# Sufficient Conditions for Solvability of
Operators of Subprincipal Type

Nils Dencker

Lund University  Centre for Mathematical Sciences, University of Lund, Box
118, 221 00 Lund, Sweden<EMAIL_ADDRESS>
###### Abstract.

In this paper we show that condition $\operatorname{Sub}_{r}(\Psi)$ on the
subprincipal symbol is sufficient for local solvability of pseudodifferential
operators of real subprincipal type. These are the operators having real
principal symbol which vanish on an involutive manifold where the subprincipal
symbol is of principal type. This condition has been shown in [5] and [6] to
be necessary for local solvability of pseudodifferential operators of real
subprincipal type.

###### 2000 Mathematics Subject Classification:

35A01 (primary) 35S05, 58J40, 47G30 (secondary)

## Acknowledgement

This project was partly funded by the Swedish Research Council grant
2018-04228.

##  1\. Introduction

In this paper we shall study the local solvability of classical
pseudodifferential operators $P\in\Psi^{m}_{cl}$, which are given by an
asymptotic expansion $p_{m}(x,\xi)+p_{m-1}(x,\xi)+\dots$ of terms
$p_{m-j}(x,\xi)$ homogeneous of degree $m-j$ in $\xi$ for $j\in\mathbf{N}$,
where $p_{m}=p$ is the principal symbol. We are going to study operators which
are not of principal type, i.e., when the principal symbol $p$ vanishes of at
least order 2, in particular the sufficiency in the case when the principal
symbol is real and has involutive double characteristics. But we will also
assume that the operator is of subprincipal type, so that the subprincipal
symbol of the operator is of principal type, see Definition 1.9.

The definition that $P$ is locally solvable at a compact subset of a manifold
$K\subseteq X$ is that the equation

(1.1) $Pu=v$

has a local solution $u\in\mathcal{D}^{\prime}(X)$ in a neighborhood of $K$
for any $v\in C^{\infty}(X)$ in a set of finite codimension. We can also
define microlocal solvability of $P$ at any compactly based cone $K\subset
T^{*}X$, see Definition 1.10.

For classical pseudodifferential operators $P\in\Psi^{m}_{cl}$ which are of
principal type, local solvability is equivalent to condition $(\Psi)$ on the
principal symbol $p$, i.e.,

(1.2) $\text{$\operatorname{Im}ap$ does not change sign from $-$ to $+$}\\\
\text{along the oriented bicharacteristics of $\operatorname{Re}ap$ when
$\operatorname{Re}ap=0$}$

for any $0\neq a\in C^{\infty}(T^{*}X)$, see [3] and [9]. The oriented
bicharacteristics are the positive flow of the Hamilton vector field
$H_{\operatorname{Re}ap}\neq 0$ on which $\operatorname{Re}ap=0$, these are
also called semibicharacteristics of $p$. Observe that if condition $(\Psi)$
is satisfied on a set, then it is trivially satisfied on any subset.

For operators which are not of principal type, the invariant subprincipal
symbol

(1.3) $p_{s}=p_{m-1}+\frac{i}{2}\sum_{j}\partial_{x_{j}}\partial_{\xi_{j}}p$

becomes important. There are several conditions corresponding to condition
$(\Psi)$ on the subprincipal symbol, several necessary conditions for
solvability are known, but not many sufficient conditions.

One of the earliest results are by Mendoza and Uhlman [16], who studied the
case when principal symbol is equal to a product $p=p_{1}p_{2}$ with $p_{j}$
of real principal type with linearly independent differentials $dp_{1}$ and
$dp_{2}$. Thus the double characteristic set $\Sigma_{2}=\\{p_{1}=p_{2}=0\\}$
is a intersection of two transversal hypersurfaces. In this case, they proved
that $P$ is not solvable if the imaginary part of the subprincipal symbol
$p_{s}$ changes sign on the bicharacteristics of $p_{1}$ or $p_{2}$ on
$\Sigma_{2}$. These are the limits of the characteristics of the principal
symbol at the double characteristic set $\Sigma_{2}$. They proved in [17] that
$P$ is solvable if the imaginary part of the subprincipal symbol $p_{s}$ does
not vanish on the double characteristics, thus there are no sign changes.

Mendoza [18] generalized the necessary condition to the case when the
principal symbol is real, vanishes of second order on an involutive
submanifold where it has an indefinite Hessian with rank equal to the
codimension of the manifold. Then Hessian gives well-defined limit
bicharacteristis on the submanifold, and $P$ is not solvable if the imaginary
part of the subprincipal symbol changes sign on any of these limit
bicharacteristics.

There are several other necessary condition for solvability of operators that
are not of principal type corresponding to condition ($\Psi$) on operators of
principal type. The following one generalizes Mendoza’s and Uhlmann’s
necessary conditions for solvability.

######  Example 1.1.

A necessary condition on the subprincipal symbol for solvability of operators
with real principal symbol $p$ vanishing of at least second order on an
involutive submanifold $\Sigma_{2}$ is condition $\operatorname{Lim}(\Psi)$:

(1.4) $\operatorname{Im}p_{s}\ \text{does not change sign from $-$ to $+$ on
the limit bicharacteristics of $p$ on $\Sigma_{2}$}$

which follows from the necessary condition (2.9) of [5].

This condition is invariant under symplectic changes of variables and
multiplication with nonvanishing real factors, since a negative factor changes
both the direction of the limit bicharacteristic and the sign of the imaginary
part. Thus, it is invariant under conjugations of the operator with Fourier
integral operators with real principal symbols.

Observe that this is a condition on the sign changes of
$\operatorname{Im}p_{s}$ at a (possibly empty) subset of directions on the
leaves of $\Sigma_{2}$. The sufficient conditions that we are going to use
will exclude any sign changes of $\operatorname{Im}p_{s}$ on the leaves of
$\Sigma_{2}$. But even this stronger condition is not sufficient, one also
needs conditions on the imaginary part of the subprincipal symbol in the
direction of the bicharacteristics if the real part of the subprincipal
symbol. For that, the operator has to be of subprincipal type, which means
that the subprincipal symbol is of principal type, with Hamilton vector field
that is tangent to $\Sigma_{2}$ at the characteristics, see Definition 2.2 in
[6] or Definition 1.9 in this paper.

######  Example 1.2.

A necessary conditions for solvability for operators of subprincipal type for
involutive $\Sigma_{2}$ is given by Definition 2.4 in [6]. It is condition
$\operatorname{Sub}(\Psi)$ which is Sub($\Psi$) on subprincipal symbol $p_{s}$
on $\Sigma_{2}$:

(1.5) $\text{$\operatorname{Im}ap_{s}$ does not change sign from $-$ to +}\\\
\text{on the oriented bicharacteristics of $\operatorname{Re}ap_{s}$ when
$\operatorname{Re}ap_{s}=0$ on $\Sigma_{2}$.}$

for any $0\neq a\in C^{\infty}$. It is known that this condition is invariant
under symplectic changes of variables and multiplication with nonvanishing
factors, since the subprincipal symbol then only get multiplied with these
factors.

Observe that condition (1.5) is empty if $p_{s}$ vanishes of second order, so
for this condition we need that the operator is of subprincipal type, see
Definition 1.9.

A stronger necessary condition for solvability involves the sign changes on
the imaginary part of the subprincipal symbol on a larger set of curves on the
double characteristic set, actually on the limits of the bicharacteristics of
the real part of the refined principal symbol

(1.6) $p_{r}=p+p_{s}$

where $p_{s}$ is the subprincipal symbol given by (1.3), see Theorem 18.1.33
in [10]. This symbol is invariant under conjugation with elliptic Fourier
integral operators, and multiplication with $a(x,D)\in\Psi^{0}$ gives the
refined principal symbol $ap_{r}$ modulo terms in $S^{m-1}$ vanishing of order
1 on $\Sigma_{2}$. In fact, the refined principal symbol of $a(x,D)P(x,D)$ is
$ap_{r}+\tfrac{i}{2}H_{p}a$ modulo $S^{m-2}$.

######  Example 1.3.

A necessary condition for solvability for operators of subprincipal type with
principal symbol vanishing of second order on $\Sigma_{2}$ is
$\operatorname{Sub}_{2}(\Psi)$. This condition is given by Definition 2.6 in
[7] and is condition ($\Psi$) on the symbol

(1.7) $p_{s,2}=J^{2}(p)+J^{0}(p_{m-1})=J^{2}(p)+p_{s}$

where $J^{2}(p)$ equal to the 2:nd jet of $p$ at $\Sigma_{2}$.

This condition is invariant under symplectic changes of variables and
multiplication with nonvanishing factors by Remark 2.3 in [7], since then
$p_{s,2}$ gets multiplied with a nonvanishing factor.

Observe that this definition gives conditions on the sign changes of
$\operatorname{Im}p_{s,2}$ on the limits of the bicharacteristics of
$\operatorname{Re}p_{s,2}$ at $\Sigma_{2}$, which are the limits of of the
bicharacteristics of $\operatorname{Re}p_{r}$, see (1.19). Condition
$\operatorname{Sub}_{2}(\Psi)$ gives (1.4) and (1.5), but the directions of
the limit characteristics depend on the sign of $\operatorname{Re}p_{s}$, see
the following example.

######  Example 1.4.

If $p=|\xi^{\prime}|^{2}-|\xi^{\prime\prime}|^{2}$ with
$(\xi^{\prime},\xi^{\prime\prime})\in\mathbf{R}^{n}\times\mathbf{R}^{m}$, then

$H_{p}=2(\xi^{\prime}\cdot\partial_{x^{\prime}}-\xi^{\prime\prime}\cdot\partial_{x^{\prime\prime}})=2|\xi|(\theta^{\prime}\cdot\partial_{x^{\prime}}-\theta^{\prime\prime}\cdot\partial_{x^{\prime\prime}})\qquad|\theta|=1$

which gives all directions in $x$ when $\xi\to 0$. If we take the limit only
when $p_{r}=0$, i.e., when $p=-\operatorname{Re}p_{s}$, then we get the limit
bicharacteristics
$\theta^{\prime}\cdot\partial_{x^{\prime}}-\theta^{\prime\prime}\cdot\partial_{x^{\prime\prime}}$
with $|\theta^{\prime}|>|\theta^{\prime\prime}|$ when
$\operatorname{Re}p_{s}<0$, the ones with
$|\theta^{\prime}|<|\theta^{\prime\prime}|$ when $\operatorname{Re}p_{s}>0$
and all directions in $\mathbf{R}^{n}\times\mathbf{R}^{m}$ when
$\operatorname{Re}p_{r}=0$.

Thus, when $\operatorname{Re}p_{s}=0$ we may obtain that the sign of
$\operatorname{Im}p_{s,2}=\operatorname{Im}p_{s}$ is constant on the leaves of
$\Sigma_{2}$ when $\operatorname{Re}p_{s}=0$, but by Example 1.5 that is not
enough to get solvability. Observe that we shall assume that
$\operatorname{Re}p_{r}$ is constant on the leaves by (1.13).

Also observe that the necessity of the conditions in Examples 1.1–1.3 only
hold under some additional conditions on the symbol, for example finite order
of the sign change. For the sufficiency, it is not enough that
$\operatorname{Sub}(\Psi)$ holds when $p_{s}=0$, by the following example.

######  Example 1.5.

Consider the PDO

(1.8) $P_{\pm}=(1+t^{2})\Delta_{x}+D_{t}\pm itD_{t}$

with symbol $(1+t^{2})|\xi|^{2}+\tau\pm it\tau=(1+t^{2})|\xi|^{2}+(1\pm
it)\tau$. This operator satisfies the condition $\operatorname{Lim}(\Psi)$ by
(1.4) and $\operatorname{Sub}(\Psi)$ by (1.5) since
$\operatorname{Im}p_{s}=\pm t\tau=0$ when $\operatorname{Re}p_{s}=\tau=0$. But
multiplication by $(1\pm it)^{-1}=(1\mp it)/(1+t^{2})$ gives the operator
$D_{t}+(1\mp it)\Delta_{x}$, and conjugation with the Schrödinger kernel
$\exp(\pm it\Delta_{x})$ gives the operator $Q_{\mp}=D_{1}\mp it\Delta_{x}$.
Here $Q_{+}$ is the Mizohata operator which is a standard example of an
unsolvable operator, and $Q_{-}$ is solvable, since
$u(t,x)=i\int_{0}^{t}\exp((s^{2}-t^{2})\Delta_{x}/2)f(s,x)\,ds$ solves
$Q_{-}u=f\in C^{\infty}_{0}$.

Observe that the condition $\operatorname{Lim}(\Psi)$ in Example 1.1 does in
general not imply that $\operatorname{Im}p_{s}$ has constant sign on the
leaves of $\Sigma_{2}$.

######  Example 1.6.

If the principal symbol of $P$ is $D_{x_{1}}D_{x_{2}}$, then the leaves of
$\/\Sigma_{2}$ have dimension $2$. Divide the leaves into a checkerboard and
index the squares with $(j,\,k)\in\mathbf{Z}^{2}$. Denote the squares with
index $(2j,2k)$ with $S_{+}$ and the ones with index $(2j+1,2k+1)$ with
$S_{-}$ and the rest with $S_{0}$. If $\operatorname{Im}p_{s}>0$ in the
interior of the squares in $S_{+}$, $\operatorname{Im}p_{s}<0$ in the interior
of the squares in $S_{-}$ and $\operatorname{Im}p_{s}=0$ on the squares in
$S_{0}$, then $\operatorname{Im}p_{s}$ has constant sign along any $x_{1}$ and
$x_{2}$ lines, but not on the whole plane.

The conditions on $P\in\Psi^{m}_{cl}$ in the present paper will be the
following. Let $p$ be the real principal symbol, $\Sigma=p^{-1}(0)$ be the
characteristics and $\Sigma_{2}=\left\\{\,p=|dp|=0\,\right\\}$ be the double
characteristics. We assume that $\Sigma_{2}$ is a nonradial involutive
submanifold and that $p$ is real and vanishes of exactly second order at
$\Sigma_{2}$ so that

(1.9) $\operatorname{Hess}p$ is nondegenerate on $\Sigma_{2}$

This implies that $p$ is of real principal type on
$\Sigma_{1}=\Sigma\setminus\Sigma_{2}$ in a sufficiently small conical
neighborhood of $\Sigma_{2}$, since it cannot vanish of second order on
$\Sigma_{1}$. Also, $\operatorname{Hess}p\big{|}_{\Sigma_{2}}$ has locally
constant rank and index. That $\Sigma_{2}$ is nonradial means that if a
function vanishes on $\Sigma_{2}$ then its Hamilton vector field does not have
the radial direction on $T^{*}X$, which is a generic condition.

######  Remark 1.7.

The invariant condition is that $p$ is proportional to a real function. This
means that the quotient $q=\operatorname{Im}p/\operatorname{Re}p$, which is
defined where $\operatorname{Re}p\neq 0$, can be extended to a $C^{\infty}$
function with values on the extended real line $\overline{\mathbf{R}}$, i.e.,
either $q$ or $q^{-1}$ is smooth. In fact, if $q\in C^{\infty}$ then
$p=(1+iq)\operatorname{Re}p$ and if $q^{-1}\in C^{\infty}$ then
$p=(q^{-1}+i)\operatorname{Im}p$.

In the following, we will assume that $P$ is on the form so that the principal
symbol $p$ is real valued. In the case when $p$ is not proportional to a real
function, condition $(\Psi)$ on the principal symbol has to be satisfied on
$\Sigma_{1}$ since it is then necessary for solvability.

Recall that the subprincipal symbol

(1.10)
$p_{s}=p_{m-1}+\frac{i}{2}\sum_{j}\partial_{x_{j}}\partial_{{\xi}_{j}}p$

is invariantly defined on $\Sigma_{2}$ under conjugation with elliptic Fourier
integral operators. In fact, $p_{s}$ is the value of the Weyl symbol of
$p+p_{m-1}$ at $\Sigma_{2}$ modulo $S^{m-2}$, see [8].

######  Remark 1.8.

When $\Sigma_{2}$ is involutive we may choose symplectic coordinates so that
$\Sigma_{2}=\left\\{\,\xi_{1}=\dots=\xi_{k}=0\,\right\\}$ and then the
subprincipal symbol $p_{s}=p_{m-1}$ at $\Sigma_{2}$.

In fact, since $\partial_{x}\in T\Sigma_{2}$ we find that $\partial_{x}p$
vanishes of second order on $\Sigma_{2}$. If $C$ is a pseudodifferential
operator with principal symbol $c={\sigma}(C)$, then the value of the
subprincipal symbol of the composition $CP$ is equal to
$cp_{s}+\frac{i}{2}H_{p}c=cp_{s}$ on $\Sigma_{2}$. Observe that the
subprincipal symbol is complexly conjugated when taking the adjoint of the
operator, see [10, Theorem 18.1.34].

Since we shall assume that $p$ is real, the real and imaginary parts of
$p_{s}$ are invariant under multiplication with elliptic pseudodifferential
operators and conjugation with elliptic Fourier integral operators if these
operators have real principal symbols. In order to study the invariants, we
need some symplectic concepts.

The symplectic annihilator to a linear space consists of the vectors that are
symplectically orthogonal to the space. Let $T{\Sigma}_{2}^{\sigma}$ be the
symplectic annihilator to $T{\Sigma}_{2}$, which spans the symplectic leaves
of $\Sigma_{2}$. If $\Sigma_{2}=\left\\{\,\xi=0\,\right\\}$,
$(x,y)\in\mathbf{R}^{d}\times\mathbf{R}^{n-d}$, then the leaves are spanned by
$\partial_{x}$. Let

(1.11) $T^{\sigma}{\Sigma}_{2}=T{\Sigma}_{2}/T{\Sigma}_{2}^{\sigma}$

which is a symplectic space over ${\Sigma}_{2}$. In these coordinates it is
parametrized by

(1.12)
$T^{\sigma}\Sigma_{2}=\left\\{\,((x_{0},y_{0};0,\eta_{0});(0,y;0,\eta))\in
T\Sigma_{2}:\ (y,{\eta})\in T^{*}\mathbf{R}^{n-d}\,\right\\}$

Thus the fiber is isomorphic to the symplectic manifold
$T^{*}\mathbf{R}^{n-d}$ with $(x_{0},y_{0};0,\eta_{0})=w_{0}\in\Sigma_{2}$ as
a parameter.

######  Definition 1.9.

If the principal symbol is real valued, then we say that the operator $P$ is
of _real subprincipal type_ if the following conditions hold:

(1.13) $H_{\operatorname{Re}p_{s}}\subset T\Sigma_{2}$

which means that $d\operatorname{Re}p_{s}\big{|}_{TL}=0$, and

(1.14) $d\operatorname{Re}p_{s}\big{|}_{T^{\sigma}{\Sigma}_{2}}\neq
0\qquad\text{}$

so that $H_{\operatorname{Re}p_{s}}$ is transversal to the leaves and we shall
assume that it does not have the radial direction. The bicharacteristics of
$\operatorname{Re}p_{s}$ with respect to the symplectic structure of
$\Sigma_{2}$ are called the subprincipal bicharacteristics for any value of
$\operatorname{Re}p_{s}$.

This definition is invariant under symplectic changes of variables and but not
by multiplication with nonvanishing real factors when
$\operatorname{Re}p_{s}\neq 0$. But it is invariant by multiplication with
nonvanishing real factors that are constant on the leaves of $\Sigma_{2}$.
When the coordinates are given as in (1.12), we find from (1.13) that
$\partial_{x}\operatorname{Re}p_{s}=0$ on $\Sigma_{2}$ and from (1.14) that
$\partial_{\eta}\operatorname{Re}p_{s}\neq 0$ or
$\partial_{y}\operatorname{Re}p_{s}\not\parallel\eta$ on $\Sigma_{2}$. Thus it
follows that

(1.15) $\operatorname{Re}p_{s}\ \text{is of real principal type}$

and thus has simple zeroes. In [6, Definition 2.1] the definition was that $P$
is of subprincipal type if Definition 1.9 hold with $\operatorname{Re}p_{s}$
replaced with $p_{s}$, but only when $p_{s}=0$, which is invariant under
multiplication with nonvanishing factors and symplectic changes of variables.
But in that case the principal symbol may not be proportional to a real symbol
ant then $\operatorname{Re}p_{s}$ is not well defined.

We shall study the microlocal solvability of the operator $P$, which is given
by the following definition from [10]. Recall that $H^{loc}_{(s)}(X)$ is the
set of distributions that are locally in the $L^{2}$ Sobolev space
$H_{(s)}(X)$.

######  Definition 1.10.

If $P\in\Psi^{m}_{cl}$ and $K\subset T^{*}X$ is a compactly based cone, then
we say that $P$ is microlocally solvable at $K$ if there exists an integer $N$
so that for every $f\in H^{loc}_{(N)}(X)$ there exists
$u\in\mathcal{D}^{\prime}(X)$ such that
$K\bigcap\operatorname{WF}(Pu-f)=\emptyset$.

Observe that solvability at a compact set $M\subset X$ is equivalent to
solvability at $T^{*}X\big{|}_{M}$ by [10, Theorem 26.4.2], and that
solvability at a set implies solvability at a subset. Also, by Proposition
26.4.4 in [10] the microlocal solvability is invariant under conjugation by
elliptic Fourier integral operators and multiplication by elliptic
pseudodifferential operators.

To prove solvability we shall use a priori estimates. Let $\|u\|_{(k)}$ be the
$L^{2}$ Sobolev norm of order $k$, $u\in C_{0}^{\infty}$. In the following,
$P^{*}$ will be the $L^{2}$ adjoint of $P$.

######  Remark 1.11.

Let $P\in{\Psi}^{m}_{cl}(X)$ and $K\subset T^{*}X$ be a compactly based cone,
and assume that there exists ${\nu}\in\mathbf{R}$ and a pseudodifferential
operator $A$ so that $K\bigcap\operatorname{WF}(A)=\emptyset$ and

(1.16) $\|u\|_{(-N)}\leq
C(\|P^{*}{u}\|_{({\nu})}+\|u\|_{(-N-n)}+\|Au\|_{(0)})\qquad u\in
C_{0}^{\infty}(Y)$

Then $P$ is microlocally solvable at $K$ and one can take this $N$ in
Definition 1.10.

Observe that if $P\in\Psi^{m}_{cl}$ then there is a loss of $\nu+m+N$
derivatives in the estimate (1.16) compared with the elliptic case. One can
have several operators $A_{j}$ in (1.16) by taking $A=(A_{1},\dots)$ vector
valued.

######  Definition 1.12.

We say that $P\in\Psi^{m}$ satisfies condition $\operatorname{Sub_{r}}(\Psi)$
if there exists a homogeneous $0\neq a\in S^{0}$ such that $ap_{r}$ has real
principal symbol that vanishes of order $2$ at an involutive manifold
$\Sigma_{2}$ with nondegenerate Hessian, $ap_{r}$ is of real subprincipal type
and satisfies condition $(\Psi)$ at the limit $\Sigma_{2}$. This means that
$\operatorname{Im}ap_{r}$ does not change sign from $-$ to $+$ on the limits
of the bicharacteristic of $\operatorname{Re}ap_{r}$ at $\Sigma_{2}$.

The refined principal symbol is equal to $p_{r}=p+p_{s}$ by (1.6). Observe
that this condition gives conditions on the sign changes for any value of
$\operatorname{Re}p_{s}$, and that is not the case in Example 1.5. This
condition is stronger than the conditions in Examples (1.1)–(1.3). Observe
that the factor $a$ makes this condition invariant under multiplication with
with nonvanishing factors. It is also invariant under symplectic changes of
variables, thus the conditions is invariant under conjugation with Fourier
integral operators and multiplication with elliptic pseudodifferential
operators having real principal symbols.

