# The backscattering problem for time-dependent potentials

Medet Nursultanov  Department of Mathematics and Statistics, University of
Helsinki, Helsinki, and Institute of Mathematics and Mathematical Modeling,
Almaty, Kazakhstan  ,  Lauri Oksanen  Department of Mathematics, University of
Helsinki, Helsinki, Finland  and  Plamen Stefanov  Department of Mathematics,
Purdue University, West Lafayette, IN 47907, USA

###### Abstract.

We study the inverse backscattering problem for time-dependent potentials. We
prove uniqueness and Lipshitz stability for the recovery of small potentials.

###### Key words and phrases:

Inverse problems, scattering theory, time dependent potentials

###### 2010 Mathematics Subject Classification:

Primary: 35P25, 35R30 Secondary: 35L05, 47F05

##  1\. Introduction

Let $q(t,x)$ be smooth and supported in the cylinder $\mathbb{R}\times\Omega$,
where $\Omega\subset B(0,\rho):=\\{|x|<\rho\\}$ with some $\rho>0$ is a fixed
domain un $\mathbb{R}^{n}$. We study the inverse back-scattering problem for
the wave equation

(1)
$(\partial_{t}^{2}-\Delta+q(t,x))u=0,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{n},$

$n\geq 3$, odd. We show that small enough potentials $q$ are stably
recoverable from the data.

Results for stationary potentials $q(x)$ have been proven in [6, 5, 20, 17,
26]. Even though stability (say, of conditional Hölder type) has not been
stated explicitly there (see also [23] for a related result), it follows from
the fact that the linearization of the problem near $q=0$ is essentially the
Fourier transform of $q$, see, e.g., [20]. In terms of uniqueness, the best
known result is generic uniqueness so far.

The inverse problem of recovery of $q(t,x)$ from “near-field” scattering data,
closely related to the inverse scattering one but not restricted to back-
scattering, has been studied in [19, 15, 18, 27, 1, 10], and other works.
Uniqueness is known, for example for potentials supported in a cylinder as
above, with a tempered growth in $t$, as shown in [19]. One of the techniques
is to extract the light-ray transform from the data, which relies on forward
scattering, and invert it, see, e.g., [24] for an even more general situation.
That transform does not see timelike singularities however, see [21, 12, 22]
which makes it unstable. In view of that, the possibility of a stable recovery
of $q$ remained unclear. In [11], it was shown that a similar boundary value
problem, with inputs plane waves as below, and the output measured at a fixed
time $t=T$ in the whole $\mathbb{R}^{n}_{x}$, provides Lipschitz stable
recovery. The proof is based on Carleman estimates.

Even though forward propagating rays do not see all singularities, broken rays
reflecting from the interior could, at least on the principal level. Back-
scattering provides such a geometry, in particular. The main reason why one
can expect a stable recovery in this case is the following. Plane waves can
only possibly detect singularities conormal to them, which are lightlike,
indeed. On the other hand, a linearization of the backscattering data near
$q=0$ is an integral over the product of one such incoming and one outgoing
wave. That product, on the principal level, is supported on the intersection
of such two hyperplanes in timespace, which is a delta on a codimension two
(vs. one) hyperplane, see Figure 1, where it looks like a line. That
hyperplane has a richer subspace of conormals and can possibly detect non-
necessarily lightlike singularities. Varying the incident direction of the
incoming wave provides a complete set of conormals. We refer to the discussion
in section 3.3 as well.

We are restricted to small potentials, and as we pointed out already, even for
stationary $q(x)$, the uniqueness without that assumption is a well-known open
problem. It seems feasible that our methods could help prove local generic
uniqueness (and stability) in line with the stationary results in [5, 6, 20].

##  2\. Main Results

We describe the scattering amplitude for (1) briefly in order to formulate the
main theorem. In Appendix A, we review the scattering theory for (1) in more
detail.

We are sending waves $\delta(t+s-x\cdot\omega)$, where $s$ is a delay
parameter, $|\omega|=1$, and $t\ll 0$; let them propagate and scatter, and
measure them at infinity at directions $\omega^{\prime}$ and delay time
$s^{\prime}$. The scattering amplitude
$A^{\sharp}(s^{\prime},\omega^{\prime},s,\omega)$, see Definition A.2 and
Proposition A.1, measures the difference between the wave we sent and the
scattered one. Taking $\omega^{\prime}=-\omega$, we measure the response in
the direction opposite of the incoming one. If we have two potentials, $q_{1}$
and $q_{2}$, we denote the corresponding quantities by the subscripts $1$ and
$2$.

To state our main results, we introduce the change of variables

(2)
$\sigma=\frac{s-s^{\prime}}{2},\qquad\sigma^{\prime}=\frac{s+s^{\prime}}{2}.$

By $\tilde{A}_{1}^{\sharp}(\sigma^{\prime},\sigma,\omega)$ and
$\tilde{A}_{2}^{\sharp}(\sigma^{\prime},\sigma,\omega)$, we denote the
functions $A_{1}^{\sharp}(s^{\prime},-\omega,s,\omega)$ and
$A_{2}^{\sharp}(s^{\prime},-\omega,s,\omega)$ in the new variables. Our main
result is the following.

######  Theorem 2.1.

There exists $\varepsilon>0$ and $k>0$ such that if

$\|q_{1}\|_{C^{k}(\mathbb{R}\times\bar{\Omega})}<\varepsilon,\qquad\|q_{2}\|_{C^{k}(\mathbb{R}\times\bar{\Omega})}<\varepsilon,$

then the identity $\tilde{A}_{1}^{\sharp}=\tilde{A}_{2}^{\sharp}$ implies
$q_{1}=q_{2}$. Moreover, under the same assumptions on $q_{1}$, $q_{2}$, there
exists a constant $C_{\Omega}>0$ such that

$\|q_{1}-q_{2}\|_{L^{\infty}(\mathbb{R};\;L^{2}(\mathbb{R}^{n}))}\leq
C_{\Omega}\|\tilde{A}_{1}^{\sharp}-\tilde{A}_{2}^{\sharp}\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\mathbb{R}_{\sigma})))}.$

Note that we could use other norms above using complex interpolation under the
assumptions of the theorem but the price for that is to make the estimate of
conditional Hölder type, i.e., to have
$\|A_{1}^{\sharp}-A_{2}^{\sharp}\|^{\mu}$ above with some $\mu\in(0,1)$.

##  3\. Proofs

###  3.1. A pseudo-linearization identity

We review the scattering theory for time-dependent potentials in Appendix A
mostly following [3, 4, 19] with some additions as well. We sketch the main
notions below.

We send a plane wave $\delta(t+s-x\cdot\omega)$ to the perturbation, and let
it interact with the potential. More precisely, we are solving

(3) $(\partial_{t}^{2}-\Delta+q(t,x))u^{-}=0,\quad
u^{-}|_{t<-s-\rho}=\delta(t+s-x\cdot\omega).$

Then we set

(4) $u_{\textrm{sc}}^{-}=u^{-}-\delta(t+s-x\cdot\omega).$

The distribution $u_{\textrm{sc}}^{-}$ (which is actually a function, see
Proposition 3.2) would be automatically outgoing by Definition A.1, since it
vanishes for $t\ll 0$. Then we could compute the asymptotic wave profile
$u^{-,\sharp}_{\textrm{sc}}(s^{\prime},\omega^{\prime};s,\omega)$ of
$u_{\textrm{sc}}^{-}(t,x;s,\omega)$, which would give us the analog of the
scattering amplitude, see section A.4. As in the stationary case, we expect
this to be “essentially” the kernel of the scattering operator minus identity.
This is true, indeed, at least when the scattering operator exists as a
bounded one as we show in Theorem A.5. One defines the scattering amplitude
$A^{\sharp}(s^{\prime},\omega^{\prime};s,\omega)$ by canceling some constant
and ignoring some $s^{\prime}$ derivatives.

We need to define the time-reversed analog of $u^{-}$ above, which we will
denote by $u^{\textrm{+}}(t,x;s,\omega)$. It solves

(5) $(\partial_{t}^{2}-\Delta+q(t,x))u^{\textrm{+}}=0,\quad
u^{\textrm{+}}|_{t>-s+\rho}=\delta(t+s-x\cdot\omega).$

We want to warn the reader about a possible confusion caused by the terms
incoming/outgoing. The solution $u$ of (3), which we denote by $u^{-}$ below,
is the response to an incident plane wave and it is neither incoming nor
outgoing by Definition A.1. On the other hand,
$u_{\textrm{sc}}^{-}=u^{-}_{\textrm{sc}}$ is outgoing. Similarly, $u^{+}$ is
neither but $u^{+}_{\textrm{sc}}$, defined as in (4) but with $u$ replaced by
$u^{+}$, is incoming.

Let $q_{1}$ and $q_{2}$ be two such potentials, and denote the corresponding
quantities with subscripts $1$ and $2$. We have the following formula, proven
also in [25] for $n=3$, generalizing that in [20], where the potentials are
time-independent.

######  Proposition 3.1.

We have

(6)
$(A_{1}^{\sharp}-A_{2}^{\sharp})(s^{\prime},\omega^{\prime};s,\omega)=\int(q_{1}-q_{2})(t,x)u_{1}^{-}(t,x,s,\omega)u_{2}^{+}(t,x,s^{\prime},\omega^{\prime})\,dt\,dx,$

where $u_{1}^{-}$ solves (3) with $q=q_{1}$, and $u_{2}^{+}$ solves (5) with
$q=q_{2}$.

###### Proof.

Start with

(7)
$U_{1}(t,s)-U_{2}(t,s)=\int_{s}^{t}U_{1}(t,\sigma)(Q_{1}(\sigma)-Q_{2}(\sigma))U_{2}(\sigma,s)\,d\sigma,$

which can be obtained by applying the Fundamental Theorem of Calculus to
$F(\sigma)=U_{1}(t,\sigma)U_{2}(\sigma,s)$ in the interval $\sigma\in[s,t]$.
Apply $U_{0}(-s)\mathbf{f}$ on the right-hand, and take the (strong) limit
$s\to-\infty$ to get

(8)
$U_{1}(t,0)\Omega_{1,-}\mathbf{f}-U_{2}(t,0)\Omega_{2,-}\mathbf{f}=\int_{-\infty}^{t}U_{1}(t,\sigma)(Q_{1}(\sigma)-Q_{2}(\sigma))\left[U_{2}(\sigma,0)\Omega_{2,-}\mathbf{f}\right]\,d\sigma.$

For the left-hand side, and for the expression in the square brackets we will
apply Theorem A.5(a). We take the asymptotic wave profile of (8) next applying
Theorem A.4. Then (6) is just that expression, written as a composition of
Schwartz kernels, eventually applied to $\mathcal{R}\mathbf{f}$. ∎

###  3.2. Progressive wave expansion

We have the following progressive wave expansion. Let $h=:h_{0}$ be the
Heaviside function and set $h_{j}(\tau)=\tau^{j}/j!$ for $\tau>0$;
$h_{j}(\tau)=0$ for $\tau\leq 0$.

######  Proposition 3.2 ([19]).

For each integer $N\geq 0$ we have

(9)
$u(t,s,x,\omega)\sim\delta(t+s-x\cdot\omega)+\sum_{j=0}^{N}a_{j}(t,x,\omega)h_{j}(t+s-x\cdot\omega)+R_{N}(t,x,s,\omega),$

where

$\begin{split}a_{0}(t,x,\omega)&=-\frac{1}{2}\int_{-\infty}^{0}q(t+\tau,x+\tau\omega)\,d\tau,\\\
a_{j}(t,x,\omega)&=-\frac{1}{2}\int_{-\infty}^{0}(\Box+q)a_{j-1}(t+\tau,x+\tau\omega,\omega)\,d\tau,\quad
j=1,\dots,N,\end{split}$

and $R_{N}\in C(\mathbb{R}_{t}\times\mathbb{R}_{s}\times
S_{\omega}^{n-1};\;H^{N+1}(\mathbb{R}^{n}_{x}))$.

The latter statement follows from the fact that $R_{N}$ solves

$(\partial_{t}^{2}-\Delta+q)R_{N}=-[(\partial_{t}^{2}-\Delta+q)A_{N}]h_{N},\quad
R_{N}|_{t<-s-\rho}=0.$

We get the same expansion for $u^{+}$ but with a different remainder $R_{N}$.

###  3.3. Sketch of the main idea

Using Proposition 3.1, and keeping the most singular terms of $u_{1}^{-}$ and
$u_{2}^{+}$ only, we get

(10) $\begin{split}\delta
A^{\sharp}(s^{\prime},\omega^{\prime};s,\omega)&\sim\int\delta
q(t,x)\delta(t+s-x\cdot\omega)\delta(t+s^{\prime}-x\cdot\omega^{\prime})\,dt\,dx,\end{split}$

where $\delta A^{\sharp}$ and $\delta q$ are formal linearizations, while the
other two deltas above are Dirac deltas.

The product of the two deltas is a delta, with the coefficient
$2(4-(1+\omega\cdot\omega^{\prime})^{2})^{-1/2}$ on the $n-1$ dimensional
hyperplane (co-dimension $2$) given by the system

(11) $-t+x\cdot\omega=s,\quad-t+x\cdot\omega^{\prime}=s^{\prime}$

with $s$, $s^{\prime}$ parameters, assuming $\omega\not=\omega^{\prime}$,
i.e., staying away from the forward scattering directions. Its conormal bundle
is the span of $(-1,\omega)$ and $(-1,\omega^{\prime})$. Those are two
lightlike covectors, and all future pointing lightlike covectors look like
this. Taking linear combinations, and varying $\omega$ and $\omega^{\prime}$,
we get all covectors. So we are really inverting the $k=(n-1)$ – Radon
transform in $\mathbb{R}^{1+n}$ (by Helgason’s terminology [7]) over all
$k$-planes; and this is stably invertible. We must stay away from
$\omega=\omega^{\prime}$ though. On the other hand, the codimension two Radon
transform is overdetermined, so we do not need all of them, and we can avoid
the bad planes. Back-scattering only ($\omega=-\omega^{\prime}$) is one case
where this works.

