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\begin{document}

\title{``Bootstrap domain of dependence'':
  bounds and time decay of solutions of
  the wave equation}
\author{Thomas G. Anderson}
\address{Applied \& Comp.\ Mathematics, California Institute of
Technology, Pasadena, CA}
\curraddr{Department of Mathematics, University of Michigan, Ann Arbor, MI}
\email{tganders@umich.edu}

\author{Oscar P. Bruno}
\address{Applied \& Comp.\ Mathematics, California Institute of
Technology, Pasadena, CA}
\email{obruno@caltech.edu}
\thanks{This work was supported by NSF and DARPA through contracts DMS-1714169
and HR00111720035, and the NSSEFF Vannevar Bush Fellowship under
contract number N00014-16-1-2808.}

\subjclass[2010]{Primary 35B40, 35B30, 35L05, 35L20, 45M05}
\date{\today}
\keywords{boundary integral equations, wave equation, decay, Huygens' principle}
\begin{abstract}
  This article introduces a novel ``bootstrap domain-of-dependence''
  concept, according to which, for all time following a given
  illumination period of arbitrary duration, the wave field scattered
  by an obstacle is encoded in the history of boundary scattering
  events for a time-length equal to the diameter of the obstacle,
  measured in time units. Resulting solution bounds provide estimates
  on the solution values in terms of a short-time history record, and
  they establish superalgebraically fast decay (i.e., decay faster
  than any negative power of time) for a wide range of scattering
  obstacles---including certain types of ``trapping'' obstacles whose
  periodic trapped orbits span a set of positive volumetric measure,
  and for which no previous fast-decay theory was available.  The
  results, which do not rely on consideration of the Lax-Phillips
  complex-variables scattering framework and associated resonance-free
  regions in the \emph{complex} plane, utilize only \emph{real-valued}
  frequencies, and follow from use of Green functions and boundary
  integral equation representations in the frequency and time domains,
  together with a certain $q$-growth condition on the frequency-domain
  operator resolvent.
\end{abstract}

\maketitle
\markright{\MakeUppercase{Bootstrap Domain-of-dependence bounds \& decay for wave
solutions}}

\section{Introduction\label{Introduction}}

We present a novel ``bootstrap domain-of-dependence'' concept
(bootstrap DoD) and associated bounds on solutions of the scattering
problem for the wave equation on an exterior region $\Omega^c$.  The
classical domain-of-dependence~\cite{John} concept for a space-time
point $({\mathbf{r}}_0, T_0)$ concerns the initial and boundary values that
determine the free-space solution at that space-time point. The
bootstrap DoD ending at a given time $T_0$, in contrast, involves
field values on the scattering boundary
$\Gamma = \partial \Omega= \partial \Omega^c$ over the length of time
$T_*$ preceding $T_0$, where $T_*$ equals the amount of time that is
sufficient for a wave to traverse the diameter of the obstacle. As
discussed in this paper, in absence of additional illumination after
time $t = T_0 - T_*$, the bootstrap DoD completely determines the
scattered field on $\Gamma$ for all times $t\geq T_0$.  Further, the
solution bounds that result from the bootstrap DoD approach provide
estimates on the solution values in terms of a short-time history
record, and they establish superalgebraically fast decay (i.e., decay
faster than any negative power of time) for a wide range of scattering
obstacles---including certain types of trapping obstacles for which no
previous fast-decay theory was available. The results, which do not
rely on consideration of the Lax-Phillips complex-variables scattering
framework and associated resonance-free regions in the \emph{complex}
plane, utilize only \emph{real-valued} frequencies, and follow from
use of Green functions and boundary integral equation representations
in the frequency and time domains, together with a certain $q$-growth
condition on the frequency-domain operator resolvent. As discussed in
\Cref{3d_decay_rmk_iii}, relying on some materials in
\Cref{sec:theory_part_i} and most of the materials in
\Cref{sec:theory_part_ii}, related but less informative
superalgebraically decaying solution bounds can also be obtained by
relying on the complete time-history of the incident field instead of
the bootstrap DoD.

(Estimates based on the {\em classical} concept of domain of the
dependence have previously been provided, including, in particular, a
``domain-of-dependence inequality'' for the problem of scattering by
obstacles~\cite[Thm.\ 5.2]{Wilcox:62}. We note however, that this is
not a decay result and, in fact, it plays an important but very
different role: by establishing that at most exponential {\em growth}
can occur, it provides the necessary stability elements in a proof of
existence and uniqueness based on energy considerations.)

In addition to the decay-problem application, the bootstrap
DoD estimates introduced in this paper provide a
valuable tool in connection with the numerical analysis of certain
frequency-time ``hybrid'' numerical methods introduced
recently~\cite{Anderson:20} for the time-domain wave equation, which
proceed by Fourier transformation from the time domain into the
frequency domain.  These hybrid solvers evaluate solutions of the wave
equation by partitioning incident wave fields into a sequence of
smooth compactly-supported wave packets and Fourier transformation in
time for each packet, and they incorporate a number of novel
approaches designed to accurately capture all high-frequency
behavior---both in time, for high frequencies, and in frequency, for
large time. The overall solution is then reconstructed as a sum of
many wave equation solutions, each with a distinct center in
time. Unlike the aforementioned complete time-history bounds, the
bootstrap DoD bounds introduced in this paper provide a natural
theoretical basis for efficient truncation of this sum while meeting a
prescribed accuracy tolerance.

Returning to the decay problem we note that, in contrast with previous
approaches, that typically rely on energy arguments and/or on analytic
continuation of the frequency-domain resolvent, the method proposed in
this paper relies on use of boundary integral equations for both
frequency-domain and time-domain problems, as well as on the Fourier
transform that relates them, and it characterizes the
multiple-scattering effects and trapping character of domain
boundaries (which is described in terms of billiard-ball trajectories,
see Section~\ref{sec:detailed_review}) in terms of the growth of the
norm of the resolvent for the frequency-domain problem as the {\em
  real} frequency $\omega$ grows without bound. In detail, without
recourse to complex-analytic methods, the new approach to the decay
problem proceeds on the basis of the bootstrap DoD formulation introduced
in Section~\ref{sec:theory_part_i}, which, as mentioned above,
captures the impact of the complete time history up to a given time
$t$ in terms of the history over the time interval, immediately
preceeding $t$, of time-length equal to the time $T_*$ required for
propagation across the diameter of the obstacle. The resulting bounds
provide superalgebraically-fast time-decay energy estimates (decaying
faster than any negative power of time) for a wide range of (both
trapping and nontrapping) obstacles satisfying a certain $q$-growth
condition on the resolvent operator as a function of the \emph{real}
frequency $\omega$. In particular, this theory establishes the first
rapid decay estimates for specific types of trapping obstacles which
are: (a)~Not equal to unions of convex obstacles; and (b)~Whose
periodic trapped orbits span a set of positive volumetric measure
(such as those depicted in \Cref{fig:3d_connected_trapping}).

\subsection{Overview}\label{sec:overview}
This paper is organized as follows. A brief but somewhat detailed
overview of previous decay results for the obstacle scattering problem
is presented in \Cref{sec:detailed_review}.  Then, after preliminaries
presented in \Cref{sec:prelim}, \Cref{sec:theory_part_i} introduces
the bootstrap domain-of-dependence formulation and it presents
\Cref{3d_decay_thm}, which shows that for obstacles satisfying the
$q$-growth condition, if the Neumann trace is ``small'' on a (slightly
extended) bootstrap DoD time interval, then, absent additional
illumination on and after that interval, it must remain ``permanently
small'', that is, small for all times subsequent to that interval.
\Cref{sec:theory_part_ii} then extends the estimates and it
establishes, in \Cref{3d_decay_thm_ii} and
Corollaries~\ref{decay_corr} and~\ref{decay_corr_energy}, various
superalgebraically fast time decay estimates for solutions
in presence of such obstacles as well
as associated local solution energies.

\subsection{Additional Background on Decay
  Theory}\label{sec:detailed_review}

In order to provide relevant background concerning decay estimates we
briefly review the literature on this long-standing problem, and we
note the significant role played in this context by the shape of the
obstacle $\Omega$. Previous studies of the decay problem establish
exponential decay of solutions for certain classes of domain
shapes---including star-shaped domains~\cite{Morawetz:61}, domains
that are ``nontrapping'' with respect to
rays~\cite{Morawetz:77,Melrose:79}, and unions of strictly convex
domains that satisfy certain spacing
criteria~\cite{Ikawa:82,Ikawa:88}. (A domain is nontrapping if each
billiard ball traveling in the exterior of $\Omega$, which bounces off
the boundary $\Gamma = \partial \Omega$ in accordance with the law of
specular reflection, and which starts within a ball $B_R$ of radius $R$ around
$\Omega$, eventually escapes $B_R$ with bounded, $R$-dependent,
trajectory length~\cite{Melrose:79}.)

Early results~\cite{Morawetz:61,Lax:63,Morawetz:66}, obtained on the
basis of energy estimates in the domains of time and frequency,
establish exponential decay for ``star-shaped'' obstacles, that is,
obstacles $\Omega$ which, for a certain $\mathbf{r}_0 \in \Omega$,
contain the line segment connecting $\mathbf{r}_0$ and any other point
$\mathbf{r}\in\Omega$. As noted in~\cite{Lax:63,Lax:67}, exponential
decay generally implies analyticity of the resolvent operator in a
strip around the real $\omega$ frequency axis. A significant
generalization of these results was achieved in~\cite{Morawetz:77}
using a hypothesis somewhat more restrictive than the non-trapping
condition, while~\cite{Melrose:79} established
exponential
local energy decay for all non-trapping obstacles.  All of these
results establish exponential decay
\begin{equation}\label{energy_decay}
  E(u, D, t) \le C{\mathrm{e}}^{-\alpha t} E(u, \infty, 0), \quad \alpha > 0,
\end{equation}
for the local energy
\begin{equation}\label{local_energy}
  E(u, D, t) = \int_D \left|\nabla u({\mathbf{r}}, t)\right|^2 + \left|u_t({\mathbf{r}},
  t)\right|^2\,\d V({\mathbf{r}})
\end{equation}
contained in a compact region $D \subset \widebar{\Omega}^c$ in terms
of the energy $E(u, \infty, 0)$ contained in all of
$\widebar{\Omega}^c$---the latter one of which is, of course,
conserved.  As reviewed in Remark~\ref{ralston}, a uniform decay estimate of
the form~\eqref{energy_decay} cannot hold for trapping obstacles.

In this connection it is tangentially relevant to consider
reference~\cite{Datchev:12}, which establishes sub-exponential decay
for the problem of scattering by a globally defined smooth
potential. The method utilized in that work relies on use of simple
resolvent manipulations for the differential scattering operator
$P =-\frac{1}{\omega^2}\Delta + V(x)$, where
$V\in C_0^\infty(\mathbb{R}^n)$ denotes the globally defined
potential. Such manipulations are not applicable in the context of the
Green function-based operators for the impenetrable-scattering
problem, for which, in particular, the frequency $\omega$ is featured
in the {\em Green function exponent} of the {\em integral} scattering
operator instead of a {\em linear factor} $\frac{1}{\omega^2}$ in the
corresponding {\em differential} operator.

As mentioned above, a decay estimate of the form~\eqref{energy_decay}
cannot hold for trapping obstacles. However, by relying on analytic
continuation of the frequency-domain resolvent into a strip
surrounding the real-axis in the complex frequency domain $\omega$,
exponential decay (of a different character than expressed
in~\eqref{energy_decay}; see \Cref{ralston}) has been established for
certain trapping geometries~\cite{Ikawa:82,Ikawa:88,Fahry:91}.  In
view of previous work leading to results of exponential decay, even
for trapping geometries, the question may arise as to whether the
results of superalgebraically fast convergence presented in this paper
could actually be improved to imply, for example, exponentially fast
decay for trapping geometries which merely satisfy a $q$-growth
condition. A definite answer in the negative to such a question is
provided in~\cite[Thm.\ 1]{Ikawa:85}. This contribution exhibits an
example for which, consistent with earlier general
suggestions~\cite[p.\ 158]{Lax:67}, the sequence of imaginary parts of
the pole-locations of the scattering matrix in the complex plane tends
to zero as the corresponding sequence of real parts tends to infinity;
clearly such a domain cannot exhibit exponential decay in view of the
Paley-Wiener theorem~\cite[Thm.\ I]{PaleyWiener:34}. In detail, the
example presented in~\cite{Ikawa:85} concerns a domain $\Omega$
consisting of a union of two disjoint convex obstacles, where the
principal curvature of each connected component vanishes at the
closest point between the obstacles.  For such a trapping domain
reference~\cite{Burq:98} provides the only previous applicable decay
result namely, the general-obstacle inverse polylogarithmic decay,
based on regions of the complex-$\Omega$ plane for which the resolvent
operator provably has no poles.  The real-$\omega$ decay analysis
presented in this paper provides significant progress, as it
establishes super-algebraic decay for a wide range of obstacles not
previously treated by classical scattering theory, including the
aforementioned vanishing-curvature example~\cite{Ikawa:85} and the
connected and significantly more strongly trapping structures depicted
in \Cref{fig:3d_connected_trapping}, for which the trapped rays form a
set of positive volumetric measure.

The aforementioned references~\cite{Ikawa:82,Ikawa:88,Fahry:91}
establish exponential decay for wave scattering for certain trapping
structures consisting of unions of disjoint convex obstacles (but see
Remark~\ref{ralston}), and thereby answer in the negative a conjecture
by Lax and Phillips~\cite[p.  158]{Lax:67} according to which
exponential decay could not occur for any trapping structure (in view
of the Lax/Phillips conjectured existence, for all trapping obstacles,
of a sequence of resonances $\lambda_j$ for which
$\mathrm{Im}\,\lambda_j \to 0^-$ as $j \to \infty$). The trapping
structures with exponential decay are taken to equal a disjoint union
of two smooth strictly convex obstacles in~\cite{Ikawa:82}, and
otherwise unions of disjoint convex obstacles in~\cite{Ikawa:88}; in
all cases the geometries considered give rise to sets of trapping rays
spanning three-dimensional point sets of zero volumetric measure:
single rays in~\cite{Ikawa:82}, and countable sets of primitive
trapped rays in~\cite{Ikawa:88} as implied by Assumption (H.2) in that
reference. To the authors' knowledge, these are the only known results
on fast decay of solutions in trapping geometries. Despite these known
exceptions to the Lax-Phillips conjecture, it has been
surmised~\cite{Ikawa:85} that ``it seems very sure that the conjecture
remains to be correct for a great part of trapping
obstacles,''---which would disallow exponential decay for most
trapping obstacles---thus providing an interesting perspective on the
main results presented in this paper---which, in particular, establish
superalgebraic decay for certain obstacles for which the set of
trapped rays span a set of positive measure.

\section{Preliminaries}\label{sec:prelim}
We consider the problem of scattering of an incident field $b$ by a
Lipschitz bounded obstacle $\Omega$, an open set $\Omega\subset \mathbb{R}^3$. More
precisely, letting
$\Box = \frac{\partial^2}{\partial t^2} - c^2\Delta$ denote the
d'Alembertian operator with wavespeed $c > 0$ and given an open set
$\Omega^\textit{inc}$ containing the closure $\widebar{\Omega}$ of
$\Omega$, $\widebar{\Omega} \subset \Omega^\textit{inc}$, we study the
solution $u$ of the initial and boundary value problem
\begin{subequations}\label{eq:w_eq}
  \begin{align}
    \Box\,u(\mathbf{r}, t)
    &= 0,\quad\mbox{for}\quad ({\mathbf{r}}, t) \in \widebar{\Omega}^c \times (0, \infty),\label{eq:w_eq_a}\\
    u(\mathbf{r},0) &= \frac{\partial u}{\partial t}(\mathbf{r}, 0)
                      = 0,\quad\mbox{for}\quad\mathbf{r} \in \Omega^c,\\
    u(\mathbf{r}, t) &=  -\gamma^+ b(\mathbf{r}, t),
                       \quad\mbox{for}\quad(\mathbf{r},t)\in\Gamma\times (0,\infty),\label{eq:w_eq_c}
    \end{align}
  \end{subequations}
  on the complement $\widebar{\Omega}^c$ of $\widebar{\Omega}$, for a
  given incident-field function
\begin{equation}\label{eq:bdef}
b: \Omega^\textit{inc}\times\mathbb{R} \to \mathbb{R}\;\mbox{satisfying}\;
b \in C^2(\Omega^\textit{inc}\times\mathbb{R})\; \mbox{and}\; \Box b = 0\;\mbox{in}\; \Omega^\textit{inc}\times\mathbb{R},
\end{equation}
(cf.\ \Cref{rem:wave_eq_sob_assump}), where
$\Gamma = \partial\Omega^c= \partial\Omega$ denotes the boundary of
the obstacle $\Omega$, and where $\gamma^+$ (\Cref{trace_def} in
\Cref{sob-boch}) denotes the exterior trace operator.  Compatibility
of the boundary values with the initial condition requires
$b({\mathbf{r}}, 0) = b_t({\mathbf{r}}, 0) = 0$ for ${\mathbf{r}} \in \Gamma$, that is to say,
that at $t = 0$ the wave has not as yet impinged on the obstacle.
Additionally, for compatibility with the integral equation
formulation~\eqref{eq:tdie_sl} below, letting
$\mathbb{R}_0^- =\{t\in\mathbb{R}\ : \ t\leq 0 \}$, we additionally
assume that
\begin{equation}\label{all_t_b}
\mbox{``The function}\; b \in C^2(\Omega^\textit{inc} \times \mathbb{R})\;\mbox{satisfies}\; b({\mathbf{r}}, t) = 0\;\mbox{for}\; ({\mathbf{r}}, t) \in \widebar{\Omega} \times \mathbb{R}_0^-\mbox{''}.
\end{equation}
The solution $u$ is the ``scattered'' component of the total field
$u^\textit{tot} = u + b$; with these conventions, we clearly have
$\gamma^+ u^\textit{tot}(\mathbf{r}, t) = 0$ for $\mathbf{r}$ on $\Gamma$.

\begin{rem}\label{rem:w_eq_ivp_ibvp_equiv}
  In the case $\Omega^\textit{inc}= \mathbb{R}^3$, the function $v = u^\textit{tot} = u + b$, satisfies
  the initial-value problem
\begin{subequations}\label{eq:w_eq_ivp}
    \begin{align}
      \Box &v(\mathbf{r}, t) = 0,\quad\mbox{for}\quad ({\mathbf{r}}, t) \in
            \widebar{\Omega}^c\times (0, \infty),\label{eq:w_eq_ivp_a}\\
        v(\mathbf{r},0) &= f({\mathbf{r}}),\, \frac{\partial
        v}{\partial t}(\mathbf{r}, 0)
            = g({\mathbf{r}}),\quad\mbox{for}\quad\mathbf{r} \in
            \Omega^c,\label{eq:w_eq_ivp_b}\\
        v(\mathbf{r}, t) &= 0,
            \quad\mbox{for}\quad(\mathbf{r},t)\in\Gamma\times (0,\infty),\label{eq:w_eq_ivp_c}
    \end{align}
\end{subequations}
in that domain, where $f(\mathbf{r}) = u^\textit{tot}({\mathbf{r}}, 0)$ and
$g(\mathbf{r}) = u^\textit{tot}_t({\mathbf{r}}, 0)$. Conversely, given smooth
data $f$ and $g$ in all of $\Omega^c \cap \Omega^\textit{inc}$, an incident field $b$ can be
obtained by first extending $f$ and $g$ to all of space~\cite[Thm.\
3.10]{Necas:11} and using the extended data as initial data for a
free-space wave equation with solution $b$. In other words, the
problems~\eqref{eq:w_eq} and~\eqref{eq:w_eq_ivp} are equivalent.
\end{rem}

\begin{rem}
  The classical literature on decay rates for the wave
  equation~\cite{Lax:67} concerns problem~\eqref{eq:w_eq_ivp} with the
  additional assumption that $f$ and $g$ are compactly supported. In
  view of strong Huygens principle~\cite{John}, which is valid in the
  present three-dimensional context, the procedure described in
  \Cref{rem:w_eq_ivp_ibvp_equiv} translates the spatial
  compact-support condition on $f$ and $g$ into a temporal
  compact-support condition for the function $b$
  in~\eqref{eq:w_eq}---i.e.\ that the incident field vanishes on the
  boundary $\Gamma = \partial \Omega$ after a certain initial
  ``illumination time period'' has elapsed.
\end{rem}

\begin{rem}\label{rem:wave_eq_sob_assump}
  Even though the main problem considered in this paper,
  problem~\eqref{eq:w_eq}, is solely driven by the Dirichlet
  data~\eqref{eq:w_eq_c}, our analysis relies on assumptions that are
  imposed not only on the function $\gamma^+b$, but also on
  $\gamma^+\partial_\mathbf{n} b$. Specifically, various results
  presented in this paper assume that, for an integer $s\geq 0$
  the incident field $b$ satisfies the ``$s$-regularity
  conditions''
  \begin{equation}\label{eq:gamma_Hs_assump}
    \gamma^+b \in H^{s+1}(\mathbb{R};\,L^2(\Gamma))\quad\mbox{and}\quad
    \gamma^+\partial_\mathbf{n} b \in H^s(\mathbb{R};L^2(\Gamma)).
  \end{equation}
  Clearly, for the most relevent incident fields $b$, such as those
  mentioned in \Cref{rem:w_eq_ivp_ibvp_equiv}, namely, solutions of
  the wave equation that are smooth and compactly supported in time
  over any compact subset of $\Omega^{\textit{inc}}$ the
  $s$-regularity conditions hold for all non-negative integers $s$.
\end{rem}

The boundary value problem~\cref{eq:w_eq} can be
reduced~\cite{HaDuong:86} to an equivalent time-domain integral
equation formulation, in which a boundary integral density $\psi$
(defined to vanish for all $t < 0$) is sought that satisfies the
boundary integral equation
\begin{equation}\label{eq:tdie_sl}
  \left(S \psi \right)(\mathbf{r}, t) = \gamma^+ b(\mathbf{r}, t)\quad\mbox{for}\quad (\mathbf{r},
  t) \in \Gamma \times [0, \infty),
\end{equation}
where $\gamma^+$ is the exterior trace operator, and where,
calling
\begin{equation}\label{eq:green_fnct_time}
  G(\mathbf{r}, t; \mathbf{r}', t') = \frac{\delta\left((t - t') - |\mathbf{r} -
  \mathbf{r}'|/c\right)}{4\pi |\mathbf{r} - \mathbf{r}'|}
\end{equation}
the Green function for the three-dimensional wave equation, $S$
denotes the time-domain single-layer boundary integral operator
\begin{equation}\label{eq:single_layer_op_time}
    (S\mu)(\mathbf{r}, t) = \int_{-\infty}^t \int_\Gamma G(\mathbf{r}, t;
  \mathbf{r}', t') \mu(\mathbf{r}', t')\,\d\sigma(\mathbf{r}')\,\d
  t',\quad{\mathbf{r}}\in\mathbb{R}^3.
\end{equation}
Once $\psi$ has been obtained, the solution $u$ of~\eqref{eq:w_eq} is given by
\begin{equation}\label{eq:kirchhoff_3d_soft}
  u(\mathbf{r}, t) = \left(S \psi \right)(\mathbf{r}, t), \quad \mathbf{r} \in
  \Omega^c.
\end{equation}
As is well known, further, the solution $\psi$ of~\eqref{eq:tdie_sl} equals the Neumann trace of the solution:
$\psi(\mathbf{r}, t) = \gamma^+ \frac{\partial u^\textit{tot}}{\partial \mathbf{n}}(\mathbf{r}, t)$, $\mathbf{r}\in\Gamma$. In view
of~\eqref{eq:kirchhoff_3d_soft}, spatio-temporal estimates and
temporal decay rates for the density $\psi$ imply similar
properties for the energy $E(u, D, t)$ in~\Cref{local_energy} over a
given open set $D \subset \overline{\Omega}^c$; see \Cref{decay_corr_energy}.

