_{ij})$ we see that
$r_{ij}\in S(\phi^{-n}\omega^{-1}\rho^{1/2},g)\subset
S(M^{-2}\kappa\phi^{-n},g)$ for $j\geq i$ because $\omega^{-1}\leq\kappa$.
Writing $\lambda_{i}\\#r_{ij}=\lambda_{i}\\#{\tilde{r}}_{ij}\\#\phi^{-n}$ with
${\tilde{r}}_{ij}\in S(M^{-2}\kappa,g)$ one obtains $\big{|}({\rm
op}({\lambda_{i}}){\rm op}({r_{ij}})V_{j},W_{i})\big{|}\leq CM^{-2}\|{\rm
op}({\kappa^{1/2}\varLambda^{1/2}})W\|^{2}$ for $j\geq i$ since
$\lambda_{i}\\#{\tilde{r}}_{ij}\in S(M^{-2}\kappa\lambda_{i},g)$. From Lemma
5.8 one has ${\tilde{a}}_{21}=\lambda_{1}\,{\mathcal{C}}(\sigma^{-1})$ hence
thanks to Lemmas 6.5 and 7.7

$\displaystyle\phi^{-n}\\#({\tilde{a}}_{21}[\xi])-\phi^{-n}{\tilde{a}}_{21}[\xi]\in
S(\sigma^{-1/2}\lambda_{1}^{1/2}\omega^{-1}\rho^{1/2}\phi^{-n},g)\subset
S(\lambda_{1}^{1/2}\kappa\phi^{-n},g)$

for $\omega^{-1}\leq\kappa$ again. Thus we have

$\displaystyle\big{|}({\rm op}({\lambda_{2}}){\rm
op}({r_{21}})V_{1},W_{2})\big{|}\leq CM^{-2}\|{\rm
op}({\kappa^{1/2}\varLambda^{1/2}})W\|^{2}$

since $\lambda_{2}^{1/2}\leq CM^{-2}$. Similarly from
${\tilde{a}}_{31}=\lambda_{1}\,{\mathcal{C}}(\sigma^{1/2})$,
${\tilde{a}}_{32}=\lambda_{2}\,{\mathcal{C}}(1)$ and Lemma 7.7 it follows that
$r_{3j}\in
S(\sigma^{2-j}\lambda_{j}^{1/2}\omega^{-1}\rho^{1/2}\phi^{-n},g)\subset
S(M^{-2}\lambda_{j}^{1/2}\kappa\phi^{-n},g)$ for $j=1,2$. Here we have used
$\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\lambda_{2}\in
S(\lambda_{2}^{1/2}\langle{\xi}\rangle_{\gamma}^{-|\beta|},g)$ for
$|\alpha+\beta|=1$ which follows from $\lambda_{2}\in{\mathcal{C}}(\sigma)$
easily. Then one obtains

$\displaystyle\big{|}({\rm op}({\lambda_{3}}){\rm
op}({r_{3j}})V_{j},W_{3})\big{|}\leq CM^{-2}\|{\rm
op}({\kappa^{1/2}\varLambda^{1/2}})W\|^{2},\quad j=1,2.$

Therefore $({\rm op}({\varLambda}){\rm op}({w^{-n}}){\rm
op}({{\mathcal{A}}})V,{\rm op}({w^{-n}})V)-({\rm op}({\varLambda}){\rm
op}({{\mathcal{A}}})W,W)$ is bounded by constant times
$M^{-2}{\mathcal{E}}_{1}(V)$.

Next study $\varLambda\\#{\mathcal{A}}-\varLambda{\mathcal{A}}=(q_{ij})$. From
Corollary 5.3 and Lemma 7.7 we have

$\displaystyle\lambda_{1}\\#({\tilde{a}}_{1j}[\xi])-\lambda_{1}{\tilde{a}}_{1j}[\xi]\in
S(\sigma^{1/2}\lambda_{1}^{1/2},g),\;\lambda_{2}\\#({\tilde{a}}_{2j}[\xi])-\lambda_{2}{\tilde{a}}_{2j}[\xi]\in
S(\lambda_{2}^{1/2},g)$

for $j\geq 1$ and $j\geq 2$ respectively. Then noting
$C\lambda_{1}^{1/2}\kappa\geq 1$ and $C\lambda_{2}\kappa\geq 1$

$\big{|}({\rm op}({q_{ij}})W_{j},W_{i})\big{|}\leq CM^{-2}\|{\rm
op}({\kappa^{1/2}\varLambda^{1/2}})W\|^{2}+CM^{2}\|{\rm
op}({\varLambda^{1/2}})W\|^{2},\;\;j\geq i.$

Repeating similar arguments one has
$\lambda_{i}\\#({\tilde{a}}_{ij}[\xi])-\lambda_{i}{\tilde{a}}_{ij}[\xi]\in
S(M^{-2}\kappa\lambda_{i}^{1/2}\lambda_{j}^{1/2},g)$ and hence $|({\rm
op}({q_{ij}})W_{i},W_{j})|\leq CM^{-2}\|{\rm
op}({\varLambda^{1/2}\kappa^{1/2}})W\|^{2}$ for $i>j$. Thus we conclude that

(10.9) $\begin{split}\big{|}({\rm op}({\varLambda}){\rm op}({w^{-n}}){\rm
op}({{\mathcal{A}}})V,{\rm op}({w^{-n}})V)-({\rm
op}({\varLambda{\mathcal{A}}})W,W)\big{|}\\\ \leq
CM^{-2}{\mathcal{E}}_{1}(V)+CM^{2}{\mathcal{E}}_{2}(V).\end{split}$

Since $\varLambda{\mathcal{A}}={\mathcal{A}}^{*}\varLambda$ we have

######  Lemma 10.4.

One can find $C>0$ such that

$\big{|}{\mathsf{Re}}\,\big{(}{\rm op}({\varLambda}){\rm op}({w^{-n}}){\rm
op}({i{\mathcal{A}}})V,{\rm op}({w^{-n}})V)\big{|}\leq
CM^{-2}{\mathcal{E}}_{1}(V)+CM^{2}{\mathcal{E}}_{2}(V).$

###  10.4 Term $({\rm op}({\partial_{t}\varLambda}){\rm op}({w^{-n}})V,{\rm
op}({w^{-n}})V)$

Start with

######  Lemma 10.5.

We have $\partial_{t}\lambda_{j}\in S(\kappa\lambda_{j},g)$, $j=1,2$.

###### Proof.

Note that Lemma 3.6 with $\epsilon=\sqrt{2}M\langle{\xi}\rangle_{\gamma}^{-1}$
implies

$|\partial_{t}\Delta_{M}|\leq
C^{*}\big{(}1/t+1/\omega\big{)}\Delta_{M}=C^{*}\kappa\Delta_{M}.$

Recalling
$\partial_{t}\lambda_{1}=-\partial_{t}q(\lambda_{1})/\partial_{\lambda}q(\lambda_{1})$
it follows from (5.6) and (5.9) that

$|\partial_{t}\lambda_{1}|\leq(1+CM^{-2})\big{(}|\partial_{t}a_{M}/a_{M}|\lambda_{1}+|\partial_{t}\Delta_{M}|/6a_{M}\big{)}.$

Since $(1+CM^{-2})\lambda_{1}\geq\Delta_{M}/6a_{M}$ by Proposition 5.1 and
$1/a_{M}\leq\kappa/e$ by Lemma 7.11 one concludes

(10.10) $|\partial_{t}\lambda_{1}|\leq(1+CM^{-2})(C^{*}+1)\kappa\lambda_{1}.$

Note that
$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial_{t}\lambda_{1}|\leq
C\sigma^{1-|\alpha+\beta|/2}\langle{\xi}\rangle_{\gamma}^{-|\beta|}\leq
C\sigma^{1/2}\langle{\xi}\rangle_{\gamma}^{-1/2}\langle{\xi}\rangle_{\gamma}^{(|\alpha|-|\beta|)/2}$
for $\partial_{t}\lambda_{1}\in{\mathcal{C}}(\sigma)$. From Lemma 7.11 and
$C\lambda_{1}\geq M\sigma\langle{\xi}\rangle_{\gamma}^{-1}$ it follows that

$\kappa\lambda_{1}\geq\kappa\sqrt{\lambda_{1}}M^{1/2}\sigma^{1/2}\langle{\xi}\rangle_{\gamma}^{-1/2}/C\geq
M^{1/2}\sigma^{1/2}\langle{\xi}\rangle_{\gamma}^{-1/2}/C^{\prime}$

which proves
$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial_{t}\lambda_{1}|\leq
CM^{-1/2}\kappa\lambda_{1}\langle{\xi}\rangle_{\gamma}^{(|\alpha|-|\beta|)/2}$
for $|\alpha+\beta|=1$. For $|\alpha+\beta|\geq 2$ it follows that

$\displaystyle|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial_{t}\lambda_{1}|\precsim\sigma^{-(|\alpha+\beta|-2)/2}\langle{\xi}\rangle_{\gamma}^{-|\beta|}\precsim
M\langle{\xi}\rangle_{\gamma}^{-1}M^{-|\alpha+\beta|/2}\langle{\xi}\rangle_{\gamma}^{(|\alpha|-|\beta|)/2}$
$\displaystyle\leq\sigma^{-1}M\sigma\langle{\xi}\rangle_{\gamma}^{-1}M^{-|\alpha+\beta|/2}\langle{\xi}\rangle_{\gamma}^{(|\alpha|-|\beta|)/2}\leq
C\kappa\lambda_{1}M^{-|\alpha+\beta|/2}\langle{\xi}\rangle_{\gamma}^{(|\alpha|-|\beta|)/2}$

because $\kappa\sigma\geq 1$. Thus $\partial_{t}\lambda_{1}\in
S(\kappa\lambda_{1},g)$. On the other hand $\partial_{t}\lambda_{j}\in
S(\kappa\lambda_{j},g)$, $j=2,3$ is clear since
$\partial_{t}\lambda_{j}\in{\mathcal{C}}(1)\subset S(1,g)\subset
S(\kappa\lambda_{2},g)$ for $C\lambda_{2}\kappa\geq 1$. This completes the
proof. ∎

Note that from (5.6), (5.9) and $|\partial_{t}\Delta_{M}|\precsim a_{M}^{2}$
we see that

(10.11)
$\big{|}\partial_{t}\lambda_{2}\big{|}\leq(1+CM^{-2})|\partial_{t}a_{M}|\lambda_{2}/a_{M}+Ca_{M}\leq\big{(}1+CM^{-2}\big{)}\kappa\lambda_{2}$

for $C\kappa\lambda_{2}\geq 1$. Since $\big{|}({\rm
op}({\partial_{t}\lambda_{3}})W_{3},W_{3})\big{|}\leq C\|{\rm
op}({\lambda_{3}})W_{3}\|^{2}$ is clear applying Lemma 8.4 one obtains from
(10.10) and (10.11) that

######  Lemma 10.6.

We have

$\displaystyle\big{|}({\rm op}({\partial_{t}\varLambda}){\rm
op}({w^{-n}})V,{\rm
op}({w^{-n}})V)\big{|}\leq(C^{*}+2+CM^{-1/2}){\mathcal{E}}_{1}(V)+C{\mathcal{E}}_{2}(V).$

###  10.5 Conclusion

In what follows we denote $\|u\|_{s}=\|\langle{D}\rangle_{\gamma}^{s}u\|$ and
by $H^{s}=H^{s}({\mathbb{R}}^{d})$ the set of tempered distribution $u$ on
${\mathbb{R}}^{d}$ such that $\|u\|_{s}<+\infty$.

######  Definition 10.2.

Denote by ${\mathcal{H}}_{-n,s}(0,\delta)$ the set of $f$ such that

$t^{-n}\langle{D}\rangle_{\gamma}^{s}f(t,\cdot)\in
L^{2}((0,\delta)\times{\mathbb{R}}^{d}).$

Consider the term ${\mathsf{Re}}\,\big{(}{\rm op}({\varLambda}){\rm
op}({w^{-n}})F,{\rm op}({w^{-n}})V)$ where $F={{}^{t}}(F_{1},F_{2},F_{3})$.
Write
$\varLambda=(\kappa^{1/2}\varLambda^{1/2})\\#R\\#(\kappa^{-1/2}\varLambda^{1/2})$
with $R\in S(1,g)$. Because of the choice of $n$ it follows from (10.4) and
Lemmas 10.1, 10.3, 10.4, 10.6 one can find $c_{i}>0$ and $M_{0}$,
$\gamma_{0}$, $\theta_{0}$ such that

(10.12) $\begin{split}\frac{d}{dt}{\mathcal{E}}(V)\leq-c_{1}e^{-\theta
t}{\mathcal{E}}_{1}(V)-c_{2}\theta e^{-\theta t}{\mathcal{E}}_{2}(V)\\\
+\big{|}{\mathsf{Re}}\,\big{(}{\rm op}({\varLambda}){\rm op}({w^{-n}})F,{\rm
op}({w^{-n}})V)\big{|}\end{split}$

for $M\geq M_{0}$, $\gamma\geq\gamma_{0}$ and $\theta\geq\theta_{0}$. Thanks
to Lemma 6.3 one has $\kappa^{-1/2}\phi^{-n}\lambda_{j}^{1/2}\in
S(M^{-n}\sqrt{t}\,\langle{\xi}\rangle_{\gamma}^{n},g)$ then we see easily

$\big{|}{\mathsf{Re}}\,\big{(}{\rm op}({\varLambda}){\rm op}({w^{-n}})F,{\rm
op}({w^{-n}})V)\big{|}\leq
CM^{-1}{\mathcal{E}}_{1}(V)+CM^{-2n+1}t^{2n+1}\|F\|_{n}^{2}.$

Since $M\langle{\xi}\rangle_{\gamma}^{-1}/C\leq\phi\leq CM^{-4}$ and
$t^{-1/2}\leq\kappa^{1/2}\leq t^{-1/2}+\omega^{-1/2}\leq
t^{-1/2}+M^{-1}\langle{\xi}\rangle_{\gamma}$ and
$\langle{\xi}\rangle_{\gamma}^{-3/2+j/2}\leq C\lambda_{j}^{1/2}$ for $1\leq
j\leq 3$ then

(10.13) $\begin{split}M^{8n}t^{-1}\|V\|^{2}_{-1}/C\leq
t^{2n}{\mathcal{E}}_{1}(V)\leq
CM^{-2n}(t^{-1}\|V\|^{2}_{n}+\|V\|^{2}_{n+1}),\\\ M^{8n}\|V\|^{2}_{-1}/C\leq
t^{2n}{\mathcal{E}}_{2}(V)\leq CM^{-2n}\|V\|^{2}_{n}.\end{split}$

Assume $D_{t}^{j}V\in{\mathcal{H}}_{-n-1/2+j/2,n+1-j}(0,M^{-4})$, $j=0,1$.
From this one sees that $\lim_{t\to+0}\|V(t)\|_{n}$ exists which is $0$. Using
this we see $\lim_{t\to+0}t^{-n}\|V(t)\|_{n}=0$. Noting that
${\mathcal{E}}(V)\leq CM^{-n}t^{-2n}\|V(t)\|_{n}^{2}$ and integrating (10.12)
in $t$ we obtain

######  Proposition 10.1.

There exist $c_{i}>0$, $C>0$ and $M_{0}$, $\gamma_{0}$, $\theta_{0}$ such that
for any $V$ with $D_{t}^{j}V\in{\mathcal{H}}_{-n-1/2+j/2,n+1-j}(0,M^{-4})$,
$j=0,1$, one has

$\displaystyle c_{1}t^{-2n}e^{-\theta
t}\|V(t)\|_{-1}^{2}+c_{2}\int_{0}^{t}e^{-\theta
s}s^{-2n-1}\|V(s)\|_{-1}^{2}ds$
$\displaystyle+c_{3}\theta\int_{0}^{t}e^{-\theta
s}s^{-2n}\|V(s)\|_{-1}^{2}ds\leq CM^{1-10n}\int_{0}^{t}e^{-\theta
s}s^{-2n+1}\|{\tilde{L}}V(s)\|_{n}^{2}ds$

for $0\leq t\leq M^{-4}$, $M\geq M_{0}$, $\gamma\geq\gamma_{0}$,
$\theta\geq\theta_{0}$.