Consider the back-scattering ($\omega^{\prime}=-\omega$) problem now. Then
(11) reduces to

(12) $-t+x\cdot\omega=s,\quad-t-x\cdot\omega=s^{\prime}$

which implies

(13) $x\cdot\omega=(s-s^{\prime})/2=\sigma,\quad
t=-(s+s^{\prime})/2=-\sigma^{\prime}.$

This is easy to visualize as two hyperplanes in time-space at angle 45 degrees
with the $t$-axis each, intersecting at a right angle, see Figure 1 below.

Figure 1. On the principal level, we integrate over the horizontal line in the
middle, which is a codimension two hyperplane, actually. The support of the
integrand, all terms considered, is inside the wedge, intersected with the
cylinder $|x|\leq\rho$.

Varying $\omega$, and $s,s^{\prime}$ so that $s+s^{\prime}$ is fixed, we get
integrals over all “lines” (codimension two hyperplanes, actually) on the
hyperplane $t=\text{const.}$, which is invertible, slice by slice. The
stability estimate we get however treats $t$ and $x$ differently, in
principle.

We need to analyze the lower order terms now. If we linearize near $q\not=0$,
and take the lower order term $a_{0}$ for $u_{1}^{-}$ in (9) into account
only, then (10) is the main term but we get the following additional terms:

$\begin{split}B_{1}(s^{\prime},\omega^{\prime};s,\omega)&:=\int\delta
q(t,x)a_{1}(t,x,\omega)h(t+s-x\cdot\omega)\delta(t+s^{\prime}-x\cdot\omega^{\prime})\,dt\,dx,\end{split}$

plus a similar $B_{2}$ term coming from $u_{1}^{+}$, plus an even more regular
term containing two Heaviside functions $h$. Each of $B_{1}$, $B_{2}$
integrates $\delta q$ over a (weighted) truncated delta on the hyperplane
$t+s^{\prime}-x\cdot\omega^{\prime}=0$, or $t+s-x\cdot\omega=0$, respectively,
see Figure 1. In other words, the Schwartz kernels are deltas on half-
hyperplanes. They can be thought of as a superposition of deltas on
codimension two hyperplanes, as in (10), all parallel to the intersection in
Figure 1, moving along the corresponding wing of the wedge there. We just need
upper bounds of $B_{1}$, $B_{2}$, and they can be done in the same norms as
those we use for (10) to get an $O(\varepsilon\|q\|)$ perturbation. The
$\varepsilon$ gain comes from $a_{1}$. We can treat the more regular terms,
coming from $a_{j}$, $j\geq 1$, and from $R_{N}$ in Proposition 3.2 similarly,
which is a tedious task but doable. An important observation is that the
support of the product $u_{1}^{-}u_{2}^{+}$ in (6) is contained in the
intersection of the half-space above the lower hyperplane (due to $u_{1}^{-}$,
see (61)) and the half-space below the upper hyperplane (due to $u_{1}^{+}$),
i.e., on the left of that the wedge in the figure. This is further intersected
with the cylinder $|x|\leq\rho$, so in particular, we integrate in (6) over a
compact set depending on the parameters.

###  3.4. Proof of the main result

Here, we will prove our main result. We begin with a few notations. Inspired
by (6), we set

$q(t,x)=q_{1}(t,x)-q_{2}(t,x)$

and

(14) $Mq(s^{\prime},s,\omega)=\int
q(t,x)u_{1}^{-}(t,x,s,\omega)u_{2}^{+}(t,x,s^{\prime},-\omega)\,dt\,dx.$

Writing $u_{1}^{-}(t,x,s,\omega)=\delta(t+s-x\cdot\omega)+u_{1,\rm sc}^{-}$,
$u_{2}(t,x,s,-\omega)=\delta(t+s+x\cdot\omega)+u_{2,\rm sc}^{+}$, we get

(15) $M=M_{00}+M_{01}+M_{10}+M_{11},$

where

(16) $\displaystyle M_{00}q(s^{\prime},s,\omega)$ $\displaystyle=\int
q(t,x)\delta(t+s-x\cdot\omega)\delta(t+s^{\prime}+x\cdot\omega)\,dt\,dx,$ (17)
$\displaystyle M_{10}q(s^{\prime},s,\omega)$ $\displaystyle=\int
q(t,x)u_{1,\rm
sc}^{-}(t,x,s,\omega)\delta(t+s^{\prime}+x\cdot\omega)\,dt\,dx,$ (18)
$\displaystyle M_{01}q(s^{\prime},s,\omega)$ $\displaystyle=\int
q(t,x)\delta(t+s-x\cdot\omega)u_{2,\rm
sc}^{+}(t,x,s^{\prime},-\omega)\,dt\,dx,$ (19) $\displaystyle
M_{11}q(s^{\prime},s,\omega)$ $\displaystyle=\int q(t,x)u_{1,\rm
sc}^{-}(t,x,s,\omega)u_{2,\rm sc}^{+}(t,x,s^{\prime},-\omega)\,dt\,dx.$

We denote by $\tilde{M}q$, $\tilde{M}_{kl}q$ the above functions in the
variables given by (2).

######  Remark 3.1.

Let

(20) $\Sigma=[-\rho,\rho].$

Note that if $\sigma\notin\Sigma$, then (13) does not hold for any
$x\in\Omega$ and $\omega\in\mathbb{S}^{n-1}$, and hence, the line of
intersection (a hyperplane) in Figure 1 does not intersect the cylinder
$\mathbb{R}\times\Omega$. Therefore,
$\tilde{M}_{00}q(\sigma^{\prime},\sigma,\omega)=0$ for $\sigma\notin\Sigma$.
If $\sigma\in\Sigma$, due to the definitions of $u_{1,sc}^{+}$ and
$u_{2,sc}^{+}$, it follows that there exists $T>0$ such that the
$(t,x)$-supports of the integrands in the definition of $\tilde{M}q$,
$\tilde{M}_{kl}q$ belongs to $(-T-\sigma^{\prime},T-\sigma)\times\Omega$.

Next, we will study $\tilde{M}_{kl}q$. We begin with $\tilde{M}_{00}q$:

######  Lemma 3.1.

Let $\Sigma=[-\rho,\rho]$. There exists a constant $C>0$ depending only on $n$
such that

(21)
$\|q\|_{L^{\infty}(\mathbb{R};\;L^{2}(\mathbb{R}^{n}))}/C\leq\|\tilde{M}_{00}q\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\Sigma_{\sigma})))}\leq
C\|q\|_{L^{\infty}(\mathbb{R};\;L^{2}(\mathbb{R}^{n}))}.$

###### Proof.

As explained in Section 3.3, the product of the deltas in (16) can also be
written as

$\delta\left(t+\frac{s+s^{\prime}}{2}\right)\delta\left(\frac{s-s^{\prime}}{2}-x\cdot\omega\right).$

Then

$M_{00}q(s^{\prime},s,\omega)=\left[Rq(-(s^{\prime}+s)/2,\cdot)\right]((s-s^{\prime})/2,\omega),$

where $Rf(p,\omega)=\int\delta(p-x\cdot\omega)f(x)\,dx$ is the Radon transform
of $f$. Using the change of variables given by (2), we rewrite

$\tilde{M}_{00}q(\sigma^{\prime},\sigma,\omega)=\left[Rq(-\sigma^{\prime},\cdot)\right](\sigma,\omega).$

For every $s$, we have the following stability estimate for the Radon
transform, see [14, Ch. 2]:

$\|f\|_{H^{s}(\mathbb{R}^{n})}/C\leq\|Rf\|_{L^{2}(\mathbb{S}_{\omega}^{n-1};H^{s+(n-1)/2}(\mathbb{R}_{\sigma}))}\leq
C\|f\|_{H^{s}(\mathbb{R}^{n})},$

for some constant $C>0$. Therefore, choosing $s=0$, we obtain

(22)
$\|q(-\sigma^{\prime},\cdot)\|_{L^{2}(\mathbb{R}^{n})}/C\leq\|\tilde{M}_{00}q(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\mathbb{R}_{\sigma}))}\leq
C\|q(-\sigma^{\prime},\cdot)\|_{L^{2}(\mathbb{R}^{n})}$

for every $\sigma^{\prime}$ with $C$ independent of it. Finally, as we noted
in Remark 3.1,

$\|\tilde{M}_{00}q\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\Sigma_{\sigma})))}=\|\tilde{M}_{00}q\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\mathbb{R}_{\sigma})))},$

and hence, (21) holds. ∎

We need the next lemmas to obtain similar results for $\tilde{M}_{10}q$ and
$\tilde{M}_{01}q$.

######  Lemma 3.2.

Let $\Sigma$ and $T$ be as in Remark 3.1. Let $a_{1,j}$, $h_{j}$, and
$R_{1,N}$ be the functions introduced in Section 3.2 with $q=q_{1}$ and let
$\bar{R}_{1,N}$ be the function such that

$\bar{R}_{1,N}(t,x,t+s-x\cdot\omega,\omega)=R_{1,N}(t,x,s,\omega).$

Then, there exists $C_{\Omega,N}>0$ such that

$\|\tilde{M}_{10}q\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\Sigma_{\sigma})))}\leq
C_{\Omega,N}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\mathbb{R}^{n}))}\times\\\
\times\left(\sum_{j=0}^{N}\|a_{1,j}\|_{L^{\infty}(\mathbb{R}\times\mathbb{S}^{n-1};C^{2n-2}(\bar{\Omega}))}+\sup_{\sigma^{\prime}\in\mathbb{R}}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\sup_{\omega\in\mathbb{S}^{n-1}}\|\bar{R}_{1,N}(t,\cdot,2t+2\sigma^{\prime},\omega)\|_{C^{2n-2}(\bar{\Omega})}dt\right).$

###### Proof.

For the sake of brevity, we denote

$U(t,x,t+s-x\cdot\omega,\omega)=\sum_{j=0}^{N}a_{1,j}(t,x,\omega)h_{j}(t+s-x\cdot\omega)+\bar{R}_{1,N}(t,x,t+s-x\cdot\omega,\omega).$

Then,

$\displaystyle M_{10}q(s^{\prime},s,\omega)$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}q(t,x)U(t,x,t+s-x\cdot\omega,\omega)\delta(t+s^{\prime}+x\cdot\omega)\,dt\,dx$
$\displaystyle=\int_{\mathbb{R}}\int_{\omega^{\perp}}q(t,-(t+s^{\prime})\omega+y)U(t,-(t+s^{\prime})\omega+y,2t+s+s^{\prime},\omega)\,dt\,dy.$

Using the change of variables given by (2), for any $\sigma\in\Sigma$, we
obtain

$\displaystyle\tilde{M}_{10}q(\sigma^{\prime},\sigma,\omega)$
$\displaystyle=\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\int_{\omega^{\perp}}q(t,-(t+\sigma^{\prime}-\sigma)\omega+y)U(t,-(t+\sigma^{\prime}-\sigma)\omega+y,2t+2\sigma^{\prime},\omega)\,dt\,dy$
$\displaystyle=\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\left[R_{U(t,\cdot,2t+2\sigma^{\prime},\cdot)}q(t,\cdot)\right](-t+\sigma-\sigma^{\prime},\omega)\,dt.$

By Theorem B.1,

$\|\tilde{M}_{10}q(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\Sigma_{\sigma}))}\\\
\leq
C_{\Omega}\sup_{t\in[-T-\sigma^{\prime},T-\sigma^{\prime}]}\|q(t,\cdot)\|_{L^{2}(\Omega)}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\sup_{\omega\in\mathbb{S}^{n-1}}\|U(t,\cdot,2t+2\sigma^{\prime},\omega)\|_{C^{2n-2}(\bar{\Omega})}dt.$

We estimate next

(23)
$\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\sup_{\omega\in\mathbb{S}^{n-1}}\|U(t,\cdot,2t+2\sigma^{\prime},\omega)\|_{C^{2n-2}(\bar{\Omega})}dt\leq
C_{N}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\sup_{\omega\in\mathbb{S}^{n-1}}\|a(t,\cdot,\omega)\|_{C^{2n-2}(\bar{\Omega})}dt\\\
+\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\sup_{\omega\in\mathbb{S}^{n-1}}\|\bar{R}_{1,N}(t,\cdot,2t+2\sigma^{\prime},\omega)\|_{C^{2n-2}(\bar{\Omega})}dt.$

The last two estimates complete the proof. ∎

######  Lemma 3.3.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded set, $\alpha$ be a multi-index,
and $K=|\alpha|+(n+1)/2$. Let $f\in
C^{N}(\mathbb{R}_{t}\times\mathbb{R}_{s}\times\mathbb{R}^{n}_{x})$ such that
$f(t,s,\cdot)$ has a compact support. Assume that $N\in\mathbb{N}$ is large
enough so that the equation

$\begin{cases}(\partial_{t}^{2}-\Delta+q(t,x))v(t,s,x)=f(t,s,x),\\\
v\arrowvert_{t<-s-\rho}=0.\end{cases}$

has the unique smooth (as much as needed) solution $v$. Then

(24) $\|D_{x}^{\alpha}v\|_{L^{\infty}(\Omega)}\leq
C_{\Omega,K}e^{C_{q}^{K}(t+s+\rho)}\int_{-s-\rho}^{t}\|f(\tau,s,\cdot)\|_{H^{K}(\mathbb{R}^{n})}d\tau,$

where

$C_{q}^{K}=2+C_{K}\|q(t,\cdot)\|_{C^{k}(\mathbb{R}^{n})}$

with some $C_{K}>0$ dependent on $K$.