######  Proposition 1.13.

If $P\in\Psi^{m}$ has real principal symbol that vanishes of order $2$ at an
involutive manifold $\Sigma_{2}$ with nondegenerate Hessian and is of real
subprincipal type, then $P$ satisfies condition $\operatorname{Sub_{r}}(\Psi)$
if and only if $\operatorname{Im}p_{r}=\operatorname{Im}p_{s}$ does not change
sign on the leaves of $\Sigma_{2}$ and the sign of $\operatorname{Im}p_{s}$ on
the leaves do not change from $-$ to $+$ on the subprincipal
bicharacteristics, i.e., the bicharacteristics of $\operatorname{Re}p_{s}$
with respect to the symplectic structure of $\Sigma_{2}$ for any value of
$\operatorname{Re}p_{s}$, see Definition 1.9.

Here, the sign on the leaves is $\pm 1$ if $\pm\operatorname{Im}p_{s}\geq 0$
and $\operatorname{Im}p_{s}\not\equiv 0$ on leaf $L$ of $\Sigma_{2}$ and equal
to $0$ if $\operatorname{Im}p_{s}\equiv 0$ on $L$, see Definition 4.1. Thus,
condition $\operatorname{Sub_{r}}(\Psi)$ implies the necessary conditions in
Examples 1.1 and 1.2, and it is not hard to show that it implies the condition
$\operatorname{Sub}_{2}(\Psi)$ in Example 1.3. In fact, the limits at
$\Sigma_{2}$ of the Hamilton vector field of the refined principal symbol only
depend on the values at $\Sigma_{2}$ of the Hessian of the principal symbol
and the gradient of the subprincipal symbol, see (1.19).

###### Proof.

By multiplying with $\langle D\rangle^{2-m}$ we assume that $P\in\Psi^{2}$,
and we may choose symplectic coordinates so that
$(x,y)\in\mathbf{R}^{d}\times\mathbf{R}^{nd}$ and
$\Sigma_{2}=\left\\{\,\xi=0\,\right\\}$. By Taylor’s formula we can write the
real principal symbol as

(1.17) $p_{2}(x,y;\xi,\eta)=\sum_{jk}a_{jk}(x,y;\xi,\eta)\xi_{j}\xi_{k}$

where the Hessian $\\{a_{jk}(x,y;0,\eta)\\}_{jk}$ is nondegenerate near
$w_{0}$ by assumption. We have that $p_{1}=p_{s}$ the subprincipal symbol on
$\Sigma_{2}$ and by extending it we may assume it is constant in $\xi$.

Now the Poisson parentheses
$\left\\{\,\xi,\operatorname{Re}p_{s}\,\right\\}\equiv\partial_{x}\operatorname{Re}p_{s}\equiv
0$ on $\Sigma_{2}$ and
$\left\\{\,x,\operatorname{Re}p_{s}\,\right\\}\equiv-\partial_{\xi}\operatorname{Re}p_{s}\equiv
0$. Thus $\partial_{x}\partial_{\xi}\operatorname{Re}p_{s}\equiv 0$ which
gives $\partial_{x}\operatorname{Re}p_{s}\equiv 0$. Since
$\operatorname{Re}p_{s}$ is homogeneous and of principal type, we can thus
complete $x$, $\xi$ and $\tau=\operatorname{Re}p_{s}\big{|}_{\Sigma_{2}}$ to a
symplectic homogeneous coordinate system $(x,t,y;\xi,\tau,\eta)$ microlocally
near $w_{0}\in\Sigma_{2}$. We then obtain that the refined principal symbol of
$P$ is equal to

(1.18)
$\sum_{jk}a_{jk}(x,t,y;\xi,\tau,\eta)\xi_{j}\xi_{k}+\tau+if(x,t,y;\tau,\eta)+C(x,t,y;\xi,\tau,\eta)\cdot\xi+p_{0}(x,t,y;\tau,\xi,\eta)$

modulo terms in $S^{-1}$, where $C$ homogeneous of degree 0 with values in
$\mathbf{R}^{m}$, $f$ is real and homogeneous of degree 1 and $p_{0}\in
S^{0}$.

The Hamilton vector field of the real part of the refined principal symbol is
given by

(1.19) $H_{\operatorname{Re}p_{r}}\cong
2\sum_{jk}a_{jk}(x,t,y;0,\tau,\eta)\xi_{j}\partial_{x_{k}}+\partial_{t}+\operatorname{Re}C(x,t,y;0,\tau,\eta)\cdot\partial_{x}$

modulo terms with coefficients that are
$\mathcal{O}((|\xi|^{2}+|\xi|)/\Lambda)$ with
$\Lambda=\sqrt{\tau^{2}+|\eta|^{2}}$ in the base of homogeneous vector fields
$V=(\partial_{t},\partial_{x},\partial_{y},\Lambda\partial_{\tau},\Lambda\partial_{\xi},\Lambda\partial_{\eta})$.
We get the limit at $\Sigma_{2}$ when $|\xi|/\Lambda\to 0$.

One limit is when $\xi\to 0$ for fixed $\Lambda$ and then the limit of (1.19)
is equal to $\partial_{t}+\operatorname{Re}C\cdot\partial_{x}$ which gives the
subprincipal bicharacteristics. We can also take $\tau=\Lambda\tau_{0}$,
$\eta=\Lambda\eta_{0}$ and $\xi=\Lambda^{1/3}\theta$ with
$|\tau_{0}|^{2}+|\eta_{0}|^{2}=|\theta|=1$. Since the coefficients $a_{jk}$
and $C$ are homogeneous, one can write (1.19) as

(1.20)
$2\Lambda^{1/3}\sum_{jk}a_{jk}(x,t,y_{0};0,\tau_{0},\eta_{0})\theta_{j}\partial_{x_{k}}+\partial_{t}+\operatorname{Re}C(x,t,y;0,\tau_{0},\eta_{0})\cdot\partial_{x}$

modulo vector fields in $V$ with coefficients that are
$\mathcal{O}(\Lambda^{-1/3})$. By dividing by $\lambda^{1/3}$ and taking the
limit $\lambda\to\infty$ we obtain the limit Hamilton vector field

(1.21)
$H_{p_{2}}=2\sum_{jk}a_{jk}(x,t,y_{0};0,\tau_{0},\eta_{0})\theta_{j}\partial_{x_{k}}$

This vector field gives for fixed $\theta_{0}=(\theta_{1},\dots,\theta_{d})$ a
foliation of the leaves of $\Sigma_{2}$. Let $\Gamma(\xi)$ with
$\xi=\varrho\theta$ be the flow-out of the Hamilton vector field $\varrho
H_{p_{2}}$ with $H_{p_{2}}$ given by (1.21). Then the Hessian
$\operatorname{Hess}\Gamma(0)$ is non-degenerate so the mapping
$\xi\mapsto\Gamma(\xi)$ is a diffeomorphism from $|\xi|<c$ to a neighborhood
of the $w_{0}$ in the leaf through $w_{0}=(x,t,y;0,\tau_{0},\eta_{0})$. Since
we can take $\pm\xi$ we obtain from condition $\operatorname{Sub_{r}}(\Psi)$
that there can be no sign changes of $f$ on the leaves of $\Sigma_{2}$ near
$w_{0}$.

We can also take the limit $(\tau,\eta)=\Lambda(\tau_{0},\eta_{0})$ and
$\xi=\varrho\cdot\theta$ with $|\tau_{0}|^{2}+|\eta_{0}|^{2}=|\theta|=1$ and
$0\leq\varrho\ll\sqrt{\Lambda}\to\infty$. As before, we obtain the limit
Hamilton vector field

(1.22)
$2\sum_{jk}a_{jk}(x,t,y_{0};0,\tau_{0},\eta_{0})\xi_{j}\partial_{x_{k}}+\partial_{t}+\operatorname{Re}C(x,t,y_{0};0,\tau_{0},\eta_{0})\cdot\partial_{x}$

As before, the orbits of (1.19) gives a foliation of $\Sigma_{2}$ for fixed
$\theta$. When $\xi=0$ the orbit $\gamma_{0}$ of (1.19) through
$w_{0}=(x_{0},t_{0},y_{0};0,\tau_{0},\eta_{0})$ is a subprincipal
bicharacteristic. When $\xi=\varrho\theta\neq 0$ then the orbits
$\Gamma(t,\xi)$ of (1.19) through $w_{0}$ with $t>0$ form a proper cone in
$\Sigma_{2}$ with the subprincipal bicharacteristic in its interior. Now
scaling gives that the Hessian of $\xi\mapsto\Gamma(\varrho t,\xi/\varrho)$ at
$t=0$ is constant in $\varrho>0$, so by letting $\varrho\to 0$ we find for
some $c>0$ that $\bigcup_{0<\varrho\leq c}\Gamma(\varrho t,\xi/\varrho)$ for
$|\xi|<c$ forms a cylindrical neighborhood of the forward subprincipal
bicharacteristic. Thus, we find from condition $\operatorname{Sub_{r}}(\Psi)$
that if $f\not\geq 0$ on a leaf in a neighborhood of
$(t_{0},y_{0};0,\tau_{0},\eta_{0})$ then $f\leq 0$ on the leafs in a
neighborhood of $(t,y_{0};\tau_{0},\eta_{0})$ for $0\leq t-t_{0}\leq c$ which
proves the proposition. ∎

######  Remark 1.14.

The requirement that condition ($\Psi$) shall hold on the refined principal
symbol for all values of the real part is only needed when
$f=\operatorname{Im}p_{r}=\operatorname{Im}p_{s}$ depends on
$\tau=\operatorname{Re}p_{s}$. In fact, if $f$ does not depend on $\tau$, then
by choosing suitable $\tau$ so that $\operatorname{Re}p_{r}=0$ we can get the
same limits at $\Sigma_{2}$ of the Hamilton vector field of
$\operatorname{Re}p_{r}$ as in the proof of Proposition 1.13. One may of
course eliminate the $\tau$ dependence of $f$ by the Malgrange preparation
theorem, but that would change the imaginary part of the principal symbol $p$,
see for example Example 1.5.

The following is the main result of the paper.

######  Theorem 1.15.

Assume that $P\in{\Psi}^{m}_{cl}(X)$ satisfies $\operatorname{Sub}_{r}(\Psi)$
microlocally near $w_{0}\in\Sigma_{2}$, then $P$ is microolocally solvable
near $w_{0}$ with a loss of $5/2$ derivatives by the a priori estimate (1.16).

Thus, by Example 1.3 condition $\operatorname{Sub_{r}}(\Psi)$ is both
necessary (under additional conditions) and sufficient for local solvability
for operators of subprincipal type with principal symbol that is real and
vanishes of exactly second order at a nonradial involutive manifold
$\Sigma_{2}$.

The solvability with a loss of $5/2$ derivatives can be compared with the loss
of $2$ derivatives when the antisymmetric part of P is bounded. In fact, by
using the normal form (2.1) with $f\equiv 0$ gives a Schrödinger type operator
that is symmetric modulo bounded operators. By using a multiplier as in Lemma
6.9 one can obtain $L^{2}$ estimates with arbitrarily small constants. For
small enough constant, these estimates may be perturbed by any bounded term.
This is similar to the case of operators of principal part, where the loss of
derivatives is $1/2$ more when the antisymmetric part if the operator is
unbounded and condition $(\Psi)$ is satisfied.

This paper treats subprincipal type operators with involutive characteristics
having nondegenerate second order vanishing of the principal symbol. For the
noninvolutive or degenerate cases, see [19], [20] and the references there.

To prove Theorem 1.15 we shall use suitable a priori estimate and Remark 1.11.
The proof will occupy most of the remaining paper.

##  2\. The Preparation

As in the proof of Proposition 1.13 we may assume that the operator
$P\in{\Psi}^{2}_{cl}(X)$ is of second order with real principal symbol,
$X=\mathbf{R}^{n}$ and the coordinates are chosen so that
$\Sigma_{2}=\left\\{\,\xi=0\,\right\\}$, $(x,y;\xi,\eta)\in
T^{*}(\mathbf{R}^{d}\times\mathbf{R}^{d-n})$ microlocally near
$w_{0}\in\Sigma_{2}$, where $x\mapsto(x,y_{0};0,\eta_{0})$ spans the leaves of
the symplectic foliation of $\Sigma_{2}$ and $d$ is the codimension of
$\Sigma_{2}$. We may multiply $P$ with an elliptic operator of order zero so
that the refined principal symbol satifies condition
$\operatorname{Sub_{r}}(\Psi)$, see Definition 1.12. Thus, we have the
operator on the normal form given by (1.18)

(2.1)
$\sum_{jk}a_{jk}(x,t,y;\xi,\tau,\eta)\xi_{j}\xi_{k}+p_{1}(x,t,y;\xi,\tau,\eta)+p_{0}(x,t,y;\xi,\tau,\eta)$

modulo terms in $S^{-1}$, where $\left\\{\,a_{jk}\,\right\\}_{jk}$ is
nondegenerate on $\Sigma_{2}$,

(2.2)
$p_{1}(x,t,y;\xi,\tau,\eta)=\tau+if(x,t,y;\tau,\eta)+C(x,t,y;\xi,\tau,\eta)\cdot\xi$

with $f$ homogeneous of degree 1, and $C$ and $p_{0}$ homogeneous of degree 0.
By condition $\operatorname{Sub_{r}}(\Psi)$ and Proposition 1.13 we find that
$f$ does not change sign on the leaves of $\Sigma_{2}$ and the sign on the
leaves do not change from $-$ to $+$ as $t$ increases by Definition 4.1.

We shall compute the symbol modulo the error terms

(2.3) $R=\\{\langle C_{2}\xi,\xi\rangle+C_{1}\cdot\xi+C_{0}:\ C_{j}\in
S^{-1}\\}$

These are sums of terms that are either in $S^{1}$ vanishing of second order
on $\Sigma_{2}$, in $S^{0}$ vanishing on $\Sigma_{2}$ or in $S^{-1}$. Observe
that homogeneous vector fields that are tangent to $\Sigma_{2}$ maps $R$ into
itself.

We shall use the Weyl quantization, which has the property that symmetric
operators have real symbols. The Weyl quantization of symbols
$a\in\mathcal{S}^{\prime}(T^{*}\mathbf{R}^{n})$ is defined by:

(2.4) $\left(a^{w}u,v\right)=(2{\pi})^{-n}\iint\exp{(i\langle
x-y,{\xi}\rangle)}a\\!\left(\tfrac{x+y}{2},{\xi}\right)u(x)\overline{v(y)}\,dxdyd{\xi}\qquad
u,v\in C_{0}^{\infty}$

Observe that $\operatorname{Re}a^{w}=(\operatorname{Re}a)^{w}$ is the
symmetric part and $i\operatorname{Im}a^{w}=(i\operatorname{Im}a)^{w}$ the
antisymmetric part of the operator $a^{w}$. Also, if $a\in S^{m}_{1,0}$ then
$a(x,D_{x})=b^{w}(x,D_{x})$ modulo ${\Psi}^{m-2}_{1,0}$ where

(2.5)
$b(x,\xi)=a(x,\xi)+\frac{i}{2}\sum_{j}\partial_{x_{j}}\partial_{\xi_{j}}a(x,\xi)$

which gives the subprincipal symbol by [10, Theorem 18.5.10]. The equality
(2.5) shows that $a\in R$ if and only if $b\in R$. I

Now by conjugating with $e^{\phi}\in S^{0}$ having phase
$\phi(x,t,y;\tau,\eta)$ that is real and homogeneous of degree 0, we may
obtain that $\operatorname{Im}p_{0}=0$ at $\Sigma_{2}$, i.e.,
$\operatorname{Im}p_{0}\in R$. In fact, we obtain this by solving the equation

(2.6)
$\partial_{t}\phi(x,t,y;\tau,\eta)+\operatorname{Re}C(x,t,y;\tau,0,\eta)\cdot\partial_{x}\phi(x,t,y;\tau,\eta)=\operatorname{Im}p_{0}(x,t,y;\tau,0,\eta)$

but this may of course change the values of $\operatorname{Im}C$ and
$\operatorname{Re}p_{0}$.

Next, we want to reduce to the case $\operatorname{Im}C=0$ on $\Sigma_{2}$.
i.e, $\operatorname{Im}C\in R$, but that can in general not be done by
conjugation. Instead we shall use symplectic changes of variables given
microlocally by Fourier integral operators. In the following, we shall for
simplicity include the variable $t$ in the $y$ variables, and the variable
$\tau$ in the $\eta$ variables. The variables $(y,\eta)$ will then parametrize
the leaves of $\Sigma_{2}$.

Let $\mathbf{R}^{d}\ni x\mapsto\chi(x,y,\eta)\in\mathbf{R}^{d}$, where
$\chi\in C^{\infty}$, homogeneous in $\eta$ and $|\partial_{x}\chi|\neq 0$ and
let

(2.7) $Fu(x,y)=(2\pi)^{-n}\iint
e^{i(\langle\chi(x,y,\eta)-z,\xi\rangle+\langle
y-w,\eta\rangle)}u(z,w)\,dzdwd\xi d\eta\qquad u\in C^{\infty}_{0}$

which is an elliptic Fourier integral operator. This correspond to the
homogeneous symplectic transformation

$(x,y;\partial_{x}\chi(x,y,\eta)\cdot\xi,\eta+\partial_{y}\chi(x,y,\eta)\cdot\xi)\mapsto(\chi(x,y,\eta),y+\partial_{\eta}\chi\cdot\xi;\xi,\eta)$

which preserves $\Sigma_{2}$, thus $|\xi|$ and $|\eta|$ are preserved modulo
multiplicative constants. In fact, when $\xi=0$ we get the mapping
$(x,y;0,\eta)\mapsto(\chi(x,y,\eta),y;0,\eta)$ which gives a homogeneous
change of $x$ variables. We put the amplitude of $F$ equal to 1 to simplify
the notation, actually the amplitude only has to equal to 1 near the wave
front set of the kernel of $F$.

By applying the operator $P$ we find

(2.8) $PFu(x,y)=(2\pi)^{-n}\iint
e^{i(\langle\chi(x,y,\eta)-z,\xi\rangle+\langle
y-w,\eta\rangle)}Q(x,y;\xi,\eta)u(z,w)\,dzdwd\xi d\eta$

so $PFu=FQu$, where

(2.9)
$Q(x,y;\xi,\eta)=\sum_{\alpha,\,\beta\in\mathbf{N}}\partial_{{\xi}}^{\alpha}\partial_{\eta}^{\beta}P(x,y;\partial_{x}\chi\cdot\xi,\eta+\partial_{y}\chi\cdot\xi){\mathcal{M}}^{\chi}_{\alpha,\beta}(x,y;\xi,\eta)/\alpha!\beta!$

with

(2.10)
${\mathcal{M}}^{\chi}_{\alpha,\beta}(x,y;\xi,\eta)=D_{z}^{\alpha}D_{w}^{\beta}e^{i\chi_{2}(x,y,z,w,\eta)\cdot\xi}\big{|}_{\begin{subarray}{c}z=x\\\
w=y\end{subarray}}$

where the phase function

(2.11) $\chi_{2}(x,y,z,w,\eta)=\chi(z,w,\eta)-\chi(x,y,\eta)\\\ +\langle
x-z,\partial_{x}\chi(x,y,\eta)\rangle+\langle
y-w,\partial_{y}\chi(x,y,\eta)\rangle$

vanishes of second order at $z=x$ and $w=y$, see [10, Th. 18.1.17] or [21,
Chapter 7, Theorem 3.1]. Thus there are no terms with $|\alpha|+|\beta|=1$ in
the expansion of (2.9). Since we only need the symbols modulo terms in $R$ it
suffices to compute the first two terms of the expansion (2.9).

We obtain from (2.1) by a straighforward computation that

(2.12) $Q(x,y;\xi,\eta)\cong
P(x,y;\partial_{x}\chi\cdot\xi,\eta+\partial_{y}\chi\cdot\xi)\\\
+\frac{1}{2i}\sum_{jk}a_{jk}(x,y;0,\eta)\partial_{x_{j}}\partial_{x_{k}}\chi(x,y,\eta)\cdot\xi$

modulo terms that are in $R$. In fact, $\partial_{{\xi}}a_{jk}\in S^{-1}$,
$\partial_{{\xi}}^{\alpha}\partial_{\eta}^{\beta}p_{2}=\mathcal{O}(|\eta|^{-1}|\xi|)$
if $\beta\neq 0$ and $|\alpha+\beta|\geq 2$, and
$\partial_{{\xi}}^{\alpha}\partial_{\eta}^{\beta}p_{1}\in S^{-1}$ if
$|\alpha+\beta|\geq 2$.

We shall first simplify by making a change of variables to diagonalize
$A=\\{a_{jk}\\}_{jk}=\operatorname{Hess}p_{2}$. Since $A$ is nondegenerate, we
can use the spectral projections to obtain either that
$A=\begin{pmatrix}A_{+}&0\\\ 0&A_{-}\end{pmatrix}$ or $A=A_{\pm}$, where $\pm
A_{\pm}$ is positive definite. Then we can use $(\pm A_{\pm})^{-1/2}$ to
construct real valued $\chi(x,y,\eta)$ so that
$(\partial_{x}\chi\cdot\xi)^{t}A\/\partial_{x}\chi\cdot\xi=|\xi^{\prime}|^{2}-|\xi^{\prime\prime}|^{2}=L(\xi)$
has constant coefficients near $w_{0}\in\Sigma_{2}$. Here
$(\xi^{\prime},\xi^{\prime\prime})=\xi$, and $L(\xi)$ is the real quadratic
form with the polarized bilinear form $L(\xi_{0},\xi)$. Of course, this may
also change the values of $p_{j}$ for $j<2$. But observe that the term
homogeneous of order 0 in the expansion (2.12) of $Q$ is equal to

$q_{0}(x,y;\xi,\eta)=p_{0}(x,y;\partial_{x}\chi_{k}(x,y,\eta)\cdot\xi,\eta+\partial_{y}\chi(x,y,\eta)\cdot\xi)$

modulo terms vanishing at $\Sigma_{2}$, so we find that
$\operatorname{Im}q_{0}(x,y;0,\eta)=\operatorname{Im}p_{0}(x,y;0,\eta)=0$.

Thus, we may assume that $A=L$ is constant in the following. Next we shall do
another change of symplectic variables to make $\operatorname{Im}C=0$ at
$\Sigma_{2}$. Assume that $\chi=(\chi_{1},\dots,\chi_{d})$ where $\chi_{j}$ is
parallel to $e_{j}$ for the standard base $e_{1},\dots,e_{d}$ of
$\mathbf{R}^{d}$, so that we can write
$\chi=(\chi_{1}e_{1},\dots,\chi_{d}e_{d})$ with scalar $\chi_{j}$. As before,
we get the expansion (2.12) with $\left\\{\,a_{k}\,\right\\}_{jk}=A=L$.