######  Corollary 10.2.

For any $V$ with $D_{t}^{j}V\in{\mathcal{H}}_{-n-1/2+j/2,n+1-j}(0,M^{-4})$,
$j=0,1$

$\displaystyle\int_{0}^{t}s^{-2n-1}\|V(s)\|_{-1}^{2}ds\leq
C\int_{0}^{t}s^{-2n+1}\|{\tilde{L}}V(s)\|_{n}^{2}ds,\;\;0\leq t\leq M^{-4}.$

Consider the adjoint operator ${\hat{P}^{*}}$ of ${\hat{P}}$. Noting
$a_{M}\in{\mathcal{C}}(\sigma)$, $b\in{\mathcal{C}}(\sigma^{3/2})$ and (4.5),
(5.1) we see that

$\displaystyle{\hat{P}^{*}}=D_{t}^{3}-a_{M}(t,x,D)[D]^{2}D_{t}-b(t,x,D)\,[D]^{3}$
$\displaystyle+b_{1}D_{t}^{2}+\big{(}{\tilde{b}}_{2}+d_{M})[D]D_{t}+{\tilde{b}}_{3}[D]^{2}+{\tilde{c}}_{1}D_{t}+{\tilde{c}}_{2}[D]$

with ${\tilde{b}}_{j}\in S(1,g)$ and ${\tilde{c}}_{j}\in S(M^{2},g)$ hence
${\tilde{c}}_{j}[D]^{-1}\in S(M^{-3},g)$ where it is not difficult to check
that ${\tilde{b}}_{3}-\big{(}b_{3}+ie\big{)}\in S(M^{-3},g)$. Denote by
${\tilde{L}}^{*}$ the corresponding first order system (which is not the
adjoint of ${\tilde{L}}$). Since the power $n$ of the weight $\phi^{-n}$
depends only on $a$, $b$ and $b_{3}$ (see (10.1)) then we can choose the same
$n$ for ${\hat{P}^{*}}$ as for ${\hat{P}}$. Then employing the weighted energy

${\mathcal{E}}^{*}(V)=e^{\theta t}\big{(}{\rm op}({\varLambda}){\rm
op}({t^{n}\phi^{n}})V,{\rm op}({t^{n}\phi^{n}})V\big{)}$

and repeating the same arguments for ${\mathcal{E}}(V)$ and making the
integration

$-\int^{\delta}_{t}\frac{d}{dt}{\mathcal{E}}^{*}\,dt,\quad 0<t<\delta=M^{-4}$

we have

######  Proposition 10.2.

There exist $c_{i}>0$, $C>0$ and $M_{0}$, $\gamma_{0}$, $\theta_{0}$ such that
for any $V$ with $D_{t}^{j}V\in{\mathcal{H}}_{n-1/2+j/2,1-j}(0,M^{-4})$,
$j=0,1$, one has

$\displaystyle c_{1}t^{2n}e^{\theta
t}\|V(t)\|_{-n-1}^{2}+c_{2}\int_{t}^{\delta}e^{\theta\tau}\tau^{2n-1}\|V(\tau)\|_{-n-1}^{2}d\tau$
$\displaystyle+c_{3}\theta\int_{t}^{\delta}e^{-\theta\tau}\tau^{2n}\|V(\tau)\|_{-n-1}^{2}d\tau\leq
CM^{-10n}\delta^{2n}e^{\theta\delta}\|V(\delta)\|^{2}$
$\displaystyle+CM^{1-10n}\int_{t}^{\delta}e^{\theta\tau}\tau^{2n+1}\|{\tilde{L}}^{*}V(\tau)\|^{2}d\tau,\quad
0\leq t\leq\delta$

for $M\geq M_{0}$, $\gamma\geq\gamma_{0}$, $\theta\geq\theta_{0}$ where
${\tilde{L}}^{*}V={\rm op}({T}){{}^{t}}({\hat{P}^{*}}u,0,0)$ and ${\rm
op}({T})V={{}^{t}}(D_{t}^{2}u,[D]D_{t}u,[D]^{2}u)$.

######  Remark 10.1.

It is clear from the proof that for any $n^{\prime}\geq n$, Propositions 10.1
and 10.2 hold.

##  11 Preliminary existence result

Let $s\in{\mathbb{R}}$ and we estimate $\langle{D}\rangle^{s}V$. In what
follows we fix $M$ and $\gamma$ (actually it is enough to choose
$\gamma=M^{5}$, see (4.1)) such that Propositions 10.1 and 10.2 hold,
therefore from now on we can assume
$\langle{D}\rangle_{\gamma}=\langle{D}\rangle=[D]$ without restrictions, while
$\theta$ remains to be free. From (10.3) one has

$\partial_{t}(\langle{D}\rangle^{s}V)=\big{(}{\rm
op}({i{\mathcal{A}}+i{\mathcal{B}}})+i[\langle{D}\rangle^{s},{\rm
op}({{\mathcal{A}}+{\mathcal{B}}})]\langle{D}\rangle^{-s}\big{)}\langle{D}\rangle^{s}V+\langle{D}\rangle^{s}F.$

######  Lemma 11.1.

For any $s\in{\mathbb{R}}$ there is $C>0$ such that

$\big{|}([\langle{D}\rangle^{s},{\rm op}({{\mathcal{A}}})]V,{\rm
op}({\varLambda})\langle{D}\rangle^{s}V)\big{|}\leq
C{\mathcal{E}}_{2}(\langle{D}\rangle^{s}V).$

###### Proof.

Denoting $T^{-1}AT=({\tilde{a}}_{ij})$ thanks to Lemmas 5.8 and 7.7 we see
that

(11.1)
$\begin{split}(({\tilde{a}}_{3j}[\xi])\\#\langle{\xi}\rangle^{s}-\langle{\xi}\rangle^{s}\\#({\tilde{a}}_{3j}[\xi]))\\#\langle{\xi}\rangle^{-s}\in
S(\sigma^{2-j}\sqrt{\lambda_{j}},g),\;\;j=1,2,\\\
(({\tilde{a}}_{21}[\xi])\\#\langle{\xi}\rangle^{s}-\langle{\xi}\rangle_{\gamma}^{s}\\#({\tilde{a}}_{21}[\xi]))\\#\langle{\xi}\rangle^{-s}\in
S(\sigma^{-1/2}\sqrt{\lambda_{1}},g)\end{split}$

where
$S(\sigma^{-1/2}\sqrt{\lambda_{1}},g)=S(\lambda_{2}^{-1/2}\sqrt{\lambda_{1}},g)$.
From Corollary 5.3 it is easy to see
$(({\tilde{a}}_{ij}[\xi])\\#\langle{\xi}\rangle^{s}-\langle{\xi}\rangle^{s}\\#({\tilde{a}}_{ij}[\xi]))\\#\langle{\xi}\rangle^{-s}\in
S(1,g)$ for $j\geq i$. Then together with (11.1) the proof follows from a
repetition of similar arguments. ∎

######  Lemma 11.2.

For any $s\in{\mathbb{R}}$ and any $\epsilon>0$ there is $C>0$ such that

$\big{|}([\langle{D}\rangle^{s},{\rm op}({{\mathcal{B}}})]V,{\rm
op}({\varLambda})\langle{D}\rangle^{s}V)\big{|}\leq\epsilon\,{\mathcal{E}}_{1}(\langle{D}\rangle^{s}V)+C{\mathcal{E}}_{2}(\langle{D}\rangle^{s}V).$

###### Proof.

Write ${\mathcal{B}}_{1}=({\tilde{b}}_{ij})$. Since ${\tilde{b}}_{ij}\in
S(1,g)$ then
$\lambda_{i}\\#(\langle{\xi}\rangle^{s}\\#{\tilde{b}}_{ij}-{\tilde{b}}_{ij}\\#\langle{\xi}\rangle^{s})\\#\langle{\xi}\rangle^{-s}$
is in $S(\langle{\xi}\rangle^{-1/2}\lambda_{i},g)$. Noting
$C\lambda_{1}\geq\sigma\langle{\xi}\rangle^{-1}$ and
$C\lambda_{2}\geq\sigma\geq\langle{\xi}\rangle^{-1}$ it is easy to see that
$S(\langle{\xi}\rangle^{-1/2}\lambda_{i},g)\subset
S(\lambda_{i}^{1/2}\lambda_{j}^{1/2},g)$ except for $(i,j)=(3,1)$. For
$(i,j)=(3,1)$ recalling ${\tilde{b}}_{31}=b_{3}+S(\sigma^{1/2})$ by (9.4) one
sees

$\displaystyle\lambda_{3}\\#(\langle{\xi}\rangle^{s}\\#{\tilde{b}}_{31}-{\tilde{b}}_{31}\\#\langle{\xi}\rangle^{s})\\#\langle{\xi}\rangle^{-s}\in
S(\sigma^{1/2}\langle{\xi}\rangle^{-1/2},g)\subset
S(\lambda_{1}^{1/2}\lambda_{3}^{1/2},g)$

(recall that $M$ has been fixed). Therefore one obtains

(11.2) $\big{|}([\langle{D}\rangle^{s},{\rm op}({{\mathcal{B}}_{1}})]V,{\rm
op}({\varLambda})\langle{D}\rangle^{s}V)\big{|}\leq
C{\mathcal{E}}_{2}(\langle{D}\rangle^{s}V).$

Next consider $T^{-1}\partial_{t}T=({\tilde{t}}_{ij})$. Recalling
${\tilde{t}}_{21}\in{\mathcal{C}}(\sigma^{-1/2})$,
${\tilde{t}}_{31}\in{\mathcal{C}}(1)$ and
${\tilde{t}}_{32}\in{\mathcal{C}}(\sigma^{1/2})$ and noting
$C\kappa\lambda_{1}\geq\langle{\xi}\rangle^{-1}$ and
$C\lambda_{2}\geq\sigma\geq\langle{\xi}\rangle^{-1}$ one has

$\lambda_{i}\\#(\langle{\xi}\rangle^{s}\\#{\tilde{t}}_{ij}-{\tilde{t}}_{ij}\\#\langle{\xi}\rangle^{s})\\#\langle{\xi}\rangle^{-s}\in
S(\sqrt{\kappa\lambda_{j}}\,\sqrt{\lambda_{i}},g),\;\;\quad i>j.$

Therefore we have

$\displaystyle\big{|}([\langle{D}\rangle^{s},{\rm
op}({T^{-1}\partial_{t}T})]V,{\rm
op}({\varLambda})\langle{D}\rangle^{s}V)\big{|}\leq
C\sqrt{{\mathcal{E}}_{1}(\langle{D}\rangle^{s}V)}\sqrt{{\mathcal{E}}_{2}(\langle{D}\rangle^{s}V)}$
$\displaystyle\leq\epsilon\,{\mathcal{E}}_{1}(\langle{D}\rangle^{s}V)+C^{2}\epsilon^{-1}{\mathcal{E}}_{2}(\langle{D}\rangle^{s}V)$

which together with (11.2) proves the assertion. ∎

Choosing $\epsilon>0$ smaller than $c_{2}$ in Proposition 10.1 and $\theta$
large we conclude

######  Proposition 11.1.

For any $s\in{\mathbb{R}}$ there exists $C>0$ such that for any $V$ with
$D_{t}^{j}V\in{\mathcal{H}}_{-n-1/2+j/2,n+s+1-j}(0,\delta)$, $j=0,1$, one has

$\displaystyle
t^{-2n}\|V(t)\|_{s-1}^{2}+\int_{0}^{t}\tau^{-2n-1}\|V(\tau)\|_{s-1}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n+1}\|{\tilde{L}}V(\tau)\|^{2}_{n+s}d\tau$

for $0\leq t\leq\delta$.