###### Proof.

For non-negative $k\in\mathbb{Z}$, we define

$E_{s}^{k}(t)=\|\partial_{t}v(t,s,\cdot)\|_{H^{k}(\mathbb{R}^{n})}^{2}+\sum_{|\gamma|\leq
k}\|\nabla_{x}D_{x}^{\gamma}v(t,s,\cdot)\|_{(L^{2}(\mathbb{R}^{2}))^{n}}^{2}+\|v(t,s,\cdot)\|_{L^{2}(\mathbb{R}^{2})}^{2}.$

Then

$\partial_{t}E_{s}^{k}(t)=2\sum_{|\gamma|\leq
k}\Re(\partial_{t}^{2}D_{x}^{\gamma}v(t,s,\cdot),\partial_{t}D_{x}^{\gamma}v(t,s,\cdot))_{L^{2}(\mathbb{R}^{2})}\\\
-2\sum_{|\gamma|\leq k}\Re(\Delta
D_{x}^{\gamma}v(t,s,\cdot),\partial_{t}D_{x}^{\gamma}v(t,s,\cdot))_{L^{2}(\mathbb{R}^{2})}+2\Re(\partial_{t}v(t,s,\cdot),v(t,s,\cdot))_{L^{2}(\mathbb{R}^{2})},$

where $\Re$ denotes the real part of the number. Since

$(\partial_{t}^{2}-\Delta)D_{x}^{\gamma}v(t,s,x)+D_{x}^{\gamma}(q(t,x)v(t,s,x))=D_{x}^{\gamma}f(t,s,x),$

the last identity becomes

$\partial_{t}E_{s}^{k}(t)=2\sum_{|\gamma|\leq
k}\Re(D_{x}^{\gamma}f(t,s,\cdot),\partial_{t}D_{x}^{\gamma}v(t,s,\cdot))_{L^{2}(\mathbb{R}^{2})}\\\
-2\sum_{|\gamma|\leq
k}\Re(D_{x}^{\gamma}(q(t,\cdot)v(t,s,\cdot)),\partial_{t}D_{x}^{\gamma}v(t,s,\cdot))_{L^{2}(\mathbb{R}^{2})}+2\Re(\partial_{t}v(t,s,\cdot),v(t,s,\cdot))_{L^{2}(\mathbb{R}^{2})}.$

Using the Cauchy–Schwarz inequality, we obtain

$\partial_{t}E_{s}^{k}(t)\leq\sum_{|\gamma|\leq
k}\|D_{x}^{\gamma}f(t,s,\cdot)\|_{L^{2}(\mathbb{R}^{n})}^{2}+(2+C_{k}\|q(t,\cdot)\|_{C^{k}(\mathbb{R}^{2})})E_{s}^{k}(t).$

By integration over $(-s-\rho,t)$, we obtain

$E_{s}^{k}(t)\leq
F_{s}^{k}(t)+C_{q}^{k}\int_{-s-\rho}^{t}E_{s}^{k}(\tau)d\tau,$

where

$F_{s}^{k}(t)=\int_{-s-\rho}^{t}\|f(\tau,s,\cdot)\|_{H^{k}(\mathbb{R}^{n})}d\tau,\qquad
C_{q}^{k}=2+C_{k}\|q(t,\cdot)\|_{C^{k}(\mathbb{R}^{n})}.$

Then, the Grönwall’s inequality implies

$E_{s}^{k}(t)\leq
F_{s}^{k}(t)+C_{q}^{k}\int_{-s-\rho}^{t}F_{s}^{k}(\tau)e^{C_{q}^{k}(t-\tau)}d\tau.$

Since $F_{s}^{k}$ is an increasing function,

(25) $E_{s}^{k}(t)\leq
F_{s}^{k}(t)\left(1+C_{q}^{k}\int_{-s-\rho}^{t}e^{C_{q}^{k}(t-\tau)}d\tau\right)\leq
F_{s}^{k}(t)e^{C_{q}^{k}(t+s+\rho)}.$

Due to the Sobolev inequality, it follows that

$\|v(t,s,\cdot)\|_{C^{|\alpha|}(\bar{\Omega})}\leq
C_{\Omega,K}\|v(t,s,\cdot)\|_{H^{K+1}(\Omega)}\leq
C_{\Omega,K}\|v(t,s,\cdot)\|_{H^{K+1}(\mathbb{R}^{n})}.$

Combining this with (25), we obtain (24). ∎

######  Lemma 3.4.

Let $L=2n-2$. There exists a sufficiently large $N\in\mathbb{R}$ such that if

(26) $\|q_{1}\|_{C^{L+\frac{n-1}{2}+3+2N}(\mathbb{R}\times\bar{\Omega})}<1.$

Then,

$\sum_{j=0}^{N}\|a_{1,j}\|_{L^{\infty}(\mathbb{R}\times\mathbb{S}^{n-1};C^{2n-2}(\bar{\Omega}))}\leq
C_{\Omega,N}\|q_{1}\|_{C^{L+2N}(\mathbb{R}\times\bar{\Omega})}$

and

$\sup_{\sigma^{\prime}\in\mathbb{R}}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\sup_{\omega\in\mathbb{S}^{n-1}}\|\bar{R}_{1,N}(t,\cdot,2t+2\sigma^{\prime},\omega)\|_{C^{L}(\bar{\Omega})}dt\leq
C_{\Omega,N}\|q_{1}\|_{C^{L+\frac{n-1}{2}+3+2N}(\mathbb{R}\times\bar{\Omega})}.$

###### Proof.

The first estimate comes directly from the definition of $a_{1,j}$. It remains
to show the second estimate. Set

$A=\sup_{\sigma^{\prime}\in\mathbb{R}}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}\sup_{\omega\in\mathbb{S}^{n-1}}\|\bar{R}_{1,N}(t,\cdot,2t+2\sigma^{\prime},\omega)\|_{C^{L}(\bar{\Omega})}dt.$

We write

$\displaystyle A$
$\displaystyle=\sup_{\sigma^{\prime}\in\mathbb{R}}\int_{-T}^{T}\sup_{\omega\in\mathbb{S}^{n-1}}\|\bar{R}_{1,N}(t-\sigma^{\prime},\cdot,2t,\omega)\|_{C^{L}(\bar{\Omega})}dt$
$\displaystyle=\sup_{\sigma^{\prime}\in\mathbb{R}}\int_{-T}^{T}\sup_{\omega\in\mathbb{S}^{n-1}}\sum_{|\gamma|\leq
L}\sup_{x\in\Omega}\left|[D_{x}^{\gamma}\bar{R}_{1,N}](t-\sigma^{\prime},x,2t,\omega)\right|dt$
$\displaystyle=\sup_{\sigma^{\prime}\in\mathbb{R}}\int_{-T}^{T}\sup_{\omega\in\mathbb{S}^{n-1}}\sum_{|\alpha|+|\beta|\leq
L}\sup_{x\in\Omega}\left|[D_{x}^{\alpha}D_{s}^{|\beta|}R_{1,N}](t-\sigma^{\prime},x,2t-(t-\sigma^{\prime})+x\cdot\omega,\omega)\right|dt.$

Then, we estimate

$\displaystyle A$ $\displaystyle\leq\sum_{|\alpha|+|\beta|\leq
L}2T\sup_{\sigma^{\prime}\in\mathbb{R}}\sup_{\omega\in\mathbb{S}^{n-1}}\sup_{x\in\Omega}\sup_{t\in[-T,T]}\left|[D_{x}^{\alpha}D_{s}^{|\beta|}R_{1,N}](t-\sigma^{\prime},x,2t-(t-\sigma^{\prime})+x\cdot\omega,\omega)\right|$
(27) $\displaystyle\leq\sum_{|\alpha|+|\beta|\leq
L}2T\sup_{t\in\mathbb{R}}\sup_{\omega\in\mathbb{S}^{n-1}}\sup_{x\in\Omega}\sup_{s\in[-2T,2T]}\left|[D_{x}^{\alpha}D_{s}^{|\beta|}R_{1,N}](t,x,s-t+x\cdot\omega,\omega)\right|.$

To estimate the last term, let us fix multi-indexes $\alpha$, $\beta$ such
that $|\alpha|+|\beta|=L$ and note that

$(\partial_{t}^{2}-\Delta+q)D_{s}^{|\beta|}R_{N}(t,x,s,\omega)=-[(\partial_{t}^{2}-\Delta+q)a_{1,N}](t,x,\omega)h_{N}^{(|\beta|)}(s+t-x\cdot\omega).$

Moreover, by employing the derivative definition, it can be verified that

$D_{s}^{|\beta|}R_{N}(t,x,s,\omega)\arrowvert_{t<-s-\rho}=0.$

We denote

(28)
$A_{1,N}(\tau,x,\omega)=-[(\partial_{t}^{2}-\Delta+q)a_{1,N}](t,x,\omega),$

Next, we will show that

(29)
$f(t,s,x,\omega)=A_{1,N}(\tau,x,\omega)h_{N}^{(|\beta|)}(s+t-x\cdot\omega)$

has a compact support as a function of the $x$ variable. In [19], it was shown
that

$a_{1,N}(t,x,\omega)=0,\quad\text{for }x\cdot\omega<-\rho\text{ and for
}|x-x\cdot\omega|>\rho.$

Hence, if $x\cdot\omega\leq 0$, for sufficiently large $|x|$, it follows that
$a_{1,N}(t,x,\omega)=0$. If $x\cdot\omega>0$, then
$h_{N}^{(|\beta|)}(s+t-x\cdot\omega)=0$ for sufficiently large $|x|$.
Therefore, for fixed $t$, $s$, and $\omega$, the function given by (29) is
compactly supported. Therefore, by Lemma 3.3,

$\sup_{x\in\Omega}|D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}(t,x,s,\omega)|\\\ \leq
C_{\Omega,K}e^{C_{q}^{K}(t+s+\rho)}\int_{-s-\rho}^{t}\|A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(s+\tau-(\cdot)\cdot\omega)\|_{H^{K}(\mathbb{R}^{n})}d\tau,$

where

(30) $K=|\alpha|+\frac{n-1}{2}+1,\qquad
C_{q}^{K}=2+C_{K}\|q(t,\cdot)\|_{C^{K}(\mathbb{R}^{n})}.$

Since $h_{N}^{(|\beta|+k)}$ is an increasing function for all $k=0,\cdots,K$,
it follows that

$\sup_{x\in\Omega}|D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}(t,x,s,\omega)|\leq
C_{\Omega,K}e^{C_{q}^{K}(t+s+\rho)}(t+s+\rho)\\\
\times\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(s+t-(\cdot)\cdot\omega)\|_{H^{K}(\mathbb{R}^{n})}.$

This is true for all $s\in\mathbb{R}$. Hence, if we choose $y\in\Omega$ and
replace $s$ by $s-t+y\cdot\omega$, the last estimate becomes

$\sup_{x\in\Omega}|[D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}](t,x,s-t+y\cdot\omega,\omega)|\leq
C_{\Omega,K}e^{C_{q}^{K}(s+y\cdot\omega+\rho)}(s+y\cdot\omega+\rho)\\\
\times\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(s+y\cdot\omega-(\cdot)\cdot\omega)\|_{H^{K}(\mathbb{R}^{n})}.$

Let

$z=\sup_{s\in[-2T,2T]}\sup_{y\in\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}(s+y\cdot\omega).$

Note that $z$ is constant depending only on $\Omega$. Then, from the last
inequality, we derive

(31)
$\sup_{x,y\in\Omega}\sup_{t\in\mathbb{R}}\sup_{s\in[-2T,2T]}|[D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}](t,x,s-t+y\cdot\omega,\omega)|\leq
C_{\Omega,K}e^{C_{q}^{K}(z+\rho)}(z+\rho)\\\
\times\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(z-(\cdot)\cdot\omega)\|_{H^{K}(\mathbb{R}^{n})}.$

Moreover, we know that

$A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(z-(\cdot)\cdot\omega)$

has a compact support with respect to $x$, which is uniformly bounded in
$\tau\in\mathbb{R}$ and $\omega\in\mathbb{S}^{n-1}$, that is, there is a
compact set $\tilde{\Omega}$ such that

$\operatorname{supp}A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(z-(\cdot)\cdot\omega)\subset\tilde{\Omega}\qquad\text{for
all }\tau\in\mathbb{R}\text{ and }\omega\in\mathbb{S}^{n-1}.$

The set $\tilde{\Omega}$ depends only on $\Omega$. Therefore,

(32)
$\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(z-(\cdot)\cdot\omega)\|_{H^{K}(\mathbb{R}^{n})}\leq\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)h_{N}^{(|\beta|)}(z-(\cdot)\cdot\omega)\|_{H^{K}(\tilde{\Omega})}\\\
\leq
C_{\Omega,N}\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)\|_{H^{K}(\tilde{\Omega})}.$

Therefore, since

$\sup_{x\in\Omega}\sup_{t\in\mathbb{R}}\sup_{s\in[-2T,2T]}|[D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}](t,x,s-t+x\cdot\omega,\omega)|\\\
\leq\sup_{x,y\in\Omega}\sup_{t\in\mathbb{R}}\sup_{s\in[-2T,2T]}|[D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}](t,x,s-t+y\cdot\omega,\omega)|,$

from (31), (30), and (26), we obtain

$\sup_{x\in\Omega}\sup_{t\in\mathbb{R}}\sup_{s\in[-2T,2T]}\sup_{\omega\in\mathbb{S}^{n-1}}|[D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}](t,x,s-t+x\cdot\omega,\omega)|\\\
\leq
C_{\Omega,N}\sup_{\omega\in\mathbb{S}^{n-1}}\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)\|_{C^{K}(\tilde{\Omega})}.$

Since

(33)
$\sup_{\omega\in\mathbb{S}^{n-1}}\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)\|_{C^{K}(\tilde{\Omega})}\leq
C_{\Omega,N}\|q_{1}\|_{C^{K+2+2N}(\mathbb{R}\times\bar{\Omega})},$

the previous estimate implies

$\sup_{x\in\Omega}\sup_{t\in\mathbb{R}}\sup_{s\in[-2T,2T]}\sup_{\omega\in\mathbb{S}^{n-1}}|[D^{\alpha}_{x}D_{s}^{|\beta|}R_{1,N}](t,x,s-t+x\cdot\omega,\omega)|\leq
C_{\Omega,N}\|q_{1}\|_{C^{K+2+2N}(\mathbb{R}\times\bar{\Omega})}.$

Then, from (3.4), it follows that

$A\leq
C_{\Omega,N}\|q_{1}\|_{C^{L+\frac{n-1}{2}+3+2N}(\mathbb{R}\times\bar{\Omega})}.$

This completes the proof. ∎

Now, we are ready to estimate $M_{10}q$. Similarly, the same holds for
$M_{01}q$.