In order to get the symmetric part of the operator we shall compute the Weyl
symbol of $Q$ which is given by $\widetilde{Q}\cong
Q+\frac{i}{2}\sum_{j}\partial_{x_{j}}\partial_{\xi_{j}}q_{2}$ modulo $S^{0}$
where

$q_{2}(\xi)=L(\partial_{x}\chi\cdot\xi)=|\partial_{x^{\prime}}\chi\cdot\xi|^{2}-|\partial_{x^{\prime\prime}}\chi\cdot\xi|^{2}=\sum_{k=1}^{\ell}(\partial_{x_{k}}\chi\cdot\xi)^{2}-\sum_{k=\ell+1}^{m}(\partial_{x_{k}}\chi\cdot\xi)^{2}$

is the principal symbol of $\widetilde{Q}$, so (2.12) gives

(2.13) $\widetilde{Q}\cong
q_{2}+i\sum_{k=1}^{m}\left(2L(\partial_{x}\chi_{k},\partial_{x}\partial_{x_{k}}\chi_{k})-\frac{1}{2}L(\partial_{x})\chi_{k}\right)\xi_{k}+p_{1}(x,y;\partial_{x}\chi\cdot\xi,\eta+\partial_{y}\chi\cdot\xi)$

modulo $S^{0}$.

Let $q_{1}$ be the terms homogeneous of order 1 in the expansion (2.13) of
$\widetilde{Q}$, then we have $q_{1}(x,y;0,\eta)=p_{1}(x,y;0,\eta)$ on
$\Sigma_{2}$. We find from (2.13) that the $\xi_{k}$ derivative of $q_{1}$ at
$\Sigma_{2}$ is equal to

(2.14)
$2iL(\partial_{x}\chi_{k}(x,y,\eta),\partial_{x}\partial_{k}\chi_{k}(x,y,\eta))-\frac{i}{2}L(\partial_{x})\chi_{k}(x,y,\eta)\\\
+\partial_{\xi}p_{1}(x,y;0,\eta)\cdot\partial_{x}\chi_{k}(x,y,\eta)+\partial_{\eta}p_{1}(x,y;0,\eta)\cdot\partial_{y}\chi_{k}(x,y,\eta)$

Here $\operatorname{Im}\partial_{\eta}p_{1}=\partial_{\eta}f$ and
$\operatorname{Im}\partial_{\xi}p_{1}=\operatorname{Im}C$ on $\Sigma_{2}$.
Observe that the terms in $q_{1}$ vanishing of second order at $\Sigma_{2}$
are in $R$.

By taking the imaginary part and ignoring the term
$\partial_{\eta}f\cdot\partial_{y}\chi_{k}$ for now, we find
$\partial_{{\xi}}\operatorname{Im}q_{1}(x,y;0,\eta)=0$ if for any $k$ we have

(2.15)
$L(\partial_{x})\chi_{k}(x,y,\eta)-4L(\partial_{x}\chi_{k}(x,y,\eta),\partial_{x}\partial_{k}\chi_{k}(x,y,\eta))\\\
-2\operatorname{Im}C(x,y;\eta)\cdot\partial_{x}\chi_{k}(x,y,\eta)=0$

which is a quasilinear second order system of PDE on the leaves of
$\Sigma_{2}$ having real coefficients. Observe that this system is completely
decoupled with one equation for each $\chi_{k}$. As before, we we find that
$\operatorname{Im}q_{0}(x,y;0,\eta)=\operatorname{Im}p_{0}(x,y;0,\eta)=0$. In
order to get a suitable change of coordinates we shall solve system (2.15) in
a neighborhood of $w_{0}=(x_{0},y_{0},\eta_{0})$ with initial data $\chi=0$
and $|\partial_{x}\chi|\neq 0$ at $w_{0}$.

######  Proposition 2.1.

For any $v_{k}\in\mathbf{R}^{d}$, $1\leq k\leq d$, the equation (2.15) with
data $\chi_{k}=0$ and $\partial_{x}\chi_{k}=v_{k}$ at
$w_{0}=(x_{0},y_{0},\eta_{0})$ has a solution $\chi_{k}(x,y,\eta)\in
C^{\infty}$ in a neighborhood of $w_{0}$.

Proposition 2.1 follows from Theorem A.1 in Appendix A, and shows that this
initial value problem has a $C^{\infty}$ solution
$\chi(x,y,\eta)=(\chi_{1}(x,y,\eta),\chi_{2}(x,y,\eta)\cdots)$ near $w_{0}$
such that $\chi(x_{0},y_{0},\eta_{0})=0$ and
$\partial_{x}\chi(x_{0},y_{0},\eta_{0})=\operatorname{Id}$. By restricting
$\chi(x,y,\eta)$ to the set $|\eta|=1$ and extending it by homogeneity in
$\eta$, we obtain a homogeneous change of coordinates so that
$\partial_{{\xi}}\operatorname{Im}p_{1}=\operatorname{Im}C=0$ and
$\operatorname{Im}p_{0}=0$ at $\Sigma_{2}$ near $w_{0}$. Thus we have proved
the following result.

######  Proposition 2.2.

Assume that $P\in\Psi_{cl}^{m}$(X) satisfies the conditions in Theorem 1.15
microlocally near $w\in\Sigma_{2}$. By conjugation with elliptic Fourier
integral operators and multiplication with symmetric elliptic
pseudodifferential operators, we may assume that $X=T^{*}\mathbf{R}^{n}$, the
coordinates are $(x,t,y;\xi,\tau.\eta)$ so that
$\Sigma_{2}=\left\\{\,\xi=0\,\right\\}$ and $P$ is on the form

(2.16) $P\cong D_{t}+A^{w}+if_{1}^{w}$

microlocally near $w\in\Sigma_{2}$ modulo terms with symbols in $R$ given by
(2.3). Here $A=A_{2}+A_{1}+A_{0}\in S^{2}_{cl}$, $A_{j}\in S^{j}$, is real
valued, with principal symbol $A_{2}$ vanishing of second order at
$\Sigma_{2}$ with $\operatorname{Hess}A_{2}$ nondegenerate on the normal
bundle $N\Sigma_{2}$, and $A_{1}=0$ at $\Sigma_{2}$. Also, $f_{1}=f+f_{0}$ is
real and homogeneous of degree $1$ where $f$ does not depend on $\xi$ and
$f_{0}=\partial_{\eta}f\cdot r\cdot\xi$ with $r\in S^{0}$. By condition
$\operatorname{Sub_{r}}(\Psi)$ we have that $f=f_{1}\big{|}_{\Sigma_{2}}$ does
not change sign on the leaves of $\Sigma_{2}$ and the sign of $f$ on the
leaves do not change from $-$ to $+$ as $t$ increases. Here the sign of $f$ is
given by Definition 4.1.

Observe that $P$ in (2.16) is an evolution operator and it is of subprincipal
type.

######  Remark 2.3.

The normal form of the $L^{2}$ adjoint $P^{*}$ is (2.16) with $f_{1}$ replaced
by $-f_{1}$. Then $P^{*}$ satisfies condition
$\operatorname{Sub_{r}}(\overline{\Psi})$ which gives the opposite conditions
on sign changes as $t$ increases given by (3.3).

##  3\. The microlocal estimate

Next, we shall microlocalize and reduce the proof of Theorem 1.15 to the
semiclassical multiplier estimate of Proposition 3.6 for a microlocal normal
form of the adjoint operator. We shall consider operators given by Proposition
2.2

(3.1) $P^{*}\cong D_{t}+A^{w}+if^{w}(x,t,y;D_{t},D_{y})$

modulo terms with symbols in $R$ given by (2.3). Here $f_{1}=f+f_{0}$ is real
and homogeneous of degree $1$ where $f_{0}=\partial_{\eta}f\cdot r\cdot\xi$
with $r\in S^{0}$, $f$ does not depend on $\xi$ and

(3.2) $A=\sum_{jk}a_{jk}\xi_{j}\xi_{k}+\sum_{j}a_{j}\xi_{j}+a_{0}$

where $a_{jk}$ and $a_{j}\in S^{0}_{1,0}$ are real and homogeneous of degree
0, and $\\{a_{jk}\\}_{jk}$ is symmetric and nondegenerate.

In the following, we shall assume that $P^{*}$ satisfies condition
$\operatorname{Sub_{r}}(\overline{\Psi})$, so that the sign of
$f(t,x,y;\tau,\eta)$ is constant in $x$ and

(3.3) $f(t,x_{0},y_{0};\tau_{0},\eta_{0})>0\quad\text{and $s>t$}\implies
f(s,x,y_{0};\tau_{0},\eta_{0})\geq 0\ \ \forall\,x$

so that the sign on the leaves cannot change from $+$ to $-$ as $t$ increases,
see Definition 4.1 for the definition of the sign. Observe that if ${\chi}\geq
0$ then ${\chi}f$ also satisfies condition
$\operatorname{Sub_{r}}(\overline{\Psi})$, so this condition can be
microlocalized.

In order to prove Theorem 1.15 we shall make a second microlocalization using
the specialized symbol classes of the Weyl calculus. We shall therefore recall
the definitions of the Weyl calculus: let $g_{w}$ be a Riemannean metric on
$T^{*}\mathbf{R}^{n}$, $w=(x,{\xi})$, then we say that $g$ is slowly varying
if there exists $c>0$ so that $g_{w_{0}}(w-w_{0})<c$ implies $g_{w}\cong
g_{w_{0}}$, i.e., $1/C\leq g_{w}/g_{w_{0}}\leq C.$ Let ${\sigma}$ be the
standard symplectic form on $T^{*}\mathbf{R}^{n}$, and assume
$g^{\sigma}(w)\geq g(w)$ where $g^{\sigma}$ is the dual metric of $w\mapsto
g({\sigma}(w))$. We say that $g$ is ${\sigma}$ temperate if it is slowly
varying and there exists $C>0$ and $N\in\mathbf{N}$ so that

$g_{w}\leq Cg_{w_{0}}(1+g^{\sigma}_{w}(w-w_{0}))^{N}\qquad\forall\ w,\
w_{0}\in T^{*}\mathbf{R}^{n}$

Actually, ${\sigma}$ temperate metrics with $g\leq g^{\sigma}$ are called
Hörmander metrics. A positive real valued function $m(w)$ on
$T^{*}\mathbf{R}^{n}$ is $g$ continuous if there exists $c>0$ so that
$g_{w_{0}}(w-w_{0})<c$ implies $m(w)\cong m(w_{0})$. We say that $m$ is
${\sigma}$, $g$ temperate if it is $g$ continuous and there exists $C>0$ and
$N\in\mathbf{N}$ so that

$m(w)\leq Cm(w_{0})(1+g^{\sigma}_{w}(w-w_{0}))^{N}\qquad\forall\ w,\ w_{0}\in
T^{*}\mathbf{R}^{n}.$

If $m$ is ${\sigma}$, $g$ temperate, then $m$ is a weight for $g$ and we can
define the symbol classes: $a\in S(m,g)$ if $a\in
C^{\infty}(T^{*}\mathbf{R}^{n})$ and

(3.4) $|a|^{g}_{j}(w)=\sup_{T_{i}\neq
0}\frac{|a^{(j)}(w,T_{1},\dots,T_{j})|}{\prod_{1}^{j}g_{w}(T_{i})^{1/2}}\leq
C_{j}m(w)\qquad\forall\ w\in T^{*}\mathbf{R}^{n}\qquad\text{for $j\geq 0$,}$

which gives the seminorms of $S(m,g)$. If $a\in S(m,g)$ then we say that the
corresponding Weyl operator $a^{w}\in\operatorname{Op}S(m,g)$. For more on the
Weyl calculus, see [10, Section 18.5].

######  Definition 3.1.

Let $m$ be a weight for the metric $g$. We say that $a\in S^{+}(m,g)$ if $a\in
C^{\infty}(T^{*}\mathbf{R}^{n})$ and $|a|_{j}^{g}\leq C_{j}m$ for $j\geq 1$.

Observe that by the mean value theorem we find that

(3.5) $|a(w)-a(w_{0})|\leq
C_{1}\sup_{{\theta}\in[0,1]}g_{w_{\theta}}(w-w_{0})^{1/2}m(w_{\theta})\\\ \leq
C^{\prime}m(w_{0})(1+g_{w_{0}}^{{\sigma}}(w-w_{0}))^{(3N+1)/{2}}$

where $w_{\theta}={\theta}w+(1-{\theta})w_{0}$, since
$w_{\theta}-w_{0}=\theta(w-w_{0})$ for some $0<\theta<1$ and

$g_{w_{\theta}}(w-w_{0})\lesssim g^{\sigma}_{w_{\theta}}(w-w_{0})\lesssim
g^{\sigma}_{w_{0}}(w-w_{0})(1+g^{\sigma}_{w_{0}}(w-w_{0}))^{N}$

Thus $m+|a|$ is a weight for $g$ and $a\in S(m+|a|,g)$, so the operator
$a^{w}$ is well-defined.

######  Lemma 3.2.

Assume that $m_{j}$ is a weight for $g_{j}=h_{j}g^{\sharp}\leq
g^{\sharp}=(g^{\sharp})^{\sigma}\leq g_{j}^{\sigma}\leq h_{j}^{-1}g^{\sharp}$
and $a_{j}\in S^{+}(m_{j},g_{j})$, $j=1$, $2$. Let $g=g_{1}+g_{2}$ and
$h^{2}=\sup g_{1}/g_{2}^{\sigma}=\sup g_{2}/g_{1}^{\sigma}=h_{1}h_{2}$, then

(3.6)
$a_{1}^{w}a_{2}^{w}-(a_{1}a_{2})^{w}\in\operatorname{Op}S(m_{1}m_{2}h,g)$

with the usual expansion of (3.6) in terms in $S(m_{1}m_{2}h^{k},g)$, $k\geq
1$. We also have that

(3.7)
$\operatorname{Re}a_{1}^{w}a_{2}^{w}-(a_{1}a_{2})^{w}\in\operatorname{Op}S(m_{1}m_{2}h^{2},g)$

if $a_{j}\in C^{\infty}$ is real and $|a_{j}|^{g_{j}}_{k}\leq C_{k}m_{j}$,
$k\geq 2$, for $j=1$, $2$. In that case we have $a_{j}\in
S(m_{j}+|a_{j}|+|a_{j}|^{g_{j}}_{1},g_{j})$.

###### Proof.

As shown after Definition 3.1 we have that $m_{j}+|a_{j}|$ is a weight for
$g_{j}$ and $a_{j}\in S(m_{j}+|a_{j}|,g_{j})$, $j=1$, 2. Thus
$a_{1}^{w}a_{2}^{w}\in\operatorname{Op}S((m_{1}+|a_{1}|)(m_{2}+|a_{2}|),g)$ is
given by Proposition 18.5.5 in [10]. We find that
$a_{1}^{w}a_{2}^{w}-(a_{1}a_{2})^{w}=a^{w}$ with

(3.8)
$a(w)=E(\tfrac{i}{2}{\sigma}(D_{w_{1}},D_{w_{2}}))\tfrac{i}{2}{\sigma}(D_{w_{1}},D_{w_{2}})a_{1}(w_{1})a_{2}(w_{2})\big{|}_{w_{1}=w_{2}=w}$

where $E(z)=(e^{z}-1)/z=\int_{0}^{1}e^{{\theta}z}\,d{\theta}$. Here
${\sigma}(D_{w_{1}},D_{w_{2}})a_{1}(w_{1})a_{2}(w_{2})\in S(MH,G)$ where
$M(w_{1},w_{2})=m_{1}(w_{1})m_{2}(w_{2})$,
$G_{w_{1},w_{2}}(z_{1},z_{2})=g_{1,w_{1}}(z_{1})+g_{2,w_{2}}(z_{2})$ and
$H^{2}(w_{1},w_{2})=h_{1}(w_{1})h_{2}(w_{2})=\sup\,G_{w_{1},w_{2}}/G_{w_{1},w_{2}}^{\sigma}$
so that $H(w,w)=h(w)$. The proof of Theorem 18.5.5 in [10] works when
${\sigma}(D_{w_{1}},D_{w_{2}})$ is replaced by
${\theta}{\sigma}(D_{w_{1}},D_{w_{2}})$, uniformly in $0\leq{\theta}\leq 1$
(when $\theta=0$ we just get the Poisson parenthesis
$\tfrac{i}{2}\\{a_{1},a_{2}\\}$). By integrating over ${\theta}\in[0,1]$ we
obtain that $a(w)$ has an asymptotic expansion in $S(m_{1}m_{2}h^{k},g)$,
which proves (3.6).

If $|a_{j}|^{g_{j}}_{k}\leq C_{k}m_{j}$, $k\geq 2$, then we have by Taylor’s
formula as in (3.5) that

$|a_{j}(w)-a_{j}(w_{0})|\leq
g_{w_{0}}(w-w_{0})^{1/2}|a_{j}|_{1}^{g}(w_{0})+C_{1}\sup_{{\theta}\in[0,1]}g_{w_{\theta}}(w-w_{0})m(w_{\theta})\\\
\leq
C^{\prime}(|a_{j}|_{1}^{g}(w_{0})+m(w_{0}))(1+g_{w_{0}}^{{\sigma}}(w-w_{0}))^{2N+1}$
$|\langle T,\partial_{w}a_{j}(w)\rangle-\langle
T,\partial_{w}a_{j}(w_{0})\rangle|\leq
C_{2}\sup_{{\theta}\in[0,1]}g_{w_{\theta}}(T)^{1/2}g_{w_{\theta}}(w-w_{0})^{1/2}m(w_{\theta})\\\
\leq
C_{3}g_{w_{0}}(T)^{1/2}m(w_{0})(1+g_{w_{0}}^{{\sigma}}(w-w_{0}))^{(4N+1)/{2}}$

thus $m_{j}+|a_{j}|+|a_{j}|^{g_{j}}_{1}$ is a weight for $g_{j}$ and clearly
$a_{j}\in S(m_{j}+|a_{j}|+|a_{j}|^{g_{j}}_{1},g_{j})$.

Now if $a_{1}$ and $a_{2}$ are real, then
$\operatorname{Re}a_{1}^{w}a_{2}^{w}-(a_{1}a_{2})^{w}=a^{w}$ with

$a(w)=\operatorname{Re}E(\tfrac{i}{2}{\sigma}(D_{w_{1}},D_{w_{2}}))(\tfrac{i}{2}{\sigma}(D_{w_{1}},D_{w_{2}}))^{2}a_{1}(w_{1})a_{2}(w_{2})/2\big{|}_{w_{1}=w_{2}=w}$

where ${\sigma}(D_{w_{1}},D_{w_{2}})^{2}a_{1}(w_{1})a_{2}(w_{2})\in
S(MH^{2},G)$, with the same $E$, $M$, $G$ and $H$ as before. The proof of
(3.7) then follows in the same way as the proof of (3.6). ∎

######  Remark 3.3.

The conclusions of Lemma 3.2 also hold if $a_{1}$ has values in
$\mathcal{L}(B_{1},B_{2})$ and $a_{2}$ in $B_{1}$ where $B_{1}$ and $B_{2}$
are Banach spaces, then $a_{1}^{w}a_{2}^{w}$ has values in $B_{2}$.

For example, if $\left\\{\,a_{j}\,\right\\}_{j}\in S(m_{1},g_{1})$ with values
in $\ell^{2}$, and $b_{j}\in S(m_{2},g_{2})$ uniformly in $j$, then
$\left\\{\,a_{j}^{w}b_{j}^{w}\,\right\\}_{j}\in\operatorname{Op}(m_{1}m_{2},g)$
with values in $\ell^{2}$.

######  Remark 3.4.

For pseudodifferential operators with the Kohn-Nirenberg quantization, we have
by Theorem 4.5 and (4.13) in [8] that $a_{1}(x,D)a_{2}(x,D)=a(x,D)$ with

(3.9) $a(x,\xi)=e^{i\langle
D_{\xi},D_{y}\rangle}a_{1}(x,\xi)a_{1}(y,\eta)\big{|}_{\begin{subarray}{c}y=x\\\
\eta=\xi\end{subarray}}$

As in the proof of Lemma 3.2 we find that $a_{1}(x,D)a_{2}(x,D)-a(x,D)=r(x,D)$
with

(3.10) $r(x,\xi)=E({i\langle
D_{\xi},D_{y}\rangle})\partial_{{\xi}}a_{1}(x,\xi)D_{y}a_{1}(y,\eta)\big{|}_{\begin{subarray}{c}y=x\\\
\eta=\xi\end{subarray}}$

where $E(z)=(e^{z}-1)/z=\int_{0}^{1}e^{{\theta}z}\,d{\theta}$.

To prove Theorem 1.15 we shall prove an estimate for the microlocal normal
form of the adjoint operator. Since the proof is rather long, we will take it
in two steps, and the first is microlocal estimate. For that, we shall use the
symbol classes $S^{m}_{1,0}$ and $S^{m}_{1/2,1/2}$.

######  Proposition 3.5.

Assume that $P$ is as in Proposition 2.2 microlocally near
$w_{0}\in\Sigma_{2}$ and that $t=t_{0}$ and $x=x_{0}$ at $w_{0}$. Then there
exist $T_{0}>0$ and a real valued symbol $b_{T}\in S^{1}_{1/2,1/2}$ with
homogeneous gradient $\nabla
b_{T}=(\partial_{z}b_{T},|{{\zeta}}|\partial_{\zeta}b_{T})\in S^{1}_{1/2,1/2}$
uniformly for $0<T\leq T_{0}$, $(z,\zeta)\in T^{*}\mathbf{R}^{n}$, such that
for every $N>0$ there exists $C_{N}>0$ so that

(3.11) $\|b_{T}^{w}u\|^{2}_{(-1/2)}+\|D_{x}u\|^{2}+\|u\|^{2}\leq
C_{N}\left(T\operatorname{Im}\left(P^{*}u,b_{T}^{w}u\right)+\|u\|_{(-N)}^{2}\right)+\|\psi^{w}u\|^{2}$

for $u\in C_{0}^{\infty}$ having support where $|t-t_{0}|\leq T$ and
$|x-x_{0}|\leq T$. Here $\psi\in S^{2}$,
$w_{0}\notin\operatorname{WF}\psi^{w}$ and the constants $T_{0}$, $C_{N}$ and
the seminorms of $b_{T}$ only depend on the seminorms of’ the symbols in $P$.

###### Proof that Proposition 3.5 gives Theorem 1.15.

We shall prove that there exists ${\phi}\in S^{0}_{1,0}$ such that ${\phi}\geq
0$ and $\phi=1$ in a conical neighborhood of $w_{0}\in\Sigma_{2}$, and $R\in
S^{3/2}_{1,0}$ with $w_{0}\notin\operatorname{WF}R^{w}$ so that for any $N>0$
there exists $C_{N}>0$ such that

(3.12) $\|{\phi}^{w}u\|\leq
C_{N}\left(\|{\phi}^{w}P^{*}u\|_{(1/2)}+\|R^{w}u\|+\|u\|_{(-N)}\right)\qquad
u\in C_{0}^{\infty}$

Here $\|u\|_{(s)}$ is the usual $L^{2}$ Sobolev norm, so by Remark 1.11 we
obtain that $P$ is solvable with a loss of $5/2$ derivatives in a conical
neighborhood of $w_{0}$ since $w_{0}\notin\operatorname{WF}(1-\phi)^{w}$ and
$m=2$.

We may assume that $m=2$ and $P\in\Psi^{2}_{1,0}$ is on the form in
Proposition 2.2 in a conical neighborhood of $w_{0}$. Let $\phi\geq 0$ have
support in a smaller conical neighborhood such that $\max{(|t|,|x|)}\leq T\leq
T_{0}$ in $\operatorname{\rm supp}{\phi}$, $\psi$ in (3.11) vanishes on
$\operatorname{\rm supp}\phi$ and $\phi=1$ in a conical neighborhood of
$w_{0}$. Then by applying the estimate (3.11) on ${\phi}^{w}u$ we obtain for
any $N>0$

(3.13)
$\|b_{T}^{w}{\phi}^{w}u\|^{2}_{(-1/2)}+\|{\phi}^{w}u\|^{2}+\|D_{x}{\phi}^{w}u\|^{2}\leq
C_{N}\left(T\operatorname{Im}\left(P^{*}{\phi}^{w}u,b_{T}^{w}{\phi}^{w}u\right)+\|u\|^{2}_{(-N)}\right)$

where $C_{N}>0$ and $b_{T}^{w}\in{\Psi}^{1}_{1/2,1/2}$ is symmetric with
homogeneous gradient $\nabla b_{T}\in S^{1}_{1/2,1/2}$.

