# On Métivier’s Lax-Mizohata theorem and extensions to weak defects of
hyperbolicity.
Part one.

Karim Ndoumajoud Institut de Mathématiques de Jussieu - Paris Rive Gauche UMR
CNRS 7586, Sorbonne Université Benjamin Texier Université Claude Bernard Lyon
1, Institut Camille Jordan UMR CNRS 5208

###### Abstract

We prove that, for first-order, fully nonlinear systems of partial
differential equations, under an hypothesis of ellipticity for the principal
symbol, the Cauchy problem has no solution within a range of Sobolev indices
depending on the regularity of the initial datum. This gives a new and greatly
detailed proof of a result of G. Métivier [Remarks on the Cauchy problem,
2005]. We then extend this result to systems experiencing a transition from
hyperbolicity and ellipticity, in the spirit of recent work by N. Lerner, Y.
Morimoto, and C.-J. Xu, [Instability of the Cauchy-Kovalevskaya solution for a
class of non-linear systems, 2010], and N. Lerner, T. Nguyen and B. Texier
[The onset of instability in first-order systems, 2018].

##  1 Introduction

We study the fully nonlinear, order-one Cauchy problem in $\mathbb{R}^{d}:$

(1.1) $\left\\{\begin{aligned} \partial_{t}u+F(t,x,u,\partial_{x}u)&=0,\\\
u_{|t=0}&=u_{in}\in H^{\sigma}(\mathbb{R}^{d})\end{aligned}\right.$

where $t\geq 0$ and $d\geq 1.$ The map
$F:\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{N}\times\mathbb{R}^{Nd}\to\mathbb{R}^{N}$
is smooth in a neighborhood of some $(0,x^{0},u^{0},v^{0}),$ where $N\geq 1.$
The principal symbol is defined as

(1.2) $A(t,x,u,v,\xi):=\xi\cdot\partial_{4}F(t,x,u,v)=\sum_{1\leq j\leq
d}\xi_{j}\partial_{4_{j}}F(t,x,u,v),$

where $\partial_{4_{j}}F$ refers to the derivative of $F$ with respect to the
$j$-th component $v_{j}$ of its fourth argument
$(v_{1},\dots,v_{d})\in(\mathbb{R}^{N})^{d}.$

Under an assumption of initial ellipticity (strong defect of hyperbolicity,
see Assumption 2.1 below) for $A(0,\omega^{0}),$ for some
$\omega^{0}=(x^{0},u^{0},v^{0},\xi^{0})\in\mathbb{R}^{d}\times\mathbb{R}^{N}\times\mathbb{R}^{Nd}\times\mathbb{S}^{d-1},$
we prove in Theorem 1 that if $s>1+d/2,$ then no local-in-time $H^{s}$
solution exists to the Cauchy problem (1.1), even on arbitrarily small time
intervals, for some data with a Sobolev degree of regularity
$\sigma\in[s,2s-1-d/2).$

Under an assumption of a transition from hyperbolicity to ellipticity (weak
defect of hyperbolicity; see Assumption 2.2 below) for $A(0,\omega^{0}),$ for
some $\omega^{0},$ we prove in Theorem 2 that if $s>3/2+3/4+d/2,$ then no
local-in-time $H^{s}$ solution exists to the Cauchy problem (1.1), even on
arbitrarily small time intervals, for some data with a Sobolev degree of
regularity $\sigma\in[s,2s-3/2-d/2).$

Our results recover and extend an elliptic theorem of G. Métivier [3]. The
analysis of weak defects of hyperbolicity was pioneered in the article of N.
Lerner, Y. Morimoto and C.-J. Xu [11]. These authors proved that for complex
scalar equations exhibiting a weak defect of hyperbolicity, no solutions could
exist above a certain level of regularity. Following [11], strong Hadamard
instabilities were proved by N. Lerner, T. Nguyen and B. Texier [12] for
systems exhibiting weak defects of hyperbolicity. Our second result (Theorem
2) extends Theorem 1.6 from [12]: while the result of [12], under an
assumption analogous to Assumption 2.2, proved a lack of regularity of an
hypothetical flow, we prove that such a flow does not exist. The functional
spaces considered in [12] are however larger: the lack of Hölder continuity is
from $W^{1,\infty}$ to $H^{m},$ any $m\in\mathbb{R},$ while here we restrict
to sufficiently small Sobolev spaces.

##  2 Assumptions and results

###  2.1 Elliptic initial-value problems

Consider the ellipticity assumption:

######  Assumption 2.1.

For some
$(x^{0},u^{0},v^{0},\xi^{0})\in\mathbb{R}^{d}\times\mathbb{R}^{N}\times\mathbb{R}^{Nd}\times\mathbb{S}^{d-1},$
the spectrum of the principal symbol $A(0,x^{0},u^{0},v^{0},\xi^{0})$ defined
in (1.2) is not entirely contained in $\mathbb{R}.$

The above ellipticity assumption implies non-existence of Sobolev solutions to
the initial-value problem (1.1), in the following sense:

######  Theorem 1.

Under Assumption 2.1, for any $s>d/2+1,$ any $s\leq\sigma<2s-1-d/2,$ for some
$u_{in}\in H^{\sigma}(\mathbb{R}^{d}),$ there is no $T>0$ and no ball
$B_{x^{0}}$ centered at $x^{0}$ such that the initial-value problem (1.1) has
a solution in $C^{0}([0,T],H^{s}(B_{x^{0}})).$

Theorem 1 recovers the ellipticity theorem of Métivier (Theorem 4.2 [3]),
which extends the classical linear ellipticity theorems of Lax [9] and
Mizohata [14]. The merit of our proof might be to clarify Métivier’s and
prepare for extensions to weak defects of hyperbolicity (see Theorem 2 below),
and to Gevrey spaces in the spirit of the work of B. Morisse [5, 6, 7]. The
proof of Theorem 1 is given in Section 3.

###  2.2 Transition to ellipticity

Weak defects of hyperbolicity were first explored for scalar equations in
[11]. The results of [11] were extended to quasi-linear systems in [12].
Roughly speaking, an assumption of a weak defect of hyperbolicity consists in
relaxing Assumption 2.1 into an assumption of initial hyperbolicity that is
instantaneously lost: the principal symbol is hyperbolic at $t=0$ but not for
any $t>0.$ This kind of assumption was given precise formulations for systems
in [12], and shown to induce strong Hadamard instabilities in initial-value
problems for quasi-linear systems.

Introduce $P$ the characteristic polynomial of $A$ evaluated along a solution
$u$ of (1.1):

(2.1)
$P(t,x,\xi,\lambda)=\mbox{det}\,(A(t,x,u(t,x),\partial_{x}u(t,x),\xi)-\lambda{\rm
Id}\big{)}.$

Via the equation (1.1), at $t=0$ the characteristic polynomial $P$ and its
partial derivatives can all be expressed in terms of the initial datum.
Denoting $v_{in}=(u_{in},\partial_{x}u_{in}),$ and
$A_{in}(x,\xi)=A(0,x,v_{in}(x),\xi),$ $G_{in}(x)=F(0,x,v_{in}(x)),$ we see, in
particular, that $P(0,x,\xi,\lambda)$ is equal to
$\mbox{det}\,\big{(}A_{in}(x,\xi)-\lambda{\rm Id}\big{)},$ and
$\partial_{t}P(0,x,\xi,\lambda)$ to

(2.2)
$\displaystyle\partial_{t}P(0,x,\xi,\lambda)=\mbox{det}^{\prime}\big{(}A_{in}(x,\xi)$
$\displaystyle-\lambda{\rm Id}\big{)}$
$\displaystyle\cdot\Big{(}\partial_{1}A(0,x,v_{in}(x),\xi)+\partial_{3}A(0,x,v_{in}(x),\xi)G_{in}(x)$
$\displaystyle\quad+\partial_{4}A(0,x,v_{in}(x),\xi)\cdot\partial_{x}G_{in}(x)\Big{)}.$

The transition from hyperbolicity to ellipticity is formulated in [12] in
terms of the jet of $P$ (the first terms in its Taylor expansion) at $t=0.$
These conditions thus bear on the datum, and involve the nonlinear terms in
the equation.

We formulate here an assumption that is analogous to Hypothesis 1.5 in [12],
and describes a time-differentiable branching of eigenvalues of $A$ from the
hyperbolic to the elliptic region. Given $U$ an open set in
$\mathbb{R}^{d}\times\mathbb{S}^{d-1},$ denote ${\mathcal{S}}_{U}$ the
spectrum at $t=0$ above $U:$

${\mathcal{S}}_{U}:=\big{\\{}(x,\xi,\lambda)\in U\times\mathbb{C},\quad
P(0,x,\xi,\lambda)=0\big{\\}}.$

######  Assumption 2.2.

For the principal symbol $A$ (1.2) we assume the following conditions:

  * (i)

initial hyperbolicity: for some open set $U,$ we have
$\omega=(x,\xi,\lambda)\in{\mathcal{S}}_{U}$ only if $\lambda\in\mathbb{R}.$

  * (ii)

Smooth diagonalizability: for some $(x^{0},\xi^{0})\in U,$ the matrix
$A(0,x,\xi)$ is smoothly diagonalizable for all $(x,\xi)$ near
$(x^{0},\xi^{0})$ in $U.$

  * (iii)

Existence of double eigenvalues: for some
$\omega^{0}=(x^{0},\xi^{0},\lambda^{0})\in{\mathcal{S}}_{U},$ we have
$\partial_{\lambda}P(0,\omega^{0})=0.$

  * (iv)

$C^{1}$ bifurcations of the spectrum away from $\mathbb{R}:$ given any
$\lambda\in\mathbb{R}$ such that
$\omega=(x^{0},\xi^{0},\lambda)\in{\mathcal{S}}_{U}$ and
$\partial_{\lambda}P(0,\omega)=0:$ we have $\partial_{t}P(0,\omega)=0$ and

(2.3)
$\big{(}\partial_{t\lambda}^{2}P(0,\omega)\big{)}^{2}<(\partial_{t}^{2}P\partial_{\lambda}^{2}P)(0,\omega),$

and $\partial_{t}P(0,\omega^{\prime})=0,$
$\partial_{\lambda}P(0,\omega^{\prime})=0$ for all $\omega^{\prime}$ near
$\omega$ in ${\mathcal{S}}_{U}.$

In condition (ii) above, by “smooth” we mean “as smooth as $A_{|t=0}$”, which
by assumption on $F$ means “as smooth as the datum”.

We describe in Section 4.1 how Assumption 2.2 paints a picture of the spectrum
of $A$ evaluated along a putative solution $u$ of (1.1) for very small time:
at $t$ goes from $0$ to positive values, some real eigenvalues of $A$ stay
real, while pairs of eigenvalues branch out of the real axis into the rest of
the complex plane, in a $C^{1}$ fashion, so that the real parts of the
branching eigenvalues is $O(t).$

######  Theorem 2.

Under Assumption 2.2, for any $s>3/2+3/4+d/2,$ any $\sigma$ such that
$3+d/2\leq\sigma<2s-3/2-d/2,$ for some $u_{in}\in H^{\sigma}(\mathbb{R}^{d}),$
there is no $T>0$ and no ball $B_{x^{0}}$ centered at $x^{0}$ such that the
initial-value problem (1.1) has a solution in $C^{0}([0,T],H^{s}(B_{x^{0}})).$

Examples of systems and data satisfying Assumption 2.2 are given in [12].
Those include in particular systems of coupled Burgers equations.

In Assumption 2.2 the instability is weaker than in Assumption 2.1. As a
consequence, the instability is slower to develop, and the condition on
$\sigma-s$ in Theorem 2 is stronger than the one in Theorem 1.

Condition $\sigma>3+d/2$ is required in order to make sense of two time
derivatives (corresponding, via the equation, to three spatial derivatives) at
$t=0,$ which we need in Assumption 2.2(iv). For the proof to go through we
only need $s>2+d/2,$ but then the conditions bearing on $\sigma$ impose that
$s$ is actually greater than $3/2+3/4+d/2.$

The proof of Theorem 2 is given in Section 4.

###  2.3 Remarks

In the companion paper [15], we get rid of the upper bound on $\sigma,$ that
is prove non-existence of Sobolev solutions above a certain level of
regularity no matter how regular the datum is, under an extra assumption of
semi-simplicity and constant multiplicity for the purely imaginary eigenvalues
of the principal symbol. We do so in the elliptic case and in the case of a
transition to ellipticity. This extends Theorem 4.4 in [3].

A lengthy appendix (pages A.1 to B) gives some standard and not-so-standard
results pertaining to pseudo- and para-differential operators. In the course
of the analysis, we indeed need an estimate for the action on Sobolev maps of
the difference between a pseudo-differential operator in semiclassical
quantization and its para-differential counterpart. This issue was examined
earlier in [8] and [16]. We correct here small errors from [8] and [16]; see
Proposition A.3. We also show how Gårding’s inequality translates into rates
of growth for the flows of para-differential operators of order zero, in
semiclassical quantization, for symbols with Hölder spatial regularity (see
Proposition B.1).

Note finally that the condition $s>1+d/2$ is critical, in the following sense:
in the favorable case of symmetric hyperbolic systems, well-posedness is known
only under that condition on the regularity of the datum; moreover, for the
related (but nonlocal, hence distinct) incompressible Euler equations in two
dimensions, well-posedness fails to hold in $H^{2}$ [1]. Here the assumption
$s>1+d/2$ means that the posited solution $u$ of (1.1) (the existence of which
we endeavor to disprove) does not have a high level of pointwise regularity.
Since $F$ depends on $(u,\partial_{x}u),$ the symbols are only Hölder
continuous in space. In particular, this forces us to rely on Gårding’s
inequality in order to derive bounds for the flows of relevant order-zero
para-differential operators, as opposed to the high-regularity approach of
[17, 13].

##  3 Ellipticity: proof of Theorem 1

We work by contradiction and assume the existence of $T>0,$ a ball $B_{x_{0}}$
centered at $x_{0}$ and $u\in C^{0}([0,T],H^{s}(B_{x_{0}}))$ which solves the
initial-value problem (1.1). The contradiction will eventually come from the
high but finite regularity of the datum $u_{in}.$

Our assumption puts forward a distinguished point
$(x^{0},u^{0},v^{0},\xi^{0})\in\mathbb{R}^{d}\times\mathbb{R}^{N}\times\mathbb{R}^{Nd}\times\mathbb{S}^{d-1}.$
We will choose the datum $u_{in}$ (see Section 3.7), so that
$u_{in}(x^{0})\simeq u^{0},$ and $\partial_{x}u_{in}(x^{0})=v^{0}.$

We identify the solution $u$ with its extension to the whole space
$\mathbb{R}^{d}$ via a linear bounded extension operator $H^{s}(B_{x_{0}})\to
H^{s}(\mathbb{R}^{d}).$

###  3.1 Hyperbolic rescaling

Define the spatial rescaling map:

(3.1) $h_{j}:\quad f\in H^{s}\to(x\to f(2^{-j}x))\in H^{s},$

for some $j$ that will eventually be chosen large, and let

(3.2) $\tilde{u}(t,x):=u(2^{-j}t,x),\qquad u_{h}:=h_{j}\tilde{u},$

where $u$ is the posited solution. Then, $u_{h}$ belongs to
$C^{0}([0,2^{j}T],H^{s}(\mathbb{R}^{d})),$ and solves the initial-value
problem

(3.3) $\left\\{\begin{aligned}
\partial_{t}u_{h}+2^{-j}F(2^{-j}t,2^{-j}x,u_{h},2^{j}\partial_{x}u_{h})&=0,\\\
u_{h}(0,x)=u_{in}(2^{-j}x).\end{aligned}\right.$

###  3.2 Paralinearization

Our second step is to para-linearize the system. Since $u_{h}$ belongs to
$H^{s}$ for $t\in[0,2^{j}T],$ we have (see for instance Theorem 5.2.4 in [4]):

(3.4)
$F(2^{-j}t,2^{-j}x,u_{h},2^{j}\partial_{x}u_{h})=T_{\partial_{3}F(u_{h},2^{j}\partial_{x}u_{h})}u_{h}+T_{\partial_{4}F(u_{h},2^{j}\partial_{x}u_{h})}2^{j}\partial_{x}u_{h}+R_{j}^{para},$

where $T_{a}b$ refers to the para-product of $b$ by $a,$ and $\partial_{3}F$
refers to the partial derivative of $F$ with respect to its third ($u$)
argument. The remainder satisfies

(3.5) $R_{j}^{para}\in L^{\infty}([0,2^{j}T],H^{2(s-1)-d/2}(\mathbb{R}^{d})).$

The proof of Theorem 5.2.4 in [4] shows moreover that

(3.6) $\|R_{j}^{para}\|_{H^{2(s-1)-d/2}(\mathbb{R}^{d})}\lesssim
C_{F}(\|u_{h},2^{j}\partial_{x}u_{h}\|_{L^{\infty}(\mathbb{R}^{d})})\|(u_{h},2^{j}\partial_{x}u_{h})\|_{H^{s-1}(\mathbb{R}^{d})},$

where $C_{F}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is nondecreasing and depends
only on $F.$ In (3.4), we omitted the arguments $(2^{-j}t,2^{-j}x)$ of
$\partial_{3}F$ and $\partial_{4}F$ in the right-hand side.

###  3.3 Semiclassical quantization

Given a symbol $a,$ our preferred notation for the para-differential operator
with symbol $a$ is ${\rm op}(a):$

(3.7) $T_{a}={\rm op}(a),$

where the spatial convolution with the partial inverse Fourier transform of
some admissible cut-off function is implicit. For details on our notation, and
classical results pertaining to para-differential operators, see Appendix A.

By definition, the para-differential operator with symbol $a$ in
$2^{-j}$-semiclassical quantization is

${\rm op}_{j}(a):=h_{j}^{-1}{\rm op}(h_{j}a)h_{j},$

where $h_{j}$ is defined in (3.1). Classical results in semiclassical
quantization are recalled in Appendix A. We remark that, using notation (3.7),
we can write (3.4) in the form

(3.8) $2^{-j}h_{j}F(\tilde{u},\partial_{x}\tilde{u})=2^{-j}{\rm
op}\big{(}h_{j}\partial_{3}F(\tilde{u},\partial_{x}\tilde{u})\big{)}h_{j}\tilde{u}+{\rm
op}\big{(}h_{j}\partial_{4}F(\tilde{u},\partial_{x}\tilde{u})\big{)}\partial_{x}(h_{j}\tilde{u})+2^{-j}R^{para}_{j}.$

That is,

(3.9) $\displaystyle 2^{-j}F(\tilde{u},\partial_{x}\tilde{u})$
$\displaystyle=h_{j}^{-1}{\rm
op}\Big{(}2^{-j}h_{j}\partial_{3}F(\tilde{u},\partial_{x}\tilde{u})\Big{)}h_{j}\tilde{u}+h_{j}^{-1}{\rm
op}\Big{(}h_{j}\partial_{4}F(\tilde{u},\partial_{x}\tilde{u})\cdot
i\xi\Big{)}h_{j}\tilde{u}$ $\displaystyle+2^{-j}h_{j}^{-1}R^{para}_{j}.$

Thus $\tilde{u}$ solves the initial-value problem

(3.10) $\left\\{\begin{aligned} \partial_{t}\tilde{u}+{\rm
op}_{j}(\tilde{A})\tilde{u}&=\tilde{R}^{para}_{j},\\\
\tilde{u}(0)&=u_{in},\end{aligned}\right.$

where $\tilde{R}^{para}_{j}$ is

(3.11) $\tilde{R}^{para}_{j}:=-2^{-j}h_{j}^{-1}R^{para}_{j},$

and the symbol $\tilde{A}$ is defined by

(3.12)
$\displaystyle\tilde{A}(t,x,\tilde{u},\partial_{x}\tilde{u},\xi):=2^{-j}\partial_{3}F(2^{-j}t,x,\tilde{u},\partial_{x}\tilde{u})+\partial_{4}F(2^{-j}t,x,\tilde{u},\partial_{x}\tilde{u})\cdot
i\xi,$

where $\tilde{u}$ is evaluated at $(t,x).$ Recall that $j$ will be chosen to
be large. Thus the symbol $\tilde{A}$ appears as a small perturbation of the
principal symbol $iA,$ with $A$ defined in (1.2).

###  3.4 Localization near the distinguished point in the cotangent space

We use a Littlewood-Paley dyadic decomposition $(\phi_{j})_{j\geq 0}.$ That
is, a smooth frequency truncation function $\phi_{0}$ is given, with
$0\leq\phi_{0}\leq 1,$ and $\phi_{0}(\xi)\equiv 1$ for
$|\xi|\leq\varepsilon_{1}$ and $\phi_{0}(\xi)\equiv 0$ for
$|\xi|\geq\varepsilon_{2},$ for some real numbers
$\varepsilon_{1},\varepsilon_{2}$ satisfying
$0<\varepsilon_{1}<\varepsilon_{2}<1<2\varepsilon_{1}.$ We let

(3.13)
$\phi_{j}(\xi):=\phi_{0}(2^{-j}\xi)-\phi_{0}(2^{-j+1}\xi),\qquad\mbox{for
$j\geq 1.$}$

In particular, the cut-off $\phi_{1}$ is supported in the neighborhood
$\\{\varepsilon_{1}\leq|\xi|\leq 2\varepsilon_{2}\\}$ of the unit disk; the
cut-off $\phi_{k}$ is supported in the annulus
$\\{2^{k}\varepsilon_{1}\leq|\xi|\leq 2^{1+k}\varepsilon_{2}\\}.$ We have the
decomposition

(3.14) $f=\sum_{j\geq 0}\phi_{j}(D)f,$

where the operator $\phi_{j}(D)$ is the Fourier multiplier with symbol
$\phi_{j}(\xi).$

We localize in $(x,\xi)$ near $(x^{0},\xi^{0}),$ within the annulus
$|\xi|\simeq 2^{-j},$ for some integer $j$ which will eventually be chosen
large. In this view, we let $\psi$ be a smooth space-frequency cut-off
supported in $(x^{0},\xi^{0})\in\mathbb{R}^{d}_{x}\times\mathbb{R}^{d}_{\xi}:$
for some $\delta>0,$ we have

(3.15) $\psi(x,\xi)=\left\\{\begin{aligned}
1,&\quad|x-x^{0}|+|\xi-\xi^{0}|\leq\delta/2,\\\
0,&\quad|x-x^{0}|+|\xi-\xi^{0}|\geq\delta.\end{aligned}\right.$

We may choose $\psi$ to be a tensor product
$\psi(x,\xi)=\psi_{1}(x)\psi_{2}(\xi).$ We also let

(3.16) $v:={\rm op}_{j}(\psi)\tilde{u}$

where $\tilde{u}$ is defined in (3.2) and was found in Section 3.3 to solve
the initial-value problem (3.10). The initial-value problem in $v$ is

(3.17) $\left\\{\begin{aligned} \partial_{t}v+{\rm op}_{j}(\tilde{A})v&=f,\\\
v(0)&={\rm op}_{j}(\psi)u_{in},\end{aligned}\right.$

where

(3.18) $f=-[{\rm op}_{j}(\psi),{\rm op}_{j}(\tilde{A})]\tilde{u}+{\rm
op}_{j}(\psi)\tilde{R}^{para}_{j}.$

We observe that the operator in (3.17) is further localized, up to a
remainder. Indeed, we have

${\rm op}_{j}(\tilde{A})v={\rm op}_{j}(\psi^{\sharp}\tilde{A})v-f_{1},$

where

(3.19) $f_{1}:={\rm op}_{j}(\tilde{A})\big{(}{\rm op}_{j}(\psi^{\sharp}){\rm
op}_{j}(\psi)-{\rm op}_{j}(\psi)\big{)}\tilde{u}+({\rm
op}_{j}(\psi^{\sharp}\tilde{A})-{\rm op}_{j}(\tilde{A}){\rm
op}_{j}(\psi^{\sharp})\big{)}v,$

and $f_{1}$ will be seen to be appropriately sense.