As mentioned in Section~\ref{Introduction}, this paper presents
estimates on the solutions $\psi$ of the integral equation
problem~\eqref{eq:tdie_sl}, including results for certain classes of
trapping obstacles (cf. the first paragraph of
Section~\ref{Introduction}). Our analysis is based on consideration of
the respective frequency-domain counterparts of problems~\ref{eq:w_eq}
and~\ref{eq:tdie_sl}, namely, for each frequency $\omega$, the
Helmholtz equation with wavenumber $\kappa(\omega)=\omega/c$ and the frequency
domain integral equation
\begin{equation}\label{CFIE_direct}
  A_\omega \psi^f = \gamma^+ \partial_\mathbf{n} B^f - \i \eta \gamma^+ B^f,
\end{equation}
for the temporal Fourier transform
$\psi^f = \psi^f(\mathbf{r}, \omega)$ of $\psi$, where we have used
the definition~\eqref{eq:fourier_transf} below of the temporal Fourier
transform. (Note that
$\psi^f(\mathbf{r}, \omega) = \mathcal{F}\lbrace \gamma^+ \frac{\partial u^\textit{tot}}{\partial
  \mathbf{n}}(\mathbf{r}, \cdot)\rbrace(\omega)$, $\mathbf{r}\in\Gamma$.)
In~\eqref{CFIE_direct} $A_\omega$ denotes the frequency-domain
combined-field scattering operator, which is given explicitly in
\Cref{Aomega_def_eqn}.
\begin{rem}\label{FT_conv}
  Throughout this article the superscript $f$ upon an upper case
  letter indicates a function of the temporal frequency
  $\omega$. Thus, for any function $f(t)$ of the time variable $t$ we
  write
\begin{equation}\label{eq:fourier_transf}
  F^f(\omega) = \mathcal{F} [f](\omega) = \int_{-\infty}^\infty f(t) {\mathrm{e}}^{-\i\omega t}\,\d t,
\end{equation}
where $f$ (and therefore also $F^f = \mathcal{F} [f]$), is either a
complex-valued scalar function or a function with values on a Banach
space $X$ over the complex numbers. In the latter case the integral on
the right-hand side of~\eqref{eq:fourier_transf} indicates integration
in the sense of Bochner~\cite{Hille,DunfordSchwartz,Weis}.
\end{rem}

Our analysis
relates the decay problem to the growth of norms of associated
frequency-domain solution operators as the frequency grows without
bound. Studies of such norm growths, which provide an indicator of the
energy-trapping character of obstacles, have been undertaken over the
last several decades---including growth estimates for geometries that
exhibit a variety of trapping
behavior~\cite{Popov:91,Cardoso:02,ChandlerWilde:09,BetckeChandlerWilde:10,Spence:16,Spence:20}.

\begin{defi}[Combined field operator]\label{Aop_def}
  Let $\Omega$ denote a Lipschitz domain with boundary $\Gamma$ and
  let $\eta$ be a nonzero real number.  Then, letting $G_\omega$
  denote the Green function for the Helmholtz equation with wavenumber $\kappa(\omega) = \omega / c$,
  $G_\omega(\mathbf{r}, \mathbf{r}') = \frac{{\mathrm{e}}^{\i\frac{\omega}{c}
      |\mathbf{r} - \mathbf{r}'|}}{4\pi|\mathbf{r} - \mathbf{r}'|}$,
  and letting $S_\omega$ and $K_\omega^*$ denote the single-layer and
  adjoint double-layer operators (see e.g.~\cite{ChandlerWilde:12}),
\begin{equation}\label{eq:single_layer_op}
    (S_\omega\mu)(\mathbf{r}) = \int_\Gamma G_{\omega}(\mathbf{r}, \mathbf{r'})
    \mu(\mathbf{r'})\,\d\sigma(\mathbf{r'}),\quad  \mathbf{r} \in \mathbb{R}^3, \quad \mbox{and}
  \end{equation}
\begin{equation}\label{eq:adjoint_double_layer_op}
    (K^*_\omega\mu)(\mathbf{r}) = \int_\Gamma \frac{\partial G_{\omega}(\mathbf{r},
    \mathbf{r'})}{\partial n(\mathbf{r})}
    \mu(\mathbf{r'})\,\d\sigma(\mathbf{r'}), \quad \mathbf{r} \in \Gamma,
  \end{equation}
the combined-field operator $A_\omega$ is defined by
\begin{equation}\label{Aop_def_eqn}
  A_{\omega,\eta} = \frac{1}{2}I + K_\omega^* - \i\eta
  S_\omega.
\end{equation}
Letting $\omega_0 > 0$ we also define for $\omega \ge 0$
\begin{equation}\label{Aomega_def_eqn}
  A_\omega = A_{\omega,\eta_0} \quad\mbox{where}\quad \eta_0 =
\begin{cases} 1, &~\mbox{if}~0 \le \omega < \omega_0\\
\omega, &~\mbox{if}~\omega \ge \omega_0,
\end{cases}
\end{equation}
where the dependence of $A_\omega$ on $\omega_0$ has been suppressed.
\end{defi}

\begin{rem}\label{A_om_invert}
  As is well-known~\cite[Thm. 2.17]{ChandlerWilde:12}, for domains
  with Lipschitz boundaries $\Gamma$, the operators $S_\omega$,
  $K_\omega^*$ and $A_\omega$ are continuous operators on
  $L^2(\Gamma)$ for each $\omega \ge 0$.  Further, for
  $\mathrm{Re}(\eta) \ne 0$ the operator $A_{\omega,\eta}$ is
  invertible for all $\omega \ge 0$, see e.g.~\cite[Thm.\
  2.27]{ChandlerWilde:12}.
\end{rem}

\begin{rem}\label{psi_omega_independent}
  The function $\psi^f = \psi^f(\mathbf{r}, \omega)$ defined above,
  which, by definition, is independent of the real parameter $\eta$
  in~\cref{Aop_def_eqn}, is a solution of the $\eta$-dependent
  equation
  $A_{\omega,\eta} \psi^f = \gamma^+ \partial_\mathbf{n} B^f - \i \eta
  \gamma^+ B^f$ for every real value of $\eta$ (see
  e.g.~\cite{ChandlerWilde:12}). This fact, which implies that
  $\psi^f$ is also a solution of~\eqref{CFIE_direct} for all real
  values of $\omega$, is exploited in our analysis of the decay
  problem (e.g., in the proofs of Lemmas~\ref{3d_decay_lemma_Ainv_freq_bounds}
  and~\ref{3d_decay_thm_recursive_bound_general}).
\end{rem}

This article utilizes a certain ``$q$-growth condition'' which, for a
given $q\in \mathbb{R}$, may or may not be satisfied by a given
obstacle $\Omega$---namely, that a constant $C$ exists such that, for
all $\omega>0$, the relation~\eqref{eq:q-nontrapp} below holds. The
$q$-growth condition is fundamentally a statement on the
high-frequency character of the operator $A_\omega^{-1}$, and it is
indeed a condition on the domain $\Omega$ which is independent on the
constant $\omega_0$ in~\eqref{Aomega_def_eqn}. This can be verified by
noting that, given any compact intervals $I_\eta$ and $I_\omega$,
where $I_\eta$ is bounded away from $0$, the norm
$\norm{A_{\omega,\eta}^{-1}}_{L^2(\Gamma)\to L^2(\Gamma)}$ is
uniformly bounded for $(\eta,\omega)\in I_\eta\times I_\omega$---as it
can be established on the basis of the
theory~\cite{ChandlerWilde:12,McLean} for the combined field operator
on a Lipschitz domain together with a compactness argument based on
Taylor expansions of the oscillatory exponential factor of the Green
function together with Neumann series for the resolvent around each
pair $(\eta,\omega)\in I_\eta\times I_\omega$.
\begin{defi}[$q$-growth condition]\label{q-nontrapp} For real $q$
  and $\omega_0 >0$ and for an obstacle $\Omega$ with Lipschitz
  boundary $\Gamma$, we say that $\Omega$, and $\Gamma$, satisfy a
  $q$-\emph{growth} condition iff for all $\omega \ge 0$, the operator
  $A_\omega$ in~\eqref{Aomega_def_eqn} satisfies either the bound
  \begin{equation}\label{eq:q-nontrapp}
    \norm{A_\omega^{-1}}_{L^2(\Gamma)\to L^2(\Gamma)} \le C (1 +
    \omega^2)^{q/2},
  \end{equation}
  for some constant $C > 0$, or, equivalently, the bound
  \begin{equation}\label{eq:q-nontrapp-cases}
    \left\|A_\omega^{-1}\right\|_{L^2(\Gamma)\to L^2(\Gamma)} \le
    \begin{cases} C_1, &~\mbox{if}~\omega \le \omega_0\\
      C_2 \omega^q, &~\mbox{if}~\omega > \omega_0
    \end{cases}
  \end{equation}
  for certain constants $C_1 > 0$ and $C_2 > 0$.
\end{defi}
Per~\cite[Lem.\ 6.2]{Spence:20}, polynomially growing bounds on the
norm of this inverse operator in the high-frequency regime can be
obtained on the basis of corresponding polynomially growing bounds as
$\omega\to\infty$ on the norm of the resolvent operator
$(\Delta + \omega^2)^{-1}$. The results of this article thus imply
superalgebraically-fast local energy decay of solutions to
\Cref{eq:w_eq} for any Lipschitz domain for which such real-axis
high-frequency resolvent bounds can be established.

\begin{rem}\label{3d_decay_generalization}
It is known that the $q$-growth condition is satisfied with a variety of
$q$-values for various classes of obstacles. For example, reference~\cite[Thm.\
1.13]{Spence:16} shows that a smooth \emph{nontrapping} obstacle
satisfies a $q$-growth condition with $q = 0$. A related $q = 0$ result is
presented in~\cite{ChandlerWilde:08} for merely Lipschitz domains, but under
the stronger assumption that the obstacle is star-shaped.
Reference~\cite{Spence:20} shows that for ``hyperbolic'' trapping regions (in
which all periodic billiard ball trajectories are unstable),
a merely logarithmic growth in $\omega$ results, while for certain
``parabolic'' trapping regions, stronger ($q = 2$ or $q = 3$) growth takes place. It is
also known that much more strongly-trapping obstacles exist, including
obstacles for which exponentially-large inverse operator norms
$\left\|A_\omega^{-1}\right\|_{L^2(\Gamma)\to L^2(\Gamma)}$
occur~\cite{Cardoso:02, BetckeChandlerWilde:10, Lafontaine:20} (and which,
therefore, do not satisfy the $q$-growth condition for any $q$).
\end{rem}

\begin{rem}\label{connected-trapping}
  There exist connected trapping obstacles containing cavities that
  satisfy the $q$-growth condition of \Cref{q-nontrapp} for some value
  of $q$. Indeed, the obstacle
  \[
\Omega = [-1,1]\times[-1,1]\times[-1,1]\setminus [-1/2,1/2]\times[-1/2,1/2]\times[0,1]
    \]
    (a cube with a smaller cube removed from one of its faces as
    displayed in \Cref{fig:3d_connected_trapping}) is an
    $(R_0, R_1, a)$ parallel trapping obstacle in the sense
    of~\cite[Def.\ 1.9]{Spence:20}, for $R_1 > {\mathrm{e}}^{1/4} R_0$,
    $R_0 \ge \sqrt{3/2}$, and $a = 1$.  According to~\cite[Cor.\ 1.14
    and Rem.\ 1.16]{Spence:20}, $\Omega$ satisfies a $q$-growth
    condition with $q=3$.  Smoothing of the corners of this obstacle
    results in a connected trapping obstacle that satisfies a
    $q$-growth condition with $q=2$.
\end{rem}

\begin{figure}
  \centering
  \includegraphics[width=0.49\textwidth]{connected-trapping-cavity.png}
  \includegraphics[width=0.49\textwidth]{elongated-connected-trapping-cavity.png}
  \caption[Example of connected trapping obstacles that satisfy a $q$-growth
  condition and for which wave equation decay rates are established.]{Examples
  of connected trapping obstacles that satisfy a $q$-growth condition ($q = 3$)
  and for which superalgebraically-fast wave equation time decay rates are
  established in this article. Left: Visualization of the obstacle given in
  \Cref{connected-trapping}, and which serves to demonstrate the existence of connected
  trapping obstacles satisfying the $q$-growth condition of \Cref{q-nontrapp}. Right: An
  elongated cavity trapping obstacle, with a vertical dimension of $12$ units
  and rectangular dimensions of $4$ units, that also satisfies a $q$-growth
  condition ($q = 3$).}\label{fig:3d_connected_trapping}
\end{figure}

\section{Uniform bootstrap domain-of-dependence boundary density estimates}\label{sec:theory_part_i}

This section presents uniform-in-time domain-of-dependence estimates
(\Cref{3d_decay_thm}) on the solution $\psi$ of the time-domain
boundary integral equation~\cref{eq:tdie_sl}. In particular, these
estimates show that if $\psi$ is ``small'' throughout $\Gamma$ for any
domain-of-dependence time interval of length
\begin{equation}\label{Tstar_def}
 T_* \coloneqq \diam(\Gamma) / c = \max_{\mathbf{r},\mathbf{r}' \in \Gamma} |\mathbf{r} - \mathbf{r}'|/c
\end{equation}
after the $\Gamma$-values of the incident field have been turned off,
then $\psi$ will remain small for all time thereafter. At a
fundamental level, the domain-of-dependence analysis presented in this
section exploits an interesting property of
equation~\eqref{eq:tdie_sl} (made precise in
Lemmas~\ref{3d_decay_lemma_2ndkind}
and~\ref{3d_decay_thm_h_equiv}), namely, that, after the incident
field $b(\mathbf{r}, t)$ has been turned off permanently throughout
$\Gamma$, the values of its solution $\psi(\mathbf{r}, t)$ over a
domain-of-dependence time length $T^*$ determine the solution uniquely
for all subsequent times: the boundary integral density over the
interval $I_{T}$ encodes the necessary information to reproduce all
future scattering events and, in particular, it encapsulates the
effect of all previously imposed boundary values, over time periods of
arbitrarily long duration.

For technical reasons we utilize domain-of-dependence intervals
$I_{T}$, as detailed in \Cref{domainofdep}, of a length slightly
larger than $T^*$---larger by a small amount $2\tau>0$. Any positive
value of $\tau$ can be used, and the selection only affects the
multiplicative constants and integration domains in the main
domain-of-dependence estimates presented in this paper. The rest of
this section is organized as follows: Definitions~\ref{domainofdep}
through~\ref{def:sob_bochner} lay down the conventions necessary to
subsequently state and prove the main result of the section, namely,
\Cref{3d_decay_thm}. After these definitions, the statement of the
theorem is introduced.  Lemmas~\ref{psi_Hp} through~\ref{sob_lemma} then
establish a series of results required in the proof of the theorem,
and then, concluding the section, the theorem's proof is presented.

\begin{defi}[Domain-of-dependence interval]\label{domainofdep}
  Using the definition~\eqref{Tstar_def},
  $T_* \coloneqq \diam(\Gamma) / c$, for a given Lipschitz boundary
  $\Gamma$, and for given real numbers $T$ and $\tau > 0$, the time
  interval
  \begin{equation}\label{time_interval_def}
    I_{T} = I_{T,T_*,\tau}= [T - T_* - 2\tau, T),
  \end{equation}
  will be referred to as the $\tau$-extended ``domain-of-dependence''
  interval ($\tau$DoD) relative to the ``observation'' time $T$.  (As
  suggested by the notations in~\eqref{time_interval_def}, the
  dependence of $I_T$ on the paramaters $T_*$ and $\tau$ will not be
  made explicit in what follows.)
\end{defi}

\begin{rem}\label{ITvsI0}
  The interval $I_T$ plays two important roles in our analysis. On one
  hand, this interval, for some given value $T=T_0$, figures
  prominently in the statements of the two main theorems, namely,
  Theorems~\ref{3d_decay_thm} and~\ref{3d_decay_thm_ii}: the estimates
  provided by these theorems are valid for times past the end of the
  interval $I_{T_0}$ for any fixed given time $T_0$ under
  consideration. To adequately account for the complex interplay of
  recentering in time and oscillation rate in the frequency variable
  $\omega$ for solutions of the wave equation, on the other hand, the
  proof of Theorem~\ref{3d_decay_thm_ii} also relies on use of the
  interval $I_T$ and the time $T_0$, but, this time, with
  $T = 0 < T_0$. Thus, the recentering idea that is exploited
  in~\cite{Anderson:20} to eliminate oscillations induced in frequency
  domain by the passage of time, is also employed here as an element
  of the proof of Theorem~\ref{3d_decay_thm_ii} (or, more precisely,
  in the proof of the preliminary Lemma~\ref{Rderiv_sobolev_bound}) to
  usefully exploit the frequency-time isometry inherent in the
  Plancherel theorem without incurring the impact of large frequency
  derivatives and resulting uncontrolled constants in Sobolev
  estimates provided in Lemma~\ref{Rderiv_sobolev_bound}. To make this
  possible, a translation in time, and thus the use of intervals $I_T$
  with $T\ne T_0$, become necessary.
\end{rem}

\begin{defi}[Time-windowed solutions]\label{timewinddens}
  For a given solution $\psi$ to~\eqref{eq:tdie_sl} and a given
  $\tau$DoD interval $I_{T}$, using non-negative window functions
  \begin{equation}\label{wtau_def}
  w_-(t) = \begin{cases} &1\mbox{~for~} t < -\tau\\
  &0\mbox{~for~} t \ge 0
  \end{cases},\quad
  w_+(t) = \begin{cases} &0\mbox{~for~} t < -\tau\\
  &1\mbox{~for~} t \ge 0,
  \end{cases}
  \end{equation}
  satisfying the ``Partition-of-unity'' relation $w_- + w_+ = 1$, we
  define the auxiliary densities
  \begin{equation}\label{psipm_def}
    \psi_{-,T}({\mathbf{r}}, t) = w_-(t - T)\psi({\mathbf{r}},
    t),\quad\psi_{+,T}({\mathbf{r}}, t) = w_+(t - T)\psi({\mathbf{r}}, t),
  \end{equation}
  whose temporal supports satisfy
  $\supp \psi_{-,T} \subset (-\infty, T]$ and
  $\supp \psi_{+,T} \subset [T -\tau, \infty)$, and for which the
  relation
\begin{equation}\label{pou_decomp}
  \psi(\cdot, t) = \psi_{-,T_0}(\cdot, t) + \psi_{+,T_0}(\cdot, t)
\end{equation}
holds for all time $t$. We further define
  \begin{equation}\label{psistar_def}
    \psi_{*,T}(\mathbf{r}, t) = w_+(t - T + T_* + \tau)w_-(t - T)\psi(\mathbf{r}, t),
  \end{equation}
  which is nonzero only in the interior of $I_{T}$
  (equation~\eqref{time_interval_def}). Similarly, for a given integer
  $p > 0$ we define the functions
  \begin{equation}\label{psipm_p_def}
    \psi_{p,-,T}({\mathbf{r}}, t) = w_-(t - T)\partial^p \psi({\mathbf{r}},
    t),\quad\psi_{p,+,T}({\mathbf{r}}, t) = w_+(t - T)\partial^p \psi({\mathbf{r}}, t)
  \end{equation}
  and
  \begin{equation}\label{psistar_p_def}
      \psi_{p,*,T}(\mathbf{r}, t) = w_+(t - T + T_* + \tau)w_-(t - T)\partial^p \psi(\mathbf{r}, t).\qedhere
  \end{equation}
\end{defi}

The smooth temporal decompositions introduced
in \Cref{timewinddens} play central roles in the derivations of the
uniform-in-time bounds and decay estimates presented in this
paper. Noting that $\psi$ is identical to $\psi_{+,T_0}$ for
$t > T_0$, to produce such bounds and estimates we first relate
$\psi_{+,T_0}$ to $\psi_{-,T_0}$ and thereby obtain a norm bound for
$\psi_{+,T_0}$ over the slightly larger time interval
$[T_0 - \tau, \infty)$---which then yields, in particular, the desired
domain-of-dependence estimate on the interval $[T_0 , \infty)$.  The
necessary bounds for $\psi_{+,T_0}$ are produced via Fourier
transformation into the frequency domain and subsequent use of
existing wavenumber-explicit
bounds~\cite{ChandlerWilde:08,Spence:14,Spence:16,Spence:20,ChandlerWilde:09}
on the inverses of frequency-domain versions of our integral
scattering operator. A similar approach is followed for the
time-derivatives of the incident field data, leading to bounds in
temporal Sobolev norms of arbitrary orders, and, in particular, via
Sobolev embeddings, to uniform bounds in time.  As suggested above,
the  approach intertwines the frequency and time domains, and it
thus incorporates frequency-domain estimates while also exploiting the
time domain Huygens' principle, cf.\ for example the relations~\cref{h_equiv} and~\cref{HtoUbound0}.

The main result of this section, \Cref{3d_decay_thm}, which is stated
in what follows, relies on the definition of Sobolev-Bochner spaces in \Cref{sob-boch}.
\begin{theorem}\label{3d_decay_thm}
  Let $p$ and $q$ denote non-negative integers, let $T_0 > 0$ and
  $\tau > 0$ be given, and assume (i) $\Gamma$ satisfies the
  $q$-growth condition (\Cref{q-nontrapp}); (ii) The incident field
  satisfies the $s$-regularity
  conditions~\eqref{eq:gamma_Hs_assump} with
  $s = p + 2q + 1$; as well as, (iii) The incident field
  $b = b({\mathbf{r}}, t)$ satisfies \Cref{all_t_b} and it vanishes for
  $({\mathbf{r}}, t) \in \widebar{\Omega} \times \left\lbrace I_{T_0} \cup [T_0, \infty)\right\rbrace$, with $I_{T_0}$ as in
  \Cref{domainofdep}. Then, the solution $\psi$ of \Cref{eq:tdie_sl}
  satisfies both the $H^p$ estimate
\begin{equation}\label{density_Hp_time_bound}
  \left\|\psi\right\|_{H^p([T_0, \infty);\,L^2(\Gamma))} \le
  C(\Gamma, \tau, p) \left\|\psi\right\|_{H^{p+q+1}(I_{T_0};\,L^2(\Gamma))}
  < \infty,
\end{equation}
and, in the case $p=1$ the time-uniform estimate
\begin{equation}\label{density_unif_time_bound}
  \left\|\psi(\cdot, t)\right\|_{L^2(\Gamma)} \le
  C(\Gamma, \tau)\left\|\psi\right\|_{H^{q+2}(I_{T_0};\,L^2(\Gamma))}\quad \mbox{for}\quad t > T_0.
\end{equation}
\end{theorem}

\begin{rem}\label{laplace_estimate_remark}
    Several results in this article, including \Cref{3d_decay_thm}, \Cref{int_eq_pderiv}, \Cref{3d_decay_thm_ii} and \Cref{decay_estimate_L2},
    utilize the (Fourier domain) $q$-growth condition-based estimate of \Cref{psi_Hp} to ensure that $\psi \in H^p(\mathbb{R}; L^2(\Gamma))$ in order to establish the less stringent requirement that
    $\psi \in H^p([r, s]; L^2(\Gamma))$, $-\infty < r < s < \infty$. It is interesting to note, however, that the regularity assumptions of all of the aforementioned results can be relaxed if, instead,
    Laplace-domain bounds are utilized. In contrast to \Cref{psi_Hp}, on the
    basis of Laplace-domain estimates (in particular~\cite[Thm.\ 4.2]{Monk:14}
    in conjunction with~\cite[Lem.\ 2]{Chen:10}, cf.\
    also~\cite{HaDuong:86,Lubich:94}) such bounds provide that if $b$ is causal
    and satisfies
    \[
        \gamma^+ b \in H^{p+1}([0, T]; L^2(\Gamma))\quad\mbox{and}\quad \gamma^+ \partial_\mathbf{n} b \in H^p([0, T]; L^2(\Gamma)),
    \]
    then $\psi \in H^p([0, T]; L^2(\Gamma))$. While we eschewed use of these bounds in order that our results be more self-contained,
    we note here how the regularity assumptions of the various lemmas and theorems
    can be relaxed by application of these auxiliary results. Assumption (ii) of \Cref{3d_decay_thm} can be relaxed to
    the $s$-regularity conditions with $s = p + q + 1$, the assumptions of \Cref{int_eq_pderiv} can be relaxed to
    the $s$-regularity conditions with $s = p + 1$, assumption (ii) of \Cref{3d_decay_thm_ii} can be relaxed to
    $s$-regularity conditions with $s = p + (n+1)(q+1)$, and, finally, the assumptions of \Cref{decay_estimate_L2} can be relaxed to
    the $s$-regularity conditions with $s = (n+1)(q+1)$.
    Note that none of the estimates are quantitatively affected---these assumptions effectively are only used to ensure that the right-hand side quantities in the desired estimates are finite.
    Also note that all of the assumptions mentioned here are immediately satisfied for data $b$ that is smooth in time---see \Cref{rem:wave_eq_sob_assump}.
\end{rem}

The proof of \Cref{3d_decay_thm} is deferred to the end of this
section, following a series of preparatory Lemmas
(Lemmas~\ref{psi_Hp} through \ref{sob_lemma}). The first of these
lemmas relates the size of the solution of equation~\eqref{eq:tdie_sl}
to the size of the imposed incident fields
$\gamma^+ b(\mathbf{r}, t)$ in various norms and under various assumptions on $\Gamma$.

\begin{rem}\label{negative_freq}
  Without loss of generality, the incident field $b(\mathbf{r}, t)$ is
  assumed to be real-valued. It follows that the solution
  $\psi(\mathbf{r}, t)$ of \Cref{eq:tdie_sl} is real-valued, which
  implies that its Fourier transform $\psi^f(\mathbf{r}, \omega)$
  satisfies the Hermitian symmetry relation
  \begin{equation}\label{negative_freq_eq}
    \psi^f(\mathbf{r}, -\omega) = \overline{\psi^f(\mathbf{r}, \omega)}.
  \end{equation}
  Our studies of frequency-domain operator norms are therefore
  restricted to the range $\omega \geq 0$, as is often the case in
  mathematical frequency-domain scattering
  theory~\cite{Spence:20,Spence:16,BetckeChandlerWilde:10}. The same
  symmetry relations apply for quantities such as $h_T(\mathbf{r}, t)$,
    $\psi_{\pm,T}(\mathbf{r}, t)$, $\psi_{*,T}(\mathbf{r}, t)$, etc., that are
  defined in terms of the real-valued $\psi(\mathbf{r}, t)$.
\end{rem}

\begin{lemma}\label{psi_Hp}
  Given an integer $q\geq 0$, let $\Gamma$ satisfy a $q$-growth
  condition, and let the incident field $b$ satisfy the
  $s$-regularity conditions~\eqref{eq:gamma_Hs_assump} with
  $s = p + q$. Then \Cref{eq:tdie_sl} admits a unique solution $\psi \in H^p(\mathbb{R};L^2(\Gamma))$ and
  \begin{equation}\label{psi_Hp_bound}
    \left\|\psi\right\|_{H^p(\mathbb{R};\,L^2(\Gamma))} \le
    C(\Gamma)\left( \left\|\gamma^+ \partial_\mathbf{n}
        b\right\|_{H^{p+q}(\mathbb{R};\,L^2(\Gamma))}\\
      + \left\|\gamma^+
        b\right\|_{H^{p+q+1}(\mathbb{R};\,L^2(\Gamma))}\right).
  \end{equation}
  If, additionally, $b$ satisfies
  the $s$-regularity conditions~\eqref{eq:gamma_Hs_assump} with
  $s = p + q + 1$, then $\psi \in C^p(\mathbb{R};L^2(\Gamma))$.
\end{lemma}
\begin{proof}
  Using equation~\cref{CFIE_direct} we obtain
\[
    (1 + \omega^2)^{p/2} \psi^f = (1 + \omega^2)^{p/2}
    A_\omega^{-1}\left(\gamma^+ \partial_\mathbf{n} B^f - \i \eta \gamma^+B^f \right),
  \]
  and, thus, in view of \eqref{eq:q-nontrapp} and taking into account
  the relation $|\eta| \le (1 + \omega^2)^{1/2}$ that follows easily
  from~\eqref{Aomega_def_eqn}, we obtain
  \begin{align*}
    &\begin{alignedat}{2}(1 + \omega^2)^p\left\|\psi^f(\cdot, \omega)\right\|^2_{L^2(\Gamma)} \le C(1
      + \omega^2)^{p+q}&
    \left(\left\|\gamma^+\partial_\mathbf{n} B^f(\cdot,
      \omega)\right\|^2_{L^2(\Gamma)}\right.\\
      &+ \left.|\eta|^2\left\| \gamma^+B^f(\cdot, \omega)\right\|^2_{L^2(\Gamma)}\right)
    \end{alignedat}\\
    &\le C\left((1 + \omega^2)^{p+q}\left\|\gamma^+\partial_\mathbf{n} B^f(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}
    + (1 + \omega^2)^{p+q+1}\left\|\gamma^+B^f(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\right).
  \end{align*}
  Integrating with respect to $\omega$ the
  estimate~\cref{psi_Hp_bound} follows.  The fact that
  $\psi \in C^p(\mathbb{R};\,L^2(\Gamma))$ results from an application
  of~\eqref{psi_Hp_bound} with $p$ substituted by $p+1$ and the use of
  the Sobolev embedding \Cref{sob_lemma}.
\end{proof}

\begin{rem}\label{rem:tilde_1}
  It will be necessary in what follows (specifically, in the proofs of
  the $H^p$ estimates in \Cref{3d_decay_thm} and
  \Cref{3d_decay_thm_ii} for $p > 0$ and associated preparatory
  lemmas) to allow for right-hand sides other than the specific wave
  solution $b(\mathbf{r}, t)$ present in the Dirichlet problem
  \Cref{eq:w_eq} and the integral equation formulation
  \Cref{eq:tdie_sl}, and we thus consider scattering problems of an
  incident field wave solution $\widetilde{b}(\mathbf{r}, t)$, similar
  to $b$, for which we have a corresponding integral equation
  formulation
\begin{equation}\label{eq:tdie_sl_generic}
    (S\widetilde{\psi})(\mathbf{r}, t) = \gamma^+ \widetilde{b}(\mathbf{r}, t)\quad\mbox{for}\quad (\mathbf{r}, t) \in
    \Gamma\times\mathbb{R},
\end{equation}
with solution $\widetilde{\psi}$, where $S$ denotes the single layer
integral operator \Cref{eq:single_layer_op_time}. We also define
$\widetilde{\psi}_{-,T}$, $\widetilde{\psi}_{+,T}$, and
$\widetilde{\psi}_{*,T}$ analogously to $\psi_{-,T}$, $\psi_{+,T}$,
and $\psi_{*,T}$, respectively, in \Cref{timewinddens}.
Of course, Lemma~\ref{psi_Hp} also holds in the context of solutions
of the integral equation~\cref{eq:tdie_sl_generic}.
\end{rem}