Since ${\tilde{L}}={\rm op}({I+K}){\rm op}({T^{-1}})\cdot L\cdot{\rm op}({T})$
with $T$, $T^{-1}\in S(1,g)$ then $\|{\tilde{L}}V\|_{s}\leq C_{s}\|L\cdot{\rm
op}({T})V\|_{s}$ and $\|{\rm op}({T})V\|_{s}\leq C_{s}\|V\|_{s}$ with some
$C_{s}>0$. Thanks to Lemma 5.7 one has $\|{\rm op}({\partial_{t}T})V\|_{s}\leq
C_{s}t^{-1/2}\|V\|_{s}$ then replacing ${\rm op}({T})V$ by $U$ one sees that
for any $U$ with $D_{t}^{j}U\in{\mathcal{H}}_{n-1/2+j/2,n+s+1-j}(0,\delta)$,
$j=0,1$ it holds

$\displaystyle
t^{-2n}\|U(t)\|_{s-1}^{2}+\int_{0}^{t}{\tau}^{-2n-1}\|U(\tau)\|_{s-1}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n+1}\|LU(\tau)\|_{n+s}^{2}d\tau.$

Since $U={{}^{t}}(D_{t}^{2}u,[D]D_{t}u,[D]^{2}u)$ and
$LU={{}^{t}}({\hat{P}}u,0,0)$ we have

######  Proposition 11.2.

For any $s\in{\mathbb{R}}$ there is $C>0$ such that for any $u$ with
$D_{t}^{j}u\in{\mathcal{H}}_{-n-1/2,n+s+3-j}(0,\delta)$, $0\leq j\leq 3$, one
has

(11.3)
$\begin{split}t^{-2n}\sum_{j=0}^{2}\|D_{t}^{j}u(t)\|_{s+1-j}^{2}+\sum_{j=0}^{2}\int_{0}^{t}{\tau}^{-2n-1}\|D_{t}^{j}u(\tau)\|_{s+1-j}^{2}d\tau\\\
\leq C\int_{0}^{t}\tau^{-2n+1}\|{\hat{P}}u(\tau)\|_{n+s}^{2}d\tau,\quad 0\leq
t\leq\delta.\end{split}$

Repeating the same arguments we conclude

######  Proposition 11.3.

For any $s\in{\mathbb{R}}$ there is $C>0$ such that for any $u$ with
$D_{t}^{j}u\in{\mathcal{H}}_{n-1/2+j/2,s+3-j}(0,\delta)$, $0\leq j\leq 3$, we
have

(11.4)
$\begin{split}t^{2n}\sum_{j=0}^{2}\|D_{t}^{j}u(t)\|_{s+1-j}^{2}+\sum_{j=0}^{2}\int_{t}^{\delta}{\tau}^{2n-1}\|D_{t}^{j}u(\tau)\|_{s+1-j}^{2}d\tau\\\
\leq
C\big{(}\sum_{j=0}^{2}\|D_{t}^{j}u(\delta)\|_{n+s+2-j}^{2}+\int_{t}^{\delta}\tau^{2n+1}\|{\hat{P}^{*}}u(\tau)\|_{n+s}^{2}d\tau\big{)},\;\;0<t\leq\delta.\end{split}$

Since (11.4) holds with ${\bar{n}}\geq n+3$ as noticed in Remark 10.1 then, in
the resulting (11.4), replacing $s$ by $-3{\bar{n}}-s-1$ we have

$\int_{0}^{\delta}t^{2{\bar{n}}-1}\|u(t)\|_{-3{\bar{n}}-s}^{2}dt\leq
C\int_{0}^{\delta}t^{2{\bar{n}}+1}\|{\hat{P}^{*}}u(t)\|_{-2{\bar{n}}-s-1}^{2}dt$

for any $u\in C_{0}^{\infty}((0,\delta)\times{\mathbb{R}}^{d})$. This implies

$\displaystyle\big{|}\int_{0}^{\delta}(f,v)dt\big{|}\leq\big{(}\int_{0}^{\delta}t^{-2{\bar{n}}+1}\|f\|_{3{\bar{n}}+s}^{2}dt\big{)}^{1/2}\big{(}\int_{0}^{\delta}t^{2{\bar{n}}-1}\|v\|_{-3{\bar{n}}-s}^{2}dt\big{)}^{1/2}$
$\displaystyle\leq
C\big{(}\int_{0}^{\delta}t^{-2{\bar{n}}+1}\|f\|_{3{\bar{n}}+s}^{2}dt\big{)}^{1/2}\big{(}\int_{0}^{\delta}t^{2{\bar{n}}+1}\|{\hat{P}^{*}}v\|_{-2{\bar{n}}-s-1}^{2}dt\big{)}^{1/2}$

for any $v\in C_{0}^{\infty}((0,\delta)\times{\mathbb{R}}^{d})$ and
$f\in{\mathcal{H}}_{-{\bar{n}}+1/2,3{\bar{n}}+s}(0,\delta)$. Using the Hahn-
Banach theorem to extend the anti-linear form in ${\hat{P}^{*}}v$;

(11.5) ${\hat{P}^{*}}v\mapsto\int_{0}^{\delta}(f,v)dt$

we conclude that there is some
$u\in{\mathcal{H}}_{-{\bar{n}}-1/2,2{\bar{n}}+s+1}(0,\delta)$ such that

$\int_{0}^{\delta}(f,v)dt=\int_{0}^{\delta}(u,{\hat{P}^{*}}v)dt,\quad v\in
C_{0}^{\infty}((0,\delta)\times{\mathbb{R}}^{d}).$

This implies that ${\hat{P}}u=f$. Since
$u\in{\mathcal{H}}_{0,2{\bar{n}}+s+1}(0,\delta)$ and
$f\in{\mathcal{H}}_{0,3{\bar{n}}+s}(0,\delta)$ it follows from [8, Theorem
B.2.9] that $D_{t}^{j}u\in{\mathcal{H}}_{0,2{\bar{n}}+s+1-j}(0,\delta)$ for
$j=0,1,2,\ldots$. Thus with $w=\langle{D}\rangle^{{\bar{n}}+s}u$ one has
$D_{t}^{j}w\in L^{2}((0,\delta)\times{\mathbb{R}}^{d})$ for
$j=0,\ldots,{\bar{n}}+1$ hence $D_{t}^{j}w(0)$ exists in
$L^{2}({\mathbb{R}}^{d})$ which is $0$ for $j=0,\ldots,{\bar{n}}$ for
$w\in{\mathcal{H}}_{-{\bar{n}}+1/2,0}(0,\delta)$. Thus one can write
$w(t)=\int_{0}^{t}(t-\tau)^{{\bar{n}}}\partial_{t}^{{\bar{n}}+1}w(\tau)d\tau/{\bar{n}}!$.
From this one concludes that
$D_{t}^{j}u\in{\mathcal{H}}_{-{\bar{n}}+j-1/2,{\bar{n}}+s}(0,\delta)$ hence
$D_{t}^{j}u\in{\mathcal{H}}_{-n-1/2,n+s+3-j}(0,\delta)$ for $0\leq j\leq 3$
because ${\bar{n}}\geq n+3$ thus (11.3) holds for this $u$. Now let
$f\in{\mathcal{H}}_{-n+1/2,n+s}(0,\delta)$. Take a rapidly decreasing function
$\rho(\xi)$ with $\rho(0)=1$ then $f_{\epsilon}=e^{-\epsilon/t}\rho(\epsilon
D)f\in{\mathcal{H}}_{-{\bar{n}}+1/2,2{\bar{n}}+s+1}(0,\delta)$ and
$f_{\epsilon}\to f$ in ${\mathcal{H}}_{-n+1/2,n+s}(0,\delta)$. As just proved
above there is $u_{\epsilon}$ satisfying ${\hat{P}}u_{\epsilon}=f_{\epsilon}$
and (11.3). Therefore choosing a weakly convergent subsequence
$\\{u_{\epsilon^{\prime}}\\}$ one can conclude

######  Theorem 11.1.

There exists $\delta>0$ such that for any $s\in{\mathbb{R}}$ and any
$f\in{\mathcal{H}}_{-n+1/2,n+s}(0,\delta)$ there exists a unique $u$ with
$D_{t}^{j}u\in{\mathcal{H}}_{-n-1/2,1+s-j}(0,\delta)$, $j=0,1,2$, satisfying
${\hat{P}}u=f$ and (11.3).

Instead of (11.5) considering the anti-linear form in ${\hat{P}}v$;

$\displaystyle{\hat{P}}v\mapsto\int_{0}^{\delta}(f,v)dt+\sum_{j=0}^{1}\big{(}w_{2-j},D_{t}^{j}v(\delta,\cdot)\big{)}+\big{(}w_{0},(D_{t}^{2}-[D]^{2}a(\delta,x,D))v(\delta,\cdot)\big{)}$

for $v\in C_{0}^{\infty}((0,\infty)\times{\mathbb{R}}^{d})$ and repeating
similar arguments adopting (11.3) we conclude

######  Theorem 11.2.

There exists $\delta>0$ such that for any $s\in{\mathbb{R}}$ and any
$f\in{\mathcal{H}}_{n+1/2,n+s}(0,\delta)$ and $w_{j}\in H^{n+s+2-j}$,
$j=0,1,2$, there is a unique $u$ with
$D_{t}^{j}u\in{\mathcal{H}}_{n-1/2,1+s-j}(0,\delta)$ satisfying
${\hat{P}^{*}}u=f$, $D_{t}^{j}u(\delta,\cdot)=w_{j}$, $j=0,1,2$ and (11.4).

Indeed we first see that there is $u\in{\mathcal{H}}_{n-1/2,1+s}(0,\delta)$
satisfying ${\hat{P}^{*}}u=f$ and $D_{t}^{j}u(\delta)=w_{j}$, $j=0,1,2$ (e.g.
[8, Chapter 23]). Since $f\in{\mathcal{H}}_{0,n+s}(\varepsilon,\delta)$ and
$u\in{\mathcal{H}}_{0,1+s}(\varepsilon,\delta)$ it follows from [8, Theorem
B.2.9] that $D_{t}^{j}u\in{\mathcal{H}}_{0,1+s-j}(\varepsilon,\delta)$, $0\leq
j\leq 2$ for any $\varepsilon>0$. Applying (11.4) with $t=\varepsilon$ we
conclude $D_{t}^{j}u\in{\mathcal{H}}_{n-1/2,1+s-j}(0,\delta)$, $j=1,2$ since
$\varepsilon>0$ is arbitrary.

##  12 Propagation of wave front set

In Section 11 we have proved an existence result of the Cauchy problem for
${\hat{P}}$, which coincides with the original $P$ only in $W_{M}$. Following
[23], [11] (also [27]) we show that the wave front set of $u(t,\cdot)$,
obtained by Theorem 11.1, propagates with finite speed. This fact enables us
to solve the Cauchy problem for the original $P$.

###  12.1 Estimate of wave front set

Let $\chi(x)\in C_{0}^{\infty}({\mathbb{R}}^{d})$ be equal to $1$ near $x=0$
and vanish in $|x|\geq 1$. Set

(12.1)
$\begin{split}d_{\epsilon}(x,\xi;y,\eta)=\big{\\{}\chi(x-y)|x-y|^{2}+|\xi\langle{\xi}\rangle^{-1}-\eta\langle{\eta}\rangle^{-1}|^{2}+\epsilon^{2}\big{\\}}^{1/2},\\\
f_{\epsilon}(t,x,\xi;y,\eta)=t-T+\nu d_{\epsilon}(x,\xi;y,\eta),\quad
T>0\end{split}$

where $(y,\eta)\in{\mathbb{R}}^{d}\times({\mathbb{R}}^{d}\setminus\\{0\\})$
and $\nu$ is a positive small parameter. Note that

(12.2)
$\big{|}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}d_{\epsilon}\big{|}\leq
C\langle{\xi}\rangle^{-|\beta|},\quad|\alpha+\beta|=1$

where $C$ is independent of $\epsilon>0$. Define $\Phi_{\epsilon}$ by

(12.3)
$\Phi_{\epsilon}(t,x,\xi)=\begin{cases}\exp{(1/f_{\epsilon}(t,x,\xi))}\quad\text{if}\;\;f_{\epsilon}<0\\\
0\quad\text{otherwise}\end{cases}$

and note that $\Phi_{\epsilon}\in S(1,g_{0})$ for any fixed $\epsilon>0$ where
$g_{0}=|dx|^{2}+\langle{\xi}\rangle^{-2}|d\xi|^{2}$. From now on to simplify
notations, we denote

${\mathcal{E}}_{1}(\langle{D}\rangle^{s}V)+{\mathcal{E}}_{2}(\langle{D}\rangle^{s}V)=t^{-2n}{\mathcal{N}}_{s}(V),\quad{\mathcal{N}}_{0}(V)={\mathcal{N}}(V).$

######  Lemma 12.1.

There is $\nu_{0}>0$ such that for any $0<\nu\leq\nu_{0}$ and any $\epsilon>0$
there is $C>0$ such that for any $V$ with
$\int_{0}^{\delta}t^{-2n}{\mathcal{N}}_{-1/4}(V)dt<+\infty$ and
${\tilde{L}}V\in{\mathcal{H}}_{-n+1/2,l}(0,\delta)$ with some $l$ we have

$\displaystyle{\mathcal{E}}_{1}({\rm
op}({\Phi_{\epsilon}})V)+\int_{0}^{t}\tau^{-2n}{\mathcal{N}}({\rm
op}({\Phi_{\epsilon}})V)d\tau\leq C\int_{0}^{t}\tau^{-2n+1}\|{\rm
op}({\Phi_{\epsilon}}){\tilde{L}}V\|_{n}^{2}d\tau$
$\displaystyle+C\int_{0}^{t}\tau^{-2n}{\mathcal{N}}_{-1/4}(V)d\tau,\quad
0<t\leq\delta.$

###### Proof.

Denote $V^{\mu}=\langle{\mu D}\rangle^{-{\bar{n}}}V$ with small $\mu>0$ where
we choose ${\bar{n}}=2n+\max\\{-l,0\\}+3$. Assume ${\tilde{L}}V=F$ so that
${\tilde{L}}V^{\mu}=F^{\mu}+R^{\mu}V^{\mu}=G^{\mu}$ where
$R^{\mu}=[\langle{\mu D}\rangle^{-\bar{n}},{\rm
op}({{\mathcal{A}}+{\mathcal{B}}})]\langle{\mu D}\rangle^{\bar{n}}$ and
$F^{\mu}=\langle{\mu D}\rangle^{-{\bar{n}}}F$. Note that $\Phi_{\epsilon
1}=f_{\epsilon}^{-1}\Phi_{\epsilon}\in S(1,g_{0})$ for any fixed $\epsilon>0$
and $\Phi_{\epsilon}-f_{\epsilon}\\#\Phi_{\epsilon 1}\in
S(\langle{\xi}\rangle^{-1},g_{0})$. Since
$\partial_{t}\Phi_{\epsilon}=-\Phi_{\epsilon 1}/f_{\epsilon}$ one can write