Above, we used notation $\chi\prec\chi^{\sharp},$ borrowed from [12]:

(3.20)
$\chi\prec\chi^{\sharp}\quad\mbox{means}\quad(1-\chi^{\sharp})\chi\equiv 0,$

with $\chi,\chi^{\sharp}\in C^{\infty}_{c}(\mathbb{R}^{2d};[0,1]).$ Whenever
we use (3.20), it is implicit that the support of the extension
$\chi^{\sharp}$ is not much bigger than the support of $\chi.$ The context
usually makes clear what is meant by “not much bigger”.

We let

(3.21) $g:=f+f_{1},$

and the initial-value problem in $v$ now appears as

(3.22) $\left\\{\begin{aligned} \partial_{t}v+{\rm op}_{j}(M)v&=g,\\\
v(0)&={\rm op}_{j}(\psi)u_{in}.\end{aligned}\right.$

The symbol $M$ is, up to a small perturbation, a localized version of the
principal symbol $A$ defined in (1.2):

(3.23) $\displaystyle M(t,x,\xi)=\psi^{\sharp}(x,\xi)\Big{(}$ $\displaystyle
A(2^{-j}t,x,\tilde{u},\partial_{x}\tilde{u},i\xi)+2^{-j}\partial_{3}F(2^{-j}t,x,\tilde{u},\partial_{x}\tilde{u})\Big{)},$

where $\tilde{u}$ is evaluated at $(t,x).$ By regularity of $F,$ the posited
regularity of $u,$ and Lemma A.1, we have $M(t)\in
C^{[\theta],\theta-[\theta]}S^{0},$ for all $t$ in the interval of definition
of $\tilde{u},$ where

(3.24) $\theta:=s-1-d/2.$

###  3.5 Spectral decomposition

Assumption 2.1 states that the spectrum of $A(0,x^{0},u^{0},v^{0},\xi^{0})$ is
not entirely contained in $\mathbb{R}.$

The datum will be chosen to satisfy $v(0,x^{0})\simeq u^{0},$
$\partial_{x}v(0,x^{0})\simeq v^{0}.$ (A precise choice is made in Section
3.7). By continuity of the spectrum (a consequence of Rouché’s theorem, see
for instance Proposition 1.1 in [18]), if

(3.25) $j$ is large enough, and
$|v(0,x^{0})-u^{0}|+|\partial_{x}v(0,x^{0})-v^{0}|$ is small enough,

then the spectrum of $M(0,x^{0},\xi^{0})$ is not entirely contained in
$i\mathbb{R}.$

Note that $M$ is defined in (3.23) in terms of $A$ evaluated at $i\xi,$ so
that it is indeed the real parts of the eigenvalues that matter in this ill-
posedness proof. Since $M$ appears in the left-hand side of (3.22), an
eigenvalue with a negative real part is susceptible to generate growth.

We denote

$\gamma:=\Re e\,\lambda_{0}<\Re e\,\lambda_{1}<\dots<\Re e\,\lambda_{p_{0}}$

the spectrum of $M(0,x^{0},\xi^{0}).$ The most unstable eigenvalue is
$\lambda_{0}.$ By reality of the coefficients of $A,$ the real part of the
spectrum of $M$ is symmetric with respect to 0. Thus we have $\gamma=\Re
e\,\lambda_{0}<0.$

The elliptic eigenvalue $\lambda_{0}$ might correspond to a coalescing point
$(0,x^{0},\xi^{0},\lambda_{0})$ in the spectrum. We denote
$\mu_{1},\dots,\mu_{p}$ the eigenvalues of $M$ which coalesce at
$z^{0}=(0,x^{0},\xi^{0}),$ with value $\lambda_{0}$ at $z^{0}:$

(3.26) $\mu_{j}(z^{0})=\lambda_{0},\qquad 1\leq j\leq p.$

We say that the eigenvalues $\mu_{1},\dots,\mu_{p}$ are elliptic, and call
hyperbolic the other eigenvalues of $M.$ The hyperbolic eigenvalues are thus
those which take values $\Re e\,\lambda_{1},\dots,\Re e\,\lambda_{p_{0}}$ at
$z^{0}.$ We denote $\mu_{p+1},\dots,\mu_{N}$ the other branches of eigenvalues
of $M.$ It may be that $\Re e\,\mu_{j}\neq 0$ for some $j\in[p+1,N],$ even
though we call $\mu_{j}$ hyperbolic, but it will be a convenient way of
referring to this part of the spectrum.

If $j\leq p<j^{\prime},$ then

(3.27) $\Re e\,\mu_{j}<\Re e\,\mu_{j^{\prime}},$

uniformly in a small neighborhood of $z^{0},$ by continuity of the
eigenvalues. We may and do assume that the support of $\psi^{\sharp}$ is small
enough so that (3.27) holds over the support of $\psi^{\sharp}.$

The eigenvalues $\mu_{j}$ may not be smooth, but satisfy the Hölder estimates
(see for instance Proposition 3.1 in [18], or [2]):

(3.28) $|\mu_{j}(z)-\mu_{j}(z^{0})|\leq c|z-z_{0}|^{1/p^{\prime}},$

for some $c>0$ and some $p^{\prime}\geq p.$

We denote $E=E(t,x,\xi)$ the direct sum of the generalized eigenspaces
associated with the eigenvalues $\mu_{1},\dots,\mu_{p}$ which coalesce at
$z^{0}=(0,x^{0},\xi^{0})$ with value $\lambda_{0},$ and $H=H(t,x,\xi)$ the
direct sum of the generalized eigenspaces associated with the other
eigenvalues, so that

(3.29) $\mathbb{C}^{N}=E(t,x,\xi)+H(t,x,\xi),\qquad\mbox{for all $(t,x,\xi)$
near $(0,x^{0},\xi^{0}).$}$

###  3.6 Projections

We denote $P_{H}$ the projector onto $H$ parallel to $E,$ and $P_{E}$ the
projector onto $E$ parallel to $H,$ so that

(3.30) ${\rm Id}=P_{E}+P_{H}.$

We have integral representations for $P_{E}$ and $P_{H}$ in terms of $M$ (see
for instance Proposition 2.1 in [18]):

(3.31)
$P_{\star}=\frac{1}{2i\pi}\int_{{\mathcal{C}}_{\star}}(z-M)^{-1}\,dz,\qquad\star\in\\{H,E\\},$

where ${\mathcal{C}}_{H}$ is a contour enclosing all the hyperbolic
eigenvalues and only those, and ${\mathcal{C}}_{E}$ is a contour enclosing the
elliptic eigenvalues. By spectral separations (3.27), we can choose non-
overlapping contours ${\mathcal{C}}_{E}$ and ${\mathcal{C}}_{H}.$ The
representations (3.31) then ensure that $P_{E}$ and $P_{H}$ are as smooth as
$M,$ so that $P_{E},P_{H}\in C^{0,\theta}S^{0},$ pointwise in time.

The symbol $M,$ and the projectors $P_{E}$ and $P_{H}$ depend on
$\partial_{x}u,$ which a priori is only continuous in time. This will be an
issue in view of the change of variable (3.32). Hence we regularize the
projectors. Let $P_{H}^{\varepsilon}$ be $P_{H}$ evaluated at
$k_{\varepsilon}\star(u,\partial_{x}u),$ where $k_{\varepsilon}$ is a
smoothing kernel, just like in (A.23). The small parameter $\varepsilon$ will
be chosen in terms of $j$ in the proof of Lemma 3.5.

The associated components of the unknown $v$ are

(3.32) $v_{H}:={\rm op}_{j}(\psi^{\flat}P_{H}^{\varepsilon})v,\qquad
v_{E}={\rm op}_{j}(\psi^{\flat}P_{E}^{\varepsilon})v,$

with

(3.33) $\psi\prec\psi\prec\psi^{\sharp},$

where we used notation (3.20). We denote

(3.34) $M_{H}:=P_{H}M,\qquad M_{E}:=P_{E}M,$

so that

(3.35) $M_{H}\equiv M_{H}P_{H},\qquad M_{E}\equiv M_{E}P_{E},$

and now derive the equations in $v_{H}$ and $v_{E}.$ We have

$\displaystyle\partial_{t}v_{H}$ $\displaystyle={\rm
op}_{j}(\psi\partial_{t}P^{\varepsilon}_{H})v+{\rm op}_{j}(\psi
P^{\varepsilon}_{H})\partial_{t}v$ $\displaystyle={\rm
op}_{j}(\psi\partial_{t}P^{\varepsilon}_{H})v-{\rm
op}_{j}(\psi^{\flat}P_{H}^{\varepsilon}){\rm op}_{j}(M)v+{\rm
op}_{j}(\psi^{\flat}P_{H}^{\varepsilon})g.$

Next, by composition of para-differential operators in semiclassical
quantization, and approximation of $P_{H}$ by $P_{H}^{\varepsilon},$

${\rm op}_{j}(\psi^{\flat}P_{H}^{\varepsilon}){\rm op}_{j}(M)={\rm
op}_{j}(\psi^{\flat}M_{H})+\mbox{a smaller term.}$

Precisely, using the composition result described in Section A.5,

${\rm op}_{j}(\psi^{\flat}P_{H}^{\varepsilon}){\rm op}_{j}(M)={\rm
op}_{j}(\psi^{\flat}M_{H})+2^{-j\theta}R_{\theta}(\psi^{\flat}P_{H}^{\varepsilon},M)+{\rm
op}_{j}(\psi^{\flat}(P_{H}^{\varepsilon}-P_{H})){\rm op}_{j}(M),$

where we recall (3.24) $\theta=s-1-d/2.$ Besides, by (3.35),

$\displaystyle{\rm op}_{j}(\psi^{\flat}M_{H})$ $\displaystyle={\rm
op}_{j}(M_{H}){\rm
op}_{j}(\psi^{\flat}P_{H}^{\varepsilon})+2^{-j\theta}R_{\theta}(M_{H},\psi^{\flat}P_{H})+{\rm
op}_{j}(M_{H}){\rm op}_{j}(\psi^{\flat}(P_{H}-P_{H}^{\varepsilon})).$

We denote $R_{H}$ the sum of the remainders in the last two equations, as we
let

(3.36) $\begin{aligned}
R_{H}&:=2^{-j\theta}\big{(}R_{\theta}(\psi^{\flat}P_{H}^{\varepsilon},M)+R_{\theta}(M_{H},\psi^{\flat}P_{H})\big{)}\\\
&\quad+{\rm op}_{j}(M_{H}){\rm
op}_{j}(\psi^{\flat}(P_{H}-P_{H}^{\varepsilon}))++{\rm
op}_{j}(\psi^{\flat}(P_{H}^{\varepsilon}-P_{H})){\rm op}_{j}(M)\end{aligned}.$

Substituting $E$ for $H,$ we obtain a system in $(v_{H},v_{E}):$

(3.37) $\left\\{\begin{aligned} \partial_{t}v_{H}+{\rm
op}_{j}(M_{H})v_{H}&=g_{H},\\\ \partial_{t}v_{E}+{\rm
op}_{j}(M_{E})v_{E}&=g_{E},\end{aligned}\right.$

with notation

(3.38) $g_{\star}:={\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})g-(R_{\star}+{\rm
op}_{j}(\psi^{\flat}\partial_{t}P_{\star}^{\varepsilon}))v,\qquad\star\in\\{H,E\\}.$

In the next Section, we choose the datum $u_{in}$ and examine what our choice
entails for the data $v_{E}(0)$ and $v_{H}(0).$

###  3.7 The initial datum

We carefully choose the polarization of the datum, and its frequency
localization. This Section ends with Corollary 3.3, where a lower bound for an
$L^{2}$ norm of $v_{E}(0)$ is proved.

####  3.7.1 Polarization

Here we choose the polarization of the initial datum $u_{in},$ meaning its
direction in the state space $\mathbb{R}^{N}.$ The most obvious choice
$u_{in}(x^{0})=u^{0}$ is not always appropriate. Indeed, it could be that the
elliptic component of $u^{0}$ (in the sense of (3.29)) is zero. For instance,
with $d=2$ and $N=3$ and

$A(u,i\xi)=\left(\begin{array}[]{ccc}0&iu_{3}\xi_{1}&0\\\
-iu_{3}\xi_{1}&0&0\\\ 0&0&iu_{1}\xi_{2}\end{array}\right),$

then $A$ is elliptic at $(u^{0},\xi^{0})$ with $u^{0}=(0,0,1)$ and
$\xi^{0}=(1,0).$ The elliptic subspace $E,$ however, satisfies
$E(u^{0},\xi^{0})=\mathbb{R}^{2}\times\\{0\\},$ so that
$P_{E}(u^{0},\xi^{0})u^{0}=0.$ Thus choosing $u_{in}(x^{0})=u^{0}$ would be an
issue in the perspective of proving a bound from below for the elliptic
component $v_{E}(0)$ of the datum. The following Lemma shows that an
appropriate choice for $u_{in}(x^{0})$ is always possible:

######  Lemma 3.1.

Under Assumption 2.1, for some $u^{1}\in\mathbb{R}^{d},$ for $u_{in}$ such
that $u_{in}(x^{0})=u^{1}$ and $\partial_{x}u_{in}(x^{0})=v^{0},$ the spectrum
of $M(0,x^{0},\xi^{0})$ is not included in $i\mathbb{R}$ and we have
$P_{E}(0,x^{0},\xi^{0})u^{1}\neq 0,$ where $P_{E}$ is the projection onto the
elliptic space and parallel to the hyperbolic space (3.29).

Recall that the symbol $M$ is defined in (3.23). It depends on $(t,x)$ through
$(u,\partial_{x}u),$ where $u$ is our posited solution. In particular, the
symbol $M$ depends on the datum $u_{in}.$ Accordingly, the symbol $P_{E}$
depends on the datum.

###### Proof.

Let $(x^{0},u^{0},v^{0},\xi^{0})$ as in Assumption 2.1. Let $M_{0}$ be the
matrix-valued symbol (3.23) at $(t,x,\xi)=(0,x^{0},\xi^{0}),$ with the choice
$u_{in}(x^{0})=u^{0},$ $\partial_{x}u_{in}(x^{0})=v^{0}.$ Let $E_{0}$ be the
associated elliptic space, as defined in Section (3.27). If
$P_{E_{0}}u^{0}\neq 0,$ then we may choose $u^{1}=u^{0}.$

Otherwise, $P_{E_{0}}u^{0}=0.$ Let $\vec{e}$ be a unitary elliptic eigenvector
of $M_{0},$ and posit

$u_{in}(x^{0})=u^{1}:=u^{0}+\alpha\vec{e},$

for some small $\alpha>0,$ and $\partial_{x}u_{in}(x^{0})=v^{0}.$ If $\alpha$
is small enough, by continuity of the spectrum, the eigenvalues of
$M(0,x^{0},\xi^{0})$ are not all purely imaginary, so that the space
$E(0,x^{0},\xi^{0})$ is not reduced to $\\{0\\}.$

By definition of $M$ in (3.23), the projector $P_{E}$ takes the form

$P_{E(0,x,\xi)}=\underline{P}_{E}(x,u(0,x),\partial_{x}u(0,x),\xi).$

Since the elliptic eigenvalues are separated from the hyperbolic eigenvalues,
the map $\underline{P}$ is smooth in its arguments (see for instance
Proposition 2.1 in [18]). We may further choose $\vec{e}$ so that

(3.39) $\vec{e}+\partial_{u}\underline{P}_{E}(0,u^{0},v^{0},\xi^{0})\cdot
u^{0}\neq 0.$

Indeed, if $\partial_{u}\underline{P}_{E}(0,u^{0},v^{0},\xi^{0})\cdot
u^{0}\neq 0$ does not generates the whole elliptic eigenspace
$E(0,x^{0},\xi^{0}),$ then we choose for $\vec{e}$ to be any other (unitary)
elliptic eigenvector. If
$\partial_{u}\underline{P}(0,u^{0},v^{0},\xi^{0})\cdot u^{0}$ happens to
generate $E(0,x^{0},\xi^{0})$ in its entirety, then we let
$\vec{e}=\lambda\partial_{u}\underline{P}_{E}(0,u^{0},v^{0},\xi^{0})\cdot
u^{0}\neq 0,$ and we may choose $\lambda\neq-1$ so that $\vec{e}$ is unitary.

Then,

$\displaystyle P_{E(0,x^{0},\xi^{0})}u^{1}$
$\displaystyle=P_{E_{0}}(u^{0}+\alpha\vec{e})+(P_{E(0,x^{0},\xi^{0})}-P_{E_{0}})(u^{0}+\alpha\vec{e})$
$\displaystyle=\alpha\vec{e}+(P_{E(0,x^{0},\xi^{0})}-P_{E_{0}})(u^{0}+\alpha\vec{e})$
$\displaystyle=\alpha\big{(}\vec{e}+\partial_{u}\underline{P}_{E}(x^{0},u^{0},\xi^{0})\cdot
u^{0}\big{)}+O(\alpha^{2}),$

which implies $P_{E(0,x^{0},\xi^{0})}u^{1}\neq 0$ by (3.39), if $\alpha$ is
small enough. ∎

####  3.7.2 Oscillations

Let $w$ be a scalar map in the Schwartz class
${\mathcal{S}}(\mathbb{R}^{d}_{x}),$ such that $\hat{w}\in
C^{\infty}_{c}(\mathbb{R}^{d}_{\xi}),$ and $w(x^{0})\neq 0.$ Associated with
$w,$ we let

(3.40) $w_{in}(x):=\sum_{j\geq 0}a_{j}e^{ix\cdot 2^{j}\xi^{0}}w,$

where $a_{j}>0$ is a $j$-dependent amplitude.

######  Lemma 3.2.

With the choice

(3.41) $a_{j}:=2^{-j\sigma}(1+j)^{-1},\qquad j>0,$

we have

$c_{1}a_{j}\leq\|\phi_{j+1}(D)v_{in}\|_{L^{2}}\leq c_{2}a_{j},$

for $j\geq j^{0}>0,$ for some $j^{0}$ and some $c_{1}>0$ and $c_{2}>0$ which
depend only $w$ and the radii $\varepsilon_{1}$ and $\varepsilon_{2}$ that
appear in the Littlewood-Paley decomposition $(\phi_{j})_{j\geq 0}$ introduced
in Section 3.4. In particular, the scalar map $w_{in}$ belongs to
$H^{\sigma}(\mathbb{R}^{d})\setminus\big{(}\cup_{\varepsilon>0}H^{\sigma+\varepsilon}(\mathbb{R}^{d})\big{)}.$
Moreover, if $|a_{0}|$ is large enough, then $w_{in}(x^{0})\neq 0.$

###### Proof.

Given $j_{0}>0,$ we examine $\phi_{j_{0}+1}(D_{x})v_{in}.$ We note that
$\phi_{j_{0}+1}\equiv\phi_{1}(2^{-j_{0}}\cdot).$ The term of index $j$ in the
sum is

$\phi_{j_{0}+1}(D)\big{(}a_{j}e^{ix\cdot 2^{j}\xi^{0}}w\big{)}=a_{j}e^{ix\cdot
2^{j}\xi^{0}}{\mathcal{F}}^{-1}\Big{(}\phi_{1}(2^{-j_{0}+j}\xi_{0}+2^{-j_{0}}\cdot)\hat{w}\,\Big{)}.$

Let $R>0$ such that the support of $\hat{w}$ is supported in $\\{|\xi|\leq
R\\}.$ Then, we observe that for
$\phi_{1}(2^{-j_{0}+j}\xi_{0}+2^{-j_{0}}\xi)\hat{w}(\xi)$ to be different from
zero, we must have

(3.42) $|\xi|\leq R,\qquad 2^{j_{0}}\varepsilon_{1}\leq|2^{j}\xi_{0}+\xi|\leq
2^{j_{0}+1}\varepsilon_{2}.$

(Recall indeed that $\phi_{1}$ is supported in
$\\{\varepsilon_{1}\leq|\xi|\leq 2\varepsilon_{2}\\}.$) If $j_{0}$ is large
enough so that

$2^{-j_{0}}R<\varepsilon_{1}-1/2,\quad\mbox{and}\quad
2^{-(j_{0}-1)}R+\varepsilon_{2}<1,$

then there is no $\xi$ such that (3.42) holds unless $j=j_{0}.$ Now

$\|\phi_{1}(2^{-j}D)\big{(}a_{j}e^{ix\cdot
2^{j}\xi^{0}}w\big{)}\|_{L^{2}}=a_{j}w_{j},$

where

$w_{j}=\|\phi_{1}(\xi^{0}+2^{-j}\cdot)\hat{w}\|_{L^{2}},$

so that $w_{j}\leq|w|_{L^{2}}.$ Besides, since $\hat{w}\neq 0$ (indeed, $w\neq
0$), for some $|\eta|\leq R,$ we have $\hat{w}(\eta)\neq 0.$ If $j$ is large
enough, then $\varepsilon_{1}\leq|\xi^{0}+2^{-j}\eta|\leq 2\varepsilon_{2},$
so that $\phi_{1}(\xi^{0}+2^{-j}\eta)\neq 0,$ hence $w_{j}>0,$ with a lower
bound which does not depend on $j.$ Indeed, we have
$|\hat{w}|\geq\gamma_{0}>0$ in the ball $B(\eta,r)$ for some $\gamma_{0}>0$
and some $r>0$ which depend only on the choice of $w.$ Then,
$\phi_{1}(\xi^{0}+2^{-j}\xi)\equiv 1$ on this ball if $j$ is large enough.
Hence the lower bound for $w_{j}$ is independent on $j.$

Finally, if $|a_{0}|$ is large enough, then

$|w_{in}(x^{0})|\geq|w(x^{0})|\Big{(}|a_{0}|-\sum_{j\geq 1}|a_{j}|\Big{)}>0,$

since $w(x^{0})\neq 0.$ ∎

####  3.7.3 Localization

The datum is defined as

(3.43) $\displaystyle u_{in}(x)$
$\displaystyle=u^{1}\frac{w_{in}(x)}{|w_{in}(x^{0})|}+(x-x^{0})\cdot(v^{0}-u^{1}\frac{\partial_{x}w_{in}(x^{0})}{|w_{in}(x^{0})|}\big{)}\psi_{1}(x),$

where the vector $u^{1}$ is introduced in Lemma 3.1 and the scalar function
$w_{in}$ in Section 3.7.2; the spatial cut-off $\psi_{1}$ is introduced in
Section 3.4: we have $\psi_{1}\in C^{\infty}_{c}(\mathbb{R}^{d};[0,1]),$ with
$\psi_{1}\equiv 1$ in a neighborhood of $0,$ and the support of $\psi_{1}$ is
appropriately small.