######  Lemma 3.5.

Let $\Sigma$ be as in (20). There exists a sufficiently large $k\in\mathbb{N}$
such that if

(34) $\|q_{1}\|_{C^{k}(\mathbb{R}\times\bar{\Omega})}\leq\varepsilon<1,$

then

(35)
$\|\tilde{M}_{10}q\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\Sigma_{\sigma})))}\leq\varepsilon
C_{\Omega}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\mathbb{R}^{n}))}.$

###### Proof.

Depending on $\Omega$, we choose $N\in\mathbb{N}$ large enough so that the
hypothesis of Lemma 3.4 is satisfied. Let $k=L+\frac{n-1}{2}+3+2N$. Then,
Lemma 3.4 and (34) give (35). ∎

Next, we obtain a similar result for $M_{11}q$.

######  Lemma 3.6.

Let $\Sigma$ be as in (20). There exists a sufficiently large $k\in\mathbb{R}$
such that if

(36) $\|q_{1}\|_{C^{k}(\mathbb{R}\times\bar{\Omega})}\leq\varepsilon<1,$

then

(37)
$\|\tilde{M}_{11}q\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\Sigma_{\sigma})))}\leq\varepsilon
C_{\Omega}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\mathbb{R}^{n}))}.$

###### Proof.

Let $N\in\mathbb{N}$ be sufficiently large. We set

$\displaystyle
Q_{j}^{k}(t,x,\omega)=q(t,x)a_{1,j}(t,x,\omega)a_{2,k}(t,x,-\omega),$
$\displaystyle Q_{j}(t,x,\omega)=q(t,x)a_{1,j}(t,x,\omega),$ $\displaystyle
Q^{k}(t,x,\omega)=q(t,x)a_{2,k}(t,x,-\omega).$

Next, we define

$\displaystyle
A_{kj}^{l}(\sigma^{\prime},\sigma,\omega)=\int_{\Omega}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}Q_{j}^{k}(t,x,\omega)h_{j}^{(l)}(t+\sigma^{\prime}+\sigma-x\cdot\omega)\delta(t+\sigma^{\prime}-\sigma+x\cdot\omega)dtdx,$
$\displaystyle
B_{kj}^{l}(\sigma^{\prime},\sigma,\omega)=\int_{\Omega}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}Q_{j}^{k}(t,x,\omega)\delta(t+\sigma^{\prime}+\sigma-x\cdot\omega)h_{k}^{(l)}(t+\sigma^{\prime}-\sigma+x\cdot\omega)dtdx,$
$\displaystyle
A_{k}^{l}(\sigma^{\prime},\sigma,\omega)=\int_{\Omega}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}Q^{k}(t,x,\omega)\partial_{\sigma}^{l}R_{1,N}(t,x,\sigma^{\prime}+\sigma,\omega)\delta(t+\sigma^{\prime}-\sigma+x\cdot\omega)dtdx,$
$\displaystyle
B_{j}^{l}(\sigma^{\prime},\sigma,\omega)=\int_{\Omega}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}Q_{j}(t,x,\omega)\delta(t+\sigma^{\prime}+\sigma-x\cdot\omega)\partial_{\sigma}^{l}R_{2,N}(t,x,\sigma^{\prime}-\sigma,\omega)dtdx$

and

$\displaystyle
A_{j}^{l_{1}l_{2}}(\sigma^{\prime},\sigma,\omega)=\int_{\Omega}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}Q_{j}(t,x,\omega)h_{j}^{(l_{1})}(t+\sigma^{\prime}+\sigma-x\cdot\omega)\partial_{\sigma}^{l_{2}}R_{2,N}(t,x,\sigma^{\prime}-\sigma,\omega)dtdx,$
$\displaystyle
B_{k}^{l_{1}l_{2}}(\sigma^{\prime},\sigma,\omega)=\int_{\Omega}\int_{-T-\sigma^{\prime}}^{T-\sigma^{\prime}}Q^{k}(t,x,\omega)\partial_{\sigma}^{l_{1}}R_{1,N}(t,x,\sigma^{\prime}+\sigma,\omega)h_{k}^{(l_{2})}(t+\sigma^{\prime}-\sigma+x\cdot\omega)dtdx,$
$\displaystyle
C^{l_{1}l_{2}}(\sigma^{\prime},\sigma,\omega)=\int_{\Omega}\int_{-T-\sigma^{\prime}}^{T+\sigma^{\prime}}q(t,x)\partial_{\sigma}^{l_{1}}R_{1,N}(t,x,\sigma^{\prime}+\sigma,\omega)\partial_{\sigma}^{l_{2}}R_{2,N}(t,x,\sigma^{\prime}-\sigma,\omega)dtdx.$

Then,

(38)
$\|\tilde{M}_{11}q(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\mathbb{S}^{n-1}_{\omega};H^{(n-1)/2}(\Sigma_{\sigma}))}\leq\sum
M_{kj}^{l_{1}l_{2}},$

where the sum is taking over all $j$, $k$, $l_{1}$, $l_{2}\leq N$ such that

$j,k\leq N,\quad l_{1}<j,\quad l_{2}<k,\quad l_{1}+l_{2}\leq\frac{n-1}{2}$

and

$\displaystyle M_{kj}^{l_{1}l_{2}}=$
$\displaystyle\|\partial_{\sigma}^{l_{1}}A_{kj}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}+\|\partial_{\sigma}^{l_{1}}B_{kj}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}$
$\displaystyle+\|\partial_{\sigma}^{l_{1}}A_{k}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}+\|\partial_{\sigma}^{l_{1}}B_{j}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}$
$\displaystyle+\|A_{j}^{l_{1}l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}+\|B_{k}^{l_{1}l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}+\|C^{l_{1}l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}.$

As we noted in the proof of Lemma 3.4, $\partial_{s}^{l_{1}}R_{1,N}$ satisfies

(39)
$(\partial_{t}^{2}-\Delta+q)D_{s}^{l_{1}}R_{N}(t,x,s,\omega)=-[(\partial_{t}^{2}-\Delta+q)a_{1,N}](t,x,\omega)h_{N}^{l_{1}}(s+t-x\cdot\omega)$

and

(40) $D_{s}^{l_{1}}R_{N}(t,x,s,\omega)\arrowvert_{t<-s-\rho}=0.$

Therefore, the same steps we used to prove Lemma 3.5, will give

(41)
$\|\partial_{\sigma}^{l_{1}}A_{kj}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}+\|\partial_{\sigma}^{l_{1}}B_{kj}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}\\\
+\|\partial_{\sigma}^{l_{1}}A_{k}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}+\|\partial_{\sigma}^{l_{1}}B_{j}^{l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{2}(\Sigma_{\sigma}\times\mathbb{S}_{\omega}^{n-1})}\leq\varepsilon
C_{\Omega}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\mathbb{R}^{n}))}$

for any $\sigma^{\prime}\in\mathbb{R}$.

Next, we estimate

$|B_{k}^{l_{1}l_{2}}(\sigma^{\prime},\sigma,\omega)|\leq\|a_{2,k}\|_{L^{\infty}(\mathbb{R}\times\Omega\times\mathbb{S}^{n-1})}\\\
\times\int_{\Omega}\|q(\cdot,x)\|_{L^{\infty}(\mathbb{R})}\int_{-T}^{T}|\partial_{\sigma}^{l_{1}}R_{1,N}(t-\sigma^{\prime},x,\sigma^{\prime}+\sigma,\omega)|h_{k}^{(l_{2})}(t-\sigma+x\cdot\omega)dtdx$

Due to Lemma 3.4, it follows

$\|B_{k}^{l_{1}l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{\infty}(\Sigma\times\mathbb{S}^{n-1})}\\\
\leq
C_{\Omega}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\Omega))}\sup_{\sigma\in\Sigma}\sup_{\omega\in\mathbb{S}^{n-1}}\sup_{x\in\Omega}\sup_{t\in[-T,T]}|\partial_{\sigma}^{l_{1}}R_{1,N}(t-\sigma^{\prime},x,\sigma^{\prime}+\sigma,\omega)|.$

To estimate the right-hand side, we repeat some steps of Lemma 3.4. Since
$\partial_{s}^{l_{1}}R_{1,N}$ satisfies (39) and (40), Lemma 3.3 gives

$\sup_{\sigma\in\Sigma}\sup_{t\in[-T,T]}|\partial_{\sigma}^{l_{1}}R_{1,N}(t-\sigma^{\prime},x,\sigma^{\prime}+\sigma,\omega)|\\\
\leq
C_{\Omega}\int_{-\sigma-\sigma^{\prime}-\rho}^{T-\sigma^{\prime}}\|A_{1,N}(\tau,\cdot,\omega)h_{N}^{l_{1}}(T+\sigma-(\cdot)\cdot\omega\|_{H^{K}(\mathbb{R}^{n})}d\tau,$

where $K=(n-1)/2+1$ and $A_{1,N}$ is the function defined by (28). Let

$z=\sup_{\sigma\in\Sigma}T+\sigma.$

Due to (32),

$\sup_{\sigma\in\Sigma}\sup_{t\in[-T,T]}|\partial_{\sigma}^{l_{1}}R_{1,N}(t-\sigma^{\prime},x,\sigma^{\prime}+\sigma,\omega)|\leq
C_{\Omega}\sup_{\tau\in\mathbb{R}}\|A_{1,N}(\tau,\cdot,\omega)\|_{H^{K}(\tilde{\Omega})}$

for some compact $\tilde{\Omega}$ which depends only on $\Omega$. Then, (33)
gives

$\sup_{\sigma\in\Sigma}\sup_{t\in[-T,T]}|\partial_{\sigma}^{l_{1}}R_{1,N}(t-\sigma^{\prime},x,\sigma^{\prime}+\sigma,\omega)|\leq
C_{\Omega}\|q_{1}\|_{C^{L+\frac{n-1}{2}+3+2N}(\mathbb{R}\times\bar{\Omega})}\leq\varepsilon
C_{\Omega},$

and hence,

$\|B_{k}^{l_{1}l_{2}}(\sigma^{\prime},\cdot,\cdot)\|_{L^{\infty}(\Sigma\times\mathbb{S}^{n-1})}\leq\varepsilon
C_{\Omega}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\Omega))}.$

Similarly, this estimate holds also for $A_{j}^{l_{1}l_{2}}$ and
$C^{l_{1}l_{2}}$. Hence, from (38) and (41), we obtain (37). ∎

Finally, we prove Theorem 2.1.

###### Proof of Theorem 2.1.

Let $C$ be a constant form Lemma 3.1 and $C_{\Omega}$ be a common constant
from Lemmas 3.5 and 3.6. Let us fix $0<\varepsilon<1$ such that
$1/C-3\varepsilon C_{\Omega}>0$. Next, we choose $k\in\mathbb{N}$ as large as
Lemmas 3.5 and 3.6 require. Then, under conditions
$\|q_{1}\|_{C^{k}(\mathbb{R}\times\bar{\Omega})}$,
$\|q_{1}\|_{C^{k}(\mathbb{R}\times\bar{\Omega})}<\varepsilon$, Lemmas 3.1,
3.5, and 3.6 imply

$\|\tilde{M}q\|_{L^{\infty}(\mathbb{R}_{\sigma^{\prime}};L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\Sigma_{\sigma})))}\geq\frac{1}{C}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\mathbb{R}^{n}))}-3\varepsilon
C_{\Omega}\|q\|_{L^{\infty}(\mathbb{R};L^{2}(\mathbb{R}^{n}))}.$

This completes the proof. ∎

######  Remark 3.2.

We want to emphasize on some subtle moment in the proof. The integration in
(6) happens inside the wedge in Figure 1, which, intersected with the cylinder
$|x|\leq\rho$, is compact. On the other hand, $M_{00}q$ integrates over the
“line segment” (a hyperplane) there only while the other integrals integrate
$q$ inside the whole wedge. In order to absorb the $\tilde{M}_{10}q$, etc.,
terms, we need them to be small in $q$, which they are but they depend on $q$
over a set larger than the one needed in $\tilde{M}_{00}q$. This arguments
still works because we actually extend the estimates to $q$ everywhere in the
$t$ variable by taking a supremum in $\sigma^{\prime}$ above. On the other
hand, if we wanted to establish local stability, like estimating $q$ for $t$
over a finite time interval having finite time back-scattering data, that
would have been be a problem.

##  Appendix A Scattering theory for time-dependent potentials

We recall the basics of the scattering theory for time-dependent perturbations
of the wave equation by restricting it to time-dependent potentials. We follow
Cooper and Strauss [3, 4], where it was introduced for moving obstacles, and
its adaptation to time-dependent potentials in [19]. Some of the statements
below are new however, like Theorem A.4 and Theorem A.5. This theory is a
natural extension of (a part of) the Lax-Phillips scattering theory. We
consider the wave equation (3) with a smooth time dependent potential $q(t,x)$
supported in the cylinder $\mathbb{R}\times\overline{B(0,R)}$. We assume
$n\geq 3$, odd to avoid working with the non-local translation representation
when $n$ is even.