(12.4) $\begin{split}\partial_{t}({\rm op}({\Phi_{\epsilon}})V^{\mu})=-{\rm
op}({f_{\epsilon}^{-1}\Phi_{\epsilon 1}})V^{\mu}+({\rm
op}({i{\mathcal{A}}+i{\mathcal{B}}})){\rm op}({\Phi_{\epsilon}})V^{\mu}\\\
+[{\rm op}({\Phi_{\epsilon}}),{\rm
op}({i{\mathcal{A}}+i{\mathcal{B}}})]V^{\mu}+{\rm
op}({\Phi_{\epsilon}})G^{\mu}.\end{split}$

Since
$\Phi_{\epsilon}\\#{\mathcal{B}}_{1}-{\mathcal{B}}_{1}\\#\Phi_{\epsilon}\in
S(\langle{\xi}\rangle^{-1/2},g)$ by Proposition 9.1 it is not difficult to see
from the proof of Corollary 10.1 that

$\displaystyle\big{|}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}})[{\rm
op}({\Phi_{\epsilon}}),{\rm op}({{\mathcal{B}}_{1}})]V^{\mu},{\rm
op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon}})V^{\mu}\big{)}\big{|}\leq
c(\epsilon){\mathcal{N}}_{-1/4}(V^{\mu})$

Denote
$\Phi_{\epsilon}\\#(T^{-1}\partial_{t}T)-(T^{-1}\partial_{t}T)\\#\Phi_{\epsilon}=(\varphi_{ij})$
hence $\varphi_{21}\in S(\sigma^{-1}\langle{\xi}\rangle^{-1},g)$ and
$\varphi_{31}\in S(\sigma^{-1/2}\langle{\xi}\rangle^{-1},g)$ from (5.12). Thus
$\lambda_{j}\\#\varphi_{j1}\in
S(\langle{\xi}\rangle^{-1/2}\sqrt{\kappa\lambda_{1}}\,\sqrt{\kappa\lambda_{j}},g)$
for $j=2,3$ because $C\lambda_{1}\geq\sigma\langle{\xi}\rangle^{-1}$,
$C\lambda_{2}\geq\sigma$ and $\kappa\sigma\geq 1$. A repetition of similar
arguments proving (10.6) shows that

$\displaystyle\big{|}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}})[{\rm
op}({\Phi_{\epsilon}}),{\rm op}({T^{-1}\partial_{t}T})]V^{\mu},{\rm
op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon}})V^{\mu}\big{)}\big{|}\leq
c(\epsilon){\mathcal{N}}_{-1/4}(V^{\mu}).$

Note that $\Phi_{\epsilon}\\#{\mathcal{A}}-{\mathcal{A}}\\#\Phi_{\epsilon}$
can be written

$\sum_{|\alpha+\beta|=1}\frac{(-1)^{|\alpha|}}{(2i)^{|\alpha+\beta|}\alpha!\beta!}\Big{(}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\Phi_{\epsilon}\partial_{x}^{\beta}\partial_{\xi}^{\alpha}{\mathcal{A}}-\partial_{x}^{\beta}\partial_{\xi}^{\alpha}\Phi_{\epsilon}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}{\mathcal{A}}\Big{)}+R_{\epsilon}=H_{\epsilon}+R_{\epsilon}$

where it follows from Corollary 5.3 that $R_{\epsilon}\in
S(\langle{\xi}\rangle^{-1/2},g)$ for $\sigma\geq\langle{\xi}\rangle^{-1}$. It
is not difficult to see from the proof of Corollary 10.1 that

$\displaystyle\big{|}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}}){\rm
op}({R_{\epsilon}})V^{\mu},{\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon}})V^{\mu}\big{)}\big{|}\leq
c(\epsilon){\mathcal{N}}_{-1/4}(V^{\mu}).$

Note that $H_{\epsilon}\in S(1,g)$ because
$\partial_{x}^{\alpha}\partial_{\xi}^{\beta}{\mathcal{A}}\in
S(\langle{\xi}\rangle^{1-|\beta|},g)$ for $|\alpha+\beta|=1$. Write
$\Phi_{\epsilon}=f_{\epsilon}\\#\Phi_{\epsilon 1}+r_{\epsilon}$ with
$r_{\epsilon}\in S(\langle{\xi}\rangle^{-1},g_{0})$. Noting
$\phi^{-n}\\#f_{\epsilon}-f_{\epsilon}\\#\phi^{-n}\in
S(\omega^{-1}\rho^{1/2}\langle{\xi}\rangle^{-1}\phi^{-n},g)\subset
S(\phi^{-n}\langle{\xi}\rangle^{-1/2},g)$ and
$f_{\epsilon}\\#\lambda_{j}-\lambda_{j}\\#f_{\epsilon}\in
S(\lambda_{j}\langle{\xi}\rangle^{-1/2},g)$ we see that

$\displaystyle\big{|}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}}){\rm
op}({iH_{\epsilon}})V^{\mu},{\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon}})V^{\mu}\big{)}$ $\displaystyle-\big{(}{\rm
op}({\varLambda}){\rm op}({\phi^{-n}}){\rm op}({f_{\epsilon}}){\rm
op}({iH_{\epsilon}})V^{\mu},{\rm op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu}\big{)}\big{|}$

is bounded by $c(\epsilon){\mathcal{N}}(\langle{D}\rangle^{-1/4}V^{\mu})$.
Here look at $iH_{\epsilon}$ more carefully;

$iH_{\epsilon}=\Big{(}\sum_{|\alpha+\beta|=1}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}({\tilde{a}}_{ij}[\xi])(\partial_{x}^{\beta}\partial_{\xi}^{\alpha}f_{\epsilon})\frac{1}{f_{\epsilon}}\Phi_{1\epsilon}\Big{)}=\big{(}h^{\epsilon}_{ij}\big{)}\frac{1}{f_{\epsilon}}\Phi_{1\epsilon}.$

Taking $h^{\epsilon}_{ij}\in S(1,g)$ and $f^{-1}_{\epsilon}\Phi_{\epsilon 1}$,
$\Phi_{\epsilon 1}\in S(1,g_{0})$ into account one can write
$f_{\epsilon}\\#(iH_{\epsilon})=\big{(}h^{\epsilon}_{ij}\big{)}\\#\Phi_{\epsilon
1}+R_{\epsilon}$ with $R_{\epsilon}\in S(\langle{\xi}\rangle^{-1/2},g)$ hence
denoting ${\tilde{H}_{\epsilon}}=\big{(}h^{\epsilon}_{ij}\big{)}$

$\displaystyle\big{|}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}}){\rm
op}({f_{\epsilon}}){\rm op}({iH_{\epsilon}})V^{\mu},{\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon 1}})V^{\mu})$ $\displaystyle-({\rm op}({\varLambda}){\rm
op}({\phi^{-n}}){\rm op}({{\tilde{H}_{\epsilon}}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu},{\rm op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu}\big{)}\big{|}$

is bounded by $c(\epsilon){\mathcal{N}}_{-1/4}(V^{\mu})$. From Corollary 5.3
we see that $h^{\epsilon}_{ij}\in{\mathcal{C}}(1)$ for $j\geq i$,
$h^{\epsilon}_{21},h^{\epsilon}_{32}\in{\mathcal{C}}(\sigma^{1/2})$ and
$h^{\epsilon}_{31}\in{\mathcal{C}}(\sigma)$ then in view of Lemma 6.5 one has
$\lambda_{i}\\#\big{(}\phi^{-n}\\#h^{\epsilon}_{ij}-h^{\epsilon}_{ij}\\#\phi^{-n}\big{)}\in
S(\kappa\lambda_{i}\langle{\xi}\rangle^{-1}\phi^{-n},g)$ for $j\geq i$ and
$\lambda_{i}\\#\big{(}\phi^{-n}\\#h^{\epsilon}_{ij}-h^{\epsilon}_{ij}\\#\phi^{-n}\big{)}\in
S(\kappa\lambda_{i}\lambda_{j}^{1/2}\langle{\xi}\rangle^{-1/2}\phi^{-n},g)$
for $i>j$. From this we see that

$\displaystyle\big{|}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}}){\rm
op}({{\tilde{H}}_{\epsilon}}){\rm op}({\Phi_{\epsilon 1}})V^{\mu},{\rm
op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon 1}})V^{\mu})$ $\displaystyle-({\rm
op}({\varLambda}){\rm op}({{\tilde{H}_{\epsilon}}}){\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon 1}})V^{\mu},{\rm op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu}\big{)}\big{|}$

is bounded by $c(\epsilon){\mathcal{N}}_{-1/4}(V^{\mu})$.

######  Lemma 12.2.

One can write

$h^{\epsilon}_{ij}=\sum_{|\alpha+\beta|=1}k^{\epsilon}_{ij\alpha\beta}\\#l_{ij\alpha\beta}+r^{\epsilon}_{ij}$

where $k^{\epsilon}_{ij\alpha\beta}\in S(1,g_{0})$ such that
$|k^{\epsilon}_{ij\alpha\beta}|\leq C\nu$ with some $C>0$ independent of $\nu$
and $\epsilon$ for any $1\leq i,j\leq 3$. As for $l_{ij\alpha\beta}$ and
$r^{\epsilon}_{ij}$ one has $l_{ij\alpha\beta}\in S(1,g)$ and
$r^{\epsilon}_{ij}\in S(\sigma^{-1/2}\langle{\xi}\rangle^{-1},g)$ for $j\geq
i$ and $l_{ij\alpha\beta}\in S(\sigma^{(i-3)/2}\sqrt{\lambda_{j}},g)$,
$r^{\epsilon}_{ij}\in
S(\sigma^{(i-3)/2}\sqrt{\lambda_{j}}\langle{\xi}\rangle^{-1/2},g)$ for $i>j$.

###### Proof.

Set
$k^{\epsilon}_{ij\alpha\beta}=\langle{\xi}\rangle^{|\alpha|}\partial_{x}^{\beta}\partial_{\xi}^{\alpha}f_{\epsilon}$
and
$l_{ij\alpha\beta}=\langle{\xi}\rangle^{-|\alpha|}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}({\tilde{a}}_{ij}[\xi])$
then the assertion for $k^{\epsilon}_{ij\alpha\beta}$ is clear from (12.2).
The assertions for $l_{ij\alpha\beta}$ follow from Corollary 5.3 and Lemmas
5.8, 7.7. Note that
$\partial_{x}^{\mu}\partial_{\xi}^{\nu}l_{ij\alpha\beta}\in
S(\sigma^{-1/2}\langle{\xi}\rangle^{-|\nu|},g)$, $|\mu+\nu|=1$ for $j\geq i$
and $\partial_{x}^{\mu}\partial_{\xi}^{\nu}l_{21\alpha\beta}$,
$\partial_{x}^{\mu}\partial_{\xi}^{\nu}l_{32\alpha\beta}\in
S(\langle{\xi}\rangle^{-|\nu|},g)$ and
$\partial_{x}^{\mu}\partial_{\xi}^{\nu}l_{31\alpha\beta}\in
S(\sigma^{1/2}\langle{\xi}\rangle^{-|\nu|},g)$ for $|\mu+\nu|=1$ which follows
from ${\tilde{a}}_{21}$, ${\tilde{a}}_{32}\in{\mathcal{C}}(\sigma)$ and
${\tilde{a}}_{31}\in{\mathcal{C}}(\sigma^{5/2})$. Then remarking that
$\sigma\geq\langle{\xi}\rangle_{\gamma}^{-1}$ and
$C\lambda_{1}\geq\sigma\langle{\xi}\rangle_{\gamma}^{-1}$ the assertions for
$r^{\epsilon}_{ij}$ are checked immediately. ∎

With $R^{\epsilon}=(r^{\epsilon}_{ij})$ and $W={\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon 1}})V^{\mu}$, recalling $\lambda_{1}\leq
C\sigma\lambda_{2}\leq C^{\prime}\sigma^{2}\lambda_{3}$, it is easy to see
$\big{|}({\rm op}({R^{\epsilon}})W,{\rm op}({\varLambda})W)\big{|}\leq
c(\epsilon)\|{\rm op}({\varLambda^{1/2}})\langle{D}\rangle^{-1/4}W\|^{2}$.
Turn to $|({\rm op}({h^{\epsilon}_{ij}})W_{j},{\rm op}({\lambda_{i}})W_{i})|$.
Thanks to Lemma 12.2 this is bounded by

$\displaystyle C\|{\rm op}({\lambda_{j}^{1/2}})W_{j}\|\|{\rm
op}({k^{\epsilon}_{ij\alpha\beta}}){\rm op}({\lambda_{i}^{1/2}})W_{i}\|$

with $C$ independent of $\epsilon$ because
$\lambda_{i}^{1/2}\\#l_{ij\alpha\beta}\in S(\lambda_{j}^{1/2},g)$ in view of
Lemma 12.2. On the other hand, taking Lemma 12.2 into account, it follows from
the sharp Gårding inequality (e.g. [8, Theorem 18.1.14]) that

$\displaystyle\|{\rm op}({k^{\epsilon}_{ij\alpha\beta}}){\rm
op}({\lambda_{i}^{1/2}})W_{i}\|\leq C\nu\|{\rm
op}({\lambda_{i}^{1/2}})W_{i}\|+C(\nu,\epsilon)\|{\rm
op}({\lambda_{i}^{1/2}})\langle{D}\rangle^{-1/2}W_{i}\|.$

Therefore applying the above obtained estimates one can find $C>0$ independent
of $\epsilon$ and $\nu$ such that