With the choice (3.43), we have, by Lemma 3.2,

(3.44) $u_{in}\in
H^{\sigma}(\mathbb{R}^{d})\setminus\big{(}\cup_{\varepsilon>0}H^{\sigma+\varepsilon}(\mathbb{R}^{d})\big{)},$

and also

(3.45) $u_{in}(x^{0})=u^{1},\qquad\partial_{x}u_{in}(x^{0})=v^{0}.$

Moreover, the elliptic component of the datum is bounded from below:

######  Corollary 3.3.

If the parameter $\delta$ measuring the size of the support of the cut-off
$\psi$ (3.15) is small enough, and if $\varepsilon>0$ is small enough, then
the datum $v_{E}(0)={\rm op}_{j}(\psi^{\flat}P_{E}^{\varepsilon}){\rm
op}_{j}(\psi)u_{in}$ for the elliptic component $v_{E}$ satisfies

(3.46) $\|{\rm op}_{j}(\tilde{\psi})v_{E}(0)\|_{L^{2}(B(x^{0},\delta))}\geq
C2^{-j\sigma}(1+j)^{-1},$

for $j$ large enough and some constant $C>0$ which depends on $\delta$ but not
on $j,$ for some cut-off $\tilde{\psi}$ such that
$\tilde{\psi}\prec\psi^{\flat},$ with notation introduced in (3.20).

Above, $\delta>0$ is the small localization radius introduced in Section 3.4.

###### Proof.

Throughout this proof, the projector $P_{E}$ (defined at the beginning of
Section 3.6) is evaluated at $t=0.$ In particular, $P_{E}$ depends on $x$
through the datum $u_{in}\in H^{\sigma},$ and
$\|P_{E}^{\varepsilon}\|_{0,0,1+[d/2]}<\infty$ since $\sigma>d/2.$ Here we
used the symbolic norms introduced in (A.9). This implies, by (A.17), the
bound

(3.47) $\|{\rm op}_{j}(\psi^{\flat}P_{E}^{\varepsilon})\|_{L^{2}\to
L^{2}}\lesssim 1.$

The elliptic datum is

$v_{E}(0)={\rm op}_{j}(\psi^{\flat}P_{E}^{\varepsilon}){\rm
op}_{j}(\psi)u_{in}.$

The second term in $u_{in}$ does not contribute much to the $L^{2}$ norm of
$v_{E}(0).$ Indeed, the difference $u_{in}-w_{in}(u^{1}/|w_{in}(x^{0})|)$ is
compactly supported in $x.$ Denote it $\tilde{\psi}_{1}.$ We have

${\rm op}_{j}(\psi)\tilde{\psi}_{1}={\rm
op}_{j}(\psi_{1})\psi_{2}(2^{-j}D)\tilde{\psi}_{1},$

and since the Fourier transform of $\tilde{\psi}_{1}$ belongs to the Schwartz
space, the $L^{2}$ norm of $\psi_{2}(2^{-j}D)\tilde{\psi}_{1}$ is controlled
by $2^{-jN^{\prime}},$ for $N^{\prime}$ arbitrarily large, up to an
$N^{\prime}$-dependent constant. Thus we may focus on ${\rm
op}_{j}(\psi^{\flat}P_{E}^{\varepsilon}){\rm op}_{j}(\psi)(w_{in}u^{1})$ in
the rest of this proof.

At this point it is convenient to use pseudo-differential operators, as
opposed to para-differential operators. By Proposition A.3 and regularity of
$\psi,$

$\|({\rm op}_{j}(\psi)-\mathrm{pdo}_{j}(\psi))\|_{L^{2}\to L^{2}}\lesssim
2^{-j(\sigma^{\prime}-d)},$

for any $\sigma^{\prime}>0,$ and

$\|({\rm
op}_{j}(\psi^{\flat}P_{E}^{\varepsilon})-\mathrm{pdo}_{j}(\psi^{\flat}P_{E}^{\varepsilon}))\mathrm{pdo}_{j}(\psi)v\|_{L^{2}}\lesssim
2^{-j(\sigma-1-d/2)}\|\mathrm{pdo}_{j}(\psi)v\|_{j,d/2},$

for any $v\in H^{d/2}.$ Since
$\mathrm{pdo}_{j}(\psi)=\psi_{1}\psi_{2}(2^{-j}D),$ and $\psi_{1}\in
C^{\infty}_{c},$ we have

$\|\mathrm{pdo}_{j}(\psi)w_{in}\|_{j,d/2}\lesssim\|\psi_{2}(2^{-j}D)w_{in}\|_{j,d/2},$

and by regularity of $w_{in}$ and localization of the support of $\psi_{2},$

$\|\psi_{2}(2^{-j}D)w_{in}\|_{j,d/2}\lesssim 2^{-j\sigma}.$

With the above and (3.47), we find

(3.48) $\|({\rm op}_{j}(\psi^{\flat}P_{E}^{\varepsilon}){\rm
op}_{j}(\psi)-\mathrm{pdo}_{j}(\psi^{\flat}P_{E}^{\varepsilon})\mathrm{pdo}_{j}(\psi))(w_{in}u^{1})\|_{L^{2}}\lesssim
2^{-j(2\sigma-1-d/2)},$

and we now focus on
$\mathrm{pdo}_{j}(\psi^{\flat}P_{E}^{\varepsilon})\mathrm{pdo}_{j}(\psi)(w_{in}u^{1}).$
We have

$\mathrm{pdo}_{j}(\psi)(w_{in}u^{1})(x)=\psi_{1}(x)\sum_{j^{\prime}\geq
0}\psi_{2}(2^{-j}D)(a_{j}e^{ix\cdot 2^{j^{\prime}}\xi^{0}}wu^{1})(x).$

Just like in the proof of Lemma 3.2, we have
$\psi_{2}(2^{-j^{\prime}}D)(e^{ix\cdot 2^{j^{\prime}}\xi^{0}}w)\neq 0$ only if
$j^{\prime}=j.$ Thus

$\displaystyle\mathrm{pdo}_{j}(\psi)w_{in}$
$\displaystyle=\psi_{1}(x)\psi_{2}(2^{-j}D)(a_{j}e^{ix\cdot 2^{j}\xi^{0}}w)$
$\displaystyle=a_{j}e^{ix\cdot
2^{j}\xi^{0}}a_{j}\psi_{1}(x)\psi_{2}(\xi^{0}+2^{-j}D)w,$

and

(3.49)
$\mathrm{pdo}_{j}(\psi^{\flat}P_{E}^{\varepsilon})\mathrm{pdo}_{j}(\psi)(w_{in}u^{1})=a_{j}e^{ix\cdot
2^{j}\xi^{0}}\psi_{1}^{\flat}(x)\mathrm{pdo}((\psi_{2}^{\flat}P_{E}^{\varepsilon})(\xi^{0}+2^{-j}\cdot))\tilde{w}(x),$

where we denote $\tilde{w}:=u^{1}\psi_{1}\psi_{2}(\xi^{0}+2^{-j}D)w.$ We
approximate

(3.50)
$\mathrm{pdo}((\psi_{2}^{\flat}P_{E}^{\varepsilon})(\xi^{0}+2^{-j}\cdot))\tilde{w}(x)\simeq
P_{E}^{\varepsilon}(x,\xi^{0})\tilde{w}.$

The error in (3.50) is

$\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}\big{(}\psi_{2}^{\flat}(\xi^{0}+2^{-j}\xi)P_{E}^{\varepsilon}(x,\xi^{0}+2^{-j}\xi)-P_{E}^{\varepsilon}(x,\xi^{0})\big{)}{\mathcal{F}}(\tilde{w})(\xi)\,d\xi.$

We have, using $\psi_{2}^{\flat}(\xi^{0})=1,$

$|\psi_{2}^{\flat}(\xi^{0}+2^{-j}\xi)P_{E}^{\varepsilon}(x,\xi^{0}+2^{-j}\xi)-P_{E}^{\varepsilon}(x,\xi^{0})|\leq
2^{-j}\|\partial_{\xi}(\psi_{2}^{\flat}P_{E}^{\varepsilon}(x,\cdot)\|_{L^{\infty}(\mathbb{R}^{d}_{\xi})}|\xi|.$

Thus the error in (3.50) is bounded, in $L^{2}(B(x^{0},1))$ norm, by

$2^{-j}\|\partial_{\xi}(\psi_{2}^{\flat}P_{E}^{\varepsilon})\|_{L^{\infty}(\mathbb{R}^{2d})}\|{\mathcal{F}}(\partial_{x}\tilde{w})\|_{L^{1}}.$

By regularity of the datum and $P_{E}^{\varepsilon},$ the norm
$\|\partial_{\xi}(\psi_{2}^{\flat}P_{E}^{\varepsilon})\|_{L^{\infty}(\mathbb{R}^{2d})}$
is finite (and independent of $j$). Besides, since $\psi_{1}$ is smooth and
compactly supported,

$\|{\mathcal{F}}(\partial_{x}\tilde{w})\|_{L^{1}}\lesssim\|\psi_{2}(\xi^{0}+2^{-j}D)w))\|_{H^{1+d/2^{+}}},$

where we used the bound $|\hat{f}\|_{L^{1}}\lesssim\|f\|_{H^{d/2^{+}}},$ where
$d/2^{+}$ is any real number strictly greater than $d/2.$ Since $\hat{w}$ is
smooth and compactly supported, the norm
$\|\psi_{2}(\xi^{0}+2^{-j}D)w))\|_{H^{1+d/2^{+}}}$ is finite; it is also
bounded independently of $j.$ Thus the error in (3.50) appears to be
$O(2^{-j})$ in the $L^{2}(B(x^{0},1))$ norm. The operator in (3.50) appears
with an extra $\psi_{1}$ factor in (3.49), which restricts the analysis to the
$B(x^{0},1)$ ball. Thus the leading term in the elliptic component of our
datum is

(3.51) $a_{j}e^{ix\cdot
2^{j}\xi^{0}}\psi_{1}^{\flat}(x)P_{E}^{\varepsilon}(x,\xi^{0})\tilde{w},$

and our goal is to prove a bound from below as in (3.46) for its
$L^{2}(B(x^{0},\delta))$ norm.

By Lemma 3.1, we have $P_{E}(x^{0},\xi^{0})u^{1}\neq 0.$ Thus
$P_{E}^{\varepsilon}(x^{0},\xi^{0})u^{1}\neq 0$ if $\varepsilon$ is small
enough. Besides, $\tilde{w}(x^{0})\neq 0.$ Indeed,

$\tilde{w}(x)=u^{1}\psi_{1}(x)w(x)+O(2^{-j}),$

in $L^{2}$ norm by regularity of $w,$ and $(\psi_{1}w)(x^{0})\neq 0$ by choice
of $w$ and assumption on $\psi_{1}.$ In conclusion, the continuous map (3.51)
is not equal to zero at $x^{0},$ thus stays away from zero in a small
neighborhood of $x^{0}.$

Finally, we put in the cut-off ${\rm op}_{j}(\tilde{\psi}).$ With the
arguments exposed above, we may consider the action of
$\mathrm{pdo}_{j}(\tilde{\psi})=\tilde{\psi}_{1}(x)\tilde{\psi}_{2}(2^{-j}D)$
on (3.51). Just like above, using the fast oscillations in (3.51) and a
regularity argument, we see that the leading term in

(3.52) $\mathrm{pdo}_{j}(\tilde{\psi})(a_{j}e^{ix\cdot
2^{j}\xi^{0}}\psi_{1}^{\flat}(x)P_{E}^{\varepsilon}(x,\xi^{0})\tilde{w})$

is $\tilde{\psi}_{1}(x)\tilde{\psi}_{2}(\xi^{0})$ times the map in (3.51). It
suffices to impose $\tilde{\psi}_{1}(x^{0})=1,$ and
$\tilde{\psi}_{2}(\xi^{0})=1.$ Then we have a map that is not equal to $0$ at
$x^{0}.$ By continuity, it says far from zero on a neighborhood of $x^{0}.$
This implies (3.46), by definition of $a_{j}$ in (3.41), if $\delta$ is small
enough, depending on $P_{E}^{\varepsilon}$ (evaluated at $t=0$) and $w.$ ∎

###  3.8 Sorting out the remainder term

We go back to system (3.37), which we reproduce here:

(3.53) $\partial_{t}v_{\star}+{\rm op}_{j}(M_{\star})v_{\star}=g_{\star}={\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})g-\big{(}R_{\star}+{\rm
op}_{j}(\psi^{\flat}\partial_{t}P_{\star}^{\varepsilon})\big{)}v,\qquad\star\in\\{H,E\\},$

where $g$ is defined in (3.21), in terms of $f$ and $f_{1}$ defined in (3.18)
and (3.19), and $R_{\star}$ is defined in (3.36). Recall that $v={\rm
op}_{j}(\psi)\tilde{u},$ and $\tilde{u}(t,x)=u(2^{-j}t).$ We say that terms in
$\tilde{u}$ in $g_{\star}$ are poorly localized, and we will label these terms
“out”. Terms in $v$ in $g_{\star}$ will be easier to handle. Those are

(3.54) $G_{\star}v:=-2^{j\theta^{\prime}}\Big{(}{\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})\big{(}{\rm
op}_{j}(\psi^{\sharp}\tilde{A})-{\rm op}_{j}(\tilde{A}){\rm
op}_{j}(\psi^{\sharp})\big{)}+(R_{\star}+{\rm
op}_{j}(\psi^{\flat}\partial_{t}P_{\star}^{\varepsilon})\Big{)}v.$

The poorly localized terms are

(3.55) $\displaystyle G_{\star}^{out}:=-{\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})\Big{(}\tilde{R}^{para}_{j}+{\rm
op}_{j}(\tilde{A})\big{(}{\rm op}_{j}(\psi^{\sharp}){\rm op}_{j}(\psi)-{\rm
op}_{j}(\psi)\big{)}+\big{[}{\rm op}_{j}(\psi),{\rm
op}_{j}(\tilde{A})\big{]}\Big{)}\tilde{u}.$

Note the $2^{j\theta^{\prime}}$ prefactor that we put in front of $G_{\star},$
where

(3.56) $\theta^{\prime}:=\frac{\theta}{1+\theta}.$

The reason for this prefactor is that we will prove in Section 3.10 that
$G_{\star}$ is $L^{2}\to L^{2}$ bounded with norm, uniformly in $j.$ The
initial-value problem for systems (3.53) takes the form

(3.57) $\left\\{\begin{aligned} \partial_{t}v_{\star}+{\rm
op}_{j}(M_{\star})v_{\star}&=2^{-j\theta^{\prime}}G_{\star}v+G^{out}_{\star},\\\
v_{\star}(0)&={\rm op}_{j}(\psi P_{\star}){\rm
op}_{j}(\psi)u_{in}\end{aligned}\right.\qquad\star\in\\{H,E\\}.$

###  3.9 Integral representation in forward time

The symbol $M_{\star},$ with $\star=E$ or $\star=H$ (3.34) is order zero. Thus
the associated operator ${\rm op}_{j}(M_{\star})$ is linear bounded $L^{2}\to
L^{2},$ and it has a solution operator $S_{\star}(t^{\prime};t),$ for $0\leq
t^{\prime}\leq t,$ defined as the map which takes a datum $z^{0}$ at time
$t^{\prime}$ to $z(t),$ where $z(t)$ is the unique solution at time $t$ to the
initial-value problem

$\partial_{t}z+{\rm op}_{j}(M_{\star})z=0,\qquad z(t^{\prime})=z^{0}.$

For the solution $v$ to (3.57), this means that we have (implicit)
representation formulas in forward time:

(3.58)
$v_{\star}(t)=S_{\star}(0;t)v_{\star}(0)+\int_{0}^{t}S_{\star}(t^{\prime};t)\big{(}2^{-j\theta^{\prime}}G_{\star}v(t^{\prime})+G_{\star}^{out}(t^{\prime})\big{)}\,dt^{\prime}.$

###  3.10 Iterations of the integral representation in forward time

Consider (3.58): it is an implicit representation of $v_{H}$ or $v_{E}$ in
terms of itself (see the $v$ term in the right-hand side), and of the
remainder terms. We may use inductively (3.58), and obtain

$\displaystyle v_{H}(t)$
$\displaystyle=S_{H}(0;t)v_{H}(0)+2^{-j\theta^{\prime}}\int_{0}^{t}S_{H}(t^{\prime};t)G_{H}(t^{\prime})(S_{E}(0;t^{\prime})v_{E}(0)+S_{H}(0;t^{\prime})v_{H}(0))\,dt^{\prime}$
$\displaystyle+2^{-2j\theta^{\prime}}\int_{0}^{t}\int_{0}^{t^{\prime}}S_{H}(t^{\prime};t)G_{H}(t^{\prime})\big{(}S_{H}(t^{\prime\prime};t^{\prime})G_{H}(t^{\prime\prime})+S_{E}(t^{\prime\prime};t^{\prime})G_{E}(t^{\prime\prime}))v(t^{\prime\prime})\,dt^{\prime}\,dt$
$\displaystyle+\int_{0}^{t}S_{H}(t^{\prime};t)G_{H}^{out}(t^{\prime})\,dt^{\prime}$
$\displaystyle+2^{-j\theta^{\prime}}\int_{0}^{t}\int_{0}^{t^{\prime}}S_{H}(t^{\prime\prime};t^{\prime})G_{H}(t^{\prime})\big{(}S_{H}(t^{\prime\prime};t^{\prime})G^{out}_{H}(t^{\prime\prime})+S_{E}(t^{\prime\prime};t^{\prime})G^{out}_{E}(t^{\prime\prime})\big{)}dt^{\prime\prime}\,dt^{\prime},$

and

$\displaystyle v_{E}(t)$
$\displaystyle=S_{E}(0;t)v_{E}(0)+2^{-j\theta^{\prime}}\int_{0}^{t}S_{E}(t^{\prime};t)G_{E}(t^{\prime})(S_{E}(0;t^{\prime})v_{E}(0)+S_{H}(0;t^{\prime})v_{H}(0))\,dt^{\prime}$
$\displaystyle+2^{-2j\theta^{\prime}}\int_{0}^{t}\int_{0}^{t^{\prime}}S_{E}(t^{\prime};t)G_{E}(t^{\prime})\big{(}S_{H}(t^{\prime\prime};t^{\prime})G_{H}(t^{\prime\prime})+S_{E}(t^{\prime\prime};t^{\prime})G_{E}(t^{\prime\prime}))v(t^{\prime\prime})\,dt^{\prime}\,dt$
$\displaystyle+\int_{0}^{t}S_{E}(t^{\prime};t)G_{E}^{out}(t^{\prime})\,dt^{\prime}$
$\displaystyle+2^{-j\theta^{\prime}}\int_{0}^{t}\int_{0}^{t^{\prime}}S_{E}(t^{\prime\prime};t^{\prime})G_{E}(t^{\prime})\big{(}S_{H}(t^{\prime\prime};t^{\prime})G^{out}_{H}(t^{\prime\prime})+S_{E}(t^{\prime\prime};t^{\prime})G^{out}_{E}(t^{\prime\prime})\big{)}dt^{\prime\prime}\,dt^{\prime}.$

Iterating, we obtain for any $k\geq 1:$

(3.59) $v_{\star}(t)=v_{\star
k}^{f}(t)+2^{-jk\theta^{\prime}}\int_{D_{k}(t)}S_{\star
k}(T_{k};t)v(t_{k})dT_{k}+G^{out}_{\star k}(t),$

where for $t>0$ we define the $k$-dimensional time domain

(3.60) $D_{k}(t)=\Big{\\{}(t_{k};t_{k-1};\dots;t_{1})\in\mathbb{R}^{k},\qquad
0<t_{k}<t_{k-1}<\dots<t_{1}<t\Big{\\}}.$

We denote $T_{k}$ a vector in $D_{k}(t):$

$T_{k}:=(t_{k};t_{k-1};t_{k-2};\dots;t_{1})\in D_{k}(t),$

and $dT_{k}$ the $k$-dimensional Lebesgue measure:

$dT_{k}=dt_{k}dt_{k-1}\cdots dt_{1}.$

The operator $S_{\star k}$ is defined by induction as

(3.61) $S_{\star
1}(t^{\prime};t):=S_{\star}(t^{\prime};t)G_{\star}(t^{\prime}),\qquad 0\leq
t^{\prime}\leq t,$

and

(3.62) $\displaystyle S_{\star k}$ $\displaystyle(T_{k};t)$
$\displaystyle=S_{\star
k-1}(t_{k-1};\dots;t)\circ\Big{(}\,S_{E}(t_{k};t_{k-1})G_{E}(t_{k})+S_{H}(t_{k};t_{k-1})G_{H}(t_{k})\,\Big{)},\qquad
k\geq 2.$

The “free” solutions $v^{f}_{Hk}(t)$ and $v^{f}_{Ek}(t)$ are defined as

(3.63) $\displaystyle v^{f}_{\star k}(t)$
$\displaystyle=S_{\star}(0;t)v_{\star}(0)$ $\displaystyle\quad+\sum_{1\leq
k^{\prime}\leq
k}2^{-jk^{\prime}\theta^{\prime}}\int_{D_{k^{\prime}}(t)}S_{\star
k^{\prime}}(T_{k^{\prime}};t)\big{(}S_{H}(0;t_{k^{\prime}})v_{H}(0)+S_{E}(0;t_{k^{\prime}})v_{E}(0)\big{)}\,dT_{k^{\prime}}$

Finally, the remainders $G^{out}_{\star k}$ are defined as

(3.64) $\displaystyle G^{out}_{\star
k}(t)=\int_{0}^{t}S_{\star}(t_{1};t)G^{out}_{\star}(t_{1})\,dt_{1}$
$\displaystyle+\sum_{2\leq k^{\prime}\leq
k}2^{-j(k^{\prime}-1)\theta^{\prime}}\int_{D_{k^{\prime}-1}(t)}S_{\star
k^{\prime}-1}(T_{k^{\prime}-1};t)\Big{(}\,S_{H}(t_{k^{\prime}-1};t_{k^{\prime}})G^{out}_{H}(t_{k^{\prime}})+S_{E}(t_{k^{\prime}-1};t_{k^{\prime}})G^{out}_{E}(t_{k^{\prime}})\,\Big{)}dT_{k^{\prime}}.$

Recall that $\theta^{\prime}$ is defined in Section 3.8 as
$\theta^{\prime}=\theta/(1+\theta).$

###  3.11 Backward-in-time elliptic solution operator

Consider now the backward-in-time solution operator for the vector field ${\rm
op}_{j}(M_{E})$ in $L^{2}.$ This solution operator is $S_{E}(t;0),$ where
$S_{E}(0;t)$ is introduced in Section 3.9 as the forward-in-time solution
operator. We apply $S_{E}(t;0)$ to the integral representation (3.59) with
$\star=E,$ and obtain

(3.65) $\displaystyle S_{E}(t;0)v^{f}_{Ek}(t)$
$\displaystyle=S_{E}(t;0)v_{E}(t)-S_{E}(t;0)G^{out}_{Ek}(t)$
$\displaystyle-2^{-jk\theta^{\prime}}S_{E}(t;0)\int_{D_{k}(t)}S_{Ek}(T_{k};t)v(t_{k})dT_{k},\qquad\mbox{for
any $k\geq 1.$}$

We will observe that

$S_{E}(t;0)v^{f}_{Ek}(t)\simeq v_{E}(0).$

(This is made precise in Corollary 3.8 in Section 3.14 below). Thus we write
(3.65) in the form

(3.66) $\displaystyle v_{E}(0)$
$\displaystyle=\big{(}v_{E}(0)-S_{E}(t;0)v^{f}_{Ek}(t)\big{)}+S_{E}(0;t)v_{E}(t)-S_{E}(t;0)G^{out}_{Ek}(t)$
$\displaystyle-2^{-jk\theta^{\prime}}S_{E}(t;0)\int_{D_{k}(t)}S_{Ek}(T_{k};t)v(t_{k})dT_{k},\qquad\mbox{for
any $k\geq 1.$}$

The end of the proof is based on identity (3.66): we have a lower bound for
the $L^{2}$ norm of $v_{E}(0)$ (this is Corollary 3.3), and the contradiction
eventually comes from upper bounds for the right-hand side of (3.66). In order
to derive these upper bounds, we first turn to bounds for the forward-in-time
propagators $S_{\star}(0;t).$

###  3.12 Bounds for the forward-in-time solution operator

Recall that $M_{H}$ and $M_{E}$ are introduced in (3.34), and that $S_{H}$ and
$S_{E}$ are the associated solution operators, introduced in Section 3.9.