By Cauchy-Schwarz,

(3.14)
$|\left(P^{*}{\phi}^{w}u,b_{T}^{w}{\phi}^{w}u\right)|\lesssim\|P^{*}{\phi}^{w}u\|^{2}_{(1/2)}+\|b_{T}^{w}{\phi}^{w}u\|^{2}_{(-1/2)}$

and
$\|P^{*}{\phi}^{w}u\|^{2}_{(1/2)}\leq\|{\phi}^{w}P^{*}u\|^{2}_{(1/2)}+\|[P^{*},{\phi}^{w}]u\|^{2}_{(1/2)}$
where the commutator $[P^{*},{\phi}^{w}]\in\Psi^{1}$ with
$w_{0}\notin\operatorname{WF}[P^{*},{\phi}^{w}]$.

Thus, for $T$ small enough, we obtain for any $N>1$ the estimate

(3.15)
$\|{\phi}^{w}u\|^{2}\leq\|b_{T}^{w}{\phi}^{w}u\|^{2}_{(-1/2)}+\|{\phi}^{w}u\|^{2}+\|D_{x}{\phi}^{w}u\|^{2}\\\
\leq
C_{N}\left(T\|{\phi}^{w}P^{*}u\|^{2}_{(1/2)}+T\|[P^{*},\phi^{w}]u\|_{(1/2)}^{2}+\|u\|^{2}_{(-N)}\right)$

This gives the estimate (3.12) with $R=T\langle
D\rangle^{1/2}[P^{*},{\phi}^{w}]\in\Psi^{3/2}$ which completes the proof of
Theorem 1.15. ∎

Next we shall derive a semiclassical estimate for the proof of Proposition
3.5. We shall assume that the coordinates are chosen as in Proposition 2.2 so
that $\Sigma_{2}=\left\\{\,\xi=0\,\right\\}$. The proof involves a second
microlocalization near
$(t_{0},x_{0},y_{0};\tau_{0},0,{\eta}_{0})=(z_{0};\zeta_{0})\in\Sigma_{2}$
using the homogeneous metric $g=g_{1,0}$. Then $\sup g/g^{\sigma}=h^{2}\leq 1$
are constant and $|\xi|\lesssim h^{-1}\cong\langle(\tau,{\eta})\rangle$. We
have $g/h=g^{\sharp}\cong g_{1/2,1/2}$ where
$g^{\sharp}=(g^{\sharp})^{\sigma}$ is constant, $S^{k}_{1,0}=S(h^{-k},g)$ and
$S^{k}_{1/2,1/2}=S(h^{-k},g^{\sharp})$ for $k\in\mathbf{R}$. Observe that now
the the symbols of the error terms $R$ can be written $\langle
R_{2}\xi,\xi\rangle+R_{1}\cdot\xi+R_{0}$ where $R_{j}\in S(h,g)$.

######  Proposition 3.6.

Assume that $P^{*}\cong D_{t}+A^{w}+if_{1}^{w}$ with real $f_{1}=f+f_{0}$
where $f\in S(h^{-1},g)$ is independent of $\xi$ and satisfies condition
$\operatorname{Sub_{r}}(\overline{\Psi})$ in (3.3),
$f_{0}=\partial_{\eta}f\cdot r\cdot\xi\in S(h^{-1},g)$ with $r\in S(1,g)$ and

(3.16) $A=\sum_{jk}a_{jk}\xi_{j}\xi_{k}+\sum_{j}a_{j}\xi_{j}+a_{0}$

where $a_{jk}$ and $a_{j}\in S(1,g)$ are real and $\\{a_{jk}\\}_{jk}$ is
symmetric and nondegenerate, here $0<h\leq 1$ and
$g^{\sharp}=(g^{\sharp})^{\sigma}$ are constant. Then there exist $T_{0}>0$
and real valued symbols $b_{T}(t,x,{\xi})\in S(h^{-1/2},g^{\sharp})\bigcap
S^{+}(1,g^{\sharp})+S(h^{-1/2},g)$ uniformly for any $0<T\leq T_{0}$ and
$|x|\leq T$, so that

(3.17) $h^{1/2}\left(\|b_{T}^{w}u\|^{2}+\|D_{x}u\|^{2}+\|u\|^{2}\right)\leq
C_{0}T\operatorname{Im}\left(P^{*}u,b_{T}^{w}u\right)+\|\Psi^{w}u\|^{2}$

when $u\in C_{0}^{\infty}$ has support where $|t|\leq T$ and $|x|\leq T$. Here
$\Psi\in S^{2}$, $\Sigma_{2}\bigcap\operatorname{\rm supp}\Psi=\emptyset$ and
$C_{0}$, $T_{0}$ and the seminorms of $b_{T}$ only depend on the seminorms of
$f$ in $S(h^{-1},g)$.

Proposition 3.6 will be proved at the end of Section 8.

###### Proof of that Proposition 3.6 gives Proposition 3.5.

First note that in the estimate (3.11), $P^{*}$ can be perturbed by operators
with symbols in $R$. In fact, if $\widetilde{R}=\langle
R_{2}D_{x},D_{x}\rangle+\langle R_{1},D_{x}\rangle+R_{0}$ with
$R_{j}\in\Psi^{-1}_{1,0}$, then
$\operatorname{Re}b_{T}^{w}\widetilde{R}=\langle
S_{2}D_{x},D_{x}\rangle+\langle S_{1},D_{x}\rangle+S_{0}$ where
$S_{j}\in\Psi^{0}_{1/2,1/2}$ is continuous on $L^{2}$. Thus, this term can be
estimated by the last two terms in the left hand side of (3.11) for small
enough $T$.

As before, we shall include $\tau$ in the variables $\eta$ and use the
coordinates $(z,\zeta)\in T^{*}\mathbf{R}^{n}$. For the localization, we shall
take ${\phi}\in S^{0}_{1,0}$ such that $0\leq{\phi}\leq 1$, $\phi=1$ in a
conical neighborhood of $w_{0}\in\Sigma_{2}$ such that $\phi$ is supported
where $|\xi|\lesssim h^{-1}$, $\Psi=0$ and $P$ is on the normal form (2.16).
By taking $\phi(z,\zeta)=\phi_{0}(z)\phi_{1}(\zeta)$ as a product, we obtain
that $\partial_{t,x}\varphi$ has support where $\max(|t|,|x|)|\geq T_{0}$ for
some $T_{0}>0$.

Next, we shall microlocalize in $\zeta=(\tau,\xi,\eta)$ with respect to the
homogeneous metric $g=g_{1,0}$ with a partition of unity
$\left\\{\,{\varphi}_{j}(\zeta)\,\right\\}_{j}\in S^{0}_{1,0}=S(1,g)$
independent of $z$ with values in $\ell^{2}$ such that
$\sum_{j}\varphi_{j}^{2}=1$, $0\leq{\varphi}_{\leq}1$ and $\varphi_{j}$ is
supported where $\langle{\zeta}\rangle\cong h_{j}^{-1}$. Then we can get a
partition of unity in a conical neighborhood of $w_{0}$ by putting
$\phi_{j}=\phi\varphi_{j}$, so that $\phi_{j}$ is supported where
$|\xi|\lesssim h_{j}^{-1}$, $\sum_{j}\phi_{j}^{2}=\phi^{2}$ and
$\partial_{t,x}\phi_{j}$ has support where $\max(|t|,|x|)|\geq T_{0}$.

Since the functions $\phi_{j}$ are real, we find from the calculus and
symmetry that $\sum_{j}\phi_{j}^{w}\phi_{j}^{w}=\phi^{w}\phi^{w}+r^{w}$ where
$r\in S^{-2}$ is real valued, which gives
$\|\phi^{w}v\|^{2}\leq\sum_{j}\|\phi_{j}^{w}v\|^{2}+C\|v\|_{(-2)}^{2}$ for
$v\in C^{\infty}_{0}$ and by continuity we have
$\sum_{j}\|\phi_{j}^{w}v\|^{2}\lesssim\|v\|^{2}$. By cutting off, we find that

(3.18)
$\|v\|^{2}_{(-2)}\lesssim\sum_{j}\|h_{j}^{2}\phi_{j}^{w}v\|^{2}+\|\langle
D\rangle^{-2}(1-\phi)^{w}v\|^{2}$

Since the cut-off functions have values in $\ell^{2}$ the calculus gives that
the operators that we obtain from these will have values in $\ell^{2}$ (or
scalar values after summation) by Remark 3.3.

By possibly shrinking $T_{0}$ we can also choose real symbols
$\left\\{\,{\psi}_{j}\,\right\\}_{j}\in S^{0}_{1,0}$ with values in
$\ell^{2}$, such that $0\leq{\psi}_{j}\leq 1$ has support in a $g$
neighborhood of $w_{j}$ of radius $2T_{0}$ so that
${\psi}_{j}{\phi}_{j}={\phi}_{j}$. If $T_{0}$ is small enough, we may assume
that $P$ is on the normal form (2.16) and $g_{1,0}\cong g=hg^{\sharp}$ is
constant in $\operatorname{\rm supp}\psi_{j}$, and that there is a fixed bound
on number of overlapping supports of $\psi_{j}$, see [10, Section 18.5]. Then
we obtain that $S^{m}_{1,0}=S(h_{j}^{-m},g_{j})$ and
$S^{m}_{1/2,1/2}=S(h_{j}^{-m},g^{\sharp})$ in $\operatorname{\rm
supp}{\psi}_{j}$ for $m\in\mathbf{R}$, where $h_{j}\leq 1$,
$g_{j}=h_{j}g^{\sharp}$.

The microlocalization of $P$ is $P_{j}=D_{t}+A_{j}^{w}+if_{j}^{w}$ where
$A_{j}=\psi_{j}A+(1-\psi_{j})A_{0,j}\in S(\langle\xi\rangle^{2},g_{j})$ with
$A_{0,j}(t,x,y;\eta,\xi)=\sum_{k\ell}a_{k\ell}(w_{j})\xi_{k}\xi_{\ell}$,
$f_{1j}={\psi}_{j}f_{1}\in S(h_{j}^{-1},g_{j})$ uniformly in $j$ satisfying
condition $\operatorname{Sub_{r}}(\Psi)$. If the support of $\psi_{j}$ is
small enough, then the Hessian $\partial_{{\xi}}^{2}A_{j}$ is nondegenerate at
$\Sigma_{2}$.

Then, by using Proposition 3.6 with $P_{j}$ and substituting ${\phi}_{j}^{w}u$
in (3.17), we obtain real $b_{j,T}\in S(h_{j}^{-1/2},g_{j}^{\sharp})\bigcap
S^{+}(1,g_{j}^{\sharp})+S(h_{j}^{1/2}\langle\xi\rangle,g_{j})$ uniformly so
that

(3.19)
$\|b_{j,T}^{w}{\phi}_{j}^{w}u\|^{2}+\|{\phi}_{j}^{w}u\|^{2}+\|D_{x}{\phi}_{j}^{w}u\|^{2}\\\
\leq
C_{0}Th_{j}^{-1/2}\operatorname{Im}\left(P_{j}^{*}{\phi}_{j}^{w}u,b_{j,T}^{w}{\phi}_{j}^{w}u\right)+C_{N}\|{\phi}_{j}^{w}u\|^{2}_{(-N)}$

for $u\in C_{0}^{\infty}$ having support where $\max(|t|,|x|)\leq T\leq
T_{0}$. We have $P_{j}^{*}\phi_{j}^{w}=\phi_{j}^{w}P_{j}^{*}+Q_{j}^{w}$, where

(3.20)
$Q_{j}^{w}=[D_{t},\phi_{j}^{w}]+[A_{j}^{w},\phi_{j}^{w}]-i[f_{1j}^{w},{\phi}_{j}^{w}]\in\operatorname{Op}S(h_{j}^{-1},g_{j})$

Since the commutator of symmetric operators is antisymmetric, the calculus
gives that $\operatorname{Im}Q_{j}\in S(h_{j}\langle\xi\rangle^{2},g_{j})$
when $\max(|t|,|x|)|\leq T_{0}$ since then $\partial_{t,x}\phi_{j}=0$ which
gives
$[A_{j}^{w},\phi_{j}^{w}]=\sum_{k\ell}[a_{k\ell}^{w},\phi_{j}^{w}]D_{x_{k}}D_{x_{\ell}}$
and $[D_{t},\phi_{j}^{w}]=0$. This also gives
$D_{x}{\phi}_{j}^{w}u={\phi}_{j}^{w}D_{x}u$ when u is supported where
$\max(|t|,|x|)|\leq T_{0}$. We also have $\operatorname{Re}Q_{j}\in
S(1,g_{j})$ since $\langle\xi\rangle\lesssim h_{j}^{-1}$ in $\operatorname{\rm
supp}\phi_{j}$.

By using the calculus we obtain that ${\phi}_{j}^{w}P={\phi}_{j}^{w}P_{j}$
modulo $\operatorname{Op}S(h_{j}^{N},g_{j}),\ \forall N$, since
${\psi}_{j}{\phi}_{j}={\phi}_{j}$. We obtain for any $N$ that

(3.21)
$\|b_{j,T}^{w}{\phi}_{j}^{w}u\|^{2}+\|{\phi}_{j}^{w}u\|^{2}+\|{\phi}_{j}^{w}D_{x}u\|^{2}\\\
\leq
C_{0}T\left(\operatorname{Im}\left(P^{*}u,B_{j,T}^{w}u\right)+h_{j}^{-1/2}\operatorname{Im}\left(Q_{j}^{w}u,b_{j,T}^{w}{\phi}_{j}^{w}u\right)\right)+C_{N}\|h_{j}^{N}{\phi}_{j}^{w}u\|^{2}\qquad\forall\,j$

if $u\in C_{0}^{\infty}$ supported where $\max(|t|,|x|)\leq T\leq T_{0}$. Here

$B_{j,T}^{w}=h_{j}^{-1/2}{\phi}_{j}^{w}b_{j,T}^{w}{\phi}_{j}^{w}\in\operatorname{Op}S(h_{j}^{-1},g_{j}^{\sharp})\bigcap\operatorname{Op}S^{+}(h_{j}^{-1/2},g_{j}^{\sharp})+\operatorname{Op}S(\langle\xi\rangle,g_{j})$

uniformly and by symmetry $B_{j,T}$ is real. Since
${\phi}_{j}^{w}b_{j,T}^{w}\cong(b_{j,T}{\phi}_{j})^{w}$ modulo
$\operatorname{Op}S(h_{j}^{1/2},g_{j}^{\sharp})$ and
$\operatorname{Re}Q_{j}\in S(1,g_{j})$ we find that
$\\{h_{j}^{-1/2}\operatorname{Im}{\phi}_{j}^{w}b_{j,T}^{w}Q_{j}^{w}\\}_{j}\in\operatorname{Op}S(\langle\xi\rangle^{2},g^{\sharp})$
with values in $\ell^{2}$ when $\max(|t|,|x|)|\leq T_{0}$. Thus we may find
$\psi\in S^{1}$ with support outside a conical neighborhood of $w_{0}$ so that

$\sum_{j}h_{j}^{-1/2}\operatorname{Im}\left(Q_{j}^{w}u,b_{j,T}^{w}{\phi}_{j}^{w}u\right)\lesssim\|u\|^{2}+\|D_{x}u\|^{2}+\|\psi^{w}u\|^{2}$

if $u\in C_{0}^{\infty}$ supported where $\max(|t|,|x|)\leq T_{0}$.

Let $b_{T}^{w}=\sum_{j}B_{j,T}^{w}$, then by the finite bound on the overlap
of the supports we find that

(3.22)
$\|b_{T}^{w}u\|_{(-1/2)}^{2}\lesssim\left\|\sum_{j}h_{j}^{1/2}B_{j,T}^{w}u\right\|^{2}=\left\|\sum_{j}{\phi}_{j}^{w}b_{j,T}^{w}{\phi}_{j}^{w}u\right\|^{2}\\\
\lesssim\sum_{j}\|b_{j,T}^{w}{\phi}_{j}^{w}u\|^{2}+\|u\|_{(-N)}^{2}$

since $\langle B_{j,T}^{w}u,B_{k,T}^{w}u\rangle=0$ if $|j-k|\gg 1$. Thus, by
summing up we obtain

(3.23) $\|b_{T}^{w}u\|^{2}_{(-1/2)}+\|u\|^{2}+\|D_{x}u\|^{2}\\\ \leq
C_{1}\left(T(\operatorname{Im}\left(P^{*}u,b_{T}^{w}u\right)+\|u\|^{2}+\|D_{x}u\|^{2}+\|\psi^{w}u\|^{2})+\|u\|_{(-N)}^{2}+\|(1-\phi)^{w}u\|_{(1)}^{2}\right)$

for $u\in C_{0}^{\infty}$ having support where $\max(|t|,|x|)\leq T\leq
T_{0}$. Here we find that $w_{0}\notin\operatorname{WF}\psi^{w}\
\bigcup\operatorname{WF}(1-\phi)^{w}$ which gives (3.11) for small enough $T$.
We also have that
$b_{T}^{w}=\sum_{j}h_{j}^{-1/2}{\phi}_{j}^{w}b_{j,T}^{w}{\phi}_{j}^{w}\in\Psi^{1}_{1/2,1/2}$
since ${\phi}_{j}\in S(1,g_{j})$ is supported where
$\langle{\zeta}\rangle\simeq h_{j}^{-1}$ and $b_{j,T}\in
S(h_{j}^{-1/2},g^{\sharp})$. The homogeneous gradient $\nabla b_{T}\in
S^{1}_{1/2,1/2}$ since $b_{j,T}\in S^{+}(1,g^{\sharp})$ and the homogeneous
gradient is equal to $h^{-1/2}$ times the gradient in coordinates which are
$g^{\sharp}$ orthonormal. This finishes the proof of Proposition 3.5. ∎

It remains to prove Proposition 3.6, which will be done at the end of Section
8. The proof involves the construction of a multiplier $b^{w}_{T}$, and it
will occupy most of the remaining part of the paper.

##  4\. The symbol classes

In this section we shall define the symbol classes we shall use. We shall
follow Section 3 in [4] with some changes due to the different conditions and
normal forms. We shall study the subprincipal symbol $p_{s}=\tau+if_{1}$ with
$f_{1}=f+f_{0}$ where $f(t,x,y,\tau,\eta)\in S(h^{-1},g)$ and
$f_{0}=\partial_{\eta}f\cdot r\cdot\xi$ where $\partial_{\eta}f\cdot r\in
S(1,g)$. The metric is localized and assumed to be constant, but the result
holds in general for $\sigma$ temperate metrics $g\leq h^{2}g^{\sigma}$.

Since we are going to study the adjoint, we shall also assume that
$f(t,x,\tau,w)\in S(h^{-1},g)$ is independent of $\xi$ and satisfies condition
$\operatorname{Sub_{r}}(\overline{\Psi})$ in (3.3). Here $g=g_{1,0}$ is the
usual homogeneous metric, $x\in\mathbf{R}^{m}$, $(t,\tau)\in T^{*}\mathbf{R}$
and $w=(y,\eta)\in T^{*}\mathbf{R}^{n-m-1}$ as in Section 2. We have that
$g=hg^{\sharp}$ where $g^{\sharp}\leq(g^{\sharp})^{\sigma}$, in the case of
the homogeneous metric we have

(4.1)
$g^{\sharp}=(dt^{2}+|dx|^{2}+|dy|^{2})/h+h(d\tau^{2}+|d\xi|^{2}+|d\eta|^{2})$

We shall construct a metric, weight and multiplier adapted to $f$, so the
symbols in this section will be independent of $\xi$ except for $f_{0}$, which
will be handled as an error term in the estimates, see Remark 4.8. We shall
suppress the $\xi$ variables and assume that we have choosen $g^{\sharp}$
orthonormal coordinates so that $g^{\sharp}$ is the euclidean metric so that
$g^{\sharp}(t,x,\tau,w)=|(t,x,\tau,w)|^{2}$. Then we have
$|f^{\prime}|=|f|^{g^{\sharp}}_{1}\lesssim h^{-1/2}$,
$|f^{\prime\prime}|=|f|^{g^{\sharp}}_{2}\lesssim 1$ and
$|f^{(k)}|=|f|^{g^{\sharp}}_{k}\lesssim h^{-1+k/2}\lesssim h^{1/2}$ for $k>2$.
By decreasing $h$ we may obtain that $|f^{\prime}|\leq h^{-1/2}$ which we
assume in what follows. Observe that after the change of coordinates
$|\partial_{\eta}f|=h^{1/2}|\partial_{\eta}f|^{g^{\sharp}}\leq
h^{1/2}|f^{\prime}|\leq 1$ and
$|\partial_{t}f|=h^{-1/2}|\partial_{t}f|^{g^{\sharp}}\leq
h^{-1/2}|f^{\prime}|\leq h^{-1}$. The results in this section are uniform in
the sense that they depend only on the seminorms of $f$ in $S(h^{-1},g)$.

Since we assume that $f=\operatorname{Im}p_{r}$ does not change sign on the
leaves of $\Sigma_{2}$, we may have the following definition of the sign of
$f$.

######  Definition 4.1.

If $f$ does not change sign on the leaves $L$ of $\Sigma_{2}$, then we define
the sign function

(4.2) $\operatorname{sgn}(f)=\left\\{\begin{aligned} \pm 1\quad&\text{ if
}\/\pm f\geq 0\text{ and }f\not\equiv 0\quad\text{on $L$}\\\ 0\quad&\text{ if
}f\equiv 0\quad\text{on $L$}\end{aligned}\right.$

which is then constant on the leaves of $\Sigma_{2}$ such that
$\operatorname{sgn}(f)f\geq 0$.

Let

(4.3) $\displaystyle X_{+}=\left\\{\,(t,\tau,w):\exists\,s\leq t,\
\max_{x}f(s,x,\tau,w)>0\,\right\\}$ (4.4) $\displaystyle
X_{-}=\left\\{\,(t,\tau,w):\exists\,s\geq t,\
\min_{x}f(s,x,\tau,w)<0\,\right\\}.$

Observe that by the definition, $X_{\pm}$ is open in $\Sigma_{2}$ and is a
union of leaves of $\Sigma_{2}$. By condition $\operatorname{Sub_{r}}(\Psi)$
we find that $\pm f(t,x,\tau,w)\geq 0$ when $(t,\tau,w)\in X_{\pm}$ and that
$X_{-}\bigcap X_{+}=\emptyset$. Let
$X_{0}=\Sigma_{2}\setminus\left(X_{+}\bigcup X_{-}\right)$ which is a union of
leaves and is relatively closed in $\Sigma_{2}$.

######  Definition 4.2.

Let

(4.5) $d(t,\tau,w)=\inf\left\\{\,|(t,\tau,w)-(s,\sigma,z)|:(s,\sigma,z)\in
X_{0}\,\right\\}$

be the $g^{\sharp}$ distance to $X_{0}$, it is constant in $x$ and equal to
$+\infty$ in the case when $X_{0}=\emptyset$. We define the signed distance
function ${\delta}(t,w)$ by

(4.6) ${\delta}=\operatorname{sgn}(f)\min(d,h^{-1/2}),$

where $d$ is given by (4.5) and $\operatorname{sgn}(f)$ by Definition 4.1.