$\displaystyle\big{|}{\mathsf{Re}}\big{(}{\rm op}({\varLambda}){\rm
op}({{\tilde{H}_{\epsilon}}}){\rm op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu},{\rm op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon 1}})V^{\mu})\big{|}$
$\displaystyle\leq C\nu\|{\rm op}({\varLambda^{1/2}}){\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon 1}})V^{\mu}\|^{2}+C^{\prime}(\nu,\epsilon)\|{\rm
op}({\varLambda^{1/2}}){\rm
op}({\phi^{-n}})\langle{D}\rangle^{-1/4}V^{\mu}\|^{2}.$

Since it follows from the same reasoning that

$\displaystyle\big{|}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}}){\rm
op}({f_{\epsilon}^{-1}\Phi_{\epsilon}})V^{\mu},{\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon}})V^{\mu}\big{)}\big{|}$ $\displaystyle-\big{(}{\rm
op}({\varLambda}){\rm op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu},{\rm op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu}\big{)}\big{|}\leq c(\epsilon){\mathcal{N}}_{-1/4}(V^{\mu}).$

We conclude finally that $-{\mathsf{Im}}\big{(}{\rm op}({\varLambda}){\rm
op}({\phi^{-n}}){\tilde{L}}({\rm op}({\Phi_{\epsilon}})V^{\mu}),{\rm
op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon}})V^{\mu}\big{)}$ is bounded by

(12.5) $\begin{split}-(1-C\nu)\|{\rm op}({\varLambda^{1/2}}){\rm
op}({\phi^{-n}}){\rm op}({\Phi_{\epsilon
1}})V^{\mu}\|^{2}+c(\nu,\epsilon){\mathcal{N}}_{-1/4}(V^{\mu})\\\
+{\mathsf{Re}}\big{(}{\rm op}({\varLambda}){\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon}})G^{\mu},{\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon}})V^{\mu}\big{)}.\end{split}$

We fix $\nu_{0}$ such that $1-C\nu_{0}\geq 0$. Since
$|\partial_{\xi}^{\alpha}\langle{\mu\xi}\rangle^{-\bar{n}}|\leq
C_{\alpha}\langle{\mu\xi}\rangle^{-\bar{n}}\langle{\xi}\rangle^{-|\alpha|}$
with $C_{\alpha}$ independent of $\mu>0$ we see that

$\big{|}({\rm op}({\Lambda}){\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon}})R^{\mu}V^{\mu},{\rm op}({\phi^{-n}}){\rm
op}({\Phi_{\epsilon}})V^{\mu})\big{|}\leq C{\mathcal{E}}_{1}({\rm
op}({\Phi_{\epsilon}})V^{\mu}).$

Therefore $\big{|}({\rm op}({\Lambda}){\rm op}({w^{-n}}){\rm
op}({\Phi_{\epsilon}})G^{\mu},{\rm op}({w^{-n}}){\rm
op}({\Phi_{\epsilon}})V^{\mu})\big{|}$ is bounded by

$\epsilon_{1}{\mathcal{E}}_{1}({\rm
op}({\Phi_{\epsilon}})V^{\mu})+C_{\epsilon_{1}}t^{-2n+1}\|{\rm
op}({\Phi_{\epsilon}})F^{\mu}\|_{n}^{2}+C{\mathcal{E}}_{1}({\rm
op}({\Phi_{\epsilon}})V^{\mu})$

for any $\epsilon_{1}>0$. Note that
$D_{t}^{j}V^{\mu}\in{\mathcal{H}}_{0,2n+7/4-j}(0,\delta)$, $j=0,1,\ldots$
hence $D_{t}^{j}V(0)$ exists in $H^{n+3/4}$ which is $0$ for $j=0,1,\ldots,n$
thus $\lim_{t\to+0}t^{-n}\|V^{\mu}(t)\|_{n}=0$ for $\mu>0$. Applying (10.12)
to ${\rm op}({\Phi_{\epsilon}})V^{\mu}$ instead of $V$, choosing
$\epsilon_{1}<c_{1}$ and then letting $\mu\to 0$ one concludes the proof. ∎

Applying $\langle{D}\rangle^{s}$ to (12.4) and repeating similar arguments one
obtains

######  Proposition 12.1.

For any $s\in{\mathbb{R}}$, any $0<\nu\leq\nu_{0}$ and any $\epsilon>0$ one
can find $C>0$ such that for any $V$ with
$\int_{0}^{\delta}t^{-2n}{\mathcal{N}}_{s-1/4}(V)dt<+\infty$ and
${\tilde{L}}V\in{\mathcal{H}}_{-n+1/2,l}(0,\delta)$ with some $l$ we have

$\displaystyle{\mathcal{E}}_{1}(\langle{D}\rangle^{s}{\rm
op}({\Phi_{\epsilon}})V)+\int_{0}^{t}\tau^{-2n}{\mathcal{N}}_{s}({\rm
op}({\Phi_{\epsilon}})V)d\tau$ $\displaystyle\leq
C\big{(}\int_{0}^{t}\tau^{-2n+1}\|{\rm
op}({\Phi_{\epsilon}}){\tilde{L}}V\|_{n+s}^{2}d\tau+\int_{0}^{t}\tau^{-2n}{\mathcal{N}}_{s-1/4}(V)d\tau\big{)},\;\;0\leq
t\leq\delta.$

###  12.2 Wave front set propagates with finite speed

######  Lemma 12.3.

Assume $V\in{\mathcal{H}}_{-n-1/2,l_{1}+1}(0,\delta)$ and
${\tilde{L}}V\in{\mathcal{H}}_{-n+1/2,l_{2}}(0,\delta)$ and that ${\rm
op}({\Phi_{\epsilon_{0}}}){\tilde{L}}V\in{\mathcal{H}}_{-n+1/2,n+s_{0}}(0,\delta)$
with some $l_{1},l_{2},s_{0}\in{\mathbb{R}}$, $\epsilon_{0}>0$. Then for every
$\epsilon>\epsilon_{0}$ we have ${\rm
op}({\Phi_{\epsilon}})V\in{\mathcal{H}}_{-n-1/2,s}(0,\delta)$ for all $s\leq
s_{0}-5/4$. Moreover

$\displaystyle\int_{0}^{t}\tau^{-2n-1}\|{\rm
op}({\Phi_{\epsilon}})V(\tau)\|_{s}^{2}d\tau\leq
C\int_{0}^{t}\big{(}\tau^{-2n-1}\|V(\tau)\|_{l_{1}+1}^{2}+\tau^{-2n+1}\|{\tilde{L}}V(\tau)\|_{l_{2}}^{2}\big{)}d\tau$
$\displaystyle+C\int_{0}^{t}\tau^{-2n+1}\|{\rm
op}({\Phi_{\epsilon_{0}}}){\tilde{L}}V(\tau)\|_{n+s_{0}}^{2}d\tau,\quad
0<t\leq\delta.$

###### Proof.

We may assume $l_{1}\leq s_{0}$ otherwise nothing to be proved. Let $J$ be the
largest integer such that $l_{1}+J/4\leq s_{0}$. Take $\epsilon_{j}>0$ such
that $\epsilon_{0}<\epsilon_{1}<\cdots<\epsilon_{J}=\epsilon$. We write
$\Phi_{\epsilon_{j}}=\Phi_{j}$ and $f_{j}=f_{\epsilon_{j}}$ in this proof.
Inductively we show that

(12.6) $\begin{split}\int_{0}^{t}\tau^{-2n}{\mathcal{N}}_{l_{1}+j/4}({\rm
op}({\Phi_{j}})V)d\tau\leq
C\int_{0}^{t}\tau^{-2n-1}\|V(\tau)\|_{l_{1}+1}^{2}d\tau\\\
+C\int_{0}^{t}\tau^{-2n+1}\big{\\{}\|{\tilde{L}}V(\tau)\|_{l_{2}}^{2}+\|{\rm
op}({\Phi_{0}}){\tilde{L}}V(\tau)\|_{l_{1}+n+j/4}^{2}\big{\\}}d\tau.\end{split}$

Choose $\psi_{j}(x,\xi)\in S(1,g_{0})$ so that ${\rm
supp}\,{\psi_{j}}\subset\\{f_{j}<0\\}$ and
$\\{f_{j+1}<0\\}\subset\\{\psi_{j}=1\\}$. Noting that ${\rm
op}({\Phi_{j+1}}){\tilde{L}}\,{\rm op}({\psi_{j}})={\rm
op}({\Phi_{j+1}\\#\psi_{j}}){\tilde{L}}+{\rm op}({\Phi_{j+1}})[\tilde{L},{\rm
op}({\psi_{j}})]$ we apply Proposition 12.1 with $s=l_{1}+(j+1)/4$,
$\Phi=\Phi_{j+1}$ and $V={\rm op}({\psi_{j}})V$. Since
$\Phi_{j+1}\\#\psi_{j}-\Phi_{j+1}\in S^{-\infty}$ then $\|{\rm
op}({\Phi_{j+1}}){\tilde{L}}\,{\rm op}({\psi_{j}})V\|_{l_{1}+(j+1)/4+n}^{2}$
is bounded by $c\|{\rm
op}({\Phi_{j+1}}){\tilde{L}}V\|_{l_{1}+(j+1)/4+n}^{2}+C(j)\|V\|_{l_{1}+1}^{2}$
and hence by

(12.7) $C(j)\big{\\{}\|{\rm
op}({\Phi_{0}}){\tilde{L}}V\|_{l_{1}+(j+1)/4+n}^{2}+\big{\\{}\|{\tilde{L}}V\|_{l_{2}}^{2}+\|V\|_{l_{1}+1}^{2}\big{\\}}$

because $\Phi_{j+1}-k_{j}\\#\Phi_{0}\in S^{-\infty}$ with some $k_{j}\in
S(1,g_{0})$. Since $\psi_{j}-{\tilde{k}}_{j}\\#\Phi_{j}\in S^{-\infty}$ with
some ${\tilde{k}}_{j}\in S(1,g_{0})$ it follows that

${\mathcal{N}}_{l_{1}+j/4}({\rm op}({\Phi_{j+1}}){\rm op}({\psi_{j}})V)\leq
C{\mathcal{N}}_{l_{1}+j/4}({\rm op}({\Phi_{j}})V)+C\|V\|_{l_{1}+1}^{2}.$

Consider ${\mathcal{N}}_{l_{1}+(j+1)/4}({\rm op}({\Phi_{j+1}}){\rm
op}({\psi_{j}})V)$. Noting that $\Phi_{j+1}\\#\psi_{j}-\Phi_{j+1}\in
S^{-\infty}$ the same reasoning shows that

(12.8) ${\mathcal{N}}_{l_{1}+(j+1)/4}({\rm op}({\Phi_{j+1}})V)\leq
C{\mathcal{N}}_{l_{1}+(j+1)/4}({\rm op}({\Phi_{j+1}}){\rm
op}({\psi_{j}})V)+C\|V\|_{l_{1}+1}^{2}.$

Multiply (12.8) and (12.7) by $t^{-2n}$ and $t^{-2n+1}$ respectively and
integrate it from $0$ to $t$ we conclude from Proposition 12.1 that (12.6)
holds for $j+1$ and hence for $j=J$. Since $l_{1}+J/4\leq s_{0}$,
$l_{1}+J/4>s_{0}-1/4$ and $\|V\|_{s-1}/C\leq{\mathcal{N}}_{s}(V)$ the
assertion follows. ∎

Let $\Gamma_{i}$ ($i=1,2,3$) be open conic sets in
${\mathbb{R}}^{d}\times({\mathbb{R}}^{d}\setminus\\{0\\})$ with relatively
compact basis such that $\Gamma_{1}\Subset\Gamma_{2}\Subset\Gamma_{3}$. Take
$h_{i}(x,\xi)\in S(1,g_{0})$ with ${\rm supp}\,h_{1}\subset\Gamma_{1}$ and
${\rm supp}\,h_{2}\subset\Gamma_{3}\setminus\Gamma_{2}$. Consider a solution
$V$ with $V\in{\mathcal{H}}_{-n-1/2,l}(0,\delta)$ to the equation

${\tilde{L}}V={\rm op}({h_{1}})F,\quad
F\in{\mathcal{H}}_{-n+1/2,s}(0,\delta).$

######  Proposition 12.2.

Notations being as above. There exists
$\delta^{\prime}=\delta^{\prime}(\Gamma_{i})>0$ such that for any
$r\in{\mathbb{R}}$ there is $C>0$ such that

(12.9) $\begin{split}&\int_{0}^{t}\tau^{-2n-1}\|{\rm
op}({h_{2}})V(\tau)\|_{r}^{2}d\tau\\\ &\leq
C\int_{0}^{t}\big{\\{}\tau^{-2n+1}\|F(\tau)\|_{s}^{2}+\tau^{-2n-1}\|V(\tau)\|_{l}^{2}\big{\\}}d\tau,\quad
0<t\leq\delta^{\prime}.\end{split}$

###### Proof.

Let $f_{\epsilon}=t-\nu_{0}{\hat{\tau}}+\nu_{0}d_{\epsilon}(x,\xi;y,\eta)$
with a small ${\hat{\tau}}>0$. It is clear that there is ${\hat{\epsilon}}>0$
such that $\\{t\geq 0\\}\cap\\{f_{\hat{\epsilon}}\leq
0\\}\cap\big{(}{\mathbb{R}}\times{\rm supp}\,h_{1}\big{)}=\emptyset$ for any
$(y,\eta)\notin\Gamma_{2}$. Take
${\hat{\epsilon}}<{\tilde{\epsilon}}<{\hat{\tau}}$. It is also clear that one
can find a finite number of
$(y_{i},\eta_{i})\in\Gamma_{3}\setminus\Gamma_{2}$, $i=1,\ldots,M$ such that
with $\delta^{\prime}=\nu_{0}({\hat{\tau}}-{\tilde{\epsilon}})/2$

$\displaystyle\Gamma_{3}\setminus\Gamma_{2}\Subset\Big{(}\bigcup_{i=1}^{M}\\{f_{\tilde{\epsilon}}(\delta^{\prime},x,\xi;y_{i},\eta_{i})\leq
0\\}\Big{)},$ $\displaystyle\\{t\geq
0\\}\cap\\{f_{\tilde{\epsilon}}(t,x,\xi;y_{i},\eta_{i})\leq
0\\}\cap\big{(}{\mathbb{R}}\times{\rm supp}\,h_{1}\big{)}=\emptyset.$

Now $\Phi_{i\epsilon}$ is defined by (12.3) with
$f_{\epsilon}(t,x,\xi;y_{i},\eta_{i})$. Then since
$\sum\Phi_{i{\tilde{\epsilon}}}>0$ on $[0,\delta^{\prime}]\times{\rm
supp}\,h_{2}$ there is $k\in S(1,g_{0})$ such that
$h_{2}-k\sum\Phi_{i{\tilde{\epsilon}}}\in S^{-\infty}$. Noting that ${\rm
op}({\Phi_{i{\hat{\epsilon}}}}){\rm
op}({h_{1}})F\in{\mathcal{H}}_{-n+1/2,r}(0,\delta)$ for any $r\in{\mathbb{R}}$
we apply Lemma 12.3 with $\Phi_{\epsilon_{0}}=\Phi_{{\hat{\epsilon}}}$,
$\Phi_{\epsilon}=\Phi_{i{\tilde{\epsilon}}}$ and $s_{0}=r+5/4$ to obtain

$\displaystyle\int_{0}^{t}\tau^{-2n-1}\|{\rm
op}({\Phi_{i{\tilde{\epsilon}}}})V(\tau)\|_{r}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n-1}\|V(\tau)\|_{l}^{2}d\tau$
$\displaystyle+\int_{0}^{t}\tau^{-2n+1}\big{(}\|{\rm
op}({\Phi_{i{\hat{\epsilon}}}}){\rm
op}({h_{1}})F(\tau)\|_{2n+r+5/4}^{2}+\|F(\tau)\|_{s}\big{)}d\tau$

for $\|{\tilde{L}}V(\tau)\|_{s}\leq C\|F(\tau)\|_{s}$. Since
$\Phi_{i{\hat{\epsilon}}}\\#h_{1}\in S^{-\infty}$ summing up the above
estimates over $i=1,\ldots,M$ one concludes the desired assertion. ∎

######  Lemma 12.4.

The same assertion as Proposition 12.2 holds for $L$.