In computing bounds for the solution operators $S_{\star},$ we may assume that
$M_{\star}$ is evaluated at $t=0.$ Indeed, this simplification results in an
extra remainder term ${\rm op}_{j}(M_{\star}(t)-M_{\star}(0))v,$ which is
acting on $v$ and is $O(2^{-j})$ because of the time rescaling. As a
consequence, in the iterated representation formula this term belongs to the
same category as $G_{\star},$ and thus comes in with a $2^{-jk}$ prefactor,
with $k$ arbitrarily large.

######  Lemma 3.4.

For some $\gamma_{E}>0$ such that $\gamma_{E}\to\Re e\,\lambda_{0}$ as
$\delta\to 0,$ some $\gamma_{H}\geq 0$ with $\gamma_{E}>\gamma_{H},$ for
$0\leq t^{\prime}\leq t,$ we have the bounds

$\|S_{\star}(t^{\prime};t)\|_{L^{2}\to L^{2}}\lesssim
e^{(t-t^{\prime})\gamma_{\star}},\qquad\star\in\\{H,E\\},$

with an implicit constant which depends on $\delta.$

Above, $\lambda_{0}$ is the most unstable eigenvalue of $M,$ as defined in
Section 3.5, and $\delta$ measures the radius of the space-frequency
truncation $\psi$ (or $\psi^{\sharp}$) which enters in the definition of
$M_{E}$ and $M_{H}$ (see Section 3.4 for the definition of $\psi$ and Section
3.6 for the definitions of $M_{H}$ and $M_{E}$).

###### Proof.

We could directly apply Proposition B.1, based on Gårding’s inequality, with
$Q=-M_{H}$ then $Q=-M_{E}.$ However, our ellipticity assumption (Assumption
2.1) bears on the real part of the spectrum of $iA,$ by extension of $M_{E}$
and $M_{H},$ but condition (B.2) that is used in Proposition B.1, bears on the
spectrum of $\Re e\,M_{\star},$ which might happen to be far from the real
part of the spectrum of $M_{\star}.$ This is an issue in view of deriving an
upper rate of growth $\gamma_{E}$ that will be close to the lower rate of
growth $\gamma_{E}^{-}$ (see Lemma 3.7).

We overcome this problem by approximately diagonalizing $M_{\star}.$ This is
reminiscent of the procedure carried out in Section 4 of [12].

The goal is to derive an upper bound for
$u_{\star}(t):=S_{\star}(t^{\prime};t)u_{0},$ given $u_{0}\in L^{2}.$ By
definition, $u_{\star}$ solves

$\partial_{t}u_{\star}+{\rm op}_{j}(M_{\star})u_{\star}=0,\qquad
u(t^{\prime})=u_{0}.$

Let $K_{0}$ be a matrix of a change of basis to upper triangular form for
$M_{\star}(0,x^{0},\xi^{0}).$ Let $K_{\delta}$ be the invertible diagonal
matrix with diagonal equal to $(1,\delta,\delta^{2},\dots,\delta^{N}).$ Then,

$\tilde{M}_{\star}(x^{0},\xi^{0}):=K_{\delta}^{-1}K_{0}^{-1}M_{\star}(0,x^{0},\xi^{0})K_{0}K_{\delta}$

is upper triangular, with entries above the diagonal which are $O(\delta).$
The support of the space-frequency $\psi^{\sharp}$ which enters in $M_{\star}$
has size $O(\delta).$ Since $M_{\star}$ has Hölder regularity in $(x,\xi),$
the symbol

$\tilde{M}_{\star}(x,\xi):=K_{\delta}^{-1}K_{0}^{-1}M_{\star}(0,x,\xi)K_{0}K_{\delta}$

is at a distance $O(\delta^{\theta})$ of $\tilde{M}_{\star}(x^{0},\xi^{0}).$
We let $\tilde{u}_{\star}=K_{\delta}^{-1}K_{0}^{-1}u_{\star}.$ Then
$\tilde{u}_{\star}$ solves

(3.67) $\partial_{t}\tilde{u}_{\star}+{\rm
op}_{j}(\tilde{M}_{\star})\tilde{u}_{\star}=0.$

By (3.28), the eigenvalues of $\Re e\,\tilde{M}_{\star}(x,\xi)$ differ from
the eigenvalues of $\Re e\,\tilde{M}_{\star}(x^{0},\xi^{0})$ at most by
$O(\delta^{1/N}).$ Now the key is that, since
$\tilde{M}_{\star}(x^{0},\xi^{0})$ is upper triangular with entries above the
diagonal that are $O(\delta),$ the spectrum of $\Re
e\,\tilde{M}_{\star}(x^{0},\xi^{0})$ is differs from the real part of the
spectrum of $M_{\star}(x^{0},\xi^{0})$ by $O(\delta^{1/N})$ at most.

Thus the spectrum of $\Re e\,\tilde{M}_{\star}(x,\xi)$ differs from the real
part of the spectrum of $M_{\star}(x^{0},\xi^{0})$ by
$O(\delta^{\min(\theta,1/N)})$ at most. The spectra of $M_{E}(x^{0},\xi^{0})$
and $M_{H}(x^{0},\xi^{0})$ are described in Section 3.5: the unique eigenvalue
of $M_{E}(x^{0},\xi^{0})$ is $\lambda_{0}$ (see (3.26)), and the eigenvalues
of $M_{H}(x^{0},\xi^{0})$ have real part strictly smaller than $\Re
e\,\lambda_{0}$ (see (3.27)).

Thus

$\Re e\,\tilde{M}_{\star}(x,\xi)\leq\gamma_{E},\qquad\mbox{for all $(x,\xi),$
with $\gamma_{E}\to\Re e\,\lambda_{0}$ as $\delta\to 0.$}$

By Proposition B.1, this implies

$\|\tilde{u}_{\star}(t)\|_{L^{2}}\lesssim e^{(t-t^{\prime})\gamma_{E}},$

the small error in the growth rate from Proposition B.1 being incorporated
into the constant. A corresponding bound for
$u_{\star}=S_{\star}(t^{\prime};t)u_{0}$ follows. ∎

###  3.13 Bounds for the iterated forward-in-time solution operators

The iterated solution operators $S_{\star k}$ are defined in (3.62) in Section
3.10. They involve the remainders $G_{\star},$ for which standard pseudo-
differential bounds, and regularity of the posited solution, give the
following:

######  Lemma 3.5.

On the time interval of existence of the posited solution $\tilde{u},$ we have
the bounds

$\|G_{\star}z\|_{L^{2}}\lesssim\|z\|_{L^{2}},\qquad\mbox{for all $z\in L^{2}$
and $\star\in\\{H,E\\},$}$

provided that the small parameter $\varepsilon$ involved in the regularization
of the eigenprojectors $P_{\star}$ in Section 3.6 satisfies
$\varepsilon=2^{-j/(1+\theta)}.$

###### Proof.

The remainder $G_{\star}$ is defined in (3.54), which we reproduce here:

$G_{\star}v:=-2^{j\theta^{\prime}}\Big{(}{\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})\big{(}{\rm
op}_{j}(\psi^{\sharp}\tilde{A})-{\rm op}_{j}(\tilde{A}){\rm
op}_{j}(\psi^{\sharp})\big{)}+R_{\star}+{\rm
op}_{j}(\psi^{\flat}\partial_{t}P_{\star}^{\varepsilon})\Big{)}v.$

The projectors $P_{\star}^{\varepsilon}$ are as regular as $M$ in $\xi$ and
$(u,\partial_{x}u)$ in $x.$ In particular, using notation for norms of symbols
introduced in (A.9),
$\|\psi^{\flat}P_{\star}^{\varepsilon}\|_{0,0,1+[d/2]}<\infty,$ since
$(u,\partial_{x}u)\in H^{s-1}\hookrightarrow L^{\infty}.$ Thus by (A.17), for
all $z\in L^{2},$ $\|{\rm
op}(\psi^{\flat}P_{\star}^{\varepsilon})z\|_{L^{2}\to
L^{2}}\lesssim\|z\|_{L^{2}}.$ By regularity of $u,$ Lemma A.1 and the
composition result of Section A.5, we have, just like in the projection step
of Section 3.6,

$\|{\rm op}_{j}(\psi^{\sharp}\tilde{A})-{\rm op}_{j}(\tilde{A}){\rm
op}_{j}(\psi^{\sharp})\|_{L^{2}\to L^{2}}\lesssim 2^{-j\theta}.$

The term $R_{\star}$ contains terms of the form $R_{\theta}(\cdot,\cdot)$
which are $L^{2}\to L^{2}$ bounded, uniformly in $j,$ by regularity of $u$ and
the composition result of Section A.5. The other terms in $R_{\star}$ involve
the difference $P_{\star}^{\varepsilon}-P_{\star},$ where
$P^{\varepsilon}_{\star}$ was defined in Section 3.6 as a regularization of
the eigenprojectors $P_{\star}.$ We use the results of Section A.3:

$\|{\rm op}_{j}(\psi^{\flat}(P_{H}-P_{H}^{\varepsilon}))\|_{L^{2}\to
L^{2}}\lesssim\varepsilon^{\theta}.$

Since ${\rm op}_{j}(M)$ and ${\rm op}_{j}(M_{\star})$ are $L^{2}\to L^{2}$
bounded, uniformly in $j,$ by (A.17), this implies

$\|R_{\star}\|_{L^{2}\to L^{2}}\lesssim 2^{-j\theta}+\varepsilon^{\theta}.$

The last term in $G_{\star}$ implies $\partial_{t}P^{\varepsilon}_{\star}.$
This is where the regularization was needed. We denote, as in the proof of
Lemma 3.1,

$P_{\star}(x,\xi)=\underline{P}_{\star}\big{(}t,x,u(2^{-j}t,x),\partial_{x}u(2^{-j}t,x),\xi\big{)},$

so that (see Section A.3),

$P_{\star}(t,x,\xi)=\underline{P}_{\star}(2^{-j}t,x,k_{\varepsilon}\star\tilde{u}(t,x),k_{\varepsilon}\star\partial_{x}\tilde{u}(t,x),\xi).$

Then, we have

$\partial_{t}P_{\star}=2^{-j}\Big{(}\partial_{1}\underline{P}_{\star}+\partial_{3}\underline{P}_{\star}\cdot
k_{\varepsilon}\star\partial_{t}u+\partial_{4}\underline{P}_{\star}\cdot
k_{\varepsilon}\star\partial_{t}\partial_{x}u\Big{)}.$

The map $\underline{P}_{\star}$ is smooth in its arguments, by spectral
separation and smoothness of $F.$ In view of the original equation (1.1), we
observe that

$\partial_{t}k_{\varepsilon}\star u+k_{\varepsilon}\star F=0,\quad\mbox{so
that}\quad
k_{\varepsilon}\star\partial_{t}\partial_{x}u=-(\partial_{x}k_{\varepsilon})\star
F,$

where $F=F(t,x,u,\partial_{x}u).$ In particular, by definition of the
regularizing kernel $k_{\varepsilon}$ (or (A.21)),

$\|k_{\varepsilon}\star\partial_{t}\partial_{x}u\|_{L^{\infty}}\lesssim\varepsilon^{-1},$

so that (by (A.17))

$\|{\rm op}(\psi^{\flat}\partial_{t}P_{\star}^{\varepsilon})\|_{L^{2}\to
L^{2}}\lesssim 2^{-j}\varepsilon^{-1}.$

It remains to choose $\varepsilon$ so that the terms are the same order of
magnitude: $\varepsilon^{\theta}=2^{-j}\varepsilon^{-1},$ leading to
$\varepsilon=2^{-j/(1+\theta)},$ and the $L^{2}\to L^{2}$ norm of $G_{\star}$
finally appears to be bounded in $j,$ since
$\theta^{\prime}=\theta/(1+\theta).$ ∎

######  Corollary 3.6.

We have the bounds

$\|S_{\star k}(T_{k};t)\|_{L^{2}\to L^{2}}\lesssim
e^{\gamma_{E}(t-t_{k})},\qquad\star\in\\{H,E\\},$

with the elliptic rate of growth $\gamma_{E}$ given in Lemma 3.4, and an
implicit constant which does not depend on $(t^{\prime};t).$

###### Proof.

Based on the definition of the iterated operators (3.62), we only need to
apply repeatedly Lemmas 3.4 and 3.5, and use $\gamma_{H}<\gamma_{E}.$ ∎

###  3.14 Bound for the backward-in-time operator

We prove here a key upper bound for the action of the backward-in-time
solution operator $S_{E}(t;0)$ associated with the most unstable eigenvalues.

######  Lemma 3.7.

For some $\gamma_{E}^{-}\in\mathbb{R},$ with $\gamma_{E}^{-}<\gamma_{E},$ we
have, so long as $t\leq t_{\star},$ with $t_{\star}$ defined in (B.6), the
bound

$\|{\rm op}_{j}(\tilde{\psi})\circ S_{E}(t;0)\|_{L^{2}\to L^{2}}\lesssim
e^{-t\gamma_{E}^{-}},$

with an implicit constant which does not depend on $t$ nor on $j.$ The cut-off
$\tilde{\psi}$ is as in Corollary 3.3. If the support of $\tilde{\psi}$ is
small enough, then the lower rate of growth $\gamma_{E}^{-}$ is positive. By
shrinking the support of $\tilde{\psi},$ we can make
$\gamma_{E}-\gamma_{E}^{-}$ arbitrarily small.

###### Proof.

We use the proof of Lemma 3.4. The spectrum of $\Re e\,\tilde{M}_{E}$ differs
from $\Re e\,\lambda_{0}$ only by $O(\delta^{1/N}).$ Thus

$\Re e\,\tilde{M}_{E}(x,\xi)\geq\gamma_{E}^{-},\qquad\mbox{for all $(x,\xi)$
on the support of $\psi,$}$

for some $\gamma_{E}^{-}$ such that $\gamma_{E}^{-}\to\Re e\,\lambda_{0}$ as
$\delta\to 0.$ Proposition B.1 then provides a lower bound for the solution
$\tilde{u}_{E}$ to (3.67):

$\|{\rm op}_{j}(\tilde{\psi})\tilde{u}_{E}(0)\|_{L^{2}}\lesssim
e^{-t\gamma_{E}^{-}}\|\tilde{u}_{E}(t)\|_{L^{2}},$

for $\tilde{u}_{\star}(0)$ such that ${\rm
op}_{j}(\tilde{\psi})\tilde{u}_{\star}(0)\neq 0,$ and for $t\leq t_{\star}.$
This translates into the announced bound for $S_{E},$ up to $\delta$-dependent
implicit constants. ∎

######  Corollary 3.8.

We have, so long as $t\leq t_{\star},$ with $t_{\star}$ defined in (B.6), the
bound

$\|{\rm
op}_{j}(\psi^{\flat}\big{(}v_{E}(0)-S_{E}(t;0)v^{f}_{Ek}(t)\big{)}\|_{L^{2}}\lesssim
2^{-j(\sigma+\theta^{\prime})}e^{t(\gamma_{E}-\gamma_{E}^{-})},$

where $\theta^{\prime}=\theta/(1+\theta).$

###### Proof.

Based on the definition of $S^{f}_{Ek}$ in (3.63), we have

$\displaystyle v_{E}(0)$ $\displaystyle-S_{E}(t;0)v^{f}_{Ek}(t)$
$\displaystyle=\sum_{1\leq k^{\prime}\leq
k}2^{-jk^{\prime}\theta^{\prime}}S_{E}(t;0)\int_{D_{k^{\prime}}(t)}S_{Ek^{\prime}}(T_{k^{\prime}};t)\big{(}S_{H}(0;t_{k^{\prime}})v_{H}(0)+S_{E}(0;t_{k^{\prime}})v_{E}(0)\big{)}\,dT_{k^{\prime}}.$

For a bound of ${\rm
op}_{j}(\psi^{\flat})S_{E}(t;0)S_{Ek^{\prime}}(T_{k^{\prime}};t)S_{E}(0;t_{k^{\prime}}),$
we use Lemma 3.4, Corollary 3.6 and Lemma 3.7:

$\|{\rm
op}_{j}(\psi^{\flat})S_{E}(t;0)S_{Ek^{\prime}}(T_{k^{\prime}};t)S_{E}(0;t_{k^{\prime}})\|_{L^{2}\to
L^{2}}\lesssim
e^{\gamma_{E}t_{k^{\prime}}}e^{\gamma_{E}(t-t_{k^{\prime}})}e^{-\gamma_{E}^{-}t}=e^{t(\gamma_{E}-\gamma_{E}^{-})}.$

Since $v_{\star}(0)$ is essentially the $j$th dyadic block of an $H^{\sigma}$
datum (up to a projection), we have $\|v_{\star}(0)\|_{L^{2}}\lesssim
2^{-j\sigma}.$ Integrating a constant over $D_{k}(t),$ with $t\leq
t_{\star}=O(j),$ we find a prefactor that is bounded by a constant times
$j^{k+1}2^{-jk\theta^{\prime}},$ which can be made less than one by choosing
$j$ large enough. ∎

###  3.15 Bound for the poorly localized remainder terms

We turn to the “out” terms $G_{\star}^{out}$ defined in (3.55), which we
reproduce here:

$\displaystyle G_{\star}^{out}$ $\displaystyle:=-{\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})\Big{(}\tilde{R}^{para}_{j}+\Big{(}{\rm
op}_{j}(\tilde{A})\big{(}{\rm op}_{j}(\psi^{\sharp}){\rm op}_{j}(\psi)-{\rm
op}_{j}(\psi)\big{)}+\big{[}{\rm op}_{j}(\psi),{\rm
op}_{j}(\tilde{A})\big{]}\Big{)}\tilde{u}.$

Unlike the “in” remainder terms handled in Lemma 3.5, the “out” terms above
require sharp bounds. The third term in $G_{\star}^{out}$ is bounded as
follows:

######  Lemma 3.9.

On a time interval of existence of the posited solution $\tilde{u},$ we have
the bound

$\|{\rm op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})\big{[}{\rm
op}_{j}(\psi),{\rm op}_{j}(\tilde{A})\big{]}\tilde{u}\|_{L^{2}}\lesssim
2^{-j(2s-1-d/2)}.$

Note that by the composition result of Section A.5, and the fact that
$\psi^{\flat}\prec\psi,$ the $L^{2}\to L^{2}$ norm of the operator ${\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})\big{[}{\rm op}_{j}(\psi),{\rm
op}_{j}(\tilde{A})\big{]}$ is $O(2^{-j(s-1-d/2)}).$ The issue here is that
$\tilde{u}$ is not localized in frequency. The proof below gives a description
of that operator which shows that, in effect, only frequencies $\sim 2^{j}$ of
$\tilde{u}$ come into play.