We say that $a(t,x,\tau,w)$ is Lipschitz continuous if it is Lipschitz with
respect to the metric $g^{\sharp}$.

######  Proposition 4.3.

The signed distance function $(t,\tau,w)\mapsto{\delta}(t,\tau,w)$ given by
Definition 4.2 is Lipschitz continuous with Lipschitz constant equal to 1. We
also find that $t\mapsto{\delta}(t,\tau,w)$ is nondecreasing, $\delta$ is
constant in $x$, $0\leq{\delta}f$, $|{\delta}|\leq h^{-1/2}$ and
$|{\delta}|=d$ when $|{\delta}|<h^{-1/2}$.

###### Proof.

Clearly ${\delta}f\geq 0$, and by the definition we have that
$|{\delta}|=\min(d,h^{-1/2})\leq h^{-1/2}$ so $|{\delta}|=d$ when
$|{\delta}|<h^{-1/2}$. Now, it suffices to show the Lipschitz continuity of
$(t,\tau,w)\mapsto{\delta}(t,\tau,w)$ locally, and thus locally on
$\complement{X_{0}}$ when $d<\infty$. Then $d(t,\tau,w)$ is the distance
function to $X_{0}$ so it is Lipschitz continuous with constant 1.

Next we show that $t\mapsto{\delta}(t,\tau,w)$ is nondecreasing. In fact, when
$t$ increases we can only go from $X_{-}$ to $X_{0}$ and from $X_{0}$ to
$X_{+}$. If $(t,\tau,w)\in X_{+}$ then $c=\delta(t,\tau,w)>0$ is the distance
to $\complement X_{+}$. If there exists $\varepsilon>0$ so that
$\delta(t+\varepsilon,\tau,w)<c$ then there would exists $(s,\sigma,z)\notin
X_{+}$ so that $|(t+\varepsilon,\tau,w)-(s,\sigma,z)|<c$. But then
$|(t,\tau,w)-(s-\varepsilon,\sigma,z)|<c$ where
$(s-\varepsilon,\sigma,z)\notin X_{+}$ which gives a contradiction. By
switching $t$ to $-t$, $\delta$ to $-\delta$ and $X_{+}$ to $X_{-}$ we
similarly find that $\delta<0$ is nondecreasing on $X_{-}$ and $\delta$ is of
course equal to $0$ on $X_{0}$. ∎

Next, we are going to define the metric that we are going to use for $f$.

######  Definition 4.4.

Let

(4.7)
$H^{-1/2}=1+|{\delta}|+\frac{|f^{\prime}|}{|{f^{\prime\prime}}|+h^{1/4}|f^{\prime}|^{1/2}+h^{1/2}}$

and $G=Hg^{\sharp}$.

######  Remark 4.5.

We have that

(4.8) $1\leq H^{-1/2}\leq 1+|{\delta}|+h^{-1/4}|f^{\prime}|^{1/2}\leq
3h^{-1/2}$

since $|f^{\prime}|\leq h^{-1/2}$ and $|{\delta}|\leq h^{-1/2}$. Moreover,
$|f^{\prime}|\leq
H^{-1/2}(|{f^{\prime\prime}}|+h^{1/4}|f^{\prime}|^{1/2}+h^{1/2})$ so by the
Cauchy-Schwarz inequality we obtain

(4.9) $|f^{\prime}|\leq 2|{f^{\prime\prime}}|H^{-1/2}+3h^{1/2}H^{-1/2}\leq
C_{2}H^{-1/2}.$

which gives that $f\in S^{+}(H^{-1},G)$, see Definition 3.1.

Since the metric $G$ does not depend on the values of $f$, we shall need a
weight to define the symbol class of $f$.

######  Definition 4.6.

Let

(4.10)
$M=|f|+|f^{\prime}|H^{-1/2}+|{f^{\prime\prime}}|H^{-1}+h^{1/2}H^{-3/2}=|f|+|f^{\prime}|^{G}_{1}+|{f^{\prime\prime}}|^{G}_{2}+h^{1/2}H^{-3/2}$

then we have that $h^{1/2}\leq M\leq C_{3}h^{-1}$ by (4.8).

In the following, we shall simplify the notation and include the variables
$x$, $t$ and $\tau$ in the $w$ variables.

######  Proposition 4.7.

We find that $H^{-1/2}$ is Lipschitz continuous, $G$ is ${\sigma}$ temperate
such that $G=H^{2}G^{\sigma}$ and

(4.11) $H(w)\leq C_{0}H(w_{0})(1+G_{w}(w-w_{0})).$

We have that $M$ is a weight for $G$ such that $f\in S(M,G)$ and

(4.12) $M(w)\leq C_{1}M(w_{0})(1+G_{w_{0}}(w-w_{0}))^{3/2}.$

In the case when $1+|{\delta}(w_{0})|\leq H^{-1/2}(w_{0})/2$ we have
$|f^{\prime}(w_{0})|\geq h^{1/2}$,

(4.13) $|f^{(k)}(w_{0})|\leq
C_{k}|f^{\prime}(w_{0})|H^{\frac{k-1}{2}}(w_{0})\qquad\text{$k\geq 1$},$

and $1/C\leq|f^{\prime}(w)|/|f^{\prime}(w_{0}|\leq C$ when $|w-w_{0}|\leq
cH^{-1/2}(w_{0})$ for some $c>0$.

######  Remark 4.8.

The term $f_{0}=\partial_{\eta}f\cdot r\cdot\xi\in S(h^{-1},g)$ in Proposition
3.6 can be written $f_{0}=r_{0}\cdot\xi$ with $r_{0}\in
S(MH^{1/2}h^{1/2},G)\subset S(1,G)$ since $|\partial_{\eta}f|\leq
h^{1/2}|f^{\prime}|$. Now, $\langle\xi\rangle$ is not a weight for
$g^{\sharp}$ near $\Sigma_{2}$ but $f_{0}\in
S(MH^{1/2}h^{1/2}\langle\xi\rangle,G_{0})$ where $G_{0}$ is given by (7.11).

Since $G\leq g^{\sharp}\leq G^{\sigma}$ we find that the conditions (4.11) and
(4.12) are stronger than the property of being ${\sigma}$ temperate (in fact,
it is strongly ${\sigma}$ temperate in the sense of [1, Definition 7.1]), and
imply that $G$ is slowly varying and $M$ is $G$ continuous. When
$1+|{\delta}|<H^{-1/2}/2$ we find from (4.13) that $|f^{\prime}|>0$ is a
weight for $G$, $f^{\prime}\in S(|f^{\prime}|,G)$ and $f^{-1}(0)$ is a
$C^{\infty}$ hypersurface. Since that surface does not depend on $x$ we find
that $\min_{x}H^{1/2}$ gives an upper bound on the curvature of $f^{-1}(0)$ by
(4.13). Proposition 4.9 shows that (4.13) also holds for $k=0$ when
$1+|{\delta}|\ll H^{-1/2}$.

###### Proof.

First we note that if $H^{-1/2}$ is Lipschitz continuous, then

(4.14) $H^{-1/2}(w_{0})\lesssim H^{-1/2}(w)+C_{1}|w-w_{0}|$

which gives (4.11) since $G=Hg^{\sharp}$. Next, we shall show that
$H_{1}^{-1/2}$ is Lipschitz continuous. Since the first terms of (4.7) are
Lipschitz continuous, we only have to prove that

${|f^{\prime}|}/(|{f^{\prime\prime}}|+h^{1/4}|f^{\prime}|^{1/2}+h^{1/2})=E/D$

is Lipschitz continuous. Since this is a local property, it suffices to prove
this when $|{\Delta}w|=|w-w_{0}|\leq 1$. Then we have that $D(w)\cong
D(w_{0})$, in fact $D^{2}\cong h+h^{1/2}|f^{\prime}|+|f^{\prime\prime}|^{2}$
so

$D^{2}(w)\leq C(D^{2}(w_{0})+|f^{\prime\prime}(w_{0})|h^{1/2}+h)\leq
C^{\prime}D^{2}(w_{0})$

when $|{\Delta}w|\leq 1$. We find that

$\left|{\Delta}\frac{E}{D}\right|=\left|\frac{E(w)}{D(w)}-\frac{E(w_{0})}{D(w_{0})}\right|\leq\frac{|{\Delta}E|}{D(w)}+\frac{E(w_{0})|{\Delta}D|}{D(w)D(w_{0})}.$

Taylor’s formula gives that

(4.15) $|{\Delta}E|\leq(|f^{\prime\prime}(w)|+Ch^{1/2})|{\Delta}w|\leq CD(w)$

when $|{\Delta}w|\leq 1$. It remains to show that $E(w_{0})|{\Delta}D|\leq
CD(w)D(w_{0})|{\Delta}w|$, which is trivial if $E(w_{0})=0$. Else, we have

$|{\Delta}|f^{\prime\prime}||\leq Ch^{1/2}|{\Delta}w|\leq
CD^{2}(w_{0})|{\Delta}w|/E(w_{0})\leq
C^{\prime}D(w_{0})D(w)||{\Delta}w|/E(w_{0})$

when $|{\Delta}w|\leq 1$ since $h^{1/2}\leq D^{2}/E$ and $D(w_{0})\leq CD(w)$.
Finally, we have

$h^{1/4}|{\Delta}|f^{\prime}|^{1/2}|\leq
h^{1/4}|{\Delta}E|/(|f^{\prime}(w_{0})|^{1/2}+|f^{\prime}(w)|^{1/2})\\\ \leq
Ch^{1/4}|f^{\prime}(w_{0})|^{1/2}D(w)|{\Delta}w|/|f^{\prime}(w_{0})|\leq
CD(w_{0})D(w)|{\Delta}w|/E(w_{0})$

when $|{\Delta}w|\leq 1$ by (4.15), which proves the Lipschitz continuity.

Next, we study the case when $1+|{\delta}(w_{0})|\leq H^{-1/2}(w_{0})/2$, then
$H^{1/2}(w_{0})\leq 1/2$. Then we find from (4.7) that

(4.16)
$|{f^{\prime\prime}(w_{0})}|+h^{1/4}|f^{\prime}(w_{0})|^{1/2}+h^{1/2}\leq
2H^{1/2}(w_{0})|f^{\prime}(w_{0})|\leq|f^{\prime}(w_{0})|.$

which gives $|f^{\prime}(w_{0})|\geq h^{1/2}$,
$|f^{\prime}(w_{0})|\geq|{f^{\prime\prime}(w_{0})}|$ and that $h^{1/2}\leq
4H(w_{0})|f^{\prime}(w_{0})|$. When $|w-w_{0}|\leq cH^{-1/2}(w_{0})$ we find
from (4.16) by using Taylors formula that

$|f^{\prime}(w)-f^{\prime}(w_{0})|\leq|f^{\prime\prime}(w_{0})|cH^{-1/2}(w_{0})+C_{3}h^{1/2}c^{2}H^{-1}(w_{0})\leq(c+4C_{3}c^{2})|f^{\prime}(w_{0})|$

which gives $1/C\leq|f^{\prime}(w)|/|f^{\prime}(w_{0}|\leq C$ for small enough
$c>0$. Now (4.13) follows from (4.16) for $k=2$. When $k\geq 3$ we have

$|f^{(k)}(w_{0})|\leq C_{k}h^{\frac{k-2}{2}}\leq
4C_{k}3^{k-3}|f^{\prime}(w_{0})|H^{\frac{k-1}{2}},$

since $h^{1/2}\leq 4H|f^{\prime}(w_{0})|$ by (4.16) and $h^{(k-3)/2}\leq
3^{k-3}H^{(k-3)/2}$ by (4.8).

Finally, we shall prove that $M$ is a weight for $G$. By Taylor’s formula we
have

(4.17) $|{f^{(k)}(w)}|\leq
C_{4}\sum_{j=0}^{2-k}|{f^{(k+j)}(w_{0})}|w-w_{0}|^{j}+C_{4}h^{1/2}|w-w_{0}|^{(3-k)}\qquad
0\leq k\leq 2,$

thus we obtain that

$M(w)\leq
C_{5}\sum_{k=0}^{2}|{f^{(k)}(w_{0})}|(|w-w_{0}|+H^{-1/2}(w))^{k}+C_{5}h^{1/2}(|w-w_{0}|+H^{-1/2}(w))^{3}.$

By switching $w$ and $w_{0}$ in (4.14) we find $H^{-1/2}(w)+|w-w_{0}|\leq
C_{0}(H^{-1/2}(w_{0})+|w-w_{0}|)$. Thus we obtain that

$M(w)\leq
C_{6}\sum_{k=0}^{2}|{f^{(k)}(w_{0})}|H^{-k/2}(w_{0})(1+H^{1/2}(w_{0})|w-w_{0}|)^{k}\\\
+C_{6}h^{1/2}H^{-3/2}(w_{0})(1+H^{1/2}(w_{0})|w-w_{0}|)^{3}\leq
C_{6}M(w_{0})(1+G_{w_{0}}(w-w_{0}))^{3/2}$

which gives (4.12). It is clear from the definition of $M$ that
$|{f^{(k)}}|\leq MH^{k/2}$ when $k\leq 2$, and when $k\geq 3$ we have
$|{f^{(k)}}|\leq C_{k}h^{\frac{k-2}{2}}\leq C_{k}3^{k-3}MH^{\frac{k}{2}}$
since $h^{1/2}\leq MH^{3/2}$ and $h^{(k-3)/2}\leq 3^{k-3}H^{(k-3)/2}$ when
$k\geq 3$. ∎

######  Proposition 4.9.

We have that

(4.18) $1/C\leq M/(|{f^{\prime\prime}}|H^{-1}+h^{1/2}H^{-3/2})\leq C$

When $|{\delta}|\leq{\kappa}_{0}H^{-1/2}$ and $H^{1/2}\leq{\kappa}_{0}$ for
$0<{\kappa}_{0}\leq 1/4$ we find that

(4.19) $1/C_{1}\leq M/|f^{\prime}|H^{-1/2}\leq C_{1}$

which implies that $f\in S(H^{-1},G)$ by (4.9).

###### Proof.

First note that when $|{\delta}|\cong h^{-1/2}$ we have $H^{-1/2}\cong
h^{-1/2}$, which gives $M\cong h^{-1}$ and proves (4.18) in this case. If
$|{\delta}(w_{0})|<h^{-1/2}$, then as before there exists $w\in f^{-1}(0)$
such that $|w-w_{0}|=|{\delta}(w_{0})|\leq H^{-1/2}(w_{0})$. Since $f(w)=0$,
Taylor’s formula gives that

(4.20)
$|f(w_{0})|\leq|f^{\prime}(w_{0}||{\delta}(w_{0})|+|{f^{\prime\prime}(w_{0})}||{\delta}(w_{0})|^{2}/2+Ch^{1/2}|{\delta}(w_{0})|^{3}.$

We obtain from (4.20) and (4.9) that

$M\leq
C\big{(}|f^{\prime}|H^{-1/2}+|{f^{\prime\prime}}|H^{-1}+h^{1/2}H^{-3/2}\big{)}\leq
C^{\prime}\left(|{f^{\prime\prime}}|H^{-1}+h^{1/2}H^{-3/2}\right)\qquad\text{at
~{}$w_{0}$,}$

which gives (4.18) since the opposite estimate is trivial.

If $|{\delta}|\leq{\kappa}_{0}H^{-1/2}<h^{1/2}$ and $H^{1/2}\leq{\kappa}_{0}$
with ${\kappa}_{0}\leq 1/4$ then $\langle\delta\rangle\leq H^{-1/2}/2$ so we
we obtain by (4.13), (4.16) and (4.20) that

$M\leq
C\big{(}|f^{\prime}|H^{-1/2}+|{f^{\prime\prime}}|H^{-1}+h^{1/2}H^{-3/2}\big{)}\leq
C^{\prime}|f^{\prime}|H^{-1/2}\qquad\text{at ~{}$w_{0}$.}$

since $h^{1/2}\leq 4H|f^{\prime}|$ by (4.16). This gives (4.19) since the
opposite estimate is trivial, which completes the proof of the proposition. ∎

######  Proposition 4.10.

Let $H^{-1/2}$ be given by Definition 4.4 for $f\in S(h^{-1},g)$. There exists
positive ${\kappa_{1}}$ and $c$ so that if
$\langle{\delta}\rangle=1+|{\delta}|\leq{\kappa}_{1}H^{-1/2}$ at $w_{0}$ then

(4.21) $f={\alpha}{\delta}\qquad\text{when $|w-w_{0}|\leq cH^{-1/2}(w_{0})$}$

where $0\leq{\alpha}\in S(MH^{1/2},G)$, such that
$\alpha\geq{\kappa}_{1}MH^{1/2}$, which implies that ${\delta}=f/{\alpha}\in
S(H^{-1/2},G)$.

###### Proof.

Let ${\kappa}_{0}>0$ be given by Proposition 4.9. If
${\kappa_{1}}\leq{\kappa}_{0}$ and
$\langle\delta\rangle\leq\kappa_{1}H^{-1/2}$ at $w_{0}$ then we find that
$|f^{\prime}(w_{0})|\cong M(w_{0})H^{1/2}(w_{0})$ by (4.19). We may change
coordinates so that $w_{0}=0$. Let $H^{1/2}=H^{1/2}(0)$ and $M=M(0)$,
$w=H^{-1/2}z$ and

$F(z)=H^{1/2}f(H^{-1/2}z)/|f^{\prime}(0)|\cong f(H^{-1/2}z)/M\in C^{\infty}$

Now ${\delta}_{1}(z)=H^{1/2}{\delta}(H^{-1/2}z)$ is the signed distance to
$F^{-1}(0)$ in the $z$ coordinates which is constant in $x$.

We have $|F(0)|\leq C_{0}$, $|F^{\prime}(0)|=1$, $|F^{\prime\prime}(0)|\leq
C_{2}$ and $|F^{(3)}(z)|\leq C_{3}$, $\forall\,z$, by (4.13) in Proposition
4.7. It is no restriction to assume that the coordinates
$z=(z_{1},z^{\prime})$ are chosen so that $\partial_{z^{\prime}}F(0)=0$, and
then $|\partial_{z_{1}}F(z)|\geq c>0$ in a fixed neighborhood of the origin.
If $|{\delta}_{1}(0)|=|{\delta}(0)H^{1/2}|\leq{\kappa}_{1}\ll 1$ then
$F^{-1}(0)$ is a $C^{\infty}$ manifold in this neighborhood, ${\delta}_{1}(z)$
is uniformly $C^{\infty}$ and $\partial_{z_{1}}{\delta}_{1}(z)\geq c_{0}>0$ in
a fixed neighborhood of the origin.

By choosing $(\delta_{1}(z),z^{\prime})$ as local coordinates and using
Taylor’s formula we find that $F(z)={\alpha}_{1}(z){\delta}_{1}(z)$ for any
$x$ since $F=0$ when $\delta_{1}=0$. Here $0\leq{\alpha}_{1}\in C^{\infty}$
and $\alpha_{1}\geq C>0$ in a fixed neighborhood of the origin. Thus

$f(w)=|f^{\prime}(0)|H^{-1/2}\alpha_{1}(H^{1/2}w)\delta_{1}(H^{1/2}w)=|f^{\prime}(0)|\alpha_{1}(H^{1/2}w)\delta(w)$

when $|w|\leq cH^{-1/2}$. Now
$\alpha(w)=|f^{\prime}(0)|\alpha_{1}(H^{1/2}w)\in S(MH^{1/2},G)$ with
$\alpha\cong MH^{1/2}$ which gives the proposition. ∎

##  5\. Properties of the symbol

In this section we shall study the properties of the symbol near the sign
changes. We shall follow Section in [4] with some minor changes, and we shall
start with a one-dimensional result.

######  Lemma 5.1.

Assume that $f(t)\in C^{3}(\mathbf{R})$ such that
$\|f^{(3)}\|_{\infty}=\sup_{t}|{f^{(3)}(t)}|$ is bounded. If

(5.1) $\operatorname{sgn}(t)f(t)\geq 0\qquad\text{when
${\varrho}_{0}\leq|t|\leq{\varrho}_{1}$}$

for ${\varrho}_{1}\geq 3{\varrho}_{0}>0$, then we find

(5.2)
$\displaystyle|f(0)|\leq\frac{3}{2}\left({\varrho}_{0}f^{\prime}(0)+{\varrho}_{0}^{3}\|f^{(3)}\|_{\infty}/2\right)$
(5.3) $\displaystyle|f^{\prime\prime}(0)|\leq
f^{\prime}(0)/{\varrho}_{0}+7{\varrho}_{0}\|f^{(3)}\|_{\infty}/6.$

###### Proof.

By Taylor’s formula we have

$0\leq\operatorname{sgn}({t})f({t})=|{t}|f^{\prime}(0)+\operatorname{sgn}({t})(f(0)+f^{\prime\prime}(0){t}^{2}/2)+R({t})\qquad{\varrho}_{0}\leq|{t}|\leq{\varrho}_{1}$

where $|R({t})|\leq\|f^{(3)}\|_{\infty}|{t}|^{3}/6$. We obtain that

(5.4) $\left|f(0)+t^{2}f^{\prime\prime}(0)/2\right|\leq
f^{\prime}(0)|{t}|+\|f^{(3)}\|_{\infty}|{t}|^{3}/6$

for any $|{t}|\in[{\varrho}_{0},{\varrho}_{1}]$. By choosing
$|{t}|={\varrho}_{0}$ and $|{t}|=3{\varrho}_{0}$, we obtain that

$4{\varrho}_{0}^{2}|f^{\prime\prime}(0)|\leq
4f^{\prime}(0){\varrho}_{0}+28\|f^{(3)}\|_{\infty}{\varrho}_{0}^{3}/6$

which gives (5.3). By letting $|t|={\varrho}_{0}$ in (5.4) and substituting
(5.3), we obtain (5.2). ∎

######  Proposition 5.2.

Let $f(w)\in C^{\infty}(\mathbf{R}^{n})$ such that
$\|f^{(3)}\|_{\infty}<\infty$. Assume that there exists $0<{\varepsilon}\leq
r/5$ such that

(5.5) $\operatorname{sgn}(w_{1})f(w)\geq 0\quad\text{when
$|w_{1}|\geq{\varepsilon}+|w^{\prime}|^{2}/r$ and $|w|\leq r$}$

where $w=(w_{1},w^{\prime})$. Then we obtain

(5.6) $|{f^{\prime\prime}(0)}|\leq
33(|\partial_{w_{1}}f(0)|/{\varrho}+{\varrho}\|f^{(3)}\|_{\infty})$

for any ${\varepsilon}\leq{\varrho}\leq r/\sqrt{10}$.