Analogous to \Cref{all_t_b} and letting $\mathbb{R}_\alpha^- =\{t\in\mathbb{R}\ : \ t\leq \alpha \}$
we assume, for some $\alpha \in \mathbb{R}$, that
\begin{equation}\label{all_t_b_generic}
\mbox{``The function}\; \widetilde{b} \in C^2(\Omega^\textit{inc} \times \mathbb{R})\;\mbox{satisfies}\; \widetilde{b}({\mathbf{r}}, t) = 0\;\mbox{for}\; ({\mathbf{r}}, t) \in \widebar{\Omega} \times \mathbb{R}_\alpha^-\mbox{''}.
\end{equation}

In preparation for Lemma~\eqref{3d_decay_lemma_2ndkind}, letting
\begin{equation}\label{umintilde_def}
  \widetilde{u}_{-,T}(\mathbf{r}, t) := (S \widetilde{\psi}_{-,T})(\mathbf{r}, t), \quad (\mathbf{r}, t) \in \mathbb{R}^3 \times \mathbb{R},
\end{equation}
we define the function
\begin{equation}\label{htildedef}
  \widetilde{h}_T(\mathbf{r}, t) := \widetilde{b}(\mathbf{r}, t) -
  \widetilde{u}_{-,T}(\mathbf{r}, t),\quad (\mathbf{r}, t) \in \Omega^{inc} \times \mathbb{R},
\end{equation}
and its Fourier transform
  \begin{equation}\label{Ht_def_generic}
    \widetilde{H}^f_T(\mathbf{r}, \omega) = \widetilde{B}^f(\mathbf{r}, \omega) - (S_\omega
      \widetilde{\psi}_{-,T}^f)(\mathbf{r}, \omega),\quad \mathbf{r}\in\Omega^{inc} \times \mathbb{R},
\end{equation}
(see~\eqref{eq:single_layer_op}.

\begin{rem}\label{rem:tilde_2}
  With reference to Remark~\ref{rem:tilde_1}, we mention that
  Lemma~\eqref{3d_decay_lemma_2ndkind} and subsequent lemmas apply in
  particular to the function $\widetilde{b} = b$ in \Cref{eq:tdie_sl};
  in that case we will accordingly use untilded notations as in e.g.
\begin{equation}\label{hdef}
  h_T(\mathbf{r}, t) := b(\mathbf{r}, t) - u_{-,T}(\mathbf{r}, t)
\end{equation}
instead of~\eqref{htildedef},
\begin{equation}\label{umin_def}
  u_{-,T}(\mathbf{r}, t) := (S \psi
  _{-,T})(\mathbf{r}, t), \quad (\mathbf{r}, t) \in \mathbb{R}^3 \times \mathbb{R},
\end{equation}
instead of~\eqref{umintilde_def}, and
\begin{equation}\label{Ht_untilded_def}
   H^f_T(\mathbf{r}, \omega) = B^f(\mathbf{r}, \omega) - (S_\omega
      \psi_{-,T}^f)(\mathbf{r}, \omega),\quad \mathbf{r}\in\Omega^{inc} \times \mathbb{R},
\end{equation}
instead of~\eqref{Ht_def_generic}.
\end{rem}

Relying on the $\tau$DoD intervals $I_{T}$ introduced in
\Cref{domainofdep} as well as the windowed time-dependent boundary
densities $\psi_{-,T}$, $\psi_{+,T}$, and $\psi_{*,T}$ in
\Cref{timewinddens}, and the trace operators $\gamma^\pm$ on $\Gamma$
in \Cref{trace_def}, Lemmas~\ref{3d_decay_lemma_2ndkind}
to~\ref{per_freq_rhs_oper_bounds} below develop estimates related to
the density $\psi_{+,T}$ as well as frequency-domain operator bounds
that are used in the proof of \Cref{3d_decay_thm}. The subsequent
\Cref{sob_lemma} is a Sobolev embedding result for functions taking
values in a Banach space (i.e. $L^2(\Gamma)$), used to establish
estimates that are uniform-in-time.
\begin{lemma}[Direct second-kind integral equations for
  $\widetilde{\psi}_{+,T}^f$]\label{3d_decay_lemma_2ndkind}
    Assume that the obstacle $\Omega$ satisfies the $q$-growth condition for some integer $q \ge 0$.
    Let $\widetilde{b}$ denote a function defined in an open set
    $\Omega^\textit{inc}$ containing $\widebar{\Omega}$, and assume that
    $\widetilde{b}$ is a solution to the wave equation in
    $\Omega^\textit{inc}$ satisfying the $s$-regularity conditions~\eqref{eq:gamma_Hs_assump} with
    $s = q$.
    Additionally, let $T$ denote a real number with associated
    time-windowed solutions $\widetilde{\psi}_{+,T}$ and
    $\widetilde{\psi}_{-,T}$ as in \Cref{timewinddens} and
    \Cref{rem:tilde_1}, where $\widetilde{\psi}$ is the solution to
    \Cref{eq:tdie_sl_generic}. Then $\widetilde{\psi}_{+,T}$ satisfies
    the integral equation
    \begin{equation}\label{int_eq_lambda}
      (S\widetilde{\psi}_{+,T})(\mathbf{r}, t) = \gamma^+ \widetilde{h}_T(\mathbf{r}, t),
    \end{equation}
    (cf. \Cref{eq:tdie_sl}) where $\widetilde{h}_T$ is given by
    \Cref{htildedef}.  Additionally, for each fixed $\omega \ge 0$ the
    Fourier transform $\widetilde{\psi}_{+,T}^f$ of
    $\widetilde{\psi}_{+,T}$ satisfies the second-kind integral
    equation
\begin{equation}\label{CFIE_proof_generic}
  \left(A_{\omega} \widetilde{\psi}_{+,T}^f \right)(\mathbf{r}, \omega) =
    \gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\mathbf{r}, \omega)
    - \i \eta \gamma^- \widetilde{H}^f_T(\mathbf{r}, \omega), \quad\mathbf{r} \in \Gamma,
\end{equation}
with $A_{\omega}$ as in \Cref{Aop_def} and $\widetilde{H}^f_T$ as in \Cref{Ht_def_generic}.
\end{lemma}
\begin{rem}\label{rem:tilde_3}
  With reference to \Cref{rem:tilde_1}, we note that, for the
  particular case $\widetilde{b} = b$, \Cref{3d_decay_lemma_2ndkind}
  tells us that the solution $\psi$ to \Cref{eq:tdie_sl} satisfies
\begin{equation}\label{CFIE_proof}
  \left(A_{\omega} \psi_{+,T}^f \right)(\mathbf{r}, \omega) =
    \gamma^-\partial_\mathbf{n} H^f_T(\mathbf{r}, \omega)
    - \i \eta \gamma^-H^f_T(\mathbf{r}, \omega), \quad\mathbf{r} \in \Gamma,
\end{equation}
where
\begin{equation}\label{Ht_def}
        H^f_T(\mathbf{r}, \omega) = B^f(\mathbf{r}, \omega) - \left(S_\omega
        \psi_{-,T}^f\right)(\mathbf{r}, \omega).\qedhere
\end{equation}
\end{rem}
\begin{proof}[Proof of \Cref{3d_decay_lemma_2ndkind}.]
  Using the partition of unity decomposition embodied
  in~\cref{psipm_def}, equation~\Cref{eq:tdie_sl_generic} may be
  re-expressed in the form
    \begin{equation}\label{int_eq_lambda_proof}
      \int_{-\infty}^f \int_\Gamma \frac{\delta\left(t - t' - |\mathbf{r} -
        \mathbf{r}'|/c\right) \widetilde{\psi}_{+,T}(\mathbf{r}', t')}{4\pi |\mathbf{r} -
        \mathbf{r}'|}\,\d\sigma(\mathbf{r}')\,\d t' = \gamma^+ \widetilde{h}_T(\mathbf{r}, t),
    \end{equation}
    where $\widetilde{h}_T$ and $\widetilde{u}_{-,T}$ are given by
    \Cref{htildedef} and \Cref{umintilde_def} respectively, and,
    thus,~\Cref{int_eq_lambda} follows.

    In order to establish~\Cref{CFIE_proof_generic}, in turn, we first
    consider the solution $v^+$ of the Dirichlet problem
    $\Delta v^+ + \kappa^2(\omega) v^+ = 0$
    ($\kappa(\omega) = \omega / c$) in
    $\mathbb{R}^3 \setminus \widebar{\Omega}$ with Dirichlet data
    $\gamma^+ v^+ = -\gamma^+ \widetilde{H}^f_T$, and we call
    $v^- = -\widetilde{H}^f_T$, which, in view of~\eqref{Ht_def_generic}
    and since $B^f$ is the Fourier transform of the solution $b$ of
    the wave equation in $\Omega^\textit{inc} \supset \Omega$,
    satisfies the Helmholtz equation
    $\Delta v^- + \kappa^2(\omega) v^- = 0$ in $\Omega$ with Dirichlet
    data $\gamma^- v^- = -\gamma^+ \widetilde{H}^f_T$. Continuing with
    the proof of~\Cref{CFIE_proof_generic}, we now derive, by
    application of Green's theorem to the functions $v^+$ and $v^-$, a
    single-layer equation that is satisfied by the solution
    $\widetilde{\psi}_{+,T}^f$ of that equation. In detail,
    per~\cite[Lem.\ 3.4]{Costabel:88} we have the representation
    formulas
  \begin{equation}\label{vs_repr_11}
    v^+(\mathbf{r}, \omega) = -\langle \gamma^+ \partial_\mathbf{n} v^+ -
    \gamma^- \partial_\mathbf{n} v^-,
    G_\omega(\mathbf{r}, \cdot) \rangle,\quad\mathbf{r} \in
    \mathbb{R}^3 \setminus \widebar{\Omega},
  \end{equation}
  and
  \begin{equation}\label{vs_repr_12}
    v^-(\mathbf{r}, \omega) = -\langle \gamma^+ \partial_\mathbf{n} v^+ -
    \gamma^- \partial_\mathbf{n} v^-,
    G_\omega(\mathbf{r}, \cdot) \rangle,\quad\mathbf{r} \in
    \Omega,
  \end{equation}
  where $\langle \cdot,\cdot \rangle$ denotes the duality pairing of the
  Sobolev spaces $H^s(\Gamma)$ and $H^{-s}(\Gamma)$. But, in view
  of~\cite[Cor.\ 2.28]{ChandlerWilde:12} we have $\gamma^+\partial_\mathbf{n} v^+ \in
  L^2(\Gamma)$ and $\gamma^-\partial_\mathbf{n} v^- \in L^2(\Gamma)$, so
  \Cref{vs_repr_11}
  and \Cref{vs_repr_12} may be re-expressed in the form
 \begin{equation}\label{vs_repr_21}
    v^+(\mathbf{r}, \omega) = -\int_\Gamma \left(\gamma^+\partial_\mathbf{n} v^+ -
    \gamma^- \partial_\mathbf{n} v^-\right)
    G_\omega(\mathbf{r}, \mathbf{r}')\,\d\sigma(\mathbf{r}'),\quad\mathbf{r} \in
    \mathbb{R}^3 \setminus \widebar{\Omega},
  \end{equation}
  and
    \begin{equation}\label{vs_repr_22}
    v^-(\mathbf{r}, \omega) = -\int_\Gamma \left(\gamma^+ \partial_\mathbf{n} v^+ -
    \gamma^- \partial_\mathbf{n} v^-\right)
    G_\omega(\mathbf{r}, \mathbf{r}')\,\d\sigma(\mathbf{r}'),\quad\mathbf{r} \in
    \Omega.
  \end{equation}
  In view of the continuity throughout $\mathbb{R}^3$ of the single-layer potential $S_\omega$ with $L^2(\Gamma)$ density~\cite[Thm.\ 6.11]{McLean}
  and setting
  \begin{equation}\label{mu_density}
    \mu = \gamma^+ \partial_\mathbf{n} v^+ - \gamma^-\partial_\mathbf{n} v^-,
  \end{equation}
  taking the interior trace
  $\gamma^-$ of
  \Cref{vs_repr_22} and using the definition that $v^- = -\widetilde{H}^f_T$
  tells us that $\mu$ satisfies the integral equation
\begin{equation}\label{single_layer_lambda}
    (S_\omega \mu) = \gamma^-\widetilde{H}^f_T, \quad \mathbf{r} \in \Gamma.
\end{equation}

  In order to obtain an integral equation which, unlike
  \Cref{single_layer_lambda}, is uniquely solvable at all frequencies,
  we follow~\cite{Burton:71} and combine \Cref{single_layer_lambda}
  with the equation
  \begin{equation}\label{psiplus_deriv_int_eq}
  \gamma^+ \partial_\mathbf{n} v^+(\mathbf{r}, \omega) =
    - \int_\Gamma \mu(\mathbf{r}', \omega) \partial_{\mathbf{n}(\mathbf{r})} G_\omega(\mathbf{r},
  \mathbf{r}')\,\d\sigma(\mathbf{r}')
      + \frac{1}{2}\mu({\mathbf{r}}, \omega),\quad\mbox{for}\quad\mathbf{r} \in
      \Gamma.
  \end{equation}
  that results as the outer normal derivative operator on $\Gamma$ is
  applied to the representation formula~\cref{vs_repr_22}.
  In detail, \Cref{psiplus_deriv_int_eq} follows from the well-known jump
  relations~\cite[p. 219]{McLean} that
  \begin{equation}
    \lim_{\varepsilon \to 0^+} \mathbf{n}(\mathbf{r}) \cdot \nabla \int_\Gamma \mu(\mathbf{r}')
    G_\omega(\mathbf{r} + \varepsilon\mathbf{n}(\mathbf{r}),
    \mathbf{r}')\,\d\sigma(\mathbf{r}') = \left(-\frac{1}{2}I +
    K_\omega^*\right)\mu(\mathbf{r}),\,\mathbf{r} \in \Gamma,
  \end{equation}
  for
  the operator equal to the normal derivative on $\Gamma$ of the single-layer
  potential with density $\mu$ in the space $L^2(\Gamma)$
  (cf.\ also~\cite[Thm.\ 1 \& Lem.\ 4.1]{Costabel:88}).
  Using~\cref{mu_density} to rewrite the
  left-hand-side of~\cref{psiplus_deriv_int_eq} gives the second-kind
  integral equation for $\mu$,
\begin{equation}\label{psiplus_2nd_kind}
  \frac{1}{2}\mu(\mathbf{r}, \omega) +
    \left(K_\omega^*\mu \right)(\mathbf{r}, \omega) =
    \gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\mathbf{r}, \omega).
\end{equation}
  posed for $\mu \in L^2(\Gamma)$~\cite[Thm.\ 1(iv)]{Costabel:88}.
    Finally, a linear combination of \cref{single_layer_lambda}
    and~\cref{psiplus_2nd_kind} yields the combined field integral
    equation
\begin{equation}
  \left(A_{\omega} \mu \right)(\mathbf{r}, \omega) =
    \gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\mathbf{r}, \omega)
    - \i \eta \gamma^- \widetilde{H}^f_T(\mathbf{r}, \omega), \quad\mathbf{r} \in \Gamma,
\end{equation}
where $A_{\omega} = \frac{1}{2}I + K_\omega^* - \i\eta S_\omega$,
which is a uniquely solvable integral equation, for all $\omega \ge 0$, for the density $\mu$ in \Cref{mu_density}~\cite[Thm.\ 2.27]{ChandlerWilde:12}.

But $\mu$ also solves the integral
equation~\Cref{single_layer_lambda}, so since $\widetilde{H}^f_T$ is
the Fourier transform of $\widetilde{h}_T$ given in~\cref{htildedef},
inverse Fourier transformation of \Cref{single_layer_lambda} yields
\Cref{int_eq_lambda} and so $\mu = \widetilde{\psi}_{+,T}^f$, the
Fourier transform of the unique solution $\widetilde{\psi}_{+,T}$ to
\Cref{int_eq_lambda}. Thus, $\mu = \widetilde{\psi}_{+,T}^f$
satisfies~\Cref{CFIE_proof_generic}, and the proof is complete.
\end{proof}

\begin{lemma}[Frequency $L^2$ bounds on the solution
  of~\cref{CFIE_proof_generic}]\label{3d_decay_lemma_Ainv_freq_bounds}
  Let $T\in\mathbb{R}$ and $q \geq 0$, let $\Gamma = \partial \Omega$
  satisfy the $q$-growth condition, and assume $\widetilde{b}$
  satisfies the conditions of \Cref{3d_decay_lemma_2ndkind}. Then
  there exists $C_1 = C_1(\Gamma) > 0$ and $C_2 = C_2(\Gamma) > 0$
  such that $\widetilde{\psi}_{+,T}$ satisfies
\begin{equation}\label{psiplus_estimate_3}
    \begin{split}
    \left\|\widetilde{\psi}_{+,T}^f\right\|^2_{L^2(\mathbb{R};\,L^2(\Gamma))}
    &\le C_1\int_0^{\omega_0}
    \left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\cdot,
    \omega) - \i \gamma^-\widetilde{H}^f_T(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega\\
    & + C_2\int_{\omega_0}^\infty\omega^{2q}
    \left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\cdot,
    \omega) - \i\omega \gamma^- \widetilde{H}^f_T(\cdot.
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega,
    \end{split}
\end{equation}
    where $\widetilde{H}^f_T$ is given by \Cref{Ht_def_generic}.
\end{lemma}
\begin{proof}
  As indicated in \Cref{A_om_invert}, $A_\omega$ is invertible for all
  $\omega \ge 0$. From~\cref{CFIE_proof_generic} we then obtain
  $\widetilde{\psi}_{+,T}^f = A_{\omega}^{-1} \left(
    \gamma^-\partial_\mathbf{n} \widetilde{H}^f_T - \i \eta
    \gamma^- \widetilde{H}^f_T\right)$, and thus
\begin{equation*}
\begin{split}
  \|\widetilde{\psi}_{+,T}^f(\cdot, \omega)\|_{L^2(\Gamma)} \le
    \|A_{\omega}^{-1}&\|_{L^2(\Gamma) \to L^2(\Gamma)}
    \left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\cdot,
    \omega) - \i\eta \gamma^-\widetilde{H}^f_T(\cdot,
    \omega)\right\|_{L^2(\Gamma)},\label{psiplus_estimate_1}
\end{split}
\end{equation*}
    for all $\omega \ge 0$.

    In view of the $q$-growth condition~\Cref{eq:q-nontrapp-cases}
    we obtain
\begin{equation}
  \|\widetilde{\psi}_{+,T}^f(\cdot, \omega)\|_{L^2(\Gamma)} \le
    \frac{C_1}{2}\omega^{2q}\left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\cdot,
    \omega) - \i\omega \gamma^-\widetilde{H}^f_T(\cdot,
    \omega)\right\|_{L^2(\Gamma)}\, (\omega > \omega_0)\label{psiplus_estimate_2}
\end{equation}
    and
\begin{equation}
  \|\widetilde{\psi}_{+,T}^f(\cdot, \omega)\|_{L^2(\Gamma)} \le
  \frac{C_2}{2}\left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\cdot,
    \omega) - \i \gamma^-\widetilde{H}^f_T(\cdot,
    \omega)\right\|_{L^2(\Gamma)}\, (0 \le \omega < \omega_0)\label{psiplus_estimate_2b}
\end{equation}
for certain constants $C_1$ and $C_2$.  Noting that by Hermitian
symmetry (see \Cref{negative_freq_eq}) we have
$\left\|\widetilde{\psi}_{+,T}^f(\cdot, \omega)\right\|_{L^2(\Gamma)}
= \left\|\widetilde{\psi}_{+,T}^f(\cdot,
  |\omega|)\right\|_{L^2(\Gamma)}$ for $\omega < 0$, it follows that
\begin{equation}
    \begin{split}
    \left\|\widetilde{\psi}_{+,T}^f\right\|^2_{L^2(\mathbb{R};\,L^2(\Gamma))}
      &= 2 \int_0^\infty \left\|\widetilde{\psi}_{+,T}^f(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega \\
    &\le C_1\int_0^{\omega_0}
      \left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\cdot,
      \omega) - \i \gamma^-\widetilde{H}^f_T(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega\\
    & + C_2\int_{\omega_0}^\infty
      \omega^{2q}\left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_T(\cdot,
      \omega) - \i\omega \gamma^-\widetilde{H}^f_T(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega,\\
  \end{split}
\end{equation}
as desired.
\end{proof}

\begin{lemma}[Relating the function $\widetilde{h}_T$ to a limited time-history density]\label{3d_decay_thm_h_equiv}

  For an observation time $T$ and associated $\tau$DoD interval
  $I_{T}$ (\Cref{domainofdep}), such that
  $\widetilde{b}(\mathbf{r}, t)$ vanishes for
  $({\mathbf{r}},t) \in \widebar\Omega \times \left\lbrace I_{T}\cup [T, \infty)\right\rbrace$ (that
  is, ${\mathbf{r}} \in \widebar\Omega$ and $t \ge T - T_* - 2\tau$), the function $\widetilde{h}_T$ defined in
  \Cref{htildedef} satisfies
\begin{equation}\label{h_equiv}
    \widetilde{h}_T(\mathbf{r}, t) =
  \left\{\begin{aligned}
    -\widetilde{u}_{*,T}(\mathbf{r}, t), \quad &t \ge T - \tau\\
    0, \quad &t < T - \tau
  \end{aligned}\right.\quad\mbox{for}\quad\mathbf{r} \in \widebar{\Omega},
\end{equation}
  where, with reference to \Cref{timewinddens} and  \Cref{rem:tilde_1},
\begin{equation}\label{ustar_def}
  \widetilde{u}_{*,T}(\mathbf{r}, t) = (S \widetilde{\psi}_{*,T})(\mathbf{r}, t).
\end{equation}
Furthermore, the function $\widetilde{h}_T(\mathbf{r}, t)$ has a bounded
temporal support:
  \begin{equation}\label{h_compactsupp}
      \supp \widetilde{h}_T(\mathbf{r}, \cdot)\subset [T - \tau, T + T_*], \quad\mbox{for}\quad \mathbf{r} \in \widebar{\Omega}.
  \end{equation}
\end{lemma}
\begin{rem}\label{rem:h_equiv_rem_1}
  It is crucial for the proof of Theorem~\ref{3d_decay_thm_ii} and
  associated lemmas, that the function
  $\widetilde{h}_T(\mathbf{r}, t) = \widetilde{b} -
  \widetilde{u}_{-,T}$ has bounded temporal support. It is interesting
  to note, further, that, in particular, the right-hand side
  $\gamma^+\widetilde{h}_T(\mathbf{r}, t)$ of~\eqref{int_eq_lambda}
  vanishes at times for which its solution $\widetilde{\psi}_{+,T}$
  does not vanish.
\end{rem}
\begin{rem}\label{rem:h_equiv_rem_2}
  In view of~\eqref{psistar_def}, and taking into account
  \Cref{rem:tilde_1}, \Cref{3d_decay_thm_h_equiv} tells us that
  $\widetilde{h}_T$
  is determined by the restriction of $\widetilde{\psi}({\mathbf{r}}, t)$ to
  the interval $I_T$, and, thus, in view of~\cref{int_eq_lambda}, the
  same is true of $\widetilde{\psi}_{+,T}$. Since,
  by~\eqref{psipm_def}, $\widetilde{\psi}({\mathbf{r}}, t)$ coincides with
  $\widetilde{\psi}_{+,T}({\mathbf{r}}, t)$ for all $t\geq T$, it follows that,
  for such values of $t$ the solution $\widetilde{\psi}({\mathbf{r}}, t)$ is
  solely determined by the restriction of $\widetilde{\psi}({\mathbf{r}}, t)$
  to the interval $I_T$. In other words, for a given time $T$ and in
  absence of illumination for $t\geq T-T_* - \tau$,
  $\widetilde{\psi}({\mathbf{r}}, t)$ is determined by the values of the same
  solution function $\widetilde{\psi}({\mathbf{r}}, t)$ in the bounded time
  interval $I_T$ immediately preceeding time $T$. Thus,
  \Cref{3d_decay_thm_h_equiv} amounts to a bootstrap domain-of-dependence
  relation for solutions of the wave equation.
\end{rem}
\begin{proof}[Proof of \Cref{3d_decay_thm_h_equiv}.]

  We first consider the function $\widetilde{u}_{-,T}$ in
  \Cref{htildedef}.  Equation~\eqref{Tstar_def} tells us that for
  $t > T - \tau$, $t' < T - T_* - \tau$,
  $\mathbf{r} \in \widebar{\Omega}$ and $\mathbf{r}' \in\Gamma$ we
  have $t - t' - |\mathbf{r} - \mathbf{r}'|/c > 0$. In view of
  \Cref{umintilde_def}, it follows that, for
  $\mathbf{r} \in \widebar{\Omega}$ and all $t > T - \tau$,
  \begin{equation}\label{umin_trunc}
    \widetilde{u}_{-,T}(\mathbf{r}, t) = \int_{T - T_* - \tau}^t \int_\Gamma \frac{\delta\left(t - t' -
        |\mathbf{r} - \mathbf{r}'|/c\right) \widetilde{\psi}_{-,T}(\mathbf{r}', t')}{4\pi |\mathbf{r}
      - \mathbf{r}'|}\,\d\sigma(\mathbf{r}')\,\d t'.
\end{equation}
But in view of \Cref{htildedef},
$\widetilde{u}_{-,T}(\mathbf{r}, t) = -\widetilde{h}_T(\mathbf{r}, t)$
for $t > T - T_* - 2\tau$ and $\mathbf{r} \in \Gamma$ since by
hypothesis $\widetilde{b}$ vanishes for $t > T - T_* - 2\tau$.  In
sum, using \Cref{timewinddens}
($\widetilde{\psi}_{-,T}(\cdot, t) = w_-(t - T)\widetilde{\psi}(\cdot,
t)$, cf.\ \Cref{rem:tilde_1}), it follows that, for all
${\mathbf{r}} \in \widebar{\Omega}$ and $t > T - \tau$,
\begin{align*}
    \widetilde{h}_T(\mathbf{r}, t) &= -\int_{T - T_* - \tau}^t \int_\Gamma \frac{\delta\left(t - t' -
    |\mathbf{r} - \mathbf{r}'|/c\right) w_-(t' - T)\widetilde{\psi}(\mathbf{r}', t')}{4\pi |\mathbf{r}
    - \mathbf{r}'|}\,\d\sigma(\mathbf{r}')\,\d t',
\end{align*}
and thus, since from \Cref{wtau_def} we have
$w_+(t' - T + T_* + \tau) = 1$ for $t' \in [T - T_* - \tau, \infty)$,
we further obtain, again for all ${\mathbf{r}} \in \widebar{\Omega}$ and
$t > T - \tau$,
\begin{equation*}
  \begin{split}
    \widetilde{h}_T(\mathbf{r}, t) &= -\int_{T - T_* - \tau}^t \int_\Gamma \frac{\delta\left(t - t' -
      |\mathbf{r} - \mathbf{r}'|/c\right)}{4\pi |\mathbf{r}
      - \mathbf{r}'|}\times \\
        &\quad\quad\quad\quad\quad\quad\quad\times w_+(t' - T + T_* + \tau) w_-(t' - T)\widetilde{\psi}(\mathbf{r}', t')\,\d\sigma(\mathbf{r}')\,\d t'\\
    &= -\int_{T - T_* - \tau}^t \int_\Gamma \frac{\delta\left(t - t' -
    |\mathbf{r} - \mathbf{r}'|/c\right) \widetilde{\psi}_{*,T}(\mathbf{r}', t')}{4\pi |\mathbf{r}
    - \mathbf{r}'|}\,\d\sigma(\mathbf{r}')\,\d t',
  \end{split}
\end{equation*}
    where the last equality follows from the definition~\Cref{psistar_def} of
    $\widetilde{\psi}_{*,T}$ (cf.\
    \Cref{rem:tilde_1}).
    We have thus shown that $\widetilde{h}_T(\mathbf{r}, t) =
    -\widetilde{u}_{*,T}(\mathbf{r}, t)$ for $(\mathbf{r}, t) \in
    \widebar{\Omega} \times [T - \tau, \infty)$ with $\widetilde{u}_{*,T}(\mathbf{r}, t)$ as defined in \Cref{ustar_def}.