###### Proof.

Recall that $\tilde{A}$ is defined in (3.12). It has the form

$\tilde{A}=A_{1}\cdot\xi+2^{-j}A_{0},\qquad A_{0},A_{1}\in H^{s-1}.$

We may focus on the $A_{1}$ term, since $A_{0}$ has the same regularity and is
smaller. By definition of para-differential operators in semiclassical
quantization (A.15),

${\rm op}_{j}(A_{1}\cdot\xi)=H_{j}^{-1}{\rm op}(h_{j}A_{1}\cdot\xi)H_{j},$

and by definition of the admissible cut-off (A.11),

$H_{j}^{-1}{\rm op}(h_{j}A_{1}\cdot\xi)H_{j}=\sum_{k\geq
0}H_{j}^{-1}\Big{(}(\phi(2^{-k+N_{0}}D_{x})h_{j}A_{1})\cdot\bar{\phi}_{k}(D_{x})\Big{)}H_{j},\qquad\bar{\phi}_{k}(\xi):=\xi\phi_{k}(\langle\xi\rangle).$

We observe the identity

(3.68) $\chi(D_{x})(h_{j}f)=h_{j}(\chi(2^{-j}D_{x})f),$

for any Fourier multiplier $\chi(D_{x})$ and any Sobolev map $f.$ We have, for
any maps $f,\chi,v:$

$H_{j}^{-1}(f\chi(D_{x})v)=(H_{j}^{-1}f)(h_{j}^{-1}\chi(D_{x})v),$

since $f$ acts on $\chi(D_{x})v$ by multiplication. Thus

$H_{j}^{-1}\Big{(}(\phi(2^{-k+N_{0}}D_{x})h_{j}A_{1})\bar{\phi}_{k}(D_{x})\Big{)}H_{j}v=h_{j}^{-1}\Big{(}\phi(2^{-k+N_{0}}D_{x})h_{j}A_{1}\Big{)}h_{j}^{-1}\Big{(}\bar{\phi}_{k}(D_{x})(h_{j}v)\Big{)}.$

As a consequence, by (3.68),

$H_{j}^{-1}\Big{(}(\phi(2^{-k+N_{0}}D_{x})h_{j}A_{1})\bar{\phi}_{k}(D_{x})\Big{)}H_{j}v=(\phi(2^{-k-j+N_{0}}D_{x})A_{1})(\bar{\phi}_{k}(2^{-j}D_{x})v).$

Recall (3.15) that $\psi(x,\xi)=\psi_{1}(x)\psi_{2}(\xi).$ We use Proposition
A.3:

(3.69) $\|\mathrm{pdo}_{j}(\psi)-{\rm op}_{j}(\psi)\|_{L^{2}\to L^{2}}\lesssim
2^{-j(s^{\prime\prime}-d)},$

for any $s^{\prime\prime}>0,$ since $\psi_{j}\in C^{\infty}_{c}.$ Thus we may
replace ${\rm op}_{j}(\psi)$ by $\mathrm{pdo}_{j}(\psi)$ in the commutator
under scrutiny, and

$\mathrm{pdo}_{j}(\psi)=H_{j}^{-1}h_{j}\psi_{1}\psi_{2}(D_{x})H_{j}.$

We have, just like above,

$H_{j}^{-1}h_{j}\psi_{1}\psi_{2}(D_{x})H_{j}=\psi_{1}\psi_{2}(2^{-j}D_{x}).$

The focus is on

$[{\rm op}_{j}(\psi),{\rm op}_{j}(A_{1}\cdot\xi)]=\sum_{k\geq 0}\Gamma_{k},$

with notation

$\Gamma_{k}:=\psi_{1}\psi_{2}(2^{-j}D_{x})A_{1}^{j+k}\bar{\phi}_{k}(2^{-j}D_{x})-A_{1}^{j+k}\bar{\phi}_{k}(2^{-j}D_{x})\psi_{1}\psi_{2}(2^{-j}D_{x}).$

up to a very small term described in (3.69), with notation

(3.70) $A_{1}^{j+k}=\phi_{0}(2^{-k-j+N_{0}}D_{x})A_{1}.$

We observe that

$\Gamma_{k}=\psi_{1}[\psi_{2}(2^{-j}D_{x}),A_{1}^{j+k}]\bar{\phi}_{k}(2^{-j}D_{x})-A_{1}^{j+k}[\bar{\phi}_{k}(2^{-j}D_{x}),\psi_{1}]\psi_{2}(2^{-j}D_{x}).$

We now take into account the truncation and projection operator ${\rm
op}_{j}(\psi^{\flat}P_{\star}\varepsilon^{)}$ to the left:

${\rm op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})=H_{j}^{-1}{\rm
op}\big{(}h_{j}(\psi^{\flat}_{1}\psi^{\flat}_{2}P_{\star}^{\varepsilon})\big{)}H_{j}.$

Since $\psi_{2}^{\flat}$ depends only on $\xi,$

${\rm
op}\big{(}h_{j}(\psi^{\flat}_{1}\psi^{\flat}_{2}P_{\star}^{\varepsilon})\big{)}={\rm
op}\big{(}h_{j}(\psi^{\flat}_{1}P_{\star}^{\varepsilon})\big{)}\psi_{2}^{\flat}(D_{x}).$

Thus

$\displaystyle{\rm op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})$
$\displaystyle=h_{j}^{-1}{\rm
op}\big{(}h_{j}(\psi^{\flat}_{1}P_{\star}^{\varepsilon}))h_{j}\psi_{2}^{\flat}(2^{-j}D_{x})$
$\displaystyle=\sum_{k^{\prime}\geq
0}h_{j}^{-1}\mathrm{pdo}\Big{(}\phi(2^{-k^{\prime}+N_{0}}D_{x})(h_{j}(\psi_{1}^{\flat}P_{\star}^{\varepsilon}))\Big{)}\phi_{k}(\langle
D_{x}\rangle)h_{j}\psi_{2}^{\flat}(2^{-j}D_{x})$
$\displaystyle=\sum_{k^{\prime}\geq
0}h_{j}^{-1}\mathrm{pdo}\Big{(}\phi(2^{-k^{\prime}+N_{0}}D_{x})(h_{j}(\psi_{1}^{\flat}P_{\star}\varepsilon^{)})\Big{)}h_{j}\phi_{k}(2^{-j}\langle
D_{x}\rangle)\psi_{2}^{\flat}(2^{-j}D_{x}).$

Now recall that in the Littlewood-Paley decomposition $(\phi_{k})_{k\geq 0},$
only $\phi_{1}$ is supported in a neighborhood of the unit sphere. Since
$\psi_{2}^{\flat}$ is supported near $\xi^{0}\in\mathbb{S}^{d-1},$ this means
that in the above sum, only $k^{\prime}=1$ subsists, and, if the support of
$\psi_{2}^{\flat}$ is chosen small enough, then
$\psi_{2}^{\flat}\phi_{1}\equiv\psi_{2}^{\flat}.$ So

(3.71) $\displaystyle{\rm op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})$
$\displaystyle=h_{j}^{-1}\mathrm{pdo}\Big{(}\phi(2^{-1+N_{0}}D_{x})(h_{j}(\psi_{1}^{\flat}P_{\star}^{\varepsilon}))\Big{)}h_{j}\psi_{2}^{\flat}(2^{-j}D_{x})$
$\displaystyle=:Q_{\star}\psi_{2}^{\flat}(2^{-j}D_{x}).$

The full operator under study now looks like

(3.72) $\sum_{k\geq 0}Q_{\star}\psi_{2}^{\flat}(2^{-j}D_{x})\Gamma_{k}.$

Consider

$\displaystyle\psi_{2}^{\flat}(2^{-j}D_{x})\Gamma_{k}$
$\displaystyle=:\sum_{1\leq\ell\leq 3}\Gamma_{k,\ell},$

with notation

$\displaystyle\Gamma_{k,1}$
$\displaystyle:=[\psi_{2}^{\flat}(2^{-j}D_{x}),\psi_{1}][\psi_{2}(2^{-j}D_{x}),A_{1}^{j+k}]\bar{\phi}_{k}(2^{-j}D_{x}),$
$\displaystyle\Gamma_{k,2}$
$\displaystyle:=\psi_{1}\psi_{2}^{\flat}(2^{-j}D_{x})[\psi_{2}(2^{-j}D_{x}),A_{1}^{j+k}]\bar{\phi}_{k}(2^{-j}D_{x})$
$\displaystyle\Gamma_{k,3}$
$\displaystyle:=\psi_{2}^{\flat}(2^{-j}D_{x})A_{1}^{j+k}[\bar{\phi}_{k}(2^{-j}D_{x}),\psi_{1}]\psi_{2}(2^{-j}D_{x}).$

We apply the operator (3.72) to $\tilde{u}\in H^{s},$ where
$\tilde{u}(t,x)=u(2^{-j}t,x).$ Once we verify the bound

(3.73) $\|Q_{\star}\|_{L^{2}\to L^{2}}\lesssim 1,$

with $Q_{\star}$ defined in (3.71), it will remain to show

(3.74) $\sum_{k\geq
0}\|\psi_{2}^{\flat}(2^{-j}D_{x})\Gamma_{k,\ell}\tilde{u}\|_{L^{2}}\lesssim
2^{-j(2s-1-d/2)},\qquad\mbox{for all $\ell\in\\{1,2,3\\}.$}$

We temporarily admit (3.73) and prove (3.74).

Consider the operator
$\psi_{2}^{\flat}(2^{-j}D_{x})[\psi_{2}(2^{-j}D_{x}),A_{1}^{j+k}]$ in
$\Gamma_{k,2}.$ We observe that

$\psi_{2}^{\flat}\partial_{\xi}^{\alpha}\psi_{2}\equiv 0,\qquad\mbox{for all
$|\alpha|>0,$}$

since $\psi_{2}^{\flat}\prec\psi_{2}$ (see (3.20)). Thus, by Lemma A.5,

$\psi_{2}^{\flat}(2^{-j}D_{x})[\psi_{2}(2^{-j}D_{x}),A_{1}^{j+k}]=\psi_{2}^{\flat}(2^{-j}D_{x})\tilde{R}_{s-1-d/2}(\psi_{2}(2^{-j}\cdot),A_{1}^{j+k}),$

where we used notation $\tilde{R}$ from the proof of Lemma A.5. Since
$A_{1}\in H^{s-1},$ we know from Lemma A.5 that the $L^{2}\to L^{2}$ operator
norm of $\tilde{R}_{s-1-d/2}(\psi_{2}(2^{-j}\cdot),A_{1}^{j+k})$ is
$O(2^{-j(s-1-d/2)}).$ Thus

$\displaystyle\|\Gamma_{k,2}\tilde{u}\|_{L^{2}}$
$\displaystyle\leq\|\psi_{1}\|_{L^{\infty}}\|\psi_{2}^{\flat}\|_{L^{\infty}}\|\tilde{R}_{s-1-d/2}(\psi_{2}(2^{-j}\cdot),A_{1}^{j+k})\|_{L^{2}\to
L^{2}}\|\bar{\phi}_{k}(2^{-j}D_{x})\tilde{u}\|_{L^{2}}$ $\displaystyle\lesssim
2^{-j(s-1-d/2)}\|\bar{\phi}_{k}(2^{-j}D_{x})\tilde{u}\|_{L^{2}}.$

The support of $\bar{\phi}_{k}(2^{-j}\cdot)$ is an annulus with frequencies
$\sim 2^{j+k}.$ Since $\tilde{u}\in H^{s},$ this implies

$\|\bar{\phi}_{k}(2^{-j}D_{x})\tilde{u}\|_{L^{2}}\lesssim 2^{k-(j+k)s},$

the $2^{k}$ factor coming from the $\xi$ factor in $\bar{\phi}_{k},$ itself a
trace of the fact that the symbol $\tilde{A}$ is order one. We can sum over
$k$ and find

$\sum_{k\geq
0}\|\psi_{2}^{\flat}(2^{-j}D_{x})\Gamma_{k,2}\tilde{u}\|_{L^{2}}\lesssim
2^{-j(2s-1-d/2)}\sum_{k\geq 0}2^{(1-s)k}\lesssim 2^{-j(2s-1-d/2)},$

which proves (3.74) in the case $\ell=2.$

Consider now $\Gamma_{k,1}.$ The commutator
$[\psi_{2}^{\flat}(2^{-j}D_{x},\psi_{1}]$ involves only smooth commutator
functions. Thus we can push the expansion in Lemma A.5 up to an arbitrarily
high order. In particular, the $L^{2}\to L^{2}$ norm of the remainder in Lemma
A.5 is smaller than $2^{-j(s-1-d/2)}$ if the expansion is pushed to a
sufficiently high order. Since $u\in H^{s},$ the operator
$[\psi_{2}(2^{-j}D_{x}),A_{1}^{j+k}],$ where $A_{1}^{j+k}$ depends on
$(u,\partial_{x}u),$ is $L^{2}\to L^{2}$ bounded. Thus it remains to handle
the first terms in the expansion of the commutator
$[\psi_{2}^{\flat}(2^{-j}D_{x},\psi_{1}].$ Those terms have the form
$(\partial_{x}^{\alpha}\psi_{1})(\partial_{\xi}^{\alpha}\psi_{2}^{\flat})(2^{-j}D_{x}),$
and we can now apply to the corresponding terms in $\Gamma_{k,1}$ the
arguments that we used for $\Gamma_{k,2}.$

We turn to $\Gamma_{k,3}.$ First we observe that the supports of
$\bar{\phi}_{k}$ and $\psi_{2}$ are disjoint unless $k=1.$ Thus only
$\Gamma_{1,3}$ remains. We have

$\Gamma_{1,3}u=\psi_{2}^{\flat}(2^{-j}D_{x})(A_{1}^{j+1}f)=(\psi_{2}^{\flat}(2^{-j}D_{x})A_{1}^{j+1})f+\tilde{\Gamma}_{1,3},$

with notation
$f:=[\bar{\phi}_{1}(2^{-j}D_{x}),\psi_{1}]\psi_{2}(2^{-j}D_{x})\tilde{u},$ and

$\tilde{\Gamma}_{1,3}u=\psi_{2}^{\flat}(2^{-j}D_{x})(A_{1}^{j+1}f)-(\psi_{2}^{\flat}(2^{-j}D_{x})A_{1}^{j+1})f.$

We bound

$\|(\psi_{2}^{\flat}(2^{-j}D_{x})A_{1}^{j+1})f\|_{L^{2}}\leq\|\psi_{2}^{\flat}((2^{-j}D_{x})A_{1}^{j+1}\|_{L^{2}}\|f\|_{L^{\infty}}.$

Since $\bar{\phi}_{1}(2^{-j}D_{x})$ and $\psi_{1}$ are $L^{\infty}\to
L^{\infty}$ bounded, uniformly in $j,$ we have

$\|f\|_{L^{\infty}}\lesssim\|\psi_{2}(2^{-j}D_{x})\tilde{u}\|_{L^{\infty}},$

and by Bernstein’s inequality (or the Sobolev embedding (A.5))

$\|\psi_{2}(2^{-j}D_{x})u\|_{L^{\infty}}\lesssim
2^{jd/2}\|\psi_{2}(2^{-j}D_{x})\tilde{u}\|_{L^{2}}\lesssim 2^{-j(s-d/2)},$

the last inequality by the posited regularity of $\tilde{u}.$ Since $A_{1}\in
H^{s-1},$ we have

$\|\psi_{2}^{\flat}(2^{-j}D_{x})A_{1}^{j+1}\|_{L^{2}}\lesssim 2^{-j(s-1)}.$

Thus we proved

$\|(\psi_{2}^{\flat}(2^{-j}D_{x})A_{1}^{j+1})f\|_{L^{2}}\lesssim
2^{-j(2s-1-d/2)},$

and we turn to $\tilde{\Gamma}_{1,3}\tilde{u}:$

$\|\tilde{\Gamma}_{1,3}\tilde{u}\|_{L^{2}}=\left\|\int_{\mathbb{R}^{d}}(\psi_{2}^{\flat}(2^{-j}(\xi)-\psi_{2}^{\flat}(2^{-j}(\xi-\xi^{\prime}))){\mathcal{F}}(A_{1}^{j+1})(\xi-\xi^{\prime})\hat{f}(\xi^{\prime})\,d\xi^{\prime}\right\|_{L^{2}}.$

We Taylor expand the frequency cut-off:

$\displaystyle\psi_{2}^{\flat}(2^{-j}\xi)-\psi_{2}^{\flat}(2^{-j}(\xi-\xi^{\prime}))$
$\displaystyle=\sum_{1\leq|\alpha|\leq
m}(\partial_{\xi}^{\alpha}\psi_{2}^{\flat})(2^{-j}(\xi-\xi^{\prime}))\cdot(-2^{-j}\xi^{\prime})^{\alpha}$
$\displaystyle+2^{-j(m+1)}R(\psi_{2}^{\flat}),$

for any $m\in\mathbb{N},$ where $R(\psi_{2}^{\flat})$ is the remainder, using
the same notational convention as in the proof of Lemma A.5. The term of index
$\alpha$ (with $|\alpha|>0$) in the above expansion has a contribution to the
$L^{2}$ norm of $\tilde{\Gamma}_{1,3}u$ that is controlled by

$\|(\partial_{\xi}^{\alpha}\psi_{2}^{\flat})(2^{-j}D_{x})A_{1}^{1+j}\|_{L^{2}}\int_{\mathbb{R}^{d}}|2^{-j}\xi|^{|\alpha|}|\hat{f}(\xi)|\,d\xi.$

Since $|\alpha|>0,$ we have, by regularity of $A_{1},$

$\|(\partial_{\xi}^{\alpha}\psi_{2}^{\flat})(2^{-j}D_{x})A_{1}^{1+j}\|_{L^{2}}\lesssim
2^{-j(s-1)}.$

Besides,

$f=\bar{\phi}_{1}(2^{-j}D_{x})(\psi_{1}\psi_{2}(2^{-j}D_{x})\tilde{u})+\psi_{1}\bar{\phi}_{1}(2^{-j}D_{x})\psi_{2}(D_{x})\tilde{u})=:f_{1}+f_{2}.$

Since $\xi\to|\xi|^{|\alpha|}\bar{\phi}_{1}(\xi)$ is bounded,

$\|(2^{-j}|\xi|)^{|\alpha|}\hat{f}_{1}\|_{L^{1}}\lesssim\|\hat{\psi}_{1}\star(\psi_{2}(2^{-j}\cdot){\mathcal{F}}(\tilde{u}))\|_{L^{1}}\lesssim
2^{-js},$

since $\hat{\psi}_{1}$ belongs to the Schwartz class and $\tilde{u}\in H^{s}.$
Similarly,

$\|(2^{-j}|\xi|)^{|\alpha|}\hat{f}_{2}\|_{L^{1}}\lesssim\int_{\mathbb{R}^{d}}|\xi-\xi^{\prime}|^{|\alpha|}|\hat{\psi}_{1}(\xi-\xi^{\prime})||2^{-j}\xi^{\prime}|^{|\alpha|}|\bar{\phi}_{1}(2^{-j}\xi^{\prime})\psi_{2}(2^{-j}\xi^{\prime}){\mathcal{F}}(\tilde{u})(\xi^{\prime})|\,d\xi^{\prime},$

so that

$\|(2^{-j}|\xi|)^{|\alpha|}\hat{f}_{2}\|_{L^{1}}\lesssim 2^{-js}.$

We finally bound the contribution to $\tilde{\Gamma}_{1,3}$ of the remainder
term in the Taylor expansion of $\psi_{2}^{\flat}.$ This is a term of the form

$\left\|\int_{\mathbb{R}^{d}}\int_{0}^{1}\tilde{\psi}_{2}(\xi-\xi^{\prime},t\xi^{\prime}){\mathcal{F}}(A_{1}^{j+1})(\xi-\xi^{\prime})(2^{-j}t\xi^{\prime})^{\alpha}\hat{f}(\xi^{\prime})\,d\xi^{\prime}\,dt\right\|_{L^{2}},$

up to a polynomial in $t,$ with $\tilde{\psi}_{2}\in L^{\infty}.$ This term
has the same form as the previous terms in the expansion of
$\psi_{2}^{\flat},$ and we may conclude that the $L^{2}$ norm of
$\tilde{\Gamma}_{1,3}$ is controlled by $2^{-j(2s-1)}.$

In a last step, we prove (3.73), where
$Q_{\star}=h_{j}^{-1}\mathrm{pdo}\Big{(}\phi_{0}(2^{-1+N_{0}}D_{x})(h_{j}(\psi_{1}^{\flat}P_{\star}^{\varepsilon}))\Big{)}h_{j}.$
We use the first bound in (A.17):

$\|Q_{\star}\|_{L^{2}\to L^{2}}\lesssim\sup_{|\alpha|,|\beta|\leq
1+[d/2]}\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}(\phi_{0}(2^{-1+N_{0}}D_{x})(h_{j}(\psi_{1}^{\flat}P_{\star}^{\varepsilon})\|_{L^{\infty}}.$

Denote $\check{\phi}$ the inverse Fourier transform of
$\phi_{0}(2^{-1+N_{0}}\cdot),$ so that $\check{\phi}$ belongs to the Schwartz
space. We have

$\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}(\phi_{0}(2^{-1+N_{0}}D_{x})(h_{j}(\psi_{1}^{\flat}P_{\star}^{\varepsilon})\|_{L^{\infty}}\lesssim\|\partial_{x}^{\alpha}\check{\phi}_{0}\|_{L^{1}}\|\psi_{1}^{\flat}\partial_{\xi}^{\beta}P_{\star}^{\varepsilon}\|_{L^{\infty}}.$

By regularity of $P_{\star}^{\varepsilon},$ we have
$\|\partial_{\xi}^{\beta}P_{\star}^{\varepsilon}\|_{L^{\infty}}\lesssim
C(\|\tilde{u},\partial_{x}\tilde{u}\|_{L^{\infty}})<\infty,$ since
$(\tilde{u},\partial_{x}\tilde{u})\in H^{s-1}\hookrightarrow L^{\infty},$
which concludes the proof. ∎

######  Corollary 3.10.

On a time interval of existence of the posited solution $\tilde{u},$ we have
the bound

$\displaystyle{\|G^{out}_{\star k}(t)\|_{L^{2}}\lesssim
2^{-j(2s-1-d/2)}e^{t\gamma_{E}}.}$

###### Proof.

First we bound $G_{\star}^{out}.$ We saw in the proof of Lemma 3.5 that ${\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})$ is $L^{2}\to L^{2}$ bounded,
uniformly in $j.$ Thus with (3.6) (where we note that
$2^{j}\partial_{x}u_{h}=\partial_{x}\tilde{u}$) and (3.11), we have

$\|{\rm
op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon})\tilde{R}^{para}_{j}\|_{L^{2}}\lesssim
2^{-j(2s-1-d/2)}.$

The term ${\rm op}_{j}(\tilde{A})\big{(}{\rm op}_{j}(\psi^{\sharp}){\rm
op}_{j}(\psi)-{\rm op}_{j}(\psi)\big{)}\tilde{u}$ remains. Here we can use the
composition result (A.34)-(A.35) up to an arbitrarily high order, since the
cut-offs $\psi^{\sharp}$ and $\psi$ are smooth. In view of
$\psi\prec\psi^{\sharp},$ this gives

${\rm op}_{j}(\psi^{\sharp}){\rm op}_{j}(\psi)-{\rm
op}_{j}(\psi)=2^{-jk}R_{k}(\psi^{\sharp},\psi),$

where $R_{k}(\psi^{\sharp},\psi)$ maps $L^{2}$ to $H^{-k}_{j},$ with a norm
that does not depend on $j.$ Thus

$\|{\rm op}_{j}(\psi^{\flat}P_{\star}^{\varepsilon}){\rm
op}_{j}(\tilde{A})\big{(}{\rm op}_{j}(\psi^{\sharp}){\rm op}_{j}(\psi)-{\rm
op}_{j}(\psi)\big{)}\tilde{u}\|_{L^{2}}\lesssim 2^{-jk},$

for any $k>0.$ With Proposition 3.9, we conclude that

$\|G_{\star}^{out}\|_{L^{2}}\lesssim 2^{-j(2s-1-d/2)}.$

With Lemma 3.6, this implies the bound for $G_{\star k}^{out}.$ ∎

###  3.16 Endgame

We use the backward-in-time representation (3.66), which we reproduce here:

(3.75) $\displaystyle v_{E}(0)$
$\displaystyle=\big{(}v_{E}(0)-S_{E}(t;0)v^{f}_{Ek}(t)\big{)}+S_{E}(t;0)v_{E}(t)-S_{E}(t;0)G^{out}_{Ek}(t)$
$\displaystyle-2^{-jk\theta}S_{E}(t;0)\int_{D_{k}(t)}S_{Ek}(T_{k};t)v(t_{k})dT_{k},\qquad\mbox{for
any $k\geq 1.$}$