###### Proof.

We shall consider the function $t\mapsto f(t,w^{\prime})$ which satisfies
(5.1) for fixed $w^{\prime}$ with

${\varepsilon}+|w^{\prime}|^{2}/r={\varrho}_{0}(w^{\prime})\leq|t|\leq{\varrho}_{1}\equiv
3r/\sqrt{10}$

and $|w^{\prime}|\leq r/\sqrt{10}$ which we assume in what follows. In fact,
then $t^{2}+|w^{\prime}|^{2}\leq r^{2}$ and $3{\varrho}_{0}(w^{\prime})\leq
9r/10\leq 3r/\sqrt{10}={\varrho}_{1}$. We obtain from (5.2) and (5.3) that

(5.7)
$\displaystyle|f(0,w^{\prime})|\leq\frac{3}{2}\partial_{w_{1}}f(0,w^{\prime}){\varrho}+3{\varrho}^{3}\|f^{(3)}\|_{\infty}/4$
(5.8)
$\displaystyle|\partial_{w_{1}}^{2}f(0,w^{\prime})|\leq\partial_{w_{1}}f(0,w^{\prime})/{\varrho}+7{\varrho}\|f^{(3)}\|_{\infty}/6\qquad\text{}$

for ${\varepsilon}+|w^{\prime}|^{2}/r\leq{\varrho}\leq r/\sqrt{10}$ and
$|w^{\prime}|\leq r/\sqrt{10}$. By letting $w^{\prime}=0$ in (5.8) we find
that

(5.9)
$|\partial_{w_{1}}^{2}f(0)|\leq\partial_{w_{1}}f(0)/{\varrho}+7{\varrho}\|f^{(3)}\|_{\infty}/6$

for ${\varepsilon}\leq{\varrho}\leq r/\sqrt{10}$. By letting
${\varrho}={\varrho}_{0}(w^{\prime})$ in (5.7) and dividing with
$3{\varrho}_{0}(w^{\prime})/2$, we obtain that

(5.10)
$0\leq\partial_{w_{1}}f(0,w^{\prime})+2\|f^{(3)}\|_{\infty}|w^{\prime}|^{2}$

when ${\varepsilon}\leq|w^{\prime}|\leq r/\sqrt{10}$ since then
${\varrho}_{0}(w^{\prime})\leq{\varepsilon}+|w^{\prime}|\leq 2|w^{\prime}|$.
By using Taylor’s formula for
$w^{\prime}\mapsto\partial_{w_{1}}f(0,w^{\prime})$ in (5.10), we find that

$0\leq\partial_{w_{1}}f(0)+\langle
w^{\prime},\partial_{w^{\prime}}(\partial_{w_{1}}f)(0)\rangle+\frac{5}{2}\|f^{(3)}\|_{\infty}|w^{\prime}|^{2}\qquad\text{}$

when ${\varepsilon}\leq|w^{\prime}|\leq r/\sqrt{10}$. Thus, by optimizing over
fixed $|w^{\prime}|$, we obtain that

(5.11)
$|w^{\prime}||\partial_{w^{\prime}}(\partial_{w_{1}}f)(0)|\leq\partial_{w_{1}}f(0)+\frac{5}{2}\|f^{(3)}\|_{\infty}|w^{\prime}|^{2}\qquad\text{when
${\varepsilon}\leq|w^{\prime}|\leq r/\sqrt{10}.$}$

By again putting ${\varrho}={\varrho}_{0}(w^{\prime})$ in (5.7), using
Taylor’s formula for $w^{\prime}\mapsto\partial_{w_{1}}f(0,w^{\prime})$ but
this time substituting (5.11), we obtain

(5.12) $|f(0,w^{\prime})|\leq
6\partial_{w_{1}}f(0)|w^{\prime}|+15\|f^{(3)}\|_{\infty}|w^{\prime}|^{3}\qquad\text{when
${\varepsilon}\leq|w^{\prime}|\leq r/\sqrt{10}$.}$

We may also estimate the even terms in Taylor’s formula by (5.12):

$\begin{split}|f(0)+\langle\partial_{w^{\prime}}^{2}f(0)w^{\prime},w^{\prime}\rangle/2|\leq\frac{1}{2}|f(0,w^{\prime})+f(0,-w^{\prime})|+\|f^{(3)}\|_{\infty}|w^{\prime}|^{3}/6\\\
\leq
6\partial_{w_{1}}f(0)|w^{\prime}|+\frac{91}{6}\|f^{(3)}\|_{\infty}|w^{\prime}|^{3}\end{split}$

when ${\varepsilon}\leq|w^{\prime}|\leq r/\sqrt{10}$. Thus, by using (5.7)
with ${\varrho}={\varepsilon}$ and $w^{\prime}=0$ to estimate $|f(0)|$ and
optimizing over fixed $|w^{\prime}|$, we obtain that

(5.13)
$|{\partial_{w^{\prime}}^{2}f(0)}||w^{\prime}|^{2}/2\leq\frac{15}{2}|\partial_{w_{1}}f(0)||w^{\prime}|+16\|f^{(3)}\|_{\infty}|w^{\prime}|^{3}$

when ${\varepsilon}\leq|w^{\prime}|\leq r/\sqrt{10}$. Thus we obtain (5.6) by
taking ${\varepsilon}\leq|w^{\prime}|={\varrho}\leq r/\sqrt{10}$ in
(5.9)–(5.13). ∎

As before, if $f\in C^{\infty}(\mathbf{R}^{n})$ then we define the signed
distance function of $f$ as ${\delta}=\operatorname{sgn}(f)d$ where $d$ is the
Euclidean distance to $f^{-1}(0)$.

######  Proposition 5.3.

Let $f_{j}(w)\in C^{\infty}(\mathbf{R}^{n})$ and ${\delta}_{j}(w)$ be the
signed distance functions of $f_{j}(w)$, for $j=1$, $2$. Assume that
$f_{1}(w)>0\implies f_{2}(w)\geq 0$. There exists positive $c_{0}$ and
$c_{1}$, such that if $|{\delta}_{j}(w_{0})|\leq c_{0}$,
$|f^{\prime}_{j}(w_{0})|\geq c_{1}$, for $j=1$, $2$, and

(5.14) $|{\delta}_{1}(w_{0})-{\delta}_{2}(w_{0})|={\varepsilon}$

then there exist $g^{\sharp}$ orthonormal coordinates $w=(w_{1},w^{\prime})$
so that $w_{0}=(x_{1},0)$ with $x_{1}={\delta}_{1}(w_{0})$ and

(5.15) $\displaystyle\operatorname{sgn}(w_{1})f_{j}(w)>0\quad\text{when
$|w_{1}|\geq({\varepsilon}+|w^{\prime}|^{2})/c_{0}$ and $|w|\leq c_{0}$}\qquad
j=1,\ 2$ (5.16)
$\displaystyle|{\delta}_{2}(w)-{\delta}_{1}(w)|\leq({\varepsilon}+|w-w_{0}|^{2})/c_{0}\qquad\text{when
$|w|\leq c_{0}$.}$

The constant $c_{0}$ only depends on the seminorms of $f_{1}$ and $f_{2}$ in a
fixed neighborhood of $w_{0}$.

###### Proof.

Observe that the conditions get stronger and the conclusions weaker when
$c_{0}$ decreases. Assume that $f_{1}$ and $f_{2}$ are uniformly bounded in
$C^{\infty}$ near $w_{0}$. For any positive $c_{0}$ and $c_{1}$ there exists
$c_{2}>0$ so that if $|f^{\prime}_{j}(w_{0})|\geq c_{1}$ and
$|{\delta}_{j}(w_{0})|\leq c_{0}$, $j=1$, $2$ then $|f^{\prime}_{j}(w)|>0$ for
$|w-w_{0}|\leq c_{2}$, thus $f_{j}^{-1}(0)$ is a $C^{\infty}$ hypersurface in
$|w-w_{0}|\leq c_{2}$. By decreasing $c_{0}$ we obtain as in the proof of
Proposition 4.10 that there exists $c_{3}>0$ so that
$w\mapsto{\delta}_{j}(w)\in C^{\infty}(\mathbf{R}^{n})$ uniformly in
$|w-w_{0}|\leq c_{3}$, $j=1$, 2. We may also choose $z_{0}\in f_{1}^{-1}(0)$
so that $|{\delta}_{1}(w_{0})|=|w_{0}-z_{0}|$, and then choose $g^{\sharp}$
orthonormal coordinates so that $z_{0}=0$, $w_{0}=({\delta}_{1}(w_{0}),0)$ and
$\partial_{w^{\prime}}{\delta}_{1}(0)=\partial_{w^{\prime}}{\delta}_{1}(w_{0})=0$,
$w=(w_{1},w^{\prime})$. If $c_{0}\leq c_{3}/3$ we find that ${\delta}_{j}\in
C^{\infty}$ in $|w|\leq c_{4}=2c_{3}/3$. Since
$\operatorname{sgn}(f_{1}(w_{0}))=\operatorname{sgn}({\delta}_{1}(w_{0}))$ we
find that $\partial_{w_{1}}f_{1}(0)>0$.

We have that $|{\partial_{w}^{2}{\delta}_{j}(w)}|\leq C_{0}$ for $|w|\leq
c_{4}$, $j=1$, 2, and ${\Delta}(w)={\delta}_{2}(w)-{\delta}_{1}(w)\geq 0$ by
the sign condition. By [10, Lemma 7.7.2] we obtain that
$|\partial_{w}{\Delta}(w)|^{2}\leq C_{1}{\Delta}(w)\leq C_{1}{\varepsilon}$
when $w=w_{0}$ by (5.14). This gives

(5.17)
$|{\Delta}(w)|\leq|{\Delta}(w_{0})|+|\partial_{w}{\Delta}(w_{0})||w-w_{0}|+C_{2}|w-w_{0}|^{2}\\\
\leq C_{3}({\varepsilon}+|w-w_{0}|^{2})\quad\text{for $|w|\leq c_{4}$}$

which proves (5.16). Since $|\partial_{w^{\prime}}{\delta}_{1}(w_{0})|=0$ we
find that $|\partial_{w^{\prime}}{\delta}_{2}(w)|\leq
C_{4}(\sqrt{\varepsilon}+|w-w_{0}|)\ll 1$ when $|w-w_{0}|\ll 1$ and
${\varepsilon}\leq 2c_{0}\ll 1$. Now $f_{2}(\overline{w})=0$ for some
$|\overline{w}|\leq{\varepsilon}$. Thus for $c_{0}\ll 1$ we obtain
$|\partial_{w^{\prime}}{\delta}_{2}(\overline{w})|\ll 1$, which gives that
$|\partial_{w_{1}}f_{2}(\overline{w})|\geq
c_{5}|\partial_{w}f_{2}(\overline{w})|\geq c_{5}^{2}>0$ for some $c_{5}>0$.
Since $\operatorname{sgn}(f_{2}(w_{1},0))=1$ when $w_{1}>0$, we obtain that
$\partial_{w_{1}}f_{2}(w)\geq c_{6}|\partial_{w}f_{2}(w)|\geq c_{6}^{2}$ when
$|w|\leq c_{6}$ for some $c_{6}>0$.

By using the implicit function theorem, we obtain $b_{j}(w^{\prime})\in
C^{\infty}$, so that that $\pm f_{j}(w)>0$ if and only if
$\pm(w_{1}-b_{j}(w^{\prime}))\geq 0$ when $|w|\leq c_{7}>0$, $j=1$, $2$. Since
$f_{1}(0)=|\partial_{w^{\prime}}f_{1}(0)|=0$ we obtain that
$b_{1}(0)=|b_{1}^{\prime}(0)|=0$. This gives that $|b_{1}(w^{\prime})|\leq
C_{5}|w^{\prime}|^{2}$ and proves the positive part of (5.15) by the sign
condition. Observe that the sign condition gives that $b_{1}(w^{\prime})\geq
b_{2}(w^{\prime})$. Now
$|\delta_{2}(w_{0})|\leq|\delta_{1}(w_{0})|+\varepsilon$, thus we find
$-\varepsilon\leq b_{2}(\overline{w}^{\prime})\leq
b_{1}(\overline{w}^{\prime})$ for some $|\overline{w}^{\prime}|\leq
C\sqrt{\varepsilon}\leq C\sqrt{2c_{0}}\leq c_{7}$ for $c_{0}\ll 1$. This gives
$b_{2}(\overline{w}^{\prime})\leq C_{5}C^{2}\varepsilon$ and
$|b_{1}^{\prime}(\overline{w}^{\prime})|\leq C_{6}\sqrt{\varepsilon}$, and we
obtain as before that
$|b_{1}^{\prime}(\overline{w}^{\prime})-b_{2}^{\prime}(\overline{w}^{\prime})|\leq
C_{7}\sqrt{{\varepsilon}}$. As in (5.17), we find

$|b_{2}(w^{\prime})|\leq
C_{8}({\varepsilon}+|w^{\prime}-\overline{w}^{\prime}|^{2})\leq
C_{9}({\varepsilon}+|w^{\prime}|^{2})\qquad\text{for $|w|\leq c_{7}$}$

which proves the negative part of (5.15) and the proposition. ∎

##  6\. The Weight function

In this section, we shall define the weight $m$ we shall use. We shall follow
Section 5 in [4] with some necessary changes because of the different
conditions and normal forms. We shall use the same notation as in Section 4,
and let ${\delta}(t,\tau,w)$, $H^{-1/2}(t,x,\tau,w)$ and $M(t,x,\tau,w)$ be
given by Definitions 4.2, 4.4 and 4.6 for $f(t,x,\tau,w)\in
S(h^{-1},hg^{\sharp})$ satisfying condition
$\operatorname{Sub_{r}}(\overline{\Psi})$ given by (3.3) such that
$|f^{\prime}|\leq h^{-1/2}$. As before, we shall include $\tau$ in $w$ but $t$
will be a parameter.

The weight $m$ will essentially measure how much $t\mapsto{\delta}(t,w)$
changes between the minima of $t\mapsto
H^{1/2}(t,x,w)\langle{\delta}(t,w)\rangle^{2}$, which will give restrictions
on the sign changes of the symbol. As before, we assume that we have choosen
$g^{\sharp}$ orthonormal coordinates so that $g^{\sharp}(w)$ is the euclidean
metric, and the results will only depend on the seminorms of $f$. The
following definition uses that $t\mapsto\delta(t,w)$ is nondecreasing and
$\delta$ is constant in $x$, and assumes that $H$ is only defined in $|x|\leq
C$.

######  Definition 6.1.

Let $H_{1}(t,w)=\min_{x}H(t,x,w)$ and

(6.1) $m(t,w)=\inf_{t_{1}\leq t\leq
t_{2}}\big{\\{}{\delta}(t_{2},w)-{\delta}(t_{1},w)\\\
+\max\big{(}H_{1}^{1/2}(t_{1},w)\langle{\delta}(t_{1},w)\rangle^{2},H_{1}^{1/2}(t_{2},w)\langle{\delta}(t_{2},w)\rangle^{2}\big{)}/2\,\big{\\}}$

where $\langle{\delta}\rangle=1+|{\delta}|\leq H_{1}^{-1/2}=\max_{x}H^{-1/2}$.
Thus $m$ is constant on the leaves of $\Sigma_{2}$.

This is actually Definition 5.1 in [4] with $H$ replaced by $H_{1}$. It will
be important in the proof that this weight is constant on the leaves of
$\Sigma_{2}$.

######  Remark 6.2.

When $t\mapsto{\delta}(t,w)$ is constant for fixed $w$, we find that $t\mapsto
m(t,w)$ is equal to the largest quasi-convex minorant of $t\mapsto
H_{1}^{1/2}(t,w)\langle{\delta}(t,w)\rangle^{2}/2$, i.e.,
$\sup_{I}m=\sup_{\partial I}m$ for compact intervals $I\subset\mathbf{R}$, see
[11, Definition 1.6.3].

######  Remark 6.3.

One can also make a local definition of $m$ by taking the infimum over $-T\leq
t_{1}\leq t\leq t_{2}\leq T$ in (6.1) for some $0<T\ll 1$. Then the results of
this section will hold when $|t|\leq T$. By making a translation in $t$ we can
of course define $m$ in a neighborhood of any point.

######  Proposition 6.4.

We have that $m\in L^{\infty}$, such that $w\mapsto m(t,w)$ is uniformly
Lipschitz continous, $\forall\,t$, and

(6.2) $h^{1/2}\langle{\delta}\rangle^{2}/6\leq m\leq
H_{1}^{1/2}\langle{\delta}\rangle^{2}/2\leq
H^{1/2}\langle{\delta}\rangle^{2}/2\leq\langle{\delta}\rangle/2.$

We may choose $t_{1}\leq t_{0}\leq t_{2}$ so that

(6.3) $\max_{j=0,1,2}\langle{\delta}(t_{j},w_{0})\rangle\leq
2\min_{j=0,1,2}\langle{\delta}(t_{j},w_{0})\rangle.$

and

(6.4) $H_{0}^{1/2}=\max(H_{1}^{1/2}(t_{1},w_{0}),\,H_{1}^{1/2}(t_{2},w_{0}))$

satisfies

(6.5)
$H_{0}^{1/2}<16m(t_{0},w_{0})/\langle{\delta}(t_{j},w_{0})\rangle^{2}\qquad\text{for
$j=0$, $1$, $2$.}$

If $m(t_{0},w_{0})\leq{\varrho}\langle{\delta}(t_{0},w_{0})\rangle$ for
${\varrho}\ll 1$, then we may choose $g^{\sharp}$ orthonormal coordinates so
that $w_{0}=(x_{1},0)$,
$|x_{1}|<2\langle{\delta}(t_{0},w_{0})\rangle<32{\varrho}H_{0}^{-1/2}$, and

(6.6) $\displaystyle\operatorname{sgn}(w_{1})f(t_{0},w)\geq 0\quad\text{when
$|w_{1}|\geq(m(t_{0},w_{0})+H_{0}^{1/2}|w^{\prime}|^{2})/c_{0}$}$ (6.7)
$\displaystyle|{\delta}(t_{1},w)-{\delta}(t_{2},w)|\leq(m(t_{0},w_{0})+H_{0}^{1/2}|w-w_{0}|^{2})/c_{0}$

when $|w|\leq c_{0}H_{0}^{-1/2}$. The constant $c_{0}$ only depends on the
seminorms of $f$.

Observe that condition (6.6) is not empty when
$m(t_{0},w_{0})\leq{\varrho}\langle{\delta}(t_{0},w_{0})\rangle$ for
${\varrho}$ sufficiently small since then
$H_{0}^{-1/2}\gtrsim\langle\delta\rangle^{2}/m\gg m$ by (6.5) .

###### Proof.

If we let

$F(s,t,w)=|{\delta}(s,w)-{\delta}(t,w)|+\max(H_{1}^{1/2}(s,w)\langle{\delta}(s,w)\rangle^{2},H_{1}^{1/2}(t,w)\langle{\delta}(t,w)\rangle^{2})/2$

then we find that $w\mapsto F(s,t,w)$ is uniformly Lipschitz continuous. Now,
it suffices to show this when $|{\Delta}w|=|w-w_{0}|\ll 1$, and we know that
$\langle{\delta}\rangle$ and $H^{-1/2}$ are uniformly Lipschitz continuous by
Proposition 4.7 which gives that $H_{1}^{-1/2}=\max_{x}H^{-1/2}$ is uniformly
Lipschitz continuous. The first term $|{\delta}(s,w)-{\delta}(t,w)|$ is
obviously uniformly Lipschitz continuous. We have for fixed $t$ that

$\left|{\Delta}(H_{1}^{1/2}\langle{\delta}\rangle^{2})\right|\leq
C(\langle{\delta}\rangle^{2}|{\Delta}H_{1}^{1/2}|+H_{1}^{1/2}\langle{\delta}\rangle|{\Delta}{\delta}|)$

where $H_{1}^{1/2}\langle{\delta}\rangle\leq 1$,
$|{\Delta}{\delta}|\leq|{\Delta}w|$ and $|{\Delta}H_{1}^{1/2}|\leq
CH_{1}|{\Delta}H_{1}^{-1/2}|\leq C^{\prime}H_{1}|{\Delta}w|$, which gives the
uniform Lipschitz continuity of $F(s,t,w)$. By taking the infimum, we obtain
(6.2) and the uniform Lipschitz continuity of $m$. In fact, $h^{1/2}/3\leq
H_{1}^{1/2}$ by (4.8) and since $t\mapsto\delta(t,w)$ is monotone, we find
that $t\mapsto\langle\delta(t,w)\rangle$ is quasi-convex. Thus
$h^{1/2}\langle{\delta}(t_{0},w_{0})\rangle/6\leq F(s,t,w_{0})$ when $s\leq
t_{0}\leq t$.

By approximating the infimum, we may choose $t_{1}\leq t_{0}\leq t_{2}$ so
that $F(t_{1},t_{2},w_{0})<m(t_{0},w_{0})+h^{1/2}/6$. Since $h^{1/2}/6\leq
m\leq H_{1}^{1/2}\langle{\delta}\rangle^{2}/2$ by (6.2), we find that

(6.8)
$\displaystyle|{\delta}(t_{1},w_{0})-{\delta}(t_{2},w_{0})|<m(t_{0},w_{0})\leq\langle{\delta}(t_{0},w_{0})\rangle/2\qquad\text{and}$
(6.9) $\displaystyle
H_{1}^{1/2}(t_{j},w_{0})\langle{\delta}(t_{j},w_{0})\rangle^{2}/2<2m(t_{0},w_{0})\qquad\text{for
$j=1$ and $2$.}$

Since $t\mapsto{\delta}(t,w_{0})$ is monotone, we obtain (6.3) from (6.8), and
(6.5) from (6.9) and (6.3).

Next assume that
$m(t_{0},w_{0})\leq{\varrho}\langle{\delta}(t_{0},w_{0})\rangle$ for some
$0<{\varrho}\leq 1$. Then we find from (6.5) that

(6.10) $1+|{\delta}(t_{j},w_{0})|<16{\varrho}H_{0}^{-1/2}\qquad\text{for
$j=0,\ 1,\ 2$}.$

We may choose $g^{\sharp}$ orthonormal coordinates so that $w_{0}=0$. Since
$\delta$, $H_{1}$, $H_{0}$ and $m$ are constant in $x$, the results will hold
for any $x$. I If we choose $x_{j}$ so that $H_{1}(t_{j},0)=H(t_{j},x_{j},0)$
then $\langle{\delta}(t_{j},0)\rangle<16{\varrho}H^{-1/2}(t_{j},x_{j},0)$ for
$j=1,2$ by (6.10) so we find from Proposition 4.7 that

(6.11)
$h^{1/2}\leq|f^{\prime}(t_{j},x_{j},0)|\cong|f^{\prime}(t_{j},x_{j},w)|\quad\text{for
}|w|\leq cH_{0}^{-1/2}\leq cH^{-1/2}(t_{j},x_{j},0)$

when ${\varrho}\ll 1$ and $j=1,\,2$. Since $H_{0}^{-1/2}\leq 3h^{-1/2}$ we
find from (6.10) that $f(t_{j},x_{j},\widetilde{w}_{j})=0$ for some
$|\widetilde{w}_{j}|<16{\varrho}H_{0}^{-1/2}$ by (6.10) when ${\varrho}<1/48$
and $j=1,\,2$. Thus, when $16{\varrho}\leq c$ we obtain from (6.11) for
$j=1,\,2$ that

$|f(t_{j},x_{j},w)|\leq
C|f^{\prime}(t_{j},x_{j},0)|H_{0}^{-1/2}\qquad\text{when $|w|<cH_{0}^{-1/2}$}$

and then (4.13) gives $f(t_{j},x_{j},w)\in
S(|f^{\prime}(t_{j},x_{j},0)|H_{0}^{-1/2},H_{0}g^{\sharp})$ since
$H^{1/2}(t_{j},x_{j},0)\leq H_{0}^{1/2}$, $j=1,\,2$. By Proposition 4.10 we
have that $f_{j}=\alpha\delta$ where $\delta\in
S(H_{0}^{-1/2},H_{0}g^{\sharp})$ and $\alpha\in
S(|f^{\prime}(t_{j},x_{j},0)|,H_{0}g^{\sharp})$ in a $H_{0}g^{\sharp}$
neighborhood of $(t_{j},0)$ such that $|\alpha|=|f^{\prime}(t_{j},x_{j},0)|$
and $|\delta^{\prime}|=1$ at $(t_{j},0)$. Now $\partial_{t}\delta\geq 0$ so if
$|\partial_{t}\delta|^{g^{\sharp}}\geq\varepsilon>0$ at $(t_{j},0)$ for $j=1$
or $2$ then $\partial_{t}\delta\geq c\varepsilon h^{-1/2}$ in a small
$H_{1}g^{\sharp}$ neighborhood. The interval $\\{(t,0):\,|t-t_{j}|\leq
c_{0}h^{1/2}H_{1}^{-1/2}\\}$ is contained in this neighborhood for small
enough $c_{0}>0$. Then we find

$|\delta(t_{0},0)-\delta(t_{j},0)|\geq cc_{0}\varepsilon
H_{1}^{-1/2}(t_{j},0)\geq cc_{0}\varepsilon H_{0}^{-1/2}\geq
cc_{0}\varepsilon\langle\delta(t_{j},0)\rangle/16\varrho$

by (6.10), which by (6.3) contradicts (6.8) for small enough $\varrho$. Thus,
we may assume that $|\partial_{w}\delta|\geq 1/2$ at $(t_{j},0)$ for
$j=1,\,2$.