###### Proof.

Assume that $U\in{\mathcal{H}}_{-n-1/2,l}(0,\delta)$ satisfies $LU={\rm
op}({h_{1}})F$ where $F\in{\mathcal{H}}_{-n+1/2,s}(0,\delta)$. Choose
${\tilde{\Gamma}}_{i}$ such that
$\Gamma_{1}\Subset{\tilde{\Gamma}}_{1}\Subset{\tilde{\Gamma}}_{2}\Subset\Gamma_{2}\Subset\Gamma_{3}\Subset{\tilde{\Gamma}}_{3}$
and ${\tilde{h}}_{i}\in S(1,g_{0})$ such that ${\rm
supp}\,{\tilde{h}}_{1}\subset{\tilde{\Gamma}}_{1}$, ${\rm
supp}\,{\tilde{h}}_{2}\subset{\tilde{\Gamma}}_{3}\setminus{\tilde{\Gamma}}_{2}$
and ${\tilde{h}}_{i}=1$ on the support of $h_{i}$. Recall that $L\,{\rm
op}({T})={\rm op}({T})\,{\tilde{L}}$ hence ${\tilde{L}}V=(I+{\rm op}({K})){\rm
op}({T^{-1}}){\rm op}({h_{1}})F$ with $U={\rm op}({T})V$. Since there is
${\tilde{T}}\in S(1,g)$ such that
$(I+K)\\#T^{-1}\\#h_{1}-{\tilde{h}}_{1}{\tilde{T}}\in S^{-\infty}$ it follows
from Proposition 12.2 (or rather its proof) that (12.9) holds with
${\tilde{h}}_{2}$ in place of $h_{2}$. Similarly since there is
${\tilde{T}}\in S(1,g)$ such that $h_{2}\\#T-{\tilde{h}}_{2}{\tilde{T}}\in
S^{-\infty}$ repeating the same arguments we conclude the assertion. ∎

Returning to ${\hat{P}}$ we have

######  Proposition 12.3.

Notations being as above. Then there exists
$\delta^{\prime}=\delta^{\prime}(\Gamma_{i})>0$ such that for any $s$,
$r\in{\mathbb{R}}$ there is $C$ such that for any $u$ with
$D_{t}^{j}u\in{\mathcal{H}}_{-n-1/2,l+2-j}(0,\delta^{\prime})$, $j=0,1,2$ with
some $l$ satisfying ${\hat{P}}u={\rm op}({h_{1}})f$ where
$f\in{\mathcal{H}}_{-n+1/2,s}(0,\delta^{\prime})$ one has

$\displaystyle\sum_{j=0}^{2}\int_{0}^{t}\tau^{-2n-1}\|{\rm
op}({h_{2}})D_{t}^{j}u(\tau)\|_{r+2-j}^{2}d\tau$ $\displaystyle\leq
C\big{(}\int_{0}^{t}\tau^{-2n+1}\|f(\tau)\|_{s}^{2}d\tau+\sum_{j=0}^{2}\int_{0}^{t}\tau^{-2n-1}\|D_{t}^{j}u(\tau)\|_{l+2-j}^{2}d\tau\big{)},\;\;0<t\leq\delta^{\prime}.$

Thanks to Theorem 11.1 for any $f\in{\mathcal{H}}_{-n+1/2,n+s}(0,\delta)$
there is a unique solution $u\in{\mathcal{H}}_{-n-1/2,s+1}(0,\delta)$ to
${\hat{P}}u=f$ satisfying (11.3). Denote this map by

${\hat{G}}:{\mathcal{H}}_{-n+1/2,n+s}(0,\delta)\ni f\mapsto
u\in{\mathcal{H}}_{-n-1/2,s+1}(0,\delta).$

From Proposition 12.3 and Theorem 11.1 we conclude

######  Proposition 12.4.

Notations being as above and let $\Gamma_{i}$ $(i=1,2,3)$ be open conic sets
in ${\mathbb{R}}^{d}\times({\mathbb{R}}^{d}\setminus\\{0\\})$ with relatively
compact basis such that $\Gamma_{1}\Subset\Gamma_{2}\Subset\Gamma_{3}$ and
$h_{i}(x,\xi)\in S(1,g_{0})$ with ${\rm supp}\,h_{1}\subset\Gamma_{1}$ and
${\rm supp}\,h_{2}\subset\Gamma_{3}\setminus\Gamma_{2}$. Then there exists
$\delta^{\prime}=\delta^{\prime}(\Gamma_{i})>0$ such that for any $r$, $s$ one
can find $C>0$ such that

$\displaystyle\sum_{j=0}^{2}\int_{0}^{t}\tau^{-2n-1}\|{\rm
op}({h_{2}})D_{t}^{j}{\hat{G}}\,{\rm op}({h_{1}})f(\tau)\|_{r-j}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n+1}\|f(\tau)\|_{s}^{2}d\tau$

for $0<t\leq\delta^{\prime}$ and for any
$f\in{\mathcal{H}}_{-n+1/2,s}(0,\delta^{\prime})$.

Denote by ${\mathcal{H}}_{n,s}^{*}(0,\delta]$ the set of all $f$ with
$t^{n}\langle{D}\rangle^{s}f(t,\cdot)\in
L^{2}((0,\infty)\times{\mathbb{R}}^{d})$ such that $f=0$ for $t\geq\delta$.
Thanks to Theorem 11.2 for any $f\in{\mathcal{H}}^{*}_{n+1/2,n+s}(0,\delta]$
there is a unique solution $u\in{\mathcal{H}}_{n-1/2,s+1}(0,\delta)$ to
${\hat{P}^{*}}u=f$ with $D_{t}^{j}u(\delta)=0$, $j=0,1,2$ satisfying (11.4)
hence $u\in{\mathcal{H}}^{*}_{n-1/2,s+1}(0,\delta]$. Denote this map by

${\hat{G}^{*}}:{\mathcal{H}}^{*}_{n+1/2,n+s}(0,\delta]\ni f\mapsto
u\in{\mathcal{H}}^{*}_{n-1/2,s+1}(0,\delta].$

Repeating similar arguments proving Proposition 12.4 one obtains

######  Proposition 12.5.

Notations being as in Proposition 12.4. There exists
$\delta^{\prime}=\delta^{\prime}(\Gamma_{i})>0$ such that for any $r$, $s$ one
can find $C>0$ such that

$\displaystyle\sum_{j=0}^{2}\int_{t}^{\delta^{\prime}}\tau^{2n-1}\|{\rm
op}({h_{2}})D_{t}^{j}{\hat{G}^{*}}\,{\rm
op}({h_{1}})f(\tau)\|_{r-j}^{2}d\tau\leq
C\int_{t}^{\delta^{\prime}}\tau^{2n+1}\|f(\tau)\|_{s}^{2}d\tau$

for $0<t\leq\delta^{\prime}$ and
$f\in{\mathcal{H}}^{*}_{n+1/2,s}(0,\delta^{\prime}]$.

######  Remark 12.1.

As already remarked in Remark 10.1, it is clear from the proof that Theorems
11.1, 11.2 and Propositions 12.4 and 12.5 hold for any $n^{\prime}$ greater
than $n$.

###  12.3 Remarks on propagation of singularities

In this section we give a more precise picture of the propagation of wave
front set $WF(u)$ of $u$ applying the same arguments in Sections 12.1 and
12.2. Denote $X=[0,\delta)\times U$ and $\overset{\circ}{X}=(0,\delta)\times
U$ and by $\Sigma$, $\Sigma_{1}$ the set of characteristics and simple
characteristics of $p$ respectively. As explained in the Introduction every
characteristic in $T^{*}{\overset{\circ}{X}}\setminus 0$ is at most double and
double characteristic is effectively hyperbolic. Let ${\mathcal{U}}$ be an
open conical set in $\subset T^{*}\overset{\circ}{X}\setminus 0$. According to
[21] we say a continuous curve $\gamma(t):(0,a]\to{\mathcal{U}}\cap\Sigma$,
parametrized by $t$, a generalized bicharacteristic if
$\gamma^{-1}(\Sigma\setminus\Sigma_{1})=\\{t_{i}\\}$ is discrete in $(0,a]$
and $\gamma$ is a parametrized smooth bicharacteristic of $p$ on
$(t_{i},t_{i+1})$. In the present case a generalized bicharacteristic is
described rather easily. Let $\rho\in{\mathcal{U}}$ be a double
characteristic. Then from [27] one can find a conical open set
${\mathcal{V}}\ni\rho$ and a smooth function $\psi(x,\xi)$, homogeneous of
degree $0$ in $\xi$, such that the double characteristics of $p$ in
${\mathcal{V}}$ are contained in $\\{t=\psi\\}$ and there are exactly two
smooth bicharacteristics enter $\rho$ transversally to $t=\psi$ in the
direction of decreasing $t$ (also exactly two in the direction of increasing
$t$) (e.g.[18]). Therefore $\gamma$ consists of segments of smooth
bicharacteristic of $p$, the only end points of theses segments lying in
$\Sigma$ and a transition to one of two segments takes place there. Let
$\gamma_{i}$, $i=1,2$ be two bicharacteristic segments entering to $\rho$ in
the direction of decreasing $t$ (or two segments in the direction of
increasing $t$) then from [24, Theorem 2.1], [21, Theorem 1.7] we have
$\rho\not\in WF(u)$ if $\gamma_{i}\not\in WF(u)$, $i=1,2$ and $\rho\not\in
WF(Pu)$.

######  Lemma 12.5.

If $\gamma(t)$ is a generalized bicharacteristic then $\lim_{t\to+0}\gamma(t)$
exists.

###### Proof.

Write $p=\prod_{j=1}^{m}(\tau-\tau_{j}(t,x,\xi))$ then thanks to [4] for any
$0<\delta_{1}<\delta$ and $U^{\prime}\Subset U$ there is $L>0$ such that

$|\nabla_{x}\tau_{j}(t,x,\xi)|/|\xi|,\;|\nabla_{\xi}\tau_{j}(t,x,\xi)|\leq
L,\;(t,x,\xi)\in[0,\delta_{1}]\times U^{\prime}\times{\mathbb{R}}^{d},\;1\leq
j\leq m.$

In each $(t_{i},t_{i+1})$ it is clear that $\gamma(t)=(t,x(t),\tau(t),\xi(t))$
is a bicharacteristic of some $\tau-\tau_{j}(t,x,\xi)$ and hence
$dx/dt=-\nabla_{\xi}\tau_{j}(t,x,\xi)$ and
$d\xi/dt=\nabla_{x}\tau_{j}(t,x,\xi)$. This shows that $x(t)$ and $\xi(t)$ are
uniformly Lipschitz continuous in $(0,a]$ with the Lipschitz constant $L$,
though $\tau(t)$ is not Lipschitz continuous in $(0,a]$ in general. Then
$\lim_{t\to+0}x(t)$ and $\lim_{t\to+0}\xi(t)$ exist. Since
$\tau(t)=\tau_{k}(t,x(t),\xi(t))$ for some $k$ and $\tau_{k}(t,x,\xi)$ are
continuous in $X\times{\mathbb{R}}^{d}$ then $\lim_{t\to+0}\tau(t)$ also
exists. ∎

Denote

$K^{-}(\rho)=\bigcup_{\gamma}\gamma(t)$

where $\gamma$ varies over all generalized bicharacteristics such that
$\gamma(a)=\rho$, extended to $t=0$ according to Lemma 12.5. Note that for any
$\varepsilon>0$ the set $K^{-}(\rho)\cap\\{t=\varepsilon\\}$ consists of
finite number of characteristic points. Thanks to the propagation results near
double effectively hyperbolic characteristics mentioned above, if

(12.10) $WF(u)\cap K^{-1}(\rho)\cap\\{t=\varepsilon\\}=\emptyset$

with some $\varepsilon>0$ then we have $\rho\not\in WF(u)$.

######  Theorem 12.1.

Notations being as above and let
$K_{0}(\rho)=\pi\big{(}K^{-}(\rho)\cap\\{t=0\\}\big{)}$ where
$\pi:(t,x,\tau,\xi)\to(x,\xi)$ is the projection. Let $Pu\in C^{\infty}(X)$
and $D_{t}^{j}u(0,\cdot)=u_{j}$. If
$K_{0}(\rho)\cap(\cup_{j=0}^{m}WF(u_{j}))=\emptyset$ then $\rho\not\in WF(u)$.