###  A.1. Lax-Phillips formalism about the wave equation

The natural Cauchy problem for the wave equation is the following

(42) $(\partial_{t}^{2}-\Delta)u=0,\quad(u,u_{t})|_{t=0}=(f_{1},f_{2}).$

We convert the wave equation into a system by setting
$\mathbf{u}(t)=(u,u_{t})$; then

(43) $\partial_{t}\mathbf{u}=A\mathbf{u},\quad A:=\begin{pmatrix}0&\textrm{\rm
Id}\\\ \Delta&0\end{pmatrix}.$

We use boldface to denote vector-valued functions not necessarily of the type
$(u,u_{t})$ if there is no background scalar function $u(t,x)$ present. In
particular, $\mathbf{u}(t)$ in Definition A.1 below is not necessarily of that
form.

The natural energy space of states of finite energy is defined as the
completion of $C_{0}^{\infty}\times C_{0}^{\infty}$ under the energy norm

$\|\mathbf{f}\|^{2}_{\mathcal{H}}=\frac{1}{2}\int\left(|\nabla
f_{1}|^{2}+|f_{2}|^{2}\right)dx,\quad\mathbf{f}:=(f_{1},f_{2}).$

In particular, the first term defines the Dirichlet space
$H_{D}(\mathbb{R}^{n})$ with norm $\|\nabla f\|_{L^{2}}$. When $n\geq 3$, they
are locally in $L^{2}$, as it follows from the Poincaré inequality. The
operator $A$ naturally extends to a skew-selfadjoint one (i.e, $\mathrm{i}A$
is self-adjoint) on $\mathcal{H}$. Then by Stone’s theorem, $U_{0}(t)=e^{tA}$
is a well-defined strongly continuous unitary group, and the solution of (43)
is given by $\mathbf{u}(t)=U_{0}(t)\mathbf{f}$. The unitarity means energy
conservation, in particular.

We define the local energy space $\mathcal{H}_{\textrm{loc}}$ in the usual
way. By the finite speed of propagation, the Cauchy problem (42) has a well
defined solution in $\mathcal{H}_{\textrm{loc}}$ if the Cauchy data
$\mathbf{f}$ is in $\mathcal{H}_{\textrm{loc}}$ only. We view those solutions
as ones with (possibly) infinite energy but locally finite one. Then
$\mathbf{u}\in C(\mathbb{R};\;\mathcal{H}_{\textrm{loc}})$ and the wave
equation is solved in distribution sense. One can easily extend this to
distributions.

###  A.2. Existence of dynamics

By [9], see also [16, X.12], the solution to

(44) $(\partial_{t}^{2}-\Delta+q(t,x))u=0,\quad(u,u_{t})|_{t=s}=(f_{1},f_{2})$

is given by $\mathbf{u}(t)=U(t,s)\mathbf{f}$, where $\mathbf{f}=(f_{1},f_{2})$
and $U(t,s)$ is a two-parameter strongly continuous group of bounded operators
with the properties

  * (i)

$U(t,s)U(s,r)=U(t,r)$ for all $t,s,r$; and $U(t,t)=\textrm{\rm Id}$,

  * (ii)

$\|U(t,s)\|\leq\exp\left\\{C|t-s|\sup_{s\leq\tau\leq
t,\;x\in\mathbb{R}^{n}}|q(\tau,x)|\right\\}$,

  * (iii)

for any $\mathbf{f}\in D(A)$, we have $U(t,s)\mathbf{f}\in D(A)$ and

(45)
$\frac{d}{dt}U(t,s)\mathbf{f}=(A-Q(t))U(t,s)\mathbf{f},\quad\frac{d}{ds}U(t,s)\mathbf{f}=-U(t,s)(A-Q(s))\mathbf{f},$

where $Q(t)\mathbf{f}=(0,q(t,\cdot)f_{1})$ (and $Q(t)$ is clearly bounded).

The two-parameter semi-group admits the expansion

(46) $U(t,s)=U_{0}(t-s)+\sum_{k=1}^{\infty}V_{k}(t,s),$

where

$\begin{split}V_{k}(t,s)\mathbf{f}&=(-1)^{k}\int_{s}^{t}ds_{1}\int_{s}^{s_{1}}ds_{k}\dots\int_{s}^{s_{k-1}}ds_{k}\\\
&\qquad\times U_{0}(t-s_{1})Q(s_{1})\dots
U_{0}(s_{k-1}-s_{k})Q(s_{k})U_{0}(s_{k}-s)\mathbf{f},\quad k\gg 1.\end{split}$

This expansion is an iterated version of the Duhamel’s formula

(47)
$\begin{split}U(t,s)&=U_{0}(t-s)+\int_{s}^{t}U(t,\sigma)Q(\sigma)U_{0}(\sigma-s)\,d\sigma\\\
&=U_{0}(t-s)+\int_{s}^{t}U_{0}(t-\sigma)Q(\sigma)U(\sigma,s)\,d\sigma.\end{split}$

The convergence of (46) follows from the estimate

$\|V_{k}(t,s)\|\leq\frac{|t-s|^{k}}{k!}\left(\sup_{s\leq\tau\leq
t}\|Q(\tau)\|\right)^{k}.$

In particular, we get that we still have the finite speed of propagation
property:

$\operatorname{supp}U(t,s)\mathbf{f}\subset\operatorname{supp}\mathbf{f}+B(0,|t-s|).$

As before, the finite speed of propagation allows us to can extend $U(t,s)$ to
the space $\mathcal{H}_{\textrm{loc}}$ by a partition of unity.

Finally, notice that when $q$ is time independent, then $U(t,s)$ depends on
the difference $t-s$ only, i.e., $U(t,s)=U(t-s)$ where $U$ is a group. It is
not unitary however (unless $q=0$) in the space $\mathcal{H}$. If we redefine
the energy norm by

(48) $\|\mathbf{f}\|^{2}_{\mathcal{H}^{q}}=\int\left(|\nabla
f_{1}|^{2}+q|f_{1}|^{2}+|f_{2}|^{2}\right)dx,$

(we need to know that it is a norm however, and $q\geq 0$ suffices for that),
then $U(t)$ is unitary in $\mathcal{H}^{q}$.

###  A.3. Plane waves, translation representation and asymptotic wave profiles
of free solutions

The plane waves

$\delta(t-\omega\cdot x)$

solve the wave equation, obviously. They can be thought of as plane waves
propagating in the direction $\omega$ with speed one. If we replace $t$ by
$t+s$ there, we can think of $s$ as the delay time. The plane wave above is
the Schwartz kernel of the Radon transform

$Rf(s,\omega)=\int\delta(s-\omega\cdot
x)f(x)\,dx=\int_{x\cdot\omega=s}f(x)\,dS_{x}.$

For any density $g(\omega,s)$ (which can be a distribution as well), the
superposition

(49) $u(t,x):=\int_{\mathbb{R}\times S^{n-1}}\delta(t+s-\omega\cdot
x)g(s,\omega)\,ds\,d\omega=\int_{S^{n-1}}g(\omega\cdot x-t,\omega)\,d\omega$

is still a solution of the wave equation. The expression above can be
recognized as the the transpose $R^{\prime}$ of the Radon transform applied to
$g_{t}(s,\omega):=g(s-t,\omega)$. It turns out that all solutions of the free
wave equation in the energy space have that form.

Indeed, in [13], Lax and Phillips defined the free translation representation
$\mathcal{R}:\mathcal{H}\to L^{2}(\mathbb{R}\times S^{n-1})$ as follows

(50)
$k(s,\omega)=\mathcal{R}\mathbf{f}(s,\omega)=c_{n}(-\partial_{s}^{(n+1)/2}Rf_{1}+\partial_{s}^{(n-1)/2}Rf_{2}),$

where $R$ is the Radon transform and $c_{n}=2^{-1}(2\pi)^{(1-n)/2}$,
$c_{n}^{-}=2^{-1}(-2\pi)^{(1-n)/2}$. The inverse is given by

(51)
$\mathcal{R}^{-1}k(x)=2c_{n}^{-}\int_{S^{n-1}}\left(-\partial_{s}^{(n-3)/2}k(x\cdot\omega,\omega),\;\partial_{s}^{(n-1)/2}k(x\cdot\omega,\omega)\right)d\omega.$

The map $\mathcal{R}$ is unitary, and
$(\mathcal{R}U_{0}(t)\mathcal{R}^{-1}k)(s,\omega)=k(s-t,\omega)$, which
explains the name. We also set

(52) $u^{\sharp}(s,\omega)=(-1)^{(n-1)/2}k(s,\omega)$

and call $u^{\sharp}$ the asymptotic wave profile of the solution
$\mathbf{u}(t)=U_{0}(t)\mathbf{f}$. This name is justified by the theorem
below, and it is the analog of the far free pattern for solutions of the free
wave equation.

######  Theorem A.1 (Lax-Phillips, [13]).

Let $\mathbf{u}(t)=U_{0}(t)\mathbf{f}$, $\mathbf{f}\in\mathcal{H}$. Then

(53)
$\int\Big{|}u_{t}-|x|^{-(n-1)/2}u^{\sharp}\Big{(}|x|-t,\frac{x}{|x|}\Big{)}\Big{|}^{2}\,dx\to
0,\quad\text{as $|t|\to\infty$}.$

######  Remark A.1.

In [13], the factor $(-1)^{(n-1)/2}$ is missing from (52), i.e.,
$u^{\sharp}=k$. Cooper and Strauss in [3] found out that this factor must be
present in (52).

###  A.4. Outgoing solutions and their asymptotic wave profiles

We follow here [2, 3]. Given $u(t,x)$ (and only then), recall the notation
$\mathbf{u}(t):=(u(t,\cdot),u_{t}(t,\cdot))$, see (43).

######  Definition A.1.

The function $\mathbf{u}(t)\in C(\mathbb{R};\;\mathcal{H}_{\textrm{\rm loc}})$
is called outgoing if $\lim_{t\to-\infty}(\mathbf{u}(t),U_{0}(t)\mathbf{g})=0$
for each $\mathbf{g}\in C_{0}^{\infty}(\mathbb{R}^{n})\times
C_{0}^{\infty}(\mathbb{R}^{n})$.

In this definition, $\mathbf{u}(t)$ does not need to be a solution of the wave
equation (anywhere). On the other hand, if $u(t,x)$ solves the wave equation
in $|x|>\rho$ for some $\rho>0$, then, see [2, 3], $u$ is outgoing if and only
if for any $T\in\mathbb{R}$, $U_{0}(t-T)\mathbf{u}(T)=0$ in the forward cone
$|x|<t-T-\rho$.

One simple example of non-trivial outgoing solutions (for $|x|>\rho)$ is the
following. Let $p\in L^{1}_{\textrm{loc}}(\mathbb{R};\;L^{2}(\mathbb{R}^{n}))$
satisfy $p=0$ for $t<t_{0}$, where $t_{0}$ is fixed. Solve

(54) $(\partial_{t}^{2}-\Delta)u=p(t,x)\quad\text{in
$\mathbb{R}\times\mathbb{R}^{n}$}.$

with Cauchy data

$(u,u_{t})|_{t=t_{0}}=(0,0).$

By Duhamel’s formula,

$\mathbf{u}(t)=\int_{t_{0}}^{t}U_{0}(t-s)\mathbf{p}(s)\,ds,\quad\mathbf{p}(s):=(0,p(s,\cdot)).$

The latter is well-defined in $\mathcal{H}_{\textrm{loc}}$ by finite speed of
propagation. The solution for $t<t_{0}$ is just zero. Then $\mathbf{u}$ is
outgoing in a trivial way. Moreover, this is the unique outgoing solution of
(54). Indeed, take the difference $v$ of any two. Then
$\mathbf{v}(t)=U_{0}(t)\mathbf{f}$, where $\mathbf{f}$ is the initial
condition. Then
$0=\lim_{t\to-\infty}(\mathbf{v}(t),U_{0}(t)\mathbf{g})=(\mathbf{f},\mathbf{g})$,
for any test function $\mathbf{g}$; therefore, $\mathbf{f}=0$ and then
$\mathbf{v}=0$.

This can be generalized as follows.

######  Theorem A.2 ([2, 3]).

Let $p\in L^{1}_{\textrm{\rm loc}}(\mathbb{R};\;L^{2}(\mathbb{R}^{n}))$ and
assume that for each $t$,

(55)
$\lim_{T\to-\infty}\int_{T}^{t}U_{0}(-s)\mathbf{p}(s)\,ds\quad\text{exists in
$\mathcal{H}_{\textrm{\rm loc}}$},\quad\mathbf{p}(s):=(0,p(s,\cdot).$

Then there exists a unique outgoing solution $\mathbf{u}\in
C(\mathbb{R};\;\mathcal{H}_{\textrm{loc}})$ of (54) given by

$\mathbf{u}(t)=\int_{-\infty}^{t}U_{0}(t-s)\mathbf{p}(s)\,ds.$

######  Remark A.2.

Clearly, $p\in L^{1}((-\infty,a);\;L^{2}(\mathbb{R}^{n}))$ for any $a$ would
guarantee the regularity assumption on $p$ and (55). Also, the assumptions on
$p$ in next theorem are enough.