Choose coordinates $z=H_{0}^{1/2}w$, we shall use Proposition 5.3 with

$f_{j}(z)=H_{0}^{1/2}f(t_{j},x_{j},H_{0}^{-1/2}z)/|f^{\prime}(t_{j},0)|\in
C^{\infty}\qquad\text{for $j=1,\,2$.}$

Let ${\delta}_{j}(z)=H_{0}^{1/2}{\delta}(t_{j},H_{0}^{-1/2}z)\in C^{\infty}$
be the signed distance function to $f_{j}^{-1}(0)$ in $z$ coordinates, then
(6.10) gives that $|\delta_{j}(0)|\leq 16\varrho$ for $=0,\,1,\,2$. Now
$|\partial_{z}\delta_{j}(t_{j},0)|\geq 1/2$, which for small enough $\varrho$
gives $|\partial_{z}f_{j}(0)|\geq c_{0}$ for some $c_{0}>0$. In fact, we have
that $f_{j}=a_{j}\delta_{j}$ where $a_{j}\in C^{\infty}$ is uniformly bounded
and $a_{j}(0)=1$. Then we obtain that
$\partial_{z}f_{j}(0)=a_{j}(0)\partial_{z}\delta_{j}(0)+\delta_{j}(0)\partial_{z}a_{j}(0)\geq
1/2-c\varrho$. Because of condition $\operatorname{Sub_{r}}(\overline{\Psi})$
given by (3.3) we find that $f_{1}(z)>0\implies f_{2}(z)\geq 0$. Since
$|{\delta}_{j}(0)|<16{\varrho}$ we find that

(6.12)
$|{\delta}_{1}(0)-{\delta}_{2}(0)|={\varepsilon}<H_{0}^{1/2}m(t_{0},0)\leq
H_{0}^{1/2}\langle{\delta}(t_{0},0)\rangle/2<8{\varrho}$

by (6.8). Thus, for sufficiently small ${\varrho}$ we may use Proposition 5.3
with this choice of $f_{j}$ to obtain $g^{\sharp}$ orthogonal coordinates
$(z_{1},z^{\prime})$ so that $w_{0}=z_{0}=(y_{1},0)$,
$|y_{1}|=|{\delta}_{1}(0)|$ and

$\left\\{\begin{aligned} &\operatorname{sgn}(z_{1})f_{j}(z)>0\quad\text{when
$|z_{1}|\geq({\varepsilon}+|z^{\prime}|^{2})/c_{0}$}\\\
&|{\delta}_{1}(z)-{\delta}_{2}(z)|\leq({\varepsilon}+|z-z_{0}|^{2})/c_{0}\end{aligned}\right.$

when $|z|\leq c_{0}$. Let $x_{1}=H_{0}^{-1/2}y_{1}$ then
$|x_{1}|<2\langle{\delta}(t_{0},0)\rangle<32{\varrho}H_{0}^{-1/2}$ by (6.3)
and (6.10). We obtain (6.6)–(6.7) by the condition
$\operatorname{Sub_{r}}(\overline{\Psi})$, since
$H_{0}^{-1/2}{\varepsilon}<m(t_{0},0)$ by (6.12). ∎

######  Proposition 6.5.

There exists $C>0$ such that

(6.13) $m(t_{0},w)\leq
Cm(t_{0},w_{0})(1+|w-w_{0}|/\langle{\delta}(t_{0},w_{0})\rangle)^{3}$

thus $m$ is a weight for $g^{\sharp}$.

###### Proof.

Since $m\leq\langle{\delta}\rangle/2$ we only have to consider the case when

(6.14) $m(t_{0},w_{0})\leq{\varrho}\langle{\delta}(t_{0},w_{0})\rangle$

for some ${\varrho}>0$. In fact, otherwise we have by (6.2) that

$m(t_{0},w)\leq\langle{\delta}(t_{0},w)\rangle/2<m(t_{0},w_{0})(1+|w-w_{0}|/\langle{\delta}(t_{0},w_{0})\rangle)/2{\varrho}$

since the Lipschitz continuity of $w\mapsto{\delta}(t_{0},w)$ gives

(6.15)
$\langle{\delta}(t,w)\rangle\leq\langle{\delta}(t,w_{0})\rangle(1+|w-w_{0}|/\langle{\delta}(t,w_{0})\rangle)\qquad\forall\,t.$

If (6.14) holds for ${\varrho}\ll 1$, then Proposition 6.4 gives $t_{1}\leq
t_{0}\leq t_{2}$ such that (6.3), (6.5) and (6.7) hold when $|w|\leq
c_{0}H_{0}^{-1/2}$ with
$H_{0}^{1/2}=\max(H_{1}^{1/2}(t_{1},w_{0}),\,H_{1}^{1/2}(t_{2},w_{0})).$

Now, for fixed $w_{0}$ it suffices to prove (6.13) when

(6.16) $|w-w_{0}|\leq{\varrho}H_{0}^{-1/2}$

for some ${\varrho}>0$. In fact, when $|w-w_{0}|>{\varrho}H_{0}^{-1/2}$ we
obtain from (6.5) that

$|w-w_{0}|^{2}/\langle{\delta}(t_{0},w_{0})\rangle^{2}>{\varrho}^{2}H_{0}^{-1}/\langle{\delta}(t_{0},w_{0})\rangle^{2}>{\varrho}^{2}\langle{\delta}(t_{0},w_{0})\rangle^{2}/256m^{2}(t_{0},w_{0})\\\
\geq{\varrho}^{2}\langle{\delta}(t_{0},w_{0})\rangle
m(t_{0},w)/64\langle{\delta}(t_{0},w)\rangle m(t_{0},w_{0})$

since $\langle{\delta}\rangle\geq 2m$. By (6.15) we obtain that (6.13) is
satisfied with $C=64/{\varrho}^{2}$. Thus in the following we shall only
consider $w$ such that (6.16) is satisfied for ${\varrho}\ll 1$. We find by
(6.5) and (6.7) that

(6.17)
$|{\delta}(t_{1},w)-{\delta}(t_{2},w)|\leq(m(t_{0},w_{0})+H_{0}^{1/2}|w-w_{0}|^{2})/c_{0}\\\
<16m(t_{0},w_{0})(1+|w-w_{0}|^{2}/\langle{\delta}(t_{0},w_{0})\rangle^{2})/c_{0}$

when $|w-w_{0}|\leq c_{0}H_{0}^{-1/2}$. Now $G$ is slowly varying, thus we
find for small enough ${\varrho{}}>0$ that

$H_{1}^{1/2}(t_{j},w)\leq CH_{1}^{1/2}(t_{j},w_{0})\qquad\text{when
$|w-w_{0}|\leq{\varrho{}}H_{0}^{-1/2}\leq{\varrho{}}H_{1}^{-1/2}(t_{j},w_{0})$}$

for $j=1$, $2$. By (6.15) and (6.3) we obtain that

(6.18) $H_{1}^{1/2}(t_{j},w)\langle{\delta}(t_{j},w)\rangle^{2}\leq
4CH_{1}^{1/2}(t_{j},w_{0})\langle{\delta}(t_{j},w_{0})\rangle^{2}(1+|w-w_{0}|/\langle{\delta}(t_{0},w_{0})\rangle)^{2}$

when $j=1,\ 2$, and $|w-w_{0}|\leq c_{0}H_{0}^{-1/2}$. Now
$H_{1}^{1/2}(t_{j},w_{0})\langle{\delta}(t_{j},w_{0})\rangle^{2}<16m(t_{0},w_{0})$
by (6.5) for $j=1$, 2. Thus, by using (6.17), (6.18) and taking the infimum we
obtain

$m(t_{0},w)\leq
C_{0}m(t_{0},w_{0})(1+|w-w_{0}|/\langle{\delta}(t_{0},w_{0})\rangle)^{2}$

when $|w-w_{0}|\leq{\varrho}H_{0}^{-1/2}$ for ${\varrho}\ll 1$. ∎

The following result will be important for the proof of Proposition 3.6 in
Section 8.

######  Proposition 6.6.

Let $M$ be given by Definition 4.6 and $m$ by Definition 6.1. Then there
exists $C_{0}>0$ such that

(6.19) $MH^{3/2}\leq C_{0}m/\langle{\delta}\rangle^{2}.$

###### Proof.

In the proof, we shall include the $t$ variable in the $w$ variables. Observe
that since $h^{1/2}\langle\delta\rangle^{2}/6\leq m$ we find that (6.19) is
equivalent to

(6.20) $|{f^{\prime\prime}}|H^{1/2}\leq Cm/\langle{\delta}\rangle^{2}$

by Proposition 4.9. First we note that if $m\geq c\langle{\delta}\rangle>0$,
then $MH^{3/2}\langle{\delta}\rangle^{2}\leq C\langle{\delta}\rangle\leq Cm/c$
since $\langle{\delta}\rangle\leq H^{-1/2}$ and $M\leq CH^{-1}$ by Proposition
4.9.

Thus, we only have to consider the case $m\leq{\varrho}\langle{\delta}\rangle$
at $w_{0}$ for some ${\varrho}>0$ to be chosen later. Then we may use
Proposition 6.4 for ${\varrho}\ll 1$ to choose $g^{\sharp}$ orthonormal
coordinates so that
$|w_{0}|<2\langle{\delta}(w_{0})\rangle<32{\varrho}H_{0}^{-1/2}$ and $f$
satisfies (6.6) with

(6.21) $h^{1/2}/3\leq
H_{0}^{1/2}<16m(w_{0})/\langle{\delta}(w_{0})\rangle^{2}\leq 8H^{1/2}(w_{0})$

by (4.8), (6.2) and (6.5). Thus it suffices to prove the estimate

(6.22) $|{f^{\prime\prime}}|H^{1/2}\leq CH_{0}^{1/2}\qquad\text{}$

at $w_{0}$. Now it actually suffices to prove (6.22) at $w=0$. In fact, (4.11)
gives

(6.23) $H(w_{0})\leq C_{0}H(0)(1+H(w_{0})|w_{0}|^{2})\leq 5C_{0}H(0)$

since $|w_{0}|<2\langle{\delta}(w_{0})\rangle\leq 2H^{-1/2}(w_{0})$. Thus
Taylor’s formula gives

(6.24)
$|{f^{\prime\prime}(w_{0})}|H^{1/2}(w_{0})\leq\left(|{f^{\prime\prime}(0)}|+C_{3}h^{1/2}|w_{0}|\right)H^{1/2}(w_{0})\leq
C_{1}(|{f^{\prime\prime}(0)}|H^{1/2}(w_{0})+h^{1/2})$

since $|{f^{(3)}}|\leq C_{3}h^{1/2}$, which gives (6.22) at $w=0$ by (6.21)
and (6.23).

By Definition 4.4 we find that

$H^{-1/2}\geq
1+|f^{\prime}|/(|{f^{\prime\prime}}|+h^{1/4}|f^{\prime}|^{1/2}+h^{1/2})\\\
\geq(|f^{\prime\prime}|+|{f^{\prime}}|+h^{1/2})/(|{f^{\prime\prime}}|+h^{1/4}|f^{\prime}|^{1/2}+h^{1/2}),$

thus (6.22) follows if we prove

(6.25)
$|{f^{\prime\prime}}|(|{f^{\prime\prime}}|+h^{1/4}|f^{\prime}|^{1/2}+h^{1/2})\leq
C\left(|f^{\prime}|+|{f^{\prime\prime}}|+h^{1/2}\right)H_{0}^{1/2}\qquad\text{at
$0$}.$

Since $h^{1/2}/3\leq H_{0}^{1/2}$ we obtain (6.25) by the Cauchy-Schwarz
inequality if we prove that

(6.26) $|{f^{\prime\prime}(0)}|\leq
C(H_{0}^{1/4}|f^{\prime}(0)|^{1/2}+h^{1/2}).$

Let $F(z)=H_{0}f(H_{0}^{-1/2}z)$, then (6.6) gives

$\operatorname{sgn}(z_{1})F(z)\geq 0\quad\text{when
$|z_{1}|\geq{\varepsilon}+|z^{\prime}|^{2}/r$ and $|z|\leq r$}$

where $r=c_{0}$ and

${\varepsilon}=H_{0}^{1/2}m(w_{0})/c_{0}\leq
16m^{2}(w_{0})/c_{0}\langle{\delta}(w_{0})\rangle^{2}\leq
16{\varrho}^{2}/c_{0}\leq c_{0}/5$

by (6.21) when ${\varrho}\leq c_{0}/4\sqrt{5}$ which we shall assume.
Proposition 5.2 then gives that

$|{F^{\prime\prime}(0)}|\leq
C_{1}\left(|F^{\prime}(0)|/{\varrho}_{0}+H_{0}^{-1/2}h^{1/2}{\varrho}_{0}\right)\qquad\varepsilon\leq{\varrho}_{0}\leq
c_{0}/\sqrt{10}$

since $\|F^{(3)}\|_{\infty}\leq C_{3}H_{0}^{-1/2}h^{1/2}$. Observe that
$|F^{\prime}(0)|\leq C_{2}$ since $H_{0}^{1/2}\leq 8H^{1/2}(w_{0})\leq
CH^{1/2}(0)$ by (6.21) and (6.23), and $|f^{\prime}(0)|\leq CH^{-1/2}(0)$ by
(4.9). Choose

${\varrho}_{0}=\varepsilon+{\lambda}|F^{\prime}(0)|^{1/2}\leq c_{0}/\sqrt{10}$

with ${\lambda}=c_{0}(\sqrt{10}-2)/10\sqrt{C_{2}}$, then we obtain that

$|{F^{\prime\prime}(0)}|\leq C_{4}(|F^{\prime}(0)|^{1/2}+h^{1/2}m(w_{0}))$

since $H_{0}^{-1/2}\leq 3h^{-1/2}$ and
${\varepsilon}=H_{0}^{1/2}m(w_{0})/c_{0}$.

If $h^{1/2}m(w_{0})\leq|F^{\prime}(0)|^{1/2}$ then we obtain (6.26) since
$F^{\prime}=H_{0}^{1/2}f^{\prime}$ and $F^{\prime\prime}=f^{\prime\prime}$. If
$|F^{\prime}(0)|^{1/2}\leq h^{1/2}m(w_{0})$, then we find

$|{f^{\prime\prime}(0)}|\leq 2C_{4}h^{1/2}m(w_{0})\leq
4C_{2}m(w_{0})/\langle{\delta}(w_{0})\rangle.$

Then (6.20) follows from (6.13), (6.15) and (6.23) since
$H^{1/2}(w_{0})\leq\langle{\delta}(w_{0})\rangle^{-1}$, which completes the
proof of the proposition. ∎

Next, we shall prove a convexity property of $t\mapsto m(t,w)$, which will be
essential for the proof.

######  Proposition 6.7.

Let $m$ be given by Definition 6.1. Then

(6.27) $\sup_{t_{1}\leq t\leq
t_{2}}m(t,w)\leq{\delta}(t_{2},w)-{\delta}(t_{1},w)+m(t_{1},w)+m(t_{2},w)\qquad\forall\,w.$

###### Proof.

By definition we find that

(6.28) $\inf_{\pm(t-t_{0})\geq
0}\left(|{\delta}(t,w)-{\delta}(t_{0},w)|+H_{1}^{1/2}(t,w)\langle{\delta}(t,w)\rangle^{2}/2\right)\leq
m(t_{0},w).$

Let $t\in[t_{1},t_{2}]$, then by taking the independent infima, we obtain that

$m(t,w)\leq\inf_{r\leq t_{1}<t_{2}\leq
s}{\delta}(s,w)-{\delta}(r,w)+H_{1}^{1/2}(s,w)\langle{\delta}(s,w)\rangle^{2}/2+H_{1}^{1/2}(r,w)\langle{\delta}(r,w)\rangle^{2}/2\\\
\leq{\delta}(t_{2},w)-{\delta}(t_{1},w)+\inf_{t\geq
t_{2}}\left(|{\delta}(t,w)-{\delta}(t_{2},w)|+H_{1}^{1/2}(t,w)\langle{\delta}(t,w)\rangle^{2}/2\right)\\\
+\inf_{t\leq
t_{1}}\left(|{\delta}(t,w)-{\delta}(t_{1},w)|+H_{1}^{1/2}(t,w)\langle{\delta}(t,w)\rangle^{2}/2\right).$

By using (6.28) for $t_{0}=t_{1}$, $t_{2}$, we obtain (6.27) after taking the
supremum. ∎

Next, we shall construct the pseudo-sign $B={\delta}+{\varrho}_{0}$, which we
shall use in Section 8 to prove Proposition 3.6 with the multiplier
$b^{w}=B^{Wick}$.

######  Proposition 6.8.

Assume that ${\delta}$ is given by Definition 4.2 and $m$ is given by
Definition 6.1. Then for $T>0$ there exists real valued ${\varrho}_{T}(t,w)\in
L^{\infty}$ with the property that $w\mapsto{\varrho}_{T}(t,w)$ is uniformly
Lipschitz continuous, and

(6.29) $\displaystyle|{\varrho}_{T}|\leq m$ (6.30) $\displaystyle
T\partial_{t}({\delta}+{\varrho}_{T})\geq m/2\qquad\text{in
$\mathcal{D}^{\prime}(\mathbf{R})$}$

when $|t|<T$.

###### Proof.

(We owe this argument to Lars Hörmander [12].) Let

(6.31) ${\varrho}_{T}(t,w)=\sup_{-T\leq s\leq
t}\left({\delta}(s,w)-{\delta}(t,w)+\frac{1}{2T}\int_{s}^{t}m(r,w)\,dr-m(s,w)\right)$

for $|t|\leq T$, then

${\delta}(t,w)+{\varrho}_{T}(t,w)=\sup_{-T\leq s\leq
t}\left({\delta}(s,w)-\frac{1}{2T}\int_{0}^{s}m(r,w)\,dr-m(s,w)\right)\\\
+\frac{1}{2T}\int_{0}^{t}m(r,w)\,dr$

which immediately gives (6.30) since the supremum is nondecreasing. Since
$w\mapsto{\delta}(t,w)$ and $w\mapsto m(t,w)$ are uniformly Lipschitz
continuous by Proposition 6.4, we find that $w\mapsto{\varrho}_{T}(t,w)$ is
uniformly Lipschitz continuous by taking the supremum. Since
$\delta(s,w)\leq\delta(t,w)$ when $s\leq t\leq T$, we find from Proposition
6.7 that

${\delta}(s,w)-{\delta}(t,w)+\frac{1}{2T}\int_{s}^{t}m(r,w)\,dr-m(s,w)\leq
m(t,w)\qquad-T\leq s\leq t\leq T.$

By taking the supremum, we obtain that $-m(t,w)\leq{\varrho}_{T}(t,w)\leq
m(t,w)$ when $|t|\leq T$, which proves the result. ∎

We shall also include a term in the multiplier to control the error terms
involving $D_{x}u$.

######  Lemma 6.9.

Assume that $A$ satisfies the conditions in Proposition 2.2 near
$w_{0}\in\Sigma_{2}$. Then there exists a matrix $L$ and constant $c_{1}>0$
such that $\\{A,\langle L(x-x_{0}),\xi\rangle\\}\geq|\xi|^{2}-c_{1}$
microlocally near $w_{0}\in\Sigma_{2}$, where $x_{0}$ is the value of $x$ at
$w_{0}$. The constants only depend on the seminorms of $A$.

###### Proof.

Let $w=(x,\xi,z)$, then we find from the conditions that

$A(x,\xi,z)=\langle C_{2}(x,\xi,z)\xi,\xi\rangle+\langle
C_{1}(x,z),\xi\rangle+C_{0}(x,z)$

where $C_{j}\in S^{0}$ is real valued, $\forall\,j$, and $C_{2}$ is a
symmetric and nondegenerate matrix microlocally near $w_{0}$. By a translation
we may assume that $x=0$ at $w_{0}$. If we take $L=C_{2}^{-1}(0,0,w_{0})$ then
we find that

(6.32) $\\{A,\langle Lx,\xi\rangle\\}=\langle
L\xi,\partial_{\xi}A\rangle-\langle Lx,\partial_{x}A\rangle\geq
c_{2}|\xi|^{2}-c_{0}$

where $c_{2}=2$ at $w_{0}$, since
$\partial_{\xi}A=2C_{2}\xi+\langle\partial_{\xi}C_{2}\xi,\xi\rangle+C_{1}$. By
continuity, we get the estimate in a neighborhood of $w_{0}$ where
$|\partial_{\xi}C_{2}\xi|\ll 1$. ∎

Because of the cut-off in the estimate (3.17) we will only need the lower
bound in a neighborhood of $w_{0}$.

######  Definition 6.10.

Let the multiplier symbol $B_{T}={\delta}_{0}+{\varrho}_{T}+\lambda_{T}$,
where ${\delta}_{0}={\delta}$ is given by Definition 4.2, ${\varrho}_{T}$ is
given by Definition 6.8 for $T>0$ so it is real valued and Lipschitz
continuous, satisfying $|{\varrho}_{T}|\leq m$ when $|t|\leq T$, with
$m\leq\langle{\delta}_{0}\rangle/2$ given by Definition 6.1, and
$\lambda_{T}=\epsilon h^{1/2}\langle L(x-x_{0}),\xi\rangle/T\in S(h^{-1/2},g)$
uniformly when $|x-x_{0}|\leq T$ and $|\xi|\lesssim h^{-1}$, where
$0<\epsilon\leq 1$ and $L$ is given by Lemma 6.9, so $\lambda_{T}$ is
Lipschitz continuous.