On the one hand, we have a lower bound for ${\rm op}_{j}(\tilde{\psi})$
applied to the left-hand side of (3.75): this is Corollary 3.3. Recall that
$\tilde{\psi}$ is a smooth truncation near $(x^{0},\xi^{0})$ with an
appropriately small support. On the other hand, we have upper bounds for the
operator ${\rm op}_{j}(\tilde{\psi})$ applied to the right-hand side of
(3.75). These show that the left-hand side of (3.75) (the datum) is much
greater than the right-hand side, yielding a contradiction. Indeed:

$\bullet$ by Corollary 3.3,

$\|{\rm op}_{j}(\tilde{\psi})v_{E}(0)\|_{L^{2}}\geq C2^{-j\sigma}(1+j)^{-1},$

for some constant $C>0;$

$\bullet$ by Corollary 3.8, for $t=O(j(\gamma_{E}-\gamma_{E}^{-})^{-1}),$

$\big{\|}{\rm
op}_{j}(\tilde{\psi})(v_{E}(0)-S_{E}(t;0)v^{f}_{Ek}(t))\big{\|}_{L^{2}}\lesssim
2^{-j(\sigma+\theta^{\prime})}e^{t(\gamma_{E}-\gamma_{E}^{-})}.$

$\bullet$ By Lemma 3.7,

$\|{\rm op}_{j}(\tilde{\psi})S_{E}(t;0)v_{E}(t)\|_{L^{2}}\lesssim
e^{-t\gamma_{E}^{-}}\|v_{E}(t)\|_{L^{2}}.$

We saw that $\|{\rm op}_{j}(\psi^{\flat}P_{E}^{\varepsilon})\|_{L^{2}\to
L^{2}}\lesssim 1.$ So $\|v_{E}(t)\|_{L^{2}}\lesssim\|v(t)\|_{L^{2}}.$ Now
$v={\rm op}_{j}(\psi)\tilde{u},$ and $\psi$ localizes around frequencies $\sim
2^{j},$ hence $\|v\|_{L^{2}}\lesssim 2^{-js}.$ (Precisely: we could use
Proposition A.3, and regularity of $\psi,$ to change ${\rm op}_{j}(\psi)$ into
$\mathrm{pdo}_{j}(\psi)=\psi_{1}(x)\psi_{2}(2^{-j}D),$ and then use
Bernstein’s inequality.) Thus

$\|{\rm op}_{j}(\tilde{\psi})S_{E}(t;0)v_{E}(t)\|_{L^{2}}\lesssim
2^{-js}e^{-t\gamma_{E}^{-}}.$

$\bullet$ By Corollary 3.10, and Lemma 3.7,

$\|{\rm op}_{j}(\tilde{\psi})S_{E}(t;0)G^{out}_{Ek}(t)\|_{L^{2}}\lesssim
2^{-j(2s-1-d/2)}e^{t(\gamma_{E}-\gamma_{E}^{-})}.$

$\bullet$ The fourth and last term in the right-hand side of (3.75) is
smaller: by Lemma 3.7, Corollary 3.6, and $\|v(t)\|_{L^{2}}\lesssim 2^{-js}$
(see above), we see that

$\Big{\|}2^{-jk}{\rm
op}_{j}(\tilde{\psi})S_{E}(t;0)\int_{D_{k}(t)}S_{Ek}(T_{k};t)v(t_{k})dT_{k}\Big{\|}_{L^{2}}\lesssim
2^{-j(s+k)}e^{t(\gamma_{E}-\gamma_{E}^{-})}.$

Putting those bounds together, we find from (3.75) the inequality

(3.76) $\displaystyle 2^{-j\sigma}(1+j)^{-1}\leq C\Big{(}$ $\displaystyle
2^{-j(\sigma+\theta^{\prime})}e^{t(\gamma_{E}-\gamma_{E}^{-})}+2^{-js}e^{-t\gamma_{E}^{-}}$
$\displaystyle+2^{-j(2s-1-d/2)}e^{t(\gamma_{E}-\gamma_{E}^{-})}+2^{-j(s+k)}e^{t(\gamma_{E}-\gamma_{E}^{-})}\Big{)},$

It is not restrictive to assume that $\sigma$ is large enough so that
$\sigma+\theta^{\prime}>2s-1-d/2.$ (Recall,
$\theta^{\prime}=\theta/(1+\theta).$) Indeed, the range of $\sigma$ in the
statement of Theorem 1 is $\sigma\in[s,2s-1-d/2).$ If we manage to prove non-
existence of an $H^{s}$ solution for any value of
$\sigma\in(2s-1-d/2-\theta^{\prime},2s-1-d/2),$ it will prove non-existence of
an $H^{s}$ solution for smaller values of $\sigma,$ since $H^{\sigma}\subset
H^{\sigma_{1}}$ for $\sigma\geq\sigma_{1}.$ Thus among the first, third and
fourth terms in the above upper bound have, the third is the largest (since
$k$ can be chosen to be arbitrarily large). We obtain, with a different
constant $C:$

(3.77) $2^{-j\sigma}(1+j)^{-1}\leq
C\Big{(}2^{-js}e^{-t\gamma_{E}^{-}}+2^{-j(2s-1-d/2)}e^{t(\gamma_{E}-\gamma_{E}^{-})}\Big{)}.$

In (3.77), the constant $C$ does not depend on $j$ nor on $t.$ (It depends on
$F,$ the datum, and the radius $\delta$ that describes the size of the support
of the space-frequency cut-off around $(x^{0},\xi^{0}).$)

We now choose $\sigma,$ depending on $s$ and $d,$ a final observation time
$t_{final},$ then $\delta,$ depending on $\sigma,$ $s,$ $d,$ and $F,$ and then
finally $j$ depending on all other parameters:

$\bullet$ We choose $\sigma$ (as we may, see the discussion above (3.77)) such
that

$2s-1-d/2-\frac{\theta}{\theta+d_{\star}}<\sigma<2s-1-d/2,$

where $\theta=s-1-d/2,$ and $d_{\star}$ is the $d$-dependent integer that
comes up in Gårding’s inequality (see Section A.6).

$\bullet$ We let an observation time $t_{final}$ be defined by

(3.78) $t_{final}=\frac{7j}{8}\cdot\frac{(2s-1-d/2-\sigma)\ln
2}{\gamma_{E}-\gamma_{E}^{-}}.$

The ratio $7/8$ is arbitrary; any number in $(0,1)$ would do. By choice of
$\sigma$ above, we observe that this $t_{final}$ is smaller than the time of
validity of the growth rates of Section B: $t_{final}<t_{\star}.$ Also, in the
original time frame, this observation time is $2^{-j}t_{final}:$ smaller than
$T,$ the posited time of existence of the solution $u,$ if $j$ is chosen large
enough (as it will be later). Thus inequality (3.77) holds at $t=t_{final}.$

$\bullet$ We choose $\delta$ so that

(3.79) $e^{-t_{final}\gamma_{E}^{-}}\leq 2^{-j(\sigma-s+1/100)}.$

For that we only need to choose $\delta$ small enough: indeed,
$\gamma_{E}^{-}$ then approaches some value $\gamma,$ which depends on $F$ and
$w,$ and simultaneously $\gamma_{E}-\gamma_{E}^{-}$ approaches zero, so that
$t_{final}$ becomes large. Thus in (3.79) we are comparing two decaying
exponentials in $j,$ one (in the right-hand side) with a fixed decay rate, and
the other (in the left-hand side) with a decay rate that we make large in the
small $\delta$ limit.

$\bullet$ With the above choice for $\sigma$ and $\delta,$ we consider (3.77)
at $t=t_{final}.$ This gives

$(1+j)^{-1}\leq C(2^{-j/100}+2^{-j(2s-1-d/2-\sigma)/8}),$

a contradiction if $j$ is chosen to be large enough, depending on $C$ and on
$2s-1-d/2-\sigma.$

This proves that, under the ellipticity Assumption 2.1, there are no
$C^{0}([0,T],H^{s})$ solutions to (1.1) issued from the datum $u_{in}\in
H^{\sigma}$ (the specific datum defined in Section 3.7), no matter how small
$T>0$ is, and concludes the proof of Theorem 1.

##  4 Transition to ellipticity: proof of Theorem 2

As in the proof of Theorem 1, we assume the existence of $T>0,$ a ball
$B_{x_{0}}$ centered at $x_{0}$ and $u\in C^{0}([0,T],H^{s}(B_{x_{0}}))$ which
solves the initial-value problem (1.1). We identify the solution $u$ with its
extension to the whole space $\mathbb{R}^{d}$ via a linear bounded extension
operator $H^{s}(B_{x_{0}})\to H^{s}(\mathbb{R}^{d}).$

For condition (iv) in Assumption 2.2 to make sense, we need the second time
derivative of the characteristic polynomial $P$ to be defined pointwise. We
saw in (2.2) that $\partial_{t}P$ involves two derivatives of the datum. The
second derivative $\partial_{t}^{2}P$ involves three derivatives of the datum,
hence the condition $\sigma>3+d/2,$ by Sobolev embedding.

###  4.1 The spectral picture

We describe here what Assumption 2.2 entails for the spectrum of the principal
symbol $A$ evaluated along the posited solution $u.$ We denote

(4.1) $\underline{A}(t,x,\xi):=A(t,x,u(t,x),\partial_{x}u(t,x),\xi).$

The map $\underline{A}$ is $C^{1}$ in $t$ and $C^{1,1/2}$ in $x$ (by
assumption on $s$ in Theorem 2, Sobolev embedding, and the original system
(1.1)) and smooth in $\xi\in\mathbb{S}^{d-1}.$

By condition (i) in Assumption 2.2, the spectrum of $\underline{A}$ at $t=0$
is real for all $(x,\xi)\in U.$

Given $(x,\xi)\in U,$ consider an eigenvalue $\mu$ of $\underline{A}$ such
that $\partial_{\lambda}P(0,x,\xi,\mu(0,x,\xi))\neq 0:$ the eigenvalue
$\mu(0,x,\xi)$ of $\underline{A}(0,x,\xi)$ is simple. As a consequence, it is
as smooth as $\underline{A},$ pointwise in $(t,x,\xi),$ and it stays real for
$(t,x^{\prime},\xi^{\prime})$ in a neighborhood of $(0,x,\xi).$ Indeed, the
coefficients of $\underline{A}$ are real, implying that the spectrum is
symmetric with respect to $\mathbb{R}:$ eigenvalues away from $\mathbb{R}$
occur in pairs, and by continuity for eigenvalues away from $\mathbb{R}$ to be
observed for $t>0,$ we need eigenvalues to coalesce on the real axis at $t=0.$

Consider now $(x^{0},\xi^{0})\in U$ given by condition (iii) in Assumption
2.2. For any $\lambda$ such that
$(0,x^{0},\xi^{0},\lambda)\in{\mathcal{S}}_{U},$ either $\lambda$ is a simple
eigenvalue, or we have a neighborhood $\Omega_{\lambda}$ of
$\omega=(0,x^{0},\xi^{0},\lambda)$ such that the conditions (iv) hold over
$\Omega_{\lambda}.$ Indeed, the conditions $\partial_{t}P=0,$
$\partial_{\lambda}P=0$ are stated to hold locally, and the inequality in
condition (iv) is an open condition. The projection of $\Omega_{\lambda}$ over
$\mathbb{R}^{2d}_{x,\xi}$ is a neighborhood of $(x^{0},\xi^{0}),$ which we
denote $V_{\lambda}.$ The finite intersection of the $V_{\lambda}$ over all
$\lambda$ in the spectrum at $(0,x^{0},\xi^{0})$ is a neighborhood of
$(x^{0},\xi^{0})$ which we denote $V.$

Given $(x,\xi)\in V$ and an eigenvalue $\mu(0,x,\xi)$ of
$\underline{A}(0,x,\xi),$ let $\omega=(0,x,\xi,\mu(0,x,\xi)).$ If we have
$\partial_{\lambda}P(0,\omega)=0,$ then the conditions (iv) in Assumption 2.2
hold at $\omega.$ The multiplicity of $\mu$ at $(0,x,\xi)$ is exactly two.
Thus $\mu$ belongs to a branch (in $(x,\xi)$) of eigenvalues of
$\underline{A}_{|t=0}$ drawn over $V$ which have constant multiplicity equal
to two.

Now consider the time evolution of these branches of eigenvalues, for fixed
$(x,\xi)\in V:$

######  Lemma 4.1.

Condition (iv) in Assumption 2.2 implies that two branches of simple
eigenvalues $\mu^{\pm}$ branch out of the eigenvalue $\mu$ described just
above, and out of the real axis at $t=0,$ such that $\mu^{\pm}$ are time-
differentiable at $t=0,$ for all $(x,\xi)\in V,$ with

(4.2) $\mu^{\pm}(0,x,\xi)=\mu(0,x,\xi),\quad\partial_{t}\Im
m\,\mu^{+}(0,x,\xi)=-\partial_{t}\Im m\,\mu^{-}(0,x,\xi)\neq 0.$

Besides, under the conditions bearing on $s$ and $\sigma$ in Theorem 2, the
functions $\mu^{\pm}$ are as regular as $\underline{A},$ and
$\partial_{t}\mu^{\pm}$ is continuous in $(t,x)$ and smooth in $\xi.$

###### Proof.

Given $(x,\xi)\in V$ and an eigenvalue $\mu(0,x,\xi)$ such that
$\partial_{\lambda}P(0,\omega)=0,$ the eigenvalue $\mu(0,x,\xi)$ has
multiplicity exactly two, by the inequality in condition (iv) from Assumption
2.2, and is separated from the other eigenvalues. Thus for $t>0,$ its
multiplicity is at most two, and $\mu$ is an eigenvalue for a size-two matrix
$B,$ with real coefficients just like $\underline{A},$ the same regularity as
$\underline{A},$ and with associated characteristic polynomial $P_{0}.$ The
characteristic polynomial $P$ of $\underline{A}$ factorizes into
$P=P_{0}P_{1},$ with $P_{1}(0,\omega)\neq 0.$ We have

$2\mu(t,x,\xi)=\mbox{tr}\,B(t,x,\xi)\pm(\det B(t,x,\xi))^{1/2},$

with $\det B(0,x,\xi)=0$ for all $(x,\xi)\in V.$ We compute, as in Proposition
1.4 and Appendix A of [12]:

$\displaystyle\partial_{t}\det B(0,x,\xi)$
$\displaystyle=2\mbox{tr}\,B(0)\partial_{t}\mbox{tr}\,B(0)-4\partial_{t}\det
B(0)=-4\partial_{t}P_{0}(0,x,\xi)=0,$

and

$\displaystyle\partial_{t}^{2}\det B(0,x,\xi)$
$\displaystyle=2(\partial_{t}\mbox{tr}\,B(0))^{2}+2\mbox{tr}\,B(0)\partial_{t}^{2}\mbox{tr}\,B(0)-4\partial_{t}^{2}\det
B(0)$
$\displaystyle=2(\partial_{t}\partial_{\lambda}P_{0})^{2}-2\partial_{\lambda}^{2}P_{0}\partial_{t}^{2}P_{0}(0)<0$

by the inequality in condition (iv) in Assumption 2.2. We may shrink $V$ if
necessary so that $\partial_{t}^{2}B(0,x,\xi)$ is bounded away from zero in
$V.$ Thus, given $(x,\xi)\in V,$

(4.3) $\mu(t,x,\xi)=\mbox{tr}\,B(t,x,\xi)\pm
t\left(\int_{0}^{1}(1-s)\partial_{t}^{2}\det B(st,x,\xi)\,ds\right)^{1/2},$

where $\partial_{t}^{2}B(t,x,\xi)<0$ is bounded away from zero in $[0,T]\times
V,$ if $T>0$ and $V$ are small enough.

Thus $\mu$ branches out of the imaginary axis, is differentiable in $t$ at
$(0,x,\xi),$ and splits into two eigenvalues $\mu^{\pm},$ with
$\partial_{t}\Im m\,\mu^{\pm}(0,x,\xi)\neq 0.$ The functions
$(x,\xi)\to\partial_{t}\mu^{\pm}(0,x,\xi)$ are continuous in $x$ and smooth in
$\xi,$ by condition on $\sigma:$ since $\sigma>3+d/2,$ the first and second
time derivatives of $P,$ hence $P_{0},$ at $t=0,$ are continuous in $x;$ the
regularity in $\xi$ comes from the smoothness in $\xi$ of $A$ and (4.3).

For $t>0,$ the eigenvalues $\mu^{\pm}(t,x,\xi)$ are simple hence as regular as
$\underline{A}.$ By condition $s>2+d/2,$ the time-derivative
$(\partial_{t}u,\partial_{t}\partial_{x}u)$ is continuous in $x,$ hence for
$t>0,$ $\partial_{t}\mu^{\pm}$ is continuous in $x;$ besides, it is also
smooth in $\xi$ just like $\underline{A}.$ ∎

Thus the spectral picture is as follows: for some $(x^{0},\xi^{0})$ in $U,$
the eigenvalues of $\underline{A}(t,x,\xi)$ for $(t,x,\xi)$ near
$(0,x^{0},\xi^{0})$ fall into two categories:

  * •

real eigenvalues with multiplicity one;

  * •

and eigenvalues which at $t=0$ and in a neighborhood $V$ of $(x^{0},\xi^{0}),$
are multiplicity-two and real, and which as $t$ goes from $0$ to $t>0$ all
simultaneously branch out of the real axis into simple eigenvalues with non-
zero (and opposite) imaginary parts which are time-differentiable, hence
$O(t).$

The smooth diagonalizability of $A$ at $t=0$ is assumed: this is condition
(ii) in Assumption 2.2. The discussion above shows that condition (iv) in
Assumption 2.2 implies furthermore the smooth diagonalizability of
$\underline{A}$ (defined in (4.1)) for small positive times. Indeed, the
bifurcating eigenvalues are semi-simple at $t=0$ and split into distinct
eigenvalues (see (4.2)) at $t>0.$ By continuity, these eigenvalues stay
distinct for small times. The non-bifurcating eigenvalues are simple at $t=0,$
hence stay simple for small times, by continuity. The diagonalization matrix
is also time-differentiable at $t=0:$

######  Lemma 4.2.

If the space-frequency domain $V$ and $T>0$ are small enough, then associated
with the bifurcation eigenvalues $\mu^{\pm}$ discussed in Lemma 4.1 we can
find eigenvectors $e^{\pm},$ which enjoy the same regularity property in
$(t,x,\xi)$ as $\mu^{\pm}.$

###### Proof.

The symbol $\underline{A}$ can be smoothly block diagonalized, for all times.
Consider $A_{0}$ a block of size two associated with a pair of bifurcating
eigenvalues as in Lemma 4.1. At $t=0,$ we have
$A_{0}(0,x,\xi)=\mu(0,x,\xi){\rm Id},$ since $\mu(0,x,\xi)$ is a double and
semi-simple eigenvalue of $A_{0}.$ Thus the matrix $A_{0}-\mu{\rm Id}$
vanishes at $t=0$ and is time-differentiable at $t=0$ (recall, $\underline{A}$
is $C^{1}$ in $t;$ the time regularity of $\mu$ was established in Lemma 4.1),
with a time derivative at $t=0$ which is continuous in $(t,x),$ and smooth in
$\xi.$ We may write $A_{0}-\mu{\rm
Id}=t\tilde{A}_{0}(t,x,\xi)=:t(\tilde{a}_{ij})_{1\leq i,j\leq 2}.$ If we had
$\tilde{a}_{12}(0,x,\xi)=0,$ then the eigenvalues of $\tilde{A}_{0}$ would
stay real. Indeed, those are

$\mu_{0}^{\pm}=\frac{1}{2}\mbox{tr}\tilde{A}_{0}\pm\frac{1}{2}\big{(}(\tilde{a}_{11}-\tilde{a}_{22})^{2}-4\tilde{a}_{12}\tilde{a}_{21}\big{)}^{1/2}.$

Thus $\tilde{a}_{12}(0,x,\xi)\neq 0,$ and we may assume that $V$ and $T$ are
small enough so that $|\tilde{a}_{12}(t,x,\xi)|$ is bounded away from 0 over
$[0,T]\times V.$ Then, the entries $(e_{1},e_{2})$ of an eigenvector of
$\tilde{A}_{0}$ take the form

$e_{2}=(\mu_{0}^{\pm}-\tilde{a}_{11})\tilde{a}_{12}^{-1}e_{1},$

which implies the announced regularity for the eigenvectors. ∎

###  4.2 Para-linearization and localization

The hyperbolic change of scales from Section 3.1 where we changed $t$ into
$2^{-j}t,$ and considered frequencies of size $2^{j},$ is not appropriate
here. We let instead

(4.4) $\tilde{u}(t,x)=u(2^{-j/2}t,x),$

and still consider frequencies of size $2^{j}.$ After para-linearization (see
Section 3.2), we arrive at

(4.5) $\left\\{\begin{aligned} \partial_{t}\tilde{u}+2^{j/2}{\rm
op}_{j}(iA)\tilde{u}&=-2^{-j/2}\big{(}{\rm
op}_{j}(\partial_{3}F)\tilde{u}+h_{j}^{-1}R^{para}_{j}\big{)},\\\
\tilde{u}(0)&=u_{in},\end{aligned}\right.$

where the principal symbol $A$ is evaluated at
$(2^{-j/2}t,x,\tilde{u},\partial_{x}\tilde{u}).$ Note that, in a difference
with Section 3.10, the leading operator ${\rm op}_{j}(iA)$ has a large
$2^{j/2}$ prefactor.

As in Section 3.2, the para-differential remainder $R^{para}_{j}$ belongs to
$H^{2(s-1)-d/2},$ uniformly in the small time interval under consideration,
and $h_{j}$ is defined in (3.1).