    To establish~\eqref{h_equiv} it remains to show that
    $\widetilde{h}_T = 0$ for ${\mathbf{r}} \in \widebar{\Omega}$ and
    $t < T - \tau$. To do this we note that, since
    $\widetilde{\psi}_{+,T}(\mathbf{r}, t) = 0$ for $t < T - \tau$,
    \cref{int_eq_lambda} tells us that
    $\gamma^+\widetilde{h}_T(\mathbf{r}, t) = 0$ for
    $(\mathbf{r}, t) \in \Gamma \times (-\infty, T - \tau)$. It
    follows that $\widetilde{h}_T$ is a solution of the wave equation on
    $\Omega$ (by~\cref{htildedef}) with vanishing boundary values for
    $t\in (-\infty, T - \tau)$. Thus, $\widetilde{h}_T(\mathbf{r}, t)$
    vanishes for
    $(\mathbf{r}, t) \in \widebar{\Omega} \times (-\infty, T - \tau)$,
    by solution uniqueness, as desired, and thus \Cref{h_equiv}
    follows.

    In view of \Cref{h_equiv}, in order to establish
    \Cref{h_compactsupp} it remains to show that
    $\widetilde{h}_T(\mathbf{r}, t) = 0$ for all $t > T + T_*$. From
    \Cref{timewinddens} we have
    $\supp \widetilde{\psi}_*(\mathbf{r}, \cdot) \subset I_T = [T -
    T_* - 2\tau, T)$. Since, in view of~\eqref{Tstar_def},
    $|\mathbf{r} - \mathbf{r}'|/c < T_*$ for all
    $\mathbf{r},\mathbf{r}' \in \widebar{\Omega}$,
    equation~\eqref{ustar_def}, which can be expressed in the form
    $\widetilde{u}_{*,T}(\mathbf{r}, t) = \int_\Gamma
    \frac{\widetilde{\psi}_{*,T}(\mathbf{r}', t - |\mathbf{r} -
      \mathbf{r}'|/c)}{4\pi |\mathbf{r} -
      \mathbf{r}'|}\,\d\sigma(\mathbf{r}')$, tells us that
    $\widetilde{h}_T(\mathbf{r}, t) = 0$ for
    $(\mathbf{r}, t) \in \widebar{\Omega} \times [T + T_*, \infty)$,
    and, thus, \Cref{h_compactsupp} follows.
  \end{proof}

\begin{lemma}\label{per_freq_rhs_oper_bounds}
  Let $\Gamma$ denote the boundary of a Lipschitz obstacle. Then there
  exist constants $C_1 = C_1(\Gamma) > 0$ and $C_2 = C_2(\Gamma) > 0$
  such that for all $\omega \ge 0$ the operator norm bounds
  \begin{equation}\label{per_freq_rhs_oper_bound_i}
    \left\|\left(\gamma^-\partial_\mathbf{n} \pm
    \i\omega\gamma^- \right)S_\omega\right\|_{L^2(\Gamma)\to L^2(\Gamma)} \le C_1(1 +
    \omega^2)^{1/2}
  \end{equation}
  and
  \begin{equation}\label{per_freq_rhs_oper_bound_ii}
    \left\|\left(\gamma^-\partial_\mathbf{n} \pm
    \i\gamma^- \right)S_\omega\right\|_{L^2(\Gamma)\to L^2(\Gamma)} \le C_2(1 +
    \omega^2)^{1/2}
  \end{equation}
  hold.
\end{lemma}
\begin{proof}
  The relations~\eqref{per_freq_rhs_oper_bound_i}
  and~\eqref{per_freq_rhs_oper_bound_ii} follow directly from the bounds
\begin{equation}\label{Somega_Kstaromega_bounds}
    \left\|\gamma^-S_\omega\right\|_{L^2(\Gamma) \to L^2(\Gamma)}
    \le D_1 ,\quad\mbox{and}\quad
    \left\|K_\omega^*\right\|_{L^2(\Gamma) \to L^2(\Gamma)}
    \le D_2 \omega + D_3,
\end{equation}
valid for all $\omega \geq 0$ (where $D_1, D_2, D_3 > 0$ are
constants), which are presented in reference~\cite[Thms.\ 3.3 and
3.5]{ChandlerWilde:09}. Indeed, in view of the jump relation
$\gamma^- \partial_\mathbf{n} S_\omega = \frac{1}{2}I + K_\omega^*$
for the normal derivative of the single layer potential as an operator on
$L^2(\Gamma)$~\cite[p. 219]{McLean}, it follows that there exist
constants $\widetilde{D}_1 > 0$ and $\widetilde{D}_2 > 0$ and $C_1$
such that
\begin{equation*}
  \begin{split}
      \left\|\left(\gamma^-\partial_\mathbf{n} \pm \i\omega\gamma^-\right)S_\omega\right\|_{L^2(\Gamma)\to L^2(\Gamma)} &= \left\|\frac{1}{2}I + K_\omega^* \pm
    \i\omega \gamma^-S_\omega\right\|_{L^2(\Gamma)\to L^2(\Gamma)}\\
      &\le (\widetilde{D}_1 + \widetilde{D}_2\omega) \le C_1(1 + \omega^2)^{1/2}.
  \end{split}
\end{equation*}
Similarly,
\begin{equation*}
  \begin{split}
      \left\|\left(\gamma^-\partial_\mathbf{n} \pm \i\gamma^- \right)S_\omega\right\|_{L^2(\Gamma)\to L^2(\Gamma)}
    &\le C_2(1 + \omega^2)^{1/2}
  \end{split}
\end{equation*}
for some constant $C_2 > 0$, and the result follows.
\end{proof}

\begin{lemma}\label{conv_H_to_psi}
  Let $q$ denote a non-negative integer, let $\Gamma = \partial\Omega$
  satisfy a $q$-growth condition (\Cref{q-nontrapp}), let $T_0$ be a
  given real number such that $\widetilde{b}$ vanishes for
  $({\mathbf{r}},t) \in \widebar{\Omega} \times \left\lbrace I_{T_0}\cup [T_0, \infty)\right\rbrace$,
  and let $\widetilde{H}_{T_0}^f$ be defined by \Cref{Ht_def_generic}
  with $T = T_0$. Then for some constant $C > 0$ independent of $T_0$ and $\widetilde{b}$ we have
\begin{equation}\label{conv_H_to_psi_eq}
  \begin{alignedat}{2}
      \int_0^{\omega_0} &\, &&
      \left\|\left(\gamma^-\partial_\mathbf{n} - \i
      \gamma^-\right) \widetilde{H}^f_{T_0}(\cdot,
      \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega \\
      +\,& &&\int_{\omega_0}^\infty \omega^{2q}
      \left\|\left(\gamma^-\partial_\mathbf{n} - \i\omega
      \gamma^-\right) \widetilde{H}^f_{T_0}(\cdot.
      \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega\\
      &\,&&\le C \int_{-\infty}^\infty (1 + \omega^2)^{q+1} \left\|
      \widetilde{\psi}_{*,T_0}^f(\cdot, \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega.
  \end{alignedat}
\end{equation}
\end{lemma}
\begin{proof}
  In order to obtain the desired bound we define the differential
  operators
\begin{equation}\label{ST_op_time}
  \mathcal{R} = \left(\gamma^-\partial_{\mathbf{n}} - \i\gamma^- \right),
  \enspace\mbox{and}\enspace\mathcal{T} = \left(-\i \frac{\partial}{\partial
    t}\right)^{q}\left(\gamma^-\partial_{\mathbf{n}} -
  \frac{\partial}{\partial t}\gamma^-\right),
\end{equation}
whose Fourier symbols are given by
\begin{equation}\label{ST_op_freq}
\hat{\mathcal{R}} =
  \left(\gamma^-\partial_\mathbf{n} - \i \gamma^-\right),\enspace\mbox{and}\enspace \hat{\mathcal{T}} =
    \omega^{q}\left(\gamma^-\partial_\mathbf{n} - \i\omega\gamma^- \right).
\end{equation}
Then, denoting by $\mathcal{Q}$ the sum of quantities on the left-hand
side of \Cref{conv_H_to_psi_eq} and using~\eqref{h_equiv} together
with Plancherel's theorem we obtain
\begin{alignat*}{2}
  &\begin{alignedat}{2}
      \mathcal{Q} =
      \int_0^{\omega_0} &\left\|\hat{\mathcal{R}}\widetilde{H}^f_{T_0}(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega
      &+ \int_{\omega_0}^\infty
      \left\|\hat{\mathcal{T}}\widetilde{H}^f_{T_0}(\cdot,
      \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega
    \end{alignedat}\\
    &\quad\quad\begin{alignedat}{2}
        \le \int_\Gamma \int_{-\infty}^\infty &\left|\hat{\mathcal{R}} \widetilde{H}^f_{T_0}(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r}) + \int_\Gamma
        \int_{-\infty}^\infty \left|\hat{\mathcal{T}} \widetilde{H}^f_{T_0}(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r})
    \end{alignedat}\\
    &\quad\quad\begin{alignedat}{2}
        = \int_\Gamma \int_{-\infty}^\infty &\left|\mathcal{R} \widetilde{h}_{T_0}(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r}) + \int_\Gamma
        \int_{-\infty}^\infty \left|\mathcal{T} \widetilde{h}_{T_0}(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r})
    \end{alignedat}\\
    &\quad\quad\begin{alignedat}{2}
        = \int_\Gamma \int_{T_0 - \tau}^\infty &\left|\mathcal{R} \widetilde{u}_{*,T_0}(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r}) + \int_\Gamma
        \int_{T_0 - \tau}^\infty \left|\mathcal{T} \widetilde{u}_{*,T_0}(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r}),\\
    \end{alignedat}
\end{alignat*}
where the last equality follows from~\eqref{conv_H_to_psi_eq} together
with the temporal locality of the operators $\mathcal{R}$ and
$\mathcal{T}$. In view of the nonnegativity of the integrands on the
right-hand side of this estimate, it follows that
\begin{equation}
\begin{split}
    \mathcal{Q} \le \int_\Gamma
    \int_{-\infty}^\infty \left|\mathcal{R} \widetilde{u}_{*,T_0}(\mathbf{r}, t')\right|^2\,\d
    t'\,\d\sigma(\mathbf{r}) + \int_\Gamma
    \int_{-\infty}^\infty \left|\mathcal{T} \widetilde{u}_{*,T_0}(\mathbf{r}, t')\right|^2\,\d
    t'\,\d\sigma(\mathbf{r}).\label{HtoUbound0}
\end{split}
\end{equation}
Using once again Plancherel's theorem, \Cref{HtoUbound0} becomes
\begin{equation}\label{HtoUbound}
  \begin{split}
      \mathcal{Q} \le
      \int_{-\infty}^\infty \left\|\hat{\mathcal{R}} \widetilde{U}_{*,T_0}^f(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega + \int_{-\infty}^\infty \left\|\hat{\mathcal{T}}
      \widetilde{U}_{*,T_0}^f(\cdot, \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega,
  \end{split}
\end{equation}
where $\widetilde{U}_{*,T_0}^f$ denotes the temporal Fourier transform of the single layer
potential $\widetilde{u}_{*,T_0}$,
\begin{equation}\label{UstarT0}
    \widetilde{U}_{*,T_0}^f({\mathbf{r}}, \omega) = \left(S_\omega \widetilde{\psi}^f_{*,T_0}\right)({\mathbf{r}}, \omega).
\end{equation}

Using \Cref{ST_op_freq}, \Cref{per_freq_rhs_oper_bound_i} and
\Cref{per_freq_rhs_oper_bound_ii} to bound the integrands on the
right-hand side of \Cref{HtoUbound}, and then using the fact that
$\omega^{2r} \le (1 + \omega^2)^{r}$ for nonnegative $r$, we obtain
\begin{equation*}
  \begin{split}
    &\begin{alignedat}{2}
    \mathcal{Q} \le
        C_1 \int_{-\infty}^\infty &\left(1 + \omega^2\right)
        \left\|\widetilde{\psi}_{*,T_0}^f(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega\\
        &+ C_2 \int_{-\infty}^\infty \omega^{2q} \left(1 + \omega^2\right)
        \left\|\widetilde{\psi}_{*,T_0}^f(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega
\end{alignedat}\\
&\quad\le C\int_{-\infty}^\infty \left(1 + \omega^2\right)^{q+1}
\left\|\widetilde{\psi}_{*,T_0}^f(\cdot,
  \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega.
  \end{split}
\end{equation*}
for a suitable constant $C$, and the result follows.
\end{proof}

In order to establish the uniform-in-time
estimates~\eqref{density_unif_time_bound}
and~\eqref{density_unif_time_bound_Tn}, we utilize a Bochner-space
version (cf. \Cref{def:sob_bochner}) below of the Sobolev embedding
lemma; it is primarily used in the case $\mathcal{S} = \Gamma$, for its other use cf.\ \Cref{decay_corr}.  The proof results by merely incorporating the Bochner space
nomenclature in the classical arguments used for the real-valued
case~\cite[Lem.\ 6.5]{Folland}.
\begin{lemma}[Sobolev embedding in Bochner spaces]\label{sob_lemma}
  Let $k$ and $n$ denote nonnegative and positive integers respectively, let $q \ge 0$, and let $\mathcal{S} \subset \mathbb{R}^n$.  Then for
  $p > k + 1/2$, $H^p(\mathbb{R}; H^q(\mathcal{S}))$ is continuously
  embedded in $C^k(\mathbb{R}; H^q(\mathcal{S}))$, or in symbols,
  $H^p(\mathbb{R}; H^q(\mathcal{S})) \hookrightarrow C^k(\mathbb{R};
  H^q(\mathcal{S}))$. In particular, there exists a constant $C$ such that
  for all functions $f \in H^p(\mathbb{R}; H^q(\mathcal{S}))$ the bound
    \begin{equation}\label{eq:sob_lemma}
      \max_{0 \le \ell \le k} \sup_{t \in \mathbb{R}}
      \left\|\frac{\partial^\ell}{\partial t^\ell} f(t)\right\|_{H^q(\mathcal{S})} \le C \left\|f\right\|_{H^p(\mathbb{R}; H^q(\mathcal{S}))},
    \end{equation}
    holds, where $\frac{\partial^\ell}{\partial t^\ell} f(t)$ denotes
    the $\ell$-th classical derivative of $f$ with respect to $t$ (see
    \Cref{sob-boch}).
\end{lemma}
\begin{proof}
  Let $f \in H^p(\mathbb{R}; H^q(\mathcal{S}))$. We first show that
  $\widehat{\partial^\ell f} \in L^1(\mathbb{R}; H^q(\mathcal{S}))$ for all
  nonnegative integers $\ell \le k$.  Indeed, using the relation
  $\widehat{\partial^\ell f}(\omega) = (\i \omega)^\ell
  \widehat{f}(\omega)$ (in accordance with the Fourier transform
  convention~\cref{eq:fourier_transf}) we obtain
    \begin{equation}\label{eq:bochner_L1_estimate}
        \begin{split}
          &\int_{-\infty}^\infty \left\|\widehat{\partial^\ell
              f}\right.\left.  (\omega)\right\|_{H^q(\mathcal{S})}\,\d\omega =
          \int_{-\infty}^\infty \left\|(\i\omega)^\ell
            \widehat{f}(\omega)\right\|_{H^q(\mathcal{S})}\,\d\omega\\
          &\le \int_{-\infty}^\infty (1 + \omega^2)^{k/2}\left\|\widehat{f}(\omega)\right\|_{H^q(\mathcal{S})}\,\d\omega\\
          &= \int_{-\infty}^\infty (1 +
          \omega^2)^{(k-p)/2}\left\|\widehat{f}(\omega)\right\|_{H^q(\mathcal{S})} (1 +
          \omega^2)^{p/2}\,\d\omega\\
          &\le \left(\int_{-\infty}^\infty (1 +
            \omega^2)^{k-p}\,\d\omega \right)^{1/2}
          \left(\int_{-\infty}^\infty (1 +
            \omega^2)^p \left\|\widehat{f}(\omega)\right\|_{H^q(\mathcal{S})}^2\,\d\omega\right)^{1/2}\\
          &\le \tilde{\tilde{C}} \left\|f\right\|_{H^p(\mathbb{R};\, H^q(\mathcal{S}))},
        \end{split}
    \end{equation}
    where we used the fact that $p > k + 1/2$ to bound the integral of
    $(1 + \omega^2)^{k-p}$ by a constant $\widetilde{C}$ independent
    of $f$, and we utilized the equivalent norm displayed
    in~\eqref{sob_bochner_fourier} in the space
    $H^p(\mathbb{R}; H^q(\mathcal{S}))$. It follows that
    $\widehat{\partial^\ell f} \in L^1(\mathbb{R}; H^q(\mathcal{S}))$, as
    claimed.

    By the Bochner-specific version of the Riemann-Lebesgue lemma~\cite[Lem.\ 2.4.3]{Weis}
    (which ensures that the Fourier transform of an $L^1(\mathbb{R}; H^q(\mathcal{S}))$ function is
    continuous), in conjunction with the fact that
    $\widehat{\partial^\ell f} \in L^1(\mathbb{R}; H^q(\mathcal{S}))$, it follows
    that
    \begin{equation}\label{fell_C}
      \widehat{\widehat{\partial^\ell f}} \in C(\mathbb{R}; H^q(\mathcal{S})), \quad 0 \le \ell \le k.
    \end{equation}

    Since by hypothesis $f\in H^p(\mathbb{R}; H^q(\mathcal{S}))$, further, it follows
    that for $0 \le \ell \le k$, $\partial^\ell f \in L^2(\mathbb{R}; H^q(\mathcal{S}))$, and
    thus by the Bochner Plancherel theorem~\cite[Thm.\ 2.20]{Karunakaran:98} we
    have, additionally,
    $\widehat{\partial^\ell f} \in L^1(\mathbb{R}; H^q(\mathcal{S})) \cap L^2(\mathbb{R};
    H^q(\mathcal{S}))$. Now, applying the Bochner Fourier inversion
    theorem~\cite{Karunakaran:98} (see also~\cite[\S 8.4]{Zemanian}),
    we obtain
    \begin{equation}\label{sob_lemma_partial_ell_f}
      \partial^\ell f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty
      \widehat{\partial^\ell f}(\omega) {\mathrm{e}}^{\i\omega t}\,\d\omega,\quad 0 \le
      \ell \le k,
    \end{equation}
    which, in view of \Cref{fell_C} shows that
    $\partial^\ell f(t) = \frac{1}{2\pi}
    \widehat{\widehat{\partial^\ell f}}(-t) \in C(\mathbb{R};
    H^q(\mathcal{S}))$ for $0 \le \ell \le k$ and, therefore, that
    $H^p(\mathbb{R}; H^q(\mathcal{S})) \subset C^k(\mathbb{R};
    H^q(\mathcal{S}))$.  Finally, using \Cref{sob_lemma_partial_ell_f} and
    then \Cref{eq:bochner_L1_estimate} we see that, for
    $0 \le \ell \le k$,
    \begin{equation*}
      \begin{split}
        \sup_{t \in \mathbb{R}} \left\|\frac{\partial^\ell}{\partial t^\ell} f(t)\right\|_{H^q(\mathcal{S})} &= \sup_{t \in
      \mathbb{R}} \frac{1}{2\pi} \left\|\int_{-\infty}^\infty
        \widehat{\partial^\ell f}(\omega)
        {\mathrm{e}}^{\i \omega (-t)}\,\d\omega\right\|_{H^q(\mathcal{S})}\\
        &\le \frac{1}{2\pi} \int_{-\infty}^\infty
        \left\|\widehat{\partial^\ell f}(\omega) \right\|_{H^q(\mathcal{S})}\,\d\omega \le C
        \left\|f\right\|_{H^p(\mathbb{R}; H^q(\mathcal{S}))}.
      \end{split}
    \end{equation*}
    This establishes \Cref{eq:sob_lemma} and concludes the proof of
    the lemma.
\end{proof}

\begin{lemma}\label{int_eq_pderiv}
  Let $p,q$ denote non-negative integers, assume $\Gamma$ satisfies the $q$-growth condition, and assume $\widetilde{b}$
  satisfies the $s$-regularity
  conditions~\eqref{eq:gamma_Hs_assump} with $s = p + q + 1$.
  Then, letting $\widetilde{\psi}$ denote the solution of
  \Cref{eq:tdie_sl_generic} we have
  $\widetilde{\psi} \in C^{p}(\mathbb{R}; L^2(\Gamma))$, and
  $\partial^p \widetilde{\psi}$ satisfies the integral equation
  \begin{equation}\label{eq:tdie_sl_generic_p}
        \left(S\partial^p \widetilde{\psi}\right)(\mathbf{r}, t) = \gamma^+ \partial^p \widetilde{b}(\mathbf{r}, t)\quad\mbox{for}\quad (\mathbf{r}, t) \in
    \Gamma\times\mathbb{R}.
  \end{equation}
\end{lemma}
\begin{proof}
  From \Cref{psi_Hp} we have
  $\widetilde{\psi} \in H^{p+1}(\mathbb{R};\, L^2(\Gamma))$, and,
  thus, by \Cref{sob_lemma} we obtain
  $\widetilde{\psi} \in C^{p}(\mathbb{R};\,L^2(\Gamma))$ as claimed.
  Similarly, by \Cref{sob_lemma}, $\widetilde{b}$ satisfies
  $\gamma^+ \widetilde{b} \in C^{p+q+1}(\mathbb{R};
  L^2(\Gamma))\subset C^{p}(\mathbb{R}; L^2(\Gamma))$. The proof is
  now completed by differentiating \Cref{eq:tdie_sl_generic} under the
  integral sign.
\end{proof}
\noindent
The proof of \Cref{3d_decay_thm} is presented in what follows.
\vspace{0.5cm}
\begin{center}
{\bf Proof of \Cref{3d_decay_thm}}
\end{center}
The proof of~\Cref{density_Hp_time_bound} follows by first deriving an
$L^2$-in-time estimate for the solution $\widetilde{\psi}$ to
\Cref{eq:tdie_sl_generic} with a generic right-hand side
$\widetilde{b}$ in suitable Sobolev-Bochner spaces, and then applying
that estimate to the particular cases $\widetilde{\psi} = \psi$ (for
$\widetilde{b} = b$) and $\widetilde{\psi} = \partial^p \psi$ (for
$\widetilde{b} = \partial^p b$, see~\Cref{def:sob_bochner}).

To derive the necesary estimates for $\widetilde{\psi}$ we develop
$L^2$-in-time estimates for the quantities $\widetilde{\psi}_{+,T_0}$
that are related to the solution $\widetilde{\psi}$ of
\Cref{eq:tdie_sl_generic} via \Cref{timewinddens} and
\Cref{rem:tilde_1}.  Since, by hypothesis, both aforementioned
selections $\widetilde{b} = b$ and $\widetilde{b} = \partial^p b$ of
$\widetilde b$ satisfy
$\gamma^+\widetilde{b} \in H^{q+1}(\mathbb{R}; H^1(\Gamma))$ and
$\gamma^+\partial_\mathbf{n} \widetilde{b} \in H^{q}(\mathbb{R};
L^2(\Gamma))$, it follows that the conditions of
\Cref{3d_decay_lemma_2ndkind} are met, and, thus
\Cref{3d_decay_lemma_Ainv_freq_bounds} tells us that
$\widetilde{\psi}_{+,T_0}^f$ satisfies the
estimate~\cref{psiplus_estimate_3}.  Using this bound in conjunction
with the relation $\widetilde{\psi} = \widetilde{\psi}_{+,T_0}$ for
$t > T_0$ (see \Cref{psipm_def}) we obtain, using Plancherel's
identity,
\begin{equation}\label{psi_to_psiplus_to_H}
  \begin{alignedat}{2}
      \left\|\widetilde{\psi}\right.\mkern-24mu &&&\left.\vphantom{\widetilde{\psi}}\right\|^2_{L^2([T_0, \infty);\,L^2(\Gamma))} \le
      \left\|\widetilde{\psi}_{+,T_0}\right\|^2_{L^2(\mathbb{R};\,L^2(\Gamma))} =
      \left\|\widetilde{\psi}_{+,T_0}^f\right\|^2_{L^2(\mathbb{R};\,L^2(\Gamma))} \\
    &\le\,&& C_1\int_0^{\omega_0}
      \left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_{T_0}(\cdot,
      \omega) - \i \gamma^-\widetilde{H}^f_{T_0}(\cdot,
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega\\
    & && + C_2\int_{\omega_0}^\infty \omega^{2q}
      \left\|\gamma^-\partial_\mathbf{n} \widetilde{H}^f_{T_0}(\cdot,
      \omega) - \i\omega \gamma^-\widetilde{H}^f_{T_0}(\cdot.
    \omega)\right\|^2_{L^2(\Gamma)}\,\d\omega.\\
  \end{alignedat}
\end{equation}
    Then using \Cref{conv_H_to_psi_eq} in \Cref{conv_H_to_psi}, the integrals on the right-hand side of
\Cref{psi_to_psiplus_to_H} can be bounded by
\begin{equation}\label{final_psiplus_estimate}
    \left\|\widetilde{\psi}\right\|^2_{L^2([T_0, \infty);\,L^2(\Gamma))} \le \tilde{C} \int_{-\infty}^\infty (1 + \omega^2)^{q+1} \left\|
    \widetilde{\psi}_{*,T_0}^f(\cdot, \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega,
\end{equation}
and, in view of the equivalence of the norms~\eqref{sob_bochner_norm}
and~\eqref{sob_bochner_fourier}, we can re-write \Cref{final_psiplus_estimate} as
\begin{equation}\label{final_psiplus_estimate_2}
    \left\|\widetilde{\psi}\right\|^2_{L^2([T_0, \infty);\,L^2(\Gamma))} \le \tilde{\tilde{C}}\left(\left\|
    \widetilde{\psi}_{*,T_0}\right\|_{L^2(\mathbb{R}; L^2(\Gamma))}^2 + \left\|
    \partial^{q+1} \widetilde{\psi}_{*,T_0}\right\|_{L^2(\mathbb{R}; L^2(\Gamma))}^2\right).
\end{equation}
Applying the Leibniz formula to the expression
\[
    \partial^{q+1} \widetilde{\psi}_{*,T_0}({\mathbf{r}}, t) = \partial^{q+1} w_+(t - T_0 + T_* + \tau)w_-(t - T_0)
    \widetilde{\psi}
\]
and noting that for all ${\mathbf{r}} \in \Gamma$ and for $1 \le i \le p$ we
have
$\supp \partial^{i} \widetilde{\psi}_{*,T_0}({\mathbf{r}}, \cdot) \subset
I_{T_0}$, using straightforward bounds on the functions $w_-$ and
$w_+$ (that do not depend on $T_0$---see \Cref{timewinddens}), we
obtain
\begin{equation}\label{final_psiplus_estimate_2a}
    \left\|\widetilde{\psi}\right\|^2_{L^2([T_0, \infty);\,L^2(\Gamma))} \le \sum_{i=0}^{q+1} C_i\left\|
    \partial^i \widetilde{\psi}\right\|_{L^2(I_{T_0}; L^2(\Gamma))}^2, \quad C_i = C_i(\Gamma, \tau).
\end{equation}
Using the continuity of the inclusion map in Sobolev spaces, it follows that
\begin{equation}\label{psiplus_estimate_L2_nontrapping}
    \lVert\widetilde{\psi}\rVert_{L^2([T_0, \infty);\,L^2(\Gamma))} \le
    C\left\|\widetilde{\psi}\right\|_{H^{q+1}(I_{T_0};\,L^2(\Gamma))} < \infty,\quad C = C(\Gamma, \tau),
\end{equation}
where the finiteness of the norm over $I_{T_0}$ in
\Cref{psiplus_estimate_L2_nontrapping} follows from \Cref{psi_Hp},
which tells us that
$\widetilde{\psi} \in H^{q+1}(I_{T_0};L^2(\Gamma))$ since, by
hypothesis,
$\gamma^+\widetilde{b} \in H^{2q+2}(\mathbb{R};L^2(\Gamma))$ and
$\gamma^+\partial_\mathbf{n} \widetilde{b} \in
H^{2q+1}(\mathbb{R};L^2(\Gamma))$ for each of $\widetilde{b} = b$ and
$\widetilde{b} = \partial^p b$. Applying
\Cref{psiplus_estimate_L2_nontrapping} with $\widetilde{b} = b$ yields
\Cref{density_Hp_time_bound} in the case $p = 0$. Furthermore,
\Cref{int_eq_pderiv} with $\widetilde{b} = b$ implies that
$\widetilde{\psi} = \psi$ satisfies \Cref{eq:tdie_sl_generic_p}. But
this equation can be expressed in the form \Cref{eq:tdie_sl_generic}
with $\widetilde{b} = \partial^p b$ and
$\widetilde \psi = \partial^p \psi$, for which the estimate
\Cref{psiplus_estimate_L2_nontrapping} becomes
\[
    \|\partial^p \psi\|_{L^2([T_0, \infty);\,L^2(\Gamma))} \le
    C\left\|\partial^p \psi\right\|_{H^{q+1}(I_{T_0};\,L^2(\Gamma))} < \infty,\quad C = C(\Gamma, \tau).
\]
which, using once again the continuity of the inclusion map in Sobolev spaces, equation
implies \Cref{density_Hp_time_bound}.