###### Proof.

The absolute convergence of the integral in ${H}^{0}_{\textrm{loc}}$ follows
from the assumptions. To show that $u$ is outgoing, for $\mathbf{g}\in
C_{0}^{\infty}\times C_{0}^{\infty}$, consider

$(\mathbf{u}(t),U_{0}(t)\mathbf{g})=\int_{-\infty}^{t}\left(U_{0}(t-s)\mathbf{p}(s),U_{0}(t)\mathbf{g}\right)ds=\int_{-\infty}^{t}\left(U_{0}(-s)\mathbf{p}(s),\mathbf{g}\right)ds.$

The latter converges to $0$, as $t\to-\infty$ by assumption. ∎

######  Theorem A.3 ([2, 3]).

Let $n\geq 3$ be odd. Let $p\in L^{1}_{\textrm{\rm
loc}}(\mathbb{R};\;L^{2}(\mathbb{R}^{n}))$ with $p(t,x)=0$ for $|x|>\rho$. Let
$u$ be the unique outgoing solution of (54).

(a) Then there is a unique function $u^{\sharp}\in L_{\textrm{\rm
loc}}^{2}(\mathbb{R}\times S^{n-1})$ such that for all $R_{1}<R_{2}$ we have

$\int_{R_{1}+t<|x|<R_{2}+t}\left|u_{t}(t,x)-|x|^{-(n-1)/2}u^{\sharp}\left(|x|-t,\frac{x}{|x|}\right)\right|^{2}dx\to
0,\quad\text{\rm as $t\to\infty$}.$

(b) If $p\in C_{0}^{\infty}$,

(56) $u^{\sharp}(s,\omega)=c_{n}^{-}\partial_{s}^{(n-1)/2}\int p(\omega\cdot
x-s,x)\,dx.$

(c) The map $p\to u^{\sharp}$ is continuous.

######  Remark A.3.

For general $p$ as in the theorem, $u^{\sharp}$ is still given by (56) but the
derivative is in distribution sense; by (b), the result is in $L_{\textrm{\rm
loc}}^{2}(\mathbb{R}\times S^{n-1})$. Another way to write (56) is

(57) $\int u^{\sharp}(s,\omega)\phi(s)\,ds=c_{n}\iint
p(t,x)(\partial_{s}^{(n-1)/2}\phi)(\omega\cdot
x-s,x)\,dt\,ds,\quad\forall\psi(s)\in C_{0}^{\infty}(\mathbb{R}).$

###### Proof.

Motivated by Theorem A.2, for fixed $R_{1}<R_{2}$, set

$\mathbf{f}=\int_{-R_{2}-\rho}^{-R_{1}+\rho}U_{0}(-\tau)\mathbf{p}(\tau)\,d\tau,\quad\mathbf{v}(t)=U_{0}(t)\mathbf{f}.$

By Huygens’ principle, $\mathbf{v}(t)=\mathbf{u}(t)$ for
$R_{1}+t<|x|<R_{2}+t$. Therefore, $\mathbf{v}$ does have an asymptotic wave
profile, and $v^{\sharp}(s,\omega)=u^{\sharp}(s,\omega)$ for $R_{1}<s<R_{2}$.
On the other hand, we have a formula for $v^{\sharp}$, (51) and (52) which
says

$\begin{split}v^{\sharp}(s,\omega)&=(-1)^{(n-1)/2}(\mathcal{R}\mathbf{f})(s,\omega)\\\
&=c_{n}^{-}\partial_{s}^{(n-1)/2}\int_{-R_{2}-\rho}^{-R_{1}+\rho}\int_{x\cdot\omega=s+\tau}p(x\cdot\omega-s,x)\,dS_{x}\,d\tau.\end{split}$

Then

$u^{\sharp}(s,\omega)|_{R_{1}<s<R_{2}}=v^{\sharp}(s,\omega)=c_{n}^{-}\partial_{s}^{(n-1)/2}\int
p(x\cdot\omega-s,x)\,dx.$

Since $R_{1}<R_{2}$ are arbitrary, this, combined with Theorem A.2, proves
(a); and (b) for $p\in C_{0}^{\infty}$.

The proof of (c) is straightforward: use (56) and take Fourier transform
w.r.t. $s$. In particular, we get that the map $p\to u^{\sharp}$ can be
extended continuously in those spaces. ∎

######  Remark A.4.

We call $u^{\sharp}$ in the theorem the asymptotic wave profile of the unique
outgoing solution of (54). Note that there are two cases where we defined such
profiles: for free solutions in the energy space, see Theorem A.2, and in
Theorem A.3 above, where $u$ is in the energy space locally only, and solves
(54) instead.

###  A.5. Scattering solutions

The scattering solutions $u^{-}$ and $u^{+}$ were introduced in section 3 as
the solutions of (3), and (5), respectively. Since they involve distributions,
not necessarily in the energy spaces (even locally), we proceed as follows. We
can think of $(u(t,x;s,\omega),u_{t}(t,x;s,\omega))$ as distribution in the
$(s,\omega)$ variables with values in $\mathcal{H}_{\textrm{loc}}$. It is more
convenient however to do the following. Let $h_{j}(t)=h(t)t^{j}/j!$,
$j=1,2\dots$, where $h$ is the Heaviside function; and we also set
$h_{-1}=\delta$ . Then $h_{j}^{\prime}=h_{j-1}$, $j=0,1,2,\dots$. To define
$u^{-}$ eventually, we solve

(58)
$(\partial_{t}^{2}-\Delta+q(t,x))\Gamma=0,\quad\Gamma|_{t<-s-\rho}=h_{1}(t+s-x\cdot\omega)$

first (notice that $h_{1}(t+s-x\cdot\omega)$ is locally in the energy space
now), set

(59) $\Gamma_{\textrm{sc}}=\Gamma-h_{1}(t+s-x\cdot\omega),$

compute the asymptotic wave profile
$\Gamma^{\sharp}(s^{\prime},\omega^{\prime};s,\omega)$ of
$\Gamma_{\textrm{sc}}$, and differentiate the result twice w.r.t. $s$ to get
the analog of the scattering amplitude. In particular, then

(60) $u(t,x;s,\omega)=\partial_{s}^{2}\Gamma(t,x;s,\omega),\quad
u_{\textrm{sc}}^{-}(t,x;s,\omega)=\partial_{s}^{2}\Gamma_{\textrm{sc}}(t,x;s,\omega).$

will be well defined as distributions.

In a similar way, one can construct the scattering solutions $u^{+}$ which
look like plane waves as $t\to+\infty$, instead of $t\to-\infty$. They would
solve (5). Performing the change of variables $\tilde{t}=-t$ (time reversal),
$\tilde{s}=-s$, $\tilde{\omega}=-\omega$, we see that
$u^{+}(t,x;s,\omega)=\tilde{u}^{+}(-t,x;-s,-\omega)$, where $\tilde{u}$ is
related to $\tilde{q}(t,x):=q(-t,x)$. The regularized version, $\Gamma^{+}$,
can be constructed as in (58) with the condition
$\Gamma^{+}=h_{1}(-t-s+x\cdot\omega)$ for $t>-s+\rho$. The right-hand side of
this condition then would be supported outside $\mathbb{R}\times B(0,R)$ for
$t>-s+\rho$. Then we define $u^{+}$ and $u^{+}_{\textrm{sc}}$ as in (60).

By the finite speed of propagation property

(61)
$\operatorname{supp}u^{-}(\cdot,\cdot;s,\omega)\subset\\{t+s-x\cdot\omega\geq
0\\},\quad\operatorname{supp}u^{+}(\cdot,\cdot;s,\omega)\subset\\{t+s-x\cdot\omega\leq
0\\}.$

Next theorem generalizes Theorem A.3.

######  Theorem A.4.

Let $p$ be as in Theorem A.3. Set

$\mathbf{u}(t):=\int_{-\infty}^{t}U(t,s)\mathbf{p}(s)\,ds.$

Then $\mathbf{u}\in C(\mathbb{R};\;\mathcal{H}_{\textrm{loc}})$ is outgoing,
and has has an asymptotic wave profile $u^{\sharp}(s,\omega)$ given by

(62) $u^{\sharp}(s,\omega)=c_{n}^{-}\partial_{s}^{(n-1)/2}\int
p(t,x)u^{+}(t,x;s,\omega)\,dt\,dx.$

###### Proof.

By (47),

(63)
$\begin{split}\mathbf{u}(t)&=\int_{-\infty}^{t}U_{0}(t-s)\mathbf{p}(s)\,ds+\int_{-\infty}^{t}\int_{s}^{t}U_{0}(t-\sigma)Q(\sigma)U(\sigma,s)\mathbf{p}(s)\,d\sigma\,ds\\\
&=\int_{-\infty}^{t}U_{0}(t-s)\mathbf{p}(s)\,ds+\int_{-\infty}^{t}\int_{-\infty}^{\sigma}U_{0}(t-\sigma)Q(\sigma)U(\sigma,s)\mathbf{p}(s)\,ds\,d\sigma.\end{split}$

Then we are in the situation of Theorem A.3 with $\mathbf{p}(t)$ there
replaced by

(64)
$\tilde{\mathbf{p}}(t):=\mathbf{p}(t)+\mathbf{p}_{1}(t),\quad\mathbf{p}_{1}(t):=Q(t)\int_{-\infty}^{t}U(t,s)\mathbf{p}(s)\,ds=Q(t)\mathbf{u}(t).$

Then $\mathbf{u}$ has an asymptotic wave profile
$u^{\sharp}(s^{\prime},\omega^{\prime})$ satisfying

(65)
$u^{\sharp}(s^{\prime},\omega^{\prime})=c_{n}^{-}\partial_{s}^{(n-1)/2}\int\tilde{p}(t,x)\delta(t+s^{\prime}-\omega^{\prime}\cdot
x)\,dx\,dt.$

The first term on the right-hand side of (63) is handled by Theorem A.3. We
analyze the second term below, which we call $\mathbf{u}_{1}(t)$. By Theorem
A.3 again, its asymptotic wave profile is given by

(66)
$\begin{split}u_{1}^{\sharp}(s^{\prime},\omega^{\prime})&=c_{n}^{-}\partial_{s}^{(n-1)/2}\int\Big{[}Q(t,x)\int_{-\infty}^{t}U(t,x;s,y)\mathbf{p}(s,y)\,ds\,dy\Big{]}_{2}\,{\delta}(t+s^{\prime}-\omega^{\prime}\cdot
x)\,dt\,dx\\\ &=c_{n}^{-}\partial_{s}^{(n-1)/2}\int
K(s^{\prime},\omega^{\prime};s,y)p(s,y)\,ds\,dy,\end{split}$

where the last identity defines $K$, i.e.,

(67) $K(s^{\prime},\omega^{\prime};s,y)=\int^{\infty}_{s}\int
q(t,x)U_{12}(t,x;s,y)\delta(t+s^{\prime}-\omega^{\prime}\cdot x)\,dx\,dt.$

By (45),

$(-\partial_{s}+A_{y}^{\prime}-Q^{\prime}(s))U^{\prime}(t,x;s,y)=0,$

where the primes denote transpose operators in distribution (not in energy
space) sense. This equality can be written also as

$\begin{pmatrix}-\partial_{s}&\Delta_{y}-q(s)\\\ \textrm{\rm
Id}&-\partial_{s}\end{pmatrix}U^{\prime}(t,x;s,y)=0.$

In particular,

(68) $(\partial_{s}^{2}-\Delta_{y}+q(s))U_{12}(t,x;s,y)=0.$

Differentiate $K$ in (67) to obtain

$\partial_{s}K(s^{\prime},\omega^{\prime};s,y)=\int^{\infty}_{s}\int
q(t,x)\partial_{s}U_{12}(t,x;s,y)\delta(t+s^{\prime}-\omega^{\prime}\cdot
x)\,dx\,dt$

because $U_{12}(s,x;s,y)=0$. Differentiate again:

$\begin{split}\partial_{s}^{2}K(s^{\prime},\omega^{\prime};s,y)&=\int^{\infty}_{s}\int
q(t,x)\partial_{s}^{2}U_{12}(t,x;s,y)\delta(t+s^{\prime}-\omega^{\prime}\cdot
x)\,dx\,dt\\\ &\qquad-q(s,x)\delta(s+s^{\prime}-\omega^{\prime}\cdot
y).\end{split}$

Then by (68) and (67),

(69)
$(\partial_{s}^{2}-\Delta_{y}+q(s))K(s^{\prime},\omega^{\prime};s,y)=-q(s,x)\delta(s+s^{\prime}-\omega^{\prime}\cdot
y).$

On the support of the integrand in (67), we have $t+s^{\prime}<\rho$, $s<t$.
Therefore,

(70) $K(s^{\prime},\omega^{\prime};s,y)|_{s>-s^{\prime}+\rho}=0.$

Therefore, $K$ solves (69), (70), which is the same problem solved by
$u^{+}_{\textrm{sc}}(s^{\prime},\omega^{\prime};s,y)$, see (5). Therefore,
$K=u^{+}_{\textrm{sc}}$.