Here the $\partial_{3}F$ term coming from the para-linearization is not seen
as a perturbation of the principal symbol, but as a remainder. This is due to
the nature of our assumptions: even a small perturbation would here
significantly perturb the spectral picture of Section 4.1, since we assume the
eigenvalue coalesce. Just like $A,$ in (4.5), the function $\partial_{3}F$ is
evaluated at $(2^{-j/2}t,x,\tilde{u},\partial_{x}\tilde{u}).$

Next we use the same localization step as in Section 3.4. We define $v={\rm
op}_{j}(\psi)\tilde{u},$ with the same cut-off $\psi.$ This leads to

(4.6) $\left\\{\begin{aligned} \partial_{t}v+2^{j/2}{\rm op}_{j}(M)v&=g,\\\
v(0)&={\rm op}_{j}(\psi)u_{in},\end{aligned}\right.$

where the leading symbol $M$ is defined as

(4.7) $M=i\psi^{\sharp}A=i\psi^{\sharp}\underline{A}(2^{-j/2}t),$

(recall that $\underline{A}$ is defined in (4.1), and note the difference with
(3.23)), and the remainder term $g$ is defined as

(4.8) $\displaystyle g$ $\displaystyle:=-[{\rm op}_{j}(\psi),{\rm
op}_{j}(iA)]\tilde{u}-2^{-j/2}{\rm op}_{j}(\psi)\big{(}{\rm
op}_{j}(\partial_{3}F)\tilde{u}+h_{j}^{-1}R^{para}_{j}\big{)}$
$\displaystyle+{\rm op}_{j}(iA)\big{(}{\rm op}_{j}(\psi^{\sharp}){\rm
op}_{j}(\psi)-{\rm op}_{j}(\psi)\big{)}\tilde{u}+({\rm
op}_{j}(i\psi^{\sharp}A)-{\rm op}_{j}(iA){\rm op}_{j}(\psi^{\sharp})\big{)}v.$

Compared to the $g$ term from the proof of Theorem 1, we find an extra
$\partial_{3}F$ term here, a different prefactor in front of the para-
differential remainder, and $iA$ instead of $\tilde{A}.$

###  4.3 Diagonalization

We now diagonalize $M:$ by the results of Section 4.1, there is a family of
symbols $Q(t)$ of order 0, such that $Q(t,\cdot)$ has the same regularity as
$M,$ the symbol $Q$ is differentiable in $t,$ with $\partial_{t}Q(t,\cdot)$
continuous in $(x,\xi),$ and such that $QMQ^{-1}$ is diagonal. We let

(4.9) $w={\rm op}_{j}(\psi^{\flat}Q)v.$

Note that $Q$ is a priori only defined locally. This is an issue in view of
defining ${\rm op}_{j}(Q),$ above, and in the perspective of inverting the
change of variable (4.9). We can extend the locally defined symbol $Q$ into a
globally defined symbol which is order zero and uniformly invertible, so that
its inverse is also a symbol of order zero. This is described for instance in
Appendix C of [12]. In the following we identify $Q$ with its extension. We
observe that

(4.10) ${\rm op}_{j}(Q^{-1}){\rm op}_{j}(Q)={\rm Id}+2^{-j}R_{Q},$

since by assumption $s-1-d/2>1$ where $R_{Q}:=R_{1}(Q^{-1},Q),$ with notation
from Section A.5, is $L^{2}\to L^{2}$ bounded, uniformly in $j$ (see
(A.34)-(A.35)).

The system in $w$ involves the term

$\displaystyle{\rm op}_{j}(\psi^{\flat}Q)$ $\displaystyle{\rm op}_{j}(M)v$
$\displaystyle={\rm op}_{j}(\psi^{\flat}Q){\rm op}_{j}(M)\Big{(}{\rm
op}_{j}(Q^{-1}){\rm op}_{j}(Q)-2^{-j}R_{Q}\Big{)}v$ $\displaystyle=\Big{(}{\rm
op}_{j}(\psi^{\flat}QMQ^{-1})+2^{-j}R_{1}(\psi^{\flat}QM,Q^{-1})+2^{-j}R_{1}(\psi^{\flat}Q,M){\rm
op}_{j}(Q^{-1})\Big{)}{\rm op}_{j}(Q)v$ $\displaystyle+2^{-j}{\rm
op}_{j}(\psi^{\flat}Q){\rm op}_{j}(M)R_{Q}v.$

We further have

$\displaystyle{\rm op}_{j}(\psi^{\flat}QMQ^{-1}){\rm op}_{j}(Q)$
$\displaystyle=\Big{(}{\rm op}_{j}(QMQ^{-1}){\rm
op}_{j}(\psi^{\flat})+2^{-j}R_{1}(QMQ^{-1},\psi^{\flat})\Big{)}{\rm
op}_{j}(Q)$ $\displaystyle={\rm op}_{j}(QMQ^{-1}){\rm
op}_{j}(\psi^{\flat}Q)+2^{-j}{\rm op}_{j}(QMQ^{-1})R_{1}(\psi^{\flat},Q)$
$\displaystyle+2^{-j}R_{1}(QMQ^{-1},\psi^{\flat}){\rm op}_{j}(Q).$

Thus the system in $w$ appears as

(4.11) $\partial_{t}w+2^{j/2}{\rm op}_{j}(QMQ^{-1})w=g_{Q},$

with

(4.12) $g_{Q}:={\rm op}_{j}(\psi^{\flat}Q)g-2^{-j/2}{\rm
op}_{j}(\psi^{\flat}\partial_{t}Q)v+2^{-j/2}R(\psi^{\flat},M,Q,Q^{-1})v,$

where $R(\psi^{\flat},M,Q,Q^{-1})$ is the sum of all five remainders terms
involving $R_{1}$ terms that appear above. These terms are exactly the same
nature as the $R_{H}$ terms from (3.36): those are order-one remainders in the
composition of an order-one and an order-zero para-differential operator, with
$(t,x)$ regularity equal to the one of $(u,\partial_{x}u).$

The new remainder terms (the last two terms in $g_{Q}$ above) all involve $v.$
Hence they do not belong to the “out” category, but rather to the $G$ term
(see Section 3.8).

Finally, the new $\partial_{3}F$ term in $g$ is decomposed into an “in” and an
“out” term, as follows:

${\rm op}_{j}(\psi^{\flat}Q){\rm op}_{j}(\psi){\rm
op}_{j}(\partial_{3}F)\tilde{u}={\rm op}_{j}(\psi^{\flat}Q){\rm
op}_{j}(\partial_{3}F)v+{\rm op}_{j}(\psi^{\flat}Q)[{\rm
op}_{j}(\psi^{\flat}),{\rm op}_{j}(\partial_{3}F)]\tilde{u},$

and the first term in the above right-hand side is handled just like the $Gv$
terms in the proof of Theorem 1 (see Lemma 3.5), while the second term is
amenable to the analysis of Lemma 3.9 (it is actually easier to handle than
the term in the proof of Lemma 3.9, for the operator involved is order-zero,
and comes with a small $2^{-j/2}$ prefactor).

###  4.4 Bounds for the forward-in-time and backward-in-time solution
operators

Let $\mu$ be one branch of eigenvalues of $\underline{A},$ as described in
Section 4.1. The operator ${\rm op}_{j}(\mu)$ is $L^{2}\to L^{2}$ bounded,
hence has a solution operator $S(t^{\prime};t),$ for $0\leq t^{\prime}\leq t,$
defined as the map which takes a datum $z^{0}$ at time $t^{\prime}$ to $z(t),$
where $z(t)$ is the unique solution at time $t$ to the initial-value problem

$\partial_{t}z+2^{j/2}{\rm op}_{j}(i\mu)z=0,\qquad z(t^{\prime})=z^{0}.$

Note that it is indeed ${\rm op}_{j}(i\mu)$ that we should focus on (and not
${\rm op}_{j}(\mu)$), since $M$ is defined in (4.7) in terms of
$i\underline{A}.$

######  Lemma 4.3.

For $0\leq\tau\leq t,$ if the eigenvalue $\mu$ is real, then

(4.13) $\|S(\tau;t)\|_{L^{2}\to L^{2}}\lesssim 1.$

If $\mu$ is a bifurcating eigenvalue, then for any $\zeta_{-},\zeta_{+}$ such
that

$\zeta_{-}<|\partial_{t}\Im m\,\mu^{\pm}(0,x^{0},\xi^{0})|<\zeta_{+},$

the following holds true. If the support of $\psi$ is small enough, for
$0\leq\tau\leq t:$

(4.14) $\|S(\tau;t)\|_{L^{2}\to
L^{2}}\lesssim\exp\big{(}(t^{2}-\tau^{2})\zeta_{+}/2+C(t-\tau)2^{-j\theta_{\star}}\big{)},$

where $\theta_{\star}=\theta/(\theta+d_{\star}),$ with a constant $d_{\star}$
that depends only on $d.$ Besides, letting

(4.15) $t_{\star}:=\left(\frac{2j(\theta_{\star}\ln
2-\varepsilon)}{\zeta_{+}-\zeta_{-}}\right)^{1/2},$

with $\varepsilon>0$ arbitrarily small, we have, for $0\leq\tau\leq t\leq
t_{\star},$ if the support of $\tilde{\psi}$ is small enough, and if $j$ is
large enough depending in $Q,$ $d$ and $\varepsilon,$ if if $\mu$ is an
unstable bifurcating eigenvalue, meaning $\Im m\,\mu<0$ for $t>0,$ then we
have the backward-in-time decay bound:

(4.16) $\|{\rm op}(\tilde{\psi})\circ S(t;\tau)\|_{L^{2}\to
L^{2}}\lesssim\exp\big{(}-(t^{2}-\tau^{2})\zeta_{-}/2+C(t-\tau)2^{-j\theta_{\star}}\big{)}.$

Above in (4.13) and (4.16), the implicit constants do not depend on $t$ nor on
$j.$ The positive constant $C$ depends only on $F$ and the space dimension
$d,$

###### Proof.

As described in Section 4.1, under Assumption 2.2, the branch of eigenvalues
$\mu$ can either be real for all times, or branch out of the real axis as time
becomes positive.

If $\mu$ is real at all time, we perform a simple $L^{2}$ estimate:

$\displaystyle 2\Re e\,\big{(}{\rm op}_{j}(i\mu)z,z,\big{)}_{L^{2}}$
$\displaystyle=\big{(}({\rm op}_{j}(i\mu)+{\rm
op}_{j}(i\mu)^{\star})z,z\big{)}_{L^{2}}$ $\displaystyle=\big{(}{\rm
op}_{j}(i\mu-i\mu^{\star})z,z\big{)}_{L^{2}}+O(2^{-j\theta}\|z\|_{L^{2}}^{2}),$

where ${\rm op}_{j}(\mu)^{\star}$ denotes the $L^{2}$ adjoint of ${\rm
op}_{j}(\mu),$ and $\mu^{\star}$ the complex conjugate of $\mu.$ Indeed, we
have for adjoint operators a result analogous to the composition result of
Section A.5. With $\mu\in\mathbb{R}$ for all times, this gives

$\partial_{t}\|z\|_{L^{2}}^{2}\lesssim 2^{-j(\theta-1/2)}\|z\|_{L^{2}}^{2},$

and the bound (4.13) then follows from the assumed regularity of the solution.

If $\mu$ is a bifurcating eigenvalue, we use the description of Section 4.1:

(4.17)
$2^{j/2}\mu(2^{-j/2}t,x,\xi)=it\mu_{1}(2^{-j/2}t,x,\xi)+2^{j/2}\mu_{2}(2^{-j/2}t,x,\xi),\qquad\mu_{1}(0,x^{0},\xi^{0})\neq
0,$

where $\mu_{1}$ and $\mu_{2}$ are real. Let $\zeta_{\pm}$ such that

$\zeta_{-}<\mu_{1}(0,x^{0},\xi^{0})<\zeta_{+}.$

Then, if the support of $\psi$ is small enough, by Lemma 4.1, the spectral
bound (B.2) holds with $\gamma_{+}(t)=t\zeta_{+}.$ The upper bound (4.13) then
follows by Proposition B.1. If the support of $\tilde{\psi}$ is small enough,
the spectral bound (B.3) holds with $\gamma_{-}(t)=t\zeta_{-}.$ The backward-
in-time upper bound (4.16) then follows from Proposition B.1. ∎

###  4.5 Conclusion: non-existence of Sobolev solutions in the transition case

System (4.11) is a collection of differential equations that are decoupled to
first order. For each of those differential equations, we have solution
operators which satisfy the bounds of Lemma 4.3. We single out an eigenvalue
$\mu=\mu_{m}$ with $\Re e\,\mu_{m}<0$ for $t>0,$ such that $|\partial_{t}\Im
m\,\mu^{\pm}_{m}(0,x^{0},\xi^{0})|$ is maximal among the bifurcating
eigenvalues. The associated solution operator is $S_{m},$ and the associated
component of $w$ is $w_{m}.$ Then, we have a representation formula analogous
to (3.75):

(4.18) $\displaystyle w_{m}(0)$
$\displaystyle=\big{(}w_{m}(0)-S_{m}(t;0)w^{f}_{mk}(t)\big{)}+S_{m}(t;0)w_{m}(t)-S_{m}(t;0)G^{out}_{mk}(t)$
$\displaystyle-2^{-jk\theta}S_{m}(t;0)\int_{D_{k}(t)}S_{mk}(T_{k};t)v(t_{k})dT_{k},\qquad\mbox{for
any $k\geq 1.$}$

Above, the iterated operator $S_{mk}$ is defined in a way analogous to
$S_{Ek}$ and $S_{Hk}$ in the proof of Theorem 1. That is, we have
$S_{m1}=S_{m}G_{m},$ where $G_{m}={\rm op}(Q_{m})G,$ where $Q_{m}$ is the
projection onto the eigenspace spanned by $\mu_{m},$ and parallel to the
direct sum of the othe eigenspaces, and
$S_{m2}=S_{m1}\circ\sum_{m^{\prime}}S_{m^{\prime}1},$ the sum running over the
finite spectrum of $A.$ Above, $G$ comprises all the remainder terms that are
not “out”, after sorting out the remainder terms just like in Section 3.8. We
saw above in the last paragraph of Section 4.2 that no “out” terms were added
in the diagonalization procedure. The “free” component of the solution
$w^{f}_{mk}$ and the iterated “out” terms $G^{out}_{mk}$ are defined and
bounded just like in the proof of Theorem 1, using Lemma 4.3 instead of the
solution operator bounds of the proof of Theorem 1.

We choose the datum to be exactly like in the proof of Theorem 1 (see Section
3.7), with $P_{E}$ replaced by the projection over the eigenspace spanned by
$\mu_{m}.$

Using Lemma 4.3 and following the proof of Theorem 1, we obtain from (4.18)
the inequality

(4.19) $\displaystyle 2^{-j\sigma}(1+j)^{-1}\leq C\Big{(}$ $\displaystyle
2^{-j(\sigma+\theta^{\prime})}e^{t^{2}(\zeta_{+}-\zeta_{-})/2}+2^{-js}e^{-t^{2}\zeta_{-}/2}$
$\displaystyle+2^{-j(2s-3/2-d/2)}e^{t^{2}(\zeta_{+}-\zeta_{-})/2}+2^{-j(s+k)}e^{t^{2}(\zeta_{+}-\zeta_{-})/2}\Big{)}.$

The differences with (3.76) are in the rates of growth, now $O(t^{2}),$ and in
the leading term in the remainder, now with a $2s-3/2-d/2$ exponent instead of
$2s-1-d/2$ in (3.76). The loss of $2^{j/2}$ comes from the weakness of the
defect of hyperbolicity, with destabilizing eigenvalues which are $O(t),$
implying that the relevant time scale is $O(2^{j/2}).$

From (4.19), we conclude as in Section 3.16. The observation time, at which we
record the contradiction, is just a bit smaller than the maximal observation
time $t_{\star}$ (4.15). In particular, it is $O(\sqrt{j})$ in the rescaled
time frame, and $O(\sqrt{j2^{-j}})$ in the original time frame, much longer
than the instability time $O(j2^{-j})$ in the elliptic case.

##  Appendix A Pseudo- and para-differential symbols and operators

###  A.1 Symbols with limited spatial regularity

Given $k\in\mathbb{N},$ and $m\in\mathbb{R},$ the class $C^{k}S^{m}$ of
matrix-valued symbols with limited spatial regularity is defined as follows: a
map $a:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{N\times N}$ with
regularity $C^{k,\theta}$ in its first (spatial) variable and $C^{\infty}$ in
its second (frequency) variable is said to belong to $C^{k,\theta}S^{m}$ if,
for all $\alpha\in\mathbb{N}^{d}$ with $|\alpha|\leq k$ and all
$\beta\in\mathbb{N}^{d},$ there exists $C_{\alpha\beta}>0$ such that

(A.1)
$\langle\xi\rangle^{|\beta|-m}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leq
C_{\alpha\beta},\qquad\mbox{for all
$(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d},$}$

where $\langle\xi\rangle:=(1+|\xi|^{2})^{1/2}$ and $|\cdot|$ is some norm in
the finite-dimensional target space. Given $k\in\mathbb{N},$ $\theta\in(0,1)$
and $m\in\mathbb{R},$ the class $C^{k,\theta}S^{m}$ of matrix-valued symbols
is defined as the space of symbols which are $C^{k}$ in $x$ and $C^{\infty}$
in $\xi,$ and such that (A.1) hold, and, for all $|\alpha|=k,$

(A.2)
$\langle\xi\rangle^{|\beta|-m}\big{|}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)-\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x^{\prime},\xi)\big{|}\leq
C_{\alpha\beta}|x-x^{\prime}|^{\theta},\qquad\mbox{for all
$(x,x^{\prime},\xi)\in\mathbb{R}^{2d}\times\mathbb{R}^{d},$}$

We also encounter the class $C^{\infty}S^{m},$ defined as follows: a map
$\sigma:\mathbb{R}^{N^{\prime}}\times\mathbb{R}^{d}\to\mathbb{R}^{N\times N},$
for some $N,N^{\prime}\in\mathbb{N},$ $m\in\mathbb{R},$ is said to belong to
$C^{\infty}S^{m}$ if it is smooth in both its arguments, and satisfies the
bounds

(A.3)
$\langle\xi\rangle^{|\beta|-m}|\partial_{u}^{\alpha}\partial_{\xi}^{\beta}\sigma(u,\xi)|\leq
C_{\alpha\beta}(|u|),\qquad\mbox{for all
$(u,\xi)\in\mathbb{R}^{N}\times\mathbb{R}^{d},$}$

for all $(\alpha,\beta)\in\mathbb{N}^{N\times d},$ for some nondecreasing
function $C_{\alpha\beta}.$

In our context, symbols with limited spatial regularity $a\in
C^{k,\theta}S^{m}$ appear in the form

(A.4) $a(x,\xi)=\sigma(u(x),\xi),\qquad\mbox{where $u\in
H^{s}(\mathbb{R}^{d}),$ with $s>d/2.$}$

######  Lemma A.1.

Given $\sigma\in C^{\infty}S^{m},$ given $u\in H^{s}(\mathbb{R}^{d})$ with
$s>d/2,$ the symbol $a(x,\xi)=\sigma(u(x),\xi)$ belongs to
$C^{k,\theta}S^{m},$ with $k=[s-d/2]$ and $\theta=s-d/2-[s-d/2].$

###### Proof.

We use the Sobolev embedding into Hölder spaces:

(A.5) $H^{s}(\mathbb{R}^{d})\hookrightarrow
C^{[s-d/2],s-d/2-[s-d/2]}(\mathbb{R}^{d}).$

Given $0<|\alpha|\leq[s-d/2],$ by Faá di Bruno’s formula, we have the bound

$\langle\xi\rangle^{|\beta|-m}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\lesssim\langle\xi\rangle^{|\beta|-m}\sum_{\begin{smallmatrix}1\leq|\alpha^{\prime}|\leq|\alpha|\\\
\alpha_{1}+\dots+\alpha_{|\alpha^{\prime}|}=\alpha\end{smallmatrix}}\langle\xi\rangle^{|\beta|-m}\partial_{u}^{\alpha^{\prime}}\partial_{\xi}^{\beta}\sigma(u(x),\xi)\prod_{1\leq
k\leq|\alpha^{\prime}|}|\partial_{x}^{\alpha_{j}}u(x)|.$

Since every $\alpha_{j}$ above has length at most $[s-d/2],$ the maps
$\partial_{x}^{\alpha_{j}}u$ are bounded, hence the bound (A.1) follows from
the assumed bound (A.3).

Given $|\alpha|=[s-d/2],$ we bound
$\langle\xi\rangle^{|\beta|-m}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)-\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x^{\prime},\xi)|$
by the sum of terms of the form

(A.6)
$\langle\xi\rangle^{|\beta|-m}|\partial_{u}^{\alpha^{\prime}}\partial_{\xi}^{\beta}\sigma(u(x),\xi)-\partial_{u}^{\alpha^{\prime}}\partial_{\xi}^{\beta}\sigma(u(x^{\prime}),\xi)|\prod_{1\leq
k\leq|\alpha^{\prime}|}|\partial_{x}^{\alpha_{j}}u(x)|,$

with $|\alpha^{\prime}|\leq[s-d/2]$ and the $\alpha_{j}$ summing up to
$\alpha^{\prime},$ and terms of the form

(A.7)
$\langle\xi\rangle^{|\beta|-m}|\partial_{u}^{\alpha^{\prime}}\partial_{\xi}^{\beta}\sigma(u(x^{\prime}),\xi)|\left(\prod_{1\leq
k\leq|\alpha^{\prime}|}|\partial_{x}^{\alpha_{j}}u(x)|-\prod_{1\leq
k\leq|\alpha^{\prime}|}|\partial_{x}^{\alpha_{j}}u(x^{\prime})|\right).$

In (A.6), the product involving the $\partial_{x}^{\alpha_{j}}u$ is again
bounded. We use the Sobolev embedding (A.5): if $[s-d/2]>0,$ then $u$ is
Lipschitz, otherwise $u\in C^{0,\theta}.$ Since $\sigma$ is Lipschitz in $u,$
and $u$ is bounded, this gives an upper bound for (A.6) in the form
$C|x-x^{\prime}|^{\theta}.$

In (A.7), the factor involving $\sigma$ is bounded by (A.3). For the product
of derivatives of $u,$ we use (A.5) again: if $|\alpha_{j}|=[s-d/2]$ for some
$j,$ then the product in (A.7) has only one term, and by the Sobolev embedding
$\partial_{x}^{\alpha_{j}}u\in C^{0,\theta}.$ Otherwise, all indices
$\alpha_{j}$ have length strictly less than $[s-d/2],$ implying that all
factors in the product are actually Lipschitz and bounded. ∎

We verify that the assumption $\sigma\in C^{\infty}S^{m}$ implies a version of
the classical nonlinear Sobolev estimate:

######  Lemma A.2.

Let $\sigma\in C^{\infty}S^{m},$ such that for all $\xi\in\mathbb{R}^{d},$
$\sigma(0,\xi)=0.$ Then, for all $s\geq 0,$ for all $u\in H^{s}\cap
L^{\infty}(\mathbb{R}^{d}),$

$\|\sigma(u,\xi)\|_{H^{s}}\leq\langle\xi\rangle^{m}C(\|u\|_{L^{\infty}})\|u\|_{H^{s}},$

for some nondecreasing function $C$ which depends on $\sigma$ and $s$ but not
on $\xi.$

###### Proof.

We review the classical proof of the Sobolev estimate of smooth nonlinear
functions of Sobolev maps, and track the dependence in $\xi.$ We will see that
it suffices to assume that $\sigma$ has $[s]+2$ continuous derivatives with
respect to $u.$

Given $s\geq 0,$ $u\in H^{s}\cap L^{\infty},$ we let
$U_{k}=\phi_{0}(2^{-k}D)u,$ and $u_{k}=\phi_{k}(D)u.$ We have

(A.8) $|\partial_{x}^{\alpha}u_{k}|_{L^{2}}\leq
C_{\alpha}2^{k|\alpha|}|u_{k}|_{L^{2}},\quad|\partial_{x}^{\alpha}u_{k}|_{L^{\infty}}\leq
C_{\alpha}2^{k|\alpha|}|u|_{L^{\infty}},\quad|\partial_{x}^{\alpha}U_{k}|_{L^{\infty}}\leq
C_{\alpha}2^{k|\alpha|}|u|_{L^{\infty}}$

as a consequence of the classical Bernstein lemma. We also have, by definition
of $U_{k},$ for $k\geq 1:$

$\displaystyle\big{|}U_{k}-u\big{|}_{L^{2}}^{2}$
$\displaystyle=\int_{\mathbb{R}^{d}}|1-\phi_{0}(2^{-k}\xi)|^{2}|\hat{u}(\xi)|^{2}\leq\int_{|\xi|\geq\varepsilon_{1}2^{k}}|\hat{u}(\xi)|^{2}\,d\xi$
$\displaystyle\leq\varepsilon_{1}^{-2s}2^{-2ks}\int_{\mathbb{R}^{d}}|\xi|^{2s}|\hat{u}(\xi)|^{2}\,d\xi\leq\varepsilon_{1}^{-2s}2^{-2ks}|u|^{2}_{H^{s}},$

hence, Taylor expanding:

$\displaystyle\big{\|}\sigma(U_{k},\xi)-\sigma(u,\xi)\big{\|}_{L^{2}}$
$\displaystyle\leq\langle\xi\rangle^{m}\int_{0}^{1}\big{|}\partial_{u}\sigma(U_{k}+tu,\xi)\big{|}_{L^{\infty}}|U_{k}-u|_{L^{2}}\,dt$
$\displaystyle\leq\langle\xi\rangle^{m}C(|u|_{L^{\infty}})2^{-ks}\|u\|_{H^{s}},$

by assumption on $\sigma,$ where we used $|U_{k}+tu|_{L^{\infty}}\leq
C|u|_{L^{\infty}},$ which follows from (A.8).