Applying~\cref{density_Hp_time_bound} with $p = 1$ together with \Cref{sob_lemma}
yields
\begin{equation}
  \begin{split}
    \sup_{t>T_0} \left\|\psi(\cdot, t)\right\|_{L^2(\Gamma)} &\le \widetilde{C}
    \left\|\psi\right\|_{H^{1}([T_0, \infty);\,L^2(\Gamma))}\\
    &\le C(\Gamma, \tau)
    \left\|\psi\right\|_{H^{q+2}(I_{T_0};\,L^2(\Gamma))},
  \end{split}
\end{equation}
and, thus, \cref{density_unif_time_bound}. The proof is now
complete. $\qed$

\section{Superalgebraic decay estimates of \\ finite-time-history boundary densities}\label{sec:theory_part_ii}
This section extends the theoretical results of
\Cref{sec:theory_part_i}: it establishes that not only is it possible
to bound the density $\psi$ in the unbounded time interval
$[T_0, \infty)$ by its values on the preceding bounded subinterval
$I_{T_0}$, as shown in \Cref{3d_decay_thm}, but also, in
\Cref{3d_decay_thm_ii} below, the main theorem of this paper, that the
{\em temporal} Sobolev and maximum norms of the solution $\psi$ on
time intervals of the form $[T_0 + t, \infty)$, $t > 0$, each decay
rapidly (e.g., superalgebraically fast for temporally smooth incident
signals) as $t\to \infty$; see also \Cref{3d_decay_rmk_iii} where a
related but somewhat modified decay result and proof strategy are
suggested.  The statement and proof of \Cref{3d_decay_thm_ii} rely on
the nomenclature introduced in Sections~\ref{sec:prelim}
and~\ref{sec:theory_part_i}. Two important corollaries to this theorem,
namely Corollaries~\ref{decay_corr} and~\ref{decay_corr_energy},
relate \Cref{3d_decay_thm_ii} to rapid decay and associated local energies
of solutions of the
wave equation. Following these corollaries a brief discussion is presented that lays
out the main lines of the proof of \Cref{3d_decay_thm_ii}; a detailed
proof of \Cref{3d_decay_thm_ii} is presented at the end of this
section, following a sequence of preparatory lemmas.

\begin{theorem}\label{3d_decay_thm_ii}
  Let $p$, $q$ and $n$ denote non-negative integers, $n>0$, let
  $T_0 > 0$ and $\tau > 0$ be given, and assume (i) $\Gamma$ satisfies
  the $q$-growth condition (\Cref{q-nontrapp}); (ii) The incident
  field $b({\mathbf{r}}, t)$ satisfies the $s$-regularity
  conditions~\eqref{eq:gamma_Hs_assump} with
  $s = p + q + (n+1)(q+1)$; as well as, (iii) The
  incident field $b = b({\mathbf{r}}, t)$ satisfies \Cref{all_t_b} and
  it vanishes for $({\mathbf{r}}, t) \in \widebar{\Omega}\times\left \{I_{T_0} \cup [T_0, \infty)\right\}$, with $I_{T_0}$ as in
  \Cref{domainofdep}. Then for $t > T_0$ the solution $\psi$ of
  \Cref{eq:tdie_sl} satisfies the $t$-decay estimate
\begin{equation}\label{density_Hp_time_bound_Tn}
  \left\|\psi\right\|_{H^p([t, \infty);\,L^2(\Gamma))} \le
    C(\Gamma, \tau, p, n) (t - T_0)^{1/2-n} \left\|\psi\right\|_{H^{p + (n+1)(q +
    1)}(I_{T_0};\,L^2(\Gamma))} < \infty.
\end{equation}
If $p\geq 1$ then $\psi$ additionally satisfies the pointwise
$t$-decay estimate
\begin{equation}\label{density_unif_time_bound_Tn}
  \left\|\psi(\cdot, t)\right\|_{L^2(\Gamma)} \le
  C(\Gamma, \tau, n) (t - T_0)^{1/2-n} \left\|\psi\right\|_{H^{(n+1)(q + 1) +
      1}(I_{T_0};\,L^2(\Gamma))} < \infty
\end{equation}
for all $t > T_0$.
\end{theorem}
\begin{corr}[Boundary-separated pointwise /  Full spatially-local-$L^2$ solution decay]\label{decay_corr}
  Let $q$ denote a non-negative integer, let $\Gamma$ satisfy the
  $q$-growth condition (\Cref{q-nontrapp}), and assume that the data
  $b$ for the problem~\cref{eq:w_eq}-\cref{eq:bdef} is such that
      \begin{equation}\label{b_smooth}
\gamma^+b \in C^\infty(\mathbb{R}; L^2(\Gamma))\quad\mbox{and}\quad
\gamma^+\partial_\mathbf{n} b \in C^\infty(\mathbb{R};L^2(\Gamma)).
\end{equation}
Further, assume that $b = b({\mathbf{r}}, t)$ satisfies~\eqref{all_t_b}, and
that, for given $T_0 > 0$ and $\tau > 0$, $b({\mathbf{r}}, t)$ vanishes for
$({\mathbf{r}}, t) \in \widebar{\Omega} \times \left\lbrace I_{T_0} \cup [T_0,
  \infty)\right\rbrace$, where $I_{T_0} = I_{T_0, T_*, \tau}$ is
defined in \Cref{domainofdep} with $T_* = \diam(\Gamma)/c$ as
indicated in eqn.~\Cref{Tstar_def}.

Then, letting $R > 0$ be such that $\Omega \subset \left\lbrace |\mathbf{r}| <
R\right\rbrace$, letting $D = \Omega^c \cap \left\lbrace |\mathbf{r}| <
R\right\rbrace$, and defining $r_{\mathrm{max}} = \sup_{\mathbf{r} \in D, \mathbf{r}' \in
\Gamma} |\mathbf{r} - \mathbf{r}'|$, for each pair of integers $n > 0$ and  $p \ge 0$ there exists a constant $C
= C(p, n, \tau, R, \Gamma) > 0$ such that the solution $u$ to \Cref{eq:w_eq} satisfies the $t$-decay estimate
\begin{equation}\label{uk_l2_sup_bound_decay}
    \left\|\partial_t^p u(\cdot, t)\right\|_{L^2(D)} \le C (t - T_0 -
    r_{\mathrm{max}}/c)^{1/2-n} \left\|\psi\right\|_{H^{p + (n+1)(q+1)+1}(I_{T_0};
    L^2(\Gamma))} < \infty,
\end{equation}
for all $t \in (T_0 + r_{\max}/c, \infty)$.

Further, $u$ also decays
superalgebraically fast with increasing time $t$ for each point
$\mathbf{r}$ outside $\widebar{\Omega}$: given any compact set
$\mathcal{R} \subset (\widebar{\Omega})^c$ and defining
$r_{\mathrm{max}} = \max_{\mathbf{r} \in \mathcal{R}, \mathbf{r}' \in
  \Gamma} |\mathbf{r} - \mathbf{r}'|$, for each pair of integers
$n > 0$ and $p \ge 0$ there exists a constant
$C = C(p, n, \tau, \mathcal{R}, \Gamma) > 0$ such that $u$ satisfies
the $t$-decay estimate
\begin{equation}\label{u_ptwise_bound_decay}
      \left|\partial_t^p u(\mathbf{r}, t)\right| \le
      C(t - T_0 - r_{\mathrm{max}}/c)^{1/2-n} \left\|\psi\right\|_{H^{p + (n+1)(q+1)+1}(I_{T_0}; L^2(\Gamma))} < \infty,
\end{equation}
for all $(\mathbf{r}, t) \in \mathcal{R}\times (T_0 + r_{\mathrm{max}}/c,
\infty)$.

\end{corr}
\begin{proof}
Except for hypothesis (ii) of \Cref{3d_decay_thm_ii}, all conditions of that Theorem follow immediately from hypotheses of the present corollary. Hypothesis (ii), in turn, follows since
by hypothesis~\eqref{b_smooth} and the assumed compact support of $b$, $b$ satisfies $\gamma^+b \in H^p(\mathbb{R}; L^2(\Gamma))$ and
$\gamma^+ \partial_{\mathbf{n}} b \in H^p(\mathbb{R}; L^2(\Gamma))$ for every integer $p\ge 0$. Thus, the conditions of \Cref{3d_decay_thm_ii} are satisfied for every integer $n > 0$ and every integer $p \ge 0$ and so we have
\begin{equation}\label{psi_decay}
  \begin{split}
    \left\|\partial_t^p \psi\right\|_{H^1([t', \infty);\,L^2(\Gamma))} &\le
  \left\|\psi\right\|_{H^{p+1}([t', \infty);\,L^2(\Gamma))}\\
  &\le C_1(p, n, \Gamma, \tau) (t' - T_0)^{1/2-n} \left\|\psi\right\|_{H^{p + (n+1)(q +
    1) + 1}(I_{T_0};\,L^2(\Gamma))},
  \end{split}
\end{equation}
for $t' > T_0$.

The estimates \Cref{uk_l2_sup_bound_decay} and
\Cref{u_ptwise_bound_decay} are obtained in what follows by relying on
a corresponding estimate on $\partial_t^p u$ in the norm of
$H^1([T_0 + r_{\max}/c, \infty); L^2(\Gamma))$, on one hand, and an
estimate on the norm of $\partial_t^p u(\mathbf{r}, \cdot)$ in
$H^1([t + r_{\max}/c, \infty))$ for each $\mathbf{r} \in \mathcal{R}$,
on the other hand.  Estimate~\eqref{uk_l2_sup_bound_decay} is
established by first differentiating $s$ times under the integral sign
in the integral representation~\eqref{eq:kirchhoff_3d_soft} for the
solution $u$ (for certain values of the integer $s$) obtaining
$\partial_t^s u(\mathbf{r}, t) = \left(S \partial_t^s
  \psi\right)(\mathbf{r}, t)$, and then applying the Cauchy-Schwarz
inequality to obtain, for all $t'$,
\begin{equation}
    \begin{split}
      \left\|\partial_t^s u(\cdot, t')\right.&\hspace{-1.5mm}\left.\vphantom{\partial_t^s u(\cdot, t')}\right\|^2_{L^2(D)} = \int_D
      \left|\partial_t^s u(\mathbf{r}, t')\right|^2\,\d V(\mathbf{r})\\
    &\le \frac{1}{16\pi^2} \int_D \left(\int_\Gamma
    \left|\partial_t^s \psi(\mathbf{r}', t' - |\mathbf{r} -
    \mathbf{r}'|/c)\right|^2\,\d\sigma(\mathbf{r}')\right)
    \left(\int_\Gamma \frac{\d\sigma(\mathbf{r}')}{|\mathbf{r} -
    \mathbf{r}'|^2}\right)\d V(\mathbf{r}).
    \end{split}
    \label{uk_L2_estimate}
\end{equation}
  Integrating this bound in time, we obtain
\begin{equation}
    \begin{split}
      \left\|\partial_t^s u\right\|^2_{L^2([t, \infty); L^2(D))} &\le
      \frac{1}{16\pi^2} \int_{D} \left[ \left(\int_t^\infty \int_\Gamma
      |\partial_t^s \psi(\mathbf{r}', t' - |\mathbf{r} - \mathbf{r}'|/c)|^2\,
      \d\sigma(\mathbf{r}')\,\d t'\right) \times\right.\\
      &\left.\quad\quad\quad\quad \times \left(\int_\Gamma \frac{\d\sigma(\mathbf{r}')}{|\mathbf{r} -
    \mathbf{r}'|^2}\right)\right]\,\d V(\mathbf{r}).
    \end{split}
    \label{uk_L2_estimate_integrated}
\end{equation}
  Letting $I_{s,1}(\mathbf{r}, t)$ and $I_2(\mathbf{r})$ denote, respectively, the first
  and second factors in the integrand of the  integral over
  $D$,
  \begin{equation}\label{I1_I2_decay_corr}
    I_{s,1}(\mathbf{r}, t) = \int_t^\infty \int_\Gamma
      |\partial_t^s\psi(\mathbf{r}', t' - |\mathbf{r} - \mathbf{r}'|/c)|^2\,
      \d\sigma(\mathbf{r}')\,\d t',\;\mbox{and}\; I_2(\mathbf{r}) = \int_\Gamma \frac{\d\sigma(\mathbf{r}')}{|\mathbf{r} -
    \mathbf{r}'|^2},
  \end{equation}
  we seek to bound each of $I_{s,1}(\mathbf{r}, t)$ and $I_2(\mathbf{r})$ by quantities independent of $\mathbf{r}$.
  Considering first $I_{s,1}(\mathbf{r}, t)$ for fixed $\mathbf{r} \in D$, a change in integration order and a slight extension of the integration domain in the $t'$ variable yields
  \begin{equation}\label{I1_decay_corr_estimate}
    I_{s,1}(\mathbf{r}, t) \le \int_\Gamma \int_{t - r_{\max}/c}^\infty |\partial_t^s \psi(\mathbf{r}', t')|^2\,\d t'\,\d\sigma(\mathbf{r}')= \left\|\partial_t^s\psi\right\|^2_{L^2([t - r_{\max}/c, \infty); L^2(\Gamma))}.
  \end{equation}
    The integral of $I_2(\mathbf{r})$ over $D$, in turn, is bounded by an $R-$ and $\Gamma$-dependent constant since by switching the order of integration
    \begin{equation*}
      \int_{D} I_2(\mathbf{r}) \,\d V(\mathbf{r}) = \int_\Gamma \left(\int_{D} \frac{1}{|\mathbf{r} - \mathbf{r}'|^2}\,\d V(\mathbf{r})\right)\d\sigma(\mathbf{r}'),
    \end{equation*}
    and the inner integral is bounded for each $\mathbf{r}'
    \in \Gamma$ by an $R-$ and $\Gamma$-dependent constant---a fact which can be
    established by use of a spherical coordinate system centered at $\mathbf{r}'
    \in \Gamma$ for integration. We have therefore shown that
    \begin{equation}
      \label{u_l2_to_psi}
      \left\| \partial_t^s u\right\|^2_{L^2([t, \infty); L^2(D))} \le C_2\left\|\partial_t^s \psi\right\|^2_{L^2([t - r_{\max}/c, \infty); L^2(\Gamma))}.
    \end{equation}
    Taking separately $s = p$ and $s = p + 1$ in \Cref{u_l2_to_psi} and adding the results we obtain
    \begin{equation}\label{u_h1_to_psi}
      \left\| \partial_t^p u\right\|^2_{H^1([t, \infty); L^2(D))} \le C_3\left\|\partial_t^p\psi\right\|^2_{H^1([t - r_{\max}/c, \infty); L^2(\Gamma))},
    \end{equation}
    Using \Cref{u_h1_to_psi} in conjunction
    with \Cref{psi_decay} and the Sobolev embedding \Cref{sob_lemma} we obtain
    \begin{equation*}
      \begin{split}
        \left\|\partial_t^p u(\cdot, t)\right\|_{L^2(D)} &\le C_4 \left\|\partial_t^p u\right\|_{H^1([t, \infty); L^2(D))} \le \sqrt{C_3} C_4\left\|\partial_t^p\psi\right\|_{H^1([t - r_{\max}/c, \infty); L^2(\Gamma))}\\
        &\le C_5(\Gamma, \tau, n) (t - T_0 - r_{\max}/c)^{1/2-n} \left\|\psi\right\|_{H^{p + (n+1)(q +
  1) + 1}(I_{T_0};\,L^2(\Gamma))},
      \end{split}
    \end{equation*}
    and, thus, \Cref{uk_l2_sup_bound_decay} follows.

    To establish \Cref{u_ptwise_bound_decay}, in turn, we once again
    use $s$-times differentiation under the integral sign in the
    variable $t$ in the representation~\Cref{eq:kirchhoff_3d_soft} and
    the Cauchy-Schwarz inequality to obtain
\begin{equation}\label{u_CS_to_psi_sup}
    \begin{split}
  |\partial_t^s u(\mathbf{r}, t')|^2 &\le \left( \int_\Gamma \frac{1}{|\mathbf{r} -
    \mathbf{r}'|^2}\,\d\sigma(\mathbf{r}')\right) \left(\int_\Gamma
    |\partial_t^s \psi(\mathbf{r}', t' - |\mathbf{r} - \mathbf{r}'|/c)|^2
    \,\d\sigma(\mathbf{r}')\right)\\
      &\le C_6^2 \int_\Gamma
    |\partial_t^s \psi(\mathbf{r}', t' - |\mathbf{r} - \mathbf{r}'|/c)|^2
    \,\d\sigma(\mathbf{r}'),
    \end{split}
\end{equation}
where
$C_6^2 = \sup_{\mathbf{r}\in \mathcal{R}} \int_\Gamma
\frac{1}{|\mathbf{r} - \mathbf{r}'|^2}\,\d\sigma(\mathbf{r}')$.
Integrating in $t'$ and using \Cref{I1_decay_corr_estimate} to estimate
the resulting quantity $I_{s,1}(\mathbf{r}, t)$, for each
$\mathbf{r} \in \mathcal{R}$ we obtain the $L^2$ bound
\begin{equation}\label{u_ptwise_l2_to_psi}
  \int_{t}^\infty |\partial_t^s u(\mathbf{r}, t')|^2\,\d t' \le C_6 \left\|\partial_t^s \psi\right\|^2_{L^2([t - r_{\max}/c, \infty); L^2(\Gamma))}.
\end{equation}
Taking separately $s = p$ and $s = p + 1$ in \Cref{u_ptwise_l2_to_psi}, adding the results, and using \Cref{psi_decay} in conjunction with the Sobolev embedding \Cref{sob_lemma}, we obtain, again for $\mathbf{r} \in \mathcal{R}$,
    \begin{equation*}
      \begin{split}
        \left|\partial_t^p u(\mathbf{r}, t)\right| &\le C_7 \left\|\partial_t^p u(\mathbf{r}, \cdot)\right\|_{H^1([t, \infty))} \le C_8\left\|\partial_t^p \psi\right\|_{H^1([t - r_{\max}/c, \infty); L^2(\Gamma))}\\
        &\le C_9(\Gamma, \tau, n) (t - T_0 - r_{\max}/c)^{1/2-n} \left\|\psi\right\|_{H^{p + (n+1)(q +
  1) + 1}(I_{T_0};\,L^2(\Gamma))} < \infty,
      \end{split}
    \end{equation*}
establishing \Cref{u_ptwise_bound_decay}. The proof is complete.
\end{proof}

While equation~\eqref{u_ptwise_bound_decay} provides a concrete
solution decay estimate at each spatial point $\mathbf{r}$ outside
$\widebar{\Omega}$, the associated constant $C$ does increase without
bound as $\mathbf{r}$ approaches $\Omega$ (cf.\
\Cref{u_CS_to_psi_sup}). The result presented next, in contrast, which
holds for arbitrary bounded subsets of $\widebar{\Omega}^c$,
establishes decay of the local energy expression~\eqref{local_energy}
considered in much of the literature---thus strengthening the previous
estimate~\eqref{uk_l2_sup_bound_decay}. The proof below relies on the
estimates provided in \Cref{decay_corr} in conjunction with standard
elliptic regularity properties.
\begin{corr}[Local energy decay estimates]\label{decay_corr_energy}
  Let $\Gamma$ and $b$ satisfy the hypotheses of \Cref{decay_corr}
  and let $u$ denote the solution to the problem~\eqref{eq:w_eq}. Then,
  the local energy $E$ (equation~\eqref{local_energy}) associated with
  the solution $u$ decays superalgebraically fast as
  $t\to\infty$. More precisely, letting $R > 0$ be such that $\partial B_R$ does not intersect $\Omega$, defining the
  compact region
  $D = \Omega^c \cap \left\lbrace |\mathbf{r}| \le R\right\rbrace$,
  and letting
  $r_{\mathrm{max}} = \sup_{\mathbf{r} \in D, \mathbf{r}' \in \Gamma}
  |\mathbf{r} - \mathbf{r}'|$, for each integer $n > 0$ there exists a
  constant $C = C(\Gamma, R, \tau, n) > 0$ such that
  \begin{equation}\label{local_energy_superalg_decay}
    E(u, D, t) \le C (t - T_0 - r_{\max}/c)^{1 - 2n} \left\| \psi \right\|^2_{H^{(n+1)(q+1)+3}(I_{T_0}; L^2(\Gamma))} < \infty,
  \end{equation}
  for all $t \in (T_0 + r_{\max}/c, \infty)$.
\end{corr}
\begin{proof}
  In view of the bound on
  $\int_D |u_t(\mathbf{r}, t)|^2\,\d V(\mathbf{r})$ provided by
  \Cref{decay_corr}, it suffices to establish a corresponding estimate
  for $\int_D |\nabla u(\mathbf{r}, t)|^2\,\d V(\mathbf{r})$ for
  $t \in (T_0 + r_{\max}/c, \infty)$. To do this, we use the fact that
  $u$ is a solution of the wave equation~\eqref{eq:w_eq_a} as well as
  the hypothesis that $\gamma^+ b(\mathbf{r}', t') = 0$ for
  $(\mathbf{r}', t') \in \Gamma \times \left\{ I_{T_0} \cup [T_0,
    \infty)\right\}$, which together imply that $u$ satisfies, for
  each $t' \in I_{T_0} \cup [T_0, \infty)$, the exterior Dirichlet
  problem
\begin{subequations}\label{u_ellipt}
  \begin{align}
    \Delta u(\mathbf{r}, t') &= \frac{1}{c^2} \partial_t^2 u(\mathbf{r}, t') \eqqcolon f(\mathbf{r}, t'), \quad\mbox{for}\quad \mathbf{r} \in \Omega^c,\label{u_ellipt_a}\\
    u(\mathbf{r}, t') &=  0,
                       \quad\mbox{for}\quad \mathbf{r}\in\Gamma = \partial\Omega,\label{u_ellipt_b}
    \end{align}
  \end{subequations}
  for the Poisson equation~\eqref{u_ellipt_a}. In order to do so, we
  first apply \Cref{decay_corr} so that from
  \Cref{uk_l2_sup_bound_decay} with $p = 2$ we have, for
  $t > T_0 + r_{\max}/c$,
  \begin{equation}\label{f_l2_decay_estimate}
    \left\|f(\cdot, t)\right\|_{L^2(D)} \le \frac{C_1}{c^2}(t - T_0 -
    r_{\mathrm{max}}/c)^{1/2-n} \left\|\psi\right\|_{H^{(n+1)(q+1)+3}(I_{T_0};
    L^2(\Gamma))} < \infty,
  \end{equation}
  in particular implying
  \begin{equation}\label{f_poisson_def}
    f(\cdot, t) \in L^2(D) \quad\mbox{for}\quad t \in (T_0 + r_{\max}/c, \infty).
  \end{equation}
  We also introduce the region $D_\varepsilon = \Omega^c \cap \left\lbrace |\mathbf{r}| \le R + \varepsilon\right\rbrace$ for some $\varepsilon > 0$ as well as the
  function $\varphi: D_\varepsilon \to \mathbb{R}$ satisfying $\varphi = u(\cdot, t)$ in a
  neighborhood of $|\mathbf{r}| = R + \varepsilon$ and satisfying $\varphi = 0$ in a
  neighborhood of $\partial\Omega$ as well as for $|\mathbf{r}| > R + 2\varepsilon$---which can be easily constructed by
  multiplication of $u$ by a smooth cutoff function in the radial variable.
  Moreover, from the representation~\eqref{eq:kirchhoff_3d_soft} for
  $\mathbf{r} \in D_\varepsilon$ bounded away from $\partial \Omega$ we have $\varphi \in
  H^1(\Omega)$.  Applying the regularity result~\cite[Thm.\ 8.9]{GilbargTrudinger} to the
  problem~\eqref{u_ellipt} on $D_\varepsilon$ with $f(\cdot, t) \in L^2(D_\varepsilon)$ and with boundary values given by $\varphi
  \in H^1(D_\varepsilon)$, we have $u(\cdot, t) \in H^1(D_\varepsilon)\cap H^2_{\mathrm{loc}}(D_\varepsilon)$. Since $D \subset D_\varepsilon$ we also have $u(\cdot, t) \in H^1(D)\cap H^2_{\mathrm{loc}}(D)$.