Going back to (65) and (64), we see that

(71)
$\begin{split}u^{\sharp}(s^{\prime},\omega^{\prime})&=c_{n}^{-}\partial_{s}^{(n-1)/2}\int\left(p(t,x)+p_{1}(t,x)\right)\delta(t+s^{\prime}-\omega^{\prime}\cdot
x)\,dx\,dt\\\ &=c_{n}^{-}\partial_{s}^{(n-1)/2}\int
p(t,x))\left(\delta(t+s^{\prime}-\omega^{\prime}\cdot
x)+u^{+}_{\textrm{sc}}(s^{\prime},\omega^{\prime};t,x)\right)\,dx\,dt\\\
&=c_{n}^{-}\partial_{s}^{(n-1)/2}\int
p(t,x))u^{+}(s^{\prime},\omega^{\prime};t,x)\,dx\,dt,\end{split}$

where we used (66) and the identity $K=u^{+}_{\textrm{sc}}$ we just derived. ∎

###  A.6. The scattering amplitude and the scattering kernel

Let $\Gamma$ solve (58). Since the Cauchy data
$(h_{1}(t+s-x\cdot\omega),h_{0}(t+s-x\cdot\omega))$, for say, $t=-s-\rho-1$,
is in $\mathcal{H}_{\textrm{loc}}$, a solution $(\Gamma,\Gamma_{t})$ with
locally finite energy exists. Then $\Gamma_{\textrm{sc}}$ is clearly outgoing.
It solves the Cauchy problem

(72)
$(\partial_{t}^{2}-\Delta)\Gamma_{\textrm{sc}}=-q\Gamma,\quad\Gamma_{\textrm{sc}}|_{t<-s-\rho}=0.$

By Theorem A.3, $\Gamma_{\textrm{sc}}$ has an asymptotic wave profile
$\Gamma_{\textrm{sc}}^{\sharp}$ given by

$\begin{split}\Gamma_{\textrm{sc}}^{\sharp}(s^{\prime},\omega^{\prime};s,\omega)&=-c_{n}^{-}\partial_{s^{\prime}}^{(n-1)/2}\int
q(x\cdot\omega^{\prime}-s^{\prime},x)\Gamma(x\cdot\omega^{\prime}-s^{\prime},x;s,\omega)\,dx\\\
&=-c_{n}^{-}\partial_{s^{\prime}}^{(n-1)/2}\int
q(t,x)\Gamma(t,x;s,\omega)\delta(t+s^{\prime}-x\cdot\omega^{\prime})\,dt\,dx.\end{split}$

Differentiate twice w.r.t. $s$, see (60), to get

$u^{-,\sharp}_{\textrm{sc}}(s^{\prime},\omega^{\prime};s,\omega)=-c_{n}^{-}\partial_{s^{\prime}}^{(n-1)/2}\int
q(t,x)u^{-}(t,x;s,\omega)\delta(t+s^{\prime}-x\cdot\omega^{\prime})\,dt\,dx.$

######  Definition A.2.

The scattering amplitude $A^{\sharp}$ is given by

(73) $A^{\sharp}(s^{\prime},\omega^{\prime};s,\omega)=\int
q(t,x)u^{-}(t,x;s,\omega)\delta(t+s^{\prime}-x\cdot\omega^{\prime})\,dt\,dx,$

where $u^{-}$ solves (3).

By the finite speed of propagation, $u^{-}(t,x;s,\omega)=0$ for
$x\cdot\omega>t+s$. Therefore, the integrand vanishes outside of the region
$x\cdot(\omega-\omega^{\prime})\leq s-s^{\prime}$. The l.h.s. has a lower
bound $-2\rho$ on $\operatorname{supp}q$; therefore,

(74) $\operatorname{supp}A^{\sharp}\subset\\{s^{\prime}\leq
s+\rho|\omega-\omega^{\prime}|\\}\subset\\{s^{\prime}\leq s+2\rho\\}.$

Note that $A^{\sharp}$ and
$u^{\sharp}_{\textrm{sc}}=-c_{n}^{-}\partial_{s^{\prime}}^{(n-1)/2}A^{\sharp}$
can be reconstructed from each other thanks to that support property.

Since the perturbed dynamics is a two-parameter group, we need to generalize
the notion of the wave operators and the scattering scattering operator.

######  Definition A.3.

The wave operators $\Omega_{-}$ and $W_{+}$ in $\mathcal{H}$ are defined as
the strong limits

$\Omega_{-}=\textrm{s\,-}\lim_{t\to-\infty}U(0,t)U_{0}(t),\quad
W_{+}\mathbf{f}=\lim_{t\to\infty}U_{0}(-t)U(t,0)\mathbf{f};\quad\mathbf{f}\in\textrm{\rm
Ran}\;\Omega_{-},$

if they exist and define continuous operators. In the latter case, the
scattering operator $S$ is defined by

$S=W_{+}\Omega_{-}.$

This definition also makes sense for
$\mathbf{f}\in\mathcal{H}_{\textrm{comp}}$ with $S\mathbf{f}$ taking values
possibly in $\mathcal{H}_{\textrm{loc}}$.

######  Theorem A.5.

(a) The wave operator $\Omega_{-}:\mathcal{H}_{\textrm{\rm
comp}}\to\mathcal{H}$ exists and

(75) $U(t,0)\Omega_{-}\mathbf{f}=2c_{n}^{-}\int_{\mathbb{R}\times
S^{n-1}}\boldsymbol{u}^{-}(t,x;s,\omega)\partial_{s}^{(n-3)/2}(\mathcal{R}\boldsymbol{f})(s,\omega)\,ds\,d\omega.$

(b) The wave operator $W_{+}:\mathcal{H}\to\mathcal{H}_{\textrm{\rm loc}}$
exists.

(c) The scattering operator $S:\mathcal{H}_{\textrm{\rm
comp}}\to\mathcal{H}_{\textrm{\rm loc}}$ exists.

###### Proof.

Choose $\mathbf{f}\in\mathcal{H}_{\textrm{\rm comp}}$, so that
$\mathbf{f}(x)=0$ for $|x|>R$ with some $R>0$. Let $k=\mathcal{R}\mathbf{f}$.
Then $k(s,\omega)=0$ for $|s|>R$. For $t<-R-\rho:=t_{0}$,
$U(0,t)U_{0}(t)\mathbf{f}=U(0,t_{0})U_{0}(t_{0})\mathbf{f}$. In particular,
the limit defining $\Omega_{-}\mathbf{f}$ exists trivially and
$U(t,0)\Omega_{-}\mathbf{f}=U(t,t_{0})U_{0}(t_{0})\mathbf{f}$. The r.h.s. of
the latter solves the perturbed wave equation and equals
$U_{0}(t_{0})\mathbf{f}=\mathcal{R}^{-1}k(\cdot-t_{0},\cdot)$ for $t=t_{0}$.
To prove (75), we need to show that the r.h.s. of (75), call it
$\mathbf{v}(t)$, has the same initial condition for $t\leq t_{0}$.

For $t\leq t_{0}$, $u(t,x;s,\omega)=\delta(t+s-x\cdot\omega)$. Then by (51),

$v(t)=2c_{n}^{-}\int_{\mathbb{R}\times
S^{n-1}}\delta(t+s-x\cdot\omega)\partial_{s}^{(n-3)/2}k(s,\omega)\,ds\,d\omega=(\mathcal{R}^{-1}k)_{1}(\cdot-t,\cdot),$

which proves (a).

To prove the existence of $W_{+}$ in (b), fix first $R>0$ and let
$\mathbf{1}_{B(0,R)}$ be the characteristic function of that ball. By (47),

(76)
$\mathbf{1}_{B(0,R)}U_{0}(-t)U(t,s)=\mathbf{1}_{B(0,R)}U_{0}(-s)+\mathbf{1}_{B(0,R)}\int_{s}^{t}U_{0}(-\sigma)Q(\sigma)U(\sigma,s)\,d\sigma.$

By Huygens’ principle, $\mathbf{1}_{B(0,R)}U_{0}(-\sigma)Q(\sigma)=0$ for
$\sigma>R+\rho$. For $t>R+\rho$ then the integral above is independent of $t$
and therefore the strong limit $\mathbf{1}_{B(0,R)}W_{+}$ exists in a trivial
way, defining a unique element in $\mathcal{H}_{\textrm{loc}}$.

Part (c) follows from (a) and (b). ∎

The scattering operator $S$ on $\mathcal{H}$ exists (as a bounded operator)
under some conditions, see the references in [19, sec. 3]. Then
$-c_{n}^{-}\partial_{s^{\prime}}^{(n-1)/2}A^{\sharp}(s^{\prime},\omega^{\prime};s,\omega)$
is the Schwartz kernel of $\mathcal{R}(S-\textrm{\rm Id})\mathcal{R}^{-1}$. In
the general case, we can consider the latter as the Schwartz kernel of the
operator mapping asymptotic wave profiles instead of translation
representations, see (52), as shown [19, Proposition 3.1].

######  Proposition A.1 ([2, 3], [19]).

Let $\mathbf{f}\in C_{0}^{\infty}\times C_{0}^{\infty}$. Let
$\mathbf{v}_{0}(t)=U_{0}(t)\mathbf{f}$, and let $\mathbf{v}(t)$ be the
solution of (1) which equals $\mathbf{v}_{0}(t)$ for $t\ll 0$. Then we have

$v^{\sharp}(s^{\prime},\omega^{\prime})=v_{0}^{\sharp}(s^{\prime},\omega^{\prime})-c_{n}^{-}\partial_{s^{\prime}}^{(n-1)/2}\int_{\mathbb{R}\times
S^{n-1}}A^{\sharp}(s^{\prime},\omega^{\prime};s,\omega)v_{0}^{\sharp}(s,\omega)\,ds\,d\omega.$

The proof of the proposition is done by taking the asymptotic wave profile of
(75) applying Duhamel’s formula (47) first.

##  Appendix B A weighted Radon transform

We recall the definition of the weighted Euclidean Radon transform

$R_{\mu}f(p,\omega)=\int_{x\cdot\omega=p}\mu(x,\omega)f(x)dx=\int_{\omega^{\perp}}\mu(p\omega+y,\omega)f(p\omega+y)dy.$

where $\mu\in C^{\infty}(\mathbb{R}\times\mathbb{S}^{n-1})$ is a weight. We
will use the next result:

######  Theorem B.1.

Let $\Omega$ be an open, bounded set and $\mu\in
C^{\infty}(\mathbb{R}\times\mathbb{S}^{n-1})$. Then, there is a constant
$C_{\Omega}>0$, which depends only on $\Omega$, such that

$\|R_{\mu}f\|_{L^{2}(\mathbb{S}_{\omega}^{n-1};H^{(n-1)/2}(\mathbb{R}_{\sigma}))}\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{2n-2}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}$

for all $f\in C_{0}^{\infty}(\Omega)$.

To prove this, we need the following auxiliary lemmas.

######  Lemma B.1.

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded set and $\nu$ be a
function on $\mathbb{R}^{n}\times\mathbb{R}^{n}$ such that for any fixed
$\xi\in\mathbb{R}^{n}$, $\nu(\cdot,\xi)\subset C^{n}(\bar{\Omega})$ with
$\mathrm{supp\,}\nu(\cdot,\xi)\subset\Omega$. Then, the operator $V$, given by

$V:f\rightarrow\int_{\mathbb{R}^{n}}e^{-ix\xi}\nu(x,\xi)f(x)dx,$

satisfies

$\|V\|_{L^{2}(\Omega)\rightarrow L^{2}(\mathbb{R}^{n})}\leq
C_{\Omega}\sup_{\xi\in\mathbb{R}^{n}}\|\nu(\cdot,\xi)\|_{C^{n}(\bar{\Omega})}.$

###### Proof.

Throughout this proof, $C_{\Omega}$ will serve as a universal positive
constant depending only on $\Omega$, which may vary from line to line. Let
$f\in C_{0}^{\infty}(\Omega)$, then

$|Vf(\xi)|=\left|\int_{\mathbb{R}^{n}}D_{x_{n}}\cdots
D_{x_{1}}\left(\int_{-\infty}^{x_{1}}\cdots\int_{-\infty}^{x_{n}}f(y)e^{-iy\xi}dy_{n}\cdots
dy_{1}\right)\nu(x,\xi)dx\right|.$

By integration by parts, we obtain

$|Vf(\xi)|\leq\int_{\mathbb{R}^{n}}\left|\int_{-\infty}^{x_{1}}\cdots\int_{-\infty}^{x_{n}}f(y)e^{-iy\xi}dy_{n}\cdots
dy_{1}\right||D_{x_{n}}\cdots D_{x_{1}}\nu(x,\xi)|dx\\\ \leq
C_{\Omega}\sup_{\xi\in\mathbb{R}^{n}}\|\nu(\cdot,\xi)\|_{C^{n}(\bar{\Omega})}\sup_{x\in\mathbb{R}^{n}}\left|\int_{\mathbb{R}^{n}}e^{-iy\xi}f(y)\chi_{\\{z:z_{k}\leq
x_{k}\\}}(y)dy\right|,$

where $\chi_{A}$ is the indicator function for a set $A$. Therefore, we
estimate

$\displaystyle\|Vf\|_{L^{2}(\mathbb{R}^{n})}$ $\displaystyle\leq
C_{\Omega}\sup_{\xi\in\mathbb{R}^{n}}\|\nu(\cdot,\xi)\|_{C^{n}(\bar{\Omega})}\sup_{x\in\mathbb{R}^{n}}\|\mathcal{F}[f\chi_{\\{z:z_{k}\leq
x_{k}\\}}]\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C_{\Omega}\sup_{\xi\in\mathbb{R}^{n}}\|\nu(\cdot,\xi)\|_{C^{n}(\bar{\Omega})}\sup_{x\in\mathbb{R}^{n}}\|f\chi_{\\{z:z_{k}\leq
x_{k}\\}}\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C_{\Omega}\sup_{\xi\in\mathbb{R}^{n}}\|\nu(\cdot,\xi)\|_{C^{n}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}.$

This completes the proof. ∎

######  Lemma B.2.

Let $\omega\subset\mathbb{R}^{n}$ be an open bounded set and $\mu$, $\nu\in
C^{\infty}(\mathbb{R}\times\mathbb{S}^{n-1})$. Then, there exists a constant
$C_{\Omega}>0$, which depend only on $\Omega$, such that

$|(R_{\mu}^{*}R_{\nu}D^{\gamma}f,f)_{L^{2}(\mathbb{R}^{n})}|\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}\sup_{\omega\in\mathbb{S}^{n-1}}\left\|\nu(\cdot,\omega)\right\|_{C^{n-1+|\gamma|}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}.$

for any $f\in C_{0}^{\infty}(\Omega)$ and multi-index any $\gamma$ with
$0\leq|\gamma|\leq n-1$.