##  7\. The Wick quantization

In order to define the multiplier we shall use the Wick quantization. We shall
start by recapitulating some results from Section 6 in [4] about the Wick
operators. As before, we shall assume that $g^{\sharp}=(g^{\sharp})^{\sigma}$
and the coordinates chosen so that $g^{\sharp}(w)=|w|^{2}$. For $a\in
L^{\infty}(T^{*}\mathbf{R}^{n})$ we define the Wick quantization:

$a^{Wick}(x,D_{x})u(x)=\int_{T^{*}\mathbf{R}^{n}}a(y,{\eta}){\Sigma}^{w}_{y,{\eta}}(x,D_{x})u(x)\,dyd{\eta}\qquad
u\in C_{0}^{\infty}$

using the orthonormal projections ${\Sigma}^{w}_{y,{\eta}}(x,D_{x})$ with Weyl
symbol

${\Sigma}_{y,{\eta}}(x,{\xi})={\pi}^{-n}\exp(-g^{\sharp}(x-y,{\xi}-{\eta}))$

(see [2, Appendix B] or [13, Section 4]). We find that $a^{Wick}$:
$\mathcal{S}\mapsto\mathcal{S}^{\prime}$ so that

(7.1) $a\geq 0\implies\left(a^{Wick}(x,D_{x})u,u\right)\geq 0\qquad u\in
C_{0}^{\infty}$

$(a^{Wick})^{*}=(\overline{a})^{Wick}$ and
$\|a^{Wick}(x,D_{x})\|_{\mathcal{L}(L^{2}(\mathbf{R}^{n}))}\leq\|a\|_{L^{\infty}(T^{*}\mathbf{R}^{n})},$
which is the main advantage with the Wick quantization (see [13, Proposition
4.2]).

We obtain from the definition that $a^{Wick}=a_{0}^{w}$ where

(7.2) $a_{0}(w)={\pi}^{-n}\int_{T^{*}\mathbf{R}^{n}}a(z)\exp(-|w-z|^{2})\,dz$

is the Gaussian regularization, thus Wick operators with real symbols have
real Weyl symbols. This convolution also maps polynomials to polynomials.

######  Remark 7.1.

Observe that $a^{Wick}=a^{w}$ if $a(x,\xi)$ is affine in $x$ for fixed $\xi$
and affine in $\xi$ for fixed $x$, for example if $a(x,\xi)=\langle
Lx,\xi\rangle$ with a constant matrix $L$ .

In fact, $a^{Wick}=a^{w}$ if and only if

${\pi}^{-n}\int_{T^{*}\mathbf{R}^{n}}(a(z)-a(w))\exp(-|w-z|^{2})\,dz=0$

This vanishes if $a$ is affine in $x$ for fixed $\xi$ and affine in $\xi$ for
fixed $x$ since

$a(x,\xi)-a(y,\eta)=a(x,\xi)-a(y,\xi)+a(y,\xi)-a(y,\eta)$

which are two odd integrands giving vanishing integrals.

In the following, we shall assume that $G=Hg^{\sharp}\leq g^{\sharp}$ is a
slowly varying metric satisfying

(7.3) $H(w)\leq C_{0}H(w_{0})(1+|w-w_{0}|)^{N_{0}}$

and $m$ is a weight for $G$ satisfying (7.3) with $H$ replaced by $m$, by
Propositions 4.7 and 6.5. This means that $G$ and $m$ are strongly ${\sigma}$
temperate in the sense of [1, Definition 7.1]. Recall the symbol class
$S^{+}(1,g^{\sharp})$ given by Definition 3.1.

######  Proposition 7.2.

Assume that $a\in L^{\infty}$ such that $|a|\leq Cm$, where $m$ is a weight
for $g^{\sharp}$, then $a^{Wick}=a_{0}^{w}$ where $a_{0}\in S(m,g^{\sharp})$
is given by (7.2). If $a\geq m$ we obtain that $a_{0}\geq c_{0}m$ for a fixed
constant $c_{0}>0$. If $a\in S(M,G)$, where $M$ is a weight for $G$, then
$a_{0}\cong a$ modulo $S(mH,G)$. If $|a|\leq Cm$ and $a=0$ in a fixed $G$ ball
with center $w$, then $a\in S(mH^{N},G)$ at $w$ for any $N$. If $a$ is
polynomial in the variable $\xi$ then $a_{0}$ is polynomial in $\xi$ with the
same degree as $a$, and if $a$ is Lipschitz continuous then we have $a_{0}\in
S^{+}(1,g^{\sharp})$.

By localization we find, for example, that if $|a|\leq Cm$ and $a\in S(m,G)$
in a $G$ neighborhood of $w_{0}$, then $a_{0}\cong a$ modulo $S(mH,G)$ in a
smaller $G$ neighborhood of $w_{0}$. Observe that the results are uniform in
the metrics and weights. The results are well known, but for completeness we
give a proof.

###### Proof.

Since $a$ is measurable satisfying $|a|\leq Cm$, where $m(z)\leq
C_{0}m(w)(1+|z-w|)^{N_{0}}$ by (7.3), we find that $a^{Wick}=a_{0}^{w}$ where
$a_{0}=\mathcal{O}(m)$ is given by (7.2). By differentiating on the
exponential factor, we find $a_{0}\in S(m,g^{\sharp})$, and similarly we find
that $a_{0}\geq m/C$ if $a\geq m$ since $m(z)\gtrsim m(w)/(1+|z-w|)^{N_{0}}$.

If $a=0$ in a $G$ ball of radius ${\varepsilon}>0$ and center at $w$, then we
can write

${\pi}^{n}a_{0}(w)=\int_{|z-w|\geq{\varepsilon}H^{-1/2}(w)}a(z)\exp(-|w-z|^{2})\,dz=\mathcal{O}(m(w)H^{N}(w))$

for any $N$ even after repeated differentiation.

If $a\in S(m,G)$ then Taylor’s formula gives

$a_{0}(w)=a(w)+{\pi}^{-n}\int_{0}^{1}\int_{T^{*}\mathbf{R}^{n}}(1-{\theta})^{2}\langle
a^{\prime\prime}(w+{\theta}z)z,z\rangle e^{-|z|^{2}}\,dzd{\theta}/2$

where $a^{\prime\prime}\in S(mH,G)$ since $G=Hg^{\sharp}$. Since
$m(w+{\theta}z)\leq C_{0}m(w)(1+|z|)^{N_{0}}$ and $H(w+{\theta}z)\leq
C_{0}H(w)(1+|z|)^{N_{0}}$ when $|{\theta}|\leq 1$, we find that $a_{0}(w)\cong
a(w)$ modulo $S(mH,G)$.

If $a$ is polynomial in the variable $\xi$ of degree $k$ then
$\partial^{\alpha}_{\xi}a\equiv 0$,$\forall|\alpha|>k$, which gives
$\partial^{\alpha}_{\xi}a_{0}\equiv 0$. Thus $a_{0}$ is of degree $\leq k$.
The Lipschitz continuity of $a$ means that $\partial a\in
L^{\infty}(T^{*}\mathbf{R}^{n})$. Since we have $\partial
a_{0}(w)={\pi}^{-n}\int_{T^{*}\mathbf{R}^{n}}\partial
a(z)\exp(-|w-z|^{2})\,dz$, we obtain the proposition. ∎

We shall need the following result about the composition of Wick operators.

######  Proposition 7.3.

Assume that $a$ and $b\in L^{\infty}$. If $|a|\leq m_{1}$ and $|\partial
b|\leq m_{2}$, where $m_{j}$ are weights for $g^{\sharp}$ satisfying (7.3),
then

(7.4) $a^{Wick}b^{Wick}=(ab)^{Wick}+r^{w}$

with $r\in S(m_{1}m_{2},g^{\sharp})$. If $a$ and $b$ are real such that
$|a|\leq m_{1}$ and $|\partial^{2}b|\leq m_{2}$, then

(7.5) $\operatorname{Re}a^{Wick}b^{Wick}=\left(ab-\frac{1}{2}\partial
a\cdot\partial b\right)^{Wick}+R^{w}$

with $R\in S(m_{1}m_{2},g^{\sharp})$. By taking the adjoints, we get these
results with $a$ and $b$ switched.

Observe that since $a\in L^{\infty}$ and $\partial b$ is Lipschitz continuous
in (7.5), we find that $\partial a\cdot\partial b$ is a well-defined
distribution. In fact, we can define it as $\partial a\cdot\partial
b(\varphi)=-\iint a\partial(\varphi\partial b)dw$. Proposition 7.3 essentially
follows from Proposition 3.4 in [14] and Lemma A.1.5 in [15] but we shall for
completeness give a proof.

###### Proof.

By Proposition 7.2 we have $a^{Wick}b^{Wick}=a_{0}^{w}b_{0}^{w}$ in (7.4)
where $a_{0}\in S(m_{1},g^{\sharp})$ and $b_{0}\in S^{+}(m_{2},g^{\sharp})$.
By Lemma 3.2 we find $a^{Wick}b^{Wick}\cong(a_{0}b_{0})^{w}$ modulo
$\operatorname{Op}S(m_{1}m_{2},g^{\sharp})$, where

(7.6) $a_{0}(w)b_{0}(w)=\pi^{-2n}\iint
a(w+z_{1})b(w+z_{2})e^{-|z_{1}|^{2}-|z_{2}|^{2}}\,dz_{1}dz_{2}.$

By using the Taylor formula we find that
$b(w+z_{2})=b(w+z_{1})+r_{1}(w,z_{1},z_{2})$ where $|r_{1}(w,z_{1},z_{2})|\leq
Cm_{2}(w)(1+|z_{1}|+|z_{2}|)^{N}$ by (7.3). Integration in $z_{2}$ then gives
(7.4).

For the proof of (7.5) we use that
$\operatorname{Re}a_{0}^{w}b_{0}^{w}\cong(a_{0}b_{0})^{w}$ modulo
$\operatorname{Op}S(m_{1}m_{2},g^{\sharp})$ by Lemma 3.2, since $a_{0}$ and
$b_{0}$ are real and $\partial^{2}b_{0}\in S(m_{2},g^{\sharp})$. We use the
Taylor formula again:

$b(w+z_{2})=b(w+z_{1})+\partial
b(w+z_{1})\cdot(z_{2}-z_{1})+r_{2}(w,z_{1},z_{2})$

where $|r_{2}(w,z_{1},z_{2})|\leq Cm_{2}(w)(1+|z_{1}|+|z_{2}|)^{N}$. The term
with $z_{2}$ is odd and gives a vanishing contribution in (7.6). Since
$\partial_{z_{1}}e^{-|z_{1}|^{2}-|z_{2}|^{2}}=-2z_{1}e^{-|z_{1}|^{2}}$ we
obtain (7.5) after an integration by parts, since $|a\partial^{2}b|\leq
m_{1}m_{2}$. ∎

######  Example 7.4.

If $a\in S(H^{-1/2},g^{\sharp})\bigcap S^{+}(1,g^{\sharp})$ and $b\in S(M,G)$,
then $\operatorname{Re}a^{Wick}b^{Wick}\cong(ab)^{Wick}$ modulo
$\operatorname{Op}S(MH^{1/2},g^{\sharp})$.

We shall compute the Weyl symbol for the Wick operator
$B_{T}^{Wick}=({\delta}_{0}+{\varrho}_{T}+\lambda_{T})^{Wick}$ given by
Definition 6.10. In the following, we shall suppress the $T$ parameter.

######  Proposition 7.5.

Let $B={\delta}_{0}+{\varrho}_{0}+\lambda$ be given by Definition 6.10, then
we have $B^{Wick}=b^{w}$ where $b={\delta}_{1}+{\varrho}_{1}+\lambda$ is real,
${\delta}_{1}\in S(H^{-1/2},g^{\sharp})\bigcap S^{+}(1,g^{\sharp})$, and
${\varrho}_{1}\in S(m,g^{\sharp})\bigcap S^{+}(1,g^{\sharp})$ uniformly when
$|t|\leq T$. Also, there exists ${\kappa}_{2}>0$ so that
${\delta}_{1}\cong{\delta}_{0}$ modulo $S(H^{1/2},G)$ when
$\langle{\delta}_{0}\rangle\leq{\kappa}_{2}H^{-1/2}$. For any $\varepsilon>0$
we find that $|{\delta}_{0}|\geq\varepsilon H^{-1/2}$ and
$H^{1/2}\leq\varepsilon/3$ imply that $|\delta_{0}+\varrho_{0}|\geq\varepsilon
H^{-1/2}/3$.

###### Proof.

Let ${\delta}_{0}^{Wick}={\delta}_{1}^{w}$ and
${\varrho}_{0}^{Wick}={\varrho}_{1}^{w}$. Since $|{\delta}_{0}|\leq
H^{-1/2}_{1}$, $|{\varrho}_{0}|\leq m$ and the symbols are real valued, we
obtain from Proposition 7.2 that ${\delta}_{1}\in S(H^{-1/2},g^{\sharp})$ and
${\varrho}_{1}\in S(m,g^{\sharp})$ are real valued. Since ${\delta}_{0}$ and
${\varrho}_{0}$ are uniformly Lipschitz continuous, we find that
${\delta}_{1}$ and ${\varrho}_{1}\in S^{+}(1,g^{\sharp})$ by Proposition 7.2.
By Remark 7.1 we have $\lambda^{Wick}=\lambda^{w}$.

If $\langle{\delta}_{0}\rangle\leq{\kappa}H^{-1/2}$ at $w_{0}$ for
sufficiently small ${\kappa}>0$, then we find by the Lipschitz continuity of
${\delta}_{0}$ and the slow variation of $G$ that
$\langle{\delta}_{0}\rangle\leq C_{0}{\kappa}H^{-1/2}$ in a fixed $G$
neighborhood ${\omega}_{\kappa}$ of $w_{0}$ (depending on ${\kappa}$). For
${\kappa}\ll 1$ we find that ${\delta}_{0}\in S(H^{-1/2},G)$ in
${\omega}_{\kappa}$ by Proposition 4.9, which implies that
${\delta}_{1}\cong{\delta}_{0}$ modulo $S(H^{1/2},G)$ near $w_{0}$ by
Proposition 7.2 after localization.

When $|{\delta}_{0}|\geq\varepsilon H^{-1/2}\geq\varepsilon>0$ at $w_{0}$,
then we find that

$|{\varrho}_{0}|\leq
m\leq\langle{\delta}_{0}\rangle/2\leq(1+H^{1/2}/\varepsilon)|{\delta}_{0}|/2.$

We obtain that $|{\varrho}_{0}|\leq 2|{\delta}_{0}|/3$ and
$|\delta_{0}+\varrho_{0}|\geq|{\delta}_{0}|/3\geq\varepsilon H^{-1/2}/3$ when
$H^{1/2}\leq\varepsilon/3$, which completes the proof. ∎

Let $m$ be given by Definition 6.1, then $m$ is a weight for $g^{\sharp}$
according to Proposition 6.5. We are going to use the symbol classes
$S(m^{k},g^{\sharp})$, $k\in\mathbf{R}$. The following proposition shows that
the operator $m^{Wick}$ dominates all operators in
$\operatorname{Op}S(m,g^{\sharp})$.

######  Proposition 7.6.

If $c\in S(m,g^{\sharp})$ then there exists a positive constant $C_{0}$ such
that

(7.7) $|\langle c^{w}u,u\rangle|\leq C_{0}\left(m^{Wick}u,u\right)\qquad u\in
C_{0}^{\infty}.$

Here $C_{0}$ only depends on the seminorms of $c\in S(m,g^{\sharp})$.

###### Proof.

We shall use an argument by Hörmander [12]. Let $0<{\varrho}\leq 1$

(7.8) $M_{\varrho}(w_{0})=\sup_{w}m(w)/(1+{\varrho}|w-w_{0}|)^{3}$

then $m\leq M_{\varrho}\leq Cm/{\varrho}^{3}$ and

(7.9) $M_{\varrho}(w)\leq
CM_{\varrho}(w_{0})(1+{\varrho}|w-w_{0}|)^{3}\qquad\text{uniformly in
$0<{\varrho}\leq 1$}$

by the triangle inequality. Thus, $M_{\varrho}$ is a weight for
$g_{\varrho}={\varrho}^{2}g^{\sharp}$, uniformly in ${\varrho}$. Take
$0\leq{\chi}\in C_{0}^{\infty}$ such that
$\int_{T^{*}\mathbf{R}^{n}}{\chi}(w)\,dw>0$ and let

$m_{\varrho}(w)={\varrho}^{-2n}\int{\chi}({\varrho}(w-z))M_{\varrho}(z)\,dz.$

Then by (7.9) we find $1/C_{0}\leq m_{\varrho}/M_{\varrho}\leq C_{0}$, and
$|\partial^{\alpha}m_{\varrho}|\leq
C_{\alpha}{\varrho}^{|{\alpha}|}m_{\varrho}$ thus $m_{\varrho}\in
S(m_{\varrho},g_{\varrho})$ uniformly in $0<{\varrho}\leq 1$. Let
$m_{\varrho}^{Wick}={\mu}_{\varrho}^{w}$ then Proposition 7.2 and (7.9) give
$m_{\varrho}/c\leq{\mu}_{\varrho}\in S(m_{\varrho},g_{\varrho})$ uniformly in
$0<{\varrho}\leq 1$. Since $m\cong m_{\varrho}$ (depending on $\varrho$) we
may replace $m^{Wick}$ with $m_{\varrho}^{Wick}={\mu}_{\varrho}^{w}$ in (7.7)
for any fixed ${\varrho}>0$.

Let $a_{\varrho}={\mu}_{{\varrho}}^{-1/2}\in
S(m^{-1/2}_{\varrho},g^{\sharp}_{\varrho})$ with $0<{\varrho}\leq 1$ to be
chosen later. Since $g_{\varrho}$ is uniformly ${\sigma}$ temperate,
$g_{\varrho}/g_{\varrho}^{\sigma}={\varrho}^{4}$, $m_{\varrho}$ is uniformly
${\sigma}$, $g_{\varrho}$ temperate, and ${\mu}_{\varrho}^{\pm 1/2}\in
S(m_{\varrho}^{\pm 1/2},g_{\varrho})$ uniformly, the calculus gives that
$a_{\varrho}^{w}(a_{\varrho}^{-1})^{w}=1+r^{w}_{\varrho}$ where
$r_{\varrho}/{\varrho}^{2}\in S(1,g^{\sharp})$ uniformly for $0<{\varrho}\leq
1$. Similarly, we find that
$a_{\varrho}^{w}{\mu}_{\varrho}^{w}a_{\varrho}^{w}=1+s^{w}_{\varrho}$ where
$s_{\varrho}/{\varrho}^{2}\in S(1,g^{\sharp})$ uniformly. We obtain that the
$L^{2}$ operator norms

$\|r^{w}_{\varrho}\|_{\mathcal{L}(L^{2})}+\|s^{w}_{\varrho}\|_{\mathcal{L}(L^{2})}\leq
C{\varrho}^{2}\leq 1/2$

for sufficiently small ${\varrho}$. By fixing such a value of ${\varrho}$ we
find that $1/2\leq a_{\varrho}^{w}{\mu}_{\varrho}^{w}a_{\varrho}^{w}\leq 2$
and

(7.10) $\frac{1}{2}\|u\|\leq\|a_{\varrho}^{w}(a^{-1}_{\varrho})^{w}u\|\leq
2\|u\|$

thus $u\mapsto a_{\varrho}^{w}(a^{-1}_{\varrho})^{w}u$ is an homeomorphism on
$L^{2}$. The estimate (7.7) then follows from

$|\langle
c^{w}a_{\varrho}^{w}(a^{-1}_{\varrho})^{w}u,a_{\varrho}^{w}(a^{-1}_{\varrho})^{w}u\rangle|\leq
C\|(a^{-1}_{\varrho})^{w}u\|^{2}\leq
2C\langle\mu_{\varrho}^{w}a_{\varrho}^{w}(a^{-1}_{\varrho})^{w}u,a_{\varrho}^{w}(a^{-1}_{\varrho})^{w}u\rangle$

which holds since
$a_{\varrho}^{w}c^{w}a_{\varrho}^{w}\in\operatorname{Op}S(1,g^{\sharp})$ is
bounded in $L^{2}$. Observe that the bounds only depend on the seminorms of
$c$ in $S(m,g^{\sharp})$, since ${\varrho}$ and $a_{\varrho}$ are fixed. ∎

We shall also need a weight to handle the calculus errors in the $\xi$
variables. Let $\mu=h^{1/2}\langle\xi\rangle^{2}$ which is a weight for
$g_{0}(dx,d\xi)=|dx|^{2}+|d\xi|^{2}/\langle\xi\rangle^{2}$ so that $\lambda\in
S(\mu,g_{0})$ uniformly in $h$. In order to handle the compositions with
$\lambda$ and $f_{0}$ we shall need the following metric:

(7.11)
$G_{0}=H|dw|^{2}+H(dt^{2}+|dx|^{2})/h+Hhd\tau^{2}+|d\xi|^{2}/\langle\xi\rangle^{2}$

which is strongly $\sigma$ temperate in the sense that

(7.12) $G_{0,z}\lesssim G_{0,z_{0}}(1+G_{0,z_{0}}(z-z_{0}))$

but since $G_{0}/G_{0}^{\sigma}\not\lesssim 1$ it is not $\sigma$ temperate.
But since $\lambda$ and $f_{0}$ are linear in $\xi$ we shall only use the
calculus on $\Sigma_{2}$, i.e., in $(t,x,w)$, and $G_{0}=G$ on $T\Sigma_{2}$.

To handle the compositions with $b_{1}=\delta_{1}+\varrho_{1}\in
S(H^{-1/2},g^{\sharp})$ we shall use the metric:

(7.13)
$g^{\sharp}_{0}=|dw|^{2}+(dt^{2}+|dx|^{2})/h+hd\tau^{2}+|d\xi|^{2}/\langle\xi\rangle^{2}$

which is strongly $\sigma$ temperate on $\Sigma_{2}$, since
$g^{\sharp}_{0}=g^{\sharp}$ on $T\Sigma_{2}$. Then $\lambda f\in
S(Mh^{1/2}\langle\xi\rangle,G_{0})$ and $b_{1}f_{0}\in
S(MH^{1/2}h^{1/2}\langle\delta_{0}\rangle\langle\xi\rangle,g^{\sharp}_{0})$.

######  Lemma 7.7.

If $|C|\lesssim\mu$ then we have

(7.14) $|\langle
C^{Wick}u,u\rangle|\lesssim\langle\mu^{Wick}u,u\rangle\lesssim
h^{1/2}\|\langle D_{x}\rangle u\|^{2}=h^{1/2}(\|D_{x}u\|^{2}+\|u\|^{2})$

where $D_{x}u=(D_{x_{1}}u,D_{x_{2}}u,\dots)$.

###### Proof.

By taking the real and imaginary part it suffices to prove the estimate for
real valued $C$. We have $|C|\lesssim\mu$ so $\pm\langle
C^{Wick}u,u\rangle\lesssim\langle\mu^{Wick}u,u\rangle$. Now
$\mu^{Wick}=\nu^{w}$ where $\nu\in S(\mu,g^{\sharp}_{0})$ by Proposition 7.2,
so $\nu_{0}^{w}=\langle D_{x}\rangle^{-1}\nu^{w}\langle
D_{x}\rangle^{-1}\in\operatorname{Op}S(h^{1/2},g^{\sharp}_{0})$ which gives

$\langle\mu^{Wick}u,u\rangle=\langle\nu_{0}^{w}\langle D_{x}\rangle u,\langle
D_{x}\rangle u\rangle\lesssim h^{1/2}\|\langle D_{x}\rangle u\|^{2}$

which proves the result. ∎

##  8\. The multiplier estimate

In this section we shall obtain a proof of Proposition 3.6 which involves
giving lower bounds on $\operatorname{Re}b_{T}^{w}f_{1}^{w}$, with the
multipler $b_{T}^{w}=B_{T}^{Wick}$ having symbol
$B_{T}={\delta}_{0}+{\varrho}_{0}+\lambda$ given by Proposition 7.5. Also,
$f_{1}=f+f_{0}$, where $f\in S(M,G)$ and $f_{0}=\partial_{\eta}f\cdot
r\cdot\xi$ with $r\in S(1,g)$ so that $f_{1}\in