  Since $u(\cdot, t) \in H^1(D)$ and $\Delta u(\cdot, t) \in L^2(D)$ we can
  apply a version~\cite[Thm.\ 4.4]{McLean} of Green's first identity to obtain, for each $t \in (T_0 + r_{\max}/c, \infty)$,
  \begin{equation}\label{gradu_grns_identity}
    \int_D \left|\nabla u(\mathbf{r}, t)\right|^2\,\d V(\mathbf{r}) = \left\langle \gamma u(\cdot, t), \gamma \partial_\mathbf{n} u(\cdot, t) \right\rangle_{\partial D} - \int_D u(\mathbf{r}, t) \Delta u(\mathbf{r}, t)\,\d V(\mathbf{r}),
  \end{equation}
  where $\left\langle \cdot, \cdot \right\rangle$ denotes the duality pairing of $H^{-1/2}(\partial D)$ and $H^{1/2}(\partial D)$ and $\gamma$ denotes the trace operator on $D$. In fact,
  since $\gamma u(\cdot, t) = 0$ on $\Gamma$ and $u(\cdot, t)$ is smooth for $\mathbf{r}$ in a neighborhood of $\partial B_R$, we have
  $\left\langle \gamma u(\cdot, t), \gamma \partial_\mathbf{n} u(\cdot, t)\right\rangle_{\partial D} = \int_{\partial B_R} u(\mathbf{r}, t) \partial_\mathbf{n} u(\mathbf{r}, t)\,\d\sigma(\mathbf{r})$.
  Using \Cref{gradu_grns_identity} we now proceed to derive bounds for
  $E(u, D, t)$. Since $u$ satisfies \Cref{u_ellipt}, using
  the Cauchy-Schwarz inequality we obtain
  \begin{equation}\label{gradu_first_bound}
    \begin{split}
      \int_D \left|\nabla u(\mathbf{r}, t)\right|^2\,\d & V(\mathbf{r}) = \int_{\partial B_R} u(\mathbf{r}, t) \partial_{\mathbf{n}} u(\mathbf{r}, t) \,\d\sigma(\mathbf{r}) - \int_D u(\mathbf{r}, t) f(\mathbf{r}, t)\,\d V(\mathbf{r})\\
      &\le \left\| u(\cdot, t)\right\|_{L^2(\partial B_R)} \left\|\partial_{\mathbf{n}} u(\cdot, t)\right\|_{L^2(\partial B_R)} + \left\|u(\cdot, t)\right\|_{L^2(D)} \left\|f(\cdot, t)\right\|_{L^2(D)},
    \end{split}
  \end{equation}
  where $B_R$ denotes the ball of radius $R$ centered at the origin.
  The first volume term in \Cref{gradu_first_bound} can be estimated
  using \Cref{decay_corr} with $p = 0$:
  \begin{equation}\label{u_volume_decay}
    \left\|u(\cdot, t)\right\|_{L^2(D)} \le C_2 (t - T_0 - r_{\max}/c)^{1/2-n} \left\|\psi\right\|_{H^{(n+1)(q+1)+1}(I_{T_0}; L^2(\Gamma))}.
  \end{equation}
  Then, using \Cref{f_l2_decay_estimate} and the continuity of the
  inclusion map in Sobolev spaces, \Cref{u_volume_decay} provides a bound for the second summand on the right-hand side of~\eqref{gradu_first_bound}:
  \begin{equation}\label{gradu_volume_bound}
    \left\|u(\cdot, t)\right\|_{L^2(D)} \left\|f(\cdot, t)\right\|_{L^2(D)} \le C_3 (t - T_0 - r_{\max}/c)^{1-2n} \left\|\psi\right\|_{H^{(n+1)(q+1)+3}(I_{T_0}; L^2(\Gamma))}^2.
  \end{equation}

  We now to turn to the first summand on the right-hand side
  of~\eqref{gradu_first_bound}, and we estimate each one of the
  corresponding boundary terms by relying on the representation
  formula~\cref{eq:kirchhoff_3d_soft}. Indeed, differentiating that
  formula once with respect to the normal $\mathbf{n}$ and $s$-times
  ($s = 0, 1$) with respect to time for
  $(\mathbf{r},t') \in \partial B_R \times \mathbb{R}$ , we obtain the
  relation
  \begin{equation}
    \begin{split}
        \partial_{\mathbf{n}} \partial_t^s u(\mathbf{r}, t') = -\int_\Gamma &\left( \frac{\hat{\mathbf{r}}\cdot (\mathbf{r} - \mathbf{r}') \partial_t^s \psi(\mathbf{r}', t' - |\mathbf{r} - \mathbf{r}'|/c)}{|\mathbf{r} - \mathbf{r}'|^3}\right.\\&\left.+ \frac{\hat{\mathbf{r}} \cdot (\mathbf{r} - \mathbf{r}') \partial_t^{s+1} \psi(\mathbf{r}', t' - |\mathbf{r} - \mathbf{r}'|/c)}{c|\mathbf{r} - \mathbf{r}'|^2}\right)\,\d\sigma(\mathbf{r}'),
    \end{split}
  \end{equation}
  for which the Cauchy-Schwarz inequality implies that, for $(\mathbf{r},t') \in \partial B_R \times \mathbb{R}$,
  \begin{equation}\label{nablau_ptwise_to_l2_bound}
    \begin{split}
      |\partial_{\mathbf{n}} \partial_t^s u(\mathbf{r}, t')|^2 &\le C_4(\Gamma, R, \varepsilon) \int_\Gamma |\partial_t^s \psi(\mathbf{r}, t' - |\mathbf{r} - \mathbf{r}'|/c)|^2\,\d\sigma(\mathbf{r}')\\
        &+ C_5(\Gamma, R, \varepsilon) \int_\Gamma |\partial_t^{s+1} \psi(\mathbf{r}', t' - |\mathbf{r} - \mathbf{r}'|/c)|^2\,\d\sigma(\mathbf{r}').
    \end{split}
  \end{equation}
  The argument we use to bound $\partial_{\mathbf{n}} u$ uniformly in time is similar
  to the one used in the proof of \Cref{decay_corr}: we obtain bounds
  on $\partial_{\mathbf{n}} u(\mathbf{r}, \cdot)$ in the norm of $H^1([t, \infty))$
  by taking $s = 0$ and $s = 1$ in \Cref{nablau_ptwise_to_l2_bound},
  summing the result, and integrating this inequality for $t' > t$;
  the uniform-in-time bound then follows from the Sobolev
  Lemma. Thus, applying \Cref{nablau_ptwise_to_l2_bound} with $s=0$ and $s=1$ and using the definition~\eqref{I1_I2_decay_corr} of
  $I_{0, 1}(\mathbf{r}, t)$ and $I_{1,1}(\mathbf{r}, t)$ and
  $I_{2,1}(\mathbf{r}, t)$,
  \begin{equation}\label{nablau_ptwise_to_l2_bound_integrated}
    \begin{split}
      \left\| \partial_{\mathbf{n}} u(\mathbf{r}, \cdot) \right\|^2_{H^1([t, \infty))} &\le C_4(\Gamma, R, \varepsilon) I_{0,1}(\mathbf{r}, t) + C_5(\Gamma, R, \varepsilon) I_{2,1}(\mathbf{r}, t)\\
      &\quad + (C_4(\Gamma, R, \varepsilon) + C_5(\Gamma, R, \varepsilon))I_{1,1}(\mathbf{r}, t).
    \end{split}
  \end{equation}
  Then, using the bound~\eqref{I1_decay_corr_estimate}, $I_{s,1}(\mathbf{r}, t)
  \le \left\|\partial_t^s \psi\right\|^2_{L^2([t - r_{\max}/c, \infty);
  L^2(\Gamma))}$, we obtain the relation
  \begin{equation}\label{nablau_H1_bound}
    \left\|\partial_{\mathbf{n}} u(\mathbf{r}, \cdot)\right\|_{H^1([t, \infty)} \le C_6(\Gamma, R, \varepsilon) \left\|\psi\right\|_{H^2([t - r_{\max}/c, \infty); L^2(\Gamma))}.
  \end{equation}
  In conjunction with the Sobolev embedding \Cref{sob_lemma} and the bound \Cref{density_Hp_time_bound_Tn} of \Cref{3d_decay_thm_ii}, the bound~\Cref{nablau_H1_bound} implies that
  \begin{equation}\label{nablau_ptwise_bound}
    \begin{split}
      \left|\partial_{\mathbf{n}} u(\mathbf{r}, t)\right| &\le C_7 \left\|\partial_{\mathbf{n}} u(\mathbf{r}, \cdot)\right\|_{H^1([t, \infty)} \le C_7 C_6(\Gamma, R, \varepsilon) \left\|\psi\right\|_{H^2([t - r_{\max}/c, \infty); L^2(\Gamma))}\\
      &\le C_8(\Gamma, R, \tau, n, \varepsilon) (t - T_0 - r_{\max}/c)^{1/2 - n} \left\|\psi\right\|_{H^{(n+1)(q+1)+2}(I_{T_0}; L^2(\Gamma))}
    \end{split}
  \end{equation}
  for
  $(\mathbf{r}, t) \in \partial B_R \times (T_0 + r_{\max}/c,
  \infty)$. Taking $L^2(\partial B_R)$ norms in both
  \Cref{u_ptwise_bound_decay} with $p=0$ and in
  \Cref{nablau_ptwise_bound} yields the desired estimate for the first
  summand on the right-hand side of~\eqref{gradu_first_bound}:
  \begin{equation}\label{unablau_ptwise_bound}
    \begin{split}
      \left\|u(\cdot, t)\right\|_{L^2(\partial B_R)} \left\|\partial_{\mathbf{n}} u(\cdot, t)\right\|_{L^2(\partial B_R)} \le C_9 &(t - T_0 - r_{\max}/c)^{1-2n} \times\\
    &\times \left\|\psi\right\|^2_{H^{(n+1)(q+1)+2}(I_{T_0}; L^2(\Gamma))},
    \end{split}
  \end{equation}
  for $t\geq T_0 + r_{\max}/c$, where we again used the continuity of
  the inclusion map in Sobolev spaces.

  To obtain the desired decay result, finally, we apply
  the bound \Cref{uk_l2_sup_bound_decay} in \Cref{decay_corr}
  with $p = 1$ to obtain, for $t \in (T_0 + r_{\max}/c, \infty)$,
  \begin{equation}\label{dt2u_decay}
    \left\|\partial_t u(\cdot, t)\right\|_{L^2(D)} \le C_{10} (t - T_0 -
    r_{\mathrm{max}}/c)^{1/2-n} \left\|\psi\right\|_{H^{(n+1)(q+1)+2}(I_{T_0};
    L^2(\Gamma))}.
  \end{equation}
  Then, using~\Cref{gradu_first_bound} for the first term in the
  definition~\eqref{local_energy} of $E(u, D, t)$, we conclude that
  \begin{equation}
    \begin{split}
      E(u, D, t) &\le \left\| u(\cdot, t)\right\|_{L^2(\partial B_R)} \left\|\partial_{\mathbf{n}} u(\cdot, t)\right\|_{L^2(\partial B_R)}\\
      &\quad + \left\|u(\cdot, t)\right\|_{L^2(D)} \left\|f(\cdot, t)\right\|_{L^2(D)} + \left\|\partial_t u(\cdot, t)\right\|^2_{L^2(D)}\\
      &\le C_9 (t - T_0 - r_{\max}/c)^{1-2n} \left\|\psi\right\|^2_{H^{(n+1)(q+1)+2}(I_{T_0}; L^2(\Gamma))}\\
      &\quad + C_3^2 (t - T_0 - r_{\max}/c)^{1-2n} \left\|\psi\right\|_{H^{(n+1)(q+1)+3}(I_{T_0}; L^2(\Gamma))}^2\\
      &\quad + C_{10}^2 (t - T_0 -
    r_{\mathrm{max}}/c)^{1-2n} \left\|\psi\right\|^2_{H^{(n+1)(q+1)+2}(I_{T_0};
    L^2(\Gamma))},
    \end{split}
  \end{equation}
  where the last inequality results by incorporating the
  estimates~\eqref{unablau_ptwise_bound}, \Cref{gradu_volume_bound}, and \Cref{dt2u_decay}.  The desired
  estimate~\eqref{local_energy_superalg_decay} follows, once again, by
  continuity of the inclusion map in Sobolev spaces and the proof is thus
  complete.
\end{proof}

\begin{rem}\label{ralston}
  The well known contribution~\cite{Ralston:69} implies that, for a
  trapping obstacle $\Omega$ and for an arbitrarily large time
  $\mathcal{T}$, initial conditions can be selected so that the local
  energy $E(u, D, t)$ in~\eqref{energy_decay} with
  $D =\Omega^c\cap \{|{\mathbf{r}}| < R\}$ remains arbitrarily close to the
  initial energy value $E(u, D, 0)$ for $0\leq t\leq \mathcal{T}$. In
  particular, this result implies that a decay bound of the
  form~\eqref{energy_decay} that is uniformly valid for all admissible
  incident initial conditions and for all compact sets
  $D\subset \Omega^c$ cannot hold for a trapping
  obstacle. References~\cite{Ikawa:82,Ikawa:88} do present, however,
  uniformly valid decay estimates relative to higher order Sobolev
  norms over the complete exterior domain, which are valid for
  trapping structures consisting a unions of convex obstacles, for
  which the trapping rays span spatial regions of zero measure
  (indeed, a single ray in the case of the structure consisting of two
  convex connected obstacles in~\cite{Ikawa:82}, and as implied by
  Assumption (H.2) in~\cite{Ikawa:88}). In the same spirit,
  \Cref{3d_decay_thm_ii} and Corollaries~\ref{decay_corr} and~\ref{decay_corr_energy} present decay results,
  including of the local energy $E(u, D, t)$,
  that are uniformly valid for all admissible incident fields in terms
  of higher-order {\em surface} Sobolev norms, which are valid for
  trapping obstacles satisfying the $q$-growth condition---which
  includes obstacles, such as those depicted in \Cref{fig:3d_connected_trapping}, for
  which the trapping rays span volumetric regions of positive mesaure.
\end{rem}

The overall approach to the proof of \Cref{3d_decay_thm_ii} relies
critically on the novel domain-of-dependence time-history technique
described in \Cref{rem:h_equiv_rem_2} (see also \Cref{3d_decay_rmk_iii} where a related but somewhat
modified decay result and proof strategy are suggested). Technically, the proof of
\Cref{3d_decay_thm_ii} proceeds on the basis of an argument resulting from
integration-by-parts with respect to the temporal frequency $\omega$,
which requires $\omega$-differentiation of a certain function
$\breve{\psi}^f_{+,0}({\mathbf{r}}, \omega)$ closely related to the boundary
integral density $\psi^f({\mathbf{r}}, \omega)$.  Certain necessary results
concerning differentiability of boundary integral operators and
associated integral densities are established in a series of lemmas
presented in the Appendix.  Some of the main elements of the proof of
\Cref{3d_decay_thm_ii}, in turn, are presented in
Lemmas~\ref{3d_decay_thm_recursive_bound_general}
through~\ref{decay_estimate_L2} below. Thus, with reference to the
Appendix, \Cref{3d_decay_thm_recursive_bound_general} provides
pointwise bounds on density derivatives. Then, the technical
\Cref{Rderiv_sobolev_bound} (of a similar character to
\Cref{conv_H_to_psi}) establishes bounds on the integrals of certain
quantities in~\Cref{3d_decay_thm_recursive_bound_general}, and
\Cref{decay_estimate_L2} provides the primary decay estimate used in
the proof of \Cref{3d_decay_thm_ii}.

\begin{remark}\label{3d_decay_rmk_iii}
  Before proceeding with the proof of \Cref{3d_decay_thm_ii} we note
  that a related but somewhat less informative decay result and proof
  strategy, which do not depend on the bootstrap DoD concept, can be
  contemplated. In the alternative approach, the decay proof results
  once again from an argument based on integration by parts (in
  frequency-domain) in the Fourier integral that represents the time-domain
  solution. In detail, starting from the given time-domain data
  $ b(r,t)$ defined in the neighborhood $\Omega^\textit{inc}$ of
  $\Omega$, a Fourier transform is performed to obtain the right-hand
  side of the integral equation~\eqref{CFIE_direct}. Since $\Gamma$ satisfies the
  $q$-growth condition, it follows that $\psi^f({\mathbf{r}},\omega)$, the solution
  of this equation, admits an
  upper bound that grows polynomially as a function of $\omega$. Using
  an argument similar to the one presented in \Cref{3d_decay_thm_recursive_bound_general} below,
  corresponding polynomially growing bounds are obtained for the
  $\omega$ derivatives of $\psi^f({\mathbf{r}},\omega)$. Thus, the decay
  result can be obtained by an integration by parts argument followed
  by a bound on the integral of the resulting integrands. Such a bound
  can be obtained by relying on smooth partitioning of the integral
  together with a Young inequality-based estimate in an argument
  similar to the one in \Cref{decay_estimate_L2}. The resulting time-decay bound
  resembles the bound~\eqref{density_Hp_time_bound_Tn}. But,
  unlike equation~\eqref{density_Hp_time_bound_Tn}, which expressed
  decay in terms of the norm of the solution $\psi$ itself over the
  bootstrap DoD $I_{T_0}$, the alternative bound expresses decay in
  terms of the norm {\em over all time} of the right-hand side
  function $b({\mathbf{r}},t)$ and its normal derivative on $\Gamma$. Thus
  \Cref{3d_decay_thm_ii} provides a significantly more precise decay estimate,
  but it does so at the cost of certain added complexity in the
  proof---which is mainly confined to \Cref{Rderiv_sobolev_bound}.
\end{remark}

Following the aforementioned proof plan for
Theorem~\ref{3d_decay_thm_ii}, we first present, in
\Cref{3d_decay_thm_recursive_bound_general}, estimates on
frequency-derivatives of certain frequency-domain density solutions.

\begin{lemma}\label{3d_decay_thm_recursive_bound_general}
  Let $\widetilde{b}$ satisfy the assumptions of \Cref{3d_decay_lemma_2ndkind}
  as well as the additional assumption that for all ${\mathbf{r}} \in \widebar{\Omega}$ the function
  $\widetilde{b}$ is compactly-supported as a function of time, and let
  $\Omega$ satisfy the $q$-growth condition for some non-negative integer $q$.
  Further, call $\widetilde{R}_T(\mathbf{r}, \omega)$ the right-hand side of
  \cref{CFIE_proof_generic} and let $\widetilde{\psi}$ and
    $\mu = \widetilde{\psi}_{+,T}^f$ denote the solutions to
  \Cref{eq:tdie_sl_generic} and \cref{CFIE_proof_generic},
  respectively.
  Then, for all nonnegative integers $p$ and all
  $\omega \neq \pm \omega_0$ (cf. \Cref{Aop_def}) we have
  \begin{equation}\label{recursion_hypothesis_squared_general}
    \begin{split}
      \left\| \partial_\omega^p \mu(\cdot, \omega)\right\|_{L^2(\Gamma)}^2
      \le \sum_{i=0}^{p} &\left(\sum_{j=0}^{(i+1)(q+1)-1} d_{ij}^p \omega^{2j}
      \left\|\partial_\omega^{p-i} \widetilde{R}_T(\cdot,
      |\omega|)\right\|^2_{L^2(\Gamma)}\right),
    \end{split}
  \end{equation}
  where $d_{ij}^p > 0$ denote certain positive constants independent of $T$. Additionally, for each nonnegative integer $p$
  there exists a constant $C_p$ such that
  \begin{equation}\label{psit_nongrowth}
    \left\|\partial_\omega^p\mu(\cdot, \omega)\right\|_{L^2(\Gamma)} \le C_p |\omega|^{(p+1)(q+1)}
  \end{equation}
  holds for all sufficiently large values of $|\omega|$.
\end{lemma}
\begin{proof}
  Let $p$ denote a nonnegative integer and let
  $\omega \ne \pm \omega_0$.  Since from \Cref{int_eq_freq_regularity}
  we have
  $\widetilde{R}_T \in C^\infty(\mathbb{R}\setminus \pm \omega_0;
  L^2(\Gamma))$ and
  $\widetilde{\psi}_{+,T}^f \in C^\infty(\mathbb{R}; L^2(\Gamma))$, it
  follows that all quantities in
  \Cref{recursion_hypothesis_squared_general} are well-defined. We
  restrict the remainder of the proof to the case $\omega > 0$; the
  full result then follows by the property of Hermitian symmetry
  satisfied by $\mu = \widetilde{\psi}^f_{+,T}$ (see
  \Cref{negative_freq}).

  In order to prove \Cref{recursion_hypothesis_squared_general} we first show that
  \begin{equation}\label{recursion_hypothesis_general}
    \begin{split}
      \left\| \partial_\omega^p \mu(\cdot, \omega)\right\|_{L^2(\Gamma)}
      \le \sum_{i=0}^{p - 1} &\left(\sum_{j=0}^{(i+1)(q+1)-1} b_{ij}^p \omega^j
      \left\|\partial_\omega^{p-i} \widetilde{R}_T(\cdot,
      \omega)\right\|_{L^2(\Gamma)}\right)\\
      &+ \sum_{i=0}^{p(q+1)} c_{i}^p \omega^i \left\|\mu(\cdot,
      \omega)\right\|_{L^2(\Gamma)},
  \end{split}
  \end{equation}
  for some positive constants $b_{ij}^p > 0$ and $c_i^p > 0$.  The
  proof of \Cref{recursion_hypothesis_general} proceeds inductively:
  assuming that, for a given nonnegative integer $p$, the relation
  \begin{equation}\label{recursion_hypothesis_general_s}
    \begin{split}
      \left\| \partial_\omega^s \mu(\cdot, \omega)\right\|_{L^2(\Gamma)}
      \le \sum_{i=0}^{s - 1} &\left(\sum_{j=0}^{(i+1)(q+1)-1} b_{ij}^s \omega^j
      \left\|\partial_\omega^{s-i} \widetilde{R}_T(\cdot,
      \omega)\right\|_{L^2(\Gamma)}\right)\\
      &+ \sum_{i=0}^{s(q+1)} c_{i}^s \omega^i \left\|\mu(\cdot,
      \omega)\right\|_{L^2(\Gamma)},
  \end{split}
\end{equation}
holds for all nonnegative integers $s \le p$, we show that
\Cref{recursion_hypothesis_general_s} holds for $s = p + 1$. (The base
case $s = 0$ is trivially satisfied as the first sum in the inequality
equals zero in that case.)  To do this we use Leibniz's product
differentiation rule (\Cref{int_eq_freq_regularity}), which tells us that
  \begin{equation}\label{leibniz}
    \left(\partial_\omega^{p+1} \mu \right)(\mathbf{r}, \omega) =
    A_\omega^{-1}\left(\partial_\omega^{p+1} \widetilde{R}_T(\mathbf{r}, \omega) -
    \sum_{k=1}^{p+1} a_k^{p+1}(\partial_\omega^k A_\omega)
    (\partial_\omega^{p+1-k} \mu)(\mathbf{r}, \omega) \right)
  \end{equation}
  for certain integers $a_k^{p+1}$; $k=1, \ldots, p+1$.  But, in
  view of the $q$-growth condition (\Cref{q-nontrapp}),
  there exist positive $C_1, C_2$ such that
  $\left\|A_\omega^{-1}\right\|_{L^2(\Gamma)\to L^2(\Gamma)}\le C_1 +
  C_2 \omega^q$. Further, the operator-norm bound
  \Cref{Aop_deriv_bound} in \Cref{omega_explicit_norms_deriv_int_op}
  tells us that, for certain constants $\alpha_{0k}$ and
  $\alpha_{1k}$, we have
  $\left\|\partial_\omega^k A_\omega\right\|_{L^2(\Gamma)\to
    L^2(\Gamma)} \le \alpha_{0k} + \alpha_{1k} \omega$ for all
  $\omega \in \mathbb{R}^+ \setminus \omega_0$. It thus follows
  from~\eqref{leibniz} that
  \begin{equation}\label{partial_p_mu_deriv_first_bound}
    \begin{split}
      \left\|\vphantom{\partial_\omega^{p+1} \mu(\cdot, \omega)}
      \partial_\omega^{p+1} \mu(\cdot, \omega)\right.&
      \left.\vphantom{\partial_\omega^{p+1} \mu(\cdot, \omega)}\right\|_{L^2(\Gamma)}
      \le
      (C_1 + C_2\omega^q) \left(\vphantom{\sum_{k=1}^{p+1}}
      \left\|\partial_\omega^{p+1}
      \widetilde{R}_T(\cdot, \omega)\right\|_{L^2(\Gamma)} \right.\\
      &+ \left.\sum_{k=1}^{p+1} |a_k^{p+1}|
      (\alpha_{0k} + \alpha_{1k} \omega)\left\|\partial_\omega^{p+1-k}
      \mu(\cdot, \omega)\right\|_{L^2(\Gamma)}\right).
    \end{split}
  \end{equation}
  Substituting \Cref{recursion_hypothesis_general_s} with
  $s = p + 1 - k $ for $k = 1, \ldots, p$ into
  \Cref{partial_p_mu_deriv_first_bound} we obtain
  \begin{alignat*}{2}
    \left\|\partial_\omega^{p+1}\mu(\cdot, \omega)\right\|_{L^2(\Gamma)} &\le
    (C_1 + C_2\omega^q)\left\|\partial_\omega^{p+1} \widetilde{R}_T(\cdot,
    \omega)\right\|_{L^2(\Gamma)}\\
    &\begin{alignedat}[t]{2}
      + \sum_{k=1}^{p+1} &|a_k^{p+1}|\left(C_1\alpha_{0k} +
      C_2\alpha_{0k}\omega^q + C_1\alpha_{1k}\omega +
      C_2\alpha_{1k}\omega^{q+1}\right)\times\\
      &\begin{alignedat}[t]{2}
      \times\left[\vphantom{\sum_{i=0}^{p+1-k}\sum_{j=0}^i}\right.&
        \sum_{i=0}^{p-k}\sum_{j=0}^{(i+1)(q+1)-1} b_{ij}^{p+1-k} \omega^j
        \left\|\partial_\omega^{p+1-k-i} \widetilde{R}_T(\cdot,
    \omega)\right\|_{L^2(\Gamma)}\\
        &\left.+ \sum_{i=0}^{(p+1-k)(q+1)} c_i^{p+1-k} \omega^i
    \left\|\mu(\cdot, \omega)\right\|_{L^2(\Gamma)}\right],
    \end{alignedat}
    \end{alignedat}
  \end{alignat*}
  from which, expanding the products, we obtain
  \begin{alignat}{2}\label{final_inequality}
    \left\|\partial_\omega^{p+1}\mu(\cdot, \omega)\right\|_{L^2(\Gamma)} &
    \le (C_1 + C_2\omega^q)\left\|\partial_\omega^{p+1} \widetilde{R}_T(\cdot,
    \omega)\right\|_{L^2(\Gamma)} + (A) + (B),
  \end{alignat}
where
\begin{alignat*}{2}
  (A) = \sum_{k=1}^{p+1}&\sum_{i=0}^{p-k}\sum_{j=0}^{(i+1)(q+1)-1}
      |a_k^{p+1}b_{ij}^{p+1-k}|\left(C_1\alpha_{0k}\omega^j +
      C_2\alpha_{0k}\omega^{q+j}\vphantom{+ C_1\alpha_{1k}\omega^{j+1} +
      C_2\alpha_{1k}\omega^{q+j+1}}\right.\\
      &\left.+\;C_1\alpha_{1k}\omega^{j+1} +
      C_2\alpha_{1k}\omega^{q+j+1}\right)\left\|\partial_\omega^{p+1-k-i}
      \widetilde{R}_T(\cdot, \omega)\right\|_{L^2(\Gamma)},
\end{alignat*}
and
\begin{alignat*}{2}
  (B) = \sum_{k=1}^{p+1}&\sum_{i=0}^{(p+1-k)(q+1)}
      |a_k^{p+1}c_{i}^{p+1-k}|\left(C_1\alpha_{0k}\omega^i +
      C_2\alpha_{0k}\omega^{q+i}\vphantom{+ C_1\alpha_{1k}\omega^{i+1} +
      C_2\alpha_{1k}\omega^{q+i+1}}\right.\\
      &\left.+\; C_1\alpha_{1k}\omega^{i+1} +
      C_2\alpha_{1k}\omega^{q+i+1}\right)\left\|
      \mu(\cdot, \omega)\right\|_{L^2(\Gamma)}.
\end{alignat*}
It is easy to show that \Cref{final_inequality} implies
\Cref{recursion_hypothesis_general_s} with $s=p+1$. Indeed, on one
hand, the first term on the right-hand side of \Cref{final_inequality}
and every term arising from the summations in $(A)$ can be expressed
as a constant multiplied by a term of the form
$\omega^\ell \left\|\partial_\omega^m \widetilde{R}_T(\cdot,
  \omega\right\|_{L^2(\Gamma)}$ for some $m \le p$ and for some
$\ell\leq (p+1)(q+1)$---all of which match corresponding terms in
\Cref{recursion_hypothesis_general_s} for $s = p +
1$. The maximal power of $\omega$ in the expression $(B)$, on the other
hand, equals $(p+1)(q+1)$ (it occurs for $k=1$ and $i=p(q+1)$), the
corresponding term in $(B)$, namely
$\omega^{(p+1)(q+1)}\left\|\mu(\cdot,
  \omega)\right\|_{L^2(\Gamma)}$, is also present
in~\cref{recursion_hypothesis_general_s} for $s = p+1$.  Similarly, it
is easily checked that all remaining terms
$\omega^\ell \left\|\partial_\omega^{p+1-k-i} \widetilde{R}_T(\cdot,
  \omega)\right\|_{L^2(\Gamma)}$ and
$\omega^i\left\|\mu(\cdot, \omega)\right\|_{L^2(\Gamma)}$
in~\eqref{final_inequality} are accounted for in
\Cref{recursion_hypothesis_general_s} with $s = p+1$, and, thus, the
validity of the inequality~\cref{recursion_hypothesis_general_s} with
$s = p + 1$ and certain constants $b_{ij}^{p+1}$ and $c_{i}^{p+1}$
follows, which completes the inductive proof
of~\Cref{recursion_hypothesis_general}.