###### Proof.

Throughout this proof, $C_{\Omega}$ will serve as a universal positive
constant depending only on $\Omega$, which may vary from line to line. Let
$\chi\in C_{0}^{\infty}(\mathbb{R}^{n})$ such that $\chi(x)=1$ for
$x\in\Omega$. Then, for $f\in C_{0}^{\infty}(\Omega)$,

$|(R_{\mu}^{*}R_{\nu}D^{\gamma}f,f)_{L^{2}(\mathbb{R}^{n})}|=|(\chi
R_{\mu}^{*}R_{\nu}D^{\gamma}(\chi f),f)_{L^{2}(\mathbb{R}^{n})}|\leq\|\chi
R_{\mu}^{*}R_{\nu}D^{\gamma}(\chi
f)\|_{L^{2}(\mathbb{R}^{n})}\|f\|_{L^{2}(\mathbb{R}^{n})}.$

Let us investigate the first multiplier on the right-hand side. By Proposition
5.8.3 in [22], $R^{*}_{\mu}R_{\nu}$ is a $\Psi\mathrm{DO}$ of order $1-n$ with
the amplitude given by

$(2\pi-1)\frac{\mu(x,\xi/|\xi|)\nu(y,\xi/|\xi|)+\mu(x,-\xi/|\xi|)\nu(y,-\xi/|\xi|)}{|\xi|^{n-1}}.$

Then,

$\chi R_{\mu}^{*}R_{\nu}D^{\gamma}(\chi f)=Af+Bf$

with

$Af(x)=C_{\Omega}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i(x-y)\xi}\chi(x)\frac{\mu(x,\xi/|\xi|)\nu(y,\xi/|\xi|)}{|\xi|^{n-1}}D_{y}^{\gamma}(\chi(y)f(y))dyd\xi$

and

$Bf(x)=C_{\Omega}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i(x-y)\xi}\chi(x)\frac{\mu(x,-\xi/|\xi|)\nu(y,-\xi/|\xi|)}{|\xi|^{n-1}}D_{y}^{\gamma}(\chi(y)f(y))dyd\xi.$

By integration by parts, we obtain

$|Af(x)|\leq
C_{\Omega}\sum_{\alpha+\beta=\gamma}\left|\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i(x-y)\xi}\chi(x)\xi^{\alpha}\frac{\mu(x,\xi/|\xi|)D_{y}^{\beta}\nu(y,\xi/|\xi|)}{|\xi|^{n-1}}\chi(y)f(y)dyd\xi\right|.$

Let $\phi\in C^{\infty}(\mathbb{R}^{n})$ be a function such that

$\phi(\xi)=\begin{cases}0&|\xi|\leq 1;\\\ 1&|\xi|\geq 2.\end{cases}$

We denote

$A_{\alpha,\beta}^{1}f(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i(x-y)\xi}(1-\phi(\xi))\chi(x)\xi^{\alpha}\frac{\mu(x,\xi/|\xi|)D_{y}^{\beta}\nu(y,\xi/|\xi|)}{|\xi|^{n-1}}\chi(y)f(y)dyd\xi$

and

$A_{\alpha,\beta}^{2}f(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i(x-y)\xi}\phi(\xi)\chi(x)\xi^{\alpha}\frac{\mu(x,\xi/|\xi|)D_{y}^{\beta}\nu(y,\xi/|\xi|)}{|\xi|^{n-1}}\chi(y)f(y)dyd\xi.$

Then, the last inequality implies

(77) $|Af(x)|\leq
C_{\Omega}\sum_{\alpha+\beta=\gamma}\left(|A_{\alpha,\beta}^{1}f(x)|+|A_{\alpha,\beta}^{2}f(x)|\right).$

Let us estimate the $L^{2}$-norm of $A_{\alpha,\beta}^{1}f$. The kernel of
$A_{\alpha,\beta}^{1}$ is given by

$K_{\alpha,\beta}(x,y)=\int_{\mathbb{R}^{n}}e^{i(x-y)\xi}(1-\phi(\xi))\xi^{\alpha}\frac{\chi(x)\chi(y)\mu(x,\xi/|\xi|)D_{y}^{\beta}\nu(y,\xi/|\xi|)}{|\xi|^{n-1}}d\xi.$

We estimate

$\sup_{y\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|K_{\alpha,\beta}(x,y)|dx\\\
\leq\|\chi\|_{L^{\infty}(\mathbb{R}^{n})}\int_{\mathbb{R}^{n}}\xi^{\alpha}\frac{(1-\phi(\xi))}{|\xi|^{n-1}}d\xi\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{L^{\infty}(\mathbb{R}^{n})}\sup_{\omega\in\mathbb{S}^{n-1}}\|\nu(\cdot,\omega)\|_{C^{|\beta|}(\mathbb{R}^{n})}.$

Hence,

$\sup_{y\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|K_{\alpha,\beta}(x,y)|dx=C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{L^{\infty}(\mathbb{R}^{n})}\sup_{\omega\in\mathbb{S}^{n-1}}\|\nu(\cdot,\omega)\|_{C^{|\beta|}(\mathbb{R}^{n})},$

and similarly,

$\sup_{x\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|K_{\alpha,\beta}(x,y)|dy=C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{L^{\infty}(\mathbb{R}^{n})}\sup_{\omega\in\mathbb{S}^{n-1}}\|\nu(\cdot,\omega)\|_{C^{|\beta|}(\mathbb{R}^{n})}.$

Therefore, by Lemma 18.1.12 in [8],

(78) $\|A_{\alpha,\beta}^{1}f\|_{L^{2}(\Omega)}\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{L^{\infty}(\mathbb{R}^{n})}\sup_{\omega\in\mathbb{S}^{n-1}}\|\nu(\cdot,\omega)\|_{C^{|\beta|}(\mathbb{R}^{n})}\|f\|_{L^{2}(\Omega)}.$

Next, we estimate the $L^{2}$-norm of $A_{\alpha,\beta}^{2}f$. We denote We
denote

$\nu_{\alpha,\beta}(y,\xi)=\frac{\xi^{\alpha}D_{y}^{\beta}\nu(y,\xi/|\xi|)}{|\xi|^{|\alpha|}}\chi(y)$

and

$v_{\alpha,\beta}(\xi)=\int_{\mathbb{R}^{n}}e^{-iy\xi}\nu_{\alpha,\beta}(y,\xi)f(y)dy,$

so that

$A_{\alpha,\beta}^{2}f(x)=\int_{\mathbb{R}^{n}}e^{ix\xi}\phi(\xi)\frac{\chi(x)\mu(x,\xi/|\xi|)}{|\xi|^{n-1-|\alpha|}}v_{\alpha,\beta}(\xi)d\xi\\\
=\int_{\mathbb{R}^{n}}e^{ix\xi}\phi(\xi)\frac{\chi(x)\mu(x,\xi/|\xi|)}{|\xi|^{n-1-|\alpha|}}\mathcal{F}[\mathcal{F}^{-1}v_{\alpha,\beta}](\xi)d\xi.$

Next, we note that

$\displaystyle\sum_{\tau\leq
n+1}\int_{\mathbb{R}^{n}}\left|D_{x}^{\tau}\left(\frac{\chi(x)\phi(\xi)\mu(x,\xi/|\xi|)}{|\xi|^{n-1-|\alpha|}}\right)\right|dx$
$\displaystyle\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\chi(\cdot)\mu(\cdot,\omega)\|_{C^{n+1}(\mathbb{R}^{n})}$
$\displaystyle\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}.$

Therefore, Theorem 18.1.11’ in [8] implies that

$\displaystyle\int_{\mathbb{R}^{n}}\Big{|}\int_{\mathbb{R}^{n}}e^{ix\xi}\frac{\chi(x)\phi(\xi)\mu(x,\xi/|\xi|)}{|\xi|^{n-1-|\alpha|}}$
$\displaystyle\mathcal{F}[\mathcal{F}^{-1}v_{\alpha,\beta}](\xi)d\xi\Big{|}^{2}dx$
$\displaystyle\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}^{2}\|\mathcal{F}^{-1}v_{\alpha,\beta}\|_{L^{2}(\mathbb{R}^{n})}^{2}$
$\displaystyle\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}^{2}\|v_{\alpha,\beta}\|_{L^{2}(\mathbb{R}^{n})}^{2}.$

Hence,

$\|A_{\alpha,\beta}^{2}f\|_{L^{2}(\mathbb{R}^{n})}\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}\|v_{\alpha,\beta}\|_{L^{2}(\mathbb{R}^{n})}.$

By Lemma B.1,

(79) $\|v_{\alpha,\beta}\|_{L^{2}(\mathbb{R}^{n})}\leq
C_{\Omega}\sup_{\xi\in\mathbb{R}^{n}}\|\nu_{\alpha,\beta}(\cdot,\xi)\|_{C^{n}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}.$

We estimate

$\displaystyle\sup_{\xi\in\mathbb{R}^{n}}\|\nu_{\alpha,\beta}(\cdot,\xi)\|_{C^{n}(\bar{\Omega})}$
$\displaystyle=\sup_{\xi\in\mathbb{R}^{n}}\left\|\frac{\xi^{\alpha}D^{\beta}\nu(\cdot,\xi/|\xi|)}{|\xi|^{|\alpha|}}\chi(\cdot)\right\|_{C^{n}(\bar{\Omega})}\leq
C_{\Omega}\sup_{\xi\in\mathbb{R}^{n}}\left\|D^{\beta}\nu(\cdot,\xi/|\xi|)\right\|_{C^{n}(\bar{\Omega})}$
$\displaystyle\leq
C_{\Omega}\sup_{\xi\in\mathbb{S}^{n-1}}\left\|\nu(\cdot,\xi)\right\|_{C^{n+|\beta|}(\bar{\Omega})}.$

Therefore,

$\|A_{\alpha,\beta}^{2}f\|_{L^{2}(\mathbb{R}^{n})}\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}\sup_{\xi\in\mathbb{S}^{n-1}}\left\|\nu(\cdot,\xi)\right\|_{C^{n+|\beta|}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}.$

Therefore, by (77) and (78), we obtain

$\|Af\|_{L^{2}(\mathbb{R}^{n})}\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}\sup_{\omega\in\mathbb{S}^{n-1}}\left\|\nu(\cdot,\omega)\right\|_{C^{n-1+|\gamma|}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}.$

Similarly, we estimate $\|Bf\|_{L^{2}(\mathbb{R}^{n})}$ and conclude that

$\|\chi R_{\mu}^{*}R_{\nu}D^{\alpha}(\chi f)\|_{L^{2}(\mathbb{R}^{n})}\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}\sup_{\omega\in\mathbb{S}^{n-1}}\left\|\nu(\cdot,\omega)\right\|_{C^{n-1+|\gamma|}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}.$

This completes the proof. ∎

Now, we prove Theorem B.1.

###### Proof of Theorem B.1.

Let $f\in C_{0}^{\infty}(\Omega)$ and $\Omega\subset\mathbb{R}^{n}$ compact.
We denote $L=(n-1)/2$ and estimate

$\|R_{\mu}f\|_{L^{2}(\mathbb{S}_{\omega}^{n-1};H^{L}(\mathbb{R}_{\sigma}))}^{2}=\sum_{l=0}^{L}\|\partial_{p}^{l}R_{\mu}f\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{n-1})}\leq\sum_{l=0}^{2L}|(R_{\mu}^{*}\partial_{p}^{l}R_{\mu}f,f)_{L^{2}(\mathbb{R}^{n})}|.$

We note that

$\displaystyle\partial_{p}^{l}R_{\mu}f(p,\omega)$
$\displaystyle=\partial_{p}^{l}\left(\int_{\omega^{\perp}}\mu(p\omega+y,\omega)f(p\omega+y)dy\right)$
$\displaystyle=\sum_{|\alpha|+|\beta|=l}C_{\alpha,\beta}\int_{\omega^{\perp}}D^{\alpha}f(p\omega+y)D^{\beta}\mu(p\omega+y,\omega)\omega^{\alpha+\beta}dy.$

Let us set

$\mu_{\alpha,\beta}(x,\omega)=\omega^{\alpha+\beta}D^{\beta}\mu(x,\omega)$

so that, the previous equality gives

$\partial_{p}^{l}R_{\mu}=\sum_{|\alpha|+|\beta|=l}C_{\alpha,\beta}R_{\mu_{\alpha,\beta}}D^{\alpha}\quad\text{and}\quad
R_{\mu}^{*}\partial_{p}^{l}R_{\mu}=\sum_{|\alpha|+|\beta|=l}C_{\alpha,\beta}R_{\mu}^{*}R_{\mu_{\alpha,\beta}}D^{\alpha}.$

By Lemma B.2, we estimate

$\displaystyle|(R_{\mu}^{*}R_{\mu_{\alpha,\beta}}D^{\alpha}f,f)_{L^{2}(\Omega)}|$
$\displaystyle\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{n+1}(\bar{\Omega})}\sup_{\omega\in\mathbb{S}^{n-1}}\|D^{\beta}\mu(\cdot,\omega)\|_{C^{n-1+|\alpha|}(\bar{\Omega})}\|f\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq
C_{\Omega}\sup_{\omega\in\mathbb{S}^{n-1}}\|\mu(\cdot,\omega)\|_{C^{2n-2}(\bar{\Omega})}^{2}\|f\|_{L^{2}(\Omega)}^{2}.$

This completes the proof. ∎

## Acknowledgment

The first author was partially supported by the Ministry of Education and
Science of the Republic of Kazakhstan under grant AP22683207. P.S. partly
supported by NSF Grant DMS-2154489. L.O. and M.N. were supported by the
European Research Council of the European Union, grant 101086697 (LoCal), and
the Research Council of Finland, grants 347715 and 353096. Views and opinions
expressed are those of the authors only and do not necessarily reflect those
of the European Union or the other funding organizations. Neither the European
Union nor the other funding organizations can be held responsible for them.

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