From the above, we deduce that for fixed $\xi,$ the sequence
$(\sigma(U_{k},\xi))$ converges in $L^{2}.$ Thus the series
$(\sigma(U_{k},\xi)-\sigma(U_{k-1},\xi))$ converges in $L^{2},$ for fixed
$\xi,$ and we may write

$\sigma(u,\xi)=\sigma(u_{0},\xi)+\sum_{k\geq 1}m_{k}u_{k},$

with notation

$m_{k}=\int_{0}^{1}\partial_{u}\sigma(U_{k-1}+tu_{k},\xi)\,dt.$

By the first inequality in (A.8),

$|\partial_{x}^{\alpha}(m_{k}u_{k})|_{L^{2}}\leq
C_{\alpha}\sum_{\alpha_{1}+\alpha_{2}=\alpha}|\partial_{x}^{\alpha_{1}}m_{k}|_{L^{\infty}}2^{k|\alpha_{2}|}|u_{k}|_{L^{2}}.$

By the Faá di Bruno formula, we observe that
$|\partial_{x}^{\alpha_{1}}m_{k}|_{L^{\infty}}$ is bounded from above, up to a
multiplicative constant depending only on $|\alpha_{1}|,$ by the sum over
$(j,(\beta_{\ell})_{1\leq\ell\leq j})$ such that $1\leq j\leq|\alpha_{1}|$ and
$\sum_{\ell}\beta_{\ell}=\alpha_{1},$ of the terms

$\int_{0}^{1}\big{|}\partial_{u}^{1+j}\sigma(U_{k-1}+tu_{k},\xi)\big{|}_{L^{\infty}}\prod_{1\leq\ell\leq
j}|\partial_{x}^{\beta_{\ell}}(U_{k-1}+tu_{k})|_{L^{\infty}}\,dt.$

By the last two bounds in (A.8), and assumption (A.3) on $\sigma,$ we then
obtain

$|\partial_{x}^{\alpha_{1}}m_{k}|_{L^{\infty}}\leq\langle\xi\rangle^{m}C(|u|_{L^{\infty}})2^{k|\alpha_{1}|}.$

Since $u\in H^{s},$ it follows from the definition of the sequence $(u_{k})$
that $|u_{k}|_{L^{2}}\lesssim 2^{-ks}\zeta_{k},$ with
$|\zeta_{k}|_{\ell_{2}}\leq C_{s}|u|_{H^{s}}$ for some $C_{s}>0.$ Summing up,
we have

$|\partial_{x}^{\alpha}(m_{k}u_{k})|_{L^{2}}\leq
2^{k(|\alpha|-s)}\zeta_{k},\qquad|\zeta_{k}|_{\ell_{2}}\leq
C(|u|_{L^{\infty}})|u|_{H^{s}},\qquad|\alpha|\leq[s]+1.$

By a classical result of Littlewood-Paley theory (see for instance Proposition
4.1.13 in [4]), this implies the result. ∎

###  A.2 Pseudo- and para-differential operators

The pseudo-differential operator $\mathrm{pdo}(a)$ associated with symbol $a$
is defined by

$\mathrm{pdo}(a)u:=\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}a(x,\xi)\hat{u}(\xi)\,d\xi,\qquad
u\in{\mathcal{S}}(\mathbb{R}^{d}),$

where $\hat{u}$ is the Fourier transform of $u$ and
${\mathcal{S}}(\mathbb{R}^{d})$ is the Schwartz space of smooth and fast
decaying maps over $\mathbb{R}^{d}.$ Given $a\in C^{k}S^{m},$ we denote for
$k_{1}\leq k:$

(A.9)
$\|a\|_{m,k_{1},k^{\prime}}:=\sup_{\begin{smallmatrix}(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\end{smallmatrix}}\sup_{\begin{smallmatrix}|\alpha|\leq
k_{1}\\\ |\beta|\leq
k^{\prime}\end{smallmatrix}}\langle\xi\rangle^{|\beta|-m}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|.$

Given $a\in C^{k,\theta}S^{m},$ we denote

(A.10)
$\|a\|_{m,k+\theta,k^{\prime}}:=\|a\|_{m,k,k^{\prime}}+\sup_{\begin{smallmatrix}(x,x^{\prime},\xi)\in\mathbb{R}^{3d}\\\
x\neq x^{\prime}\end{smallmatrix}}\sup_{\begin{smallmatrix}|\alpha|=k\\\
|\beta|\leq
k^{\prime}\end{smallmatrix}}\langle\xi\rangle^{|\beta|-m}\frac{|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)-|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|}{|x-x^{\prime}|^{\theta}}$

We use a Littlewood-Paley decomposition $(\phi_{j})_{j\geq 0}$ (see the first
paragraph of Section 3.4). An admissible cut-off is
$\phi_{adm}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$ defined by111Note
the slight abuse of notation in the definition of $\phi_{adm}:$ the maps
$\phi_{j}$ are defined in terms of the cut-off
$\phi_{0}:\mathbb{R}^{d}\to\mathbb{R}_{+},$ but we evaluate $\phi_{j}$ at
$\langle\xi\rangle$ in $\phi_{adm}.$ We would actually need another
Littlewood-Paley decomposition
$\tilde{\phi}_{j}:\mathbb{R}_{+}\to\mathbb{R}_{+},$ and let
$\phi_{adm}(\eta,\xi)=\sum_{j\geq
0}\phi_{0}(2^{-j+N}\eta)\tilde{\phi}_{j}(\langle\xi\rangle).$ It is easy to
check that the abuse of notation in (A.11) is harmless.

(A.11) $\phi_{adm}(\eta,\xi)=\sum_{j\geq
0}\phi_{0}(2^{-j+N_{0}}\eta)\phi_{k}(\langle\xi\rangle),$

for $N_{0}\in\mathbb{N}$ with $N_{0}\geq 3.$ The admissible cut-off
$\phi_{adm}$ satisfies

(A.12) $\phi_{adm}(\eta,\xi)\equiv\left\\{\begin{aligned} 1,&\quad|\eta|\leq
2^{-N_{0}}\langle\xi\rangle,\\\ 0,&\quad|\eta|\geq
2^{-N_{0}}\langle\xi\rangle.\end{aligned}\right.$

Given $a\in C^{k,\theta}S^{m},$ we call para-differential symbol associated
with $a$ the symbol

$a_{\phi_{adm}}(x,\xi):={\mathcal{F}}^{-1}\Big{(}\phi_{adm}(\cdot,\xi)\hat{a}(\cdot,\xi)\Big{)}(x)=\Big{(}\big{(}{\mathcal{F}}^{-1}\phi_{adm}(\cdot,\xi)\big{)}\star
a(\cdot,\xi)\Big{)}(x),$

where convolution takes place in the spatial variable $x,$ and the smooth
function ${\mathcal{F}}^{-1}\phi_{adm}(\cdot,\xi)(x)$ is the inverse Fourier
transform of $\phi_{adm}$ in its first variable $\eta.$ The pseudo-
differential operator

(A.13) $\mathrm{pdo}(a_{\phi_{adm}})=:{\rm op}(a)$

is said to be the para-differential operator associated with $a$ in classical
quantization.

Thus $\mathrm{pdo}(a)$ refers to the pseudo-differential operator with symbol
$a,$ and ${\rm op}(a)$ refers to the para-differential operator (with respect
to some admissible cut-off) with symbol $a.$

Consider the maps $h_{j}$ introduced in (3.1):

$(h_{j}u)(x):=u(2^{-j}x),\qquad\mbox{and define}\qquad H_{j}:=2^{-jd/2}h_{j}.$

We denote $\mathrm{pdo}_{j}(a)$ the operator

$\mathrm{pdo}_{j}(a)u)(x):=\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}a(x,2^{-j}\xi)\hat{u}(\xi)\,d\xi,\qquad
a\in C^{k,\theta}S^{m},\quad u\in{\mathcal{S}}(\mathbb{R}^{d}).$

We observe that

(A.14) $\mathrm{pdo}_{j}(a)=H_{j}^{-1}\mathrm{pdo}(h_{j}a)H_{j},$

and define

(A.15) ${\rm op}_{j}(a):=H_{j}^{-1}{\rm op}(h_{j}a)H_{j}.$

Associated with the change of spatial scale $h_{j},$ we have weighted Sobolev
norms

(A.16) $\|u\|_{j,s}:=\|\langle
2^{-j}\xi\rangle^{s/2}\hat{u}(\xi)\|_{L^{2}(\mathbb{R}^{d}_{\xi})}.$

We sometimes denote $H^{s}_{j}$ the Sobolev space $H^{s}$ equipped with the
norm $\|\cdot\|_{j,s}.$

Given $a\in C^{k,\theta}S^{m}:$ if $k\geq 1+[d/2],$ the pseudo-differential
operator $\mathrm{pdo}_{j}(a)$ maps $L^{2}$ to $H^{-m};$ if $k\geq 0,$ the
para-differential operator ${\rm op}_{j}(a)$ maps $L^{2}$ to $H^{-m},$ and

(A.17) $\|\mathrm{pdo}(a)\|_{L^{2}\to
H_{j}^{-m}}\lesssim\|a\|_{m,1+[d/2],1+[d/2]},\qquad\|{\rm
op}_{j}(a)\|_{L^{2}\to H_{j}^{-m}}\lesssim\|a\|_{m,0,1+[d/2]},$

the implicit constant depending only on dimensions. For a proof of the
statement on pseudo-differential operators in (A.17), see for instance Theorem
1.1.4 and its proof in [10]; for a proof of the statement on para-differential
operators in (A.17), see for instance Theorem 4.3.5 in [4]. The extension to
semiclassical operators follow from the definitions (A.14) and (A.15), and the
property

(A.18) $\|H_{j}u\|_{H^{s}}=\|u\|_{j,s}.$

###  A.3 Regularized symbols

Let $a\in C^{0,\theta}S^{m},$ with $m\in\mathbb{R}$ and $0<\theta<1.$ We
regularize $a$ into $a^{\varepsilon}\in C^{\infty}S^{m},$ by spatial
convolution with a regularizing kernel
$k_{\varepsilon}(x)=\varepsilon^{-d}k(x/\varepsilon),$ with $k\in
C^{\infty}_{c}(\mathbb{R}^{d}),$ such that $k\geq 0$ and the integral of $k$
over $\mathbb{R}^{d}$ is equal to one. For all $f\in C^{0,\theta},$ we have

$\|f-k_{\varepsilon}\star f\|_{L^{\infty}}\leq
C_{\theta}\varepsilon^{\theta}\|f\|_{C^{0,\theta}},\qquad
C_{\theta}:=\int_{\mathbb{R}^{d}}|y|^{\theta}k(y)\,dy.$

We apply the above bound pointwise in $\xi$ to $\partial_{\xi}^{\alpha}a,$ for
any $\alpha,$ and obtain

(A.19)
$\langle\xi\rangle^{m-|\alpha|}\|\partial_{\xi}^{\alpha}(a-a^{\varepsilon})(\cdot,\xi)\|_{L^{\infty}}\leq
C_{\theta}\varepsilon^{\theta}\|a\|_{m,\theta,|\alpha|},$

where we used notation (A.10) for the symbolic norm. Applying the above to all
$|\alpha|\leq 1+[d/2],$ we deduce by (A.17) the bound

(A.20) $\|{\rm op}_{j}(a-a^{\varepsilon})\|_{H_{j}^{m}\to
L^{2}}\lesssim\varepsilon^{\theta}.$

Spatial derivatives of $k_{\varepsilon}$ bring out powers of
$\varepsilon^{-1}:$

(A.21) $\|\partial_{x}^{\beta}(k_{\varepsilon}\star f)\|_{L^{\infty}}\leq
C_{\beta}\varepsilon^{-|\beta|}\|f\|_{L^{\infty}},\qquad\mbox{for some
$C_{\beta}>0.$}$

By (A.21), norms of $a^{\varepsilon}$ are large in $\varepsilon:$

(A.22) $\|a^{\varepsilon}\|_{m,k,k^{\prime}}\lesssim\varepsilon^{-k},\qquad
k>0.$

For $a$ in the form (A.4), a variant on the above regularization procedure is
given by

(A.23) $a^{\varepsilon}(x,\xi):=\sigma(k_{\varepsilon}\star u,\xi).$

By regularity of $\sigma$ in $u,$ the above bounds (A.20) and (A.22) hold also
in this case.

###  A.4 Para-differential approximation to a semiclassical pseudo-
differential operator

For $a$ in the form (A.4), we can estimate how small the difference ${\rm
op}_{j}(a)-\mathrm{pdo}_{j}(a)$ is:

######  Proposition A.3.

Given $a$ in the form (A.4), the difference $\mathrm{pdo}_{j}(a)-{\rm
op}_{j}(a)$ maps the space $H^{m+d/2}(\mathbb{R}^{d})$ into
$L^{2}(\mathbb{R}^{d})$ and we have

(A.24) $\big{\|}\big{(}\mathrm{pdo}_{j}(a)-{\rm
op}_{j}(a)\big{)}v\big{\|}_{L^{2}}\lesssim
2^{-j(s-d/2)}C(|u|_{L^{\infty}})\|u\|_{H^{s}}\|v\|_{j,m+d/2}$

for all $v\in H^{m+d/2},$ where the weighted Sobolev norm
$\|\cdot\|_{j,m+d/2}$ is defined in (A.16), and $C$ is nondecreasing. If in
(A.4) the Sobolev index $s$ is strictly greater than the spatial dimension
$d,$ then we have for all $v\in H^{m}:$

(A.25) $\big{\|}\big{(}\mathrm{pdo}_{j}(a)-{\rm
op}_{j}(a)\big{)}v\big{\|}_{L^{2}}\lesssim
2^{-j(s-d)}C(|u|_{L^{\infty}})\|u\|_{H^{s}}\|v\|_{j,m}.$

The semiclassical estimate (A.24) is found in [16], based on the classical
version found in [8]. However, both proofs in [8] and [16] suffer from
(relatively minor) algebraic mistakes, which we correct here.

######  Remark A.4.

The proof of Proposition A.3 tells us a bit more. For any symbol $a$ of order
$m,$ with an adequate degree of spatial regularity, the difference
$\mathrm{pdo}(a)-{\rm op}(a)$ (classical quantization) maps $H^{m+d/2}$ to
$H^{s^{\prime}}$ with norm given by the upper bounds in (A.30) and (A.32):

$\displaystyle\|{\rm op}(a)-\mathrm{pdo}(a)\|_{H^{m+d/2}\to H^{s^{\prime}}}$
$\displaystyle\lesssim\sup_{\begin{smallmatrix}|\beta|\leq 2[d/2]+2\\\
\xi\in\mathbb{R}^{d}\end{smallmatrix}}\langle\xi\rangle^{|\beta|-m}\|\partial_{\xi}^{\beta}((1-\tilde{\phi}_{0}(D_{x}))a)\|_{H^{s^{\prime}}}$
$\displaystyle\quad+\|(1-\tilde{\phi}_{0}(D_{x}))\phi_{0}(2^{-N_{0}}D_{x})a(\cdot,\xi)\|_{m,1+[d/2],1+[d/2]}.$

It also maps $H^{m-m^{\prime}}$ to $L^{2}$ with norm given by the upper bounds
in (A.31) and (A.32):

$\displaystyle\|{\rm op}(a)-\mathrm{pdo}(a)\|_{H^{m-m^{\prime}}\to L^{2}}$
$\displaystyle\lesssim\sup_{\begin{smallmatrix}|\beta|\leq 2[d/2]+2\\\
\xi\in\mathbb{R}^{d}\end{smallmatrix}}\langle\xi\rangle^{|\beta|-m}\|\partial_{\xi}^{\beta}((1-\tilde{\phi}_{0}(D_{x}))a)\|_{H^{d/2+m^{\prime}}}$
$\displaystyle+\|(1-\tilde{\phi}_{0}(D_{x}))\phi_{0}(2^{-N_{0}}D_{x})a(\cdot,\xi)\|_{m,1+[d/2],1+[d/2]}.$

Above $\|\cdot\|_{m,k,k^{\prime}}$ is our notation for symbolic norms,
introduced in (A.9), and $\phi_{0}$ and $\tilde{\phi}_{0}$ are low-frequency
cut-offs that are identically equal to one in a neighborhood of $0.$ By
“adequate” degree of regularity, we mean any measure of regularity such that
the upper bounds are finite.

####  A.4.1 Proof of Proposition A.3

We prove here Proposition A.3, based on the results of [8].

First we compute

(A.26) $\sum_{0\leq j^{\prime}\leq
j_{0}}\phi_{j^{\prime}}=\phi_{0}+\phi_{1}+\dots+\phi_{j_{0}}=\phi_{0}(2^{-j_{0}}\xi),\qquad\mbox{for
all $j_{0}\in\mathbb{N}.$}$

where $(\phi_{j})_{j\geq 0}$ is the Littlewood-Paley decomposition introduced
in Section 3.4. Next we decompose any symbol $a$ into

$a(x,\xi)=\sum_{\begin{smallmatrix}p\geq 0\\\ q\geq
0\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi),$

and

$a(x,\xi)=\sum_{\begin{smallmatrix}|p-q|<N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi)+\sum_{\begin{smallmatrix}|p-q|\geq
N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi).$

The sum over $\\{(p,q)\in\mathbb{N}\times\mathbb{N},\quad|p-q|\geq N_{0}\\}$
decomposes into a sum over $p\geq N_{0}$ with $q\leq p-N_{0}$ and the
symmetrical sum:

$\displaystyle\sum_{\begin{smallmatrix}|p-q|\geq
N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi)=\sum_{\begin{smallmatrix}p\geq
N_{0}\\\ q\leq
p-N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi)+\sum_{\begin{smallmatrix}q\geq
N_{0}\\\ p\leq
q-N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi).$

We use (A.26) in order to express the second term in the above right-hand
side:

$\displaystyle\sum_{\begin{smallmatrix}q\geq N_{0}\\\ p\leq
q-N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi)$
$\displaystyle=\sum_{q\geq N_{0}}\Big{(}\sum_{0\leq p\leq
q-N_{0}}\phi_{p}(D_{x})\Big{)}a(\cdot,\xi)\phi_{q}(\xi)$
$\displaystyle=\sum_{q\geq
N_{0}}\phi_{0}(2^{-q+N_{0}}D_{x})a(\cdot,\xi)\phi_{q}(\xi).$

Thus by definition of the admissible cut-off $\phi_{adm}:$

$\sum_{\begin{smallmatrix}q\geq N_{0}\\\ p\leq
q-N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi)=\phi_{adm}(D,\cdot)a(x,\xi)-\sum_{0\leq
q\leq N_{0}-1}\phi_{0}(2^{-q+N_{0}}D_{x})a(\cdot,\xi)\phi_{q}(\xi).$

We have symmetrically

$\sum_{\begin{smallmatrix}p\geq N_{0}\\\ q\leq
p-N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi)=\sum_{p\geq
0}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{0}(2^{-p+N_{0}}\xi)-\sum_{0\leq p\leq
N_{0}-1}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{0}(2^{-p+N_{0}}\xi).$

In the end we obtain the decomposition

(A.27)
$a(x,\xi)=\phi_{adm}(D_{x},\xi)a(\cdot,\xi)(x)+a_{R}(x,\xi)+a_{2}(x,\xi)+a_{LF}(x,\xi),$

with notation

(A.28) $\left\\{\begin{aligned}
a_{R}(x,\xi)&:=\sum_{\begin{smallmatrix}|p-q|<N_{0}\end{smallmatrix}}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{q}(\xi),\\\
a_{2}(x,\xi)&:=\sum_{p\geq
0}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{0}(2^{-p+N_{0}}\xi),\\\
a_{LF}(x,\xi)&:=-\sum_{0\leq q\leq
N_{0}-1}\phi_{0}(2^{-q+N_{0}}D_{x})a(\cdot,\xi)\phi_{q}(\xi)-\sum_{0\leq p\leq
N_{0}-1}\phi_{p}(D_{x})a(\cdot,\xi)\phi_{0}(2^{-p+N_{0}}\xi).\end{aligned}\right.$

Decomposition (A.27) corrects decomposition (2.8) in [8], where the low-
frequency term $a_{LF}$ is missing.

By property (A.12) of $\phi_{adm},$ and definition of the low-frequency cut-
off $\phi_{0}$ in Section 3.4, we observe that

$\phi_{0}(2^{N_{0}}\varepsilon_{2}\eta))(1-\phi_{adm}(\eta,\xi))=0,\qquad\mbox{for
all $\eta,\xi,$}$

or, equivalently,

$1-\phi_{adm}(\eta,\xi)\equiv(1-\phi_{0}(2^{N_{0}}\varepsilon_{2}\eta))(1-\phi_{adm}(\eta,\xi)),\qquad\mbox{for
all $\eta,\xi.$}$

In particular, we may write (A.27) is the form

(A.29)
$a(x,\xi)=\phi_{adm}(D_{x},\xi)a(\cdot,\xi)(x)+(1-\tilde{\phi}_{0}(D_{x}))\big{(}a_{R}(\cdot,\xi)+a_{2}(\cdot,\xi)+a_{LF}(\cdot,\xi)\big{)}(x),$

with the notation