Using \Cref{recursion_hypothesis_general} together with the $q$-growth
condition
$$\left\|\mu(\cdot, \omega)\right\|_{L^2(\Gamma)} \le (C_1 + C_2
\omega^q) \left\|\widetilde{R}_T(\cdot,
  \omega)\right\|_{L^2(\Gamma)}\quad (C_1,C_2\in\mathbb{R})$$ for
the solution $\mu$ of the equation $A_\omega \mu = \widetilde{R}_T$,
we obtain the bound
\begin{equation}\label{dpmu}
    \begin{split}
      \left\| \partial_\omega^p \mu(\cdot, \omega)\right\|_{L^2(\Gamma)}
      \le \sum_{i=0}^{p} &\left(\sum_{j=0}^{(i+1)(q+1)-1} f_{ij}^p \omega^{j}
      \left\|\partial_\omega^{p-i} \widetilde{R}_T(\cdot,
      \omega)\right\|_{L^2(\Gamma)}\right)
    \end{split}
  \end{equation}
  for some positive constants $f_{ij}^p > 0$. The desired
  inequality~\Cref{recursion_hypothesis_squared_general} follows
  directly from \Cref{dpmu} using the relation
  $\left\|\sum_{i=1}^p f_i\right\|^2 \le p\sum_{i=1}^p
  \left\|f_i\right\|^2$.

  To establish the relation \Cref{psit_nongrowth}, finally, for each
  integer $\ell = p - i$, $0\leq \ell\leq p$ in~\eqref{dpmu} we
  estimate the expression
  $\left\|\partial^{\ell}_\omega \widetilde{R}_T(\cdot,
    \omega)\right\|_{L^2(\Gamma)}$ where, by assumption for
  $\omega \ge \omega_0$ (cf.~\Cref{Aop_def_eqn}),
  $\widetilde{R}_T = \left(\gamma^- \partial_\mathbf{n} -
    \i\omega\gamma^- \right) \widetilde{H}^f_T$.  To do this we note
  that, in view of~\eqref{Ht_def_generic} and the jump relations for
  the normal derivative of the single layer
  potential~\cite[p. 219]{McLean}, for $\omega > \omega_0$ we have
  \begin{equation}\label{Rtderiv_def}
    \partial^{\ell}_\omega \gamma^- \widetilde{R}_T = \partial^{\ell}_\omega \gamma^- \widetilde{B}^f - \partial^{\ell}_\omega \left(\frac{1}{2}I + K_\omega^* -
    \i\omega \gamma^-S_\omega\right) \widetilde{\psi}_{-,T}.
  \end{equation}
  It therefore suffices to show that each one of the two terms on the
  right-hand side of~\Cref{Rtderiv_def} is bounded by a power of
  $\omega$.  Since $\gamma^-\widetilde{B}^f$ equals the temporal
  Fourier transform of the compactly-supported function
  $\gamma^- \widetilde{b} = \gamma^+ \widetilde{b}\in H^1(\mathbb{R};
  L^2(\Gamma)) \subset L^2(\mathbb{R}; L^2(\Gamma))$ (see the $s$-regularity hypotheses of the present lemma), differentiation under the Fourier-transform
  integral sign, which is justified since $b$ is compactly-supported
  tells us that $\partial^{\ell}_\omega \gamma^- \widetilde{B}^f$
  equals the temporal Fourier transform of the compactly-supported
  function
  $(-\i t)^\ell\gamma^+ \widetilde{b}\in L^2(\mathbb{R};
  L^2(\Gamma))$. Further, for each $0 \le \ell \le p$ and for all
  $\omega \in \mathbb{R}$, using the triangle and Cauchy-Schwarz
  inequalities we obtain
  \begin{equation}
    \label{paley_der_b}
    \left\|\partial^{\ell}_\omega \gamma^-\widetilde{B}^f(\cdot,
      \omega)\right\|_{L^2(\Gamma)} \le C'
  \end{equation}
  for some constant $C'$ (that depends on $\widetilde{b}$ and $p$),
  which provides the needed estimate of the first right-hand term.

  In order to obtain a bound for the second term on the right-hand
  side of~\Cref{Rtderiv_def}, in turn, we note that since
  $\widetilde{\psi}_{-,T}$ is compactly-supported
  (cf.~\Cref{psipm_def}) and satisfies
  $\widetilde{\psi}_{-,T} \in L^2(\mathbb{R}; L^2(\Gamma))$
  (cf. \Cref{psi_Hp}), and, thus, an argument similar to the one leading
  to~\eqref{paley_der_b} yields the bound
  \begin{equation}\label{eq:dpi}
    \left\|\partial_\omega^{\ell} \widetilde{\psi}^f_{-,T}(\cdot, \omega)\right\|_{L^2(\Gamma)} \le C''
  \end{equation}
  for all $\ell$, $0 \le \ell \le p$ and all $\omega > 0$,
  where $C''$ denotes a constant that, once again, depends on
  $\widetilde{b}$ and $p$.  Then, employing the operator norm bounds
  presented in \Cref{omega_explicit_norms_deriv_SK} for
  $\partial_\omega^\ell S_\omega$ and
  $\partial_\omega^\ell K_\omega^*$ ($\ell = 0, \ldots, p$) together
  with \Cref{eq:dpi} we obtain the second-term bound
\begin{equation}\label{eq:dpii}
  \left\|\partial^{\ell}_\omega \left(\left(\frac{1}{2}I + K_\omega^* -
    \i\omega \gamma^-S_\omega\right) \widetilde{\psi}_{-,T}(\cdot, \omega)\right)\right\|_{L^2(\Gamma)} \le C'''(1 + |\omega|)
\end{equation}
for all integers $\ell$, $0 \le \ell \le p$ and for all
$\omega > 0$, where $C'''$ denotes a constant independent of
$\omega$.

Finally, in view of~\Cref{Rtderiv_def} and incorporating
\Cref{paley_der_b} and \Cref{eq:dpii} we obtain
\begin{equation}\label{eq:dpiii}
  \left\|\partial^\ell \gamma^- \widetilde{R}_T(\cdot, \omega)\right\|_{L^2(\Gamma)} \le C''''(1 + \omega)
\end{equation}
for all integers $\ell$, $0 \le \ell \le p$ and for all
$\omega > \omega_0$, where $C''''$ denotes a constant independent of
$\omega$. The estimate \Cref{psit_nongrowth} follows directly from
\Cref{dpmu} and~\Cref{eq:dpiii}.
\end{proof}

\begin{rem}\label{rem:breve}
  In order to obtain the estimates in \Cref{3d_decay_thm_ii} that are
  uniform with respect to $T_0$ it will be necessary to perform
  temporal shifts on the data $\widetilde{b}$ (see \Cref{rem:tilde_1})
  and the solution $\widetilde{\psi}$---see \Cref{ITvsI0}.  Given a
  real number $T_0$ we define for a given $\widetilde{b}$ the
  ``breve'' quantities
  \begin{equation}\label{def:breve}
    \breve{b}({\mathbf{r}}, t) = \widetilde{b}({\mathbf{r}}, t + T_0),\quad\mbox{and}\quad \breve{\psi}({\mathbf{r}}, t) = \widetilde{\psi}({\mathbf{r}}, t + T_0).
  \end{equation}
  With reference to \Cref{rem:tilde_1}, note that
  $\breve{\psi}({\mathbf{r}}, t)$ equals the solution
  $\widetilde{\psi}({\mathbf{r}}, t)$ of \Cref{eq:tdie_sl_generic} with
  $\widetilde{b}$ substituted by $\breve{b}$. Note also that if
  $\widetilde{b}$ satisfies \Cref{all_t_b_generic} for some $\alpha$
  then $\breve{b}$ satisfies \Cref{all_t_b_generic} with $\alpha$
  substituted by $\alpha - T_0$.  Our consideration of $\breve{\psi}$
  will be limited to the choice $I_T$ with $T = 0$, and we define
  $\breve{\psi}_{\pm,0}$, $\breve{\psi}_{*,0}$,
  $\breve{\psi}_{p,\pm,0}$ and $\breve{\psi}_{p,*,0}$ analogously to
  $\psi_{\pm,T}$, $\psi_{*,T}$, $\psi_{p, \pm, T}$ and
  $\psi_{p, *, T}$ respectively, in \Cref{timewinddens}, but with
  $T=0$.
\end{rem}

  Consistent with the conventions laid out in
  \Cref{timewinddens} and \Cref{rem:tilde_1} and letting
    \begin{equation}\label{breveupmstardef}
        \breve{u}_{\pm,0}({\mathbf{r}}, t) = (S \breve{\psi}_{\pm,0})({\mathbf{r}}, t)\quad\mbox{and}\quad \breve{u}_{*,0}({\mathbf{r}}, t) = (S \breve{\psi}_{*,0})({\mathbf{r}}, t),
    \end{equation}
    we define also the function $\breve{h}_0$ (cf. \Cref{htildedef})
    \begin{equation}\label{brevehdef}
        \breve{h}_0({\mathbf{r}}, t) = \breve{b}({\mathbf{r}}, t) - \breve{u}_{-,0}({\mathbf{r}}, t),\quad (\mathbf{r}, t) \in \Omega^{inc} \times \mathbb{R},
    \end{equation}
    and its Fourier transform
    \begin{equation}\label{Ht_def_breve}
        \breve{H}^f_0(\mathbf{r}, \omega) = \breve{B}^f(\mathbf{r}, \omega) - \left(S_\omega \breve{\psi}_{-,0}^f\right)(\mathbf{r}, \omega),\quad \mathbf{r}\in\Omega^{inc} \times \mathbb{R}.
    \end{equation}
    Finally, we will denote the right-hand side of
    \Cref{CFIE_proof_generic} in the present setting by
    \begin{equation}\label{Rt_def_breve}
        \breve{R}_0({\mathbf{r}}, \omega) = \gamma^- \partial_\mathbf{n} \breve{H}^f_0({\mathbf{r}}, \omega) - \i\eta\gamma^- \breve{H}^f_0({\mathbf{r}}, \omega), \quad ({\mathbf{r}},\omega) \in \Gamma\times \mathbb{R}.
    \end{equation}
\begin{lemma}\label{Rderiv_sobolev_bound}
  Let $T_0$ and $\tau$ denote given real numbers, let $q$ denote a
  non-negative integer, let $\breve{b}$ be defined as in
  \Cref{rem:breve}, and assume that $\breve{b}({\mathbf{r}}, t)$ vanishes for
  all $({\mathbf{r}}, t) \in \widebar\Omega\times \left\lbrace I_0\cup \left[0, \infty\right) \right\rbrace$
    (where $I_0 = I_T$ as in \Cref{domainofdep} for $T=0$). Additionally, let $\Omega$ satisfy
  the $q$-growth condition and assume $\breve{b}$ satisfies the $s$-regularity conditions
    \Cref{eq:gamma_Hs_assump} with $s = q$.
  Finally, assume that for a given nonnegative integer
  $n$, $\breve{\psi}_{*,0}$ satisfies
  $\breve{\psi}_{*,0} \in H^{n+1}(I_0;L^2(\Gamma))$.  Then for all
  integers $m \ge 0$ and all integers $j$ such that $0 \le j \le n$ we
  have
  \[
    \int_0^\infty \omega^{2j} \left\| \partial_\omega^m \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega \le C \left\| \breve{\psi}_{*,0}
    \right\|_{H^{j+1}(I_0;\,L^2(\Gamma))}^2,
  \]
  where $C$ is a constant independent of $T_0$ and $b$.
\end{lemma}
\begin{rem} In order to obtain an upper bound on integrals
  containing powers of the temporal variable $t$, the proof of
  \Cref{Rderiv_sobolev_bound} relies in an essential manner on the
  domain-of-dependence relations~\eqref{h_equiv}
  and~\eqref{h_compactsupp} for the function $\breve{h}_0$ (see
  \Cref{rem:h_equiv_rem_2})---which, limiting the integration of the
  aforementioned powers of $t$ to a bounded interval, yields
  meaningful integral estimates necessary for the proof of the lemma.
\end{rem}
\begin{proof}[Proof of \Cref{Rderiv_sobolev_bound}.]
  Since $\breve{b}$ satisfies the $s$-regularity conditions with $s = q$, by \Cref{psi_Hp} the quantity $\psi$ that enters the definition \Cref{Rt_def_breve} of $\breve{R}_0$ (cf.\ \Cref{Ht_def_breve}) satisfies $\psi \in L^2(\mathbb{R}; L^2(\Gamma))$.
  In view of~\eqref{Rt_def_breve} we have
  \begin{equation}\label{w2j_proof_first_eqn}
      \int_0^\infty \omega^{2j} \left\| \partial_\omega^m \breve{R}_0(\cdot,
      \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega = \int_0^\infty \int_\Gamma \left|\widehat{S}_{jm}
    \breve{H}^f_0(\mathbf{r}, \omega)\right|^2\,\d\sigma(\mathbf{r})\,\d\omega,
  \end{equation}
  where the operator $\hat{S}_{jm}$ is defined as
  \begin{equation}\label{Sdef_def}
    \widehat{S}_{jm} = \omega^{j}
    \partial_\omega^m\left(\gamma^-\partial_{\mathbf{n}} - \i \eta\gamma^-\right).
  \end{equation}
  In view of the definition~\Cref{Aomega_def_eqn} of the quantity
  $\eta$ in~\eqref{Sdef_def}, which depends on whether
  $0\leq\omega< \omega_0$ or
  $\omega>\omega_0$,~\cref{w2j_proof_first_eqn} can be re-expressed in
  the form
  \begin{equation}\label{Rderiv_split}
    \begin{split}
        \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &= \int_\Gamma\left(\int_0^{\omega_0} + \int_{\omega_0}^\infty\right)\left|\widehat{S}_{jm}
    \breve{H}^f_0(\mathbf{r}, \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r}).
    \end{split}
  \end{equation}
  Then defining the operators
  \begin{equation}\label{S_split_omega}
    \widehat{S}_{jm}^1 = \omega^j \partial_\omega^{m}
    (\gamma^-\partial_{\mathbf{n}} - \i\omega\gamma^-),\quad\mbox{and}\quad
    \widehat{S}_{jm}^2 = \omega^j \partial_\omega^{m}
    (\gamma^-\partial_{\mathbf{n}} - \i\gamma^-),
  \end{equation}
  which clearly satisfy $\widehat{S}_{jm} = \widehat{S}_{jm}^1$ for
  $\omega > \omega_0$ and $\widehat{S}_{jm} = \widehat{S}_{jm}^2$ for
  $\omega < \omega_0$,~\eqref{Rderiv_split} yields
  \begin{align*}
    \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
                                \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega \\
    &= \int_\Gamma
    \int_{0}^{\omega_0} \left|\widehat{S}_{jm}^1 \breve{H}^f_0(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r})
    +\int_\Gamma\int_{\omega_0}^\infty \left|\widehat{S}_{jm}^2 \breve{H}^f_0(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r})\\
    &\le \int_\Gamma
    \int_{-\infty}^\infty \left|\widehat{S}_{jm}^1 \breve{H}^f_0(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r})
    +\int_\Gamma\int_{-\infty}^\infty \left|\widehat{S}_{jm}^2 \breve{H}^f_0(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r}).
  \end{align*}
  Thus, utilizing the time-domain operators
  \begin{equation}\label{S_split_t}
    S_{jm}^1 = (-\i\frac{\partial}{\partial t})^j (-it)^{m}
    (\gamma^-\partial_{\mathbf{n}} - \frac{\partial}{\partial t}\gamma^-) \quad\mbox{and}\quad
    S_{jm}^2 = (-\i\frac{\partial}{\partial t})^j (-it)^{m}
    (\gamma^-\partial_{\mathbf{n}} - \i\gamma^-)
  \end{equation}
  corresponding to~\eqref{S_split_omega} together with Plancherel's
  theorem and equations~\eqref{brevehdef} and~\eqref{Ht_def_breve} we
  obtain
  \begin{align*}
      \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
                                  \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega \\
    &\le \int_\Gamma
    \int_{-\infty}^\infty \left|S_{jm}^1 \breve{h}_0(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r})
    +\int_\Gamma\int_{-\infty}^\infty \left|S_{jm}^2 \breve{h}_0(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r})\\
    &\begin{alignedat}{2}=\int_\Gamma
        \int_{-\tau}^{T_*} &\left|S_{jm}^1 \breve{u}_{*,0}(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r})\\
        &+\int_\Gamma\int_{-\tau}^{T_*} \left|S_{jm}^2 \breve{u}_{*,0}(\mathbf{r},
    t')\right|^2\,\d t'\,\d\sigma(\mathbf{r}),
    \end{alignedat}
  \end{align*}
  where, in view of \Cref{rem:breve}, which, in particular, tells us
  that $T=0$, and since
  $\Gamma = \partial\Omega \subset \widebar\Omega$, the last equality
  follows by using~\Cref{h_equiv} and~\Cref{h_compactsupp}.  Letting
  \begin{equation}\label{v1v2_def}
  v_1 = (\gamma^-\partial_{\mathbf{n}} -
    \frac{\partial}{\partial t}\gamma^-)\breve{u}_{*,0}, \quad\mbox{and}\quad
  v_2 = (\gamma^-\partial_{\mathbf{n}} -
  \i\gamma^-)\breve{u}_{*,0},
  \end{equation}
  and calling $a_\ell = {j \choose \ell}$, by Leibniz' product rule we
  have
  \begin{alignat*}{2}
    \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &\begin{alignedat}{2}\le \int_\Gamma&
      \int_{-\tau}^{T_*} \left| (-\i \frac{\partial}{\partial t'})^j (-\i t')^m
    v_1(\mathbf{r}, t')\right|^2\,\d t'\,\d\sigma(\mathbf{r})\\
      &+ \int_\Gamma \int_{-\tau}^{T_*} \left| (-\i \frac{\partial}{\partial
      t'})^j (-\i t')^m
    v_2(\mathbf{r}, t')\right|^2\,\d t'\,\d\sigma(\mathbf{r})
    \end{alignedat}\\
    &\begin{alignedat}{2}=
      \int_\Gamma&
      \int_{-\tau}^{T_*} \left|\sum_{\ell=0}^j a_\ell \left(
      \left(\frac{\partial}{\partial t'}\right)^{\ell} (t')^m\right)
      \left(\left(\frac{\partial}{\partial t'}\right)^{j-\ell}
      v_1(\mathbf{r}, t')\right)\right|^2\,\d t'\,\d\sigma(\mathbf{r})\\
      &+ \int_\Gamma
      \int_{-\tau}^{T_*} \left|\sum_{\ell=0}^j a_\ell \left(
      \left(\frac{\partial}{\partial t'}\right)^\ell (t')^m\right)
      \left(\left(\frac{\partial}{\partial t'}\right)^{j-\ell}
      v_2(\mathbf{r}, t')\right)\right|^2\,\d t'\,\d\sigma(\mathbf{r}).
    \end{alignedat}
  \end{alignat*}
  Substituting the exact value of the derivative
  $\frac{\partial^\ell}{\partial t^\ell} (\i t')^m$ in these
  expressions, we further obtain
  \begin{alignat*}{2}
      \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &\begin{alignedat}{2}\le
      \int_\Gamma&
      \int_{-\tau}^{T_*} \left|\sum_{\ell=0}^j \tilde{a}_\ell (t')^{m-\ell}
      \left(\left(\frac{\partial}{\partial t'}\right)^{j-\ell}
      v_1(\mathbf{r}, t')\right)\right|^2\,\d t'\,\d\sigma(\mathbf{r})\\
      &+ \int_\Gamma
      \int_{-\tau}^{T_*} \left|\sum_{\ell=0}^j \tilde{a}_\ell (t')^{m-\ell}
      \left(\left(\frac{\partial}{\partial t'}\right)^{j-\ell}
      v_2(\mathbf{r}, t')\right)\right|^2\,\d t'\,\d\sigma(\mathbf{r}),
    \end{alignedat}
  \end{alignat*}
  where $\tilde{a}_\ell = \frac{m!}{(m-\ell)!} a_\ell$ for
  $m - \ell \ge 0$ and $\tilde{a}_\ell = 0$ for $m - \ell < 0$.  Since
  the $t'$-integration is limited to the bounded region
  $[- \tau, T_*]$ the quantities $|t'|^{m-\ell}$ are bounded by a
  constant (which, importantly, is independent of $T_0$---see \Cref{ITvsI0}),
  and thus
  \begin{alignat*}{2}
      \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &\le C_1\sum_{\ell=0}^j \int_\Gamma \int_{-\tau}^{T_*}
    \left(\left|\left(\frac{\partial}{\partial t'}\right)^\ell v_1(\mathbf{r},
      t')\right|^2 +\left|\left(\frac{\partial}{\partial t'}\right)^\ell v_2(\mathbf{r},
    t')\right|^2 \right)\,\d t'\,\d\sigma(\mathbf{r})\\
    &\le C_1\sum_{\ell=0}^j \int_\Gamma \int_{-\infty}^{\infty}
    \left(\left|\left(\frac{\partial}{\partial t'}\right)^\ell v_1(\mathbf{r},
      t')\right|^2 +\left|\left(\frac{\partial}{\partial t'}\right)^\ell v_2(\mathbf{r},
    t')\right|^2 \right)\,\d t'\,\d\sigma(\mathbf{r}),
  \end{alignat*}
  where the last inequality simply bounds the space-time norm on the
  finite temporal region $[- \tau, T_*]$ by the full time integral on
  $\mathbb{R}$. In view of~\Cref{v1v2_def}, using once again the Plancherel
  theorem, and denoting, per \Cref{FT_conv},
  $\breve{U}_{*,0}^f$ the Fourier transform of $\breve{u}_{*,0}$, we
  estimate
  \begin{alignat*}{2}
      \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &\begin{alignedat}{2}%
      \le C_1\sum_{\ell=0}^j &\left(\int_\Gamma \int_{-\infty}^{\infty}
      \left|\omega^\ell(\gamma^-\partial_\mathbf{n} -
        \i\omega\gamma^-)\breve{U}_{*,0}^f(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r})\right.\\
      &\left.+ \int_\Gamma \int_{-\infty}^{\infty}
      \left|\omega^\ell(\gamma^-\partial_\mathbf{n} - \i\gamma^-)\breve{U}_{*,0}^f(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r})\right)
    \end{alignedat}\\
    &\begin{alignedat}{2}%
      \le \tilde{C}_1&\int_\Gamma \int_{-\infty}^{\infty}
      \left|(1 + \omega^2)^{j/2}(\gamma^-\partial_\mathbf{n} -
        \i\omega\gamma^-)\breve{U}_{*,0}^f(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r})\\
      &+ \tilde{C}_1\int_\Gamma \int_{-\infty}^{\infty}
      \left|(1 + \omega^2)^{j/2}(\gamma^-\partial_\mathbf{n} -
        \i\gamma^-)\breve{U}_{*,0}^f(\mathbf{r},
    \omega)\right|^2\,\d\omega\,\d\sigma(\mathbf{r}).
    \end{alignedat}\\
  \end{alignat*}

  We thus have established that
  \begin{equation}\label{Rderiv_sobolev_norm_estimate}
  \begin{split}
      \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &\le \tilde{C}_1\int_{-\infty}^\infty (1 + \omega^2)^j \left\| (\gamma^- \partial_\mathbf{n} -
      \i\omega\gamma^-) \breve{U}_{*,0}^f(\cdot, \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &+ \tilde{C}_1\int_{-\infty}^\infty (1 + \omega^2)^j \left\| (\gamma^-\partial_\mathbf{n} -
      \i\gamma^-) \breve{U}_{*,0}^f(\cdot, \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega.
  \end{split}
  \end{equation}
  To complete the proof we now use \Cref{per_freq_rhs_oper_bounds},
  which provides the frequency-wise bounds
  \begin{equation}\label{Rderiv_normbound_1}
      \left\| (\gamma^-\partial_\mathbf{n} - \i\omega\gamma^-) \breve{U}_{*,0}^f(\cdot, \omega)\right\|_{L^2(\Gamma)}
    \le D(1 + \omega^2)^{1/2}\left\|\breve{\psi}_{*,0}^f(\cdot,
    \omega)\right\|_{L^2(\Gamma)},
  \end{equation}
  and
  \begin{equation}\label{Rderiv_normbound_2}
      \left\| (\gamma^-\partial_\mathbf{n} - \i\gamma^-) \breve{U}_{*,0}^f(\cdot, \omega)\right\|_{L^2(\Gamma)}
      \le E (1 + \omega^2)^{1/2}\left\|\breve{\psi}_{*,0}^f(\cdot, \omega)\right\|_{L^2(\Gamma)},
  \end{equation}
  where $D, E > 0$ are constants independent of $\omega$, $\breve{b}$
  and $\breve{\psi}$.  Substituting~\cref{Rderiv_normbound_1}
  and~\cref{Rderiv_normbound_2} in~\Cref{Rderiv_sobolev_norm_estimate}
  we conclude that
  \begin{alignat*}{2}
      \int_0^\infty \omega^{2j} &\left\| \partial_\omega^m \breve{R}_0(\cdot,
      \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
      &\begin{alignedat}{2}
        \le \int_{-\infty}^\infty C_2(1 + &\omega^2)^{j+1}
          \left\|\breve{\psi}_{*,0}^f(\cdot,
          \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
          &+ \int_{-\infty}^\infty C_3(1 + \omega^2)^{j+1}\left\|\breve{\psi}_{*,0}^f(\cdot,
        \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega.
    \end{alignedat}\\
    &\le
    C\left\|\breve{\psi}_{*,0}\right\|_{H^{j+1}(\mathbb{R};\,L^2(\Gamma))}^2
    =
    C\left\|\breve{\psi}_{*,0}\right\|_{H^{j+1}(I_0;\,L^2(\Gamma))}^2,
    \end{alignat*}
    as desired.
\end{proof}