We give a sketch of the proof. Since characteristics are at most triple we may
assume that $m=3$. It suffices to show that (12.10) holds with some
$\varepsilon>0$. In (12.1) we take $T=2\nu$. Since $K_{0}(\rho)$ is compact
one can find $0<\nu<\nu_{0}$ and finitely many $(y_{i},\eta_{i})$,
$i=1,\ldots,k$ such that
$K_{0}(\rho)\subset\cup_{i=1}^{k}\\{f_{0}(0,x,\xi,y_{i},\eta_{i})<0\\}$ and
$\\{f_{0}(0,x,\xi,y_{i},\eta_{i})\leq
0\\}\cap\big{(}\cup_{j=0}^{2}WF(u_{j})\big{)}=\emptyset$. Let
$\Phi_{i,\epsilon}$ be the symbol defined by (12.3) with
$f_{\epsilon}(t,x,\xi,y_{i},\eta_{i})$ then we have

(12.11) ${\rm op}({\Phi_{i,\epsilon}(0)})u_{j}\in H^{s},\quad 0\leq j\leq
2,\;\;1\leq i\leq k,\;\;\forall s\in{\mathbb{R}}$

for enough small $\epsilon>0$. Using the same reduction as in the proof of
Theorem 1.2 it suffices to study $Pu=f$ where
$D_{t}^{j}u\in{\mathcal{H}}_{-n-1/2,l_{1}-j}(0,\delta)$, $0\leq j\leq 2$ and
$f\in{\mathcal{H}}_{-n-1/2,l_{2}}(0,\delta)$ with some $l_{j}\in{\mathbb{R}}$
such that

${\rm op}({\Phi_{i,\epsilon}})f\in{\mathcal{H}}_{-n-1/2,s}(0,\delta),\quad
1\leq i\leq k,\;\;\forall s\in{\mathbb{R}}$

for enough small $\epsilon>0$ which follows from (12.11). Repeating the same
arguments as in Sections 12.1 and 12.2 we conclude that ${\rm
op}({\Phi_{i,0}})D_{t}^{j}u\in L^{2}((0,\delta^{\prime}),H^{s})$ for any
$1\leq i\leq k$, $0\leq j\leq 2$ and any $s\in{\mathbb{R}}$ with some
$\delta^{\prime}>0$. Finally, using the equation we have

${\rm op}({\Phi_{i,0}})D_{t}^{j}u\in L^{2}((0,\delta^{\prime}),H^{s}),\quad
1\leq i\leq k,\;\;\forall s\in{\mathbb{R}},\;\;\forall j\in{\mathbb{N}}.$

Since $\Phi_{i,0}(t,y_{i},\eta_{i})<0$ for $t<\nu$ this proves (12.10) with
some $\varepsilon>0$. ∎

##  13 Proof of Theorem 1.1

Applying Proposition 12.4 we prove Theorem 1.1 following [23], [27].

###  13.1 Solution operator with finite speed of propagation

Consider

(13.1)
$P=D_{t}^{m}+\sum_{j=1}^{m}a_{j}(t,x,D)\langle{D}\rangle^{j}D_{t}^{m-j}$

which is a differential operator in $t$ with coefficients $a_{j}\in S^{0}$. We
say that $G$ is a solution operator for $P$ with finite propagation speed of
wave front set (which we abbreviate to “solution operator with finite speed of
propagation” from now on) with loss of $(n,l)$ derivatives if $G$ satisfies
the following conditions:

  1. (i)

There exists $\delta>0$ such that for any $s\in{\mathbb{R}}$ there is $C>0$
such that for $f\in{\mathcal{H}}_{-n+1/2,s}(0,\delta)$ we have $PGf=f$ and

$\displaystyle\sum_{j=0}^{m-1}\int_{0}^{t}{\tau}^{-2n-1}\|D_{t}^{j}Gf(\tau)\|_{-l+s+m-j}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n+1}\|f(\tau)\|_{s}^{2}d\tau.$

  2. (ii)

For any $h_{j}(x,\xi)\in S(1,g_{0})$, $j=1,2$ with ${\rm
supp}\,h_{2}\Subset({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})\setminus{\rm
supp}\,h_{1}$ there exists $\delta^{\prime}>0$ such that for any $r$,
$s\in{\mathbb{R}}$ there is $C>0$ such that

$\displaystyle\sum_{j=0}^{m-1}\int_{0}^{t}\tau^{-2n-1}\|{\rm
op}({h_{2}})D_{t}^{j}G\,{\rm op}({h_{1}})f(\tau)\|_{r-j}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n+1}\|f(\tau)\|_{s}^{2}d\tau$

for $f\in{\mathcal{H}}_{-n+1/2,s}(0,\delta^{\prime})$ and
$0<t\leq\delta^{\prime}$.

Let $P_{1}$ and $P_{2}$ be two operators of the form (13.1). We say

$P_{1}\equiv P_{2}\quad\text{at}\;\;({\hat{x}},{\hat{\xi}})$

if there exist $\delta^{\prime}>0$ and a conic neighborhood $W$ of
$({\hat{x}},{\hat{\xi}})$ such that

(13.2)
$P_{1}-P_{2}=\sum_{j=1}^{m}R_{j}(t,x,D)\langle{D}\rangle^{j}D_{t}^{m-j}$

with $R_{j}\in S^{0}$ which are in $S^{-\infty}(W)$ uniformly in $0\leq
t\leq\delta^{\prime}$.

######  Theorem 13.1.

Assume that for any $({\hat{x}},\eta)$, $|\eta|=1$ one can find $P_{\eta}$ of
the form (13.1) for which there is a solution operator with finite speed of
propagation with loss of $(n,\ell(\eta))$ derivatives such that $P\equiv
P_{\eta}$ at $({\hat{x}},\eta)$. Then there exist $\delta>0$,
$\ell=\sup\ell(\eta)$ and a neighborhood $U$ of ${\hat{x}}$ such that for
every $f\in{\mathcal{H}}_{-n+1/2,s+\ell}(0,\delta)$ there exists $u$ with
$D_{t}^{j}u\in{\mathcal{H}}_{-n,s+m-j}(0,\delta)$, $0\leq j\leq m-1$,
satisfying $Pu=f$ in $(0,\delta)\times U$.

###### Proof.

By assumption $P_{\eta}$ has a solution operator $G_{\eta}$ with finite speed
of propagation with loss of $(n,\ell(\eta))$ derivatives. There are finite
open conic neighborhood $W_{i}$ of $({\hat{x}},\eta_{i})$ such that
$\cup_{i}W_{i}\supset\Omega\times({\mathbb{R}}^{d}\setminus\\{0\\})$, where
$\Omega$ is a neighborhood of ${\hat{x}}$, and $P\equiv P_{\eta_{i}}$ at
$({\hat{x}},\eta)$ with $W=W_{i}$ in (13.2). Now take another open conic
covering $\\{V_{i}\\}$ of $\Omega\times({\mathbb{R}}^{d}\setminus\\{0\\})$
with $V_{i}\Subset W_{i}$, and a partition of unity $\\{\alpha_{i}(x,\xi)\\}$
subordinate to $\\{V_{i}\\}$ so that $\sum_{i}\alpha_{i}(x,\xi)=\alpha(x)$
where $\alpha(x)=1$ in a neighborhood of ${\hat{x}}$. Define

$G=\sum_{i}G_{\eta_{i}}\alpha_{i}$

then denoting $P-P_{\eta_{i}}=R_{i}$ we have

$PGf=\sum_{i}PG_{\eta_{i}}\alpha_{i}f=\sum_{i}P_{\eta_{i}}G_{i}\alpha_{i}f+\sum_{i}R_{i}G_{i}\alpha_{i}f=\alpha(x)f-Rf$

where $R=\sum_{i}R_{i}G_{\eta_{i}}\alpha_{i}$. Then

$\int_{0}^{t}\tau^{-2n-1}\|Rf(\tau)\|_{s+\ell}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n+1}\|f(\tau)\|_{s+\ell}^{2}d\tau$

for $0\leq t\leq\delta^{\prime\prime}$ with some $\delta^{\prime\prime}>0$ by
the condition (ii) where $\ell=\max_{i}\ell(\eta_{i})$. Choosing
$0<\delta_{1}\leq\delta^{\prime\prime}$ such that $\delta_{1}^{2}C\leq 1/2$
one has

$\int_{0}^{t}\tau^{-2n+1}\|Rf(\tau)\|_{s+\ell}^{2}d\tau\leq\frac{1}{2}\int_{0}^{t}\tau^{-2n+1}\|f(\tau)\|_{s+\ell}^{2}d\tau,\quad
0<t\leq\delta_{1}$

for $f\in{\mathcal{H}}_{-n+1/2,s+\ell}(0,\delta)$. With
$S=\sum_{k=0}^{\infty}R^{k}$ one has
$Sf\in{\mathcal{H}}_{-n+1/2,s+\ell}(0,\delta_{1})$ and
$\int_{0}^{t}\tau^{-2n+1}\|Sf(\tau)\|_{s+\ell}^{2}d\tau\leq
2\int_{0}^{t}\tau^{-2n+1}\|f(\tau)\|_{s+\ell}^{2}d\tau$. Let $\gamma(x)\in
C_{0}^{\infty}({\mathbb{R}}^{d})$ be equal to $1$ near ${\hat{x}}$ such that
${\rm supp}\,\gamma\Subset\\{\alpha=1\\}$. Since
$\gamma(\alpha-R)S=\gamma(I-R)S=\gamma$ it follows that
$\gamma(x)PGSf=\gamma(x)f$, that is

$P\big{(}GSf\big{)}=f\quad\text{on}\;\;\\{\gamma(x)=1\\}.$

With $u=GSf$ one has

$\displaystyle\sum_{j=0}^{m-1}\int_{0}^{t}\tau^{-2n-1}\|D_{t}^{j}u(\tau)\|_{s+m-j}^{2}d\tau\leq
C\int_{0}^{t}\tau^{-2n+1}\|Sf(\tau)\|_{s+\ell}^{2}d\tau$

which proves the assertion. ∎

We define a solution operator with finite speed of propagation for $P^{*}$
with obvious modifications.

######  Theorem 13.2.

Assume that for every $({\hat{x}},\eta)$, $|\eta|=1$ one can find
$P^{*}_{\eta}$ of the form (13.1) such that $P^{*}\equiv P^{*}_{\eta}$ at
$({\hat{x}},\eta)$ for which solution operator with finite speed of
propagation with loss of $(n,\ell(\eta))$ derivatives exists. Then there exist
$\delta>0$, $\ell=\sup\ell(\eta)$ and a neighborhood $U$ of ${\hat{x}}$ such
that for every $f\in{\mathcal{H}}^{*}_{n+1/2,s+\ell}(0,\delta]$ there exists
$u$ with $D_{t}^{j}u\in{\mathcal{H}}^{*}_{n-1/2,s+m-j}(0,\delta]$, $0\leq
j\leq m-1$, satisfying $P^{*}u=f$ in $(0,\delta)\times U$.

###  13.2 Local existence and uniqueness

First consider a third order operator $P$ of the form (2.1). To reduce $P$ to
the case $a_{1}(t,x,D)=0$ we apply a Fourier integral operator, which is
actually the solution operator $S(t^{\prime},t)$ of the Cauchy problem

$D_{t}u+a_{1}(t,x,D)u=0,\quad u(t^{\prime},x)=\phi(x)$

such that $S(t^{\prime},t)\phi=u(t)$ then it is clear that
$S(t,0)(D_{t}+a_{1})S(0,t)=D_{t}$. Then ${\tilde{P}}=S(t,0)PS(0,t)$ has the
form (2.1) with $a_{1}=0$ (e.g. [5], [33]). Assume that ${\tilde{P}}$ has a
solution operator with finite speed of propagation ${\tilde{G}}$ with loss of
$(n,\ell)$ derivatives. Then one can show that $G=S(0,t){\tilde{G}}S(t,0)$ is
a solution operator for $P$ with finite speed of propagation with loss of
$(n,\ell)$ derivatives.

Let $|\eta|=1$ be given. Assume that $p$ has a triple characteristic root
${\bar{\tau}}$ at $(0,0,\eta)$ and $(0,0,{\bar{\tau}},\eta)$ is effectively
hyperbolic. Theorem 11.1 and Proposition 12.4 imply that ${\hat{P}}$, which
coincides with the original $P$ in $W_{M}$, given by (4.3), has a solution
operator with finite speed of propagation with loss of $(n,n+2)$ derivatives.

Next assume that $p$ has a double characteristic root ${\bar{\tau}}$ at
$(0,0,\eta)$ such that $(0,0,{\bar{\tau}},\eta)$ is effectively hyperbolic
characteristic if it is a singular point. Note that one can write

$p(t,x,\tau,\xi)=\big{(}\tau+b(t,x,\xi)\big{)}\big{(}\tau^{2}+a_{1}(t,x,\xi)\tau+a_{2}(t,x,\xi)\big{)}=p_{1}p_{2}$

in a conic neighborhood of $(0,0,\eta)$ where
$p_{1}(0,0,{\bar{\tau}},\eta)\neq 0$. There exist ${\hat{P}}_{i}$ such that
$P\equiv{\hat{P}}_{1}\cdot{\hat{P}}_{2}$ at $(0,\eta)$ where the principal
symbol of ${\hat{P}}_{i}$ coincides with $p_{i}$ in a conic neighborhood of
$(0,0,\eta)$. If ${\hat{P}}_{i}$ has a solution operator with finite speed of
propagation $G_{i}$ with loss of $(n,\ell_{i})$ derivatives then one can see
that $G_{2}G_{1}$ is a solution operator with finite speed of propagation for
${\hat{P}}_{1}\cdot{\hat{P}}_{2}$ with loss of $(n,\ell_{1}+\ell_{2})$
derivatives. Consider the case that $(0,0,{\bar{\tau}},\eta)$ is a singular
point. Then $F_{p}(0,0,{\bar{\tau}},\eta)=c\,F_{p_{2}}(0,0,{\bar{\tau}},\eta)$
with some $c\neq 0$ and $(0,0,{\bar{\tau}},\eta)$ is effectively hyperbolic
characteristic of $p_{2}$. Write $p_{2}$ as

(13.3) $p_{2}(t,x,\tau,\xi)=\tau^{2}-a(t,x,\xi)|\xi|^{2}$

such that ${\bar{\tau}}=0$ is a double characteristic and $(0,0,0,\eta)$ is a
singular point. To apply earlier results [26, 27] on operators with double
effectively hyperbolic characteristics we need some modifications because
$a(t,x,\xi)$ is assumed to be nonnegative only in $t\geq 0$ side in the
present case. One can improve [27, Lemma 1.2.2] to

######  Proposition 13.1.

Assume that $a(t,x,\xi)$ is smooth in some conic neighborhood of $(0,0,\eta)$,
homogeneous of degree $0$ in $\xi$, and nonnegative in $t\geq 0$ and
$(0,0,0,\eta)$ is a effectively hyperbolic singular point of $p_{2}=0$. Then
there exist a smooth function $\psi(x,\xi)$ in a conic neighborhood $V$ of
$(0,\eta)$ and constants $0<\kappa<1$, $c>0$ such that

(13.4) $\\{\psi,a\\}^{2}\leq 4\,\kappa\,a,\quad a(t,x,\xi)\geq
c\,\min{\big{\\{}t^{2},(t-\psi(x,\xi))^{2}\big{\\}}}$

for $(x,\xi)\in V$, $t\geq 0$ where $\psi(x,\xi)$ satisfies
$\big{|}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\psi\big{|}\precsim\langle{\xi}\rangle^{-|\beta|}$.