\Cref{Rderiv_sobolev_bound} is used in what follows to establish the
main building block in the proof of \Cref{3d_decay_thm_ii}, namely,
\Cref{decay_estimate_L2} below. The proof of \Cref{decay_estimate_L2},
in turn, incorporates a smooth windowing procedure which relies on use
of compactly-supported time-domain window functions. For definiteness
we utilize the time-window functions introduced in what follows.
\begin{definition}\label{def_w}
  Letting $v(u) = \exp(\frac{2e^{-1/u}}{u-1})$, we define
\begin{equation}\label{wTdef}
  w(s) =
    \begin{cases}
      1 - v(\frac{s + 2s_0}{s_0}), & -2s_0 \le s \le -s_0\\
      1, & -s_0 < s < s_0\\
      v(\frac{s - s_0}{s_0}), & s_0 \le s \le 2s_0\\
      0, & |s| > 2s_0,
    \end{cases}
    \qquad\mbox{and}\qquad w_\varphi(s) = w(s - \varphi),
\end{equation}
where $\varphi\in\mathbb{R}$ denotes an important ``time-shift''
parameter, and where $s_0 > 0$ is a given parameter that can be
selected arbitrarily and remains unchanged throughout the remainder of
this paper.
\end{definition}

Clearly, the functions $w$ and $w_\varphi$ (i)~Satisfy
$w,w_\varphi\in C^\infty(\mathbb{R})$; (ii)~Equal $1$ in an interval
of length $2s_0$; (iii)~Increase/decrease from $0$ to $1$ in intervals
of length $s_0$; (iv)~Satisfy $0\leq w(s) \le 1$ and
$0\leq w_\varphi(s) \le 1$ for all $s\in\mathbb{R}$.
It is easy to check that for every $\varphi \in\mathbb{R}$ we have
\begin{equation}
  \label{eq:POU2}
  \begin{split}
   & w_{\varphi + s_0}(s)  + w_{\varphi + 4 s_0}(s) =1\\
   & w_{\varphi + s_0 + 3\ell s_0}(s) = 0\quad (\ell \not\in \{0,1 \})
  \end{split}
  \qquad\mbox{for} \quad s \in [\varphi, \varphi + 3s_0].
\end{equation}
In particular, the functions $w_{3\ell s_0}(s)$ with $\ell \in\mathbb{Z}$
form a partition of unity, wherein at most two functions in the family do not vanish at any given $s\in\mathbb{R}$.

\begin{lemma}\label{decay_estimate_L2}
  Let $n$ and $q$ denote non-negative integers, $n>0$, let $T_0>0$,
  let $\widetilde{b}$ be defined as in \Cref{rem:tilde_1},  and assume that
  $\widetilde{b}$ vanishes for all $({\mathbf{r}}, t) \in \widebar{\Omega}\times\left \{I_{T_0} \cup [T_0, \infty)\right\}$, with $I_{T_0}$ as in
  \Cref{domainofdep}. Additionally, let $\Omega$ satisfy the
  $q$-growth condition and assume $\widetilde{b}$ satisfies
  the $s$-regularity conditions~\eqref{eq:gamma_Hs_assump} with $s = (n+1)(q+1) + q$.
Then, the functions $\breve{\psi}_{+,0}$ and $\breve{\psi}$ defined in~\Cref{rem:breve} satisfy
\begin{equation}\label{L2_rchidecay}
  \left\|w_\varphi \breve{\psi}_{+,0}\right\|_{L^2(\mathbb{R};\,L^2(\Gamma)}^2 \le
    C(\Gamma, \tau, n, s_0) \varphi^{-2n}
    \left\|\breve{\psi}\right\|_{H^{(n+1)(q+1)}(I_0;\,L^2(\Gamma))}^2,
\end{equation}
for all $\varphi > 0$, where $C(\Gamma, \tau, n, s_0)$ denotes a
constant independent of $\varphi, T_0,$ and $\widetilde{b}$.
\end{lemma}
\begin{proof}
  It is useful to first note that since by hypothesis $\widetilde{b}({\mathbf{r}}, t)$ vanishes for
  $({\mathbf{r}},t) \in \widebar\Omega \times \left\lbrace I_{T_0}\cup \left[T_0, \infty\right)\right\rbrace$,
  $\breve{b}({\mathbf{r}}, t)$ vanishes for all $({\mathbf{r}},t) \in \widebar\Omega \times \left\lbrace I_0\cup \left[0, \infty\right)\right\rbrace$.
  Similarly, the $s$-regularity condition hypotheses also are satisfied for $\breve{b}$.

  In order to establish the desired decay
  estimate~\eqref{L2_rchidecay}, by Plancherel's theorem we may
  estimate, instead, the $L^2$ norm of the Fourier transform:
  \begin{equation}\label{conv_w_2}
   \left\|w_\varphi \breve{\psi}_{+,0}\right\|_{L^2(\mathbb{R};\,L^2(\Gamma)}^2 =   \left\|\widehat{w_\varphi \breve{\psi}_{+,0}}\right\|_{L^2(\mathbb{R};\,L^2(\Gamma)}^2 =
    \left\|\hat{w}_\varphi \ast
    \breve{\psi}_{+,0}^f\right\|_{L^2(\mathbb{R};\,L^2(\Gamma)}^2.
  \end{equation}
  Using the relation
  $\hat{w}_\varphi = {\mathrm{e}}^{-\i\omega \varphi} \hat{w}(\omega)$ we obtain
  \begin{equation}\label{conv_w}
    \left(\hat{w}_\varphi \ast
    \breve{\psi}_{+,0}^f\right)({\mathbf{r}}, \omega)   =  \int_{-\infty}^\infty {\mathrm{e}}^{-\i \tau \varphi} \hat{\omega}(\tau) \breve{\psi}_{+,0}^f({\mathbf{r}}, \omega - \tau)\,\d \tau,
   \end{equation}
   and we proceed to integrate by parts this integral $n$ times with
   respect to $\tau$. To do this we note that, (i)~Both
   $\hat{w}_\varphi(\omega)$ and
   $\breve{\psi}_{+,0}^f (\mathbf{r}, \omega)$ are infinitely
   differentiable functions of $\omega$ (in view
   of \Cref{def_w}, \Cref{rem:breve} and
   \Cref{int_eq_freq_regularity}); (ii)~The Fourier transform
   $\hat w(\tau)$ and all of its derivatives tends to zero faster than
   any positive power of $\tau$ as $\tau\to\pm\infty$, as it befits
   the Fourier transform of an infinite differentiable and compactly
   supported function; and (iii)~We have
   $\left\|\partial_\omega^m \breve{\psi}_{+,0}^f(\cdot, \omega -
     \tau)\right\|_{L^2(\Gamma)} \le C_m|\omega - \tau|^N$ for some
   integer $N$ as $|\tau| \to \infty$, as it follows directly from
   \Cref{psit_nongrowth} in
   \Cref{3d_decay_thm_recursive_bound_general}. Thus integrating by
   parts~\eqref{conv_w} and using Leibniz' product rule, we see that
   all of the resulting boundary terms at $\tau = \pm\infty$ vanish
   and~\eqref{conv_w} becomes
   \begin{equation}\label{nIntByParts}
\begin{split}
  \left( \hat{w}_\varphi \ast \breve{\psi}_{+,0}^f \right)(\mathbf{r}, \omega) =
    \left(\frac{1}{i\varphi}\right)^n \int_{-\infty}^\infty
    {\mathrm{e}}^{-i\tau \varphi}  \left(\vphantom{\sum_{m=0}^n} \right. & \sum_{m=0}^n a_m \left( \partial^{n-m}_\tau
    \hat{w}(\tau)\right) \times\\
    &\left.\times \left( \vphantom{\sum_{m=0}^n} \partial^m_\omega \breve{\psi}_{+,0}^f(\mathbf{r}, \omega -
    \tau)\right)\right)d\tau,
\end{split}
\end{equation}
where $a_m = {n \choose m}$.

In view of~\eqref{conv_w_2} and \Cref{nIntByParts}, calling, for $0 \le m \le n$,
\begin{equation}\label{w_psi}
    F_{nm}^\varphi(\tau) = {\mathrm{e}}^{-\i \tau \varphi}\partial^{n-m}_\tau \hat{w}(\tau) \quad\mbox{and}\quad G_m({\mathbf{r}},\tau) = \partial^m_\tau \breve{\psi}_{+,0}^f({\mathbf{r}}, \tau),
\end{equation}
and using the relation $\| \sum_{i=1}^n f_i \|^2 \le n \sum_{i=1}^n \| f_i\|^2$
we obtain
\begin{equation}
\begin{split}
  \left\| w_\varphi \breve {\psi}_{+,0} \right.&\negmedspace\left.\vphantom{w_\varphi
  \breve{\psi}_{+,0}}\right\|_{L^2(\mathbb{R};\,L^2(\Gamma))}^2
  \le (n+1)\varphi^{-2n} \sum_{m=0}^n |a_m|^2 \left\|
  F_{nm}^\varphi \ast G_m\right\|_{L^2(\mathbb{R}; L^2(\Gamma))}^2.\label{pre_young}
\end{split}
\end{equation}

In order to obtain a bound on the norms on the right-hand side
of~\eqref{pre_young} we rely on the Bochner version of Young's
convolution inequality~\cite[Lem.\ 1.2.30]{Weis}, and we thus first
establish that relevant hypotheses hold, namely, that
\begin{equation}\label{w_psi_spaces}
    F_{nm}^\varphi \in L^1(\mathbb{R})\quad\mbox{and}\quad G_m \in L^2(\mathbb{R}; L^2(\Gamma)) \quad\mbox{for}\quad  0 \le m \le n.
\end{equation}
The first of these relations is easily established: since $w$ is
smooth and compactly supported, it is in the Schwartz class, and
thus~\cite{Folland}, its Fourier transform is also in the Schwartz
class---and, in particular, $\hat{w}$ and all of its derivatives are
elements of $L^1(\mathbb{R})$.  To verify the second relation
in~\eqref{w_psi_spaces}, on the other hand, we first note from
\Cref{3d_decay_lemma_2ndkind} that, for $\omega\geq 0$,
$\breve{\psi}_{+,0}^f$ satisfies the integral equation
\[
    \left(A_\omega \breve{\psi}_{+,0}^f\right)(\mathbf{r}, \omega) = \breve{R}_0(\mathbf{r}, \omega),
\]
where $\breve{R}_0(\mathbf{r}, \omega)$ is defined by
\Cref{Rt_def_breve}.  It then follows
from~\cref{recursion_hypothesis_squared_general} in
\Cref{3d_decay_thm_recursive_bound_general} that, for $m \le n$, there
exist constants $d_{ij}^m > 0$ such that for all
$\omega \in \mathbb{R}$, $\omega \neq \pm \omega_0$,
\begin{equation}\label{bound_on_deriv_psiplus}
    \begin{split}
      \left\| G_m(\cdot, \omega)\right\|_{L^2(\Gamma)}^2
      \le \sum_{i=0}^{m} &\sum_{j=0}^{(i+1)(q+1)-1} d_{ij}^m \omega^{2j}
      \left\|\partial_\omega^{m-i} \breve{R}_0(\cdot,
      |\omega|)\right\|^2_{L^2(\Gamma)}.
    \end{split}
\end{equation}
Integrating, we obtain
\begin{equation}\label{fixmeplease}
\begin{split}
    \int_{-\infty}^\infty \left\|\vphantom{G_m(\cdot,
    \omega)}\right.&\hspace{-1.4mm}\left.G_m(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2 \,\d\omega\\
    &\le \sum_{i=0}^{m} \sum_{j=0}^{(i+1)(q+1)-1} 2d_{ij}^m \int_0^\infty \omega^{2j}
  \left\|\partial_\omega^{m-i} \breve{R}_0(\cdot,
    \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega.
\end{split}
\end{equation}
We now use \Cref{Rderiv_sobolev_bound} to estimate each term on the
right-hand side of \Cref{fixmeplease}. The main hypothesis of that lemma,
which, in the present context amounts to the statement
$\breve{\psi}_{*,0} \in H^{(n+1)(q+1)}(I_0;L^2(\Gamma))$, follows from
\Cref{psi_Hp} since $\breve{b}$ satisfies the $s$-regularity
conditions~\eqref{eq:gamma_Hs_assump} with $s = (n+1)(q+1)+q$ (which
hold in view of the $s$-regularity condition hypotheses for
$\widetilde{b}$ and the definition~\cref{def:breve} of $\breve{b}$).
For $m \le n$ we may therefore write
\begin{equation}\label{partialomega_psibreve}
\begin{split}
  \int_{-\infty}^\infty \left\| G_m\right.&\hspace{-1.4mm}\left.
  (\cdot,
  \omega)\right\|_{L^2(\Gamma)}^2\,\d\omega\\
    &\le C_1\left\| \breve{\psi}_{*,0}\right\|_{H^{(m+1)(q+1)}(I_0;\,L^2(\Gamma))}^2 \le C_2\left\|
    \breve{\psi}_{*,0}\right\|_{H^{(n+1)(q+1)}(I_0;\,L^2(\Gamma))}^2,
\end{split}
\end{equation}
with $C_1$ and $C_2 > 0$ independent of $\widetilde{b}$ and $T_0$, which, in
particular, establishes the second relation in \Cref{w_psi_spaces}.

Having established \Cref{w_psi_spaces} we may apply Young's
convolution inequality~\cite[Lemma\ 1.2.30]{Weis} to each of the terms
in the right-hand sum in \Cref{pre_young} and obtain
\begin{equation}\label{3d_decay_thm_main_estimate}
  \begin{split}
    \left\| w_\varphi \breve{\psi}_{+,0} \right\|_{L^2(\mathbb{R};\,L^2(\Gamma))}^2
      &\le C(n) \varphi^{-2n} \sum_{m=0}^n |a_m|^2
  \left\|F_{nm}^\varphi \right\|_{L^1}^2
  \left\|G_m \right\|_{L^2(\mathbb{R}; L^2(\Gamma))}^2\\
  &\le C(n, s_0) \varphi^{-2n}
  \sum_{m=0}^n \left\|G_m \right\|^2_{L^2(\mathbb{R};\,L^2(\Gamma))}
  \end{split}
\end{equation}
(since, of course,
$\left\| F_{nm}^\varphi\right\|_{L^1(\mathbb{R}} =
\left\|\partial^{n-m}\hat{w}\right\|_{L^1(\mathbb{R})}$), which using
\Cref{partialomega_psibreve} yields
\begin{equation*}
  \begin{split}
    \left\|w_\varphi \breve{\psi}_{+,0}\right\|_{L^2(\mathbb{R};\,L^2(\Gamma)}^2 \le C(\Gamma, \tau, n, s_0)\varphi^{-2n}\sum_{i=0}^n C_i\left\|
    \breve{\psi}_{*,0}\right\|_{H^{(i+1)(q+1)}(I_0;\,L^2(\Gamma))}^2.
  \end{split}
\end{equation*}
Appropriately adjusting the constant, in view of the continuity of the
inclusion map
$\iota : H^{(i+1)(q+1)}(I_0; L^2(\Gamma)) \to H^{(n+1)(q+1)}(I_0;
L^2(\Gamma))$ for all $i \le n$ we obtain
\begin{equation*}
    \left\|w_\varphi \breve{\psi}_{+,0}\right\|_{L^2(\mathbb{R};\,L^2(\Gamma)}^2 \le C(\Gamma, \tau, n, s_0) \varphi^{-2n}
    \left\|\breve{\psi}_{*,0}\right\|_{H^{(n+1)(q+1)}(I_0;\,L^2(\Gamma))}^2.
\end{equation*}
The claimed relation \Cref{L2_rchidecay} follows from this inequality
since, according to \Cref{rem:breve},
$\breve{\psi}_{*,0}({\mathbf{r}}', t') = w_-(t' + T_* +
\tau)w_+(t')\breve{\psi}({\mathbf{r}}',t')$, and since each derivative of the
functions $w_-$ and $w_+$ is uniformly bounded.
\end{proof}

On the basis of the preparatory lemmas established above in this
section we now present the proof of \Cref{3d_decay_thm_ii}.
\begin{proof}[Proof of \Cref{3d_decay_thm_ii}]
  Letting (cf.\ \Cref{def:breve})
  \begin{equation}\label{psi_check}
    \breve{b}({\mathbf{r}}, t) = b({\mathbf{r}}, t + T_0)\quad\mbox{and}\quad \breve{\psi}({\mathbf{r}}', \theta) = \psi({\mathbf{r}}', \theta + T_0),
  \end{equation}
  instead of \Cref{density_Hp_time_bound_Tn}
  we establish the equivalent $\theta$-decay ($\theta = t - T_0 > 0$) estimate
\begin{equation}\label{density_Hp_time_bound_Tn_shifted}
  \left\|\breve{\psi}\right\|_{H^p([\theta, \infty); L^2(\Gamma))} \le C(\Gamma,
  \tau, p, n) \theta^{1/2-n} \left\|\breve{\psi}\right\|_{H^{p + n(q+1)}(I_0;
    L^2(\Gamma))} < \infty,
\end{equation}
where $I_0$ (i.e. $I_T$ for $T=0$) is defined per
\Cref{domainofdep}. Clearly, $\breve{\psi}$ in \Cref{psi_check} satisfies the integral equation
\begin{equation}\label{brevepsi_int_eq}
  \left(S\breve{\psi}\right)(\mathbf{r}, t) = \gamma^+ \breve{b}(\mathbf{r}, t)\quad\mbox{for}\quad (\mathbf{r}, t) \in
  \Gamma\times\mathbb{R},
\end{equation}
which coincides with \Cref{eq:tdie_sl_generic} for
$\widetilde{b}({\mathbf{r}}, t) = \breve{b}({\mathbf{r}}, t) = b({\mathbf{r}}, t + T_0)$
and $\widetilde{\psi} = \breve{\psi}$.

To establish \Cref{density_Hp_time_bound_Tn_shifted} we first obtain
certain decay results on a sequence of bounded time intervals and then
produce the final estimate by summing over the sequence. Using
\Cref{eq:POU2} with $\varphi = \theta + 3\ell s_0$ together with the
identity $\breve{\psi}(\cdot, t') = \breve{\psi}_{+,0}(\cdot, t')$ for
$t' > 0$ (see \Cref{rem:breve}) and the relation
$(A +B)^2\leq 2A^2 +2 B^2$ we obtain
\begin{equation}\label{ptwise_tprime_bound}
  \begin{split}
    \left\|\breve{\psi}\right\|^2_{L^2([\theta, \infty); L^2(\Gamma))} &=  \int_\theta^\infty\hspace{-2mm} \int_\Gamma \left( \sum_{\ell =0}^\infty (w_{\theta + 3\ell s_0 + s_0} + w_{\theta + 3(\ell+1) s_0 + s_0}) \breve{\psi}_{+,0} ({\mathbf{r}}', t')\hspace{-1mm}\right)^2 \hspace{-2mm}\d\sigma({\mathbf{r}}') \d t'\\
    &\le 4 \sum_{\ell=0}^\infty \left\|w_{\theta +
    3\ell s_0 + s_0} \breve{\psi}_{+,0}\right\|^2_{L^2(\mathbb{R}; L^2(\Gamma))}.
    \end{split}
\end{equation}
The same argument applied to $\partial^p \breve{\psi}$ tells us that
\begin{equation}\label{Wt_to_sum_ii}
    \left\|\partial^p \breve{\psi}\right\|^2_{L^2([\theta, \infty); L^2(\Gamma))} \le 4
    \sum_{\ell=0}^\infty \left\|w_{\theta +
    3\ell s_0 + s_0} \breve{\psi}_{p,+,0}\right\|^2_{L^2(\mathbb{R}; L^2(\Gamma))},
\end{equation}
Combining \Cref{ptwise_tprime_bound} and
\Cref{Wt_to_sum_ii} and using \Cref{def:sob_bochner} we obtain
\begin{equation}\label{psik_to_wT}
\begin{split}
\left\|\breve{\psi}\right\|^2_{H^p([\theta,
        \infty);\,L^2(\Gamma))} &\le  4 \sum_{\ell = 0}^\infty \left(\left\| w_{\theta + 3\ell s_0 + s_0} \breve{\psi}_{+,0} \right\|_{L^2(\mathbb{R};
        L^2(\Gamma))}^2\right.\\
        &\quad\quad\quad\quad\quad\left. + \left\| w_{\theta + 3\ell s_0 + s_0} \breve{\psi}_{p,+,0} \right\|_{L^2(\mathbb{R};
        L^2(\Gamma))}^2\right).
\end{split}
\end{equation}
To complete the proof we now estimate the $L^2$ norms on the
right-hand side of~\eqref{psik_to_wT}, that is to say, the norms
$\left\| w_{\varphi} \breve{\psi}_{+,0} \right\|_{L^2(\mathbb{R};
  L^2(\Gamma))}$ and
$\left\| w_{\varphi} \breve{\psi}_{p,+,0} \right\|_{L^2(\mathbb{R};
  L^2(\Gamma))}$ for $\varphi = \theta + 3\ell s_0 + s_0$,
$\ell = 0, 1, \cdots$; the desired result then follows by addition of
the resulting estimates.

\paragraph{\bf Decay estimate for
  $\left\| w_{\varphi} \breve{\psi}_{+,0} \right\|_{L^2(\mathbb{R};
    L^2(\Gamma))}^2$.}

Since by assumption $b$ satisfies the $s$-regularity conditions~\eqref{eq:gamma_Hs_assump}
with $s=(n+1)(q+1)+q$, and since, by hypothesis, $\breve{b}({\mathbf{r}}', t')$ vanishes for
$({\mathbf{r}}',t') \in \widebar{\Omega} \times \left\lbrace I_0\cup[0, \infty)\right\rbrace$,
\Cref{decay_estimate_L2} with $\widetilde{b} = b$ yields
\begin{equation}\label{rchipsi_L2_estimate}
  \left\|w_\varphi \breve{\psi}_{+,0}\right\|_{L^2(\mathbb{R};\,L^2(\Gamma)}^2 \le
    C(\Gamma, \tau, n, s_0) \varphi^{-2n}
    \left\|\breve{\psi}\right\|_{H^{(n+1)(q+1)}(I_0;\,L^2(\Gamma))}^2
\end{equation}
for arbitrary $\varphi > 0$, where $C$ a constant independent of
$\varphi$, $T_0$, and $b$.

\paragraph{\bf Decay estimate for
  $\left\| w_{\varphi} \breve{\psi}_{p,+,0} \right\|_{L^2(\mathbb{R};
    L^2(\Gamma))}$ ($p > 0$).}
\Cref{int_eq_pderiv} tells us that
\begin{equation}\label{eq:pderiv_brevepsi}
    \left(S \partial^p \psi\right)({\mathbf{r}}, t) = \gamma^+ \partial^p b({\mathbf{r}}, t),\quad\mbox{for}\quad
    ({\mathbf{r}}, t) \in \Gamma \times \mathbb{R}.
\end{equation}
We now apply \Cref{decay_estimate_L2} to obtain decay estimates for
$w_\varphi \breve{\psi}_{p,+,0}$. Since (i) $b({\mathbf{r}}', t')$ vanishes for
$({\mathbf{r}}',t') \in \widebar{\Omega} \times \left\lbrace I_{T_0} \cup [T_0, \infty)\right\rbrace$
so does $\partial^p b$ and (ii) $b$ satisfies
the $s$-regularity conditions~\eqref{eq:gamma_Hs_assump}
with $s=(n+1)(q+1)+q$ (by the hypotheses on $b$),
\Cref{decay_estimate_L2} with $\widetilde{b} = \partial^p b$ yields
the estimate
\begin{equation}\label{L2_deriv_rchidecay}
    \left\| w_\varphi \breve{\psi}_{p,+,0} \right\|_{L^2(\mathbb{R}; L^2(\Gamma))}^2 \le
    C(\Gamma, \tau, n, s_0) \varphi^{-2n} \left\|\partial^p
    \breve{\psi}\right\|_{H^{(n+1)(q+1)}(I_0;\,L^2(\Gamma))}^2,
\end{equation}
for arbitrary $\varphi > 0$, where $C$ is again a constant independent of
$\varphi$, $T_0$, and $b$.

\paragraph{\bf Combined decay estimate.}

Using  \Cref{psik_to_wT}, \Cref{rchipsi_L2_estimate}, and \Cref{L2_deriv_rchidecay} we obtain
\begin{equation}\label{Hp_tdecay_i}
    \begin{split}
    \left\|\breve{\psi}\right\|^2_{H^p([\theta,
        \infty);\,L^2(\Gamma))} &\le 4 \sum_{\ell = 0}^\infty \left(\left\| w_{\theta + 3\ell s_0 + s_0} \breve{\psi}_{+,0} \right\|_{L^2(\mathbb{R};
        L^2(\Gamma))}^2\right.\\
        &\quad\quad\quad\left. + \left\| w_{\theta + 3\ell s_0 + s_0} \breve{\psi}_{p,+,0} \right\|_{L^2(\mathbb{R};
        L^2(\Gamma))}^2\right)\\
        &\le C_1 \sum_{\ell = 0}^\infty  \left( (\theta + 3\ell s_0 + s_0)^{-2n}
        \left\|\breve{\psi}\right\|^2_{H^{(n+1)(q+1)}(I_0;\,L^2(\Gamma))}\right.\\
        &\quad\quad\quad\left. + (\theta + 3\ell s_0 + s_0)^{-2n}
        \left\|\partial^p \breve{\psi}\right\|^2_{H^{(n+1)(q+1)}(I_0;\,L^2(\Gamma))} \right),
    \end{split}
\end{equation}
where $C_1$ is a constant dependent only on $\Gamma$,
$\tau$, $n$ and $s_0$.  It follows that
\begin{equation}\label{Hp_tdecay_ii}
    \begin{split}
        \left\|\breve{\psi}\right\|^2_{H^p([\theta, \infty);\,L^2(\Gamma))} &\le C \left\|\breve{\psi}\right\|^2_{H^{p + (n+1)(q+1)}(I_0;\,L^2(\Gamma))}
        \sum_{\ell = 0}^\infty (\theta + 3\ell s_0 + s_0)^{-2n},
    \end{split}
\end{equation}
where, again, $C = C(\Gamma, \tau, n, s_0) > 0$.  Since $\theta > 0$
the term $(\theta + 3\ell s_0 + s_0)^{-n}$ is a strictly decreasing and
positive function of $\ell$, and, thus, estimating the sum by an
integral it is easy to check that
\[
    \sum_{\ell=0}^\infty (\theta + 3\ell s_0 + s_0)^{-2n} \le \widetilde{C} \theta^{-2n+1},
\]
which in conjunction with \Cref{Hp_tdecay_ii} establishes
\Cref{density_Hp_time_bound_Tn_shifted} which, as indicated at the
beginning of this proof, is equivalent to the desired
estimate~\cref{density_Hp_time_bound_Tn}.  The
result~\cref{density_unif_time_bound_Tn}, finally, follows from use
of~\cref{density_Hp_time_bound_Tn} with $p=1$ and the use of
\Cref{sob_lemma}. The proof is now complete.
\end{proof}

redpajama_set_name: RedPajamaArXiv