Indeed that the same time function given in [25] under the assumption
$a(t,x,\xi)\geq 0$ in a full neighborhood in $t$, denoted by $Y(t,x,\xi)$
there, satisfies (13.4) can be proved applying [25, Theorem 1.1]. Then
repeating similar arguments as in [26, 27] we conclude that there is a
solution operator with finite speed of propagation for ${\hat{P}}_{2}$. Since
${\hat{P}}_{1}$ is a first order operator with real principal symbol $p_{1}$
it is easy to see that ${\hat{P}}_{1}$ has a solution operator with finite
speed of propagation. Therefore $P$ has a solution operator with finite speed
of propagation. Turn to the case that $(0,0,0,\eta)$ is not a singular point.
It is easy to see that $(0,0,0,\eta)$ is not a singular point implies
$\partial_{t}a(0,0,\eta)>0$, which is the case that ${\hat{P}}_{2}$ is a
hyperbolic operator of principal type and some detailed discussion is found in
[8, Chapter 23.4]. It is easily proved that ${\hat{P}}_{2}$ has a solution
operator with finite speed of propagation, because it suffices to employ the
weight $t^{-n}$ ($\phi^{-n}$ is now absent) in order to obtain weighted energy
estimates.

Turn to the general case. Let $|\eta|=1$ be arbitrarily fixed. Write
$p(0,0,\tau,{\eta})=\prod_{j=1}^{r}\big{(}\tau-\tau_{j})^{m_{j}}$ where $\sum
m_{j}=m$ and $\tau_{j}$ are real and distinct from each other, where
$m_{j}\leq 3$ which follows from the assumption. There exist $T>0$ and a conic
neighborhood $U$ of $(0,\eta)$ such that one can write

$\displaystyle p(t,x,\tau,\xi)=\prod_{j=1}^{r}p^{(j)}(t,x,\tau,\xi),$
$\displaystyle
p^{(j)}(t,x,\tau,\xi)=\tau^{m_{j}}+a_{j,1}(t,x,\xi)\tau^{m_{j}-1}+\cdots+a_{j,m_{j}}(t,x,\xi)$

for $(t,x,\xi)\in(-T,T)\times U$ where $a_{j,k}(t,x,\xi)$ are real valued,
homogeneous of degree $k$ in $\xi$ and
$p^{(j)}(0,0,\tau,\eta)=(\tau-\tau_{j})^{m_{j}}$ and $p^{(j)}(t,x,\tau,\xi)=0$
has only real roots in $\tau$ for $(t,x,\xi)\in[0,T)\times U$. If
$(0,0,\tau_{j},\eta)$ is a singular point of $p$, and necessarily $m_{j}\geq
2$, then $(0,0,\tau_{j},\eta)$ is a singular point of $p^{(j)}$ and it is easy
to see $F_{p}(0,0,\tau_{j},\eta)=c_{j}F_{p^{(j)}}(0,0,\tau_{j},\eta)$ with
some $c_{j}\neq 0$ and hence $F_{p^{(j)}}(0,0,\tau_{j},\eta)$ has non-zero
real eigenvalues if $F_{p}(0,0,\tau_{j},\eta)$ does and vice versa. It is well
known that one can find $P^{(j)}$ such that

$P\equiv P^{(1)}P^{(2)}\cdots P^{(r)}\quad\text{at}\;\;(0,\eta)$

where $P^{(j)}$ are operators of the form (13.1) with $m=m_{j}$ whose
principal symbol coincides with $p^{(j)}$ in some conic neighborhood of
$(0,0,\eta)$. Since each $P^{(j)}$ has a solution operator with finite speed
of propagation with loss of $(n_{j},n_{j}+2)$ derivatives thanks to Theorem
11.1 and Proposition 12.4 hence so does $P$ with loss of $(n,r(n+2))$
derivatives with $n=\max_{j}n_{j}$ noting Remark 12.1. Therefore Theorem 1.1
results from Theorem 13.1.

Repeating a parallel arguments to the existence proof for $P$ above we obtain

######  Theorem 13.3.

Under the same assumption as in Theorem 1.1 there exist $\delta>0$, a
neighborhood $U$ of the origin and $n>0$, $\ell>0$ such that for any
$s\in{\mathbb{R}}$ and any $f\in{\mathcal{H}}^{*}_{n+1/2,s}(0,\delta]$ there
exists $u$ with
$D_{t}^{j}u\in{\mathcal{H}}^{*}_{n-1/2,-\ell+s+m-j}(0,\delta]$, $0\leq j\leq
m-1$, satisfying $P^{*}u=f$ in $(0,\delta)\times U$.

Now we prove a local uniqueness result for the Cauchy problem for $P$ applying
Theorem 13.3. From the assumption one can find a neighborhood $W$ of the
origin of ${\mathbb{R}}^{d}$ and $T>0$ such that every multiple characteristic
of $p$ on $(t,x,\xi)\in(0,T)\times W$ is at most double and double
characteristic is effectively hyperbolic. Thanks to [17, Main Theorem] there
exists ${\hat{c}}>0$ such that for any solution $v$ to $P^{*}v=f$ vanishing in
$t\geq\delta^{\prime}$ with $f\in
C_{0}^{\infty}((0,\delta^{\prime})\times\\{|x|<\varepsilon\\})$
($\delta^{\prime}\leq T$) one has

${\rm
supp}_{x}v(t,\cdot)\subset\\{|x|\leq\varepsilon+{\hat{c}}\,\delta^{\prime}\\},\quad
0<t\leq\delta^{\prime}.$

Now assume that $u$ satisfies $Pu=0$ in $(0,\delta)\times U$ and
$\partial_{t}^{k}u(0,x)=0$ for all $k$. Choose $\varepsilon>0$ and
$\delta^{\prime}>0$ such that
$\\{|x|\leq\varepsilon+{\hat{c}}\,\delta^{\prime}\\}\subset U$,
$\delta^{\prime}\leq\delta$. Then we see

$\displaystyle
0=\int_{0}^{\delta^{\prime}}\big{(}Pu,v\big{)}dt=\int_{0}^{\delta^{\prime}}\big{(}u,P^{*}v\big{)}dt=\int_{0}^{\delta^{\prime}}\big{(}u,f\big{)}dt.$

Since $f\in C_{0}^{\infty}((0,\delta^{\prime})\times\\{|x|<\varepsilon\\})$ is
arbitrary, we conclude that

$u(t,x)=0,\quad(t,x)\in(0,\delta^{\prime})\times\\{|x|\leq\varepsilon\\}.$

######  Theorem 13.4.

Assume (1.2) and that every singular point $(0,0,\tau,\xi)$, $|(\tau,\xi)|\neq
0$ of $p=0$ is effectively hyperbolic. If $u(t,x)\in
C^{\infty}([0,\delta)\times U)$ satisfies $Pu=0$ in $[0,\delta)\times U$ and
$\partial_{t}^{k}u(0,x)=0$ for all $k$ then $u=0$ in a neighborhood of
$(0,0)$.

Acknowledgment. This work was supported by JSPS KAKENHI Grant Number
JP20K03679.

## References

  * [1] R. Beals: Characterization of pseudodifferential operators and applications, Duke Math. J., 44 (1977), 45-57.
  * [2] E.Bernardi, A.Bove, V.Petkov: Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity, Journal Hyper. Differ. Equ., 12 (2015), 535-579.
  * [3] E.Bernardi, T.Nishitani: Counterexamples to $C^{\infty}$ well posedness for some hyperbolic operators with triple characteristics, Proc. Japan Acad., 91, Ser. A (2015), 19-24.
  * [4] M.D.Bronshtein: Smoothness of roots of polynomials depending on parameters, Sib. Mat. Zh., 20 (1979), 493-501; English translation: Sib. Math. J., 20 (1980), 347-352.
  * [5] J.V.Egorov: Canonical transformations and pseudodifferential operators, Trudy Moskov Mat. Obsc., 24 (1971), 3-28; English translation: Trans. Moscow Math. Soc. 24 (1971), 3-28.
  * [6] L.Hörmander: The Cauchy problem for differential equations with double characteristics, J. Anal. Math., 32 (1977), 118-196.
  * [7] L.Hörmander: The Analysis of Linear Partial Differential Operators, I, Springer, Berlin, 1990.
  * [8] L.Hörmander: The Analysis of Linear Partial Differential Operators, III, Springer, Berlin, 1985.
  * [9] V.Ivrii and V.Petkov: Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well posed, Uspekhi Mat. Nauk, 29 (1974), 3-70, English translation: Russ. Math. Surv., 29 (1974), 1-70.
  * [10] V.Ivrii: Sufficient conditions for regular and completely regular hyperbolicity, Tr. Mosk. Mat. Obs., 33 (1975), 3-65 (in Russian), English translation: Trans. Mosc. Math. Soc., 33 (1978), 1-65.
  * [11] V.Ivrii: The well-posed Cauchy problem for non-strictly hyperbolic operators, III: The energy integral, Tr. Mosk. Mat. Obs., 34 (1977), 151-170 (in Russian), English translation: Trans. Mosc. Math. Soc., 34 (1978), 149-168.
  * [12] V.Ivrii: Linear Hyperbolic Equations, In: Encyclopaedia of Mathematical Science, Partial Differential Equations IV, Yu.V.Egorov, M.A.Shubin (Eds.) pp. 149-235 (1993).
  * [13] N.Iwasaki: The Cauchy problem for effectively hyperbolic equations (a special case), J. Math. Kyoto Univ., 23 (1983), 503-562.
  * [14] N.Iwasaki: The Cauchy problem for effectively hyperbolic equations (a standard type), Publ. Res. Inst. Math. Sci., 20 (1984), 551-592.
  * [15] N.Iwasaki: The Cauchy problem for effectively hyperbolic equations (general case), J. Math. Kyoto Univ., 25 (1985), 727-743.
  * [16] E.Jannelli: The hyperbolic symmetrizer: theory and applications, in: Advances in Phase Space Analysis of Partial Differential Equations, in: Progr. Nonlinear Differential Equations Appl., vol. 78, Birkhäuser Boston, 2009, pp. 113-139.
  * [17] K.Kajitani, S.Wakabayashi, T.Nishitani; The Cauchy problem for hyperbolic operators of strong type, Duke Math. J., 75 (1994), 353-408.
  * [18] G.Komatsu, T.Nishitani: Continuation of bicharacteristics for effectively hyperbolic operators, Proc. Japan Acad. 65 (1989), 109-112.
  * [19] P.D.Lax: Asymptotic solutions of oscillatory initial value problem, Duke Math. J., 24 (1957), 627-646.
  * [20] N.Lerner, Metrics on the Phase Space and Non-selfadjoint Pseudo-Differential Operators, Birkhäuser, Basel, 2010.
  * [21] R.Melrose: The Cauchy problem for effectively hyperbolic operators, Hokkaido Math. J., 12 (1983), 371-391.
  * [22] S.Mizohata: Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 (1961), 109-127.
  * [23] T.Nishitani: On the finite propagation speed of wave front sets for effectively hyperbolic operators, Sci. Rep. College Gen. Ed. Osaka Univ., 32, No.1 (1983), 1-7.
  * [24] T.Nishitani: On wave front sets of solutions for effectively hyperbolic operators, Sci. Rep. College Gen. Ed. Osaka Univ., 32, No.2 (1983), 1-7.
  * [25] T.Nishitani: A note on reduced forms of effectively hyperbolic operators and energy integrals, Osaka J. Math., 21 (1984), 843-850.
  * [26] T.Nishitani: Local energy integrals for effectively hyperbolic operators. I, II, J. Math. Kyoto Univ., 24 (1984), 623-658, 659-666.
  * [27] T.Nishitani: The effectively hyperbolic Cauchy problem, in The Hyperbolic Cauchy Problem, Lecture Notes in Math. 1505, Springer-Verlag (1991), pp. 71-167.
  * [28] T.Nishitani, V.Petkov: Cauchy problem for effectively hyperbolic operators with triple characteristics, J. Math. Pures Appl., 123 (2019), 201-228.
  * [29] T.Nishitani: Notes on symmetrization by Bézoutian, Bollettino dell’Unione Matematica Italiana, 13 (2020), 417-428.
  * [30] T.Nishitani, V.Petkov: Cauchy problem for effectively hyperbolic operators with triple characteristics, Osaka J. Math. 57 (2020), 597-615.
  * [31] T.Nishitani: Diagonal symmetrizers for hyperbolic operators with triple characteristics, Math. Ann. (2021). https://doi.org/10.1007/s00208-021002153-2
  * [32] O.A.Oleinik: On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math., 23 (1970), 569-586.
  * [33] M.E.Taylor: Pseudodifferential Operators, Princeton Math. Ser. 34, Princeton Univ. Press, Princeton, NJ, 1981.